<%BANNER%>

Two-Dimensional Modeling of a Chemically Reacting Boundary Layer Flow in a Catalytic Reactor

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20110331_AAAACC INGEST_TIME 2011-03-31T16:18:38Z PACKAGE UFE0014866_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 87603 DFID F20110331_AABVJJ ORIGIN DEPOSITOR PATH griffin_p_Page_021.jp2 GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
7380c8c577f97cf5c45d4f0ec98255cb
SHA-1
99278589fea59caef9e581cc86415e7943f59ee5
97795 F20110331_AABVIV griffin_p_Page_089.jp2
280e968cbf1dff63cae60c750b4f3079
62f20ab87af4660ca0c1417dee85d678235e670c
93677 F20110331_AABVJK griffin_p_Page_043.jpg
0e54d546b67d93ce2da71057f38287d0
4cfd418c30bf123388009b9068e104f38ea64a6f
92252 F20110331_AABVIW griffin_p_Page_074.jpg
32d2def1c04a8bf4235d9214a3a06b9e
665b4889e95f3a83513d76c9a969c67323e4720d
47705 F20110331_AABVJL griffin_p_Page_033.pro
4814d2f68307b3998b15af5fd57b6416
523bb1b31b67089d46f2961feddebfbea8c7295f
1820 F20110331_AABVIX griffin_p_Page_089.txt
b891fda3d6d8d25ef3ce099e9afb380b
6f45dbf40520f118288435580303c149c636f00b
44062 F20110331_AABVJM griffin_p_Page_104.jpg
71085255ed14b250e2ba454df409a7b3
0474a22e3a0cdd934d9e3097662a2a02d809a31a
1130 F20110331_AABVIY griffin_p_Page_009.txt
544c3e988c58f9c2a37e5a7fa9082317
a51679965d15998b6c3ef29ff28c44718aaa7823
76469 F20110331_AABVKA griffin_p_Page_090.jpg
c2785fbe4bcf22be2b9763087bb7a830
1acd1c8ca6eee604295510de7dd0f646465bbc66
43813 F20110331_AABVJN griffin_p_Page_028.pro
b9b1bf961fb55fed8aa07bcdadb7ec80
ee12e316ae579fd43287750d6e9e7646220ca124
1654 F20110331_AABVIZ griffin_p_Page_086.txt
6df6aaae0440d21001e33d17b004fce8
d9eae0ec7973dfe1fba078cf5626c3933e4e54a8
1053954 F20110331_AABVKB griffin_p_Page_027.tif
bedfb75dbbc6bdd9a51102b9e94f3a22
a3a9228471628d300ebf9ebeca411845b7338064
746143 F20110331_AABVKC griffin_p_Page_084.jp2
e92fba2c31699b4456b0fa10a4b381df
d49094b89e0b522cc7818cbaea0744818c0f4291
78847 F20110331_AABVJO griffin_p_Page_025.jpg
c8c46efec3dff4cd0e842c54aacd2261
68fdd29123a0ca5d7f407d67915eb4fa5efe7069
F20110331_AABVKD griffin_p_Page_019.tif
5c04f1b7ab057d37a60c0a434363a669
98e6dfa8b7867b4a2042162aa7cc0e103211ba8f
78939 F20110331_AABVJP griffin_p_Page_042.jpg
4604d76ff632216032bb74d55fd1ced4
f224f169c0db234c419292703c2cc296d5a9df6a
33618 F20110331_AABVKE griffin_p_Page_084.pro
59a20210e49a95244ba428b28a029f80
96aea766ec6e8b13dfad5ba02dab43aa7e3fa968
5623 F20110331_AABVJQ griffin_p_Page_023thm.jpg
40db31536d82a5de605ebab5691f9b6b
4569202d8c2421e75e90f44ca264cb4b990dab5e
8125 F20110331_AABVKF griffin_p_Page_014thm.jpg
8f7500616e6aa7c195af05d2c6712ab6
6fccdc30d44b2a21bc12646341e0f87ee2665726
114162 F20110331_AABVJR griffin_p_Page_102.jpg
af797770bb169927ce42bc8032caadce
ff57b09c1e53342a53e2a2b73515a3053327a0a8
6837 F20110331_AABVKG griffin_p_Page_086thm.jpg
cb4b40ba675571c4b9b7dff65acc698e
b56c22f12b70f084a120b1b182b4d7ba2b5b4cea
26813 F20110331_AABVJS griffin_p_Page_001.jpg
44e50eac24dfe5449d8b4c4464ed0c06
d410a311d0881aa9f5a7616cb1807a0d21cdfbb3
1817 F20110331_AABVKH griffin_p_Page_051.txt
851d92eaf2622a2601e1d94bcdf12478
06553b3e9742841d2dff0871654be76081dc70b7
F20110331_AABVJT griffin_p_Page_025.tif
9f2fc5627956f0c10ba14497e01e71bc
53eda09aa172253d590b12b2d9ea02940ef9d8ba
40938 F20110331_AABVKI griffin_p_Page_069.pro
47a85c1716884a1297c5d29fb6de1e2b
00f77a3eb43bedf412c804a0e568bc365f2c3314
6855 F20110331_AABVJU griffin_p_Page_042thm.jpg
245aefd057525ddb479fa05f6afabfbc
63121a9c24fe1911257daeb57c5d0ae49b3c66f8
52067 F20110331_AABVKJ griffin_p_Page_062.pro
0c6084037fb2e488935de9d79006b1d9
2c9043b11749abec3c8b97697f25d748fa9c8b62
78601 F20110331_AABVJV griffin_p_Page_040.jpg
4ac1e05f198cbd9b181df2216da9c75b
3c563c36edff088fc2199d91ed4bcb8558bb789b
34178 F20110331_AABVKK griffin_p_Page_055.QC.jpg
6f41e774b1d6098f1d7912bbf92a71c1
ec1af5f13faf142c26d2e09d120a48b532a7f03f
F20110331_AABVJW griffin_p_Page_060.tif
40de540beb82e4b3822cd32af1564720
28366126bc0710ddeee020b9d420bcc935cb9134
101452 F20110331_AABVKL griffin_p_Page_057.jpg
1260a8bb3b480ba0c6ed6f4435c0768d
f5b08d99f8d1ea997bbf8ec9d0eaf06963015d1b
F20110331_AABVJX griffin_p_Page_009.tif
6bc9716ed24eed40be33a48862cffdc5
8d460c56bcee2e804c0f53bd9ffd2c4420aa81a1
2075 F20110331_AABVLA griffin_p_Page_053.txt
97437d049dd408876910faab85c74bb5
5cd64a347858e12be3c598aa8f64516c5598e6ab
27526 F20110331_AABVKM griffin_p_Page_083.QC.jpg
e29498e00ce335bdbcad1779bd45c147
6d289fbbd2411a5940ec7ae8e2afd36471d8e577
8457 F20110331_AABVJY griffin_p_Page_001.pro
6dda3316aa9302acdc4be198351b1e29
4ccda19b9e9feb0fb4df98a8490cc87169ec7620
F20110331_AABVLB griffin_p_Page_008.tif
1c3f4e44242e29ae8b7558d2a18b2698
012fe35e850b9847a7616d178f9c43ae51e747d7
103839 F20110331_AABVKN griffin_p_Page_059.jpg
c129e5a61670db6e943c3f1c1c738d71
0ee10d218569ce9539dfa11abea46e7eef4a5e63
30260 F20110331_AABVJZ griffin_p_Page_080.QC.jpg
84f632577f01fafd6bb5d6a3335ac270
8f1e8992a5052022fccee641c0bb8456e4102c7d
8423998 F20110331_AABVLC griffin_p_Page_090.tif
283ec3691867cf36d11e110a4447bd45
9b7aadb7e9bf3f1de070600ac6fe78157864788e
1762 F20110331_AABVKO griffin_p_Page_039.txt
1da73410b5de847ba713aa68e6824ae4
62b0d5b3eee9a037e016cbda56ff02278319211a
31074 F20110331_AABVLD griffin_p_Page_051.QC.jpg
5a3bcd959e93459fb87f8e8f0af15fe0
3d1d63aef71b44194b466b1fa1b5c11b35d98c32
82110 F20110331_AABVKP griffin_p_Page_076.jpg
0cb22ea313f7fa997faf2d378c15c707
0e8e7281fd6ae83518b29e843bc48740a1c5df9f
54365 F20110331_AABVLE griffin_p_Page_049.pro
d66ae8d862f18193bab183e5f6efbe32
e147ac89a7f1da61a442daf3bea268705a9db7cb
2120 F20110331_AABVKQ griffin_p_Page_054.txt
82cd0c2d646a95610f6af1e9a1af2600
4e50c7a653ffe53ed50f2918a9102cd79fa778fb
F20110331_AABVLF griffin_p_Page_014.tif
32e9ea8b353956da5f22d25b96f3223d
e28b3c6235624d9f070ebc6e16b81357d33d936c
93324 F20110331_AABVKR griffin_p_Page_038.jp2
259c00a0588dac0c158a0ccd1783d295
dcf8418f9f929447d40912e0f65a394eb3d340f7
2046 F20110331_AABVLG griffin_p_Page_013.txt
63e5db1cc10d8e51102d7346e7c9f5e5
9880b427297359c84b9ef054d11593e250437200
7023 F20110331_AABVKS griffin_p_Page_039thm.jpg
7ad03b672b0eb83842a086572c14b10d
e35337ab35f2d5443bde84182d582f4ea0020aaa
8046 F20110331_AABVLH griffin_p_Page_098.jpg
70dd12be29e288dd6612aaed46ff19e1
cb2b9eae972e55b585acaf059a1288f992653517
37474 F20110331_AABVKT griffin_p_Page_083.pro
2eddd0fad3fc102369980cc0f5f372d0
fd09de9f3586c2e78515f50142ec61cee4b6afeb
F20110331_AABVLI griffin_p_Page_057.tif
126e011495fbf5f63bf8bd8d98221d1b
d02e49392e108ddb153d91f5638bcfc3e2a93bd9
34181 F20110331_AABVKU griffin_p_Page_053.QC.jpg
adc15d78b2196af8d6def6c7457fa3d3
1c284096e6d9a93d5121c30bbd5185bed583154a
F20110331_AABVLJ griffin_p_Page_049.tif
e13841c2f91283ded95346bffc009192
630dccdee7c1ed08efe47df258af7eb2be62c505
F20110331_AABVKV griffin_p_Page_079.tif
bddcdf9aa49776c0c6e8b67299118dd9
abd6bbda855be673fc15db7d3577c3712ade1502
8339 F20110331_AABVLK griffin_p_Page_062thm.jpg
5cad0e9c0435a6fd876ee6bac2d5c47f
ad4d66fa4006fb2812ef7d7033ee3c5762cd21a7
104923 F20110331_AABVKW griffin_p_Page_058.jpg
e9061fa2fb947d856277831ca191c112
a5d0aff244e72ec72c72c6a1e817a75bb1ebc4ce
F20110331_AABVLL griffin_p_Page_073.tif
9a5d3d16e706f2119203b0790962f9a9
62b9346458cf1ffca3800824899d92b2e9c56530
2085 F20110331_AABVKX griffin_p_Page_043.txt
7bfb9156e7ef5a6429e202633fbce54b
7f4e0df1119ea995889c91c60efeb28839f2a449
8043 F20110331_AABVLM griffin_p_Page_028thm.jpg
d6f57bc63fe467050962865d9ac42baa
bc0b5b9ca0f2614ab53aa6b693d5c973f706b552
105461 F20110331_AABVKY griffin_p_Page_013.jpg
84d519307337d491cb20a68f50f698ae
5fd5f006e670d67cad4216b8e2d656a746a1243e
50839 F20110331_AABVMA griffin_p_Page_063.pro
939d3b6156bd2fac53a82192f77fd61e
4a4ffc926af0ae3028a47804eff8df7cc43d4e48
1797 F20110331_AABVLN griffin_p_Page_093.txt
9ddcf27a5d02774b75b51109e21dcf2c
98082375c656df8e88beeeec1a812cdc054abb51
49694 F20110331_AABVKZ griffin_p_Page_087.jpg
d12a600c13907902f9f0aad677c9ab37
47fcecddbda612babf44b9e91cc46a10403cd4bd
76121 F20110331_AABVMB griffin_p_Page_041.jpg
e847f374d35bf50df8d8ef90ffede1a4
cc9635cf9864685648da6b4efec8f9c04f0ff13f
7387 F20110331_AABVLO griffin_p_Page_037thm.jpg
6caec15d8fe15ea40b945ff8afb74c34
fd9c1dcd3cbc985cced47833e007ba494b790c87
F20110331_AABVMC griffin_p_Page_088.tif
1edbdc2656928467354678bd29578d92
381c1adc24f47b68b5f1ebc328459fdfbcf91cb7
1927 F20110331_AABVLP griffin_p_Page_064.txt
0d483ce0321f7f211de237f68b3eb9fe
ab68b877ac4c4f3dc99294123e4470f70b7f0bcd
99051 F20110331_AABVMD griffin_p_Page_049.jpg
a860151b57d891041f5e0fe3fd021056
fc632730249641623a07643825d1353a19378fb1
4919 F20110331_AABVLQ griffin_p_Page_101.QC.jpg
4846a93c71757d1ec2fda5c7eaacfbfd
cb26ac472cebe6a8961de8258a2be6e42ae419e8
F20110331_AABVME griffin_p_Page_061.tif
45355eff04076c23b28e210ccce68db3
9fbdff3fed319bc22acb8907fb26a852676d2ebf
102405 F20110331_AABVLR griffin_p_Page_063.jpg
5bead9462cef95ad0063ef8e74e25fc3
50749830d8a019ef0a1882fdf46b2fa60e9604cb
6867 F20110331_AABVMF griffin_p_Page_025thm.jpg
93baf267d9ae8167a24e4ddfc692321c
0b16e61f6b4832fb50a0050b131b39cda5f15c10
31791 F20110331_AABVLS griffin_p_Page_014.QC.jpg
84b0c4fe941ebe8cab3d67348f1dd7b7
b4aef6130f3c8e07c53e5e71a9d19d6c90068ef7
2126 F20110331_AABVMG griffin_p_Page_052.txt
85880db65ff290046c4eefc6fd81954d
72c6a4d1a178ffdfb8827ce69fb777c51276cc22
2125 F20110331_AABVLT griffin_p_Page_017.txt
3647913be63009ef6c8f0d64606b5317
ddae92d63852ae4a4b9b1af026ee41b2f518dddb
1254 F20110331_AABVMH griffin_p_Page_002.pro
fbe5848dfd84fb77f967210c08914c53
abc7800466f19918fa7424a16564fef8325482d2
F20110331_AABVLU griffin_p_Page_023.tif
0713d8e05000f68ab90067d239aefd5c
4a29c7820b7cd3dc07d1bbc01f941053902334ff
7139 F20110331_AABVMI griffin_p_Page_030thm.jpg
117376bd0a7585f0c512b0dccead4bd5
c7cb70ca093c9f0cc63a0dacb5702aa252e8607c
F20110331_AABVLV griffin_p_Page_082.tif
ffafc9ae5512fb7b43f6883dc3a4d1eb
dbab820df391bca5c9b41b2c260dbabf9ddc3b7d
45262 F20110331_AABVMJ griffin_p_Page_071.pro
d2beaac7cca0e049137bab9fc2ca6ea9
e14972c6913f0d7a32fb258729a7818cfc07deff
91047 F20110331_AABVLW griffin_p_Page_036.jp2
382cde280bf360b7b7efae9e8683937c
ba24a91d194bebae1ccf608286e9d06939344019
373 F20110331_AABVMK griffin_p_Page_003.txt
541740bcb228a1b4f0ad6eb91d503850
e5338a7f070a5d35eb30ff6d29896bc410c66c5c
98600 F20110331_AABVLX griffin_p_Page_028.jp2
e24c384c9085bdae52d0ec6a3afe9685
9ffda0b9f5eb43adeab41ecd2b6a5c42c965537e
23498 F20110331_AABVML griffin_p_Page_007.QC.jpg
6a25d7042ecd1b3d1c55c7e1cce9f74a
f2981d942726a9c5a5515cb33234578388b18402
693 F20110331_AABVLY griffin_p_Page_015.txt
28cfd1a54894df1cbe01e41e08a40914
efd4cef7fe06eb31e095e2680528439843a633a9
6528 F20110331_AABVNA griffin_p_Page_077thm.jpg
3d6720cb039395a9872837b57b89cbf1
34f1eb30f231a5b13022f0965a57b6dedfbe78aa
34177 F20110331_AABVMM griffin_p_Page_059.QC.jpg
468715e6dcdc39d4ccd0c65c43a60ed2
8fc42b0c59c2d3af1b32e445b49638b861dfa13e
4965 F20110331_AABVLZ griffin_p_Page_101.pro
29d2af31f5c8aaa1ed1ddc2c2f5290fc
a681d7ffe171df562df353368497b87cff796b6e
14014 F20110331_AABVNB griffin_p_Page_101.jp2
36e9f437365721bafa9736bc3034ce73
4a683410b02d774f09b1125f42928f557bab66ef
94807 F20110331_AABVMN griffin_p_Page_082.jpg
2aaad73f291bf620773185c75581e27e
66eab267d85d2540210236717261742bd96f9002
2255 F20110331_AABVNC griffin_p_Page_102.txt
e25a7e2715283f98415235f09de53f94
ab9ce4ecf4991703dffc638f6e766df9936952bd
F20110331_AABVMO griffin_p_Page_033.tif
b02d8c79a15e5de2fe1c65893b60a17d
6878ed9bc25789fbf5a9f9517f10fdb10268e007
826 F20110331_AABVND griffin_p_Page_006.txt
3e42c83df0e6ee7a7f254c122e0c7d07
b4d4e3e4e584d26eef9f4230ab5783feaadcc451
F20110331_AABVMP griffin_p_Page_085.tif
2d56e1be8526b0b0b874a0d1124d7074
6fa9a691f34299492f843930a12d0d6d406eb5ac
1167 F20110331_AABVNE griffin_p_Page_091.txt
23448361882c449de2548bb699ac9a6f
55d34fc75c9a7354fc885c2f6d6c44993b639734
99646 F20110331_AABVMQ griffin_p_Page_060.jpg
3640d74986e378f0a866c573a0e17b22
a2e5826c5df9e39b5aab5c55fd6bff39e1d7e8b3
37726 F20110331_AABVNF griffin_p_Page_008.pro
38fdb60f53f7cb1cdaac931842eede06
e8ece4aaf62fe8c3c6e6bdd52b3b0b94cb3cc92b
1813 F20110331_AABVMR griffin_p_Page_024.txt
19ab52b8d5d6d69b9bf4d881a423daa8
825915f30d4de4ddbdaf070738bb6fd41975df81
8164 F20110331_AABVNG griffin_p_Page_056thm.jpg
2302b5ae90fee54d48849d552fcf8989
3b670771f9469a898ca8b22d9f23da265b3f522f
38188 F20110331_AABVMS griffin_p_Page_093.pro
eab610c8ffc055a6f8b6ec706afb9cfc
b9c8ab8863aabfdb011ec3252baac7cc5b6f51d8
101394 F20110331_AABVNH griffin_p_Page_062.jpg
792979046e068ce0f891bdfc9584c511
0e32382f7b405576a745e41f93c24e0db3e0ca39
F20110331_AABVMT griffin_p_Page_046thm.jpg
bc5f22457d21bb00632313f0457a3253
88b9c23485838d6de23a7e724cf6b1b333aca7ac
26203 F20110331_AABVNI griffin_p_Page_076.QC.jpg
305a9544e3d34a869637851d858cd124
2aa4136cc8c6783577ef0bde4fff2d5ba8196564
2022 F20110331_AABVMU griffin_p_Page_011.txt
1f9e07ea2f6411d4e9e8d5a3cf2efce2
aa9cf9751f6dcb647f8b092de3246ef640a1da07
19460 F20110331_AABVNJ griffin_p_Page_009.QC.jpg
046eed13b9c6df13fbb56c6e7c3fefba
938d0a4fbafc77a120b09a52d3d105595883548e
1844 F20110331_AABVMV griffin_p_Page_069.txt
105bc9c426da79b8e685d4a4f9474a2b
cd9ad740d1b3804d825872896dbd445538ad7340
8058 F20110331_AABVNK griffin_p_Page_064thm.jpg
1421966448ef59806c9a3986e8f37119
bda80195a443b0ae65aef3081742051483bc793a
37144 F20110331_AABVMW griffin_p_Page_090.pro
66bbe748f6af712128ed755ad2797e6e
db035af069ad008b621cb29d6f946b42ec77ee9a
46130 F20110331_AABVNL griffin_p_Page_096.pro
f87d043ffd61d48078011e09dc9c0c0d
3e49e06f2d1ae049e7f2e8ffc0a08aca43076088
F20110331_AABVMX griffin_p_Page_100.tif
968c9aeff3d9e23223019902dd14d175
2b0dd8345c9c557c729caedf425200ce07d71931
101139 F20110331_AABVOA griffin_p_Page_033.jp2
4e3cd1cb827a4bc20f3334531665c2fa
5c37a2fb5fe5b0a4bbeaf9b9d3914c57797ecc8d
34771 F20110331_AABVNM griffin_p_Page_052.QC.jpg
7da10d45af38e837e68a342e7fd56c91
521fbf58f234ca73912e606b2f367b3e269680d6
1785 F20110331_AABVMY griffin_p_Page_031.txt
219311561690c8369641fb0dc1bb5dfe
03a27461ad9e80969c330570280d511514342b43
F20110331_AABVOB griffin_p_Page_092.tif
e0410f23a5ebcd290473ee2c6b0e87ba
f1784720a362f61b0d3f824f19706bb2156903d8
42270 F20110331_AABVNN griffin_p_Page_099.pro
60838674d7482cad4a27de2032d786ff
c54e896f8fa65dbc70ea8cf4b33fb032354131f3
8148 F20110331_AABVMZ griffin_p_Page_003.pro
dd95153772e24a9eb968a3b239db7cca
2776e7521375558ee311f46e08e2783211ff6e88
92486 F20110331_AABVOC griffin_p_Page_085.jpg
d196336afb7a0c215af3c39016000b60
3677e3173815cc527a0471ab785124089d4d8eb3
1807 F20110331_AABVNO griffin_p_Page_045.txt
92691d25b10750265dff675122cdc814
7a3342f56d597019d2dacd51cc1d48001de07d7e
7555 F20110331_AABVOD griffin_p_Page_050thm.jpg
187508cbf839e206f7ca4bf8ea7f1248
749707b9a14aa527e6508c5c164de8da9192d7d5
79976 F20110331_AABVNP griffin_p_Page_041.jp2
a5ac8800bcb844a7391247b08512ea6d
325b031275fddae55becf60e7f7cef3726164beb
8153 F20110331_AABVOE griffin_p_Page_061thm.jpg
9fc2f4f18815e1267ca049a781fdbc88
8163aef64f1c25aa7058794c7e216d6452bd4731
39093 F20110331_AABVNQ griffin_p_Page_040.pro
91ef15445d8a0ff867fd108b2d1dad43
755431116bedb52a6db8cfe519435d5225326e41
84096 F20110331_AABVOF griffin_p_Page_031.jp2
01fd83805a1f0dab66951d373c7ec8aa
f893890a5b36dd23edbf0483fbb912292f5ed125
631 F20110331_AABVNR griffin_p_Page_002thm.jpg
0162dfb887b90b0d685ed2a5342aa516
8d8283c22a9e40eae0a43ac213f39cda4d1f672c
F20110331_AABVOG griffin_p_Page_035.tif
fede051d06d686cbf794249454a4a585
519a6472d6a3432f35352272c2dbf0b28b4b5ec1
87099 F20110331_AABVNS griffin_p_Page_026.jp2
6a5906c4fefc1cb6c06783cbe3b1310b
6fe14b93c5540b692d2fa545e6b5a2ded88faf14
18977 F20110331_AABVOH griffin_p_Page_006.pro
3d47046bdc7fe443570fe9b11127e31b
2f91a361ce1804a3af8bbef17aeb8c38ad242755
7534 F20110331_AABVNT griffin_p_Page_032thm.jpg
52a217403d0e3baa66dc078a8cfbcd73
5c7bcf71c6f5f9ce43dfc725f48beead9d61fdfd
38860 F20110331_AABVOI griffin_p_Page_025.pro
81d87977e1988ba09bfd4c47a72c0807
2d4f34edeccc5fd3e31d0a0d5b621e4be41e8744
25271604 F20110331_AABVNU griffin_p_Page_007.tif
176bf4eb3ec99e35e1c4ceacef713aa0
0152a9c7e9548849062b0f879c84678d3b418760
101936 F20110331_AABVOJ griffin_p_Page_092.jp2
7faac443659d26d179f8be7653fbc011
b6fd39ac9fda3ab230cdd76f5ac25f65a9a2e19c
29824 F20110331_AABVNV griffin_p_Page_066.QC.jpg
d7802d542a8ec8f794f32fa622d37eaf
dd251838e88228ebdaa719b3e9c4be1bf535508b
F20110331_AABVOK griffin_p_Page_075.tif
a59622066496a0141ee00b27e29355ed
92dad552b396c2d9a02b2f317d43410376dcf17c
2023 F20110331_AABVNW griffin_p_Page_081.txt
d890d12f9393d3b7489ee2fb417f8535
b931ef1ba07ed46d92dd5a5b20aa63a57cb418f6
82502 F20110331_AABVNX griffin_p_Page_040.jp2
9741d7e1a76ec949240dc0046f84e458
1260a19894405c92c83335cd0d6e0d73ef10d6e2
95980 F20110331_AABVPA griffin_p_Page_033.jpg
8033a159e2dd9024812bef9912b0f281
fc506e43495f082aab21c14163d02bbd325f4da7
64323 F20110331_AABVOL griffin_p_Page_009.jp2
7bc3d2397d007de6f77ec1a3689e8778
e9a8a0860ba162d6f45de1def1d9b24a7a3406ce
7768 F20110331_AABVNY griffin_p_Page_068thm.jpg
55a88010641fd7e69b712df567a10f08
d3e2b1a6bba8e7fd35f6113dfdb8171ca2c9486a
F20110331_AABVOM griffin_p_Page_044.tif
cb6647bfeb8ae6889ac247903a40a678
508fc8ff5f779da4c2ab2c45be36143a96b3b430
2040 F20110331_AABVNZ griffin_p_Page_026.txt
3ca7b53a3b99c5a2ade192aaaee9ccb6
0b698835d9d105641315103d6ab094eca772d130
44275 F20110331_AABVPB griffin_p_Page_100.pro
bb0a4ace7581e0a7206023997eedc74a
2f0c45a5bc6d8685c7611b323bee288f1168454d
974 F20110331_AABVON griffin_p_Page_095.txt
5b2971a419ce5830ba794d7763de9751
eea0d96dd2cd249a2d193dceab5ccc28d0af795c
28665 F20110331_AABVPC griffin_p_Page_037.QC.jpg
1d0cd3e8cb3c94563a79b8b2d72e1cb4
0cb78bf91cdf7d47ccfc4d453d44a469720bab20
71645 F20110331_AABVOO griffin_p_Page_005.jpg
183102899ca5f6cc8bca4462561d102d
6c3c8faf39f48620b4bf0a0922326eb56129a83c
238 F20110331_AABVPD griffin_p_Page_101.txt
4b57a87175f9732b802fdb7e49577e48
f6bbe3312444d455ba4a10d43bdbb970bdd378e2
85913 F20110331_AABVOP griffin_p_Page_076.jp2
d74b96f25eeab2b0ee37a37f93d863af
f7f8276584e58b2c9e465cbdf929236fa4ea9bc3
52778 F20110331_AABVPE griffin_p_Page_053.pro
cbeb0259bca8da31c4a4746fafde781f
68d9e0df74fbd782bcabfc9abc135bbebbf6dbfa
7868 F20110331_AABVOQ griffin_p_Page_092thm.jpg
bc204729a3f8e454c09b8fef6648a429
a2322cf271f3efedc6e963bd83bf4e7f573d42eb
F20110331_AABVPF griffin_p_Page_031.tif
5f7ca76a4caaf1cd9d715f31a8783863
3f02a84a125d3eb7938eb5518c9f3761230a55da
7017 F20110331_AABVOR griffin_p_Page_069thm.jpg
1fdba37fd3114a0bb3f3846898c47376
cacd715cfcb5b428ecacf8031028299fa33053f1
100076 F20110331_AABVPG griffin_p_Page_088.jp2
bc0cb345531d19d1464fbd8ddecf6a6a
06c996f539a180687efdc66f736eee3313fddb85
29691 F20110331_AABVOS griffin_p_Page_016.QC.jpg
8845810a65cb2a4452541f14192f9e4b
078a88e9ed856e034fb73aead412c894f17992fd
8369 F20110331_AABVPH griffin_p_Page_011thm.jpg
996c55fea67334a985867b585b57e6fb
82545de4aa4229f4740b505c4190f8ca1e116af6
46961 F20110331_AABVOT griffin_p_Page_104.jp2
c335f01c945d07bf420200d3c876b619
4378b7bcb41081d85b288aa7c75615420d56265a
6861 F20110331_AABVPI griffin_p_Page_048thm.jpg
de89a7e092a62f7d6cb54784c3c6b6ce
5d31f8b018912dfe5d3a81435456b162cd43930f
80729 F20110331_AABVOU griffin_p_Page_048.jp2
0c3f3763ddefc15dcf2abda6c1c7ec46
005c33d3147c5641b0ae26ca0377abe717e5ee55
79050 F20110331_AABVPJ griffin_p_Page_027.jpg
e1fe1adb8051cd99d26ee6d43a1958be
57a3b199809cb73ff29acc2f883f3cf41305cdb1
2060 F20110331_AABVOV griffin_p_Page_055.txt
475f03c6c75c420c4369326c51b618dc
6eae95b14303ac96e5129e960a36bc5a590c2d05
91511 F20110331_AABVPK griffin_p_Page_034.jp2
240bdcdd22bff4bfd4ec00233ed9920f
180a41373450ec1a9232ab2fdeaa584767b8f4de
40715 F20110331_AABVOW griffin_p_Page_045.pro
d2c996d63b5dbbc3926af708563fa670
9beb0c520c0c62612266b8c4211cfb7b2d85a92a
6501 F20110331_AABVQA griffin_p_Page_090thm.jpg
4f43ddf35fa8b4cb2db4481383d52984
b8e024be3847947613eca7c1a3a7a14be183fae9
41842 F20110331_AABVPL griffin_p_Page_064.pro
ffa3b09e7434aee77a1952db80e8e94b
7704a610648f18392e48430c9bb52d6d87f3ac23
F20110331_AABVOX griffin_p_Page_096.tif
96c05f030243159e5412691b756cc1a4
f9ff477478a78e05c8722a521efa819d8d62ddda
97303 F20110331_AABVQB griffin_p_Page_073.jp2
b92420b4a0484b73664e9b0d62ef2a8b
2aa01304b35d1e46e495f5e99c194a0559987b6c
27033 F20110331_AABVPM griffin_p_Page_069.QC.jpg
5bbfcd3f9fca5005680575f2bbf138dc
98b177802289958b532c4e00fd01cb45442f6914
7638 F20110331_AABVOY griffin_p_Page_082thm.jpg
a0d41ec934950358c1a4633401a5275a
a3135eef4ee795ff1b5e8949fd81c5d811f198dd
27479 F20110331_AABVPN griffin_p_Page_072.QC.jpg
535a2ccb915571c361a0cae7d9f257c0
1708d992c4e0e92aee2691e82c784de60272a63f
2655 F20110331_AABVOZ griffin_p_Page_005.txt
ee208338d6f0b460905f593850606b17
4483633a7cf6150c424abe48f98ae0d9773f45d9
864384 F20110331_AABVQC griffin_p.pdf
510e3c729689ac5ad6b3e304970088e8
7a3d5c913f79d018ece6649896c91f18e2e0374a
BROKEN_LINK
www.cantera.org
F20110331_AABVPO griffin_p_Page_003.tif
8836d388df6124925d5ff42caa422d41
3655d6b52c2f0fe066db8e81f06f218e5b19125f
825 F20110331_AABVQD griffin_p_Page_104.txt
29c376ed9cd5803fc90cc172691c07e7
d091a33fe3c6fb0b4f26880136690dec6f367ecb
F20110331_AABVPP griffin_p_Page_016.tif
7d5f76e05d05a1c28bcfeb3dea582dbd
910b416104ac7a6df567637de799a486006215aa
1949 F20110331_AABVQE griffin_p_Page_044.txt
cfa63cf0b921eacc111b860b8f84180a
4b831983764635ce9c9fab5c56a21574b88112f7
101211 F20110331_AABVPQ griffin_p_Page_081.jpg
001415a0e905c6a994a3c0200fe68f4c
5a6d913b83d2983c1b53f9bddaaa9e79573ef1d7
38079 F20110331_AABVQF griffin_p_Page_015.jpg
d40157321210b611145bb8e9f2327158
4125fb85d8a04344e026d5c1486ca08aa6be75cb
1548 F20110331_AABVPR griffin_p_Page_084.txt
94431776515743b7d391a3528fe28072
3e7b9c1d03a7ba35845c5737a678b032d339e715
27673 F20110331_AABVQG griffin_p_Page_035.QC.jpg
cc584beec8c8d3cd08c2a8ea599f50d1
a63af8948086d63851a622b025f73e31a239d455
1872 F20110331_AABVPS griffin_p_Page_066.txt
b72dd0a7e8f343c31b0a4489d6195034
17567931c4e928b617289627f1f1a85110cc007b
29743 F20110331_AABVQH griffin_p_Page_075.QC.jpg
29a24391cf1827c9872fd0d64044c98f
579c6bb72bfd84dc37ed13d64db76ad86d0d5b05
F20110331_AABVPT griffin_p_Page_101.tif
931c08727cd936f322b8d73e34284c4d
46ac5960ce39b6fa959086640af7f1d1dc41b603
7616 F20110331_AABVQI griffin_p_Page_038thm.jpg
f0e7dd8ef78cc374116679dc55c4c325
8419543e38c859f3c04d03c69d6bffeeda4a87db
F20110331_AABVPU griffin_p_Page_056.tif
cae1fa742ef41363c0d45b12e80c164b
ef1bb69249dd59524bde4e95c87a4a134cbbf33f
33584 F20110331_AABVQJ griffin_p_Page_057.QC.jpg
35d08dcb33b2a888a0570aa926d37d9c
9d8aaf82330f74c75c8a42974bc73a2d041599fd
7782 F20110331_AABVPV griffin_p_Page_043thm.jpg
1cdbe1c8cd13cf2606af16c89030a584
7a6de05d7ea3e0e22fdf41cc8cf9140b1f4573ae
1972 F20110331_AABVQK griffin_p_Page_046.txt
95cc17369768b063d86a52b6e1453db7
e2c45b969dfafc801a4dc4f09a697c036167f0a8
1908 F20110331_AABVPW griffin_p_Page_080.txt
9774eb0cca8228583335dd2bc8308798
8b016018afef64d99375e2218a0df80b32726ffb
32449 F20110331_AABVQL griffin_p_Page_102.QC.jpg
1a5bdab17dfb4679fb2f1186241c8b90
196775fdf3b96ba5cc9579614ce103c43f60176b
2296 F20110331_AABVPX griffin_p_Page_007.txt
974a46dd6c9c8d4892e15be376914a13
2b31d1e0cccdc1a5ab1d8952531f88d51d3fc5a6
89679 F20110331_AABVRA griffin_p_Page_068.jpg
46192ac17b6c32cfd11f266bb76d54b7
98ee0fef24f18b790f5fa6bbfe3466ef56f5bd67
36990 F20110331_AABVQM griffin_p_Page_077.pro
bd0b20b144aca0bce81ff3ef732cbf49
ce31f2f2ae650a372655f9ac76e64519db44d3f7
24692 F20110331_AABVPY griffin_p_Page_094.QC.jpg
8a4e01639eb96370630c133ff92cd845
71724f48533e51dd26d92b4ea67cfb01594ea0a8
28471 F20110331_AABVRB griffin_p_Page_009.pro
18de76e3fcc062137609b14dfdc2f1e0
86c3a6a6b58a3aacf06f522543e02e6b2a89f7fc
8367 F20110331_AABVQN griffin_p_Page_012thm.jpg
ec36bde904f086b3ade3aac77442885d
8944c250f00a96101348560d9901c58665be629e
1051980 F20110331_AABVPZ griffin_p_Page_102.jp2
379f34718e4e952fa8107d6c95970c63
4211ac8a5240c2297f62c28aab01e70e0bbab120
6815 F20110331_AABVRC griffin_p_Page_076thm.jpg
36df20fdb31e77bf09856467e60e29e7
0e98e010defa31132e00ec1c747d7562ce5c55ad
33012 F20110331_AABVQO griffin_p_Page_063.QC.jpg
86840b57e883b5c4c546bf5881167ef0
b11e23bc5ceb72456ef6ae3a09b763aba5d12a05
7491 F20110331_AABVQP griffin_p_Page_018thm.jpg
ac88986d8c88c2ae490048516c7ddb06
b87d03119ec90ed6f7de10dd6d424f7d9fa3f88d
478 F20110331_AABVRD griffin_p_Page_001.txt
5ed346bbea14a0a82b77f5797f9a3ec9
6361529caf6afada07da6bb9818a99287596d756
101349 F20110331_AABVQQ griffin_p_Page_061.jpg
e8f816d714c0bf3cdaefd9fdeee522a2
8faa4d21c6899f7be48aaa0a722f9ffa5861f633
105941 F20110331_AABVRE griffin_p_Page_052.jpg
6c461fbe0b75c1f9e01d4cb551f4dcd1
07d048bcfe96cbad00d9cd59859c29c421de0b3c
7071 F20110331_AABVQR griffin_p_Page_045thm.jpg
ae9acfce6b9478555603d908031ebafe
69ed09e6853b32ff9f963da5df0643a388331d11
7540 F20110331_AABVRF griffin_p_Page_067thm.jpg
48de6bd1245d1457296db5ac4e02e9b5
c25ded03240ef9787f7d9c7f9eee8c51db6ff084
53308 F20110331_AABVQS griffin_p_Page_058.pro
f12678723be7b4a3f56c68e937fc41ad
2a86331c3305c7be48f1e84b2b8e52c62bac02de
87868 F20110331_AABVRG griffin_p_Page_037.jpg
615fb54b20b9a8124f22697a0aa1a994
a674201b527005e60293f45fb131f127dd00f6fe
8407 F20110331_AABVQT griffin_p_Page_081thm.jpg
8befa03b771dbbbbf435638352a44934
810394de8099414861fd5dd5a77304fae1f195ef
37690 F20110331_AABVRH griffin_p_Page_029.pro
df34e8bd72facfc3c2aa00972d0a3ee9
4e8ef66b3c38d4a58236bd7f5642c17277b2f92d
F20110331_AABVQU griffin_p_Page_066.tif
6da26199e4c82187e0c2599da357196e
b9d19d736d9c1e5bd54890cb0fa126e05763c5e4
94856 F20110331_AABVRI griffin_p_Page_071.jp2
310e856ad87ee9647130236ebe31269b
0d2e68e49442442a5764973916484625cc075fac
91130 F20110331_AABVQV griffin_p_Page_100.jpg
3bdbce54cb0330ad43b498db297a2e71
0a5553e9d47919476de2373476df0932ffc3f9c4
2948 F20110331_AABVRJ griffin_p_Page_049.txt
5011ededf035055fc6aaddc2be21ec0e
c281228e76eb228570f5f2a4d15fdf00fe583dad
F20110331_AABVQW griffin_p_Page_095.tif
a99c17633878204eefe026d15ab97df7
4ebc18bf83a66e6bee71a0b8dfa5abf90aeb8ba8
7947 F20110331_AABVRK griffin_p_Page_088thm.jpg
8a9a04501c21f2ea023826289fcb734e
e20c7b2b73e6043817cf3f99e34bcc38fe57156f
85382 F20110331_AABVQX griffin_p_Page_008.jp2
feb9ed9c624027f0566d09b9d193e242
14e7a1c2af090ee8498dabe1dbb7d25e999f700d
106059 F20110331_AABVSA griffin_p_Page_097.jpg
4810546ef4961f87e9456be9c7f5380e
23c50634378481cd55f5f2316a0695fbfef8c3bf
115139 F20110331_AABVRL griffin_p_Page_012.jp2
8ea0f522c09554c03040e1f76a59fbef
6cc26cc2f5868af414fc70239b3b88a2a0b3e96b
96543 F20110331_AABVQY griffin_p_Page_016.jp2
77d9e0e85cfabef7eb47ce2e754c0d16
ec65fdb29288c57db526e87a1f02e1103991d2c7
F20110331_AABVSB griffin_p_Page_036.tif
fab0e84f887c65dcf990b3d1e8700b97
863fa5bbd0bd1bb5ae9d0d1890aeaabddf2c7f75
1729 F20110331_AABVRM griffin_p_Page_099.txt
04a396aa47321e5e37832478545430e0
d92b58d19f0c107da3d94518d1c8d7d61803f4b5
41537 F20110331_AABVQZ griffin_p_Page_047.pro
93c868918a29354266543d90173d8c4d
84dd9b5a7035083b363705613531f3cba4551992
105972 F20110331_AABVSC griffin_p_Page_056.jp2
ff975ea1251267916edaa69146d78c5c
e374438c4a583b1e045526d49713b2a2b317523e
20679 F20110331_AABVRN griffin_p_Page_023.QC.jpg
eac3a36dfb3bb77f89ae261ab8cf80ab
92601f3cf4660af3b5c84a6b5a7a268303000bc8
95661 F20110331_AABVSD griffin_p_Page_032.jp2
2b2d4966d7bd32dc411fa7ab6ed6511c
a2205e7539e216661b84a0f083adf248531313a0
7358 F20110331_AABVRO griffin_p_Page_085thm.jpg
d7df68c50605ec6631aac35745a929cc
a7990bcfd44ad0f920f36ecc6ee45419b135b3ba
103094 F20110331_AABVRP griffin_p_Page_011.jpg
2b5564984a87e6902dec87a5df30c734
4160b065955ec5a3470e54e62dd3eeddd79c1e87
48200 F20110331_AABVSE griffin_p_Page_082.pro
eef98f617e39c08648e43bf99ee1444d
596ebdc6f4fb682675ac32afc8f11dfff9e07982
13661 F20110331_AABVRQ griffin_p_Page_087.pro
ca0d5f6df2a81f1eecb1e5a4772ecee1
fcb18766d6b3d35236b5a4eec394c321bd78246c
2008 F20110331_AABVSF griffin_p_Page_056.txt
d07940852db078279973e3f678f7fe45
17baae8f80180a270e0073c748f3ac7d6997a8ce
106699 F20110331_AABVRR griffin_p_Page_081.jp2
f83cc8f1dc47173a732eeec4cee92a57
bd0b79179f590922a7715d6a97cf6740d7ec2086
24703 F20110331_AABVSG griffin_p_Page_001.jp2
34cbf938febc07108a4e12e09200ebef
397aa5311519a0e926d03dccdd2580f4bf1d5a79
95002 F20110331_AABVRS griffin_p_Page_049.jp2
6ed1a8dda6505cb3b72faa7fe76539bf
0fb5b8015837044094ff5459e8baee5888fcd8ad
7039 F20110331_AABVSH griffin_p_Page_083thm.jpg
99aa9789c991cb5b48ed96133b82a863
9842216ba0ba9e35a98ede290bdfa9d50dec0b57
F20110331_AABVRT griffin_p_Page_042.tif
344bdb0c194d694f6086f060b0224702
c52e26ac98bfe45c39ee60a4aa3b3c219ff8b4e6
110646 F20110331_AABVSI griffin_p_Page_011.jp2
ad9450582499351043a6a0c78b610fb4
9f75d9edf0aecffcd8d43fb34044062fd497a86a
25103 F20110331_AABVRU griffin_p_Page_008.QC.jpg
e6187b6b95e4f5b133d686a74343152d
3ce1d9962d0ef4ff493742530e9200354599cf19
34134 F20110331_AABVSJ griffin_p_Page_097.QC.jpg
b54ae580a6e8426379da6eb3447d6bb0
e9314eb09f801a72c96bd6c3eae575a8178a8c77
27831 F20110331_AABVRV griffin_p_Page_096.QC.jpg
acb2351e6fa66b57707e4f768495792d
922d4ccf50e7251c89be0a4ed24b93ebcf0d8ed0
8082 F20110331_AABVSK griffin_p_Page_049thm.jpg
7ed8bb525c0ded4349a4eca9195ceb2b
5a0bf1093761047acccd22d6e3fe91cbbed64160
93774 F20110331_AABVRW griffin_p_Page_080.jpg
f189a4d38756c82dd404709393f193a6
5abf9294b84e40add0d53918ffa49e834f481757
19586 F20110331_AABVTA griffin_p_Page_104.pro
599091435439848db9866d867e518911
939f87b06dc25d1b91014070c020a25ab94b1731
77565 F20110331_AABVSL griffin_p_Page_024.jpg
7690af0cbc0a13bec87c91d2743bc2d6
ac6c06ded8f3294d9dbbd1b84abf49dcbf4abe3a
5577 F20110331_AABVRX griffin_p_Page_002.jp2
7cbc131ecd0a0f1eece40dc206f1747f
deda126129276c1c981cb541f1ea1e49644797dd
2069 F20110331_AABVTB griffin_p_Page_073.txt
19d50d6bcaa75c810181a54490ff53f0
eed6304a075d92b65c549490e94e95bbee6947ea
F20110331_AABVSM griffin_p_Page_021.txt
74cc53a14e04a2857fb24cc25207f29e
ce013a465fac8eb866f4cf4e2d79f0bd44ee506d
F20110331_AABVRY griffin_p_Page_022.tif
e6cdd1f1f57f159b2e55712a9524235c
507585993f4fdf280b8213a0f4e7023bd07532e1
82757 F20110331_AABVTC griffin_p_Page_026.jpg
d119f257c9ca9b4dd7ecb0dc7ea45a53
a55ca54c50a6c7b0a3d0f14c5627a696e51b73c2
1753 F20110331_AABVSN griffin_p_Page_010.txt
c00a1a3463405a631208374145eadf25
6cc66f46ef4f9b64828dab06dd7622d608e807fa
13255 F20110331_AABVRZ griffin_p_Page_101.jpg
8c4bb7b13e6eede88261b52b486f3174
93af3a4ea059fe36b3143747f55bfd7bf2a9ebbf
F20110331_AABVTD griffin_p_Page_097.tif
806a9123ff17ff9d8d093d8dbc1bf50f
9d16e290430e66716ae06909443043a2081210f6
114 F20110331_AABVSO griffin_p_Page_002.txt
7029d26cb782fe337784f9a20f3011b4
273176de49bbd6a99edc970a909db39b42a3a59b
53344 F20110331_AABVTE griffin_p_Page_052.pro
b9e01ea60932e226e44a8752fea725aa
a6875030ef52641586b4fe2ba5c50b711b5e57ac
25677 F20110331_AABVSP griffin_p_Page_040.QC.jpg
fc07c27c3de1d1a577fa109646136d61
26ae154427d14578c7c39e67e856efacb02babee
F20110331_AABVSQ griffin_p_Page_091.tif
9a4842f6f490be2e9f04a7a9c4bdbf9d
3e99116caa1a77c0167a79fb10b1ab2a653d737d
1453 F20110331_AABVTF griffin_p_Page_023.txt
75fca22b150178e776bbccd80ada930f
11a5670182fcbd9dbdd11e1bd4f0cc87f1249996
35214 F20110331_AABVSR griffin_p_Page_054.QC.jpg
0bc17f4dada99594f6a4be9328b1c8ed
0b38b9ac0d1e136260ecd0ce247f35d9bf3b7cc8
48891 F20110331_AABVTG griffin_p_Page_014.pro
b9b716b130c936a916c96056ba85235a
289eb6de88ffb349a0119cb45b0256224885fe06
7092 F20110331_AABVSS griffin_p_Page_027thm.jpg
0f220ae8598423d74bf666ace842ec7d
086560d44da933649b6cec059ce98631e827ccbb
6465 F20110331_AABVTH griffin_p_Page_024thm.jpg
8d58a731524f4c103bf81d47d5087f91
3575a9158cc499f9891c56d0d1c77aed74ad7c85
34072 F20110331_AABVST griffin_p_Page_012.QC.jpg
7a7eb2d44c6078139bf4cc3d5eeb4dc2
159adaa45c5c4592dd048e191ce1d1484b8f6001
49021 F20110331_AABVSU griffin_p_Page_088.pro
0b7fa98244517f85042b674ee5f7a2ce
a85baf5d22c37be1dc97e9e23cfa57f103876970
90605 F20110331_AABVTI griffin_p_Page_038.jpg
0d297d92ba4693f492d307ca0af81ff4
4345a8c161ca8bd6943f5e107171f794b15afc8f
97461 F20110331_AABVSV griffin_p_Page_100.jp2
dd5bd2fc2f325e63fb7ff48924ba6c26
c4c5f7e36bb021b10fea7349a09880587a6743ce
76949 F20110331_AABVTJ griffin_p_Page_093.jp2
38affb344a491d2baa7454fbcd1efd6c
ae59c423512044ee2f94c733a960932a199f3984
44969 F20110331_AABVSW griffin_p_Page_068.pro
215178a7851f456ace9ffa404fd5d4e1
387f5d9a9f0c18e81c5f15738231c4ee359e2b80
4733 F20110331_AABVTK griffin_p_Page_002.jpg
574cd512bba2043585b20c84842cc6e5
1bf19c5d68daa188824fc509f9b17b9ab1cc5e5b
F20110331_AABVSX griffin_p_Page_067.tif
f1316acbdd408daaabe096f7b71efc97
9d547642f1c5408ea1794c1f6db7e410c70e4efe
33976 F20110331_AABVUA griffin_p_Page_062.QC.jpg
75b9cffd8adbee8e9ab4ca2f62158185
08ccce3ca286c0a77b461bd7a14b045145f9cd4b
29865 F20110331_AABVTL griffin_p_Page_071.QC.jpg
35c75d91fad4a44024fdb160ed501e3b
77fa754603273ddd8eb14f3c25232e888ce8fd7a
90630 F20110331_AABVSY griffin_p_Page_089.jpg
fdea98732c3df8948228188f4ff01f06
8e47f639c5d4dca50817ba9829be1386f7893a72
110918 F20110331_AABVUB griffin_p_Page_055.jp2
dc3d6b5a4c1393e7a1679716d29dbd77
d10a5f0d29375231a6df2635bf4b2868b2192877
89489 F20110331_AABVTM griffin_p_Page_019.jp2
fda745e6d5afba74396be925107a80e3
28c0c8dd3f80d3da86ae009018d598304a33ab8c
786174 F20110331_AABVSZ griffin_p_Page_090.jp2
d941ca34c24c72bcccd8112b230e15e8
459fec2309609c38615a84438a98e3c0076b33fc
80207 F20110331_AABVUC griffin_p_Page_083.jpg
ed54fa83a516b838bc04ebe7592d148c
524a987efed51e66d6653fc4e9f2c551cfd4fbac
18632 F20110331_AABVTN griffin_p_Page_091.QC.jpg
6d2ded6f2af76da2906e01b69714ba1d
7c2cee0208da6a31817bd9025b3d05d08381327e
2433 F20110331_AABVUD griffin_p_Page_092.txt
868ae33cceae3a2cad1eb554b3537101
9971ffe187fe89c2edbeedcb756fb9628337acd1
78371 F20110331_AABVTO griffin_p_Page_093.jpg
7340dbbbf4cf3921535e0e6da29590a9
d4a4f375b17616a0d80fb40ea39255213a91744f
F20110331_AABVUE griffin_p_Page_041.tif
59f051851ef4240ade15cb86bab94e06
110421593ae2862fbd5c9aed92d353f8d7c41e8a
F20110331_AABVTP griffin_p_Page_065.txt
dd3f67f8daad360554e3325d6d118387
7e3d4a41677da13d0cf8b4cd0e66b53d97ddce93
30098 F20110331_AABVUF griffin_p_Page_088.QC.jpg
07e7a89189ffe934c10a0de566074087
c2fd02815144e9a09f38ec2f071c9500db9a2cb7
1051983 F20110331_AABVTQ griffin_p_Page_005.jp2
b7963d4ee744af4be3ba5709a333dacd
f7c6c43fa2a036132c71afd0b83ec0b6df7f7a43
1711 F20110331_AABWAA griffin_p_Page_076.txt
2324dd42635520309e25419b842c06ba
22369a5d40fa2897fca04c2b7ec691b6c350ec05
30964 F20110331_AABVTR griffin_p_Page_043.QC.jpg
ec8da394672276cc26abc76d0dca2968
8564201b2002bf0371ecb9f1d215e721f283d16e
1653 F20110331_AABWAB griffin_p_Page_077.txt
70e9d1995c929f803371a400d86cd766
684e4219ff8a731cf3d3decf5c4d7e42fde28603
36565 F20110331_AABVUG griffin_p_Page_103.pro
e65fa7792a85721224187b7967ec4464
bf6429e0c5c31a82d8b52956c067f73bc4b2fbba
24320 F20110331_AABVTS griffin_p_Page_077.QC.jpg
cc33d9559fe7569e51ad0751428de3ce
6ac2aeaa2525ab08d5150f8e48be9722ce9ba0de
1732 F20110331_AABWAC griffin_p_Page_078.txt
1aa3f5e966c4b3af0a563be2e16ae0d4
2f584b81171a82d6442a94b86b549856f14906fb
7519 F20110331_AABVUH griffin_p_Page_080thm.jpg
b68f5776269d5f6700feff67de9de164
5164c56a56503f40255f668f4cf032345f4d5236
78854 F20110331_AABVTT griffin_p_Page_086.jpg
35251c81bce46b69d1dd25b000e18e78
7d3aec1fd8d3f43f83c16657f0ee5fa2a93ed5cc
1850 F20110331_AABWAD griffin_p_Page_083.txt
f70a273333e3ea92bdfb57b12e2b98fb
75f35a412a6c463d4275d9f11d0a079a6a6b9bd9
77844 F20110331_AABVUI griffin_p_Page_077.jp2
8156c46ae9014b21d0a9b36012c563f2
59af8fdba54ea3eb081119b646a6753488c904d3
F20110331_AABVTU griffin_p_Page_074.tif
16ea15fc61cea2a4797badbe885d5bd9
d070894f064921dc6ad1065a146eccbe5aabda57
2254 F20110331_AABWAE griffin_p_Page_085.txt
3f3a076c73b0b937f52c2286f978843f
7ff19a797c1ff83259f03c83ffd6d282041026c1
5574 F20110331_AABVUJ griffin_p_Page_091thm.jpg
c1c280b3395f46ed677c111ab5c6e564
96091ebb774980f2a077cd0197c7681f7c288c23
44055 F20110331_AABVTV griffin_p_Page_046.pro
4568007cd7e0cb4205a1fee69ca3149e
e83afcf584feb975793b116a8480f3eefd3e71ae
2172 F20110331_AABWAF griffin_p_Page_088.txt
b1ecbabb8f8e025cbdf242a519f14af9
1ad1b1eeb29929e4b79eaae6222665b7129e0304
33210 F20110331_AABVUK griffin_p_Page_056.QC.jpg
a80a57658a018cec0a5ae33fb8e276a5
22dfe94d504b4089db6d3aae3c36d9d6b981a7a7
F20110331_AABVTW griffin_p_Page_094.tif
adcaee1089994b151e799e3bb4b486e8
cf06e50b29b751eaf2891491159abfa4a4a8d714
48288 F20110331_AABVVA griffin_p_Page_085.pro
1150f4677c0b96e4e1822120dac4a0b9
bf7a79f0203c9a1f81e16bd4973cef3f294fef17
F20110331_AABVUL griffin_p_Page_082.txt
cb7689ba880748b39aa89c7c953da9e2
3c0c965f2777875af25b7b77a31d8c23cbc7825b
112919 F20110331_AABVTX griffin_p_Page_013.jp2
9973ef1fb2c58ab69a193dc992aa639c
a3d2314630aab4dc810efe38ea92c3022f2b0a7f
1896 F20110331_AABWAG griffin_p_Page_090.txt
14cc86513dbca8f014a4ebc4e52bae99
6eac4a9a37a40783bec71454daf53f9625674e98
52571 F20110331_AABVVB griffin_p_Page_055.pro
d1c9683823708e5983550272a8bdaf9f
c02965e5c241510b2b84570b8f01c5545abe0d65
F20110331_AABVUM griffin_p_Page_002.tif
bb9f2c837e6593293ebc25ad2dd9966d
bc725fe68bad655f97430d89a3ca26799ce45cf3
96557 F20110331_AABVTY griffin_p_Page_088.jpg
d314a70040c604f8baa1e0e0e7f69f88
dd5d2fa0908625a86579897fbf764c9d6f73dd6c
2090 F20110331_AABWAH griffin_p_Page_096.txt
70a87169c1d1248aef007893f14bd197
efe1bbddef45bf6472001a46e18daf1428dc3ba0
7438 F20110331_AABVVC griffin_p_Page_044thm.jpg
48cc5b43d7ea1f7597d10b6519c26df8
adf98d2abfabe19474da7067db6525e4e5a3ab38
F20110331_AABVUN griffin_p_Page_001.tif
39ec5181b853d4a09e2c899625b96bc8
e23848c2f08d54ea338a61f0b5b35b33349ffcfb
28564 F20110331_AABVTZ griffin_p_Page_034.QC.jpg
825474561250bd6c955334c61c148803
9c16c8f1a3a0377528dfe435fa008b721ea02746
2105 F20110331_AABWAI griffin_p_Page_097.txt
74217dc7880d040b769f338a32ebd460
86ed3a6b094817495bce087805a59101573ccc63
92449 F20110331_AABVVD griffin_p_Page_075.jp2
7e000bdae52ba994e60bea0f49f36bb5
d2cd5d35de670f13f8fdecd8d42bd5d52c38411b
84567 F20110331_AABVUO griffin_p_Page_072.jpg
56b28346624acf96aaac4ca6d99ca2d8
ffde9091428a89b1bc06f49cefcdbf0b678df0b1
1823 F20110331_AABWAJ griffin_p_Page_100.txt
3e98601b945626b39e4221809e2f4e6f
4b1609b97c1c0519184bc3bef071d3f491f2ea6b
7857 F20110331_AABVVE griffin_p_Page_070thm.jpg
400eb24dd0031946dc47c447366f1703
4f6def870b545437a639cf94767b1ce321651bf4
32028 F20110331_AABVUP griffin_p_Page_070.QC.jpg
6d52322e4ca8790ec7981896be656cbe
b945eb2c74aa2a8ffb39cbc4e98eca585db35023
1517 F20110331_AABWAK griffin_p_Page_103.txt
0bb74d08725b8c91de83fe2609a3ec54
abe2774e6ce7913147ef1a1e549c7bdc65fa17c3
1830 F20110331_AABVVF griffin_p_Page_025.txt
ef86971ec05fdbfd3db9cdaa99234570
7f37611881d3d1ae57b8dd3caa2b4ac4c178defc
8333 F20110331_AABVUQ griffin_p_Page_060thm.jpg
177885466cf924876373d39040768bab
be173933478dd50cdc59d248d5a820786118398b
81677 F20110331_AABWAL griffin_p_Page_004.pro
e656831ce461c22d2602c614a44f5532
40c78a2b1d3897a77c0e3cd5f4326e9e46d80d9b
F20110331_AABVVG griffin_p_Page_076.tif
a8064467f9593a5b8b7863ff250b9932
ec671e1751fe7f3a050a8e765782f26db86791f0
91877 F20110331_AABVUR griffin_p_Page_066.jpg
941906548f3cd769465171c53fd4020f
04d65ef23ed8841d9c54c814fdbd3618b2f760a1
39943 F20110331_AABWBA griffin_p_Page_030.pro
c4ef2f32b7e3ea9a9c790993c778bea5
c0cb648de66423c034cb6def46c4453c5fdd16c7
63455 F20110331_AABWAM griffin_p_Page_005.pro
e7755ce52e349c1f07d2b7ce5d5ca406
b95b6f3b0310822a2df5a6ca9495d2f351e90eb6
87125 F20110331_AABVUS griffin_p_Page_044.jpg
d77a7d0b22fc4ef85935899c15f87dd4
1c12c0ba39040a135e7c1d25769baf8ee68f0905
40288 F20110331_AABWBB griffin_p_Page_031.pro
97b351ebc175f7eb385217e1dfe01fab
6b96c6b61c85fd9d66d65599030040cdb5d8ff67
42260 F20110331_AABWAN griffin_p_Page_010.pro
77135263ba1357061407890f9347cd90
64458b416482d307d84589684480f25fe2551404
F20110331_AABVVH griffin_p_Page_086.tif
9a54d8aec8e47a622d616ba2ceca44af
e4f0d14badcb933ba7b406ff3138e2978ba64034
159 F20110331_AABVUT griffin_p_Page_098.txt
0d5ef787f840ab37fe3bd022fdf8e0c7
5350cbb8e6bb50b01993ff0700fc761180c4e4bb
43420 F20110331_AABWBC griffin_p_Page_032.pro
27977fd98baa0adf6783c285af66ec95
8a0bc9df8cc3c3430532350535f8376770262f5a
51446 F20110331_AABWAO griffin_p_Page_011.pro
12f4ea2321b0633af8c3368f6d188d0e
111bf41504f30bb9880efd4eccedc8b018184655
94142 F20110331_AABVVI griffin_p_Page_028.jpg
177a50bc19ddb3a79d57ace74537537d
11dd767cd6cede1fc142a10e2e8600402a96ea10
42532 F20110331_AABVUU griffin_p_Page_022.pro
8d64540e8f785e4059bdb70e78fd83b9
c49c91fa78cf77ed6ac70a690ac983954afb64a9
42356 F20110331_AABWBD griffin_p_Page_034.pro
b3aaf437cc7bee0f0742092e76c7849e
1e107a85cbbaf1f9548bdd9e73edfbd0a696b640
53898 F20110331_AABWAP griffin_p_Page_012.pro
3249663bf4fd97f07383b35fab167a2d
bbbc87c17fee58859a7132086e879da98dadc229
91818 F20110331_AABVVJ griffin_p_Page_046.jp2
0d1068e6c9baf11dac08ce2e238a7756
9396497a23d2b79ea8da0607069a20a848cda4be
24649 F20110331_AABVUV griffin_p_Page_090.QC.jpg
a7f8d360dffb9c0438eb9e2231689c5e
5250ea12c35a317b9f9b89532c70ce46acdf9a4c
41158 F20110331_AABWBE griffin_p_Page_035.pro
73fe87238bfb5152149ce10129233a3c
2224736e1a3dd714a2d596b403b5b9fb74b50b54
17423 F20110331_AABWAQ griffin_p_Page_015.pro
3e6d9f2424a17310eeb379211401f1b2
f7732dd4eb7c3785889e6befe590f2b8630e0b38
F20110331_AABVVK griffin_p_Page_039.tif
e0a888c310a9aaf0a547a259a6c1e172
57613cb0a7addadc9872a87f4be9ff72d39d7695
2506 F20110331_AABVUW griffin_p_Page_006thm.jpg
d37307adc9604f365fa7f7f386eabb79
dadb68d15608fe4dbac194f1bd66ead57c273658
42802 F20110331_AABWBF griffin_p_Page_036.pro
800f6cd414e63aaacd00f44d9e789628
321390a30b032f979a403c655fa171c77cddfb3d
F20110331_AABVVL griffin_p_Page_012.tif
7f70357184a8d63f87a33ab52b99c2ac
3e3359bdccc110265e9c0ce7fb92e468cb49ebe1
90751 F20110331_AABVUX griffin_p_Page_071.jpg
ac44760e70dcdef1864a7bbc3ad74593
9137f2365e00197d2fa20bcb34f92582e8fd0f41
45796 F20110331_AABWBG griffin_p_Page_037.pro
961fb2657f82807799a48467de65f452
8206fde0f5fb88b0661ec55d2d8a80f0a62594bd
F20110331_AABVWA griffin_p_Page_078.tif
419d67ee46e3bc66bcf0c4c0ff6c1f0b
fd42603f8bdde6956caf9dfdc38a67b65f7b6ea7
44470 F20110331_AABWAR griffin_p_Page_016.pro
a2583bf426d6ee915c214b3d20eeb624
67e326e84d4ac8fd6b64d87e4a8c2ccc1ac77612
F20110331_AABVVM griffin_p_Page_015.tif
a3cb2861f324c2af53298783d2dec921
23f1f72b6ec8bccedd54291d60e18037303732e5
86416 F20110331_AABVUY griffin_p_Page_045.jp2
7726853a0fa822236cd2046dc3027c8f
2b506b7af757d1f7bde8b0b00c4f2ce9b4d9710b
44143 F20110331_AABWBH griffin_p_Page_038.pro
feecd69ee189fea7b79e016a953ae556
8c8ad4ab77e91e8608d9f472a7efd4f047654a9c
40127 F20110331_AABVWB griffin_p_Page_076.pro
0d0881845f1869065a51b8bb2a6df09c
17f1fa9d3c8dd12acd0b5992aee29cfe56548456
53373 F20110331_AABWAS griffin_p_Page_017.pro
f0e515f828a1bd2812e4aefaf14a5b74
4250bf70d3d6faff59c8c91baa1c2ba89ea1c6ca
30925 F20110331_AABVVN griffin_p_Page_028.QC.jpg
212e13b123668d0335b27df89d9c1dca
ab730e11b18cfb0be412a36d32a72f1c10e0e13a
40762 F20110331_AABVUZ griffin_p_Page_039.pro
ef7afb77da14f0a13bf54f8610d49d13
8c05cb75130ad02473c41cc26f8b0653edf71918
37187 F20110331_AABWBI griffin_p_Page_041.pro
ed835dd5837815dff7bc1510af8872e7
c121ba010ca70909daf46dd6a330927734660c97
8451 F20110331_AABVWC griffin_p_Page_052thm.jpg
c668bbef255c24d16f7693fe5ad89ae1
bbdc6c55c1f90da41928b2de39e413d8fe7d75a1
44894 F20110331_AABWAT griffin_p_Page_018.pro
2245cc839aaad1f48ac0008574c4fed1
61de254ecac9dfbb6bda7c2d4317084cfb887a5c
86935 F20110331_AABVVO griffin_p_Page_046.jpg
762f0692c2a680db89b48b4775327a8a
c6d430accd57959a7a7f443be4938e9702ce5251
37757 F20110331_AABWBJ griffin_p_Page_042.pro
2873ae4ed1c6add8656aadc5d955d085
1458ad6f832b1c39ba7e061db9feaa932f865860
100516 F20110331_AABVWD griffin_p_Page_050.jp2
0be4930b6f81b75463e414d66d76f3c1
7e2aaa9d683ac61fec36dba57801126efcc091a2
42530 F20110331_AABWAU griffin_p_Page_019.pro
3e0027aa1901dc842eb9680cb5c8f95e
975282d3d3460046095ade174e53d48375db3e4e
85971 F20110331_AABVVP griffin_p_Page_010.jpg
eb5fd2eb6daba6245eb533c3943a0956
af0f950800203dcc396dccad2d816d6662baf76c
48184 F20110331_AABWBK griffin_p_Page_043.pro
f731cc07978b544a9f4eede39cb8a508
08447db65b69b5bfd8918f91316611a134d49dd5
92130 F20110331_AABVWE griffin_p_Page_029.jp2
3193828d600bfd27bc85c6196acc443b
f9366b9f38bc1b41ec46c4b5587689ad58b2fd71
39720 F20110331_AABWAV griffin_p_Page_020.pro
a2d4063487978b499df42ecb9ccd6509
669ff79fd9c1a110f8eed75404aea326e4f2aa00
1900 F20110331_AABVVQ griffin_p_Page_038.txt
d6c2ae52518c359c5c667d12678b3ee0
a6233a8a3120d12317f96dfd5ecca3fd0f83ed74
42574 F20110331_AABWBL griffin_p_Page_044.pro
7aecd95e1aa7cf7b8dfc7a4a9145a26f
14350ebacbd55d3c2e1c87f10e8544d08e318b7e
30953 F20110331_AABVWF griffin_p_Page_073.QC.jpg
aef041e307212f239bd1950ad13e4043
4c05efed2d0cc3aa9af662c5be797ac1eed3a9e0
40487 F20110331_AABWAW griffin_p_Page_021.pro
2c972c0ed489b3a1e75055408af5f2ed
5fc5b169d890f69064fe43cfae31e1f8644c5570
83114 F20110331_AABVVR griffin_p_Page_042.jp2
0a06fbfac314d89f47754eb9638e40e7
964565b4178c84b84541af19003007689b766947
44956 F20110331_AABWCA griffin_p_Page_074.pro
f1cb9759311b69b1913dc701ec6d7bcf
b795b817940c94955339e5710d2b34090a6272d9
38543 F20110331_AABWBM griffin_p_Page_048.pro
b8c5d3deb7b85b1e2c93f901a42bcb6a
8de8b50d964a6ef8c837b496225c7c57310a5c1d
51991 F20110331_AABVWG griffin_p_Page_013.pro
0ddf69ea2aa2cdfaa4341911fe18966f
5ca89bc1770a93ea5ab6374e6e338382aeeff1f6
38620 F20110331_AABWAX griffin_p_Page_024.pro
f4798ab48d324ed3fa77bc6d9a648d7a
a2e9da1c856846d2a4a3600a5cec39bc1d9dc86c
30286 F20110331_AABVVS griffin_p_Page_089.QC.jpg
8e24b4e1a8475f95b06687173d8f8f01
f4f71a4637f6db84b2bc56dd8f696824d9f2ccda
43962 F20110331_AABWCB griffin_p_Page_075.pro
adaa5e7a121e83e6bbe7483795405cff
4d80a30cc173f28b9dc0a75b8f25941f499215b1
46522 F20110331_AABWBN griffin_p_Page_050.pro
e5fca97ef2d2456657265972cd542636
4b5917b2e58589cb607e078bea26ec2f23d1eee2
1939 F20110331_AABVWH griffin_p_Page_022.txt
a2eb0c917ff81d362c4b499daca8233e
d34307cc78197e6f6b7f408d5f077232c5f5ce44
44117 F20110331_AABWAY griffin_p_Page_026.pro
f043cd8e1c3099da9c42a58abb52d2a7
87a99e3c04ae83fa6a0d1d45ba69a1c1a791d241
92019 F20110331_AABVVT griffin_p_Page_067.jp2
4361075cc2b635a73006d79c2d89e07d
bccd6508283d89cbed38aaab67fcea3443d70417
37953 F20110331_AABWCC griffin_p_Page_079.pro
af93a547a7c31dce3229849d8ed82deb
fc3ef3cffa4be7db61727de35c94c38aa1b3f998
45986 F20110331_AABWBO griffin_p_Page_051.pro
6bf28b7006c8ea958df7a4f0505c1f71
b06bff0802d20c1256444c730060028cab9d072f
40073 F20110331_AABWAZ griffin_p_Page_027.pro
9a1c86fe072c0ec85dad5e32bad6b547
f5047bf3e34c45f2209ee26db1826b1aed281fe9
5314 F20110331_AABVVU griffin_p_Page_087thm.jpg
15ac99bec604c50d1667444fe4dd0a19
51c9f4a8e5fc19c34844bf94548ed3a0fcdc5635
46519 F20110331_AABWCD griffin_p_Page_080.pro
a64175b5af760fd052529f26b93cecd3
f24ff7c01adb9a9356400ef3099ed350ad6d2745
53295 F20110331_AABWBP griffin_p_Page_054.pro
bef8ca4948b4a2d044959b040156dd4e
007cde69b292f81c2409e98342d46de499bdaa8f
F20110331_AABVWI griffin_p_Page_083.tif
7bacfd6e1be1c78d7da605e932403fb1
651aa9875bde9120df8a292806b639157874c053
7220 F20110331_AABVVV griffin_p_Page_036thm.jpg
643ffa3ac205280bfdde346fc83f3965
b46a6d3cf70ddc75da11926d42d2a3f6cb51df06
35038 F20110331_AABWCE griffin_p_Page_086.pro
c7d067570a9930cbc487383fc5732b5f
9a38e237e59797110adac29d35ae2fcffb69c0b9
51124 F20110331_AABWBQ griffin_p_Page_057.pro
e9f93e698825f135f28d483c02f5c530
b241b1c0cb21de6b2bc29048c3e52023c43ec9a7
87644 F20110331_AABVWJ griffin_p_Page_075.jpg
cc6196fc7b9bd2557e07b13125952671
76c2488ee91095a9924ae6426d6af1d1fae51f64
1051985 F20110331_AABVVW griffin_p_Page_007.jp2
c79d3b7a63deb48eb4803d49372480a0
c092e88692f208783a5e5e830ad2d051f14ea06f
45094 F20110331_AABWCF griffin_p_Page_089.pro
6f5bb67ca8a3a584cb41b21d6e32dc06
b8ecb56d14d5c04fb75acb842b6da300f2933761
53037 F20110331_AABWBR griffin_p_Page_059.pro
6ec97200143b4dd8814ed21c613cf523
2ad439e177c13391e8951bbcaf7c13b5705b1e2f
33315 F20110331_AABVWK griffin_p_Page_060.QC.jpg
21b97ad8f5cada08c0a1acfbd8a1fd61
06f6b98c4e66aeafb1dd02be73bf5221be6474d1
1510 F20110331_AABVVX griffin_p_Page_094.txt
3a054ca9ef33007d23ec94d69af6e870
b72900b8ff9dc2c1d397648aa6f2bfa406524d13
17795 F20110331_AABWCG griffin_p_Page_091.pro
f9564a94a676b3e8fa344eba14f4a757
e0aacedbeb939a5db1a9b25474fcb79308d87dc4
F20110331_AABVXA griffin_p_Page_026.tif
17fe728439c3b91a7bcdd8a4147f6f0c
fb580ae9085e6443e01c6cda2296ae25e7b3a338
84427 F20110331_AABVWL griffin_p_Page_022.jpg
d8b9da0b5b877987c79310facc3cb836
37fd35ccf74ecf4a62e788749756955d69e68534
791 F20110331_AABVVY griffin_p_Page_087.txt
5c565be524a7b1648d4ff5fdd02b105c
2e5397b142cf68bff347a08222cfe9ba200875f3
51288 F20110331_AABWCH griffin_p_Page_092.pro
428bbd69b652fb1f0dc8beb0e988be3d
013364647f1226b07d809ba5260ccc2f238a1428
F20110331_AABVXB griffin_p_Page_028.tif
6d1eab6bba21f331abb40cf5ae811f98
4120b2e0fec103e6e6510de38f9caafa62e53883
49906 F20110331_AABWBS griffin_p_Page_060.pro
50b30d139ba1871672aec5dc3e8b93d8
d31a0c7cfe6e94421fc97413f923f2cb87513838
169184 F20110331_AABVWM UFE0014866_00001.xml FULL
8fe45a65b7d069f7de0f4b598d5605c8
9563a7691c4100972188a58a02d466eaba08dee6
F20110331_AABVVZ griffin_p_Page_034.tif
8d7bb9addb4471bb8b7217acb147b21a
5a9a1e2fec710fb4de574c6fdcf81c4044ef278c
33565 F20110331_AABWCI griffin_p_Page_094.pro
44ed261a3105c39180d4615425064fae
f152febf13d9828c974c8a95b7ae1c04e948e39a
F20110331_AABVXC griffin_p_Page_029.tif
c4e92dc5240ab22badce36beea863330
cf1b0584674b8f6d29a27fb1f37ed408c06d467a
50247 F20110331_AABWBT griffin_p_Page_061.pro
c79b3fee8c19f483ce1c85427357d6e4
2104584e77eda2365463f147389012d4ff6ef582
17244 F20110331_AABWCJ griffin_p_Page_095.pro
3bdd33108969066ab944508417460589
6a02fb73e85615a70e13d3452f9f196cfdb23aed
F20110331_AABVXD griffin_p_Page_030.tif
876d5c7589fd2d0746720db2f7c6b63f
548e845c152123cc324280532e37196e482a9d8f
48952 F20110331_AABWBU griffin_p_Page_065.pro
fcb1820d5ff551bd6092824acc66132f
851aafdcd9465b4d8729954a20be93368d535066
53563 F20110331_AABWCK griffin_p_Page_097.pro
0d81ff777654f4a413a0bc95c5a3d93c
c04d78f116a29b99cbfc463fb5fa84ccd4737c31
F20110331_AABVXE griffin_p_Page_032.tif
62c57c5b633268025e870bd5d10eb605
a55b950444446ee2d080f50f9f29434e5d6765c2
45816 F20110331_AABWBV griffin_p_Page_066.pro
0ababc7c2b9eebc8b040656821934f8e
76d08bb221b02836217bfefebeb6fbac4568f048
F20110331_AABVWP griffin_p_Page_004.tif
129cff2a5016f470e7c9ae0992e3e44f
11886dac8bc77a55e233c30d1fb9eff4bfeabf3e
2885 F20110331_AABWCL griffin_p_Page_098.pro
af7d3d323bea8095040ca4273261bc27
6da7a0b01ddd614fb44858eea410bece681048aa
F20110331_AABVXF griffin_p_Page_037.tif
8e53c6d90b26da2c2953877ec6d48fd5
50a0edab67e8a83f872a8c9fb13396a04d7a4b06
44334 F20110331_AABWBW griffin_p_Page_067.pro
80eda7fc724844433db341cfafc71d61
5faf115a2aacdde923ad39896a5da2d061640aa8
F20110331_AABVWQ griffin_p_Page_005.tif
7b779d3ce94a878e4715a5e95b0818c1
a632411811e5d3d44974dcf3e1798f0583e2ca71
54702 F20110331_AABWCM griffin_p_Page_102.pro
a9fc2bb19004615c1a3634023403923c
eac2cb669d9f2c4c430bddc0c063d6afe4bcdee2
F20110331_AABVXG griffin_p_Page_038.tif
59d8a5d6837549a470b986161b659031
21c7247171aa32c693db2cd3483ce27e3048399a
48013 F20110331_AABWBX griffin_p_Page_070.pro
70f592bcd65e2aa8eaae54aeec139244
7004c0ef0e5b45308748b8d7f6de233d4004da65
F20110331_AABVWR griffin_p_Page_006.tif
0c8a7d63469cfd8c94143c978422b0cf
4f64aca15a39754751fe3eee27642a72decc2bf4
33160 F20110331_AABWDA griffin_p_Page_011.QC.jpg
06ce306033b4f04363e1538e2974e8bd
bed2c5c4676f3d78637fc280072b1f340e78a6e1
8486 F20110331_AABWCN griffin_p_Page_001.QC.jpg
dcbe9ffa937f79e30aeed07d1e6abf20
eae63679b5927ada5f07960aba2b6f9841c3e329
F20110331_AABVXH griffin_p_Page_040.tif
568e9ea2b4ada7ff311631b82bb15df8
0ea3066f76c193d6271e198f24fa2800b6fae160
41010 F20110331_AABWBY griffin_p_Page_072.pro
137d235c70f8653b410bf3ed6af4182b
7f7343a1e2979254e353b07e2f85ab923e23908a
F20110331_AABVWS griffin_p_Page_010.tif
788ead4e4ad030efc145bd06cbf8f0fa
868681d0766f00fd32f5ee6cf6436c88d96b6a51
34024 F20110331_AABWDB griffin_p_Page_013.QC.jpg
43f1896d50e71ca76e8c87ce079b41ff
f578dcfafc5caaf63a00b3e89fef378f452b0b26
1602 F20110331_AABWCO griffin_p_Page_002.QC.jpg
708e3b5e4b2b3541dddc43bbf13d2a27
0841c3d5e27fb98aca2cf47dae384b06a12acc94
F20110331_AABVXI griffin_p_Page_043.tif
e7d7648d59ea19f9a9947751992afc9e
0de990e18d962695e5a6253c2b09c973721087d3
47734 F20110331_AABWBZ griffin_p_Page_073.pro
90f3d3f1100e271cb1acdd179bc4e8f6
e9c312239a3854d07edfc7d4990e5c31d909ea65
F20110331_AABVWT griffin_p_Page_011.tif
57c316d3e4263e555ac6777a40937678
ffbb1a72efca3218c20b6d575f738bb3d6bf5c3b
99685 F20110331_AABWDC griffin_p_Page_014.jpg
c225e22827f0228be3e47ae491da642e
6894e4b1737e936c1745d6c9d724e025b6532997
20913 F20110331_AABWCP griffin_p_Page_003.jpg
45f6693e279545bb144b8ae97cc65ee5
de8d2a1b220a4ba8899f2fd17efef78455445746
F20110331_AABVWU griffin_p_Page_013.tif
31236276f94da97909d14b179029e1d1
b1cd891ef867aac343c857c0f7a6c8ded6537b20
12929 F20110331_AABWDD griffin_p_Page_015.QC.jpg
084835133c40819ebf7fa74913017e0c
5a68f6ecc0362188bd6cc0583c93cfc134051a6c
6592 F20110331_AABWCQ griffin_p_Page_003.QC.jpg
996bcb6538be3d3c9e87cb69db82c212
d08d4785ceffc2d0a1c98a8e587fba7fb0ed9ca2
F20110331_AABVXJ griffin_p_Page_045.tif
eb68b29a4bbe15d11c359ce6b50efbd8
5b515f26b77fc2d1cdae64e48541b12cb4c307e9
F20110331_AABVWV griffin_p_Page_017.tif
369f8e4e270ea86aafa6e3582482f9d3
461cbfea4da1872a3341fb1943dc8626a00df9a8
90884 F20110331_AABWDE griffin_p_Page_016.jpg
f06fb48bcfba069cffe8485b4095f313
91468d79422fff4894c08cc0daa08798b9111829
94998 F20110331_AABWCR griffin_p_Page_004.jpg
ffc164dfc0964f804c6fd4fc9abd7b80
dfb0a7b9677eb3d8b650c4961f33f9f3643b36da
F20110331_AABVXK griffin_p_Page_047.tif
981de7218c2fe0abf8f4409d201e278b
89b2cc3f6223da2f701d4390181d061115ac7e1a
F20110331_AABVWW griffin_p_Page_018.tif
24277bbc487cc444166b316fa1cc1c3f
2d26cfa244ad2d9277febe5218da04ae7d8f6779
107233 F20110331_AABWDF griffin_p_Page_017.jpg
82ef1b03e53cab08b94948c83f3d1596
e682be15b3ac14e0f22bb1bb6527ec3009f967f9
F20110331_AABVYA griffin_p_Page_071.tif
bab2896da5b7a04b9a2afcdee80bdbb8
6294e064055ed4169cdac954e8cd4e43e417a638
21289 F20110331_AABWCS griffin_p_Page_004.QC.jpg
c9be7c5e139bebb95e8eca99e44d59c9
4a92b1fbed69d0b811244efec0e1b50adeef8ef7
F20110331_AABVXL griffin_p_Page_048.tif
a4958eaf4937ce0c9caa428f6091b92b
f5b9fd398b47bbc2fc0cf5aa350d6245837bb408
F20110331_AABVWX griffin_p_Page_020.tif
bf88c7a8434f5afdd7aeb99d9508bbba
1c6f7a3ee3ed72ef004aabaa92b006a14794eb01
34714 F20110331_AABWDG griffin_p_Page_017.QC.jpg
b684f578012a0df502e0b5a093d9034e
2a9a5691641e7210e54ad251aa8df8bb3c01aa7f
F20110331_AABVYB griffin_p_Page_072.tif
3105cfb9170bd868cb04d321217dc7ee
d2f8039812edddf7c9077f81586d0f64d0678104
F20110331_AABVXM griffin_p_Page_050.tif
6e5acb3d1ec37abe684044e3f97dc973
ec382d7b9c9e56e8aba281e13ec22710c16a797b
F20110331_AABVWY griffin_p_Page_021.tif
de89ddab3d20196a1b3c24faba7f28df
ed6fd18a51cce8fbd36c02f21d6f2673fb06d439
86868 F20110331_AABWDH griffin_p_Page_018.jpg
4851e088ed8b0e46762a6abf9767c35d
2fb3b512ca9a98d84024207ba5cdbf028665764b
F20110331_AABVYC griffin_p_Page_077.tif
94f0478a1a0ae462c05b206affdea343
9c25b3497a6cd101569a25dcbca8703bcdadba34
15770 F20110331_AABWCT griffin_p_Page_005.QC.jpg
320d53db4f765b68215ecfb60e600ee7
57bc23044d4c33f994f9824e3312a452f9895a0b
F20110331_AABVXN griffin_p_Page_051.tif
0eacf0d27d5cc614be89422919116326
e3c5c6d862d9b72c129cb79bf2434a0c8869be7d
F20110331_AABVWZ griffin_p_Page_024.tif
fd0fa99cc5b38d69a20a8daf156a5cef
184b92088aeef174b770c6da08eb3a8c128deacf
29504 F20110331_AABWDI griffin_p_Page_018.QC.jpg
08f662d63041ec8cf2821b129ce29dc2
b38ee3f3b2937998a093c687c3431716a2185b67
F20110331_AABVYD griffin_p_Page_080.tif
fe812d2a21b5ca7596dbec0dea4cb123
bbb3e5add7e0fb8b6d3a72c216d840ee8bc7fc7f
31070 F20110331_AABWCU griffin_p_Page_006.jpg
60fccb279c0d8aab93a43cd6f9ed1562
0bed39df28389892db7ee8498f5f83201115ed9f
F20110331_AABVXO griffin_p_Page_052.tif
f403db8a1c1e963d910ad00941fff4ee
b302a01d88fae59096f5b20bda6f66f8c9d771f5
86291 F20110331_AABWDJ griffin_p_Page_019.jpg
1183c1867b57e20e29a0481865478ec5
695fd2f20ee133a41b2a322a0c9744d36687002b
F20110331_AABVYE griffin_p_Page_081.tif
a37a0c2e1be59972d832448a21bf376c
c30a46e541d6cc32bb36d74f26ba65600737071c
9224 F20110331_AABWCV griffin_p_Page_006.QC.jpg
853e7d1235051c233c8cc538e1e43e2e
ef4af623c2ca757d2dc4b0711c06afed53435de7
F20110331_AABVXP griffin_p_Page_053.tif
2766b670f0373026d1bc4d78e36ddef3
c0237becb58d8938ac759373ce569b361258f2eb
26626 F20110331_AABWDK griffin_p_Page_019.QC.jpg
69b21d6efdaaeefb0d403b19fcc89f4e
4012e3c52ed3cd2fe02fc8c9ef369b0288c78f52
F20110331_AABVYF griffin_p_Page_084.tif
6455dac52d5672d7074781dea8bfa58f
f43117b683fa74e94f6cbf81c42606b4443fc623
82550 F20110331_AABWCW griffin_p_Page_007.jpg
1892687ed9e029f81e0456011606b030
b185e95f96b468a3eae5bd34aa846594c081b293
F20110331_AABVXQ griffin_p_Page_055.tif
bdd6ce4ad809d4252726904dbf8a388e
d428ea7c9809cdb5a5701dec77c420c80583631e
78946 F20110331_AABWDL griffin_p_Page_020.jpg
7a0e2ea7574e2443af707add612aa18b
c741b0de144bcb14d1e5ef2dc4f46776333db7e2
81105 F20110331_AABWCX griffin_p_Page_008.jpg
f64c4e2e8a464ebedf73fa6ebeda255b
976ebfa25983f38f11d51e5520e109fcf3ab5ba1
F20110331_AABVXR griffin_p_Page_058.tif
8edb0a8afef4ef0338fd4114791fee8a
8da4a024cf1a06157f8e57b98ceb04f97bb6e6b5
91179 F20110331_AABWEA griffin_p_Page_032.jpg
b45734e9314c70427d1b3511d572e04e
fb8f1696a0277ddb9c315d4eed3c27626899529b
26838 F20110331_AABWDM griffin_p_Page_020.QC.jpg
d50260ff1be401d406aeae039ee04d81
cb89c7c65fdb4bd21a5bca612a84900292d11cc4
F20110331_AABVYG griffin_p_Page_087.tif
1c4a4b4d584cea58b44a1b1b0218cba3
d35cc9c2fc0ad899f6c2c3daf10985a039dd4ff2
60607 F20110331_AABWCY griffin_p_Page_009.jpg
e8c6651cd2fe99859cf261a151aca66b
262a6f2439e4eb2c1d41ce88f05a83ad3fcc0edf
F20110331_AABVXS griffin_p_Page_059.tif
ad5d4a4da4c7f68fd8345d70c0ee0032
d2b24bddbc769ade6eeaa31254e3b9b31096c2c0
32051 F20110331_AABWEB griffin_p_Page_033.QC.jpg
2d639962d7e2ce151c892a85ad3f17d7
435cf97868b334fd53b416266b245c4e44864e54
28090 F20110331_AABWDN griffin_p_Page_021.QC.jpg
58e18bef4bbb27e001d1e4df19c12702
44a988d384012216be8b4e7d5d889088a04c6bfd
F20110331_AABVYH griffin_p_Page_089.tif
7464cc8b3628659fc9175da713df9b02
863c7f27fc856cd0bd52a2b6c97efa1ee33a4256
27661 F20110331_AABWCZ griffin_p_Page_010.QC.jpg
00295646770938955aac5fa72606d450
23c9ebe73d8bd67ae5c465cfe31bdbe86be29e4d
F20110331_AABVXT griffin_p_Page_062.tif
8306ac89aedba4f5e7e8ab5e059b0bf4
9868f6ddb2726897054f140c13dd49035062dd98
88202 F20110331_AABWEC griffin_p_Page_034.jpg
0ddafe5ebac786b0f8c11e28d571ba65
4a21ff8fb4afba8e3de978470e9533234129c686
28592 F20110331_AABWDO griffin_p_Page_022.QC.jpg
0e2c6bed782d7feb9e349916e9056554
91a1b4115efc3ef1f60ebd275fc8c503a7465b59
F20110331_AABVYI griffin_p_Page_093.tif
35c21ea563dad2fdf8dddd8b2a6e1da8
30e8d38c06162dfe3411eba3a13e56637dedf0e0
F20110331_AABVXU griffin_p_Page_063.tif
bfbfb8b48cddc088be9a43072dac544c
339952220970930357dd0c6ccd19d451eebd2590
86056 F20110331_AABWED griffin_p_Page_035.jpg
dec8c474bf701bfef3990919191d5e0d
0c48b2dfe31ad4188df1e90580670c00104dba53
60964 F20110331_AABWDP griffin_p_Page_023.jpg
ae802f36c0ac620a627ece0757241ea7
248b141ed623f771b85a40a62e896306f2b18c4e
F20110331_AABVYJ griffin_p_Page_098.tif
f4e67759d4a66637b4ab424cf0389c30
9820d5ee8b39ba8024a15997b377a1483b9ab2be
F20110331_AABVXV griffin_p_Page_064.tif
77cccb24a263524ad696015b82677cc3
037ce2fb022815dd182691f73a0d56b4560d1bd4
28663 F20110331_AABWEE griffin_p_Page_036.QC.jpg
75b6629232f820f0e6fba651554b0ce0
e6511277c491cd03144c3ae980735e2223fb0efc
23717 F20110331_AABWDQ griffin_p_Page_024.QC.jpg
47483a8d725599c6b0f1e64e2b9dfb89
70fd7692dc5b4b2b7afca496113c221f7d098ba0
F20110331_AABVXW griffin_p_Page_065.tif
120d1e132808960ff03153236d757539
c0c5b10fbb0788f8a02b1d6b53df9fdb111019e7
30235 F20110331_AABWEF griffin_p_Page_038.QC.jpg
8183013092290b828f8153ae585ad895
a743c23c74037f5db99907e1ace5d024392e3cbc
25960 F20110331_AABWDR griffin_p_Page_025.QC.jpg
2f1ffcfe6b022416d4209636abbc0b25
26b823ae60d816c64ba639a5e4ce5e5dedad575a
F20110331_AABVYK griffin_p_Page_099.tif
a65e61d786c1fbaa1aed38be75e9d239
a26a7337736302acf3ab4815e2c71c507f25364c
F20110331_AABVXX griffin_p_Page_068.tif
f8c4576161338547b9abb748a614cf4f
dfe1006d861270b198a45fa7fed4b76447482a1f
83484 F20110331_AABWEG griffin_p_Page_039.jpg
c645ac158869b966f4536a4bdac22c31
1fa77583fc3597ef448856a55c8c6445e732495b
1858 F20110331_AABVZA griffin_p_Page_032.txt
395895dc85e9855629ba382a38cfe3c2
b9431327646d63591d5e72e3703ab684bb5309cb
27231 F20110331_AABWDS griffin_p_Page_026.QC.jpg
5a7fc0da3d52285e75724dba6247c214
000afd4624b7ab44ed11959d0336be2ec8e7ba85
F20110331_AABVYL griffin_p_Page_102.tif
bce926a0773739f5a6c29914542e55a5
1de446e17e37cbb8fe134d83a8e70d45236e8f87
F20110331_AABVXY griffin_p_Page_069.tif
7f6268e443e29cb7badf27be1cc0c091
82ec5a2c071a6234131c9ba56cf43e65cee70b7a
27404 F20110331_AABWEH griffin_p_Page_039.QC.jpg
6a0864aa209081b876d2b68177a1e9d1
3d21dbb828daf95fe1cabebc6036301e352e7d9b
1933 F20110331_AABVZB griffin_p_Page_033.txt
1d343c4c30156c136f103099ba23336a
7c2bc7458e5953ecf302dfc5fc212336598590a2
25855 F20110331_AABWDT griffin_p_Page_027.QC.jpg
1f768848214f21f826a28f943323aa1d
3fe371cd66d1404a1e929c72c1d4ca22cb0adf5c
F20110331_AABVYM griffin_p_Page_103.tif
98f6e4a49e2f3557b1a2e81cb9f86eb1
ce01d14862e35d5ca87fd59ccc6e7caeb75b7067
F20110331_AABVXZ griffin_p_Page_070.tif
c854164e3dae8c9d082e14d0d4a7f510
da69edc6bfc04ad1c9f4487bad64523d85d2e0e1
25069 F20110331_AABWEI griffin_p_Page_041.QC.jpg
dfbb1f264a2194b16c0a6eabb277e5fe
e6e9e714e28a80a3bd3a640bb832dad2a52366da
1809 F20110331_AABVZC griffin_p_Page_034.txt
0585202ff7f4d7df8aeba16b4541d883
0269cfcee84195f1812ce8c2ea25623117b7c7d9
F20110331_AABVYN griffin_p_Page_104.tif
1122b3e7c7d07de17f35040a5bd80bf9
51fb561e5d675030a45cdf0a8afdc761d8fb4fc3
25439 F20110331_AABWEJ griffin_p_Page_042.QC.jpg
41001ca8c0a8f75e18d8cfebe3175212
cabd0866745ea6d67f35134d32399972c61e9a9b
1833 F20110331_AABVZD griffin_p_Page_035.txt
c63e7019fb7017c908c63ea8dd720991
e6308ff04476f50bcdd62742553b7284fd5c686b
89637 F20110331_AABWDU griffin_p_Page_029.jpg
8ecf52616c825e730901e5bb8830f2de
603d5713e601c177bbb961340edca3d10b403f1f
3311 F20110331_AABVYO griffin_p_Page_004.txt
36b9e60ebd39ad45cb869cc3b7489c89
833bf7a4330fb7117231f5778054b745caebdd7d
27674 F20110331_AABWEK griffin_p_Page_044.QC.jpg
dfff78516fa85713d63586eae2c8fd5d
a93bfdab11a9a97002a10ad0c4c96cf5f3e47707
1852 F20110331_AABVZE griffin_p_Page_036.txt
f22c885c61a4bb442b5b48015844a7de
9f0911ba3f6dc8d25f281629832f25572717f466
27888 F20110331_AABWDV griffin_p_Page_029.QC.jpg
02a385d4886e7ee53cd3a3a3b65082b4
31de0211cb508a837317c2577c76a78b748a50fa
1676 F20110331_AABVYP griffin_p_Page_008.txt
bed2876c58a734dd8266ff5491a399d1
00a772c9b1a9bee3d750b9fb973967b89b25dcbd
81026 F20110331_AABWEL griffin_p_Page_045.jpg
507a5849a8de33eed1b797ca4cefaf87
729c9bd59b54a1a488fcf73f9afaa6f17d77badc
F20110331_AABVZF griffin_p_Page_037.txt
e7c96814c4648ff16a4366ec4d55d4fc
da1470bc53ef47307d8862c0f0e9407724be524f
84272 F20110331_AABWDW griffin_p_Page_030.jpg
23198c839c6d44610eb2c8d181b75a03
4115ef2ab71360bfb39e6f0f39c571a2dd252311
2146 F20110331_AABVYQ griffin_p_Page_012.txt
4932c3116c06f394e8e43cdac55a68fd
ea08b3eaec89eceb715da951dc2fe6a2e5b03d9d
32080 F20110331_AABWFA griffin_p_Page_061.QC.jpg
4965f1b1f4fc05295a51d475342ea84f
983f73582b7df70bd8592c7a60c5d977676ddc01
26118 F20110331_AABWEM griffin_p_Page_045.QC.jpg
951419b7d58c887137666f1aa3fb51d2
1c1a9d30108ff57baa43462380a5afa93872a4f0
1763 F20110331_AABVZG griffin_p_Page_040.txt
3445f907e37287f8cea1639d2ad5c7fb
c8168d51d937101b3b9563ac71961f8f793cf5c5
27776 F20110331_AABWDX griffin_p_Page_030.QC.jpg
cbddf4c80e8a053ba7c293608a88695d
1a8a24dcb533a5471d2cc3ebde28dd95fea8c263
1930 F20110331_AABVYR griffin_p_Page_014.txt
e6fb17d3ed8b61c6ebbdf00fa7ebd150
d47803e9ec5c6e519a4944b1fcddeb7207c27f70
88032 F20110331_AABWFB griffin_p_Page_064.jpg
69e21694c6d0ba7a395770a910041752
f4a419004f11942c7668a404310faf9146f7e28d
28162 F20110331_AABWEN griffin_p_Page_046.QC.jpg
4e14e711de5f24258c8323bb35107977
d08f297f2e5655ebda39b0c294fb84a7839bb10b
1735 F20110331_AABVZH griffin_p_Page_041.txt
1cef7cc1bbad8ba5e075e8ff42b2630d
c4e10d038f3f9a849e5d322c4b529ab4cf4d1c76
79446 F20110331_AABWDY griffin_p_Page_031.jpg
f1c6de0b8ec1ec8db5291beb4e46a117
65a816fb1ba96976c2c65f7706b928644da251af
F20110331_AABVYS griffin_p_Page_016.txt
d830c65ed6c4595c737d64ab7a053609
3a526dacb97554d4e962c7b84111296e53487ed8
29107 F20110331_AABWFC griffin_p_Page_064.QC.jpg
238fdd590748b075aa44ff9dcedfdc59
d14f61c0b22b03eb1effbe58d831c965f819a695
87609 F20110331_AABWEO griffin_p_Page_047.jpg
a560cdb4478cd584ad41272356d74c16
40e910b3a3ce4037553b8decf066b86847eb9a61
F20110331_AABVZI griffin_p_Page_042.txt
cfd5ef59b75119954c51ac8db19711a4
3268a50bdd1ac7f73d6ae207d01c4aaf3f12b805
26692 F20110331_AABWDZ griffin_p_Page_031.QC.jpg
99cb85e0d82a33a91bb31ccb750f9ab3
ab32a2dcb5237cc5a87f01cd4433074608d9c7bb
F20110331_AABVYT griffin_p_Page_018.txt
a88b926b02750801984bfbc5da36c447
3234810937ae705a09bca15aa8e05c3cda3ff00e
28002 F20110331_AABWEP griffin_p_Page_047.QC.jpg
026f568b896514aeb6113d8783a1074a
4bcaf76d6bc7e7ae265abb1205e3b84ad2363039
F20110331_AABVZJ griffin_p_Page_047.txt
07e068e9f67bddb7d35dd1230c442196
afa82738c549ed0ab2ca77e3d1829c17b7e3117b
1806 F20110331_AABVYU griffin_p_Page_019.txt
4042407c2142375503a0ef90f7a08803
309666e9a31c0da33cd1d4231a8da6c89514fc78
100490 F20110331_AABWFD griffin_p_Page_065.jpg
fe124c6b869268e5598bbf933d7ec34f
ab99996c2a70bf65b61b8d8e5c090867223f570e
75052 F20110331_AABWEQ griffin_p_Page_048.jpg
a5be1a2d9032f2cdf82454826d4654ee
0aa8b6111ecdb6e00f0bd609f53ed2d4b37c0874
1742 F20110331_AABVZK griffin_p_Page_048.txt
c3f2fb88aec62714f4d04a7b4856b4a7
dc7e589de9c582767c2dd5c431ccb8fce81cff00
1885 F20110331_AABVYV griffin_p_Page_020.txt
6437c2c07d379a7c31988a00581e2c8e
f1f6587c300f172c408d803e00dbb344d6e66692
31900 F20110331_AABWFE griffin_p_Page_065.QC.jpg
add2e39f40bc444ffd5cd663973f83d2
8bbdf0d4ea36e6d118f2faa10667e24bc7d7e710
24483 F20110331_AABWER griffin_p_Page_048.QC.jpg
1de1eb64c00ec0647c88e6acd2ca6716
4a9e4e91ee505c5323d432220d31e5b21efebf57
1937 F20110331_AABVYW griffin_p_Page_027.txt
7dff8afbb7078372f1ca1dfe345ba2b4
f18201e6c32e3f0fb7d4bc2b10473bfdcb292c3a
87662 F20110331_AABWFF griffin_p_Page_067.jpg
56fd34d7704de89d8f49e296c55e153d
0ceb20b28d4c86b299aa18ff12f4e4d822d7c31b
31610 F20110331_AABWES griffin_p_Page_049.QC.jpg
86a3360670b737a8d83e7224d37b629e
a746d082c4b1541ea9b6714d78d2edac8003ed09
1897 F20110331_AABVZL griffin_p_Page_050.txt
694b813b92554f772c8d5287602304bd
ce99073097c16a0106eccb9246f91d7c49971a5b
1895 F20110331_AABVYX griffin_p_Page_028.txt
f522309f78380bb6d295fd9deb42d33c
f8e0b71e678f94842b5aba75b4b83fb926d89ec3
28237 F20110331_AABWFG griffin_p_Page_067.QC.jpg
3bfdc9067fc910f914cc809296706539
920c2d4a5dca45465fd95b1b5e490b26572df2fe
95389 F20110331_AABWET griffin_p_Page_050.jpg
d767d3aff3aafcb8013c1ef50ba6f1a8
222f34887bc8b86581293764c65c63d8c3297b2e
2007 F20110331_AABVZM griffin_p_Page_057.txt
0aa3e53c94a511fd0c4bd94ca14a913a
42ccd9dab2f67ee3fc19dc0752dfbfa244c44812
1760 F20110331_AABVYY griffin_p_Page_029.txt
fcd505a80b791b09cc862a6e7f1d1822
6883ef411ee1dc8fcccf19afbecfa237e379a305
29387 F20110331_AABWFH griffin_p_Page_068.QC.jpg
9e8cf5c0812d0d2118e23d88b1ef223e
4399ca665bdd0ab799adc5d525b54cc8ea2a925c
30865 F20110331_AABWEU griffin_p_Page_050.QC.jpg
48b6f404239b1591845373ffdd606906
992a5089401e3e5e79c27832e632147abbfa365b
2113 F20110331_AABVZN griffin_p_Page_058.txt
c785b4656788aa82cd998b1be2fe5cf3
e0f3c755aff4d392a6386ebc3c7e4eea0d62d465
1841 F20110331_AABVYZ griffin_p_Page_030.txt
a160c3b518666157a6872ecc107c1c63
cf944b961a62faf67a21a685cdaa79182598937a
81809 F20110331_AABWFI griffin_p_Page_069.jpg
dbdea849c290bf84e4b35f5ef0e32131
928f8e03203580796048667241e5c0ef10ccb523
2080 F20110331_AABVZO griffin_p_Page_059.txt
8bef36acf6362fe3e3081e137eaad93f
f3a9b8ffded5b9d1400e2ab6944d2609962968a8
97490 F20110331_AABWFJ griffin_p_Page_070.jpg
f07eaeea29c94e17baa4ce055b460d24
d45025159bb88b07209c39541d3be353624fc9e0
105206 F20110331_AABWEV griffin_p_Page_053.jpg
39b1d6f80b51a548c14290d05971d2cb
dc95520f1008fc3c7ece4795d8493d0b68929369
F20110331_AABVZP griffin_p_Page_060.txt
c7bdbbeca5b18fcf95b8d213b9d3f481
a4690bb3490690a878e08920d3ef362c6ab690d3
92499 F20110331_AABWFK griffin_p_Page_073.jpg
fe04a523ab7c4bede909d3fc5515ed59
bc4fc1a34aa14073fac40ec725cd1883404f9bf6
105076 F20110331_AABWEW griffin_p_Page_054.jpg
4606127e49dab6bf4125f9a5c8915489
4ffbfcd1b365319d01a7b8f857245c89fdd206a1
1976 F20110331_AABVZQ griffin_p_Page_061.txt
fa770058628af8e6d48acd31a2815050
46dbdd70178d7363c7db2d4c8a5beac03ebaa09b
29385 F20110331_AABWFL griffin_p_Page_074.QC.jpg
bba71a88ef488c4bc43ad7cf5a6b4995
e2f9fdcc6a874e9d90eefd54f3bafe566be476b7
103881 F20110331_AABWEX griffin_p_Page_055.jpg
f829255cd0b1f338fd3498b890178ffb
d42556bb84661920b21e38a8bd7aaf23c9c0066b
2076 F20110331_AABVZR griffin_p_Page_062.txt
216c43817995802336efbff22cf3fad9
dff3ad9a8fea30aa2f392d37cfbf4ac19e059c4a
26038 F20110331_AABWGA griffin_p_Page_093.QC.jpg
0ba4cea40c364b587af9d4ab88e035dd
1cd72d33954485b9ea83e84de95288ddd4c165a0
75215 F20110331_AABWFM griffin_p_Page_077.jpg
0476a66105d7061cdae45788c45768b4
8b18a04d36c8dc9e4873b4406e03a4afdf0e3d21
100182 F20110331_AABWEY griffin_p_Page_056.jpg
c3d9fd195afcf22eed8750975a8d7782
0fb9d213b1da5e9c4d0415e0f2f2afb9ca59436e
1995 F20110331_AABVZS griffin_p_Page_063.txt
6964d726e9ae68def3f8123fe8d7b475
766944cfd2e52fa205a05f7fcdaa199667ea6692
71361 F20110331_AABWGB griffin_p_Page_094.jpg
3f56686905180d89c66225f0c2e34a65
632a8bbe909848c31c86297fec4d4f3ad6512a49
81866 F20110331_AABWFN griffin_p_Page_078.jpg
1d02d4dbd3de4a102bf9238df184b3ec
56f34e9b4bdb5d8f45ab8de0e2e1fc379160ba06
34690 F20110331_AABWEZ griffin_p_Page_058.QC.jpg
882e3fb681f4b5624c93fbe610540337
58265161e66f800cab7d02ebe46542dbc741776c
1877 F20110331_AABVZT griffin_p_Page_067.txt
cc290085a722a4b04c89766916521408
5a3e82b138d0e24b5632051c67d8592cd902bf5c
54856 F20110331_AABWGC griffin_p_Page_095.jpg
83bfade4e3e592e4a5fe8cbb53fcaa73
076a4b946d00845ebfd063a50bf2381cbd16050d
27496 F20110331_AABWFO griffin_p_Page_078.QC.jpg
a86bbe4d57369d7116c3f8973ae87efe
35d9a366f554778e953f952dd2070c91c2dc3aae
1865 F20110331_AABVZU griffin_p_Page_068.txt
ad658b5ac373fdfc5db88446435528c5
5795d53be1ad00197c25bc9524cec2b43673812a
17713 F20110331_AABWGD griffin_p_Page_095.QC.jpg
4825d7a4e32eae089e4da1bb45301905
3e901a81a3fce54f3dc027188776248911abe5b7
77459 F20110331_AABWFP griffin_p_Page_079.jpg
719632ba7fbc8aa5048ab36b39416518
d455069b84c251d740ba6941dc6f6fbb32193384
F20110331_AABVZV griffin_p_Page_070.txt
8fdc066fd87ec9e28e3967abdf5b7994
01c204889a49c6558b9305b283e68bbc4522b43f
86238 F20110331_AABWGE griffin_p_Page_096.jpg
5e6f788a046900967651f8522f309b4c
87ec61e168695b0a1cf8ac1b2a8ab776ea36759c
25130 F20110331_AABWFQ griffin_p_Page_079.QC.jpg
f317a804655f6f9ef8e6c42ab4ac69ed
a3ac47dfa77385a31eb24a7c4df8b838ea2e7191
1905 F20110331_AABVZW griffin_p_Page_071.txt
49516f5641b3b2984786aa2a33a81faf
f5c206def085d5cf2873885b271a3297ffcbf583
3048 F20110331_AABWGF griffin_p_Page_098.QC.jpg
cca98f9fa89f20f3396501755b41a521
f36139c94a15275b4998669a271a0caec1842fe2
32360 F20110331_AABWFR griffin_p_Page_081.QC.jpg
10e0afe6e2e35d65ff51f7c591bf6c4a
4d5461c1e92283a094b3516d47305f9edb0c4345
1768 F20110331_AABVZX griffin_p_Page_072.txt
2db94d04c7b6278219f2d505fa490710
f6c6b1a394076b58802dc766670eb86abfdcbdf6
86847 F20110331_AABWGG griffin_p_Page_099.jpg
067f278fdab20371fc7a075a84f01632
b8175a4cfc0be3ae14878b5b5ccc9d3fb5c22850
31500 F20110331_AABWFS griffin_p_Page_082.QC.jpg
dd722c9096533e80d267596d911f6c84
83508c5e95c4a0aa42fb27015c3169b5e6644b84
1943 F20110331_AABVZY griffin_p_Page_074.txt
67963dc710c88493448c6974a84aa489
e87549d0cc0d5c1b1789de20869b70667e477671
28470 F20110331_AABWGH griffin_p_Page_099.QC.jpg
9786dcfd3e6890aa8903b2505afc202a
ba772a9a1e01fcce8cbb2897a66841e2dae20f13
73042 F20110331_AABWFT griffin_p_Page_084.jpg
cea315302672f4d4b787ef25b76e8432
fb68589c658551a8ba36dd1c1e7a19798e0165e4
1814 F20110331_AABVZZ griffin_p_Page_075.txt
86426bd06df1003f9b8ed77d3ebeabae
225de66c423366a147df98fec9c88622b132fb39
29931 F20110331_AABWGI griffin_p_Page_100.QC.jpg
9850d1e3a5f903589461b86295df84b3
62c8494e1ede9192d5afdc87766a1539a122b541
24461 F20110331_AABWFU griffin_p_Page_084.QC.jpg
d7d0b713475f6bd11a8cc250bb8d0e59
2abf22cd8f74d1b29352b8cd3f3c8574bbda453e
83065 F20110331_AABWGJ griffin_p_Page_103.jpg
d2b086cd24b981e2dcd5eb5301cbb1fc
d844603772f54cdcefaf2b43297873482ed8b1b6
30149 F20110331_AABWFV griffin_p_Page_085.QC.jpg
65c224e68732c92f91e764bbfcd16d2a
1400d134fb43dd56819cb08afc6c717780e703b7
24283 F20110331_AABWGK griffin_p_Page_103.QC.jpg
ae0b2ee15de25a833cdd87675b529c55
6df37addd94388277c315badbe7d67b9106a8196
14229 F20110331_AABWGL griffin_p_Page_104.QC.jpg
d5f71be968cea6e3250bd714948f9763
5bedc1a0577e4aca0c3dcd529654d70bb592b637
26466 F20110331_AABWFW griffin_p_Page_086.QC.jpg
c040a4300dd4d3c67841d689c33546f0
9697933515f9eabf0df1d2251fe732704f2edd3b
91678 F20110331_AABWHA griffin_p_Page_037.jp2
4a77471cef0078def7c1de49935a439a
047a2bf1689c4a33d80d8d504585aac60526928f
21486 F20110331_AABWGM griffin_p_Page_003.jp2
1e41309971b91069f63745f56eea0d3b
186ca511f441d8152a53ab27c4d161509eb6745f
18407 F20110331_AABWFX griffin_p_Page_087.QC.jpg
4988feea9d404d34db7c922db8abcacf
8d0d3a4902b5d9ede82270b3d583932f079521f6
89585 F20110331_AABWHB griffin_p_Page_039.jp2
d0f5f044af9d59d74987d3a2b7e1d2d6
f0b11b187e5ab76f462e9805b9113355da61a14f
1051974 F20110331_AABWGN griffin_p_Page_004.jp2
6ea0e7f922dea80c907a0ac8901dc935
defc4638a4aa7efd2bff9bd6ea1f7f2e6ce680e3
97606 F20110331_AABWFY griffin_p_Page_092.jpg
462c79c9961387e0cb8968ad15c2c311
fa9d5c369a1b548a070538d4bb2eb301194920e0
98434 F20110331_AABWHC griffin_p_Page_043.jp2
814a4131d1d62405a590d6ed2024f401
574cefa095f28992e083e820e0fd905e538b01c9
515458 F20110331_AABWGO griffin_p_Page_006.jp2
546a8c9df140210a4294955eae85ff0c
334e4d2a6f7371964b72a62966bf772ab356c8db
30643 F20110331_AABWFZ griffin_p_Page_092.QC.jpg
80a7757074f78ec774b0c4dd08c8ef5e
841d5c4c595e441d89d70347c41200003da8fc90
90877 F20110331_AABWHD griffin_p_Page_044.jp2
5591e2b00e29d4fde138df4ab41aa9f7
90271c12d91876be60501848043686cf8e4db4e9
91436 F20110331_AABWGP griffin_p_Page_010.jp2
349bffb85bdce09078cf784ecb99f106
d943a7893959f8545338a6750fee266d0790fccf
91755 F20110331_AABWHE griffin_p_Page_047.jp2
93dbfa9c3a0fa6b852c80f51fee1ce5b
1b746f9479756c9b4031f1ec623d495cab354ee2
106580 F20110331_AABWGQ griffin_p_Page_014.jp2
b0795fc08d30a6dfd1b382eb657ee91b
2f86cd6028475b2327783b43bf6793f412ec658b
101581 F20110331_AABWHF griffin_p_Page_051.jp2
f5cdfc7e08cf649ad847a35863e74cfd
ed48e230db5cc7392db0aa822375c7575b6f4fc7
41714 F20110331_AABWGR griffin_p_Page_015.jp2
292860cc584040442f94f39759c25e19
d4f07e6a645f9ee573631189c52b8a9693d70c72
112400 F20110331_AABWHG griffin_p_Page_052.jp2
01676b3d76a01cbe5ff48930b4a53da2
9fc7135338fb328f0d181225ce480930ae4d03fc
113846 F20110331_AABWGS griffin_p_Page_017.jp2
1e7c56dd09d5ea744a5a0c059d064d4b
91121aa6061f9587dc0e0a84d699ee2919205b9f
111202 F20110331_AABWHH griffin_p_Page_053.jp2
4307ce08043c2ad28bd76fa2a9ba4c40
cca1865b4c9ddf575c5be0c11fc2929a14688099
94078 F20110331_AABWGT griffin_p_Page_018.jp2
3aa3fe20e788aee5d12ba2c0d5fe649c
99fe6523023acfc5f23b08760a854f6c03b31ad3
108477 F20110331_AABWHI griffin_p_Page_057.jp2
3bca2e5039a333bd9c54b9d1878d7e08
bc2255f7b7701ed5005126fa5d1836e4c3f03bd4
84536 F20110331_AABWGU griffin_p_Page_020.jp2
b3d47c3bcbbc6977c847f754862f928e
9bc760537b3839d2b32c70aaa8be1ae1806f64f9
111791 F20110331_AABWHJ griffin_p_Page_059.jp2
7e8200a1c222166c310ca1654ae278c0
1ce1f97db429fa110d9a36459ac9f30aeeb8b38f
64732 F20110331_AABWGV griffin_p_Page_023.jp2
48fbfc80112e6652a92b715582a99619
d2306992bd0f0ca2d36b163b65b74a93c174b419
105325 F20110331_AABWHK griffin_p_Page_060.jp2
84893f9bc16a80f0be65005ea0450e2b
b07306ea330f42ed0f18d8fd3fd8d4b7a797518e
78795 F20110331_AABWGW griffin_p_Page_024.jp2
4ae1a4b8ed5bdb9011110c9e2d4a45e0
45d668f0ff6792c9fb65d49809d67c8c15cc479b
107048 F20110331_AABWHL griffin_p_Page_061.jp2
5da878cd8e4f05a25bfa90d9f8ea2f71
7f1c2f6dc61c07ca3024a4acfe2fb53ad5ade52e
109741 F20110331_AABWHM griffin_p_Page_062.jp2
c2e6d037e77caffe8f72623576019d18
46c8207be5d536e106d72976bb10ec5a3eab2297
81252 F20110331_AABWGX griffin_p_Page_025.jp2
1f769b4354e58b79062ababda48f0153
807617c80a66ed02dccbc4cbbc49a308c95b9aea
94790 F20110331_AABWIA griffin_p_Page_085.jp2
4c4485a636c7727036e9c4f9597caf5a
d43994102084075b8f5b0c133404a4750379b69d
109440 F20110331_AABWHN griffin_p_Page_063.jp2
b0dc5931b283dd21de5e951083b4d0d1
ad917c9a477108234fe9e789c3d0b7b1f09261ae
85314 F20110331_AABWGY griffin_p_Page_030.jp2
6dfe469ef1591424ef82eb9311849470
40a22c4c12b8dfdbcde27ecb0fb8aca9d405370d
423110 F20110331_AABWIB griffin_p_Page_087.jp2
2ebceecc2ea0bb80ba02169aa526aafd
a32f1b9418e0bc778f02742fcedbd16861fcd41f
881542 F20110331_AABWHO griffin_p_Page_064.jp2
3a5d9f952bcb3ae6eb6e70d03f518f78
b82eb408f406d2d50e15f261a1d22a8c9f20898e
89228 F20110331_AABWGZ griffin_p_Page_035.jp2
e1158712bebaa68b5d916aee9be07fb8
bc4cf9577f8d01381eb2b356c2f189cb06ecd6a7
433494 F20110331_AABWIC griffin_p_Page_091.jp2
2e4a5746731e9623257888655dfd922f
8c3313c2062824aa8e5030fa14976f93d25497c8
96971 F20110331_AABWHP griffin_p_Page_066.jp2
baad5affa6ef5fc89c56df310d264064
aa8c3ea82b0058242a67aefa0490ade9bdbc30e7
70211 F20110331_AABWID griffin_p_Page_094.jp2
48a4af41f9a7235bc587f6c62c42585e
f81312f5f53d8f9dee91b297e770bc7f500153b0
94265 F20110331_AABWHQ griffin_p_Page_068.jp2
8eedc266e515897618fd99217873c861
54c2331d9e6d7f054b7a06fab651e27583bc25cc
112324 F20110331_AABWIE griffin_p_Page_097.jp2
0da9fe26fde5466af6225335f5fff99f
096e29e2d39eec14862ea38f7abaf4ee666c3988
85278 F20110331_AABWHR griffin_p_Page_069.jp2
e7f54aa5e317da97e6ef471b3427f208
359fe8533957ea48c5b453aee8a78cd6574ea40d
92411 F20110331_AABWIF griffin_p_Page_099.jp2
5783ba3c05dd137463156461f04fb46b
5aad4faf0776f5b4a1d0085afd98d31d27be087f
104646 F20110331_AABWHS griffin_p_Page_070.jp2
a9047b9d1c56cc1793b01dc3a8955fdd
7048dd5fad89bbefdb356924891b59698e361bf2
883280 F20110331_AABWIG griffin_p_Page_103.jp2
a2fa4a273a5924c5cfc4213e37462214
54010c7921e905028795e2a2f4d0e97dd5d8c7e4
89275 F20110331_AABWHT griffin_p_Page_072.jp2
3f9f2872dc5bf40bba2a0f4d7f3022e0
79e3f0c44561d52d3463c55cd61d23520d09791d
2245 F20110331_AABWIH griffin_p_Page_001thm.jpg
99928c9e871a51faa18fb76d3b4228e0
d815ae6fffaee6a8f7d4952219a0d634bca0f531
96141 F20110331_AABWHU griffin_p_Page_074.jp2
101bee60980b25ebaac433e270f26ec9
55c5a84a897004fe0bedbf5691d54669f1a922b6
1956 F20110331_AABWII griffin_p_Page_003thm.jpg
6fa15cc318d4c1f9178b63b3f3e9c7cf
b2042b951f04ec407e049c52fdd57ac3081e74fe
85510 F20110331_AABWHV griffin_p_Page_078.jp2
5a0e9043111ceedb5bc063cffbefdaf9
5df109e3b14971054e185c772149b86350e90e50
5040 F20110331_AABWIJ griffin_p_Page_004thm.jpg
97fb4d6a55659939361ab3d9dde6365c
cdccf94692fe02f08a74c0af21589a728c06fbd4
79887 F20110331_AABWHW griffin_p_Page_079.jp2
a2f340b34759fb47e4013f4778443abd
f64d6ebebf834ff1304a7b2e9b9b8109e39cd938
3860 F20110331_AABWIK griffin_p_Page_005thm.jpg
fa3d4a76617ec009bb466117a236a020
9b027d4685663518659aea26ee596f7ce758a1fa
100200 F20110331_AABWHX griffin_p_Page_080.jp2
1f828b587836ec642b05633a9eb6fa20
2c5f85badf3ad057b9ae5e2311cb4c68e6c8e094
5995 F20110331_AABWIL griffin_p_Page_007thm.jpg
c42a9243c233098e002c81e09e35c756
3a7ae7206a82f5757e407b172aa0b520d271ee09
7870 F20110331_AABWJA griffin_p_Page_033thm.jpg
3ce8d092e2631766fb4713fb65ceb4d1
d46147ddcbc7b3c08b00af7fa5ce2bc423e516c0
6568 F20110331_AABWIM griffin_p_Page_008thm.jpg
f501acc21e4cb8219764049e747f577f
73323812bafd88590d01e48d525841b9a5dd2137
98146 F20110331_AABWHY griffin_p_Page_082.jp2
92dc41f32214c31d5d79624da6c5f57b
be383cb11d8a5dbb3a4d3976d8271024e0d76051
7200 F20110331_AABWJB griffin_p_Page_034thm.jpg
a69169676fb701c475e3b533b23b0775
0dab86d0ccf2ac493f05666185b365cec300e3b7
4834 F20110331_AABWIN griffin_p_Page_009thm.jpg
ed4960ccbd97c7ae0ea14beadfc09488
d6826ca5d427f12247858bfaec9297bb0830b52f
807872 F20110331_AABWHZ griffin_p_Page_083.jp2
854d7f2de6e94a6e3f01728eaba8013c
1d2db0486750ac1f5c3602cef8ae141cc58bfbef
7428 F20110331_AABWJC griffin_p_Page_035thm.jpg
0f2133888ed3720b1731b20b76abd5ae
47a5ce4ddbdaff0b7d44e1e40ec2636920eafc14
7240 F20110331_AABWIO griffin_p_Page_010thm.jpg
755a827ac680757567d297b5a1cf5cb3
bb76492c5accc17e6206969f87b587e8fcbb48c6
6685 F20110331_AABWJD griffin_p_Page_041thm.jpg
e434f716875be99160195788ef0b7ce4
370ea37f5ffcdf953c81c765f0a3919c17c465e2
8479 F20110331_AABWIP griffin_p_Page_013thm.jpg
da6a9384a211b744a78c47bb8b0af5b4
5fac5944043816f790201d5d0a961aa4d2b34a6e
7716 F20110331_AABWJE griffin_p_Page_047thm.jpg
d243904e7ac741f06b05067744e9a548
5e4d93a72f8813b8628d0aca86a478d16cfa4463
3235 F20110331_AABWIQ griffin_p_Page_015thm.jpg
0757a62cc4f50ace6c6859afef0223de
d5b8d9dd3c232ede556c550c97c81ab351620b4a
7668 F20110331_AABWJF griffin_p_Page_051thm.jpg
cd899bb981e5745e7cf33b8f4af82ff3
d13a4e97125f18f6b845a673f86aba9f004bd72a
7498 F20110331_AABWIR griffin_p_Page_016thm.jpg
232241b6ae6ad6c8e7f114b6de49a413
927edbdfdd2a7ffa322c3a1203753c6bb127d311
8718 F20110331_AABWJG griffin_p_Page_053thm.jpg
dbd274f069c23b18c26229c7e1f444a8
d20465ff50da5e126dff7dfcaa9374beea031d99
8435 F20110331_AABWIS griffin_p_Page_017thm.jpg
84a67212785ba89e734ff327de59a89a
eadce70fef9f0d971b8dad090e4ec61ab2fc4d72
8519 F20110331_AABWJH griffin_p_Page_054thm.jpg
4988a4a8da8296081e8ef4a1e25683eb
1ca185ee8fcf87a012bd4000929fb3d2167d6157
7366 F20110331_AABWIT griffin_p_Page_019thm.jpg
3d0bcdd596ccb852877610dff72cf991
9f0c270252905798847429ad6acc1444f751750a
8514 F20110331_AABWJI griffin_p_Page_055thm.jpg
2f83257a181c8775c7658c5832cf0880
6ba2ce13bbc90612748b43242e12ff98a7ea036a
6974 F20110331_AABWIU griffin_p_Page_020thm.jpg
3ccfa116078972cf2b7fee4950050347
d568762c5bde590429b7b8ed4350866bf2fcfc22
8248 F20110331_AABWJJ griffin_p_Page_058thm.jpg
eff6be292de6cd6011d3e12172953930
49573e75ab3ff00174d68d9acc65ecf8f4059a50
7031 F20110331_AABWIV griffin_p_Page_021thm.jpg
d46e83113a4676c4c2c0c79f90a66e3a
f98fbfaf15b420880dd65dc0441570448439f1d2
8221 F20110331_AABWJK griffin_p_Page_059thm.jpg
5021253dc03a92efd88856da4e26209f
5744a21cd563afb32b5a1b40c9450a1c480c77b1
7179 F20110331_AABWIW griffin_p_Page_022thm.jpg
05e56ac7e12dbc25ba2a3ba243ba2fa5
c79a5ba1284ec2c21f16a7e97d857c6b515e51a6
8323 F20110331_AABWJL griffin_p_Page_063thm.jpg
e2f95657d62899f36179425a985ff4b2
6141452a94734a8095d959f40476b320ea71f5ef
7003 F20110331_AABWIX griffin_p_Page_026thm.jpg
95eb2317e73aa751d77046cc4ca32425
12993d1a6ab80e95f3f024ea28197ad07b2a5049
8114 F20110331_AABWJM griffin_p_Page_065thm.jpg
1e26d318fd79d0e3d68c2539176e2f88
a94633ae68d5de7f9df02223c67c192f62f7ff50
7509 F20110331_AABWIY griffin_p_Page_029thm.jpg
443a864fac4e2e5f1f9d688107fb8c29
a7d4f6a8d03483599a8741f39d5badfe6656890e
7102 F20110331_AABWKA griffin_p_Page_096thm.jpg
13b53ee4db89a0f23929f71f9836bf77
5b4bdc408be21c41a5dbf2b0d050140015f42013
7532 F20110331_AABWJN griffin_p_Page_066thm.jpg
5758ea4a3b9eb5564e4f611129c483e3
100aa0df3890504e5d32dc178fb97e7a0d6dd55a
8600 F20110331_AABWKB griffin_p_Page_097thm.jpg
ac1398ed700c34df5eb4feeb334c3e00
51de04e5c5d72e8a52a8af1ffd30ef077f9ca05e
7522 F20110331_AABWJO griffin_p_Page_071thm.jpg
86157e99b970c1bc732151c1e955c78f
77a6644c21261951997ea2c6ab5b58af3018f59d
6979 F20110331_AABWIZ griffin_p_Page_031thm.jpg
4804356e768cee731ed7e4739eb04c04
74e1f79522147fb4104bd46a528813eef349c24b
916 F20110331_AABWKC griffin_p_Page_098thm.jpg
2bfbddf7c8e0c1e1fd285771c2eac9d2
6fda018abcb183553d7b4f91b9958a61e903cb90
7138 F20110331_AABWJP griffin_p_Page_072thm.jpg
3663a35b50dd8bac389fed6f392ccd1f
abb9e32b73d91a4d65f769083d031f2d5a26c6e8
7108 F20110331_AABWKD griffin_p_Page_099thm.jpg
d0b28758eafb2abd9497f567f3e85927
3b35fe10e4198efb2924d9a7d8ab721d63e928c8
7507 F20110331_AABWJQ griffin_p_Page_073thm.jpg
ecded75ed16fd3cc06ad2b7e3d9dc780
57c0cef5f2b787acf88cf67be0686538c5447091
7265 F20110331_AABWKE griffin_p_Page_100thm.jpg
8f6195252a8dee052dc9c25f29a0c6c0
e42403522bdbcc018570272160cdbc7efb286b60
7597 F20110331_AABWJR griffin_p_Page_074thm.jpg
925a175ab0d5b767dd34074ef15f25f9
b02ffea7d9b37cbde519b8946ebea4a324f805f3
1295 F20110331_AABWKF griffin_p_Page_101thm.jpg
431fca401aba40ef1331a2ce99de4ab1
83d2f9160621af97841ff2954a11ca0e987e0ef4
7441 F20110331_AABWJS griffin_p_Page_075thm.jpg
a6867137508b2459f2378c2ba018103b
ff26084bbf8bd022bfa01936d2cb6d3afb602d4b
7869 F20110331_AABWKG griffin_p_Page_102thm.jpg
f279172434bbc2b91811b5a137bc5a76
d7e298869140a99d8cbd86ccc8460074917c9900
7166 F20110331_AABWJT griffin_p_Page_078thm.jpg
9943bb5dc5f08b1a9a481060447135ea
e3544eedf443b91e85560f28bbbdb1bb6c4272d7
5824 F20110331_AABWKH griffin_p_Page_103thm.jpg
a5bf48c2c421b2a9a27d28ab2a00044e
91ade8c8925618d3b9ed8fd86a257d3214bb05b6
6426 F20110331_AABWJU griffin_p_Page_079thm.jpg
7ca7ba0fc9dd25012b0a6e06c73abd3a
0ffb3d4d56b6f47c4bb355b49c5b1d1bd8406ede
3802 F20110331_AABWKI griffin_p_Page_104thm.jpg
ab3030422cb2e974d2515d562e051d7b
86411d408d8cbfac259c3ecfd7d7d4d21b5e18f9
6745 F20110331_AABWJV griffin_p_Page_084thm.jpg
d44e5da8d83acdd4c1102b4fb29e24dc
5a8e390bc8c1e4b8c0f19859ef8a1ecb0965bb1b
123250 F20110331_AABWKJ UFE0014866_00001.mets
f760f6bf0ac2ebec73aee82c18bbef23
501471d04ba3c0ca727906b89706e9885d80d6e0
7760 F20110331_AABWJW griffin_p_Page_089thm.jpg
8fddec56eb90b26b57f2e8286194d011
abbb67abf0d4f52c003d77dc780846cef1c1572b
6862 F20110331_AABWJX griffin_p_Page_093thm.jpg
220490b2bfb45165291a5a59ef43117d
5b5f69403e281f05bdeb7031f540d21911c8d8c0
6669 F20110331_AABWJY griffin_p_Page_094thm.jpg
9a3fce5eae221b1fa6aaac8f6e67e842
9cdaabd408a84596036256dd8df52915c3297f7d
5142 F20110331_AABWJZ griffin_p_Page_095thm.jpg
8f3dfd37f5ff2ad4fd690d6c31b55050
5372765cb6135ecd2c266bec36e946a9fa9eea8f
112287 F20110331_AABVIF griffin_p_Page_054.jp2
fa0d1fc097d5a1c05381532814bacd67
c0941821241aed917898f2256740002e87c3ea32
1631 F20110331_AABVIG griffin_p_Page_079.txt
3368b35d3c9d6f1b691efbac19b8f742
ff30221d55981eae6f796ce09d5936f518dc02fd
F20110331_AABVIH griffin_p_Page_054.tif
5b8d8dc93de2b4810f5ffc9d8e4289d5
c18b08489bd95ee4d1fab6f887aae92eae9e0594
39442 F20110331_AABVII griffin_p_Page_078.pro
b10397813af14d5641bbf09c93848faa
d381a1daf03b62daadf1e1fe22d19682500763b6
88907 F20110331_AABVIJ griffin_p_Page_036.jpg
869bc369748a1bfe6beb8bbced45c511
c8f2bf82fde292b7c24e0c2a1e7a61c13ec567c8
95994 F20110331_AABVIK griffin_p_Page_051.jpg
fbd66c41d9d1e98ca22202db94704ff1
490fea774038823817dc3ff5565e04fb6c59ae98
F20110331_AABVIL griffin_p_Page_046.tif
61cf37879a5087d744b488846884c558
40edcfe18c6c9334276ec10b9325f84312e5660c
50309 F20110331_AABVJA griffin_p_Page_056.pro
c34c6acfec68021d29b35d615e61a5a5
010e82015829f03a1887f9f5aed6fe4b274b8a3a
49653 F20110331_AABVIM griffin_p_Page_091.jpg
da4faf18e9d05cfcaf46a64bfb6f6113
99c00f47530ab971e6db72ae0f28b794c9bd237f
8425 F20110331_AABVJB griffin_p_Page_057thm.jpg
4220e1d096c1f36ba34167be9d6d8c65
37285133446c7d9642d64c7c11d46697938e5ce8
103589 F20110331_AABVIN griffin_p_Page_065.jp2
8d9c11f0cadcbced0db756c4c589fbfc
c909f4fa41f8b398eb80e1de6e733296bd3453b5
89085 F20110331_AABVJC griffin_p_Page_022.jp2
fd0a8c97f48df49ca7c58f9126e291c9
2aad07b7279d4bf01d1f79917e6da86c02d036c0
9687 F20110331_AABVIO griffin_p_Page_098.jp2
a5a1f0762b30781814974d905002aea1
0b81bc8838d9d728e8a66322eab94370d4ce55d6
83784 F20110331_AABVJD griffin_p_Page_021.jpg
8a8180ee7b483d609aa28fbcb2913b94
3ea766ba4647aee30a8b55fc756bac915496cfb5
81002 F20110331_AABVIP griffin_p_Page_027.jp2
cdb596afb30e59a8ad918580c04092cf
6e2cc20179ce3becd23edfd8611f2465f5a581e1
111490 F20110331_AABVJE griffin_p_Page_058.jp2
a637426315f2f7405d96ae0a94253332
de54c5c3d3a2eb8445cf5628d3823cb5cd77f8bf
30270 F20110331_AABVIQ griffin_p_Page_023.pro
9f7d46785f01ac17d72f2a1c2ab3b3bb
5878c1af94ad3d9fdc576e508c0c87963ee00e66
50520 F20110331_AABVJF griffin_p_Page_081.pro
eb8b99fb1c55887b752710d79f3515c3
48aaef3d582e51d12801f4630137e0af88bdf6aa
107243 F20110331_AABVIR griffin_p_Page_012.jpg
95b46d9ad8cdd38ae4bdb2b89e42d545
42640ec6fcef616d740787287f743c196082db6d
29329 F20110331_AABVJG griffin_p_Page_032.QC.jpg
00f619c13059250049bde52b689b0632
3774f9e3c94e33ed417d88946eb2423f76793788
885310 F20110331_AABVIS griffin_p_Page_096.jp2
f3e08c8f9b0d50dd87938d9c4495f399
3c301429ff777c9479333f60142be64d60ad960a
6886 F20110331_AABVJH griffin_p_Page_040thm.jpg
ec1a4eef1a8ac5083f8fee6112a7117f
def79b6e7c065c2db3dea1128e8a85071b243d4b
800093 F20110331_AABVIT griffin_p_Page_086.jp2
cb9b642d8aa5224f5fc787ed1f3b5ac1
ca4c8543cfed7270e2938789a7ac158201f1ce23
44061 F20110331_AABVJI griffin_p_Page_095.jp2
17de38da202e10d6df89ebf207a542ec
03092c067a08129dfe9ef4cb4df76661f91e2c9f
56626 F20110331_AABVIU griffin_p_Page_007.pro
eae5e8ffdf73ec65bba74c6359ecf141
c1997e19f209f6a720a59d390832fe789006ae97



PAGE 1

TWO-DIMENSIONAL MODELING OF A CHEMICALLY REACTING, BOUNDARY LAYER FLOW IN A CATALYTIC REACTOR By PATRICK D. GRIFFIN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006

PAGE 2

Copyright 2006 by Patrick D. Griffin

PAGE 3

iii ACKNOWLEDGMENTS Siemens and the National Aeronautics a nd Space Administration supported this research. I thank Dr. David Mikolaitis, Dr. David Hahn, and Dr. Corin Segal for their assistance. I also thank my parents for th eir continued support a nd involvement in and out of my educational development.

PAGE 4

iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES..........................................................................................................vii ABSTRACT.....................................................................................................................vi ii CHAPTER 1 INTRODUCTION........................................................................................................1 2 REDUCTION OF CONSERVATION EQUATIONS.................................................7 Applying Assumptions.................................................................................................8 Continuity Equation...............................................................................................9 Species Continuity Equations..............................................................................10 Momentum Equations.........................................................................................12 Energy Equation..................................................................................................15 Order of Magnitude Analysis.....................................................................................21 Continuity Equation.............................................................................................22 Species Continuity Equations..............................................................................23 Axial Momentum Equation.................................................................................25 Vertical Moment um Equa tion.............................................................................27 Energy Equation..................................................................................................30 Unit Analysis..............................................................................................................34 Continuity Equation.............................................................................................34 Species Continuity Equations..............................................................................35 Momentum Equation...........................................................................................36 Energy Equation..................................................................................................37 Summary of Governing Equations.............................................................................40 3 PROGRAM METHODOLOGY................................................................................41 Discretization..............................................................................................................43 Parameters and Conditions.........................................................................................45 Input and Output Files................................................................................................47 Initial Conditions of a Stage.......................................................................................49

PAGE 5

v Stage One.............................................................................................................50 Blasius Solution...................................................................................................51 Subsequent Stages...............................................................................................53 Solving Governing Equations.....................................................................................53 Momentum Equation...........................................................................................57 Continuity Equation.............................................................................................59 Species Continuity Equations..............................................................................61 Energy Equation..................................................................................................63 Species/Energy System of Equations..................................................................67 4 TESTING....................................................................................................................71 Case One.....................................................................................................................72 Results of Case One....................................................................................................73 Case Two....................................................................................................................75 Results of Case Two...................................................................................................76 Case Three..................................................................................................................79 Results of Case Three.................................................................................................80 Case Four....................................................................................................................83 Results of Case Four...................................................................................................83 5 PROGRAM LIMITATIONS AND IMPROVEMENTS............................................90 6 CONCLUSION...........................................................................................................91 LIST OF REFERENCES...................................................................................................93 BIOGRAPHICAL SKETCH.............................................................................................95

PAGE 6

vi LIST OF TABLES Table page 2-1 Equations modeling the flow....................................................................................40 2-2 Units of the governing equations.............................................................................40 4-1 Parameters and conditions of case one.....................................................................73 4-2 Parameters and conditions of case two.....................................................................76 4-3 Parameters and conditions of case three...................................................................79 4-4 Parameters and conditions of case four....................................................................83

PAGE 7

vii LIST OF FIGURES Figure page 2-1 Dimensionless variables...........................................................................................21 3-1 Flow chart for single stage modeling.......................................................................55 3-2 Flux components in the species/energy system........................................................68 3-3 Source components in th e species/energy system....................................................69 3-4 Boundary conditions of the species/energy system..................................................70 4-1 Axial velocity profiles of case one...........................................................................74 4-2 Pressure plot of case one..........................................................................................75 4-3 Axial velocity profiles of case two...........................................................................77 4-4 Pressure plot of case two..........................................................................................78 4-5 Reduction in methane concentrations of case two...................................................78 4-6 Axial velocity pr ofiles of case three.........................................................................81 4-7 Pressure plot of case three........................................................................................82 4-8 Reduction in methane concentrations of case three.................................................82 4-9 Axial velocity pr ofiles of case four..........................................................................84 4-10 Pressure plot of case four.........................................................................................85 4-11 Temperature profiles of case four............................................................................86 4-12 Methane concentra tions of case four........................................................................86 4-13 Hydrogen concentrations of case four. A) Mass fractions of atomic hydrogen. B) Mass fractions of diatomic hydrogen..................................................................87

PAGE 8

viii Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science TWO-DIMENSIONAL MODELING OF A CHEMICALLY REACTING, BOUNDARY LAYER FLOW IN A CATALYTIC REACTOR By Patrick D. Griffin August 2006 Chair: David Mikolaitis Major Department: Mechanic al and Aerospace Engineering Problems associated with fossil fuels are in creasing interest in alternative forms of energy production. Hydrogen is quickly b ecoming a popular option, but the efficient, affordable production of hydrogen is needed for it to become a viable source of energy. Catalytic reformation of hydrocarbons and alc ohols appears to be a promising means of hydrogen production, but little is known about the surface chemistry. Research on heterogeneous catalyst and their reacti on mechanisms is growing. A greater understanding of the surface chemistry could yi eld cheaper, more effective catalysts. The evolving chemistry of the surface catalyst is in need of a flexible software program to test new surface mechanisms. A program is developed to model chemi cally reacting flow through a catalytic reactor. The reactor is represented in two-dimensional Cartesian coordinates with negligible body forces acting on the fluid. The flow is characte rized as a steady, low Mach number, boundary layer flow of a Newt onian fluid. Basic principles of mass,

PAGE 9

ix species mass, momentum, and energy conser vation are expressed mathematically and simplified. These principles are transformed into the equations cont rolling the behavior of the fluid and its motion through a pro cess of applying assumptions, an order magnitude analysis, and a unit analysis. A code is written to numerically solve the resulting system of coupled governing equa tions. The methodology of constructing the program is decomposed into developing an orthogonal computational mesh, quantitatively defining the r eactor and flow, locating chem ical data and solutions, establishing initial boundary conditions, and solving the governing equations. The program is used to model four different fl ows: one with no chemistry, the second with only gas chemistry, and the third and fourth with gas and surface chemistry. Calculated solutions from each case are examined to conf irm that the software produces reasonable results and is operational. The software is found to predict the point of ignition when the initial temperature is great e nough to cause catalytic combustion.

PAGE 10

1 CHAPTER 1 INTRODUCTION The world is becoming increasingly aware of its dependence on fossil fuels. This fuel is meeting over eighty-five percent of our countrys energy demands, which includes everything from electricity to transportation [1]. The power of fossil fuels lies in the atomic bonds of the hydrocarbons that make up these fuels. Energy is released by breaking these bonds in the process of com bustion. The burning of fossil fuels also releases harmful byproducts that include: carbon monoxide, carbon dioxide, and nitrogen oxides. The carbon released into the atmos phere is originally trapped underneath the earths surface, leading to an overall increase of carbon oxide s in the atmosphere. Some believe these byproducts are leading to weat her changes and health problems around the world. Energy extraction from fossil fuels is a re latively easy process and the fuel is readily available in deposits beneath the eart hs surface. For these reasons, fossil fuels have become the main source of the world s energy production. The finite source is nonrenewable and will eventually run out. Decreasing supplies wi ll lead to a rise in fuel cost and alternative forms of energy will become cheaper than fossil fuels. Economics involved with the decrease in fuel supplies wi ll dictate that the worl d turn to alternative forms of energy. Whether for ecological or economical r easons, the world will need to find alternative forms of energy. Some look to the most abundant element in the universe, hydrogen. Hydrogen is a clean, renewable source of energy that can be used in

PAGE 11

2 combustion engines and fuel cells. Fuel cells are very efficient at producing electricity from hydrogen with the byproduct being water. A major obstacle in th is alternative fuel is the affordable production of the energy carrier. Hydrogen rarely stands alone in its pure form. Most of the earths hydrogen is bonded to oxygen and carbon, in the form of water, alcohols, and hydrocarbons. Water is an extremely stable molecule and takes a great deal of energy to extract hydrogen atom s. This energy must come from renewable sources if we wish to address the problems associated with fossil fuels. Hydrogen extraction from alcohols and hydr ocarbons is much easier. However, fossil fuels are currently the main source of hydrocarbons. About ninety-five percent of the hydrogen supply comes from the catalytic steam reforming of natural gas according to the US Department of Energy [2]. Natural gas is a relatively clean fossil fuel consisting mostly of methane. But natural gas is still a finite resource that will eventually run out. A very promising renewable source of hydrogen comes from ethanol. Ethanol is an alcohol that can be derived from biomass such as corn. Fuels produced from biomass release carbon into the atmosphere that is originally in the atmosphere leading to zero net-production of carbon oxides [3]. There are many promising energy alternatives to fossil fuels. However, fossil fuels are so entrenched in our way of life, economi cally and politically, that few expect a quick transition away from fossil fuels. Most believe that hydrogen production will initially come from fossil fuel s, with a gradual transition to renewable sources of hydrogen production. The worlds attraction to the hydrogen economy is leading to an increased interest in heterogeneous catalyst for converting hydrocarbons and alcohols into the energy carrier. Surface catalysts are useful in incr easing the reaction rates in combustors and

PAGE 12

3 reformers. Catalytic combustors burn the fuel over a catalyst. This burns fuel at a lower temperature, which decreases the amount of n itrogen oxides produced in the exhaust [4]. Catalytic reformers transform complex hydrocarbons and alcohols into hydrogen by stripping the fuel of their hydrogen atoms. In either case, the fuel molecule is adsorbed by the catalytic surface. The molecule form s a bond with the surface, usually through an oxygen or carbon atom. This weakens the ad jacent bonds between the oxygen or carbon atom and the hydrogen atoms. The hydrogen atom s now begin to break off the molecule. The product molecule will detach from the surf ace once it is finish ed reacting with the catalyst. This leaves the surface free to adsorb a new reactant molecule. The catalyst provides reaction pathways with lower activation energies. In effect, the catalyst lowers the energy needed to break a molecule apart [5]. The efficient, affordable production of hydroge n is needed for this alternative fuel to become a viable source of energy. The efficiency of a metal to catalyze a given molecule is defined by how well the catalys t adsorbs the reactants and desorbs the products. Silver, for example, isn't a good catalyst because it doesn't form strong enough attachments with reactant molecules. Tungsten, on the other hand, isn't a good catalyst because it adsorbs too strongly. Metals like platinum and nickel make good catalysts because they adsorb strongly enough to hold and activate the reactants, but not so strongly that the products can't break away. [6] The efficiency of the catalyst has no aff ect on the metals price. The price is dependant on the demand and rarity of the metal. As mentioned above, platinum and nickel are two common metals used in catalyst. Plati num cost approximately $1000 per ounce, where nickel cost around $0.4 per ounce [7 ]. With such a large disparity between efficiency and price, a greater understandi ng of the surface chemistry could lead to cheaper, more effective catalyst.

PAGE 13

4 Catalytic reformation of hydrocarbons and alcohols appears to be a promising means of hydrogen production, but little is know about the surface reactions. Surface catalysts are not fully understood because the chemistry around the surface is difficult to measure, especially in normal operating conditio ns. In the past, catalysts were treated as a black box. The black box representation of the catalyst usually consists of one global surface reaction or a small series of reduced mechanisms. Modifications to the black box can be made until the model accurately reproduc es the experimental data. While this method is adequate for engineering applica tions, it does not accura tely represent the chemistry involved [4]. Many studies have re cently taken place in attempts to understand the surface reaction mechanisms of the heterogeneous catalys t [8-11]. The studies are mostly concerned with determining the reacti on pathways and the st ep-by-step chemical degradation process of molecules. This is leading to new chemical reactions being added to the surface chemistry. The evolving chemis try of the surface cataly st is in need of a flexible software program to test the new mechanisms being added. A program adaptable to the changing surface chemistry is developed in this study. The program models a two-dimensional, ch emically reacting flow though a catalytic reactor. The fluid motion is characterized as a steady, low Mach number, boundary layer flow. The catalytic reactor c onsists of a heterogeneous ca talyst covering the inside surface of a pipe or channel. The fluid mo tion is modeled as a flow through two flat plates with a pressure gradient The two flat plates are mode led as catalytic surfaces and are identical. Basic principles of ma ss, species mass, momentum, and energy conservation are employed to generate the model. These principles are expressed mathematically and simplified for this speci fic problem. The process of reducing the

PAGE 14

5 principles into the governing equations c onsists of applying assumptions, an order magnitude analysis, and a unit analysis. A software code is written to numerically solve the resulting governing equations. The calculated solutions thermodynamically a nd kinetically define the fluid and its motion. The code is written in MA TLAB, a programming language created by MathWorks that is used in many chemical flow simulations. MATLAB provides several built-in capabilities that make the softwa re well suited for this problem. Its compatibility with Cantera being one such cap ability. Cantera is a free software package developed by Professor David Goodwin at th e California Institute of Technology to solve problems concerning chemical r eactions [12]. The program u tilizes Cantera software to manage the chemistry. The methodology of the programs development includes the creation of an orthogonal computational mesh to resolve the equations. Then the establishment of parameters and conditions th at quantify the reactor and fluid flow is performed. The location of input and output data is defined and initial boundary conditions are set. Finally the governing e quations are solved and the flow in the catalytic reactor is modeled. The program is tested with four different cases: one with no chemistry, another with only gas chemistry, and two with gas and surface chemistry. Each case is modeled and the re sults are examined to confir m that the software produces reasonable results and is operational. In the future, calculated solutions can be compared to experimental measurements. New surface mechanisms can be tested with the program resulting from this study. Improved chemical kinetic data are updated in Cantera. The new chemistry is processed by Cantera and incorporated into the program. Any change in the chemistry being used

PAGE 15

6 to model the flow is done so inside the separate software of Cantera and not the main program. Because the data is stored separate ly, the program is able to remain flexible with the type of catalyst and fuel being used. This also allows the type of reaction pathways to change as our understanding of cat alyst grows without altering the code. As a result the program is adaptable to the varying surface reaction pathways. Comparing the two-dimensional model to experimental data provides a means of validating the accuracy of the new chemistry. With a bett er understanding, heterogeneous catalyst might be the key to the clea n, renewable source of energy.

PAGE 16

7 CHAPTER 2 REDUCTION OF CONSERVATION EQUATIONS The chemically reacting flow through th e reactor is modeled by numerically solving the governing equations. Four of the equations are derived on the principles of mass, momentum, and energy conservation [ 13]. The velocity field, pressure, and temperature in the reactor are determined with these four equations. Knowing two independent thermodynamic properties would ad equately model the fl ow if it were not for the chemistry taking place. The catalytic surface is expected to induce chemical activity changing the fluid composition. A set of equations is needed to determine this changing chemical composition. The species co ntinuity equations satisfy this need and are called upon to calculate th e composition of the flow. One equation is needed to determine the mass fraction of a single atom or molecule. As a result, the number of equations inside this set is e qual to the number of species used to model the flow, denoted as N The mass, momentum, species and energy e quations along with an equation of state are all that is needed to determine flow properties through out the reactor. These governing equations are coupled to one another in several different ways. All of them contain properties dependent on the flow variables. For example, the momentum, species, and energy equations contain transp ort properties such as viscosity, diffusion coefficients, and thermal conductivity. Most of these properties are dependent on the pressure, temperature, and composition of th e flow. Properties dependent on the flow variables indirectly couple the equations to one another. Th e equations are also directly

PAGE 17

8 coupled to one another. Vertical gradients of the species mass fraction not only appear in the set of N species continuity equations, but also the final form of the energy equation. In addition to this, the velocity and velocity gradients can be found in all of the equations. This makes for a group of highly coupled equati ons that control the be havior of the flow. Equations of mass, momentum, species, and energy conservation are broken down and modified to reflect this specific model while Cantera processes the equation of state for an ideal gas. Conservation equations ar e transformed into th e governing equations by applying assumptions characte rizing the reactor and flow. Governing equations are individually examined in an order magnitude analysis after the assumptions are made. Dominant terms in a given equation are found by comparing their magnitude to the magnitude of other terms in the equation. Neglecting the weak terms and retaining the strong terms further reduce the equations. A un it analysis or unit chec k is applied to the resulting system of equations to ensure the validity of the equations. The process also establishes the units of each va riable, property, and solution. Applying Assumptions Turns goes through a similar process of si mplifying the conservation equations for a steady one-dimensional flow [14]. Instead of one dimension, the computational space of the catalytic reac tor is modeled in two-dimensiona l orthogonal space. These two dimensions are the rectangular coordinates x and y which represent the axial direction and vertical direction respectfully. Once th e governing equations ar e reduced to their two-dimensional form, they are simplified by making assumptions about the fluid and its motion. The chemically changing fluid is al ways considered a Newtonian fluid, which carries many assumptions with it. Most importa ntly of which is that the shear stress is linearly proportional to the ra te of deformation. Another important assumption concerns

PAGE 18

9 the fluids motion. The flow is modeled as a steady-state flow meaning all fluid properties are independent of tim e. As a result, a partial de rivative of any quantity with respect to time is zero. More assumptions are made in order to reduce the governing equations and are discussed duri ng that process below. For the most part, the analysis mirrors that of a boundary layer flow. Howe ver, the catalytic surface creates density variations in the flow and comp ressibility must not be ignored. Continuity Equation The analysis begins with the reduction of the conti nuity or mass equation. The continuity equation is a mathematical repr esentation of the cons ervation of mass that states that mass cannot be crea ted or destroyed. In a Euleri an method of description, the conservation of mass is described as the time ra te of change of mass in a control volume being equal to the net flux of mass through the control surface. Equation 2-1 is the vector form of the continuity equation. 0 V t (Equation 2-1) The steady flow assumption leads to the partial derivative with respect to time being zero. The first term in Equation 2-1 is dropped as a result, leaving only the mass flux in vector notation. Th e catalytic reactor is being modeled in a two-dimensional space. Therefore, the mass flux is written out into its two-dimensional form with u and v representing the x and y component of the veloci ty, respectfully [15]. 0 uv xy (Equation 2-2) Further simplification is restricted due to the fact that density (defined as ) variations occur in the flow. Equation 2-2 represents the c ontinuity equation for the flow

PAGE 19

10 field being modeled. This equation proves to be very important in the reduction of the other governing equations. However, it is not the form used by the program. The computer code uses the mass equation to de termine the vertical velocity and some mathematical manipulation is needed before the equation reaches its final form below. u v v yxy (Equation 2-3) Species Continuity Equations Much like the continuity equation, the speci es continuity equati on requires that the rate of gain of a single species mass in a c ontrol volume equals the net flux of the species mass in through the control surface. Dissimilar ity in the two equations arises due to the chemistry. Instead of equaling zero, it equals the net chemical produc tion of that species in the control volume. The continuity equation of species i is shown as Equation 2-4 and the set consists of one equation for each speci es. The time rate of change of the species mass is zero because the flow is steady. The species mass flux is expanded out into its two-dimensional Cartesian coordinate fo rm and the result is Equation 2-5. i i im m t Y (Equation 2-4) , iy ix im m m xy (Equation 2-5) On the right hand side of the species continuity equation lies the net chemical production of species i in the control volume (im ). This is determined using Cantera, which gives the chemical production in moles. Therefore, the species mass chemical production is replaced with the molar chemi cal production times the molecular weight of

PAGE 20

11 the species. The species mass entering th e control volume, know as the species mass flux, transpire as a result of two modes, bulk flow and diffusion [14]. ,iy ix iim m M W xy (Equation 2-6) ,,, ixiixDiffmYum (Equation 2-7a) ,,, iyiiyDiffmYvm (Equation 2-7b) The first term in Equations 2-7a and 2-7b is the mass flux due to the bulk flow. It is equal to the product of the density, species mass fraction, and fluid velocity component corresponding to the direction of the mass flux. The second term is the mass flux due to diffusion. The species mass flux can now be placed in the reduced species continuity equation and the two modes separated from each other. ,,,,iiixDiffiyDiffiiYuYvmmMW xyxy (Equation 2-8) The chain rule is applied to the bulk flow terms and the process is shown in Equation 2-9. This leaves the continuity equation being multiplied by the mass fraction plus two mass fraction gradient terms being multiplied by the density and velocity. The continuity equation is equal to zero via Equation 2-2. After dropping the mass equation and replacing the two bulk flow terms with Equation 2-9, the species continuity equation reduces to Equation 2-10. ii iii ContinuityEquationuv YY YuYvYuv x yxyxy (Equation 2-9) ,,,, ii ixDiffiyDiffiiYY uvmmMW xyxy (Equation 2-10)

PAGE 21

12 Diffusion is a result of concentration grad ients, temperature gradients, pressure gradients, and uneven body forces. Ordinary diffusion from concentration gradients is the only mode of diffusion considered in this model. The species mass diffusion is approximated using a mixture-averaged diffu sion coefficient [14]. The mass diffusion terms are replaced with Equations 2-11a a nd 2-11b inside the governing equation. Equation 2-12 is the species continuity equati on after all the assumptions are applied. Further simplification is possible w ith an order magnitude analysis. ,, i ixDiffimY mD x (Equation 2-11a) ,, i iyDiffimY mD y (Equation 2-11b) iiii imimiiYYYY uvDDMW xyxxyy (Equation 2-12) Momentum Equations The momentum or Navier-Stokes equa tion is analogous to Newtons law of momentum conservation. The momentum equation states that the rate of change of linear momentum per unit volume equals the net mo mentum flux through that volume plus the sum of forces acting on the volume. This is mathematically written in vector form as Equation 2-13. Forces acting on the control volume are broken up into the surface forces and body forces. Surface forces are defined as the divergence of the stress tensor. The stress tensor is shown below in its recta ngular, two-dimension form as Equation 2-14. BF V V t V (Equation 2-13) xxxyxxxy yxyyyxyyp p (Equation 2-14)

PAGE 22

13 The stress tensor is made up of the shear stress acting on the surface plus the pressure acting normal to the volume. Agai n, the flow is considered steady with negligible body forces. Therefore, the time derivative and body force terms are dropped. The vector form of the momentum equation is separated into its x-component and ycomponent equations. Equation 2-15a repr esents the two-dimensional Cartesian momentum equations in the x-direction, while Equation 2-15b is in the y-direction. yx xxuuuv x yxy (Equation 2-15a) xyyyvuvv x yxy (Equation 2-15b) The x-momentum equation is used as one of the governing equations in the computer program. Equation 2-15b, on the othe r hand, is not explic itly used in the computer code. It is used to gain some insi ght into the behavior of the pressure. Both equations simplify in a similar manner so both are reduced collectively. In this process, the x-component equation is given first followed by the y-component equation. The chain rule is applied to the momentum convec tion on the left hand side of the two NavierStokes equations. _yx xx ContinuityEquationuv uu uuv x yxyxy (Equation 2-16a) xyyy ContinuityEquationuv vv vuv x yxyxy (Equation 2-16b) This results in the continuity equation being multiplied by the x-velocity in Equation 2-16a and by the y-velocity in Equation 2-16b. Th e continuity equation is equal

PAGE 23

14 to zero and the first term in these equations is dropped accordingly. The two momentum equations are reduced to Equations 2-17a and 2-17b. yx xxuu uv x yxy (Equation 2-17a) xyyyvv uv x yxy (Equation 2-17b) The flow through the reactor consists of a Newtonian fluid. The stress acting on a Newtonian fluid has no preferred direction, meaning the stress tensor matrix is symmetric and its components are defined below. By definition the shear stress of a Newtonian fluid is proportional to the rate of deformati on. Shear stresses of a Newtonian fluid are written below as functions of the velocity gradients [15]. xxxx p (Equation 2-18a) yyyy p (Equation 2-18b) xyyxxyyx (Equation 2-18c) 2 2 3xxuuv x xy (Equation 2-19a) 2 2 3yyvuv yxy (Equation 2-19b) xyuv yx (Equation 2-19c) Replacing the stress components with thei r definitions above and separating the pressure gradient, the two momentum equations become,

PAGE 24

15 2 2 3 uup uv xyx uuvuv x xxxyyyx (Equation 2-20a) 2 2 3 vvp uv xyy uvvuv xyxyyyxy (Equation 2-20b) Equations 2-20a and 2-20b represent the momentum equation in the x-direction and y-direction respectfully. Both are reduced to their final form via an order of magnitude comparison. Energy Equation The Energy Equation requires that the rate of change per unit volume is equal to the net energy flux into that volume due to convection, heat, and work [13]. BF V V q V e V V e t 2 22 2 (Equation 2-21) From left to right in Equa tion 2-21, the first term is the time rate of change of energy, which is zero because of the steady fl ow assumption. The second term represents the flux of energy due to convection and equa ls the heat transferred into the control volume plus the work done by the surface forces and the body forces. The work done by the surface forces is determined using the st ress tensor of Equation 2-14. The body force is assumed to be negligible ; therefore, the work done by the body force is neglected. Equation 2-21 is written out in its two-dime nsional Cartesian form with the assumed simplifications.

PAGE 25

16 2222y x xxxyyxyyVV ueve xy q q uvuv xyxy (Equation 2-22) The stress components are replaced with the shear stress and pressure. The pressure is separated from the shear stress te rms, leaving two pressure work terms at the end of the energy equation. 2222y x xxxyxyyyVV ueve xy q q uvuvupvp xyxyxy (Equation 2-23) Placing the x and y pressure work terms on the far right of Equation 2-23 into the corresponding x and y energy convection terms on the left, Equation 2-23 becomes Equation 2-24. The internal energy and pressu re is replaced by the enthalpy, defined in Equation 2-25 as the internal energy plus th e product of the pressure and the specific volume. The energy transfer due to the shear stress work is replaced by a variable called ( _work) to save space. Simplification of this term is possible via an order of magnitude analysis of the governing equation, but first the energy convection a nd heat flux terms are reduced. As of now the energy equation can be written out as Equation 2-26. 22_ 22y xq q pVpV uevework xyxy (Equation 2-24) p he (Equation 2-25) 22_ 22y xq q VV uhvhwork xyxy (Equation 2-26)

PAGE 26

17 First the two energy convection terms on th e left hand side of Equation 2-26 are simplified. These expressions consist of an enthalpy flux and kinetic energy flux, both due to bulk flow. The two convection terms are separated into en thalpy convection and kinetic energy convection. Performing th e chain rule on the two kinetic energy convection terms produces four separate terms. Equation 2-28 illustrates the process. Two of these terms are combined to form the kinetic energy multiplying the continuity equation, which equals zero. After dropping th is term, the kinetic energy flux is replaced with the last two terms of the equation above and the ener gy equation now takes the form of Equation 2-29. 2222 _y xuhvh VV uv xyxy q q work xy (Equation 2-27) 22 2 22 _22 222ContinuityEquationVV uv xy uv Vuv VV xyxy (Equation 2-28) 2222 _y xuhvh uv VV xyxy q q work xy (Equation 2-29) Moving over to the right hand side of the energy equation, the heat flux terms are now simplified. The heat flux is determined using Fouriers Law of Heat conduction plus the flux of enthalpy [14]. The enthalpy flux he re is due only to diffusion. The flux of enthalpy from the bulk flow is already accounted for in the convection term. The vector equation of the heat flux is broken up into the two Cartesian c oordinate components, x

PAGE 27

18 and y. Equations 2-7a and 2-7b are used to re place the species diffusion mass flux inside the sum of the heat flux. 1 N ii Diff iqkTmh (Equation 2-30) ,,, 111 NNN xixDiffiixiii iiiTT qkmhkmhuYh xx (Equation 2-31a) ,,, 111 NNN yiyDiffiiyiii iiiTT qkmhkmhvYh yy (Equation 2-31b) Taking the partial derivative of the two equations above produces Equation 2-32a and 2-32b. The partial derivative is not aff ected by the species sum and therefore can be moved inside the sum. Similarly, the mass flux due to the bulk flow is not affected by the sum and can be moved outside of the sum. Th e sum located inside the partial derivative of the last term contains th e product of the species mass fr action and species enthalpy. Since the species enthalpy is given on a mass basis, the sum is equal to the specific enthalpy of the flow. The last term can be rewritten as the partial derivative of the density, velocity component, and enthalpy product. 11 NN x ixiii iiq T kmhuYh xxxxx (Equation 2-32a) 11 NN y iyiii iiq T kmhvYh yyyyy (Equation 2-32b) 1 N x ixi iq T kmhuh xxxxx (Equation 2-33a) 1 N y iyi iq T kmhvh yyyyy (Equation 2-33b)

PAGE 28

19 The two heat flux terms of the energy e quation are replaced with Equations 2-33a and 2-33b above. Once this is complete the energy equation takes the form of Equation 2-34 shown below. uh x vh y 22 ,, 1122NN ixiiyi iiVV uv xy TT kkmhmh xxyyxy uh x vh y work (Equation 2-34) The enthalpy convection cancels with the modified enthalpy diffusion of the heat flux. The chain rule is performed on the e xpressions inside the two sums. The process, shown in Equations 2-35a and 2-35b, leaves the species enthalpy times the partial derivative of the species mass flux plus the sp ecies mass flux times the enthalpy gradient. Equations 2-7a and 2-7b are used again to replace this specie s mass flux. This procedure takes the original partia l derivatives and splits it into three terms each. ,, ,,,, ixix iii ixiiixiiixDiffmm hhh mhhmhuYm x xxxxx (Equation 2-35a) ,, ,,,,iyiy iii iyiiiyiiiyDiffmm hhh mhhmhvYm yyyyyy (Equation 2-35b) Replacing the two partial deri vatives with their expande d expressions above, the process of simplifying the sums can begin. For the last two expressions in Equations 236a and 2-36b, the gradient of the species enthalpy is equal to the product of the species specific heat and the species temperature grad ient. Every species comprising the fluid at a given point in the flow is assumed to ha ve the same temperature, which means the temperature gradient can be moved outside of th e sums. This process is done for all four of the terms containing enthalpy gradients.

PAGE 29

20 ,,, 1111 NNNN ix ii ixiiiixDiff iiiim hh mhhuYm x xxx (Equation 2-36a) ,, 1111yNNNN iy ii iyiiiiDiff iiiim hh mhhvYm yyyy (Equation 2-36b) 11iNN i iipp iih TT uYuYcuc x xx (Equation 2-37a) 11iNN i iipp iih TT vYvYcvc yyy (Equation 2-37b) ,,,, 111iiNNN ii ixDiffixDiffpimp iiihY TT mmcDc x xxx (Equation 2-38a) ,,,, 111iiNNN ii iyDiffiyDiffpimp iiihY TT mmcDc yyyy (Equation 2-38b) In Equations 2-37a and 2-37b, the produc t of the species mass fractions and specific heats summed over every species equals the specific heat of the flow. This is multiplied by the mass flux, which equals the flow density times the proper velocity component. The diffusion mass flux inside the sum of the last two expressions is approximated using the mixture-averaged di ffusion equation, Equation 2-11. After replacing the four terms with four equations above, Equations 236a and 2-36b are added together and rearranged before bein g placed into the energy equation. ,, 11 , 11 _iNN ixiiyipp ii NN iy ix ii iimp ii SpeciesContinuityTT mhmhucvc xyxy m m YY TT hDc x yxxyy (Equation 2-39) The two dimensional gradient of the speci es mass flux is replaced with the species chemical production via the species continu ity equation, Equation 2-6. The energy

PAGE 30

21 equation is reduced to the Equation 2-40. Fu rther simplification is performed with an order magnitude analysis in the next section. 22 1 122 _iN ppiii i N ii imp iVV uvTT kk xyxxyy TT ucvchMW xy YY TT Dcwork xxyy (Equation 2-40) Order of Magnitude Analysis An order of magnitude comparison between terms in a given equation determines which terms must be reserved and which te rms can be neglected. Governing equations that are modified for this specific model are simplified further by eliminating the insignificant terms. It is necessary to nondi mensionalize the equation prior to comparing terms. Variables are nondimensionalized with the uniform properties of the flow entering the reactor. Most of the prope rties are chosen such that the resulting magnitudes are on the order of one. The dimensionless variab les and their magnitude are shown below in Figure 2-1. *1 *1 *? u u U v v U *1 *1 *1 T T T p p U *1 *1 x x L y y H H LH L **1 1 1iim im p p pk k k D D D c c c Figure 2-1. Dimensionless variables. All but two of the dimensionless parameters have a magnitude on the order of one. The unknown magnitude of the vert ical velocity is found with the continuity equation. The characteristic distance in the axial direction, L, is much greater than the characteristic

PAGE 31

22 distance in the vertical direction, H. A dimensionless parameter with a very small magnitude, denoted by ( ), is produced when the characte ristic height is divided by the characteristic length. Continuity Equation Equation 2-2 is the two-dimensional contin uity equation that is reduced based on the steady flow assumption. Flow propert ies are replaced with their appropriate dimensionless variables. After some algebrai c rearranging, the mass equation is rewritten in its dimensionless form. 0 uv xy (Equation 2-2) **** 0 **uv L xHy (Equation 2-41) All of the known dimensionless variables ha ve an order magnitude of one. It has already been noted that the characteristic leng th is much larger than the characteristic height. This produces a relatively small value that divides the vertical mass flux term. In order to balance the mass equation, the dime nsionless y-velocity must have the same order magnitude as the division of the height by the length. 1 1* 0 11 v (Equation 2-42) v (Equation 2-43) While the continuity equation remains unc hanged, the comparison of terms reveals that the vertical veloci ty of the flow is small compared to axial velocity. This is a common result in boundary layer flow analysis. Growth of the boundary layer is dictated by the viscosity, or momentum transfer, and does not affect the entire flow until farther

PAGE 32

23 downstream. The flow does not consist entirely of a boundary layer flow. However, a vertical velocity does not exist at the entran ce of the reactor, on the surface, or at the centerline of the pipe or channel. The vert ical velocity remains much smaller than the axial velocity through out the reacto r because of these boundary conditions. Species Continuity Equations The species continuity equation is reduced based on the assumptions of a steady, two-dimensional flow, with or dinary diffusion being the only mode of diffusion. This equation is shown below. iiii imimiiYYYY uvDDMW xyxxyy (Equation 2-12) The dimensionless form of the species mass equation is obtained by replacing the flow variables with their proper dimensionles s counterpart. The species mass fraction is exempt from this part of the process because it is already a dimensionless quantity that varies between zero and one. The unknown magn itude of the mass fraction does not pose a problem since it is found in every term on the left hand side of the equation. As a result it affects the magnitude of each term equally. After some algebraic manipulation, the left hand side is rewritten in its dimensionless form as Equation 2-44. The right side of the species equation, the species chemical production, is not co mpared to the rest of the equation. Neglecting this term would resu lt in the modeling of a non-reacting flow. Therefore, the convection and diffusion terms are the only terms considered. 2 ** 2**** ** ** ****ii ii imimYY L uv xHy YY DL DD ULxxHyy (Equation 2-44)

PAGE 33

24 Note that the mass-averaged diffusion co efficients are nondimensionalized by an arbitrary value. This value is chosen such that the quantity of the dimensionless property is roughly one. This ensures that the dimens ionless diffusion coefficients have an order magnitude of one, but the size of the valu e relative to the product of the incoming velocity and characteristic length is unknow n. The species diffusion terms cannot be compared to the species bulk flow terms as a result. However, the comparison between the diffusion terms inside the brackets is still possible. 211 11 D UL (Equation 2-45) Both of the bulk flow terms have a magnit ude on the order of one. The first term inside the brackets, corresponding to diffusion in the axial dir ection, also has a magnitude of one. The second term corresponds to the di ffusion in the vertical direction and has a magnitude much greater than one. The or der magnitude comparison of the species continuity equation shows that the x -component of the species diffusion is much smaller than the vertical diffusion and can be neglec ted. Information about the size of the characteristic diffusion coefficient relative to the product of the char acteristic velocity and length is needed to determine the parame ter multiplying the vertical diffusion inside the brackets. The parameter must be very small, on the order of ( )2, in order for the vertical diffusion to be of the same magnitude as the two bulk flow terms. This means that the species mass transfer from the bulk flow is much greater than the species mass transfer due to diffusion. Though this is most likely the case for the flow being modeled, further restricting the flow to this assumpti on does not simplify the equation. The species

PAGE 34

25 continuity equation for the flow through the reactor is now reduced down to Equation 246 after dropping the axial diffusion term. iii imiiYYY uvDMW xyyy (Equation 2-46) Axial Momentum Equation Analysis of the axial momentum equation is performed in the same manner as the other equations. Flow properties are nondimensi onalized by their charact eristic variables. The comparison begins with the mome ntum equation govern ing a steady, twodimensional flow of a Newtonian fluid in the x -direction. Characteristic variables are rearranged, and the dimensionless form of the x -momentum equation is shown as Equation 2-47. The viscous term inside the brackets is compared separately from the momentum flux and pressure gr adient terms due to the lengt h of the expression. The comparison of the dimensionless momentum fl ux and pressure gradient terms is now possible. Excluding the vertical velocity, all of the dimensionless variables have a magnitude on the order of one. The divisi on of the characteri stic length by the characteristic height produces a relatively large value. This value is multiplied by the dimensionless vertical velocity, which is a small quantity. The overall effect produces a momentum flux and pressure gr adient terms that all have the same order magnitude of one. As a result, none of these terms is less important than the other and none of the three can be ignored. 2 2 3 uup uv xyx uuvuv xxxxyyyx (Equation 2-20a)

PAGE 35

26 2*** ****_ ***xuLupL uvterm xHyxU (Equation 2-47) 2111_xL term U (Equation 2-48) Viscous terms inside the brackets are tran sformed into dimensionless variables and compared to each other. Equation 2-49 repres ents the dimensionless form of the viscous term. The characteristic properties are reorganized a nd the viscous term is now multiplied by the inverse of the Reynolds number. The Reynolds number is a common dimensionless parameter used to compare inerti al forces to viscous forces. The Reynolds number in Equation 2-50 is based on the le ngth of the reactor and therefore is a comparison of these two forces in the axial direction. 2 2_ 2 2 3xL term U Luuvuv Uxxxxyyyx (Equation 2-49) 2 2*2** 2** ***3** ** *** uuLv xxxxHy UL LuLv yHyHx (Equation 2-50) Magnitudes of each term that comprise th e viscous momentum transfer expression can now be compared to one another. Every expression inside the br ackets is of the order of one, except for a single term. This term is underlined twice in Equation 2-51 and has a magnitude much greater than one. The result is a significant reduction of the viscous term. With the exception of the highlighted term, every ex pression is neglected and the complex viscous expression is simplified to just one term. Information about the

PAGE 36

27 magnitude of the Reynolds number is needed to compare the viscous term with the rest of the momentum equation. The inverse of the Reynolds number must have a magnitude of ( )2 for the remaining viscous term to be of a similar size as the momentum flux and pressure gradient. A large Reynolds number assumption forces the viscous term to balance with the other terms in the equation. It also forces the inertial forces of the flow to be more significant than the viscous forces. This is a reasonable assumption because the Reynolds number is based on the axial directi on, where inertial forc es are expected to be greater than the viscous forces [16]. 211 11 Re (Equation 2-51) The order of magnitude comparison of th e Navier-Stokes equation in the axial direction produces a couple of useful results. The complex momentum transfer due to viscosity is simplified to a single term. This reduces the x -momentum equation to its final form used in the comput er code. In addition, the Re ynolds number must be large for the viscous momentum transfer to be of a si ze comparable to the rest of the momentum equation. The large Reynolds number result is used later in the magnitude comparison of the y -momentum equation. uupu uv x yxyy (Equation 2-52) 21 Re (Equation 2-53) Vertical Momentum Equation The momentum equation in the vertical di rection is not used directly in the computer program. It is used to gain some insight into the behavior of the pressure

PAGE 37

28 through out the flow. The proce ss of comparing terms in the y -momentum equation is the same as the process for the x -momentum equation. The Navier-Stokes equation for a steady, two-dimensional flow of a Newt onian fluid is nondimensionalized by the characteristic scales. The resulting dimens ionless equation takes the form of Equation 254 after some algebraic manipulation. Again, the viscous term is broken down separately because of the length of the expression. 2 2 3 vvp uv xyy uvvuv xyxyyyxy (Equation 2-20b) 2*** ****_ ***yvLvLpL uvterm xHyHyU (Equation 2-54) Comparison of the momentum flux and pr essure terms is postponed until the magnitude of the viscous momentum transfer is known. After the flow properties are converted to their dimensionl ess form, the viscous term b ecomes Equation 2-55. Similar to the viscous term in the axial Navier-St okes equation, the parameter multiplying the viscous term is the Reynolds number. Analysis of the axial momentum equation determined that the Reynolds number is on the order of 1/ ( )2. Knowing the magnitude of the Reynolds number allows the viscous term to be compared to the momentum flux and pressure gradient terms. But first an order magnitude comparison must be performed on all the terms inside the brackets. 2 2*** *2* ***** 2** *3**LuvLv xHyxHyy UL LuLv HyxHy (Equation 2-55)

PAGE 38

29 1111 1 Re (Equation 2-56) As a reminder, all of the dimensionless pr operties, excluding the vertical velocity, have a magnitude of one. The dimensionless vertical velocity and the division of the characteristic height by the le ngth have a magnitude much less than one. The result is the magnitude of all but a single term inside th e brackets of Equati on 2-55 reducing to an order of 1/ ( ). The single remaining term, under lined twice in Equation 2-56, has a magnitude of ( ). This is much smaller than the ot her terms and can be neglected. As a result, the dimensionless viscous term in th e vertical momentum equation has an order magnitude of 1/ ( ). The process of nondimensionalizi ng the equation is complete. An analysis of the vertical mo mentum equation is now possibl e with the knowledge of the viscous terms magnitude. ***1 ****_ ***ReyvLvLp uvterm xHyHy (Equation 2-57) 211 (Equation 2-58) The magnitude of every term, with the exception of one, is a very small quantity, ( ). The exception is underscored in Equation 2-58 and is relatively large compared to the rest of the equation. The highlighted valu e corresponds to the pressure gradient in the y-direction. The pressure gradient is cons iderably larger than the other terms and dominates this equation. Neglecting all of the irrelevant terms, the dimensionless pressure gradient is the only term remaining. This results in the pressure gradient in the y-direction being essentially zero.

PAGE 39

30 0 p y (Equation 2-59) The order of magnitude comparison of the vertical momentum equation reveals that pressure through out the flow is a weak function of the vertical positi on. Pressure can be treated as strictly a functi on of the axial direction, x. Energy Equation The magnitude comparison of the energy equation begins with Equation 2-40. The equation is simplified based on the assumptions of a steady, two-dimensional flow with negligible work done on the fluid by the body forces. Magnitudes of each similar expression are compared to one another se parately. First, the two kinetic energy convection terms are compared to each othe r. Then the two heat conduction terms are compared and so on. The work done by the shear stress is evaluated last. 22 1 122 _iN ppiii i N ii imp iVV uvTT kk xyxxyy TT ucvchMW xy YY TT Dcwork xxyy (Equation 2-40) Analysis of the energy convection on the left hand side of Equation 2-40 is a qualitative process. The couple of conv ection terms are not nondimensionalized, but a simplification is possible with the understandi ng of the behavior of the velocity field. The energy being transferred consists of ki netic energy, which is proportional to the magnitude of the flow velocity squared. Balancing the continuity equation proved that the vertical component of th e velocity is much smaller than the axial component. Squaring both components only make this difference more pronounce.

PAGE 40

31 22 222222222**1 VuvUuvUVu (Equation 2-60) The contribution of the verti cal component to the magnitude of the flow velocity is neglected. The kinetic energy is calculated using only the axial component of the flow velocity and the kinetic energy convection is rewritten as Equation 2-61. The two partial derivatives of the x-velocity squared are performed to obtain the final form of the kinetic energy transfer. 22 222uu uvuu uuv x yxy (Equation 2-61) Moving over to the right hand side of E quation 2-40, the two heat conduction terms are now compared in a much more quantitativ e procedure. Variables are replaced with their dimensionless counterparts in Equati on 2-62 so a dominant term can be found. These terms cannot be compared to the rest of the energy equation because the magnitude of the parameter outs ide of the brackets of Equation 2-63 is unknown. Comparing the magnitudes of the two expressions reveals the dominant term. It is underlined twice and corresponds to the heat conducti on in the vertical direction. Being much smaller than vertical conduction, the heat conduction in the axial directi on is neglected in the final form of the energy equation. 2 22** ** ****TT kk xxyy kTTLT kk LxxHyy (Equation 2-62) 2 21 1kT L (Equation 2-63)

PAGE 41

32 The energy transfer due to the diffusion of en thalpy is the next term to be reduced. Every variable and property is replaced with its dimensionless representation in Equation 2-64. The order magnitude analysis reve als that the second term, underscored in Equation 2-65, is much larger than the other. The lesser of the two terms is the enthalpy diffusion in the axial direction and is neglected in the energy equation. 1 2 ** 22 1** ****i iN ii imp i N p ii imp iYY TT Dc xxyy YY DcTTLT Dc LxxHyy (Equation 2-64) 2 2 11 11N p iDcT L (Equation 2-65) The last expression to be analyzed in th e energy equation is th e energy transfer due to work done by the shear stress. It is gr eatly simplified by comparing the magnitude of each term that comprises the work. Shear st ress is defined as Equations 2-19a-c for a Newtonian fluid. The energy transfer expr ession becomes Equation 2-67 after the shear stress definitions are substituted. The flow properties are replaced with their proper dimensionless variables a nd characteristic scales. _xxxyxyyyworkuvuv xy (Equation 2-66) 2 2 3 2 2 3 uuvuv uuv xxxyyx vuvuv vvu yyxyyx (Equation 2-67)

PAGE 42

33 2 2 22 22*2** 2**** *3** ** ** ** *2** 2**** *3** ** ** ** uuLv uu xxHy U Lx Luv v Hyx vHuv vv yLxy UL LHy uHv u yLx (Equation 2-68) The dimensionless form of the viscous work can now be used to determine which terms dominate the energy transfer expre ssion. Most of the terms have an order magnitude of one. There are two terms that do not have this magnitude. One is very small with a magnit ude on the order of ( )2, while the other has a large magnitude and is underlined twice. The highlighted term is substantially greater than the other terms inside the brackets and can be considered the only dominant term. Neglecting all the other weak terms, the energy tr ansfer due to the shear stress work is reduced to Equation 2-70. 2 2 2 2222 2222211 1 U L (Equation 2-69) u u yy (Equation 2-70) The order magnitude analysis of the en ergy equation has greatly simplified the governing equation. The kinetic energy convect ion, heat conduction, enthalpy diffusion, and shear stress work are analyzed individually These four modes of energy transfer are not compared to one another because no a ssumptions are made about which mode is

PAGE 43

34 more important. Analysis of the velocity fi eld has revealed that the axial component can be used to determine the magnitude of the ve locity at any point of the flow. The heat conduction in the x-direction and many of the terms that comprise the shear stress work are neglected due to the analysis. As a re sult of all this simplification, the energy equation becomes Equation 2-71. 2 11ipp NN i iiiimp iiuuTTT uuvkucvc xyyyxy Y Tu hMWDcu yyyy (Equation 2-71) Unit Analysis Units are substituted into the governing equa tions to ensure each expression in an equation balances with the other expression s. Replacing variables and properties with their units reveals several important aspects of these quantities. Th e inspection validates that the equations were reduced without mispl acing any variables or properties. It also locates properties that require a unit conversion and determin es units of the calculated solutions. Note that Cantera calculates the properties with the Inte rnational System (SI) of measurement [17]. In order to keep unit conversions to a minimum, the variables also use this system of measurement. Continuity Equation The main program calculates the verti cal velocity component with the mass equation. Solving the mass equation for the y-velocity produces Equation 2-3. Each variable and property is repla ced with its units. The density is given in kilograms per meter cubed by Cantera. Therefore, the velo city and differential distances are measured in meters per second and meters, respectfully. From Equation 2-72, it is clear that the mass equation balances with equivalent units on both side of the equation.

PAGE 44

35 u v v yxy (Equation 2-3) 3kgm m 1 s m 3kgm m 1 s m m 31 kg sm m 33kgkg msms (Equation 2-72) Unit analysis of the mass equation reveals that no unit conversion of the density is necessary and the two differential step sizes shou ld be given in similar units. Units of the calculated vertical velocity depend on the units of the axial velocity, which is defined by the initial condition. Although units cancel each other out in the mass equation, the other governing equations prove that SI units should be used for the variables. Species Continuity Equations The reduced species continuity equation is algebraically reorganized in a form the program can solve. This form is discussed more in section Solving Governing Equations. Equation 2-46 is transformed into Equation 273. International Sy stem of measurement is used for the velocity and differential dist ances, and their units are meters per second and meters, respectfully. The density is still kilograms per me ter cubed and the mass fraction is dimensionless. Cantera gives a mixture-averaged diffusion coefficient in meters squared per second. The unit of the net production rate is kilomoles per second meter cubed, and the molecular weight is give n in kilograms per kilo mole. All of these units are placed into Equation 2-73 to complete the analysis. iii imiiYYY uDvMW xyyy (Equation 2-73) 3kgm m 1 s m 1 m 2 3kgm m 1 s m 3kgm m 1 s m kmol 3kg ms kmol 33kgkg msms (Equation 2-74)

PAGE 45

36 The unit check of the species continui ty equation shows that converting the molecular production rate into a mass production rate is indeed necessary. No other unit conversion is required if SI units are used for the velocity com ponents and differential step sizes. The process proves the equation is reduced correctly from a unit analysis point of view, and its solution is dimensionless. Momentum Equation The reduced momentum equation is reorgani zed into a form similar to Equation 273. Equation 2-75 is the form of the mome ntum equation solved by the program. Units of the velocity components, differential di stances, and density remain unchanged. The pressure and dynamic viscosity is also given in SI units. The SI unit of measurement for pressure is the Pascal, which equals a kilo gram per meter per second squared. Cantera reports the dynamic viscosity in units of Pas cal-second. A Pascal-second is equivalent to a kilogram per meter-second. Variables and pr operties are replaced with these units and result in balanced Equation 2-76. uupu uv x yyxy (Equation 2-75) 3kg m 2m m s 1 s m 1 kgm mms 1 s m 2 31 kgkg msm m 2m m s 1 s m 2222kgkg msms (Equation 2-76) The analysis finds that none of the prope rties determined by Ca ntera necessitate a unit conversion. Units of the simplified mo mentum equation balance accurately and its solution, the axial velocity, is ca lculated in meters per second.

PAGE 46

37 Energy Equation Like the last two governing equations, th e reduced energy equation is organized into the form of Equation 2-77. 2 11ipp NN i impiii iiTTuT uckuvc xyyyy Y Tuu DchMWuuv yyxy (Equation 2-77) Units of the flow properties already disc ussed in the previous governing equations remain the same. Several new properties are encountered in the energy equation. The temperature, thermal conductivit y, species enthalpy, and specif ic heat of the fluid and species i are used exclusively by this equation. The temperature is measured in degrees Kelvin, and the thermal conductivity is given in units of watts per meter-Kelvin. A watt per meter-Kelvin is equivalent to a Joule pe r meter-Kelvin-second. Specific heat of the fluid can be determined on a mass basis in Cant era. The unit of the fluids specific heat is Joules per kilogram-Kelvin. Each expres sion is analyzed indivi dually with the units established. Moving left to right in E quation 2-77, the axial energy convection is analyzed first. Substituting units into the axial energy convection s hows that the term has units of Joules per cubic meter second. Units of the other expression s must reduce to this unit to balance the energy equation. pkg T uc x 3m m J s kg K K m 3J ms (Equation 2-78) The next expression evaluated is the en ergy conduction along with the viscous term. Units of the viscous term are equiva lent to the energy conduction at a Joule per cubic meter second and Equation 279 balances with Equation 2-78.

PAGE 47

38 1 TuJ ku yyym mK K s m m kg s m s m s 1 m 3J ms (Equation 2-79) Units of the second expression reduce to the same units of the axial energy convection. The vertical energy convection al so reduces to these units and is shown below in Equation 2-80. pkg T vc y 3m m J s kg K K m 3J ms (Equation 2-80) Cantera reports the species specific heat in a column vector that has been nondimensionalized by the universal gas cons tant. Multiplying the vector by the universal gas constant produces specific heat s with the units of Joules per kilomoleKelvin. The sum of the energy diffusion expr ession includes the species specific heat. The term highlighted is added to convert the sp ecies specific heat fr om a molar basis to a mass basis. It is the invers e of the species molecular weight and must be added to the enthalpy diffusion expression for the units to co nform to the rest of the energy equation. Equation 2-82 illustrates the modification. 2 1iN i imp ikgm Y T Dc yy 3JK ms kmol 2Km kmol kg 3J ms (Equation 2-81) 11i iNN p ii impim ii ic YY TT DcD yyMWyy (Equation 2-82) Much like the species specific heat, Canter a reports the enthalpy of each species in a dimensionless column vector. The vector is nondimensionalized by the universal gas constant and the temperature of the fluid. After multiplying the vector by these two properties, the resulting enthalpy has the units of Joules per kilomole. Unit analysis of

PAGE 48

39 the enthalpy production expression is done in Equation 2-83. Because the species enthalpy is reported on a molar basis, the chem ical production rate does not need to be converted to a mass basis. The molecular weight term is underscored and is dropped such that the unit of this expression is c onsistent with the other terms of the energy equation. 1 N iii iJ hMW kmol kmol 33kgJkg mskmolmskmol (Equation 2-83) 11 NN iiiii iihMWh (Equation 2-84) The last expression in the unit check is the kinetic en ergy convection. Equation 285 shows that the expression needs no modification. 2kg uu uuv xy 2 3m m 2s m 1 s m 2Js kg 2m 3J ms (Equation 2-85) Analysis of the energy equation reveal s that energy diffusion and enthalpy production expressions required modification. Molecular weights are added to the energy diffusion term and removed from the enthalpy production term. After these modifications, the energy equation become s Equation 2-86 where the units of each expression are equal and the equation bala nces. Temperature being the dependent variable of the energy equation is calculated in degrees Kelvin. 2 11ipp NN p i imii ii iTTuT uckuvc xyyyy c Y Tuu Dhuuv M Wyyxy (Equation 2-86)

PAGE 49

40 Summary of Governing Equations In conclusion the governing equations are applied to the two-dimensional modeling of the catalytic reactor. Each equation is reduced based on assumptions describing the fluid and its motion. Through an order magnit ude analysis these equations are simplified further. The units of each term are verified and the equations are balanced. The resulting equations solved by the program are summarized in Table 2-1. Table 2-1. Equations modeling the flow. Principle Equation Equation number Mass Conservation u v v yxy 2-3 Species Mass Conservation iii imiiYYY uDvMW xyyy 1,2,3,... iN 2-73 Momentum Conservation uupu uv x yyxy 2-75 Energy Conservation 2 11ipp NN p i imii ii iTTuT uckuvc xyyyy c Y Tuu Dhuuv M Wyyxy 2-86 The unit analysis also determines the units of properties and variables found in the governing equations. Units of each flui d property are compiled in Table 2-2. Table 2-2. Units of the governing equations. Property Variable Units Property Variable Units Differential step sizes dx, dy m Production rates i kmol/m3s Diffusion coefficients Dim m2/s Thermal conductivity k W/m2K Density kg/m3Specific heats cp J/kgK Enthalpies hi J/kmolTemperature T K Mass fractions Yi dimensionless Velocity components U, v m/s Viscosity Pas Molecular weights MWi kg/kmol Pressure p Pa

PAGE 50

41 CHAPTER 3 PROGRAM METHODOLOGY Flow variables are calculate d throughout the catalytic r eactor via a step-by-step process of solving the governing equation. The process begins by creating a discrete mesh of points to numerically solve the equa tions. Solutions to the flow variables are recorded at these points. Th e next step involves establis hing parameters and conditions of the reactor and fluid. These values charac terize the reactor and in itial conditions of the flow. Folders to import and export data must also be defined. Once the first three steps are completed, the code can begin to find th e solutions. The program sets the initial conditions and solves the simplified governing equations in Table 2-1. The code is written in MATLAB and cons ists of a main program with three subprograms. One of the subprograms finds the initial velocity components. The main program creates the mesh, finds fluid propertie s, and sets conditions needed to solve the equations. Information from the main program is sent to the other two subprograms. Solutions are found by the subprograms and sent back to the main program where it is saved in the solution variables. MATLAB is chosen above other programming languages because of its built-in ability to handle vectors, vector operations, and partial differential equations. MATLAB incorporates several computational tools capable of solving partial differential equations. A function called pdep e is used to solve the momentum equation as a single equation. It is also used to solve the energy equation and species continuity equations as a set of coupled partial diffe rential equations. This makes MATLAB well suited for modeling chemically reacting flows. Another useful property of MATLAB is

PAGE 51

42 its compatibility with Cantera. Cantera is a free software package developed by Professor David Goodwin at the California Institute of Technology to solve problems concerning chemical reactions. The main MA TLAB program calls upon this software to determine the thermodynamic, transport and ch emical kinetic properties of the flow and catalytic surface. Cantera is able to cons truct objects of differe nt phases and tie the phases together through an interface. This al lows for the chemical interaction between the gas and surface [17]. Several studies attempt to model catalytic combustion similar to this model. A study by the National Institute for Advanced Transportation Technology at the University of Idaho modifies an existing code. Lawr ence Livermore National Laboratory provides the existing Hydrodynamics, Combustion, a nd Transport (HCT) code. The finitedifference code, HCT, utilizes the same prin ciples of conservation for its calculations. Dissimilarity occurs in the application of the governing equations to the one-dimensional time-dependent catalytic combustion, opposed to the two-dimens ional steady-state catalytic reactor modeled by this program. Still, the study offers some insight into the chemistry and equations involved with modeli ng a catalytic combustor [5]. In a second study, Chou et al. [4] uses CURRENT with CHEMKIN and SURFACE CHEMKIN software to model a two-dimensional mono lith catalytic combustor. CURRENT is a code developed by Winters et al. [18] for lo w Mach number chemically reacting flows. The study discusses the chemistry and boundary conditions of the model and compares the calculations to experimental data. Th is program uses a similar symmetric boundary condition at the centerline.

PAGE 52

43 Discretization The two-dimensional computational sp ace of the reactor is broken down and discretized before the equations can be numerica lly solved. The mesh is that of a planar geometry with the height determined by th e reactors radius. The upper boundary is moved to the centerline and the lower boundary is still the ca talytic surface. This reduces the height of the computational space and in turn reduces comput ation time and memory used by the computer. Now is a good time to me ntion that the centerlin e is assumed to be a streamline and symmetric conditions are as sumed to exist at this boundary due to the two identical plates modeling th e surface of the pipe or channel. This assumption affects the boundary conditions discussed in the se ction Solving Governing Equations. The length of the reactor is broken down into stages, the first stage being the entrance. This is also intended to reduce the time needed to calculate a solution. It is expected that the flow changes relatively fast in the beginning of the reactor when the catalyst is first encountered. This corresponds to the first few stages of the computational space. To help resolve the solution in these stages a smaller differential step size in the x -direction is chosen. Once the properties reach a quasi-ste ady state, the step size can be increased to help lower the computation time. An orthogonal mesh is created for every stag e. Each stage has its axial direction discretized in a linear manner, where every poin t is an equal distance apart. The distance is set for a given stage but can change from stag e to stage. This allows the user to adjust the axial step size of a stage if the progr am cannot converge on a solution. A possible source of this problem is a significant change of flow properties in the x -direction. Recall that the governing equations are simplified ba sed on the assumption that the characteristic length is much larger than the characteristic he ight. In other words, the vertical gradients

PAGE 53

44 are much larger than the axial gradients. Wh ile this assumption is still applicable, there may be areas where a change in step size is needed, such as the reactors entrance. The point separation in the y -direction, in contrast to the axial point placement, is the same throughout the reactor. Although the vertical point placement must remain the same for every stage, it is not restricted to only a linear displacement. The point displacement is set as a power of the point location. For example, setting the power to one would position the points linearly. Setting the power to two creates a quadratic point displacement, leading to more points near th e surface. A larger power places more points near the surface. Varying the power allows th e user to control the location of the points in the vertical direct ion. This aids the program in resolving the varying chemical composition near the surface. The catalytic surface serves as the main source of the chemical reaction in the flow. Therefore, it is expected that most of the chemical change will occur near the surface. More points are needed near the surface to determine the change in the chemical composition in the vicin ity of the catalyst. A tight mesh near the surface also helps resolve the fl uid velocity boundary layer. Velocity, temperature, and composition vari ables are not found for the entire stage at once. Instead the stages mesh is broke n up further into mini-meshes. A mini-mesh contains all the vertical points for a group of three axial locations. Governing equations are solved one mini-mesh at a time due to the coupling of the equations. The pressure, temperature and mass fraction of a mini-mesh mu st be approximated prior to solving the equations. Jumping ahead might seem prem ature because the governing equations meant to calculate the variables have not been solv ed yet. However, fluid properties dependent on the solution are imbedded inside the equations These properties must be established

PAGE 54

45 in order to solve the equations. One can now being to appreciate the complex coupling of the governing equations. On ce the equations are solved, the program updates the variables and moves downstream to the next mini-mesh. Parameters and Conditions Parameters and conditions of the catalytic reactor and incoming flow are set inside the code of the main program. All of the values, composition being the only exception, must equal a real scalar. The computer code begins by setting parameters of the catalytic reactor, such as the radius, stage length, stage number, surface temperature, and the distance of the non-reactive surface. Dime nsions of the computational space are constructed with the height a nd length of the stage. The stage number is simply the sequential numbering of each stage for which a solution is calculated. The value of this number determines the data used to set the incoming conditions and the output folder in which the export files are stored. This is discussed further in the sections Initial Conditions of a Stage and Input and Output Files. In the energy equation, the temperature at the wall or surface boundary is held constant at the value entered. The distance of the non-reactive surf ace refers to the entrance of the reactor where there is no catalyst on the surface. This is only important for the first stage and can be ignored for any other stage. The differentia l step sizes in the vertical a nd axial direction are also set at this point, along with the power used to discretize the verti cal direction. These values are used to construct the twodimensional mesh described in the section Discretization. After the parameters of the reactor are en tered, the conditions of the incoming flow are defined. The speed, temperature, pressu re, and composition of the flow entering the reactor are established. The incoming flow is assumed to be a uniform flow where the velocity is purely in the axial direction. Initially there is no vert ical component to the

PAGE 55

46 fluids velocity and the speed of the x -velocity is the same at every point. Therefore, only a single quantity is needed to define the velo city vector entering th e reactor. The flows chemical composition is initially modeled as a well-mixed fluid. This simply means that the species mass fractions are also the same at every point entering the reactor. The composition is the only value entered as a stri ng variable. This stri ng contains the name and mass faction of the species present in the incoming flow. Cantera reads the string to set the composition of the gas. The program us es these values to se t the initia l conditions of the variables. The last parameter to set is the PC variable, also a scalar. This variable controls whether the program iterates on a solution and if so, how many times the iteration takes place. An inherent delay in the solution process exists because the governing equations are decoupled. The delay is exaggerated by properties that are de pendent on the solution inside the equation. Some of these propert ies include density, vi scosity, and diffusion coefficients. With no iteration (PC equal to one), the program numerically solves the governing equations for one mini-mesh. Th en the program updates the variables and properties and moves one differe ntial step downstream to solv e the equations at the next three axial locations. The program is con tinually updating the prope rties prior to moving downstream; therefore the delay is expected to be small. To improve the calculation one may choose to iterate on a solution using a pred ictor/corrector type method. To iterate on a solution the PC variable is set to quantity greater than one. Fo r example, the program iterates once on a calculation if the variable is equal to two. Iteration occurs by solving the equations and updating the properties with th e known solution. This could be seen as a predictor step, now to correct the calculation. Instead of moving one differential step

PAGE 56

47 downstream the program recalcul ates the solution for the same three axial locations based on the updated properties. If the PC variable is three, the iteration occurs twice, and so on. Input and Output Files Input and Output file names are given pr ior to operating the program to direct import and export data. The input text file co ntains the chemical data Cantera require to model the gas and solid of th e catalytic reactor. Naming the input file informs the program where the chemical data are located to import. Data determine properties found within the governing equations. Only the filename of the i nput file is needed if it is located in Canteras current working directory. This directory is initially set as the data folder inside Canteras main folder, which is installed with th e free software. The pathname of the output folder provides the program the location of the export folders. Export folders must be created inside the output folder and given the name Stage1, Stage2, Stage3 etc. The solutions of a st age are recorded in the folder with the corresponding number. Therefore, the export fo lder of a stage must exist before seeking the solution of that stage. The entire pathname is stored in the string variable saveFile. Not only is this string variable used to export solutions of a st age; it is also used to import initial conditions for most of the stages. This is discussed further in the section Initial Conditions of a Stage. Considerable amounts of da ta are required to model the gas and solid of the catalytic reactor. Cantera accesses this data vi a the input text file specified. These files contain information on the chemical kineti cs, thermodynamics, and transport properties of many different species. Data consiste nt with the modified Arrhenius function determines the chemical kinetic properties of the gas phase reactions. This data include

PAGE 57

48 activation energy, pre-exponential coefficients and temperature exponent. In addition to this, surface reactions apply reactive sticki ng probability. The thermodynamic properties are determined using a NASA polynomial para meterization or Shomate parameterization. Coefficients of either parame terization are incorpor ated in the data of the input file. Information needed to calculate transport pr operties based on either a multi-component or mixture-averaged transport model is also included. The multi-component transport model provides a more accurate solution than th e mixture-averaged model. However, the multi-component model requires more data a nd computation time than its counterpart [17]. The program saves several variables to the ou tput folder for every stage. The value of each variable is saved as a double precision sc alar, vector, or matrix in an ASCII file. The axial location, axial velo city, pressure, temperature, mass fraction of each species, pressure gradient, and vertical velocity are all stored in the export folder. The axial location is saved in order to keep track of which discretized points in the mesh the various solutions correspond. The x -location is saved as a vect or that begins at zero and ends at the length of the stage. The axial velo city is recorded at every point in the stage and the variable is saved as a matrix. It is possible to record the vertical velocity as a matrix in a similar manner with little addition to computation time. This is due to the fact that the variable is already determined to solve the governing equations. However, the y component of the velocity is so small when compared to the x -component that it is not recorded as part of the solution. This will help retain memory space for the other properties. Two independent therm odynamic properties are recorded to thermodynamically define the fluid. One of these properties is the pressure, which does

PAGE 58

49 not vary in the vertical direction. Being only one-dimensional, the pressure at each axial location is recorded and the variable is saved as a vector. The other thermodynamic property is temperature and it rema ins a function of both dimensions, x and y Temperature at every discretized point is calculated and saved as a matrix in the stages export folder. The composition of the fluid is recorded as mass fractions of each species. Like the temperature, the mass fractions ar e a function of both dimensions. The mass fraction of each species is saved as a matrix into its own file. As a result, the number of mass fraction files saved in the export folder is equal to the number of species, N All the properties needed to kinetically and therm odynamically define the flow are recorded. The only other variables saved are the pressu re gradient and vertical velocity. The pressure gradient is recorded as a scalar and the vertical velocity is saved as a vector. Both variables correspond to the last axial po sition of the stage and are used as initial conditions of the next stage. Initial Conditions of a Stage The program can operate once all of the pa rameters, conditions, and file names are designated. Initial boundary conditions of the stages firs t mini-mesh are established before solving the governing equations. Veloci ty components at all vertical points in the first axial location ar e required to define the momentum equation and its initial boundary condition. The pressure, temperature, and co mposition in the first mini-mesh must also be defined to estimate the properties inside the governing equations. The model requires only one pressure value per x-location, because the pressure is independent of the vertical direction. The result is only three scalars being required to define the pressure in the mesh. Temperature and species mass fractions are two-dimensional and must be set for every point in the stages first mini-mesh. The process used to define these variables

PAGE 59

50 depends on the stage number. Conditions of the initial stage or stage one are based on the values discussed in the section Parameters and Conditions. Every other stage uses the solution of the previous stage to set these initial boundary conditions. The program can begin to solve the governing equations once the initial cond itions are set. Cantera creates a gas object and surface objec t prior to defining initial conditions. The gas is adjusted to the pressure, temperat ure, and composition ente ring the reactor and the two objects are connected thr ough an interface. The gas ob ject is created at this time because Cantera provides a simple means to set the composition variable of the first stage. Only the compositions string variable is needed to establish the mass fraction of all the species initially present. Cantera can take the composition of the gas object and return the mass fraction of every species. This is much easier than searching for the species not present and setting their mass fractio n to zero. Cantera also ensures that the sum of the mass fractions equals one. Stage One The reactor is characterized by the absence of a catalytic surface at its entrance. The catalyst does not begin until further downstream. This is where stage one begins and the initial boundary conditions of the first mi ni-mesh are determined. Minimal change in the conditions should occur over the non-reac tive surface with the exception of the two velocity components. Therefore, the initia l conditions of the temperature, composition, and pressure remain the flow conditions en tering the reactor. Temperature and mass fractions at the first three axial locations ar e approximated by the values entered as initial conditions. The surface temperature is set to th e value entered as a pa rameter. The initial pressure is equal to the pressure of the in coming flow, and the pressure at the next two differential steps is calculated with the pre ssure gradient. In c ontrast to the other

PAGE 60

51 variables, the surface affects the velocity vector. A boundary layer develops changing the profile of the axial velocity, which produces a vertical velocity. Blasius solution is used to model the boundary layer and de termine the two velocity components. Blasius Solution Axial velocity is quantified by two values at a point in the beginni ng of the reactor. The singularity point is located on the front edge of the reac tor, where the incoming flow first encounters the surface. A finite value is given to the uniform velocity entering the reactor. The velocity at this point must also equal zero due to the boundary conditions of the velocity. To overcome the singularity point Blasius solution is us ed to calculate the two velocity components at the end of th e non-reactive surface, wh ere the first stage begins. H. Blasius is well known for obtaining an exact solution to a laminar boundary layer flow over a flat plate. Blasius is able to find a similarity solution to the continuity and momentum equations through proper sca ling and nondimensionalization of the two equations. In his solution, the dimensionles s stream function repla ces the two velocity components as the dependent variable. The two coordinates, x and y are also combined into one dimensionless independent variab le. Blasius transforms the two partial differential equations into one ordinary differential equation. A power series expansion or numerical methods can then be used to so lve the third order, nonlinear equation. The dimensionless stream function and its deriva tive are used to calculate the axial and vertical velocity [15]. Blasius solution describes a two-dimensiona l, steady, incompressible flow with no pressure gradient. Recall that the assumption of constant density is not applicable to this model due to the chemistry involved. A pre ssure gradient equal to zero is also not

PAGE 61

52 accurate because the flow is assumed to have pressure changes in the axial direction. However, Blasius solution is used as a reason able estimate to the velocity profile over the non-reactive surface. The change in density is caused mostly by the catalytic surface, and the catalyst is not present in the region that Blasius solution is employed. A change in density from species diffusing upstream is possi ble, but the effect should be negligible. The production of a new species with a diffusion velocity great enough to overcome the axial velocity is needed for this to occur. The behavior of the pressure gradient also permits the use of Blasius solution. The pr essure slowly decreases as the flow moves downstream. This produces a decreasing pressure gradient that has an initial value of zero. The change in pressure is small and a ze ro pressure gradient should be a reasonable approximation at the entr ance of the reactor. Fortunately a non-reactive surface is located in the region of the singularity point. Blasius solution can be used to generate the velocity profile at th e end of the non-reactive surface. The main program calls on one of the subprograms, a function called Blasius. The function imports the axial differential step size, stage length, y -coordinate vector, location where the catalytic surface begins, vi scosity, and initial speed of the flow. A shooting method determines the dimensionles s stream function and its derivative. Because the equation developed by Blasius is di mensionless, the calculated values of the stream function are independent of the values imported. The axial velocity vector and vertical velocity vector are determined using the dimensionless variables with the imported variables. The axial velocity is trea ted as the initial condition of stage one. The vertical velocity is used in the momentum equation. Note that the function Blasius is

PAGE 62

53 only used in the first stage, where the non-reactive surface is located. Other stages use the solutions of the previous stage to set the velocity profile. Subsequent Stages If the stage number is greater than one, initial boundary cond itions are taken from the export folder of the previous stage. Th e program locates and lo ads initial quantities with the saveFile variable. The last valu e of the preceding stages pressure vector becomes the initial pressure of the current stage. The vertical velocity vector and pressure gradient (scalar) are loaded from the preceding stages output files. Initial values of the current stages axial velocity are set to equal the x -velocity at the end of the last stage. Temperature and composition variab les are all that remain to import. Recall that the temperature and mass fraction must be set for all three axial locations. The first axial location is equal to the last axial locati on in the previous stages matrix, similar to the axial velocity variable. For the ne xt two differential steps downstream, the x -location vector of the last stage is imported. Th is vector along with the temperature and mass fractions of the previous stage determine the variables gradient at the end of the last stage. A second-order backward-difference formul a is used to estimate this gradient [19]. Values of the next two axial locations in th e variable of the current stage are linearly extrapolated using the gradient. With the in itial conditions of the stage set, the program is prepared to solve the equations co ntrolling the behavior of the flow. Solving Governing Equations Governing equations are solved for three axial locations at a time. Remember that the equations contain properties dependent on the solution. Solving one mini-mesh at a time allows the program to update properties in side the equations pr ior to calculating the solution one step downstream. For the same reason, the pressure, temperature, and

PAGE 63

54 composition at these three places must be appr oximated before solving the equations. For the stages first mini-mesh, the method for these approximations is described in the section Initial Conditions of a Stage. A pproximations of the other mini-meshes are calculated with the solution of the previous mini-mesh. Note that two of the axial locations of the next mini-mesh are repeated locations of the previous mini-mesh because the program moves only one differential step. Two MATLAB functions, or subprograms, are written to solve the momentum equation and the species/energy equations se parately. These two functions are named Momentum and Species_Energy. The process of the solving the governing equations for a stage is illustrated in the fl ow chart of Figure 3-1. Once the initial approximations for a single mini-mesh are established, the program begins with the momentum equation in the x -direction. The main program calls on Ca ntera to find properties embedded in the momentum equation. Inform ation is exported to the subprogram Momentum, which is then used to solve the equati on. The subprogram sends the solution, the ax ial velocity, back to the main program. The mass or continuity equation cal culates the vertical velocity and confirms that the solution of the momentum equation is accurate. The pressure gradient is updated and the momentum equation is solved again if the boundary condition of the vertical velocity at th e centerline is not reacted. If the y -velocity equals zero, the species continuity and energy equations are solved together with the Species_Energy function. Again, the data n eeded to solve the system of equations are provided by Cantera and sent to the s ubprogram. This time the composition and temperature are sent back as the solution. Once the solutions are calculated and any iteration correction is performed, the program up dates the variables, saves the data in the

PAGE 64

55 export folder, and moves one differential step downstream. The temperature and composition of the next axial lo cation are predicted using a linear extrapolation from the previous two x -locations. The program loops back a nd solves the equations for the next mini-mesh. This continues until the end of the stage is reached or the program cannot resolve the solutions. Figure 3-1. Flow chart for single stage modeling. The subprograms Momentum and Species _Energy use a partial differential equation (PDE) solver provided by MATLAB to numerically solve the equations. The solver numerically computes the momentum equa tion as a single equation. It is also used to solve the energy equation and species continuity equations as a set of coupled partial differential equations. The solver, named pdepe, calculates the solution of partial differential equations with the form shown be low in Equation 3-1. The gradient of the dependent variable with respect to x is multiplied by a coupling term, c This term along with the flux term, f and the source term, s are functions of the two independent variables, the dependent variab le and its vertical gradient.

PAGE 65

56 ,,,,,,,,,mmzzzz cxyzxxfxyzsxyz yxyyy (Equation 3-1) The symmetry parameter, m along with the coupling, flux, and source term define the PDE. The mesh spacing of the two inde pendent variables, one initial condition, and two boundary conditions are all that remain to solve the equation. The mesh spacing is simply the mini-mesh discussed in the secti on Discretization. Func tion pdepe selects the x -mesh dynamically to resolve the solution, but only reports the answer at the mesh points specified. Strictly speaking the initia l condition is a boundary condition. It is the value of the dependent variable at the firs t of the three axial locations. The initial boundary condition of the dependent variable needs to be given as a function of y The other two boundaries are found at the catalytic surface and centerline. Both must fit the form shown in Equation 3-2. Boundary conditions are expressed in terms of p q and f The flux term f is already defined in the PDE above, so only p and q are needed to establish the boundary conditions. ,,,,,,0 z pxyzqxyfxyz y (Equation 3-2) Some fluid properties, such as densit y, are converted in to functions of y to conform to Equation 3-1 above. Most of these prope rties are functions of both dimensions. However, properties dependent only on the ve rtical direction should be an acceptable representation for several reasons Fluid properties vary more in the vertical direction than the axial direction a nd are a stronger function of y In addition to this, the functions only need to represent th e properties for the three x -locations of the mini-mesh. Initial boundary conditions also need to be turned into functions of y Built-in MATLAB functions spline and unmkpp gene rate the function representations. Properties are found

PAGE 66

57 at every vertical point in the middle of the three x -locations and saved in a vector. In the case of the initial condition, the vector c ontains the initial values of the dependent variable. The function spline uses the vect or to create twenty separate piecewise polynomials of the form of the cubic sp line. Function unmkpp extracts the four coefficients of the each polynomial and saves it into a four-by-twen ty matrix for each representation. The matrices are export ed to either the Momentum function or Species_Energy subprogram to reconstruct th e piecewise polynomial. The coefficients and a heavyside step function connect the piecewise polynomials inside the subprogram. The result is a smooth function representation of the initial conditions or fluid properties embedded in the governing equation. Momentum Equation Solving the momentum equation begins by guessing the pressure gradient. Pressure at the three axial locations is dete rmined with the guessed pressure gradient. The pressure along with the approximated temperature and composition are used to determine the density and viscosity. Thes e are the properties f ound in the simplified momentum equation. uupu uv x yyxy (Equation 2-75) Properties are determined at every vertical point in the middle of the three x locations and transforme d into functions of y for the coupling, flux, and source terms. Comparing the momentum equation to the form used by the pdepe function, it is evident that the axial velo city replaces the dependent variable, z The symmetry parameter, m is zero. The coupling, flux, and source terms equal Equation 3-3, Equation 3-4, Equation 35, respectfully.

PAGE 67

58 ,,, u cxyuu y (Equation 3-3) ,,, uu fxyu yy (Equation 3-4) ,,, upu sxyuv yxy (Equation 3-5) The coupling term, c contains the density and axial velocity. This term is allowed to be a function of the dependent variable. As a result, only the density must be transformed into a function. The flux term, f equals the viscosity multiplied by the vertical gradient of the axial velocity. To fit Equation 3-1, the gradient will remain but the viscosity is represented by a function of the vertical direction. The last two terms in the momentum equation are combined into th e source term. These two terms consist of the pressure gradient and the product of the density, y -velocity, and y -gradient of the dependent variable. Recall that the pressure is only a function of the axial location. Therefore, the pressure gradient remains constant at a given x -location and does not need to be transformed into a function of y The vertical gradient of the x -velocity is allowed inside the source term. The density and y -velocity product is the only element of the source term transformed into a function. Th e axial velocitys in itial boundary condition is also transformed into a function of y Boundary conditions at the su rface and centerline are all that remain to solve the momentum equation. Appling the no-slip a ssumption, the axial velocity is zero on the catalytic surface. The centerline of the reac tor is assumed to be a streamline with a vertical gradient of the x -velocity equal to zero. Equation 3-6a and 3-6b show these two conditions in a form recognized by the pdepe function.

PAGE 68

59 000 uuf (Equation 3-6a) 0010 uu yy (Equation 3-6b) Boundary conditions of the momentum equa tion are defined inside the subprogram Momentum. At the surface, or y equal to zero, p equals the dependent variable and q is zero. The centerline condition dictates that p equals zero and q equals one. This sets the flux term to zero at the boundary. The flux term is the product of the viscosity and vertical gradient of the axial velocity. Since the viscosity is finite, the gradient must equal zero, which is the condition sought. The Momentum subprogram can now be used to solve the momentum equation. The function imports the discretized mesh and guessed pressure gradient. Coefficients of the initial boundary condition, coupling, flux, and source terms are also imported. These are the coefficients of the piecewise cubic polynomial. The second-order, nonlinear PDE is solved and the axial velocity at the thre e axial locations is returned to the main program. Continuity Equation The axial velocity solution must be verified because the value of the pressure gradient is assumed. This value directly affects the momentum e quation by being part of the source term. It also indirectly a ffects the solution by changing the properties dependent on the pressure. Equation 2-3 is the mass or continuity equation that calculates the vertical velocity. At the sa me time, the solution of the mass equation acts as a check to the momentum equation. First, the gradient of the density and x -velocity product is determined at every vertical point at the e nd of the mini-mesh. This partia l derivative is calculated with a

PAGE 69

60 second-order backward-difference formula. The y -gradient of the dens ity is found with a second-order central difference formula with varying spacing. Once these two gradients are found, the partial derivativ e of the vertical velocity is approximated with another second-order central difference formula with varying spacing in Equation 3-7 [19]. Solving for the velocity at the next mesh point produces Equation 3-8. u v v yxy (Equation 2-3) 22 11 11 1jjj jj jj j jvavav u v x y aayy (Equation 3-7) where 1 1jj jjyy a yy 1 1 22 1 11 1 1j j j jj j j jjjj jju v xy aayy v avav aayy (Equation 3-8) Equation 3-8 is used to find the vertical ve locity component at th e last of the three x -locations. This new vertical velocity beco mes the variable used by the next mini-mesh downstream. The species and energy equations still use the original y -velocity for their calculations. Once the y -velocity is found at every vertical point, its value at the centerline is checked. Being a streamline boundary condition, there should be no flow across the boundary and the y -velocity should roughly equal zer o. If the velocity does not meet this requirement, the pressure gradient is adjusted and the program loops back to the momentum equation. The amount of the adjustment is proportional to the size of the y -velocity at the centerline. A weighted corr ection modifies the pre ssure gradient. This

PAGE 70

61 continues until the centerline y -velocity is less than one tenthousandths. At which point the axial velocity of the mini-mesh is saved or spliced to the axial velocity variable of the entire stage. The program then moves on to the remaining two governing equations. Now is an excellent moment to discu ss the reasoning behind breaking apart the momentum and mass equations from the other governing equations. It has already been shown that all of the equations are highly coupled and should be solved as such. However, that approach leads to a very problematic and time-consuming calculation due to the unknown pressure gradient. Solving th e entire group of equations until the correct pressure gradient is found would take a grea t amount of computing time. Decoupling the momentum and mass equations significantly re duces the time of the calculation. This method does not come without its disadvant ages. Separating the governing equations creates a delay in the solution. This dela y can be overcome with the iteration process already discussed in the sect ion Parameters and Conditions. Species Continuity Equations The remaining equations are not decoupled, but instead are solved simultaneously by the function pdepe. The set of species eq uations are solved for the mass fraction of each atom or molecule. The number of equations in this set is equal to the total number of species in the model, defined as N Equation 2-46 shows the simplified species equation. 1,2,3,...iii imiiYYY uDvMWiN xyyy (Equation 2-46) Mass fraction of species i is the dependent variable of the species equations. Again, the symmetry parameter, m is zero. Cantera determines the density, diffusion coefficients, net gas production rates, and molecular weights found in the equation.

PAGE 71

62 These quantities and the two ve locity components are used to create the coupling, flux, and source terms shown in th e three equations below. ,,,i iY cxyYu y (Equation 3-9) ,,,ii iimYY fxyYD yy (Equation 3-10) ,,,ii iiiYY sxyYvMW yy (Equation 3-11) The density and axial velocity make up th e coupling term. Ax ial velocity is no longer the dependent variable as it is in the momentum equation. This means that the multiple of the density and axial velocity must be transformed into a function of y The flux term, f can be found inside the parenthesis of Equation 2-46. It equals the density multiplied by the species mixture-averaged di ffusion coefficient and the vertical gradient of the mass fraction. The flux term is allowe d to be a function of dependent variables vertical gradient. Therefor e, only the density and diffu sion coefficient product is represented by a function. Th e source term becomes the co mbination of the last two terms of the species equation. This term equals the species mass production minus the product of the density, y -velocity, and vertical gradient of the dependent variable. The product of the density and y -velocity is transformed into a function representation, while the vertical gradient is left unaltered. This y -velocity is the origin al vector and not the velocity found from the mass equation. Th e species mass production and initial boundary condition are also changed to a function of the y -direction. Two boundary conditions of the species equa tions can be connected to the species mass flux. At the surface bounda ry is a heterogeneous cata lyst where species can be

PAGE 72

63 created or destroyed. Assuming a steady-stat e model, the species flux can be equated to the production rate on the catalyst. Speci es mass flux into the surface equals the destruction rate and the flux away from the su rface is the creation ra te. Mathematically written in Equation 3-12 and reorganized into Equation 3-13a to fit the form defined by the PDE solver. Cantera determines thes e production rates for each species. A symmetric boundary condition is applied to th e upper boundary. Resulting in the vertical gradient of a species mass fraction approxi mately equaling zero at the centerline. Equation 3-13a and 3-13b show these two c onditions in a form recognized by the pdepe function. ,i isurfaceiimY MWD y (Equation 3-12) ,10i isurfaceiimY MWD y (Equation 3-13a) 0010ii imYY D yy (Equation 3-13b) Parameter p equals the mass production rate and q is one for the lower boundary at y equal zero. The upper boundary condition has p equal to zero while q equals one. This sets the gradient equal to zero because neither the density nor the di ffusion coefficient of the flux term equal zero. The system of par tial differential equation is defined along with their initial and boundary conditi ons. Before the species equa tions are solved, the energy equation is added to the group. Energy Equation The energy equation contains kinetic energy terms defined by the velocity field. Kinetic energy terms are in the form of x and y gradients of the axial velocity. The

PAGE 73

64 solution of the momentum equation is used for the x -velocity. Both partial derivatives are calculated at every vertical point in th e mini-mesh. A simple second-order centraldifference formula is applied to estimate the ax ial gradient. The vertical gradient is a little more complicated because the spacing in the y -direction may vary. A second-order central-difference formula that is modified for varying point spacing is used for the core of the calculations. The y -gradient at the surface is found with either a first-order or second-order forward-difference formula for equal spacing. If the y spacing is linear, then the second-order formula is used. The fi rst-order equation is used if the spacing is non-linear [19]. Although it is fi rst order, the error should be small because the spacing near the surface is tight. The gradient at the centerline equals zero due to the boundary condition of the momentum equation. Equa tion 2-86 is the simplified energy equation with the two kinetic energy terms at the end of the equation. 2 11ipp NN p i imii ii iTTuT uckuvc xyyyy c Y Tuu Dhuuv M Wyyxy (Equation 2-86) Equation 2-86 contains many thermodynamic a nd transport properties that need to establish. Cantera retrieves the properties at every vertical point in the middle of the three x -locations and saves them to vectors. Th e main program uses the vectors to create the function representations of the coupling, flux, and source terms. It is apparent that the temperature is now the dependent variable of the PDE. The symmetry parameter is zero and the coupling, flux, and s ource term are listed below. ,,,pT cxyTuc y (Equation 3-14)

PAGE 74

65 ,,, TTu fxyTku yyy (Equation 3-15) 1 2 1,,,iN p i pim i i N ii ic Y TTT sxyTvcD yyMWyy uu huuv x y (Equation 3-16) The coupling term is the multiple of the density, axial velocity, and specific heat. The entire expression is tr ansformed into a function y The flux term, found inside the brackets, is made up of two parts. The fi rst equals the therma l conductivity times the temperatures axial gradient. Second is the combination of the viscosity, x -velocity and its y -gradient. The two parts must be kept sepa rate for the flux term to remain a function of the temperature gradient. Thermal conductiv ity is turn into one function, while the second part is turned into another. The re maining five terms are grouped into the source term. For the first representation, the product of the density, y -velocity, and specific heat are changed to a function of y and the temperature gradient remains a variable. The second term consists of a complicated sum c ontaining the species mass fraction gradient. This is where the coupling between the gove rning equations direc tly takes effect. Calculation of the sum is addressed in th e section Species/Energy System. The third expression is the other sum in the equation. However, it is not nearly as difficult as the last because it does not contain any of the systems dependent variables. This sum is simply calculated in the main program and adde d with the last two kinetic energy terms. The last three terms in the energy equation ar e combined into a function representation. Initial boundary condition of the temperature is transfor med into a function for the subprogram.

PAGE 75

66 Boundary conditions are established fo r the temperature at the surface and centerline. The temperature at the catalytic surface (y=0) is held constant. This lower boundary condition is shown in Equation 317a. The upper boundary condition is characterized by no heat flux. Temperature of the flow is uniform when it enters the reactor. At which point, the catalyst induces chemistry in the flow and heat production occurs at the surface. The temperature be gins to increase at the surface and slowly expand up to the centerline. A thermal boundary layer is created and heat flux across the streamline is zero until the layer reaches the cent erline. A long distance is needed for this to occur and the heat flux remains zero at the streamline for the short distance of the reactor. Equations 3-17a and 3-17b show the conditions in a form recognized by the pdepe function. 00surfacesurfaceTTTTf (Equation 3-17a) 0010 TTu ku yyy (Equation 3-17b) In Equation 3-17a, p is the dependent variable minus the temperature at the surface and q equals zero. This produces the constant value at the surface. The upper condition is created with parameter p equal to zero and q equal to one. This sets the flux term, which consists of two parts, to zero at the bo undary. The first part is the heat flux and the second contains the gradient of the axial velo city. A problem arises because only the heat flux should be zero. However, the second te rm vanishes at the centerline due to the boundary condition of the momentum equati on. The result is the proper symmetry condition at the centerline boundary.

PAGE 76

67 Species/Energy System of Equations The species and energy equations are comb ined for the pdepe function to solve. The resulting system of ( N +1) equations is shown below. The flux and source terms are split into three parts to accommodate the various expressions in each equation. 123123 ZZZZ cfffsss xyyyy (Equation 3-18) where 12,,,,N Z YYYT Components of the dependent variable vector, Z consist of each species mass fractions and the temperature. Coefficients of the cubic spline gene rated for the coupling, flux, and source terms are grouped togeth er. Each component of the vectors c f 1, f 3, s 1, and s 3 represents the cubic spline function of th at component. The other two expressions ( f 2 and s 2) generate the sums involving the system s dependent variables. The coupling term of each equation is combined into one group, c Separating this term is not necessary because it does not contain any mass fractions or temperature variables. pu u c u uc (Equation 3-19) The flux term is broken up into three separate collections. The first group, f 1, is the combination of the species equations flux terms and the energy equations thermal conductivity expression. These terms are multiplied by the axial gradient of the dependent variable. The Nth component of f 1 is set to zero and replaced with a sum in f 2. Group f 3 is a result of the additional flux te rm in the energy equation. The first N

PAGE 77

68 components of this group are zero because none of the species equations contain an additional term. 1 2 1,1 0m m NmD D f D k 0 0 21, 0 1 0 Z fdotFn y 0 0 3 0 f u u y Figure 3-2. Flux components in the species/energy system. The group, f 2, is a result of the fact that th e sum of all the species diffusion fluxes must be zero [13:227]. This fact is rear ranged and shown as the sum in Equation 3-18. It suggested that this equation be applie d to the species in excess, which in many combustion systems is N2 [13:227]. Diatomic nitrogen is the last species listed in the input file used in the tests. This corresponds to the Nth component in the dependent variable vector. Note that the Nth component in f 1 is zero and negative one in f 2. The result is this sum replacing the mass-averaged diffusion flux in the last species equation. MATLABs built-in dot product is utilized to create the sum. The last two values of Fn 1 are zero because the sum does not include the Nth specie (diatomic nitrogen) or temperature. 1 11,N i im iY Z DdotFn yy (Equation 3-20) where 121,1,,,0,0mmNmFnDDD The source term is also grouped into three expressions. Group s 1 is a combination of the terms multiplying the gradient of the dependent variable. The second group, s 2

PAGE 78

69 produces the complicated sum discussed in the section Energy Equation of this chapter. All the remaining terms are compiled into s 3. 1pv v s v vc 0 0 22, 0 1 Z sdotFn y 11 22 2 13NN N ii iMW MW s MW uu huuv x y Figure 3-3. Source components in the species/energy system. The sum in the energy equation, shown in Equation 3-21, is a function of the species mass fractions. The energy equation is not decoupled from the species equation and the mass fractions are dependent variables in the system. To reproduce this sum, all of the properties (this excludes the species mass fractions) are transformed into functions of y for each species and stored in a vector. The dot product of this vector and the dependent variable vector gradient r ecreates the sum inside the subprogram Species_Energy. The sum is then multiplied by the temperature gradient. Note that the temperature gradient is not part of th e sum, consequently the last value in Fn 2 is zero. 12,N piim i i icD Y Z dotFn M Wyy (Equation 3-21) where 1122 122,,,0pmpmpNNm NcDcDcD Fn MWMWMW The boundary conditions of both equations are also grouped together. Figure 3-4 shows the conditions for the system of e quations. For the lower boundary condition, parameter p is broken up into two groups, because the dependent variable is part of the temperatures lower boundary condition.

PAGE 79

70 1,1 2,2 ,01 01 0 01 10surface surface NsurfaceN surfaceMW MW Zf MW T 01 01 0 01 01 f Figure 3-4. Boundary conditions of the species/energy system. Function Species_Energy imports information to solve this system of governing equations. The subprogram imports the disc retized mini-mesh and all the polynomial coefficients. Parameters p and q of the systems boundary conditions, which are compiled in the main program, are also sent to the subprogram. The function then calculates the solution of each component in the dependent variable vector and returns it to the main program. A mass fraction less than 1E-20 is treated as error and the value is set to zero. Solutions for the mini-mesh ar e spliced to their corre sponding variables for the entire stage. The program moves one di fferential step downstream and loops back to the momentum equation. Recall that before solving the equations for a given mini-mesh the temperature and composition must be de fined at all three axial locations. The program linearly extrapolates thes e values from the previous two x -locations. The same process solves the governing equations for the next mini-mesh. Its solution is spliced to the stages solution variable and the program moves on. This continues until the end of the stage is reached or the differential x -step is not small enough to resolve the solution.

PAGE 80

71 CHAPTER 4 TESTING A process of running the code for severa l cases and examining the solutions is performed in order to test the program. Fo ur cases are used to test the software. Beginning with no chemistry in the first case, chemistry is slowly introduced to the other cases. The second case involves only gas chem istry while the last two tests include both gas and surface chemistry. Gas and surface chemical reactions are modeled at five hundred and seven hundred degrees Kelvin. Slowly introducing chemistry to the model will aid in locating errors during the debugging process should any problem arise. Each case uses the input file named ptcombust.cti, wh ich is provided by Cantera as part of the software package. This file contains data for the methane/oxygen surface mechanism on platinum developed by O. Deutschmann. The input file ptcombust.cti calls on the file gri30.cti, also part of the Cantera package, to manage the gas reactions. The file gri30.cti contains data for the optimized GRI-Mech mechanism and for this program calculates transport properties based on a mixture-aver aged transport model. Once the program finds the solution for a given case, the result s are examined. No experimental data is available at this time to compare to the programs solutions. However the different tests can confirm that the software produces r easonable results and is operational. Several parameters and conditions that ch aracterize the reactor and incoming flow are similar for the four cases. The reactor has a radius or thickness of two centimeters and a length of thirty centimeters. Therefore, the height of the mesh is two centimeters and the sum of stage lengths equals thirty cen timeters. The distance of the non-reactive

PAGE 81

72 surface at the entrance of the r eactor is given the variable na me Lnocat and is equal to one centimeter. In case one, a catalytic su rface is not present and the one-centimeter value only determines the location of the initial velocity cond ition found with Blasius solution. The differential step size in the ver tical direction is set at four-hundredths of a centimeter. This mesh spacing in the y -direction is not linear. In order to place more points near the surface, the power discussed in the section Discretizati on is set to four. The PC variable is equal to one so no iterati on occurs. Temperature of the surface is set to four hundred degrees Kelvin for case one a nd two. The initial temperature of the flow is also four hundred degrees Kelvin for the first two cases. A mixture of air and methane at one atmosphere of pressure comprise th e fluid entering the reactor of every test. Case One Case one models a chemically inactive gas passing through a reactor with no catalytic surface. The flow is essentially a non-reactive flow through a pipe or channel with a pressure gradient. With no chemical reactions taking place, density remains the same and the entire system of governing equati ons is altered. All of the equations could be simplified for an incompressible flow an d the set of species equations could be removed all together. Although modifying the code in this way woul d defeat the purpose of the test. To test the program only fluid properties are changed, while the code remains unaltered. Turning the gas chemistry off is achieved by equating the species mass production rate to zero. This only affects the source term, s 3 in Figure 3-3, in the species continuity equations. Creating a surface with no catalyst also exclusively affects the species continuity equations. Production rate s at the surface are forced to zero changing the lower boundary conditions in Figure 3-4.

PAGE 82

73 One stage is used to model the flow of the first case. Recall that the computational space of the reactor can be br oken up into stages. Bei ng able to change the axial differential step size of each stage allo ws the program to resolve the changing composition. However there is no varying composition because the chemistry is removed in this test. The reactor does not need to be split into stages for this reason. The fluid mixture, given in mass fractions in Tabl e 4-1, enters with a velocity of one meter per second. Other parameters and conditions of this test run are listed in Table 4-1. Table 4-1. Parameters and conditions of case one. Reactor Parameters Initial Flow Conditions Radius 0.02 mVelocity 1 m/s Lnocat 0.01 mTemperature 400 K Stage length 0.30 mPressure 101325 Pa Surface temp 400 K dy 0.0004 m power 4 Composition (mass fractions) CH4:0.004, O2:0.23, N2:0.752, AR:0.014 No. of Stages 1PC 1 dx (Stage1) 0.01 mInput file ptcombust.cti Results of Case One The solution should mirror that of a visc id two-dimensional laminar flow through two flat plates with the pressure slowly decreasing. Figure 4-1 illustrates the axial velocity profile at four different locations As expected the axial velocity solution resembles a boundary layer flow increasing from zero at the surface to the centerline velocity. The centerline veloci ty increases to compensate for the loss of mass flux near the surface. This is shown in Figure 4-1, where the centerline velocity increasing downstream as the boundary layer grows. Note the overshoot in the velocity profile at x equal zero. This is due to the program crea ting a function representation of the initial

PAGE 83

74 velocity condition. Other than the overshoot the velocity solution is a smooth continuous model of what is expected for flow over a flat plate with changing pressure. 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.00.20.40.60.81.01.2 x-velocity (m/s) y (m) x=0 x=0.1 x=0.2 x=0.3 Figure 4-1. Axial veloci ty profiles of case one. Increase in the centerline velocity should lead to a decrease in pressure. The pressure change of Figure 4-2 shows this to be the condition. The pressure slowly decreases downstream from its initial value of one atmosphere. The incompressibility of case one allows the use of Bernoullis Equation to calculate the change in pressure. This provides an alternate means of finding the pr essure with the axial velocity and ensures that the solutions of the program are consis tent. The velocity at the streamline or centerline, where viscous effects are not presen t, is used in Bernoullis Equation. The pressure difference calculated from both the program and Bernoullis Equation is graphed in Figure 4-2. The change in pressure pr edicted by the program and Bernoullis Equation is very similar and the behavior is typically found in the beginning stages of a pipe or channel flow.

PAGE 84

75 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 00.050.10.150.20.250.3 x (m) P -Pi(Pa) program Bernoulli Figure 4-2. Pressure plot of case one. Variations in the species mass fractions sh ould not exist because all chemistry is neglected. Temperature should also have mi nor changes for the same reason in addition to the low Mach number of the flow. This is the case for the first test of the program. As expected, the calculated composition and temperature remain constant. The software produces the expected solutions for all variables in case one. Case Two A flow characterized by gas reactions and no surface reactions is modeled in case two. The flow is that of a chemically r eacting fluid passing over a non-catalytic surface. The only difference between case two and case one is the presence of gas chemistry in the flow. Cantera determines the value of th e species mass production rates. Unlike case one, these values are not forced to equal zero. Removing the surface chemistry is achieved by altering the lower boundary conditio ns of the species continuity equations. Surface production rates ar e set to zero just as they are in case one.

PAGE 85

76 Three stages are used to m odel the flow in case two. Because the temperature of the flow is relatively low, little change in the composition is expected. However, using more than one stage will test the process invo lved with multiple stages. This includes the saving and loading of variables and the smoo th connection of the stages. The length of each stage is one centimeter. The same fl uid composition enters the reactor, but the initial velocity is now half a meter per second Parameters and conditions of this test run are listed in Table 4-2. Table 4-2. Parameters a nd conditions of case two. Reactor Parameters Initial Flow Conditions Radius 0.02 mVelocity 0.5 m/s Lnocat 0.01 mTemperature 400 K Stage length 0.30 mPressure 101325 Pa Surface temp 400 K dy 0.0004 m power 4 No. of Stages 3 dx (Stage1) 0.01 m Composition (mass fractions) CH4:0.004, O2:0.23, N2:0.752, AR:0.014 dx (Stage2) 0.01 mPC 1 dx (Stage3) 0.01 mInput file ptcombust.cti Results of Case Two The axial velocity calculated in the second test is gra phed in Figure 4-3 for three x locations. Similar to the first test, the velocity is recognized as a typical pipe or channel flow solution. The initial ve locity is half a meter per s econd and the centerline velocity increases from this value as the flow becomes fully developed. The presence of gas chemistry does not appear to affect the solu tion of the momentum equation. Variation in the composition is not anticipated and the velocity profile is comparable to that in case one. The reduction in the initial velocity doe s remove the overshoot found in Figure 4-1.

PAGE 86

77 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.00.10.20.30.40.50.6 x-velocity (m/s) y (m) x=0 x=0.1 x=0.2 x=0.3 Figure 4-3. Axial velocity profiles of case two. Minor variations in the density do not ch ange the velocity, meaning the pressure should also behave the same. Figure 4-4 illustrates that the pressure decreases downstream much like the pressure in case one. A difference in the programs solution and Bernoullis solution is not iceable and there is almost a twenty-two percent difference between the two. The general behavior of th e pressure is consistent with expectations; however, the software produces values that are not validated by Bernoullis Equation. As expected, the temperature remains cons tant at four hundred degrees Kelvin. Some changes in the temperature do occur but are very small and can be considered numerical error. Some of the mass fractions also contain small fluc tuations. Figure 4-5 shows the change in the mass fraction of the sp ecies methane. While some of the species mass fractions behave oddly, it is most likel y a product of numerical error. The program has a second-order accuracy and the larges t step size equals one centimeter. The

PAGE 87

78 resulting error has the size of on e ten-thousandths, which is greater than the error seen in Figure 4-5. -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 00.050.10.150.20.250. 3 x (m) P-Pi (Pa) program Bernoulli Figure 4-4. Pressure plot of case two. 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -3E-12-2E-12-1E-1201E-122E-12 reduction of CH4 mass fraction y (m) x=0 x=0.1 x=0.2 x=0.3 Figure 4-5. Reduction in methan e concentrations of case two.

PAGE 88

79 The software produces reasonable solutions for a chemically r eacting flow without a catalyst. The low temperature in this test results in little gas reactions and the mass fractions remain nearly constant. A large incoming temperature will lead to combustion of the fuel/air mixture. Care is take n to avoid combustion because the governing equations are reduced based on the expectati on that characteristic length scales in the axial direction are large. The smooth connection of multiple stages is also confirmed by the second test. This can be seen in Fi gure 4-4 where the pressure is a continuous function of x At this point the code calculates expected values for a flow with and without chemical gas reactions. Case Three A complete test of the software is perfo rmed in case three where gas and surface chemistry both exist. As it is originally intended, the model is that of a chemically reacting fluid flow over a catalytic surface. Cantera finds the gas and surface production rates used by the set of species continuity equations. Unlike the previous two tests, these values are not forced to equal zero. Table 4-3. Parameters and conditions of case three. Reactor Parameters Initial Flow Conditions Radius 0.02 mVelocity 0.5 m/s Lnocat 0.01 mTemperature 500 K Stage length 0.30 mPressure 101325 Pa Surface temp 500 K dy 0.0004 m power 4 No. of Stages 2 Composition (mass fractions) CH4:0.004, O2:0.23, N2:0.752, AR:0.014 dx (Stage1) 0.001 mPC 1 dx (Stage2) 0.001 mInput file ptcombust.cti Minimal change in the composition is encountered due to the low temperature and only two stages are applied. The temperature of the gas and catalytic surface is increased

PAGE 89

80 to five hundred degrees Kelvin and the diffe rential step size in the axial direction decreases to one millimeter. The other parameters and conditions are similar to the second test and all are found in Table 4-3. Results of Case Three At first the original software does not obta in a solution for the entire flow in case three and changes are made accordingly. Two centimeters into the reactor the program is unable to resolve the changing composition and the code prematurely terminates. This problem is found to be associated with the application of the dot products in the subprogram Species_Energy. Once these dot pr oducts are removed, the program is able to solve the entire computatio nal space. The dot product of Equation 3-20 is replaced with the mixture-averaged diffusion coefficient. The Nth species equation is now similar to the rest of the species equations. Th e other sum, Equation 3-21, is no longer performed by the dot product but is found by adding the function representations of each species. These two dot products create the su ms involving the gradients of the species mass fractions. The problem is not noticeable in case one because the change in the mass fractions is zero. This problem might be the source of the odd behavior seen in some of the mass fractions and pressure difference of the second test. The software is able to model the entire re actor after the corrections are made. The velocity profiles graphed in Figure 4-6 are typical of boundary-layer growth in the presence of a pressure gradient and are consiste nt with the models of the first two tests. Again, the velocity increases at the center line and the boundary layer grows as the fluid moves downstream.

PAGE 90

81 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.000.100.200.300.400.500.600.70 x-velocity (m/s) y (m) x=0 x=0.05 x=0.1 x=0.2 x=0.3 Figure 4-6. Axial velocity profiles of case three. Figure 4-7 shows the pressure change cal culated from the program and Bernoullis Equation. Little difference is seen between the two solutions and both agree favorably. It is evident that after the corrections are made, the program produces reasonable values for the pressure in case three. The general behavior is also consistent with that of the other two tests. Temperature of the computational space re mains constant at five hundred degrees Kelvin. Like case two the low temperature means gas reactions are at a minimum, but the presence of the catalytic surface generates chemical reac tions. The reactions produce a slight increase in the temperature just abov e the surface but the change is minimal. The chemical decomposition of methane il lustrated in Figure 4-8 also seems logical, but the values are small enough to be considered numerical error. The species mass fraction decreases from its initial value at the catalyst and the effect diffuses to the

PAGE 91

82 centerline as the flow moves downstream. The decomposition is not sufficient to generate any significant chemical activity. -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.000.050.100.150.200.250.30 x (m) P-Pi (Pa) program Bernoulli Figure 4-7. Pressure plot of case three. 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -3.5E-11-3E-11-2.5E-11-2E-11-1.5E-11-1E-11-5E-1205E-12 reduction of CH4 mass fraction y (m) x=0 x=0.1 x=0.2 x=0.3 Figure 4-8. Reduction in methan e concentrations of case three.

PAGE 92

83 The third case test reveals that the use of dot products in the subprogram, Species_Energy, leads to resolution problems in the code. After removing the dot products, the software produces good result s. However, case three does not produce a considerable amount of chemical activity an d a higher temperature is use in case four. Case Four Similar to the third test, case four is an other complete test of the software where gas and surface chemistry both exist. The incoming gas temperature and surface temperature is increased to seven hundred de grees Kelvin in an attempt to generate chemical reactions. Three stages are applie d in an attempt to resolve the changing gas composition. Parameters and conditions are listed in Table 4-4. Table 4-4. Parameters a nd conditions of case four. Reactor Parameters Initial Flow Conditions Radius 0.02 mVelocity 0.5 m/s Lnocat 0.01 mTemperature 700 K Stage length 0.30 mPressure 101325 Pa Surface temp 700 K dy 0.0004 m power 4 No. of Stages 3 dx (Stage1) 0.001 m Composition (mass fractions) CH4:0.004, O2:0.23, N2:0.752, AR:0.014 dx (Stage2) 0.00001 mPC 1 dx (Stage3) 0.000001 mInput file ptcombust.cti Results of Case Four The program is not able to obtain a solution for the entire flow in case four. Nearly five centimeters into the catalytic reactor, rapid change in the fluids composition is followed by a large increase in temperature. It appears that an initial temperature of seven hundred degrees Kelvin is sufficient to cause ignition of the air/fuel mixture over the catalytic surface. The software is unab le to resolve the rapidly changing flow variables after this point. This is due to the fact that the code being tested is not designed

PAGE 93

84 to model a combustion process. Governing equations are reduced based on the assumption of a relatively large characteristic length. Large axial gradients involved with the ignition of the fuel will cause the code to terminate at the point of ignition. The velocity profiles, Figure 4-9, behave si milarly to the other test and do not show any error prior to ignition. Inaccuracy in the axial velocity at the point of combustion is visible just above the surface in the boundary la yer. The combustion of the fuel leads to a temperature increase in this same region. The large temperature change causes the density, found in the momentum equation, to change rapidly leading to error in the velocity solution. 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.000.100.200.300.400.500.60 x-velocity (m/s) y (m) x=0 x=0.025 ignition Figure 4-9. Axial veloci ty profiles of case four. The change in pressure is graphed in Figure 4-10. This plot show s that the pressure decreases from its initial value of one atmos phere and appears more linear than the other pressure graphs. Although combustion occu rs, significant change in density is not present until ignition and Bernoullis Equatio n calculates a pressu re difference nearly

PAGE 94

85 identical to the pressure change found by the program. Ignition is predicted just before five centimeters into the reac tor where the pressure gradient becomes very steep. The pressures behavior at this point is unexpected and is attributed to the resolution problems associated with combustion. -0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 00.010.020.030.040.05 x (m) P-Pi(Pa) program Bernoulli Figure 4-10. Pressure plot of case four. Temperature is graphed at four axial lo cations in Figure 4-11. The temperature continues to increase just above the catalytic surface as the flow mo ves downstream. The exothermic reactions induced by the catalyst lead to the temperature increase in the boundary layer. This variable becomes large and unstable just before the code terminates, which is visible in Figure 4-11. At the point of ign ition, the temperature increases to over eight thousand degrees Kelv in. This value cannot be viewed as an accurate representation of the temperature. However, it appears that the ignition of the fuel is occurring just above the catalytic surface.

PAGE 95

86 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 650700750800850900950100010501100 Temp (K) y (m) x=0 x=0.02 x=0.04 ignition Figure 4-11. Temperature profiles of case four. 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 00.00050.0010.00150.0020.00250.0030.00350.0040.0045 CH4 mass fraction y (m) x=0 x=0.025 ignition Figure 4-12. Methane concentrations of case four. The chemical decomposition of methane in case four, before resolution problems arise, behaves much like the mass fraction reduction graphed in Figure 4-8. The mass

PAGE 96

87 fraction slowly decreases at the catalyst and the reduction effect diffuses up, away from the surface. Significant meth ane decomposition occurs right before combustion ends the program. This can be seen in the plot of the methane mass fraction in Figure 4-12. Many species are produced once significant amounts of methane are broken down. Two such species are atomic and diatomic h ydrogen and their mass fractions are graphed in Figure 4-13. It seems that the greater initial temperature (700K) produces the desired effect of chemical activity. However, the large temperature also produces other species such as OH radicals, and temperature continua lly grows to the point of ignition. The code is not designed to process combustion and consequently ends at this point. As expected the catalyst aids in the production of hydrogen and the mass fraction of both the hydrogen atom and molecule increase at the su rface. The wiggle found in the graph of Figure 4-13 at the last axial position is a result of the abso lute value of an overshoot. Cantera cannot process negative mass fractions and any overshoot into the negative must be adjusted. Figure 4-13. Hydrogen concentrations of case four. A) Mass fractions of atomic hydrogen. B) Mass fractions of diatomic hydrogen.

PAGE 97

88 The fourth test predicts ignition of the fu el just above the cata lytic surface nearly five centimeters into reactor. The greater in itial temperature reveals that the program is unable to model catalytic combustion, but can fo recast the point of ignition. At this point the software is unable to resolve the rapidly ch anging flow variables. However, solutions past this point are no longer physically realistic because the assumptions made to simplify the governing equations are not valid. Characteristic length scales in the axial direction become much shorter in the comb ustion process, which result in very large gradients. The programs resolution problems can be attributed to these large gradients found in some of the variables being dete rmined. The incoming temperature of five hundred degrees Kelvin in case three is too low to produce any significant chemical activity. While the initial temperature of case four is too great and causes combustion. Two additional tests are preformed to better understand the temperature dependence of chemical activity in the reactor. Both tests are similar to case three and four, but use an initial temperature of five hundred fifty and six hundred degrees Kelvin respectfully. The solution of the case using a temperature of five hundred fifty is very similar to the solution of case three. The temperature and composition of the flow remain nearly constant. The other solution, using an initial temperature of six hundred degrees Kelvin, is similar to case four. The temperat ure continues to increa ses as the flow moves downstream until ignition is reacted. Comparab le to case four, the composition begins to change at this point with the decompositi on of methane and the production OH radicals and other species. The point of ignition is further downstream than case four due to the lower initial temperature. It is clear that the chemical activity is highly dependent on the initial temperature. An initial temperatur e in the range of five hundred fifty to six

PAGE 98

89 hundred degrees Kelvin is the temperature n eeded to cause ignition in the reactor being modeled.

PAGE 99

90 CHAPTER 5 PROGRAM LIMITATIONS AND IMPROVEMENTS The program possesses several limiting ch aracteristics when modeling a reacting flow. Calculated solutions are second-ord er estimates due to the finite difference equations. Error from these estimates could propagate into the governing equations causing inaccuracies in the cal culated solutions. The software uses a mixture-averaged transport model in order to minimize the time needed to solve the sy stem of equations. The temperature at the catalytic surface is held constant. Initial conditions of a minimesh and fluid properties embedded in the eq uations are transformed into smooth cubic spline functions. Errors are undoubtedly produc ed in this process and rapid changes are not converted to smooth functions very well. Consequently, this program cannot model past the point of combustion and is only physically accurate for a relatively slow reformation process. To improve the program, the mini-mesh could be enlarged to include more than three axial locations and the use of higher-o rder finite difference equations would be possible. Increasing the size of the mini-mes h worsens the effect of the delay discussed in the section Parameters and Conditions an d iteration would probably be needed. The code could also be modified to support a multi-component transport model. Both changes would improve the accuracy of the so lution but greatly increase the computation time. The lower boundary condition of th e temperature could also be modified to represent a more realistic adiabatic su rface or a surface with heat transfer.

PAGE 100

91 CHAPTER 6 CONCLUSION A program is created to validate new surface mechanisms of heterogeneous catalysts. The adaptable program models a chemically reacting flow over a catalytic surface. The catalytic reactor is represente d in two-dimensional Cartesian coordinate form with negligible body forces acting on the fluid. The flow is characterized as a steady, low Mach number, boundary layer flow of a Newtonian fluid. The principles of mass, species mass, momentum, and energy co nservation are expressed mathematically and simplified into the governing equations. The model is constructed by numerically solving the system of coupled partial differen tial equations. The code, which consists of a main program with three subprograms, is written in MATLAB and uses Cantera to calculate chemical properties based on a mixt ure-averaged transport model. Allowing Cantera to manage the chemistry independent of the main code allows the program to remain flexible with the varying reaction path ways. Four different cases are utilized to test the program. Calculated solutions from each case are examined to confirm that the software produces reasonable results and is operational. The software is found to predict the point of ignition in the fourth test wh ere the initial temperature is great enough to cause catalytic combustion. Calculated values need to be compared to experimental data to truly determine the accuracy of the program. If the comparison between experimental data and the model reveals error in the program, improvements could be made to the code. Sacrificing computation time for accuracy might be necessary. Once the solutions of the program

PAGE 101

92 are proven acceptable, the program can begin to test surface mechanis ms of catalyst. The program could aid in the development of cheap er, more efficient heterogeneous catalyst.

PAGE 102

93 LIST OF REFERENCES 1. U.S. Department of Energy. Fossil Fu els. Retrieved February 2006, from http://www.energy.gov/energysources/fossilfuels.htm 2. U.S. Department of Energy. (2004, Ju ne 1). Nuclear Plants May Be Clean Hydrogen Source. Retrieved February 2006, from http://www.eurekalert.org/features/doe/2004-06/dnl-npm061404.php 3. Fatsikostas, A., Kondarides, D., & Very kios, X. (2001). Steam Reforming of Biomass-derived Ethanol for the Production of Hydrogen for Fuel Cell Applications [Electronic versi on]. Chem. Commun., 2001, 851-852. 4. Chou, C., Chen, J., Evans, G., & Winters, W. (2000). Numerical Studies of Methane Catalytic Combustion inside a Monolith Honeycomb Reactor Using Multi-Step Surface Reactions. Combustion Science and Technology, 150, 27-58. 5. Steciak, J., Beyerlein, S., Jones, H., Klei n, M., Kramer, S. and Wang, X. National Institute for Advanced Transportation Technology University of Idaho. (2001, September). Catalytic Reactor Studie s. Retrieved November 2005, from http://www.webs1.uidaho.edu/niatt/pub lications/Reports/KLK317_files/KLK317.ht m 6. Clark, J. (2002). Types of Catalysi s. Retrieved February 2006, from http://www.chemguide.co.uk/physi cal/catalysis/introduction.html#top 7. KITCO. (2002, March). New York Spot Price. Retrieved March 2006, from http://www.kitco.com/market/ 8. Aghalayam, P., Park, Y., Fernandes, N ., Papavassiliou, V., Mhadeshwar, A., & Vlachos, D. (2003). A C1 Mechanism for Methane Oxidation on Platinum [Electronic version]. Journal of Catalysis, 213, 23-38. 9. Di Cosimo, J., Apesteguia, C., Gines, M., & Iglesia, E. (2000). Structural Requirements and Reaction Pathways in Condensation Reactions of Alcohols on MgyAlOx Catalysts. Journal of Catalysis, 190, 261-275. 10. Deutschmann, O., Schwiedernoch, R., Maie r, L., & Chatterjee, D. (2001). Natural Gas Conversion in Monolithic Catalysts: Interaction of Chemical Reactions and Transport Phenomena [Electronic version] Natural Gas Conversion VI, Studies in Surface Science and Catalysis, 136, 251-258.

PAGE 103

94 11. Hirschl, R., Eichler, A., & Hafner, J. (2004). Hydrogenation of Ethylene and Formaldehyde on Pt (111) and Pt80 Fe20 (111): A Density-functional Study. Journal of Catalysis, 226, 273-282. 12. CANTERA. (2006, April 29). CANTERA, Object-Oriented Software for Reacting Flows. Retrived May 1, 2006, from http://www.cantera.org 13. Aeronautics Learning Laboratory for Science Technology and Research. (2004 March 12). Aeronautics Fluid Dynamics Level 3, Flow Equations. Retrieved February 28, 2006, from http://www.allstar.fiu.edu/aero/Flow2.htm 14. Turns, Stephen R. (2000). An Introduction to Combustion: Concepts and Applications, second edition. Boston: McGraw Hill. 15. Fox, R., & McDonald, A. (1998). Introduc tion to Fluid Mechanics, fifth edition. New York: John Wiley & Sons, Inc. 16. Panton, R. (1996). Incompressible Flow, second edition. New York: John Wiley & Sons, Inc. 17. Goodwin, D. (2003). Defining Phases a nd Interfaces, Cantera 1.5. California Institute of Technology, Pasadena, CA. 18. Winters, W., Evans, G., & Moen, C. (1996). CURRENT A Computer Code for Modeling Two-Dimensional, Chemically Reacting, Low Mach Number Flows. Sandia Report SAND97-8202, Sandia Na tional Laboratories, Livermore, CA. 19. Tannehill, J., Anderson, D., & Pletcher, R. (1997). Computational Fluid Mechanics and Heat Transfer, second edition. Philadelphia: Taylor & Francis.

PAGE 104

95 BIOGRAPHICAL SKETCH Patrick D. Griffin is a graduate student at the University of Florida, Department of Mechanical and Aerospace Engineering, where he is studying fluid mechanics. He was accepted to the University of Florida in 2000 where he received his Bachelor of Science degree in aerospace engineering in 2003 with the honor of summa cum laude. He has tutored a variety of engineering courses, including the Thermodynamics and Fluid Mechanics Lab, as a teaching assistant from 200 2 to 2004. He is a member of Tau Beta Pi Engineering Honor Society, Phi Kappa Phi and Phi Theta Kappa Honor Society. As a graduate research assistant, he has studied the fluid mechanics and chemistry involved with catalytic reformation and combustion.


Permanent Link: http://ufdc.ufl.edu/UFE0014866/00001

Material Information

Title: Two-Dimensional Modeling of a Chemically Reacting Boundary Layer Flow in a Catalytic Reactor
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0014866:00001

Permanent Link: http://ufdc.ufl.edu/UFE0014866/00001

Material Information

Title: Two-Dimensional Modeling of a Chemically Reacting Boundary Layer Flow in a Catalytic Reactor
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0014866:00001


This item has the following downloads:


Full Text












TWO-DIMENSIONAL MODELING OF A CHEMICALLY REACTING, BOUNDARY
LAYER FLOW IN A CATALYTIC REACTOR

















By

PATRICK D. GRIFFIN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Patrick D. Griffin















ACKNOWLEDGMENTS

Siemens and the National Aeronautics and Space Administration supported this

research. I thank Dr. David Mikolaitis, Dr. David Hahn, and Dr. Corin Segal for their

assistance. I also thank my parents for their continued support and involvement in and

out of my educational development.
















TABLE OF CONTENTS



A C K N O W L E D G M E N T S ................................................................................................. iii

LIST OF TABLES .............. ............................................... ........ vi

L IST O F F IG U R E S .... ...... ................................................ .. .. ..... .............. vii

A B S T R A C T .......................................... .................................................. v iii

CHAPTER

1 IN T R O D U C T IO N ............................................................................. .............. ...

2 REDUCTION OF CONSERVATION EQUATIONS ...........................................7

A p p ly in g A ssu m option s ...................................................................... .....................8
C ontinuity E quation ............................................ ............................. ...........9
Species C continuity E quations.................................... ......................... .. ......... 10
M om entum E qu ation s ........................................... ........................................ 12
Energy Equation .................. .................................... ................. 15
O rder of M agnitude A analysis ......................................................... ............... 21
C ontinuity E quation ........... ................................................ ....... .... .... .... ... 22
Species C continuity Equations.................................... ......................... .. ......... 23
A xial M om entum Equation ........................................ ........................... 25
V vertical M om entum Equation ................................... ............................. ....... 27
Energy Equation .................. .................................... ................. 30
U nit A naly sis ........................................... ........................... 34
C ontinuity E quation ........... ................................................ ....... .... .... .... ... 34
Species C continuity E quations................................... .......................... ... ........ 35
M om entum Equation ............................................................. ............... 36
Energy Equation .................. .................................... ................. 37
Sum m ary of Governing Equations ........................................ ........................ 40

3 PROGRAM METHODOLOGY ........................................ .......................... 41

D iscretization ..................................... ................................ ......... 43
P aram eters and C conditions .............................................................. .....................45
Input and O utput Files ........................................ ................... ..... .... 47
Initial C conditions of a Stage ............................................... ............................ 49










S ta g e O n e ........................................................................................................ 5 0
B lasius Solution................................................ 51
Subsequent Stages ............................. .................... .. ........ .. .............53
Solving G governing Equations........................................... ............... .. ............. 53
M om entum Equation ............................................................. ..................57
C ontinuity E quation.......... ..... .................................... .... .... ...... .. 59
Species C continuity Equations.................................... ............................. ....... 61
E energy E qu action ................................................................ ............. .... 63
Species/Energy System of Equations ............................................... ...............67

4 T E S T IN G .......................................................................................7 1

C a se O n e ................................................................................................................ 7 2
R results of C ase O ne ................................................................ .. .......... ......73
C a se T w o ............................................................................. 7 5
Results of Case Tw o ......................... ..... ......... ........... ....... 76
C ase T h ree ............................................................................ 7 9
R results of C ase T three ....................................................................... ... .... ...80
C a se F o u r ...................................... ..................................................... 8 3
R esu lts of C ase F ou r ................................................................... ........ .......... 83

5 PROGRAM LIMITATIONS AND IMPROVEMENTS ............................................90

6 C O N C L U S IO N ................................................................................................. 9 1

L IST O F R E F E R E N C E S ...................................... .................................... ....................93

B IO G R A PH IC A L SK E T C H ...................................................................... ..................95

























v
















LIST OF TABLES

Table page

2-1 Equations m odeling the flow ......................................................... ............... 40

2-2 Units of the governing equations. ........................................ ....................... 40

4-1 Param eters and conditions of case one.......................................... ............... 73

4-2 Param eters and conditions of case two.......................................... ............... 76

4-3 Parameters and conditions of case three........................................ ............... 79

4-4 Parameters and conditions of case four.................. ............................................. 83
















LIST OF FIGURES

Figure page

2-1 D im ensionless variables. ................................................ ............................... 21

3-1 Flow chart for single stage modeling. ........................................... ............... 55

3-2 Flux components in the species/energy system......................................................68

3-3 Source components in the species/energy system.................................................69

3-4 Boundary conditions of the species/energy system............... ........ ............... 70

4-1 Axial velocity profiles of case one. ........................................ ....................... 74

4-2 Pressure plot of case one. ............................................... .............................. 75

4-3 Axial velocity profiles of case tw o................................................ ........ ....... 77

4-4 Pressure plot of case tw o. ............................................... ............................... 78

4-5 Reduction in methane concentrations of case two. .............................................78

4-6 Axial velocity profiles of case three.......................... ...................... .................. 81

4-7 Pressure plot of case three. .............................................. .............................. 82

4-8 Reduction in methane concentrations of case three. ..........................................82

4-9 Axial velocity profiles of case four. ........................................ ...................... 84

4-10 Pressure plot of case four. ............................................................. .....................85

4-11 Temperature profiles of case four. ........................................ ....................... 86

4-12 Methane concentrations of case four.....................................................................86

4-13 Hydrogen concentrations of case four. A) Mass fractions of atomic hydrogen.
B) Mass fractions of diatomic hydrogen. ...................................... ...............87















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

TWO-DIMENSIONAL MODELING OF A CHEMICALLY REACTING, BOUNDARY
LAYER FLOW IN A CATALYTIC REACTOR

By

Patrick D. Griffin

August 2006

Chair: David Mikolaitis
Major Department: Mechanical and Aerospace Engineering

Problems associated with fossil fuels are increasing interest in alternative forms of

energy production. Hydrogen is quickly becoming a popular option, but the efficient,

affordable production of hydrogen is needed for it to become a viable source of energy.

Catalytic reformation of hydrocarbons and alcohols appears to be a promising means of

hydrogen production, but little is known about the surface chemistry. Research on

heterogeneous catalyst and their reaction mechanisms is growing. A greater

understanding of the surface chemistry could yield cheaper, more effective catalysts. The

evolving chemistry of the surface catalyst is in need of a flexible software program to test

new surface mechanisms.

A program is developed to model chemically reacting flow through a catalytic

reactor. The reactor is represented in two-dimensional Cartesian coordinates with

negligible body forces acting on the fluid. The flow is characterized as a steady, low

Mach number, boundary layer flow of a Newtonian fluid. Basic principles of mass,









species mass, momentum, and energy conservation are expressed mathematically and

simplified. These principles are transformed into the equations controlling the behavior

of the fluid and its motion through a process of applying assumptions, an order

magnitude analysis, and a unit analysis. A code is written to numerically solve the

resulting system of coupled governing equations. The methodology of constructing the

program is decomposed into developing an orthogonal computational mesh,

quantitatively defining the reactor and flow, locating chemical data and solutions,

establishing initial boundary conditions, and solving the governing equations. The

program is used to model four different flows: one with no chemistry, the second with

only gas chemistry, and the third and fourth with gas and surface chemistry. Calculated

solutions from each case are examined to confirm that the software produces reasonable

results and is operational. The software is found to predict the point of ignition when the

initial temperature is great enough to cause catalytic combustion.














CHAPTER 1
INTRODUCTION

The world is becoming increasingly aware of its dependence on fossil fuels. This

fuel is meeting over eighty-five percent of our country's energy demands, which includes

everything from electricity to transportation [1]. The power of fossil fuels lies in the

atomic bonds of the hydrocarbons that make up these fuels. Energy is released by

breaking these bonds in the process of combustion. The burning of fossil fuels also

releases harmful byproducts that include: carbon monoxide, carbon dioxide, and nitrogen

oxides. The carbon released into the atmosphere is originally trapped underneath the

earth's surface, leading to an overall increase of carbon oxides in the atmosphere. Some

believe these byproducts are leading to weather changes and health problems around the

world.

Energy extraction from fossil fuels is a relatively easy process and the fuel is

readily available in deposits beneath the earth's surface. For these reasons, fossil fuels

have become the main source of the world's energy production. The finite source is

nonrenewable and will eventually run out. Decreasing supplies will lead to a rise in fuel

cost and alternative forms of energy will become cheaper than fossil fuels. Economics

involved with the decrease in fuel supplies will dictate that the world turn to alternative

forms of energy.

Whether for ecological or economical reasons, the world will need to find

alternative forms of energy. Some look to the most abundant element in the universe,

hydrogen. Hydrogen is a clean, renewable source of energy that can be used in









combustion engines and fuel cells. Fuel cells are very efficient at producing electricity

from hydrogen with the byproduct being water. A major obstacle in this alternative fuel

is the affordable production of the energy carrier. Hydrogen rarely stands alone in its

pure form. Most of the earth's hydrogen is bonded to oxygen and carbon, in the form of

water, alcohols, and hydrocarbons. Water is an extremely stable molecule and takes a

great deal of energy to extract hydrogen atoms. This energy must come from renewable

sources if we wish to address the problems associated with fossil fuels. Hydrogen

extraction from alcohols and hydrocarbons is much easier. However, fossil fuels are

currently the main source of hydrocarbons. About ninety-five percent of the hydrogen

supply comes from the catalytic steam reforming of natural gas according to the US

Department of Energy [2]. Natural gas is a relatively clean fossil fuel consisting mostly

of methane. But natural gas is still a finite resource that will eventually run out. A very

promising renewable source of hydrogen comes from ethanol. Ethanol is an alcohol that

can be derived from biomass such as corn. Fuels produced from biomass release carbon

into the atmosphere that is originally in the atmosphere leading to zero net-production of

carbon oxides [3]. There are many promising energy alternatives to fossil fuels.

However, fossil fuels are so entrenched in our way of life, economically and politically,

that few expect a quick transition away from fossil fuels. Most believe that hydrogen

production will initially come from fossil fuels, with a gradual transition to renewable

sources of hydrogen production.

The world's attraction to the hydrogen economy is leading to an increased interest

in heterogeneous catalyst for converting hydrocarbons and alcohols into the energy

carrier. Surface catalysts are useful in increasing the reaction rates in combustors and









reformers. Catalytic combustors burn the fuel over a catalyst. This bums fuel at a lower

temperature, which decreases the amount of nitrogen oxides produced in the exhaust [4].

Catalytic reformers transform complex hydrocarbons and alcohols into hydrogen by

stripping the fuel of their hydrogen atoms. In either case, the fuel molecule is adsorbed

by the catalytic surface. The molecule forms a bond with the surface, usually through an

oxygen or carbon atom. This weakens the adjacent bonds between the oxygen or carbon

atom and the hydrogen atoms. The hydrogen atoms now begin to break off the molecule.

The product molecule will detach from the surface once it is finished reacting with the

catalyst. This leaves the surface free to adsorb a new reactant molecule. The catalyst

provides reaction pathways with lower activation energies. In effect, the catalyst lowers

the energy needed to break a molecule apart [5].

The efficient, affordable production of hydrogen is needed for this alternative fuel

to become a viable source of energy. The efficiency of a metal to catalyze a given

molecule is defined by how well the catalyst adsorbs the reactants and desorbs the

products.

Silver, for example, isn't a good catalyst because it doesn't form strong enough
attachments with reactant molecules. Tungsten, on the other hand, isn't a good
catalyst because it adsorbs too strongly. Metals like platinum and nickel make
good catalysts because they adsorb strongly enough to hold and activate the
reactants, but not so strongly that the products can't break away. [6]

The efficiency of the catalyst has no affect on the metal's price. The price is

dependant on the demand and rarity of the metal. As mentioned above, platinum and

nickel are two common metals used in catalyst. Platinum cost approximately $1000 per

ounce, where nickel cost around $0.4 per ounce [7]. With such a large disparity between

efficiency and price, a greater understanding of the surface chemistry could lead to

cheaper, more effective catalyst.









Catalytic reformation of hydrocarbons and alcohols appears to be a promising

means of hydrogen production, but little is know about the surface reactions. Surface

catalysts are not fully understood because the chemistry around the surface is difficult to

measure, especially in normal operating conditions. In the past, catalysts were treated as

a black box. The black box representation of the catalyst usually consists of one global

surface reaction or a small series of reduced mechanisms. Modifications to the black box

can be made until the model accurately reproduces the experimental data. While this

method is adequate for engineering applications, it does not accurately represent the

chemistry involved [4]. Many studies have recently taken place in attempts to understand

the surface reaction mechanisms of the heterogeneous catalyst [8-11]. The studies are

mostly concerned with determining the reaction pathways and the step-by-step chemical

degradation process of molecules. This is leading to new chemical reactions being added

to the surface chemistry. The evolving chemistry of the surface catalyst is in need of a

flexible software program to test the new mechanisms being added.

A program adaptable to the changing surface chemistry is developed in this study.

The program models a two-dimensional, chemically reacting flow though a catalytic

reactor. The fluid motion is characterized as a steady, low Mach number, boundary layer

flow. The catalytic reactor consists of a heterogeneous catalyst covering the inside

surface of a pipe or channel. The fluid motion is modeled as a flow through two flat

plates with a pressure gradient. The two flat plates are modeled as catalytic surfaces and

are identical. Basic principles of mass, species mass, momentum, and energy

conservation are employed to generate the model. These principles are expressed

mathematically and simplified for this specific problem. The process of reducing the









principles into the governing equations consists of applying assumptions, an order

magnitude analysis, and a unit analysis.

A software code is written to numerically solve the resulting governing equations.

The calculated solutions thermodynamically and kinetically define the fluid and its

motion. The code is written in MATLAB, a programming language created by

MathWorks that is used in many chemical flow simulations. MATLAB provides several

built-in capabilities that make the software well suited for this problem. It's

compatibility with Cantera being one such capability. Cantera is a free software package

developed by Professor David Goodwin at the California Institute of Technology to solve

problems concerning chemical reactions [12]. The program utilizes Cantera software to

manage the chemistry. The methodology of the program's development includes the

creation of an orthogonal computational mesh to resolve the equations. Then the

establishment of parameters and conditions that quantify the reactor and fluid flow is

performed. The location of input and output data is defined and initial boundary

conditions are set. Finally the governing equations are solved and the flow in the

catalytic reactor is modeled. The program is tested with four different cases: one with no

chemistry, another with only gas chemistry, and two with gas and surface chemistry.

Each case is modeled and the results are examined to confirm that the software produces

reasonable results and is operational. In the future, calculated solutions can be compared

to experimental measurements.

New surface mechanisms can be tested with the program resulting from this study.

Improved chemical kinetic data are updated in Cantera. The new chemistry is processed

by Cantera and incorporated into the program. Any change in the chemistry being used









to model the flow is done so inside the separate software of Cantera and not the main

program. Because the data is stored separately, the program is able to remain flexible

with the type of catalyst and fuel being used. This also allows the type of reaction

pathways to change as our understanding of catalyst grows without altering the code. As

a result the program is adaptable to the varying surface reaction pathways. Comparing

the two-dimensional model to experimental data provides a means of validating the

accuracy of the new chemistry. With a better understanding, heterogeneous catalyst

might be the key to the clean, renewable source of energy.














CHAPTER 2
REDUCTION OF CONSERVATION EQUATIONS

The chemically reacting flow through the reactor is modeled by numerically

solving the governing equations. Four of the equations are derived on the principles of

mass, momentum, and energy conservation [13]. The velocity field, pressure, and

temperature in the reactor are determined with these four equations. Knowing two

independent thermodynamic properties would adequately model the flow if it were not

for the chemistry taking place. The catalytic surface is expected to induce chemical

activity changing the fluid composition. A set of equations is needed to determine this

changing chemical composition. The species continuity equations satisfy this need and

are called upon to calculate the composition of the flow. One equation is needed to

determine the mass fraction of a single atom or molecule. As a result, the number of

equations inside this set is equal to the number of species used to model the flow, denoted

as N.

The mass, momentum, species and energy equations along with an equation of state

are all that is needed to determine flow properties through out the reactor. These

governing equations are coupled to one another in several different ways. All of them

contain properties dependent on the flow variables. For example, the momentum,

species, and energy equations contain transport properties such as viscosity, diffusion

coefficients, and thermal conductivity. Most of these properties are dependent on the

pressure, temperature, and composition of the flow. Properties dependent on the flow

variables indirectly couple the equations to one another. The equations are also directly









coupled to one another. Vertical gradients of the species mass fraction not only appear in

the set of N species continuity equations, but also the final form of the energy equation.

In addition to this, the velocity and velocity gradients can be found in all of the equations.

This makes for a group of highly coupled equations that control the behavior of the flow.

Equations of mass, momentum, species, and energy conservation are broken down

and modified to reflect this specific model while Cantera processes the equation of state

for an ideal gas. Conservation equations are transformed into the governing equations by

applying assumptions characterizing the reactor and flow. Governing equations are

individually examined in an order magnitude analysis after the assumptions are made.

Dominant terms in a given equation are found by comparing their magnitude to the

magnitude of other terms in the equation. Neglecting the weak terms and retaining the

strong terms further reduce the equations. A unit analysis or unit check is applied to the

resulting system of equations to ensure the validity of the equations. The process also

establishes the units of each variable, property, and solution.

Applying Assumptions

Turns goes through a similar process of simplifying the conservation equations for

a steady one-dimensional flow [14]. Instead of one dimension, the computational space

of the catalytic reactor is modeled in two-dimensional orthogonal space. These two

dimensions are the rectangular coordinates x and y, which represent the axial direction

and vertical direction respectfully. Once the governing equations are reduced to their

two-dimensional form, they are simplified by making assumptions about the fluid and its

motion. The chemically changing fluid is always considered a Newtonian fluid, which

carries many assumptions with it. Most importantly of which is that the shear stress is

linearly proportional to the rate of deformation. Another important assumption concerns









the fluid's motion. The flow is modeled as a steady-state flow, meaning all fluid

properties are independent of time. As a result, a partial derivative of any quantity with

respect to time is zero. More assumptions are made in order to reduce the governing

equations and are discussed during that process below. For the most part, the analysis

mirrors that of a boundary layer flow. However, the catalytic surface creates density

variations in the flow and compressibility must not be ignored.

Continuity Equation

The analysis begins with the reduction of the continuity or mass equation. The

continuity equation is a mathematical representation of the conservation of mass that

states that mass cannot be created or destroyed. In a Eulerian method of description, the

conservation of mass is described as the time rate of change of mass in a control volume

being equal to the net flux of mass through the control surface. Equation 2-1 is the vector

form of the continuity equation.

+ V (p)= 0 (Equation 2-1)
at

The steady flow assumption leads to the partial derivative with respect to time

being zero. The first term in Equation 2-1 is dropped as a result, leaving only the mass

flux in vector notation. The catalytic reactor is being modeled in a two-dimensional

space. Therefore, the mass flux is written out into its two-dimensional form with u and v

representing the x and y component of the velocity, respectfully [15].


( + )=p 0 (Equation 2-2)
Dx Qy

Further simplification is restricted due to the fact that density (defined asp)

variations occur in the flow. Equation 2-2 represents the continuity equation for the flow









field being modeled. This equation proves to be very important in the reduction of the

other governing equations. However, it is not the form used by the program. The

computer code uses the mass equation to determine the vertical velocity and some

mathematical manipulation is needed before the equation reaches its final form below.

Qv 8(pu) Sp
-v -(pu)- p (Equation 2-3)
ay x y

Species Continuity Equations

Much like the continuity equation, the species continuity equation requires that the

rate of gain of a single species mass in a control volume equals the net flux of the species

mass in through the control surface. Dissimilarity in the two equations arises due to the

chemistry. Instead of equaling zero, it equals the net chemical production of that species

in the control volume. The continuity equation of species i is shown as Equation 2-4 and

the set consists of one equation for each species. The time rate of change of the species

mass is zero because the flow is steady. The species mass flux is expanded out into its

two-dimensional Cartesian coordinate form and the result is Equation 2-5.

a(pY ) = m
+V = i"'" "(Equation 2-4)


S+ = "' (Equation 2-5)
9x 9y

On the right hand side of the species continuity equation lies the net chemical

production of species i in the control volume (ih"). This is determined using Cantera,

which gives the chemical production in moles. Therefore, the species mass chemical

production is replaced with the molar chemical production times the molecular weight of









the species. The species mass entering the control volume, know as the species mass

flux, transpire as a result of two modes, bulk flow and diffusion [14].


S+ '= AbMW (Equation 2-6)


mhx = pYu + miD (Equation 2-7a)

m" = Pv + m1yD (Equation 2-7b)


The first term in Equations 2-7a and 2-7b is the mass flux due to the bulk flow. It

is equal to the product of the density, species mass fraction, and fluid velocity component

corresponding to the direction of the mass flux. The second term is the mass flux due to

diffusion. The species mass flux can now be placed in the reduced species continuity

equation and the two modes separated from each other.


(pu)+ (pv)+ iD + (lf yD ) = AczMW (Equation 2-8)


The chain rule is applied to the bulk flow terms and the process is shown in

Equation 2-9. This leaves the continuity equation being multiplied by the mass fraction

plus two mass fraction gradient terms being multiplied by the density and velocity. The

continuity equation is equal to zero via Equation 2-2. After dropping the mass equation

and replacing the two bulk flow terms with Equation 2-9, the species continuity equation

reduces to Equation 2-10.

a a 8a(pu) a(pv) (qai
-(pYu)+-(pYv)= Y+ + pv (Equation 2-9)
ax ay Ox ay ax Qy
Coninuity Equanon

pa +a (I,,f ) ( ,D' ) D= bMW (Equation 2-10)
Dx ) + Qy
C)C C








Diffusion is a result of concentration gradients, temperature gradients, pressure

gradients, and uneven body forces. Ordinary diffusion from concentration gradients is

the only mode of diffusion considered in this model. The species mass diffusion is

approximated using a mixture-averaged diffusion coefficient [14]. The mass diffusion

terms are replaced with Equations 2-11 a and 2-1 lb inside the governing equation.

Equation 2-12 is the species continuity equation after all the assumptions are applied.

Further simplification is possible with an order magnitude analysis.

xDff = -pD, -- (Equation 2-11 a)


rhyf,Df = -PD, (Equation 2-1 lb)
ay

pu + ~+- -D +a- -P = aMW, (Equation 2-12)
ox dy dx^ ax } y oy

Momentum Equations

The momentum or Navier-Stokes equation is analogous to Newton's law of

momentum conservation. The momentum equation states that the rate of change of linear

momentum per unit volume equals the net momentum flux through that volume plus the

sum of forces acting on the volume. This is mathematically written in vector form as

Equation 2-13. Forces acting on the control volume are broken up into the surface forces

and body forces. Surface forces are defined as the divergence of the stress tensor. The

stress tensor is shown below in its rectangular, two-dimension form as Equation 2-14.

i + VV+V. ( = V + pFB (Equation 2-13)
at


0 = j (Equation 2-14)
1^I Ua~ T\ I~ ^p)









The stress tensor is made up of the shear stress acting on the surface plus the

pressure acting normal to the volume. Again, the flow is considered steady with

negligible body forces. Therefore, the time derivative and body force terms are dropped.

The vector form of the momentum equation is separated into its x-component and y-

component equations. Equation 2-15a represents the two-dimensional Cartesian

momentum equations in the x-direction, while Equation 2-15b is in the y-direction.

8(puu,) 8(puv) )d 8a
+uu- + + (Equation 2-15a)
ax ay ax ay

8(pvu) 8(pwv) o- 9o-
+ ) + (Equation 2-15b)


The x-momentum equation is used as one of the governing equations in the

computer program. Equation 2-15b, on the other hand, is not explicitly used in the

computer code. It is used to gain some insight into the behavior of the pressure. Both

equations simplify in a similar manner so both are reduced collectively. In this process,

the x-component equation is given first followed by the y-component equation. The

chain rule is applied to the momentum convection on the left hand side of the two Navier-

Stokes equations.

F(pu) 9(pv) au au Qao ao
S+ + pu-+ pv- +- (Equation 2-16a)
ax ay ] x Qy ax y
Conhnurty Equahon

0(pu) 8(pv) av av BC7 ao,
v +pu-(+pvj- + (Equation 2-16b)

Conhnurty Equahon

This results in the continuity equation being multiplied by the x-velocity in

Equation 2-16a and by they-velocity in Equation 2-16b. The continuity equation is equal









to zero and the first term in these equations is dropped accordingly. The two momentum

equations are reduced to Equations 2-17a and 2-17b.

pu + pv -+ (Equation 2-17a)
ax ay Ox ay

8v 8v Oa 8a
pu-+ pv + (Equation 2-17b)


The flow through the reactor consists of a Newtonian fluid. The stress acting on a

Newtonian fluid has no preferred direction, meaning the stress tensor matrix is symmetric

and its components are defined below. By definition the shear stress of a Newtonian

fluid is proportional to the rate of deformation. Shear stresses of a Newtonian fluid are

written below as functions of the velocity gradients [15].

,x = rT -p (Equation 2-18a)

Oy = Ty p (Equation 2-18b)

O, = -y = ry = ry (Equation 2-18c)


= u 2 Pa u +av
c- 3 OaxOy

av 2(au~av 1
, = 2u -2P +
y3 Ox O


IX= U-+-
YOX


(Equation 2-19a)


(Equation 2-19b)


(Equation 2-19c)


Replacing the stress components with their definitions above and

pressure gradient, the two momentum equations become,


separating the









au au ap
pu-+ pv --+
ax ay 8x
a r[2p u + i (Equation 2-20a)
8 8u 8F 2(au 8v 8 8u a v
2pj- p-- -+- +- p -+-
8x 8x 8x 3 8x 8y 8y By 8x

av 8v ap
pu-+ pv- -- +
8x ay ay
Sau v + v] a 2(u+v] (Equation 2-20b)
S-+- +-[ 2au 0)- a-+-
8x L ay ax ) y 8y Ly 3 a8x ay

Equations 2-20a and 2-20b represent the momentum equation in the x-direction and

y-direction respectfully. Both are reduced to their final form via an order of magnitude

comparison.

Energy Equation

The Energy Equation requires that the rate of change per unit volume is equal to the

net energy flux into that volume due to convection, heat, and work [13].

8 V2 V2
p- pe+- +V- pV e+V- =-V-q+ -+pV-F (Equation 2-21)
at 2 )] 2

From left to right in Equation 2-21, the first term is the time rate of change of

energy, which is zero because of the steady flow assumption. The second term represents

the flux of energy due to convection and equals the heat transferred into the control

volume plus the work done by the surface forces and the body forces. The work done by

the surface forces is determined using the stress tensor of Equation 2-14. The body force

is assumed to be negligible; therefore, the work done by the body force is neglected.

Equation 2-21 is written out in its two-dimensional Cartesian form with the assumed

simplifications.









8 [ V2 8 V2
Spu e+- +- pv e+- =
LLv (Equation 2-22)
Oq< Oq y a u + 9 +
(x a Vu" + x 8)

The stress components are replaced with the shear stress and pressure. The

pressure is separated from the shear stress terms, leaving two pressure work terms at the

end of the energy equation.


pu e+- +- v e+- =
c 2 Sy 2 J
[ 2 [) { y 2] (Equation 2-23)
S +- U(r +VT) ( +- Ur, +VT') -() (Vp)


Placing the x and y pressure work terms on the far right of Equation 2-23 into the

corresponding x and y energy convection terms on the left, Equation 2-23 becomes

Equation 2-24. The internal energy and pressure is replaced by the enthalpy, defined in

Equation 2-25 as the internal energy plus the product of the pressure and the specific

volume. The energy transfer due to the shear stress work is replaced by a variable called

(r work) to save space. Simplification of this term is possible via an order of magnitude

analysis of the governing equation, but first the energy convection and heat flux terms are

reduced. As of now the energy equation can be written out as Equation 2-26.


pI e+p2+- + pI e++- = workor) (Equation 2-24)
aL \ p 2} Oy \ p 2} a '


h=e+p (Equation 2-25)
P

Lpu[ 2 a\+ +- pv 2 h+- -- --+(z work) (Equation 2-26)
8x 2 y 2 O8x 8y -









First the two energy convection terms on the left hand side of Equation 2-26 are

simplified. These expressions consist of an enthalpy flux and kinetic energy flux, both

due to bulk flow. The two convection terms are separated into enthalpy convection and

kinetic energy convection. Performing the chain rule on the two kinetic energy

convection terms produces four separate terms. Equation 2-28 illustrates the process.

Two of these terms are combined to form the kinetic energy multiplying the continuity

equation, which equals zero. After dropping this term, the kinetic energy flux is replaced

with the last two terms of the equation above and the energy equation now takes the form

of Equation 2-29.

8(puh) 8(pvh) a rv2+ 8 VV2
+ +- pu\- +- pv =
S ay ax Ly2 ay J (Equation 2-27)
qx aqy + (+r work)
x ay

a [Pu V2 % a [ V2K ]
pu +- pv =
8x 2 J y 2
v2 r(pu) (pv) +L 2 +Pv V2 (Equation 2-28)
V2 \8 pu) 8pv)\ pu 8 ^ ^ pv 8 ^
+ + V + V
2 [ x 8y 2 ax 2 ay
ConhnuityEquaton


a(puh) + (pvh) pu 8 (v+ pv = (V2)
8x 8y 2 8x 2 8y
x y 2 x 2(Equation 2-29)
aqx -aq + ( work)
ax ay

Moving over to the right hand side of the energy equation, the heat flux terms are

now simplified. The heat flux is determined using Fourier's Law of Heat conduction plus

the flux of enthalpy [14]. The enthalpy flux here is due only to diffusion. The flux of

enthalpy from the bulk flow is already accounted for in the convection term. The vector

equation of the heat flux is broken up into the two Cartesian coordinate components, x








and y. Equations 2-7a and 2-7b are used to replace the species diffusion mass flux inside

the sum of the heat flux.
N
q = -kVT + ( D, ) (Equation 2-30)
i=1

NT aTN N
q- = -k -+ ( T ) -k- + ( h)- (puYh (Equation 2-31a)
x D1i x ,= ,=i

T N N N
y = -k + (1"yDh) =-k + (,yh )- (pvy~) (Equation 2-3 1b)
o z=1 y 1=1 1=1

Taking the partial derivative of the two equations above produces Equation 2-32a

and 2-32b. The partial derivative is not affected by the species sum and therefore can be

moved inside the sum. Similarly, the mass flux due to the bulk flow is not affected by the

sum and can be moved outside of the sum. The sum located inside the partial derivative

of the last term contains the product of the species mass fraction and species enthalpy.

Since the species enthalpy is given on a mass basis, the sum is equal to the specific

enthalpy of the flow. The last term can be rewritten as the partial derivative of the

density, velocity component, and enthalpy product.
__, aT_ N(a >h aF
aqx a L k dT]+," h 1 pu (Y ) (Equation 2-32a)
Ox Ox x x LX) x. '
aq= a aT ] a
k-+ 8q_,"- yh, pv (Yh,) (Equation 2-32b)
9y 8y 9y O y

ak + ,xh [+ --puh] (Equation 2-33a)
ax ax L x]x axL Ax

8qy [k aT] + a, hi:hj, [pvh] (Equation 2-33b)
ay a ay 8yy ) ay








The two heat flux terms of the energy equation are replaced with Equations 2-33a

and 2-33b above. Once this is complete the energy equation takes the form of Equation

2-34 shown below.

( p 8, pu(V2 PV ,(V2)
/-x ./2 8x 2 By

[k 2]+ [k~l Fm h [mFh,,] + (Equation 2-34)
ax 8x ay ay ,8ax1x j jy j

--uh]+- vh] +(r work)


The enthalpy convection cancels with the modified enthalpy diffusion of the heat

flux. The chain rule is performed on the expressions inside the two sums. The process,

shown in Equations 2-35a and 2-35b, leaves the species enthalpy times the partial

derivative of the species mass flux plus the species mass flux times the enthalpy gradient.

Equations 2-7a and 2-7b are used again to replace this species mass flux. This procedure

takes the original partial derivatives and splits it into three terms each.

a[,xa] =hO ax Ox+ =a h ,x + puY +hxiff" (Equation 2-35a)
)x ax ax axx

Lk, h] +'h + a =h h + Pv +, h Df,y h (Equation 2-35b)
-y Oy ,"dy -- Y

Replacing the two partial derivatives with their expanded expressions above, the

process of simplifying the sums can begin. For the last two expressions in Equations 2-

36a and 2-36b, the gradient of the species enthalpy is equal to the product of the species

specific heat and the species temperature gradient. Every species comprising the fluid at

a given point in the flow is assumed to have the same temperature, which means the

temperature gradient can be moved outside of the sums. This process is done for all four

of the terms containing enthalpy gradients.








Oa1 h h 1 O aha OX(
hj=- puY a AxDif (Equation 2-36a]

].,, h m+ pyYh l + (Equation 2-36b)
\ ax 9 ax

OpuY = pu (Yc =p (Equation 2-37a)

pv Y = pv Y(c =pV aTc (Equation 2-37b)

Sx =p = -D c (Equation 2-38a)
f OX j (1 Oax NP OX X 9a



(rnDif -yDf (" )p p(D, c (Equation 2-38b)


In Equations 2-37a and 2-37b, the product of the species mass fractions and


multiplied by the mass flux, which equals the flow density times the proper velocity
component. The diffusion mass flux inside the sum of the last two expressions is
approximated using the mixture-averaged diffusion equation, Equation 2-11. After
replacing the four terms with four equations above, Equations 2-36a and 2-36b are added
together and rearranged before being placed into the energy equation.
,a aT
D h, h = Pc ---+ pVC --+
S1- c (Equation 2-39)

h am + a j pJ D, K c +m L A -1
1=1ax Sy a Lx ( x o y y

Species _Continuty
The two dimensional gradient of the species mass flux is replaced with the species
chemical production via the species continuity equation, Equation 2-6. The energy
chemical production via the species continuity equation, Equation 2-6. The energy


I









equation is reduced to the Equation 2-40. Further simplification is performed with an

order magnitude analysis in the next section.

pu V2 ) pv a(v2) T a T_
2 8x 2 By 8xL 8xx By By

puc T- pvc (hMW)+ (Equation 2-40)
ax 8y '-I


S8x 8x y y

Order of Magnitude Analysis

An order of magnitude comparison between terms in a given equation determines

which terms must be reserved and which terms can be neglected. Governing equations

that are modified for this specific model are simplified further by eliminating the

insignificant terms. It is necessary to nondimensionalize the equation prior to comparing

terms. Variables are nondimensionalized with the uniform properties of the flow entering

the reactor. Most of the properties are chosen such that the resulting magnitudes are on

the order of one. The dimensionless variables and their magnitude are shown below in

Figure 2-1.

p*- 0,o (1) T*= = 0(1) 0(1) k* k 0(1)


u*=-- 0(1) p*- P =0(1) y*=Y= (1) D,= = D = 0(1)
U pmU H D
v* H c
v*= =0(?) P = 0(1) L] H- =0(S) c =- =0(1)
U Lo L CP
Figure 2-1. Dimensionless variables.

All but two of the dimensionless parameters have a magnitude on the order of one.

The unknown magnitude of the vertical velocity is found with the continuity equation.

The characteristic distance in the axial direction, L, is much greater than the characteristic









distance in the vertical direction, H. A dimensionless parameter with a very small

magnitude, denoted by 0(6), is produced when the characteristic height is divided by the

characteristic length.

Continuity Equation

Equation 2-2 is the two-dimensional continuity equation that is reduced based on

the steady flow assumption. Flow properties are replaced with their appropriate

dimensionless variables. After some algebraic rearranging, the mass equation is rewritten

in its dimensionless form.


a( + =( 0 (Equation 2-2)


8(p*u*) L 8(p*v*)
+ = 0 (Equation 2-41)
ax* H Qy*

All of the known dimensionless variables have an order magnitude of one. It has

already been noted that the characteristic length is much larger than the characteristic

height. This produces a relatively small value that divides the vertical mass flux term. In

order to balance the mass equation, the dimensionless y-velocity must have the same

order magnitude as the division of the height by the length.

O(1) 1 v*
0(+ = 0 (Equation 2-42)
0(1) 0(3)0(1)

v* = 0(3) (Equation 2-43)

While the continuity equation remains unchanged, the comparison of terms reveals

that the vertical velocity of the flow is small compared to axial velocity. This is a

common result in boundary layer flow analysis. Growth of the boundary layer is dictated

by the viscosity, or momentum transfer, and does not affect the entire flow until farther









downstream. The flow does not consist entirely of a boundary layer flow. However, a

vertical velocity does not exist at the entrance of the reactor, on the surface, or at the

centerline of the pipe or channel. The vertical velocity remains much smaller than the

axial velocity through out the reactor because of these boundary conditions.

Species Continuity Equations

The species continuity equation is reduced based on the assumptions of a steady,

two-dimensional flow, with ordinary diffusion being the only mode of diffusion. This

equation is shown below.


pu +pv ~+ -pD -y -PDm = MW (Equation 2-12)


The dimensionless form of the species mass equation is obtained by replacing the

flow variables with their proper dimensionless counterpart. The species mass fraction is

exempt from this part of the process because it is already a dimensionless quantity that

varies between zero and one. The unknown magnitude of the mass fraction does not pose

a problem since it is found in every term on the left hand side of the equation. As a result

it affects the magnitude of each term equally. After some algebraic manipulation, the left

hand side is rewritten in its dimensionless form as Equation 2-44. The right side of the

species equation, the species chemical production, is not compared to the rest of the

equation. Neglecting this term would result in the modeling of a non-reacting flow.

Therefore, the convection and diffusion terms are the only terms considered.

S a8Y L( _Y
p*u*-- +-L p*v* \+
8x*) v (Equation 2-44)
-p*D 2 a *K-p*
UL Lx dx*) H2 ay* D" Dy *J









Note that the mass-averaged diffusion coefficients are nondimensionalized by an

arbitrary value. This value is chosen such that the quantity of the dimensionless property

is roughly one. This ensures that the dimensionless diffusion coefficients have an order

magnitude of one, but the size of the value relative to the product of the incoming

velocity and characteristic length is unknown. The species diffusion terms cannot be

compared to the species bulk flow terms as a result. However, the comparison between

the diffusion terms inside the brackets is still possible.


O(1)+ 0 )+D- (1)+f (Equation 2-45)
0(8) UL 0(S)

Both of the bulk flow terms have a magnitude on the order of one. The first term

inside the brackets, corresponding to diffusion in the axial direction, also has a magnitude

of one. The second term corresponds to the diffusion in the vertical direction and has a

magnitude much greater than one. The order magnitude comparison of the species

continuity equation shows that the x-component of the species diffusion is much smaller

than the vertical diffusion and can be neglected. Information about the size of the

characteristic diffusion coefficient relative to the product of the characteristic velocity

and length is needed to determine the parameter multiplying the vertical diffusion inside

the brackets. The parameter must be very small, on the order of 0(6)2, in order for the

vertical diffusion to be of the same magnitude as the two bulk flow terms. This means

that the species mass transfer from the bulk flow is much greater than the species mass

transfer due to diffusion. Though this is most likely the case for the flow being modeled,

further restricting the flow to this assumption does not simplify the equation. The species









continuity equation for the flow through the reactor is now reduced down to Equation 2-

46 after dropping the axial diffusion term.

pa + a +- -pD -- = cbMW, (Equation 2-46)
tx ay ay ay)

Axial Momentum Equation

Analysis of the axial momentum equation is performed in the same manner as the

other equations. Flow properties are nondimensionalized by their characteristic variables.

The comparison begins with the momentum equation governing a steady, two-

dimensional flow of a Newtonian fluid in the x-direction. Characteristic variables are

rearranged, and the dimensionless form of the x-momentum equation is shown as

Equation 2-47. The viscous term inside the brackets is compared separately from the

momentum flux and pressure gradient terms due to the length of the expression. The

comparison of the dimensionless momentum flux and pressure gradient terms is now

possible. Excluding the vertical velocity, all of the dimensionless variables have a

magnitude on the order of one. The division of the characteristic length by the

characteristic height produces a relatively large value. This value is multiplied by the

dimensionless vertical velocity, which is a small quantity. The overall effect produces a

momentum flux and pressure gradient terms that all have the same order magnitude of

one. As a result, none of these terms is less important than the other and none of the

three can be ignored.

pu-+ pv- a --+
8x 8y &x

\ 8 S u 8 2 u 8v 8 u v
ra 2-- a avx (Equation 2-20a)
8x x L x x 38x 8y 8y y y 8x









au* L au*
p*u* + p*v*
Ox* H ay*


(1)+ ( (1) =-0(1)
0(3)


ap* L
= -+ 2rT term}
+ P.U x

LPU
+ {2\r term )
pm2U ~mx


(Equation 2-47)



(Equation 2-48)


Viscous terms inside the brackets are transformed into dimensionless variables and

compared to each other. Equation 2-49 represents the dimensionless form of the viscous

term. The characteristic properties are reorganized and the viscous term is now

multiplied by the inverse of the Reynolds number. The Reynolds number is a common

dimensionless parameter used to compare inertial forces to viscous forces. The Reynolds

number in Equation 2-50 is based on the length of the reactor and therefore is a

comparison of these two forces in the axial direction.


L (
L r term}\=
pmU2 -
L a u u 2 u av + u +v
--2 2-- -+- +- -+-


9 L ]_u* a 9 (2Su* +L av* +
2*-- -- -+-- +
P a* a* 3-* H y*)
P, UL a *( L2 u* L av*)
--*H *\ a-+--*
9y* H2 9y* H 9x*)


(Equation 2-49)






(Equation 2-50)


Magnitudes of each term that comprise the viscous momentum transfer expression

can now be compared to one another. Every expression inside the brackets is of the order

of one, except for a single term. This term is underlined twice in Equation 2-51 and has a

magnitude much greater than one. The result is a significant reduction of the viscous

term. With the exception of the highlighted term, every expression is neglected and the

complex viscous expression is simplified to just one term. Information about the








magnitude of the Reynolds number is needed to compare the viscous term with the rest of

the momentum equation. The inverse of the Reynolds number must have a magnitude of

0(6)2 for the remaining viscous term to be of a similar size as the momentum flux and

pressure gradient. A large Reynolds number assumption forces the viscous term to

balance with the other terms in the equation. It also forces the inertial forces of the flow

to be more significant than the viscous forces. This is a reasonable assumption because

the Reynolds number is based on the axial direction, where inertial forces are expected to

be greater than the viscous forces [16].


1 O 0(3) 1 O (3)
1 (1)- O(1)+ +0 (Equation 2-51)
Re O (3)1+ () 0 + ()

The order of magnitude comparison of the Navier-Stokes equation in the axial

direction produces a couple of useful results. The complex momentum transfer due to

viscosity is simplified to a single term. This reduces the x-momentum equation to its

final form used in the computer code. In addition, the Reynolds number must be large for

the viscous momentum transfer to be of a size comparable to the rest of the momentum

equation. The large Reynolds number result is used later in the magnitude comparison of

the y-momentum equation.

pu- + p- --+-/ (Equation 2-52)


1
Re = (Equation 2-53)
0(3)2

Vertical Momentum Equation

The momentum equation in the vertical direction is not used directly in the

computer program. It is used to gain some insight into the behavior of the pressure









through out the flow. The process of comparing terms in they-momentum equation is the

same as the process for the x-momentum equation. The Navier-Stokes equation for a

steady, two-dimensional flow of a Newtonian fluid is nondimensionalized by the

characteristic scales. The resulting dimensionless equation takes the form of Equation 2-

54 after some algebraic manipulation. Again, the viscous term is broken down separately

because of the length of the expression.

8v 8v 8p
PU- + PV =--+
O8x y Dy
-( [-- ] _- [ 22u + -v ) (Equation 2-20b)
a u 8v} a 8v
\- \l )-+- +-- 2/I/-- --- _- --+--
O8x \y 8x y 8y 8y 3\ 8x 8y

8v* L 8v* L 8p* L
p*u*- +-p*v* + (Equation 2-54)
8x* H Dy* H Dy* p.U2 term

Comparison of the momentum flux and pressure terms is postponed until the

magnitude of the viscous momentum transfer is known. After the flow properties are

converted to their dimensionless form, the viscous term becomes Equation 2-55. Similar

to the viscous term in the axial Navier-Stokes equation, the parameter multiplying the

viscous term is the Reynolds number. Analysis of the axial momentum equation

determined that the Reynolds number is on the order of 1/0(6)2. Knowing the magnitude

of the Reynolds number allows the viscous term to be compared to the momentum flux

and pressure gradient terms. But first an order magnitude comparison must be performed

on all the terms inside the brackets.

08 ~ J L 0u 8v* ] L2 8 a 2,u *
---+- +H2-- 2/*-
SL ; "a H + :J]kA; "t L *" I (Equation 2-55)
PUL L 8 [ 2Su* L v*)
I-H O -- +---H
H 8y 38x H Dy*









-1 + 0(5)]+ 0 )[O(1)+o0)1} (Equation 2-56)
Re O (5) O ([5) 0(5) 0(5)

As a reminder, all of the dimensionless properties, excluding the vertical velocity,

have a magnitude of one. The dimensionless vertical velocity and the division of the

characteristic height by the length have a magnitude much less than one. The result is the

magnitude of all but a single term inside the brackets of Equation 2-55 reducing to an

order of 1/0(6). The single remaining term, underlined twice in Equation 2-56, has a

magnitude of 0(6). This is much smaller than the other terms and can be neglected. As a

result, the dimensionless viscous term in the vertical momentum equation has an order

magnitude of 1/0(6). The process ofnondimensionalizing the equation is complete. An

analysis of the vertical momentum equation is now possible with the knowledge of the

viscous term's magnitude.

,v* L v* L 9p* 1
p*u*- +- + v* O + r term}) (Equation 2-57)
Ox* H 9y* H 9y* Re


0( ()+ 0(3)= +0()20 )} (Equation 2-58)
S0(,) 0(,) 0(,)

The magnitude of every term, with the exception of one, is a very small quantity,

0(6). The exception is underscored in Equation 2-58 and is relatively large compared to

the rest of the equation. The highlighted value corresponds to the pressure gradient in the

y-direction. The pressure gradient is considerably larger than the other terms and

dominates this equation. Neglecting all of the irrelevant terms, the dimensionless

pressure gradient is the only term remaining. This results in the pressure gradient in the

y-direction being essentially zero.









O 0 (Equation 2-59)
8y

The order of magnitude comparison of the vertical momentum equation reveals that

pressure through out the flow is a weak function of the vertical position. Pressure can be

treated as strictly a function of the axial direction, x.

Energy Equation

The magnitude comparison of the energy equation begins with Equation 2-40. The

equation is simplified based on the assumptions of a steady, two-dimensional flow with

negligible work done on the fluid by the body forces. Magnitudes of each similar

expression are compared to one another separately. First, the two kinetic energy

convection terms are compared to each other. Then the two heat conduction terms are

compared and so on. The work done by the shear stress is evaluated last.

pu (V) V pva(V2) 8 FkT] a FkT
2 8x 2 -y 8xL 8xx y By

pu8T- pvcp (h, cMW) + (Equation 2-40)


PDrcp + O] + workr)
pDc x 8Ox Ox ay ay

Analysis of the energy convection on the left hand side of Equation 2-40 is a

qualitative process. The couple of convection terms are not nondimensionalized, but a

simplification is possible with the understanding of the behavior of the velocity field.

The energy being transferred consists of kinetic energy, which is proportional to the

magnitude of the flow velocity squared. Balancing the continuity equation proved that

the vertical component of the velocity is much smaller than the axial component.

Squaring both components only make this difference more pronounce.









V2 =2 V2 2 2 = U2 [ *2 *2] = U2 [O(1)2 +0()2] V2 = 2 (Equation 2-60)

The contribution of the vertical component to the magnitude of the flow velocity is

neglected. The kinetic energy is calculated using only the axial component of the flow

velocity and the kinetic energy convection is rewritten as Equation 2-61. The two partial

derivatives of the x-velocity squared are performed to obtain the final form of the kinetic

energy transfer.

puS C(2) pvS(U2) 2u 8u
pUa(2+ = (u2) p + puv- (Equation 2-61)
2 Ox 2 Oy Ox Sy

Moving over to the right hand side of Equation 2-40, the two heat conduction terms

are now compared in a much more quantitative procedure. Variables are replaced with

their dimensionless counterparts in Equation 2-62 so a dominant term can be found.

These terms cannot be compared to the rest of the energy equation because the magnitude

of the parameter outside of the brackets of Equation 2-63 is unknown. Comparing the

magnitudes of the two expressions reveals the dominant term. It is underlined twice and

corresponds to the heat conduction in the vertical direction. Being much smaller than

vertical conduction, the heat conduction in the axial direction is neglected in the final

form of the energy equation.


a k- a k =O
kk*T* 2 [k (Equation 2-62)
k. T{ ( ( T1 2 ( 2T*
L2 [ H2 y*


k--T O (1)+ 1 (Equation 2-63)
L 0(,)2









The energy transfer due to the diffusion of enthalpy is the next term to be reduced.

Every variable and property is replaced with its dimensionless representation in Equation

2-64. The order magnitude analysis reveals that the second term, underscored in

Equation 2-65, is much larger than the other. The lesser of the two terms is the enthalpy

diffusion in the axial direction and is neglected in the energy equation.

"F(Y ~ T aY8Ty
Z\PDLmPC a x OT]2
pD=1 c 8 8x x y oy)
(Equation 2-64)
Dp, c,T, c. [ Y OT +* L2 O OT* *
L2 H- 2 yx* y *H
s" a' __H_


DpocPT No
D^^cT 0(1) o(1)+ 1
v ~ -( )


(Equation 2-65)


The last expression to be analyzed in the energy equation is the energy transfer due

to work done by the shear stress. It is greatly simplified by comparing the magnitude of

each term that comprises the work. Shear stress is defined as Equations 2-19a-c for a

Newtonian fluid. The energy transfer expression becomes Equation 2-67 after the shear

stress definitions are substituted. The flow properties are replaced with their proper

dimensionless variables and characteristic scales.


(- _work) = (urxx + VT ) + (ux + VT )
ax *y

8F -vu 2 ( QPu ov a(u + iv1+
Sx2up aup u+-.+vI-+- +x
ax ax 3 ax V y ay ax
Q v 2 u 8v u v
-2v--v --+- \ + up --+--
ay ay 3 ax ay ay ax


(Equation 2-66)


(Equation 2-67)









S* u* 2 u* u* L av*>
2u*u*---u** -+-- +
U2P. a dx* 3 dx* H dy*
L2 ax* *( *L du 8v*
v* *\ ----+--
H Oy* 8x*
___ -v* +
9 2 OH du 8v*
2v* *---v** --+- +
U2P, L2 8 y2* ~ 3 L 8x* y*
L2 H2 Iy (yu H 8v*)
8y L dx*


(Equation 2-68)


The dimensionless form of the viscous work can now be used to determine which

terms dominate the energy transfer expression. Most of the terms have an order

magnitude of one. There are two terms that do not have this magnitude. One is very

small with a magnitude on the order of 0(6)2, while the other has a large magnitude and

is underlined twice. The highlighted term is substantially greater than the other terms

inside the brackets and can be considered the only dominant term. Neglecting all the

other weak terms, the energy transfer due to the shear stress work is reduced to Equation

2-70.


( o(1)- o() 0o() o() o()2

u 0(2)2 0()2 0(8)21 0 ( 8)2
L0(8)2 0(5)2 0(g)2 0(g)2 0(g)2


(Equation 2-69)


y up-y (Equation 2-70)


The order magnitude analysis of the energy equation has greatly simplified the

governing equation. The kinetic energy convection, heat conduction, enthalpy diffusion,

and shear stress work are analyzed individually. These four modes of energy transfer are

not compared to one another because no assumptions are made about which mode is









more important. Analysis of the velocity field has revealed that the axial component can

be used to determine the magnitude of the velocity at any point of the flow. The heat

conduction in the x-direction and many of the terms that comprise the shear stress work

are neglected due to the analysis. As a result of all this simplification, the energy

equation becomes Equation 2-71.

2 u au 8 T aT aT
pu -+puv-- k- p -puc P
ox By By By ox By
L (Equation 2-71)
N N\ ]a YaT7 8 au
S(hc)A'MW)+ pDc -- +- I upu-
zK1 1L ayay y) ay L ay j

Unit Analysis

Units are substituted into the governing equations to ensure each expression in an

equation balances with the other expressions. Replacing variables and properties with

their units reveals several important aspects of these quantities. The inspection validates

that the equations were reduced without misplacing any variables or properties. It also

locates properties that require a unit conversion and determines units of the calculated

solutions. Note that Cantera calculates the properties with the International System (SI)

of measurement [17]. In order to keep unit conversions to a minimum, the variables also

use this system of measurement.

Continuity Equation

The main program calculates the vertical velocity component with the mass

equation. Solving the mass equation for the y-velocity produces Equation 2-3. Each

variable and property is replaced with its units. The density is given in kilograms per

meter cubed by Cantera. Therefore, the velocity and differential distances are measured

in meters per second and meters, respectfully. From Equation 2-72, it is clear that the

mass equation balances with equivalent units on both side of the equation.









av 8(pu) Sp
p-v -(u) (Equation 2-3)
y x y

kg X 1 kg 6 1 6 kg 1 kg kg
A^ __, (Equation 2-72)
mn s mn s s m3 ) m s nm3 .s

Unit analysis of the mass equation reveals that no unit conversion of the density is

necessary and the two differential step sizes should be given in similar units. Units of the

calculated vertical velocity depend on the units of the axial velocity, which is defined by

the initial condition. Although units cancel each other out in the mass equation, the other

governing equations prove that SI units should be used for the variables.

Species Continuity Equations

The reduced species continuity equation is algebraically reorganized in a form the

program can solve. This form is discussed more in section Solving Governing Equations.

Equation 2-46 is transformed into Equation 2-73. International System of measurement

is used for the velocity and differential distances, and their units are meters per second

and meters, respectfully. The density is still kilograms per meter cubed and the mass

fraction is dimensionless. Cantera gives a mixture-averaged diffusion coefficient in

meters squared per second. The unit of the net production rate is kilomoles per second

meter cubed, and the molecular weight is given in kilograms per kilomole. All of these

units are placed into Equation 2-73 to complete the analysis.


pu =- pD, -- pv + aAMW (Equation 2-73)
ax 9y Qy j y


kg 1 1 kg W/ 1 kg X 1 I6 kg
S3 S 3 S (Equation 2-74)
kg kg
m3 S m3 S









The unit check of the species continuity equation shows that converting the

molecular production rate into a mass production rate is indeed necessary. No other unit

conversion is required if SI units are used for the velocity components and differential

step sizes. The process proves the equation is reduced correctly from a unit analysis

point of view, and its solution is dimensionless.

Momentum Equation

The reduced momentum equation is reorganized into a form similar to Equation 2-

73. Equation 2-75 is the form of the momentum equation solved by the program. Units

of the velocity components, differential distances, and density remain unchanged. The

pressure and dynamic viscosity is also given in SI units. The SI unit of measurement for

pressure is the Pascal, which equals a kilogram per meter per second squared. Cantera

reports the dynamic viscosity in units of Pascal-second. A Pascal-second is equivalent to

a kilogram per meter-second. Variables and properties are replaced with these units and

result in balanced Equation 2-76.


pu- / du pv (Equation 2-75)
Ox oay ) Ox ay

kg j 1 1 kg X 1 kg 1 kg / 1
na s s i mnm- s / n -s n na s s i / -
m/2 S (Equation 2-76)
kg kg
2 2 2 2
m *s ms *ss

The analysis finds that none of the properties determined by Cantera necessitate a

unit conversion. Units of the simplified momentum equation balance accurately and its

solution, the axial velocity, is calculated in meters per second.









Energy Equation

Like the last two governing equations, the reduced energy equation is organized

into the form of Equation 2-77.

8T 8 k T + u 8T
pucP -k p+ual vc- +
SY L y (Equation 2-77)
Z OY--T A' Ou Ou
\pDc -[ (hcMW)-pu- --puv-
D' "Ic y 8y x 8y

Units of the flow properties already discussed in the previous governing equations

remain the same. Several new properties are encountered in the energy equation. The

temperature, thermal conductivity, species enthalpy, and specific heat of the fluid and

species i are used exclusively by this equation. The temperature is measured in degrees

Kelvin, and the thermal conductivity is given in units of watts per meter-Kelvin. A watt

per meter-Kelvin is equivalent to a Joule per meter-Kelvin-second. Specific heat of the

fluid can be determined on a mass basis in Cantera. The unit of the fluid's specific heat

is Joules per kilogram-Kelvin. Each expression is analyzed individually with the units

established. Moving left to right in Equation 2-77, the axial energy convection is

analyzed first. Substituting units into the axial energy convection shows that the term has

units of Joules per cubic meter second. Units of the other expressions must reduce to this

unit to balance the energy equation.

aT /J / J
pucp -- (Equation 2-78)
SOx m s S ./ M 3 S

The next expression evaluated is the energy conduction along with the viscous

term. Units of the viscous term are equivalent to the energy conduction at a Joule per

cubic meter second and Equation 2-79 balances with Equation 2-78.









8 F 8T 8u~ 1 J __ 1_\ J___
[L O +up O = J -+- = (Equation 2-79)
oy Oy oy m m-/ *-sm s / / m S

Units of the second expression reduce to the same units of the axial energy

convection. The vertical energy convection also reduces to these units and is shown

below in Equation 2-80.


PVCT J J
ay in3 s 3S 9./,M/ 4 3.


(Equation 2-80)


Cantera reports the species specific heat in a column vector that has been

nondimensionalized by the universal gas constant. Multiplying the vector by the

universal gas constant produces specific heats with the units of Joules per kilomole-

Kelvin. The sum of the energy diffusion expression includes the species specific heat.

The term highlighted is added to convert the species specific heat from a molar basis to a

mass basis. It is the inverse of the species molecular weight and must be added to the

enthalpy diffusion expression for the units to conform to the rest of the energy equation.

Equation 2-82 illustrates the modification.


Z pDDC -3-- I- = (Equation 2-81)
ay ay In S 3 S

N NT
PLDc [Y OT]pD -, LLmOT (Equation 2-82)


Much like the species specific heat, Cantera reports the enthalpy of each species in

a dimensionless column vector. The vector is nondimensionalized by the universal gas

constant and the temperature of the fluid. After multiplying the vector by these two

properties, the resulting enthalpy has the units of Joules per kilomole. Unit analysis of









the enthalpy production expression is done in Equation 2-83. Because the species

enthalpy is reported on a molar basis, the chemical production rate does not need to be

converted to a mass basis. The molecular weight term is underscored and is dropped

such that the unit of this expression is consistent with the other terms of the energy

equation.


NJ 61 kg J kg
0(h)MW) = 3 M3
,(=1 ht m3 s kmol m3 *s kmol

N N
I(haAMW4T ) -:> Y(h0 )


(Equation 2-83)



(Equation 2-84)


The last expression in the unit check is the kinetic energy convection. Equation 2-

85 shows that the expression needs no modification.


pu2 puv -- ---- (Equation 2-85)
)x C y m3 / f m3. -

Analysis of the energy equation reveals that energy diffusion and enthalpy

production expressions required modification. Molecular weights are added to the energy

diffusion term and removed from the enthalpy production term. After these

modifications, the energy equation becomes Equation 2-86 where the units of each

expression are equal and the equation balances. Temperature being the dependent

variable of the energy equation is calculated in degrees Kelvin.

8T 8 k T + u~1 8T
pucp- k- +u --I pvcp -+
8x 8y 8y 8y 8 By
OX YL OY Y (Equation 2-86)

p D. c, LY,' T O a8u au
P\D -\--- (hco -pu--puv-
I Iy 8y y 8x By









Summary of Governing Equations

In conclusion the governing equations are applied to the two-dimensional modeling

of the catalytic reactor. Each equation is reduced based on assumptions describing the

fluid and its motion. Through an order magnitude analysis these equations are simplified

further. The units of each term are verified and the equations are balanced. The resulting

equations solved by the program are summarized in Table 2-1.

Table 2-1. Equations modeling the flow.
Equation
Principle Equation number
number
Mass av a(pu) Sp
p- v 2-3
Conservation 2y ax y

Species Mass a ai"
SpeciespuMass pD MW i= 1,2,3,...N 2-73
Conservation Ox Sy[ 9 9y
Momentum p u a u P u-75
P pva 2-75


Conservation a c ( /T ,
COT __ FOT Ou PCT
Energy Ox OY -- + Y 1 2-8+
Conservation P. j i P, i Yr O~ T _I U 2 _u 7-cu
S DAm (ht, P 2 pu puv-
-1 MW 8y By 8x 8y

The unit analysis also determines the units of properties and variables found in the

governing equations. Units of each fluid property are compiled in Table 2-2.

Table 2-2. Units of the governing equations.
Property Variable Units Property Variable Units
Differential Production km
dx, dy m re, kmol/m3S
step sizes d dy rates Ckmo
Diffusion Thermal W/2
D,, m2/s k W/m -K
coefficients m conductivity
Density kg/m3 Specific heats c J/kg-K
Enthalpies h, J/kmol Temperature T K
Mass Velocity
facts Y, dimensionless Veloci, m/s
fractions components U,
Molecular A k l Viscosity Pa-s
weights PMW kg/kmol
weights Pressure p Pa














CHAPTER 3
PROGRAM METHODOLOGY

Flow variables are calculated throughout the catalytic reactor via a step-by-step

process of solving the governing equation. The process begins by creating a discrete

mesh of points to numerically solve the equations. Solutions to the flow variables are

recorded at these points. The next step involves establishing parameters and conditions

of the reactor and fluid. These values characterize the reactor and initial conditions of the

flow. Folders to import and export data must also be defined. Once the first three steps

are completed, the code can begin to find the solutions. The program sets the initial

conditions and solves the simplified governing equations in Table 2-1.

The code is written in MATLAB and consists of a main program with three

subprograms. One of the subprograms finds the initial velocity components. The main

program creates the mesh, finds fluid properties, and sets conditions needed to solve the

equations. Information from the main program is sent to the other two subprograms.

Solutions are found by the subprograms and sent back to the main program where it is

saved in the solution variables. MATLAB is chosen above other programming languages

because of its built-in ability to handle vectors, vector operations, and partial differential

equations. MATLAB incorporates several computational tools capable of solving partial

differential equations. A function called pdepe is used to solve the momentum equation

as a single equation. It is also used to solve the energy equation and species continuity

equations as a set of coupled partial differential equations. This makes MATLAB well

suited for modeling chemically reacting flows. Another useful property of MATLAB is









its compatibility with Cantera. Cantera is a free software package developed by

Professor David Goodwin at the California Institute of Technology to solve problems

concerning chemical reactions. The main MATLAB program calls upon this software to

determine the thermodynamic, transport and chemical kinetic properties of the flow and

catalytic surface. Cantera is able to construct objects of different phases and tie the

phases together through an interface. This allows for the chemical interaction between

the gas and surface [17].

Several studies attempt to model catalytic combustion similar to this model. A

study by the National Institute for Advanced Transportation Technology at the University

of Idaho modifies an existing code. Lawrence Livermore National Laboratory provides

the existing Hydrodynamics, Combustion, and Transport (HCT) code. The finite-

difference code, HCT, utilizes the same principles of conservation for its calculations.

Dissimilarity occurs in the application of the governing equations to the one-dimensional

time-dependent catalytic combustion, opposed to the two-dimensional steady-state

catalytic reactor modeled by this program. Still, the study offers some insight into the

chemistry and equations involved with modeling a catalytic combustor [5]. In a second

study, Chou et al. [4] uses CURRENT with CHEMKIN and SURFACE CHEMKIN

software to model a two-dimensional monolith catalytic combustor. CURRENT is a

code developed by Winters et al. [18] for low Mach number chemically reacting flows.

The study discusses the chemistry and boundary conditions of the model and compares

the calculations to experimental data. This program uses a similar symmetric boundary

condition at the centerline.









Discretization

The two-dimensional computational space of the reactor is broken down and

discretized before the equations can be numerically solved. The mesh is that of a planar

geometry with the height determined by the reactor's radius. The upper boundary is

moved to the centerline and the lower boundary is still the catalytic surface. This reduces

the height of the computational space and in turn reduces computation time and memory

used by the computer. Now is a good time to mention that the centerline is assumed to be

a streamline and symmetric conditions are assumed to exist at this boundary due to the

two identical plates modeling the surface of the pipe or channel. This assumption affects

the boundary conditions discussed in the section Solving Governing Equations. The

length of the reactor is broken down into stages, the first stage being the entrance. This is

also intended to reduce the time needed to calculate a solution. It is expected that the

flow changes relatively fast in the beginning of the reactor when the catalyst is first

encountered. This corresponds to the first few stages of the computational space. To

help resolve the solution in these stages a smaller differential step size in the x-direction

is chosen. Once the properties reach a quasi-steady state, the step size can be increased to

help lower the computation time.

An orthogonal mesh is created for every stage. Each stage has its axial direction

discretized in a linear manner, where every point is an equal distance apart. The distance

is set for a given stage but can change from stage to stage. This allows the user to adjust

the axial step size of a stage if the program cannot converge on a solution. A possible

source of this problem is a significant change of flow properties in the x-direction. Recall

that the governing equations are simplified based on the assumption that the characteristic

length is much larger than the characteristic height. In other words, the vertical gradients









are much larger than the axial gradients. While this assumption is still applicable, there

may be areas where a change in step size is needed, such as the reactor's entrance.

The point separation in the y-direction, in contrast to the axial point placement, is

the same throughout the reactor. Although the vertical point placement must remain the

same for every stage, it is not restricted to only a linear displacement. The point

displacement is set as a power of the point location. For example, setting the power to

one would position the points linearly. Setting the power to two creates a quadratic point

displacement, leading to more points near the surface. A larger power places more points

near the surface. Varying the power allows the user to control the location of the points

in the vertical direction. This aids the program in resolving the varying chemical

composition near the surface. The catalytic surface serves as the main source of the

chemical reaction in the flow. Therefore, it is expected that most of the chemical change

will occur near the surface. More points are needed near the surface to determine the

change in the chemical composition in the vicinity of the catalyst. A tight mesh near the

surface also helps resolve the fluid velocity boundary layer.

Velocity, temperature, and composition variables are not found for the entire stage

at once. Instead the stage's mesh is broken up further into mini-meshes. A mini-mesh

contains all the vertical points for a group of three axial locations. Governing equations

are solved one mini-mesh at a time due to the coupling of the equations. The pressure,

temperature and mass fraction of a mini-mesh must be approximated prior to solving the

equations. Jumping ahead might seem premature because the governing equations meant

to calculate the variables have not been solved yet. However, fluid properties dependent

on the solution are imbedded inside the equations. These properties must be established









in order to solve the equations. One can now being to appreciate the complex coupling of

the governing equations. Once the equations are solved, the program updates the

variables and moves downstream to the next mini-mesh.

Parameters and Conditions

Parameters and conditions of the catalytic reactor and incoming flow are set inside

the code of the main program. All of the values, composition being the only exception,

must equal a real scalar. The computer code begins by setting parameters of the catalytic

reactor, such as the radius, stage length, stage number, surface temperature, and the

distance of the non-reactive surface. Dimensions of the computational space are

constructed with the height and length of the stage. The stage number is simply the

sequential numbering of each stage for which a solution is calculated. The value of this

number determines the data used to set the incoming conditions and the output folder in

which the export files are stored. This is discussed further in the sections Initial

Conditions of a Stage and Input and Output Files. In the energy equation, the

temperature at the wall or surface boundary is held constant at the value entered. The

distance of the non-reactive surface refers to the entrance of the reactor where there is no

catalyst on the surface. This is only important for the first stage and can be ignored for

any other stage. The differential step sizes in the vertical and axial direction are also set

at this point, along with the power used to discretize the vertical direction. These values

are used to construct the two-dimensional mesh described in the section Discretization.

After the parameters of the reactor are entered, the conditions of the incoming flow

are defined. The speed, temperature, pressure, and composition of the flow entering the

reactor are established. The incoming flow is assumed to be a uniform flow where the

velocity is purely in the axial direction. Initially there is no vertical component to the









fluid's velocity and the speed of the x-velocity is the same at every point. Therefore, only

a single quantity is needed to define the velocity vector entering the reactor. The flow's

chemical composition is initially modeled as a well-mixed fluid. This simply means that

the species mass fractions are also the same at every point entering the reactor. The

composition is the only value entered as a string variable. This string contains the name

and mass faction of the species present in the incoming flow. Cantera reads the string to

set the composition of the gas. The program uses these values to set the initial conditions

of the variables.

The last parameter to set is the PC variable, also a scalar. This variable controls

whether the program iterates on a solution and if so, how many times the iteration takes

place. An inherent delay in the solution process exists because the governing equations

are decoupled. The delay is exaggerated by properties that are dependent on the solution

inside the equation. Some of these properties include density, viscosity, and diffusion

coefficients. With no iteration (PC equal to one), the program numerically solves the

governing equations for one mini-mesh. Then the program updates the variables and

properties and moves one differential step downstream to solve the equations at the next

three axial locations. The program is continually updating the properties prior to moving

downstream; therefore the delay is expected to be small. To improve the calculation one

may choose to iterate on a solution using a predictor/corrector type method. To iterate on

a solution the PC variable is set to quantity greater than one. For example, the program

iterates once on a calculation if the variable is equal to two. Iteration occurs by solving

the equations and updating the properties with the known solution. This could be seen as

a predictor step, now to correct the calculation. Instead of moving one differential step









downstream the program recalculates the solution for the same three axial locations based

on the updated properties. If the PC variable is three, the iteration occurs twice, and so

on.

Input and Output Files

Input and Output file names are given prior to operating the program to direct

import and export data. The input text file contains the chemical data Cantera require to

model the gas and solid of the catalytic reactor. Naming the input file informs the

program where the chemical data are located to import. Data determine properties found

within the governing equations. Only the filename of the input file is needed if it is

located in Cantera's current working directory. This directory is initially set as the data

folder inside Cantera's main folder, which is installed with the free software. The

pathname of the output folder provides the program the location of the export folders.

Export folders must be created inside the output folder and given the name Stagel,

Stage2, Stage3... etc. The solutions of a stage are recorded in the folder with the

corresponding number. Therefore, the export folder of a stage must exist before seeking

the solution of that stage. The entire pathname is stored in the string variable saveFile.

Not only is this string variable used to export solutions of a stage; it is also used to import

initial conditions for most of the stages. This is discussed further in the section Initial

Conditions of a Stage.

Considerable amounts of data are required to model the gas and solid of the

catalytic reactor. Cantera accesses this data via the input text file specified. These files

contain information on the chemical kinetics, thermodynamics, and transport properties

of many different species. Data consistent with the modified Arrhenius function

determines the chemical kinetic properties of the gas phase reactions. This data include









activation energy, pre-exponential coefficients and temperature exponent. In addition to

this, surface reactions apply reactive sticking probability. The thermodynamic properties

are determined using a NASA polynomial parameterization or Shomate parameterization.

Coefficients of either parameterization are incorporated in the data of the input file.

Information needed to calculate transport properties based on either a multi-component or

mixture-averaged transport model is also included. The multi-component transport

model provides a more accurate solution than the mixture-averaged model. However, the

multi-component model requires more data and computation time than its counterpart

[17].

The program saves several variables to the output folder for every stage. The value

of each variable is saved as a double precision scalar, vector, or matrix in an ASCII file.

The axial location, axial velocity, pressure, temperature, mass fraction of each species,

pressure gradient, and vertical velocity are all stored in the export folder. The axial

location is saved in order to keep track of which discretized points in the mesh the

various solutions correspond. The x-location is saved as a vector that begins at zero and

ends at the length of the stage. The axial velocity is recorded at every point in the stage

and the variable is saved as a matrix. It is possible to record the vertical velocity as a

matrix in a similar manner with little addition to computation time. This is due to the fact

that the variable is already determined to solve the governing equations. However, the y-

component of the velocity is so small when compared to the x-component that it is not

recorded as part of the solution. This will help retain memory space for the other

properties. Two independent thermodynamic properties are recorded to

thermodynamically define the fluid. One of these properties is the pressure, which does









not vary in the vertical direction. Being only one-dimensional, the pressure at each axial

location is recorded and the variable is saved as a vector. The other thermodynamic

property is temperature and it remains a function of both dimensions, x and y.

Temperature at every discretized point is calculated and saved as a matrix in the stage's

export folder. The composition of the fluid is recorded as mass fractions of each species.

Like the temperature, the mass fractions are a function of both dimensions. The mass

fraction of each species is saved as a matrix into its own file. As a result, the number of

mass fraction files saved in the export folder is equal to the number of species, N. All the

properties needed to kinetically and thermodynamically define the flow are recorded.

The only other variables saved are the pressure gradient and vertical velocity. The

pressure gradient is recorded as a scalar and the vertical velocity is saved as a vector.

Both variables correspond to the last axial position of the stage and are used as initial

conditions of the next stage.

Initial Conditions of a Stage

The program can operate once all of the parameters, conditions, and file names are

designated. Initial boundary conditions of the stage's first mini-mesh are established

before solving the governing equations. Velocity components at all vertical points in the

first axial location are required to define the momentum equation and its initial boundary

condition. The pressure, temperature, and composition in the first mini-mesh must also

be defined to estimate the properties inside the governing equations. The model requires

only one pressure value per x-location, because the pressure is independent of the vertical

direction. The result is only three scalars being required to define the pressure in the

mesh. Temperature and species mass fractions are two-dimensional and must be set for

every point in the stage's first mini-mesh. The process used to define these variables









depends on the stage number. Conditions of the initial stage or stage one are based on the

values discussed in the section Parameters and Conditions. Every other stage uses the

solution of the previous stage to set these initial boundary conditions. The program can

begin to solve the governing equations once the initial conditions are set.

Cantera creates a gas object and surface object prior to defining initial conditions.

The gas is adjusted to the pressure, temperature, and composition entering the reactor and

the two objects are connected through an interface. The gas object is created at this time

because Cantera provides a simple means to set the composition variable of the first

stage. Only the composition's string variable is needed to establish the mass fraction of

all the species initially present. Cantera can take the composition of the gas object and

return the mass fraction of every species. This is much easier than searching for the

species not present and setting their mass fraction to zero. Cantera also ensures that the

sum of the mass fractions equals one.

Stage One

The reactor is characterized by the absence of a catalytic surface at its entrance.

The catalyst does not begin until further downstream. This is where stage one begins and

the initial boundary conditions of the first mini-mesh are determined. Minimal change in

the conditions should occur over the non-reactive surface with the exception of the two

velocity components. Therefore, the initial conditions of the temperature, composition,

and pressure remain the flow conditions entering the reactor. Temperature and mass

fractions at the first three axial locations are approximated by the values entered as initial

conditions. The surface temperature is set to the value entered as a parameter. The initial

pressure is equal to the pressure of the incoming flow, and the pressure at the next two

differential steps is calculated with the pressure gradient. In contrast to the other









variables, the surface affects the velocity vector. A boundary layer develops changing

the profile of the axial velocity, which produces a vertical velocity. Blasius solution is

used to model the boundary layer and determine the two velocity components.

Blasius Solution

Axial velocity is quantified by two values at a point in the beginning of the reactor.

The singularity point is located on the front edge of the reactor, where the incoming flow

first encounters the surface. A finite value is given to the uniform velocity entering the

reactor. The velocity at this point must also equal zero due to the boundary conditions of

the velocity. To overcome the singularity point, Blasius solution is used to calculate the

two velocity components at the end of the non-reactive surface, where the first stage

begins.

H. Blasius is well known for obtaining an exact solution to a laminar boundary

layer flow over a flat plate. Blasius is able to find a similarity solution to the continuity

and momentum equations through proper scaling and nondimensionalization of the two

equations. In his solution, the dimensionless stream function replaces the two velocity

components as the dependent variable. The two coordinates, x and y, are also combined

into one dimensionless independent variable. Blasius transforms the two partial

differential equations into one ordinary differential equation. A power series expansion

or numerical methods can then be used to solve the third order, nonlinear equation. The

dimensionless stream function and its derivative are used to calculate the axial and

vertical velocity [15].

Blasius solution describes a two-dimensional, steady, incompressible flow with no

pressure gradient. Recall that the assumption of constant density is not applicable to this

model due to the chemistry involved. A pressure gradient equal to zero is also not









accurate because the flow is assumed to have pressure changes in the axial direction.

However, Blasius solution is used as a reasonable estimate to the velocity profile over the

non-reactive surface. The change in density is caused mostly by the catalytic surface, and

the catalyst is not present in the region that Blasius solution is employed. A change in

density from species diffusing upstream is possible, but the effect should be negligible.

The production of a new species with a diffusion velocity great enough to overcome the

axial velocity is needed for this to occur. The behavior of the pressure gradient also

permits the use of Blasius solution. The pressure slowly decreases as the flow moves

downstream. This produces a decreasing pressure gradient that has an initial value of

zero. The change in pressure is small and a zero pressure gradient should be a reasonable

approximation at the entrance of the reactor.

Fortunately a non-reactive surface is located in the region of the singularity point.

Blasius solution can be used to generate the velocity profile at the end of the non-reactive

surface. The main program calls on one of the subprograms, a function called Blasius.

The function imports the axial differential step size, stage length, y-coordinate vector,

location where the catalytic surface begins, viscosity, and initial speed of the flow. A

shooting method determines the dimensionless stream function and its derivative.

Because the equation developed by Blasius is dimensionless, the calculated values of the

stream function are independent of the values imported. The axial velocity vector and

vertical velocity vector are determined using the dimensionless variables with the

imported variables. The axial velocity is treated as the initial condition of stage one. The

vertical velocity is used in the momentum equation. Note that the function Blasius is









only used in the first stage, where the non-reactive surface is located. Other stages use

the solutions of the previous stage to set the velocity profile.

Subsequent Stages

If the stage number is greater than one, initial boundary conditions are taken from

the export folder of the previous stage. The program locates and loads initial quantities

with the saveFile variable. The last value of the preceding stage's pressure vector

becomes the initial pressure of the current stage. The vertical velocity vector and

pressure gradient (scalar) are loaded from the preceding stage's output files. Initial

values of the current stage's axial velocity are set to equal the x-velocity at the end of the

last stage. Temperature and composition variables are all that remain to import. Recall

that the temperature and mass fraction must be set for all three axial locations. The first

axial location is equal to the last axial location in the previous stage's matrix, similar to

the axial velocity variable. For the next two differential steps downstream, the x-location

vector of the last stage is imported. This vector along with the temperature and mass

fractions of the previous stage determine the variable's gradient at the end of the last

stage. A second-order backward-difference formula is used to estimate this gradient [19].

Values of the next two axial locations in the variable of the current stage are linearly

extrapolated using the gradient. With the initial conditions of the stage set, the program

is prepared to solve the equations controlling the behavior of the flow.

Solving Governing Equations

Governing equations are solved for three axial locations at a time. Remember that

the equations contain properties dependent on the solution. Solving one mini-mesh at a

time allows the program to update properties inside the equations prior to calculating the

solution one step downstream. For the same reason, the pressure, temperature, and









composition at these three places must be approximated before solving the equations. For

the stage's first mini-mesh, the method for these approximations is described in the

section Initial Conditions of a Stage. Approximations of the other mini-meshes are

calculated with the solution of the previous mini-mesh. Note that two of the axial

locations of the next mini-mesh are repeated locations of the previous mini-mesh because

the program moves only one differential step.

Two MATLAB functions, or subprograms, are written to solve the momentum

equation and the species/energy equations separately. These two functions are named

Momentum and SpeciesEnergy. The process of the solving the governing equations for

a stage is illustrated in the flow chart of Figure 3-1. Once the initial approximations for a

single mini-mesh are established, the program begins with the momentum equation in the

x-direction. The main program calls on Cantera to find properties embedded in the

momentum equation. Information is exported to the subprogram Momentum, which is

then used to solve the equation. The subprogram sends the solution, the axial velocity,

back to the main program. The mass or continuity equation calculates the vertical

velocity and confirms that the solution of the momentum equation is accurate. The

pressure gradient is updated and the momentum equation is solved again if the boundary

condition of the vertical velocity at the centerline is not reacted. Ifthey-velocity equals

zero, the species continuity and energy equations are solved together with the

Species_Energy function. Again, the data needed to solve the system of equations are

provided by Cantera and sent to the subprogram. This time the composition and

temperature are sent back as the solution. Once the solutions are calculated and any

iteration correction is performed, the program updates the variables, saves the data in the










export folder, and moves one differential step downstream. The temperature and

composition of the next axial location are predicted using a linear extrapolation from the

previous two x-locations. The program loops back and solves the equations for the next

mini-mesh. This continues until the end of the stage is reached or the program cannot

resolve the solutions.

IninitiaInitial Approximations (next mini-mesh) Continue until -) End of Stage
Conditions to a mini-mesh end of Stage
of a Stage I



T -e1 Momentum Ia ,
II c Solutions saved
SI Modify to export folder
(x-velocity) dp dxeotode

CANTERA
ass y-velocity 01

(y-velocity) Test Boundary
Condition
y-velocity = 0

[ Species/Energy -(comp/temp)-


Figure 3-1. Flow chart for single stage modeling.

The subprograms Momentum and Species_Energy use a partial differential

equation (PDE) solver provided by MATLAB to numerically solve the equations. The

solver numerically computes the momentum equation as a single equation. It is also used

to solve the energy equation and species continuity equations as a set of coupled partial

differential equations. The solver, named pdepe, calculates the solution of partial

differential equations with the form shown below in Equation 3-1. The gradient of the

dependent variable with respect to x is multiplied by a coupling term, c. This term along

with the flux term, J and the source term, s, are functions of the two independent

variables, the dependent variable and its vertical gradient.









a z}az a Q at z 8 z'
c x,y, x f x,y,z +s x,y, z, (Equation 3-1)


The symmetry parameter, m, along with the coupling, flux, and source term define

the PDE. The mesh spacing of the two independent variables, one initial condition, and

two boundary conditions are all that remain to solve the equation. The mesh spacing is

simply the mini-mesh discussed in the section Discretization. Function pdepe selects the

x-mesh dynamically to resolve the solution, but only reports the answer at the mesh

points specified. Strictly speaking the initial condition is a boundary condition. It is the

value of the dependent variable at the first of the three axial locations. The initial

boundary condition of the dependent variable needs to be given as a function ofy. The

other two boundaries are found at the catalytic surface and centerline. Both must fit the

form shown in Equation 3-2. Boundary conditions are expressed in terms ofp, q, andf

The flux termfis already defined in the PDE above, so only p and q are needed to

establish the boundary conditions.


p(x,y,z)+q(x,y)f x,y,z,2 =0 (Equation 3-2)


Some fluid properties, such as density, are converted into functions ofy to conform

to Equation 3-1 above. Most of these properties are functions of both dimensions.

However, properties dependent only on the vertical direction should be an acceptable

representation for several reasons. Fluid properties vary more in the vertical direction

than the axial direction and are a stronger function ofy. In addition to this, the functions

only need to represent the properties for the three x-locations of the mini-mesh. Initial

boundary conditions also need to be turned into functions ofy. Built-in MATLAB

functions spline and unmkpp generate the function representations. Properties are found









at every vertical point in the middle of the three x-locations and saved in a vector. In the

case of the initial condition, the vector contains the initial values of the dependent

variable. The function spline uses the vector to create twenty separate piecewise

polynomials of the form of the cubic spline. Function unmkpp extracts the four

coefficients of the each polynomial and saves it into a four-by-twenty matrix for each

representation. The matrices are exported to either the Momentum function or

SpeciesEnergy subprogram to reconstruct the piecewise polynomial. The coefficients

and a heavyside step function connect the piecewise polynomials inside the subprogram.

The result is a smooth function representation of the initial conditions or fluid properties

embedded in the governing equation.

Momentum Equation

Solving the momentum equation begins by guessing the pressure gradient.

Pressure at the three axial locations is determined with the guessed pressure gradient.

The pressure along with the approximated temperature and composition are used to

determine the density and viscosity. These are the properties found in the simplified

momentum equation.

9u 9 Su1 9p 9u
pu- /- u-- pv- (Equation 2-75)
Ox dy y Ox dy

Properties are determined at every vertical point in the middle of the three x-

locations and transformed into functions ofy for the coupling, flux, and source terms.

Comparing the momentum equation to the form used by the pdepe function, it is evident

that the axial velocity replaces the dependent variable, z. The symmetry parameter, m, is

zero. The coupling, flux, and source terms equal Equation 3-3, Equation 3-4, Equation 3-

5, respectfully.










c x,y,uI = pu (Equation 3-3)



fxy, \= =P- (Equation 3-4)



s xyu- =--- pv- (Equation 3-5)
Oy ) x oy

The coupling term, c, contains the density and axial velocity. This term is allowed

to be a function of the dependent variable. As a result, only the density must be

transformed into a function. The flux term,f equals the viscosity multiplied by the

vertical gradient of the axial velocity. To fit Equation 3-1, the gradient will remain but

the viscosity is represented by a function of the vertical direction. The last two terms in

the momentum equation are combined into the source term. These two terms consist of

the pressure gradient and the product of the density, y-velocity, and y-gradient of the

dependent variable. Recall that the pressure is only a function of the axial location.

Therefore, the pressure gradient remains constant at a given x-location and does not need

to be transformed into a function ofy. The vertical gradient of the x-velocity is allowed

inside the source term. The density and y-velocity product is the only element of the

source term transformed into a function. The axial velocity's initial boundary condition

is also transformed into a function ofy.

Boundary conditions at the surface and centerline are all that remain to solve the

momentum equation. Appling the no-slip assumption, the axial velocity is zero on the

catalytic surface. The centerline of the reactor is assumed to be a streamline with a

vertical gradient of the x-velocity equal to zero. Equation 3-6a and 3-6b show these two

conditions in a form recognized by the pdepe function.









u= O-u+(0).f =0 (Equation 3-6a)


= 0 (0)+(1)- p- =0 (Equation 3-6b)
ay 9y

Boundary conditions of the momentum equation are defined inside the subprogram

Momentum. At the surface, ory equal to zero, p equals the dependent variable and q is

zero. The centerline condition dictates that equals zero and q equals one. This sets the

flux term to zero at the boundary. The flux term is the product of the viscosity and

vertical gradient of the axial velocity. Since the viscosity is finite, the gradient must

equal zero, which is the condition sought.

The Momentum subprogram can now be used to solve the momentum equation.

The function imports the discretized mesh and guessed pressure gradient. Coefficients of

the initial boundary condition, coupling, flux, and source terms are also imported. These

are the coefficients of the piecewise cubic polynomial. The second-order, nonlinear PDE

is solved and the axial velocity at the three axial locations is returned to the main

program.

Continuity Equation

The axial velocity solution must be verified because the value of the pressure

gradient is assumed. This value directly affects the momentum equation by being part of

the source term. It also indirectly affects the solution by changing the properties

dependent on the pressure. Equation 2-3 is the mass or continuity equation that

calculates the vertical velocity. At the same time, the solution of the mass equation acts

as a check to the momentum equation.

First, the gradient of the density and x-velocity product is determined at every

vertical point at the end of the mini-mesh. This partial derivative is calculated with a









second-order backward-difference formula. The y-gradient of the density is found with a

second-order central difference formula with varying spacing. Once these two gradients

are found, the partial derivative of the vertical velocity is approximated with another

second-order central difference formula with varying spacing in Equation 3-7 [19].

Solving for the velocity at the next mesh point produces Equation 3-8.

(v Q(pu) 8p
-- -V) (Equation 2-3)
ay 8x ay


v l+(a2 -1)V -a2 -1
P a(a +)(y y -1)


where a (= yJ)
y -y )


a(pu) ap
ax 8y


(Equation 3-7)


a(pu) _ap
8x yJ 8 a(a +1)(y y) (Equation 3-8)
(Equation 3-8)
I P(a2l )v p-a2v-1 P
a(a+l)(yJ- yJ _1)

Equation 3-8 is used to find the vertical velocity component at the last of the three

x-locations. This new vertical velocity becomes the variable used by the next mini-mesh

downstream. The species and energy equations still use the original y-velocity for their

calculations. Once the y-velocity is found at every vertical point, its value at the

centerline is checked. Being a streamline boundary condition, there should be no flow

across the boundary and the y-velocity should roughly equal zero. If the velocity does

not meet this requirement, the pressure gradient is adjusted and the program loops back to

the momentum equation. The amount of the adjustment is proportional to the size of the

y-velocity at the centerline. A weighted correction modifies the pressure gradient. This









continues until the centerline y-velocity is less than one ten-thousandths. At which point

the axial velocity of the mini-mesh is saved or spliced to the axial velocity variable of the

entire stage. The program then moves on to the remaining two governing equations.

Now is an excellent moment to discuss the reasoning behind breaking apart the

momentum and mass equations from the other governing equations. It has already been

shown that all of the equations are highly coupled and should be solved as such.

However, that approach leads to a very problematic and time-consuming calculation due

to the unknown pressure gradient. Solving the entire group of equations until the correct

pressure gradient is found would take a great amount of computing time. Decoupling the

momentum and mass equations significantly reduces the time of the calculation. This

method does not come without its disadvantages. Separating the governing equations

creates a delay in the solution. This delay can be overcome with the iteration process

already discussed in the section Parameters and Conditions.

Species Continuity Equations

The remaining equations are not decoupled, but instead are solved simultaneously

by the function pdepe. The set of species equations are solved for the mass fraction of

each atom or molecule. The number of equations in this set is equal to the total number

of species in the model, defined as N. Equation 2-46 shows the simplified species

equation.


pU -- pD \pv +a, iMW = 1,2,3,...N (Equation 2-46)
ax oy oy oy

Mass fraction of species i is the dependent variable of the species equations. Again,

the symmetry parameter, m, is zero. Cantera determines the density, diffusion

coefficients, net gas production rates, and molecular weights found in the equation.









These quantities and the two velocity components are used to create the coupling, flux,

and source terms shown in the three equations below.


c x,y,Y, \= pu (Equation 3



f x, y, =pDm y (Equation 3-1



s x, y,Y Y = -pV +aMW, (Equation 3-1
S 9y } 9y


-9)


0)


1)


The density and axial velocity make up the coupling term. Axial velocity is no

longer the dependent variable as it is in the momentum equation. This means that the

multiple of the density and axial velocity must be transformed into a function ofy. The

flux term,f, can be found inside the parenthesis of Equation 2-46. It equals the density

multiplied by the species mixture-averaged diffusion coefficient and the vertical gradient

of the mass fraction. The flux term is allowed to be a function of dependent variable's

vertical gradient. Therefore, only the density and diffusion coefficient product is

represented by a function. The source term becomes the combination of the last two

terms of the species equation. This term equals the species mass production minus the

product of the density, y-velocity, and vertical gradient of the dependent variable. The

product of the density and y-velocity is transformed into a function representation, while

the vertical gradient is left unaltered. This y-velocity is the original vector and not the

velocity found from the mass equation. The species mass production and initial boundary

condition are also changed to a function of the y-direction.

Two boundary conditions of the species equations can be connected to the species

mass flux. At the surface boundary is a heterogeneous catalyst where species can be









created or destroyed. Assuming a steady-state model, the species flux can be equated to

the production rate on the catalyst. Species mass flux into the surface equals the

destruction rate and the flux away from the surface is the creation rate. Mathematically

written in Equation 3-12 and reorganized into Equation 3-13a to fit the form defined by

the PDE solver. Cantera determines these production rates for each species. A

symmetric boundary condition is applied to the upper boundary. Resulting in the vertical

gradient of a species mass fraction approximately equaling zero at the centerline.

Equation 3-13a and 3-13b show these two conditions in a form recognized by the pdepe

function.

8Y
S,sufaceMW = -pDm __ (Equation 3-12)


(,suaceW )+ (1). pD,- = 0 (Equation 3-13a)


Y( )Y
= 0 (0)+ (1) .D 0 (Equation 3-13b)


Parameterp equals the mass production rate and q is one for the lower boundary at

y equal zero. The upper boundary condition hasp equal to zero while q equals one. This

sets the gradient equal to zero because neither the density nor the diffusion coefficient of

the flux term equal zero. The system of partial differential equation is defined along with

their initial and boundary conditions. Before the species equations are solved, the energy

equation is added to the group.

Energy Equation

The energy equation contains kinetic energy terms defined by the velocity field.

Kinetic energy terms are in the form ofx andy gradients of the axial velocity. The









solution of the momentum equation is used for the x-velocity. Both partial derivatives

are calculated at every vertical point in the mini-mesh. A simple second-order central-

difference formula is applied to estimate the axial gradient. The vertical gradient is a

little more complicated because the spacing in the y-direction may vary. A second-order

central-difference formula that is modified for varying point spacing is used for the core

of the calculations. They-gradient at the surface is found with either a first-order or

second-order forward-difference formula for equal spacing. If the y spacing is linear,

then the second-order formula is used. The first-order equation is used if the spacing is

non-linear [19]. Although it is first order, the error should be small because the spacing

near the surface is tight. The gradient at the centerline equals zero due to the boundary

condition of the momentum equation. Equation 2-86 is the simplified energy equation

with the two kinetic energy terms at the end of the equation.

OT 8 F T + u OT
puc k-+up pc -+
X Y L (Equation 2-86)
C LY IT a 2u Bu
p1 D, -- (ho pu puv-
=1 1 By 8y 8x By

Equation 2-86 contains many thermodynamic and transport properties that need to

establish. Cantera retrieves the properties at every vertical point in the middle of the

three x-locations and saves them to vectors. The main program uses the vectors to create

the function representations of the coupling, flux, and source terms. It is apparent that the

temperature is now the dependent variable of the PDE. The symmetry parameter is zero

and the coupling, flux, and source term are listed below.


c x,y,T, = puc (Equation 3-14)
I qy )









( 9T' T ku
f x, y,T,- k-+ (Equation 3-15)


OT) OT c OY BT
s x,y,T, = VC P +P D Cv-+m -- d
S(Equation 3-16)

Y (how) pu puv-

The coupling term is the multiple of the density, axial velocity, and specific heat.

The entire expression is transformed into a function y. The flux term, found inside the

brackets, is made up of two parts. The first equals the thermal conductivity times the

temperature's axial gradient. Second is the combination of the viscosity, x-velocity and

its y-gradient. The two parts must be kept separate for the flux term to remain a function

of the temperature gradient. Thermal conductivity is turn into one function, while the

second part is turned into another. The remaining five terms are grouped into the source

term. For the first representation, the product of the density, y-velocity, and specific heat

are changed to a function of y, and the temperature gradient remains a variable. The

second term consists of a complicated sum containing the species mass fraction gradient.

This is where the coupling between the governing equations directly takes effect.

Calculation of the sum is addressed in the section Species/Energy System. The third

expression is the other sum in the equation. However, it is not nearly as difficult as the

last because it does not contain any of the system's dependent variables. This sum is

simply calculated in the main program and added with the last two kinetic energy terms.

The last three terms in the energy equation are combined into a function representation.

Initial boundary condition of the temperature is transformed into a function for the

subprogram.









Boundary conditions are established for the temperature at the surface and

centerline. The temperature at the catalytic surface (y=0) is held constant. This lower

boundary condition is shown in Equation 3-17a. The upper boundary condition is

characterized by no heat flux. Temperature of the flow is uniform when it enters the

reactor. At which point, the catalyst induces chemistry in the flow and heat production

occurs at the surface. The temperature begins to increase at the surface and slowly

expand up to the centerline. A thermal boundary layer is created and heat flux across the

streamline is zero until the layer reaches the centerline. A long distance is needed for this

to occur and the heat flux remains zero at the streamline for the short distance of the

reactor. Equations 3-17a and 3-17b show the conditions in a form recognized by the

pdepe function.

T= ,ace (T- ac)+(0). f = 0 (Equation 3-17a)

OT OT Ou
T= 0 (0)+(1). k -+u/ = 0 (Equation 3-17b)


In Equation 3-17a, p is the dependent variable minus the temperature at the surface

and q equals zero. This produces the constant value at the surface. The upper condition

is created with parameterp equal to zero and q equal to one. This sets the flux term,

which consists of two parts, to zero at the boundary. The first part is the heat flux and the

second contains the gradient of the axial velocity. A problem arises because only the heat

flux should be zero. However, the second term vanishes at the centerline due to the

boundary condition of the momentum equation. The result is the proper symmetry

condition at the centerline boundary.









Species/Energy System of Equations

The species and energy equations are combined for the pdepe function to solve.

The resulting system of (N+1) equations is shown below. The flux and source terms are

split into three parts to accommodate the various expressions in each equation.


[c] '= [fl]z+ [f2]+ [f3] [sl] Z+ [s2] + [s3] (Equation 3-18)
ax Oy y y Jy ay


where Z= (Y,Y2,...,YN, T)


Components of the dependent variable vector, Z, consist of each species mass

fractions and the temperature. Coefficients of the cubic spline generated for the coupling,

flux, and source terms are grouped together. Each component of the vectors c,fl,J3, sl,

and s3 represents the cubic spline function of that component. The other two expressions

(f2 and s2) generate the sums involving the system's dependent variables. The coupling

term of each equation is combined into one group, c. Separating this term is not

necessary because it does not contain any mass fractions or temperature variables.

pu
pu
[c]= (Equation 3-19)
pu
pUCp

The flux term is broken up into three separate collections. The first group,fl, is the

combination of the species equations' flux terms and the energy equation's thermal

conductivity expression. These terms are multiplied by the axial gradient of the

dependent variable. The Nh component offl is set to zero and replaced with a sum inJ2.

Group j3 is a result of the additional flux term in the energy equation. The first N









components of this group are zero because none of the species equations contain an

additional term.

pD, 0 0
PD2, 0 0

[f1]= [f2] dotFnl, I [f3]=

0 -1 Ou
k 0 _

Figure 3-2. Flux components in the species/energy system.

The group,J2, is a result of "the fact that the sum of all the species diffusion fluxes

must be zero" [13:227]. This fact is rearranged and shown as the sum in Equation 3-18.

It suggested that this equation "be applied to the species in excess, which in many

combustion systems is N2" [13:227]. Diatomic nitrogen is the last species listed in the

input file used in the tests. This corresponds to the Nth component in the dependent

variable vector. Note that the Nth component infl is zero and negative one inJ2. The

result is this sum replacing the mass-averaged diffusion flux in the last species equation.

MATLAB's built-in dot product is utilized to create the sum. The last two values ofFnl

are zero because the sum does not include the Nh specie (diatomic nitrogen) or

temperature.


SPDmj = dot Fn l, (Equation 3-20)


where Fnl =(pDm, pDm, ..pD, ,m0, 0,O)

The source term is also grouped into three expressions. Group sl is a combination

of the terms multiplying the gradient of the dependent variable. The second group, s2









produces the complicated sum discussed in the section Energy Equation of this chapter.

All the remaining terms are compiled into s3.

PV 0 r n -MWI
pv 0 1 1
pv 02 2
sl= [s2]= dot Fn2 s3=
pv 0
pvc, 1 ,)- PU2 u puv


Figure 3-3. Source components in the species/energy system.

The sum in the energy equation, shown in Equation 3-21, is a function of the

species mass fractions. The energy equation is not decoupled from the species equation

and the mass fractions are dependent variables in the system. To reproduce this sum, all

of the properties (this excludes the species mass fractions) are transformed into functions

ofy for each species and stored in a vector. The dot product of this vector and the

dependent variable vector gradient recreates the sum inside the subprogram

Species_Energy. The sum is then multiplied by the temperature gradient. Note that the

temperature gradient is not part of the sum, consequently the last value in Fn2 is zero.

yPCprm = dotl Fn2, l (Equation 3-21)



( pcpCDlm pcp2D2m pcpNDNm
where Fn2= -PC ,-PCp2D ..p N,0


The boundary conditions of both equations are also grouped together. Figure 3-4

shows the conditions for the system of equations. For the lower boundary condition,

parameterp is broken up into two groups, because the dependent variable is part of the

temperature's lower boundary condition.










)l,surface ~ 0 1 0 1
2surfaceMW 0 1 0 1
+ : Z+ f=0 + f=0
NsufaceMIW 0 1 0 1
Tuace 1 0 0 1

Figure 3-4. Boundary conditions of the species/energy system.

Function Species_Energy imports information to solve this system of governing

equations. The subprogram imports the discretized mini-mesh and all the polynomial

coefficients. Parametersp and q of the system's boundary conditions, which are

compiled in the main program, are also sent to the subprogram. The function then

calculates the solution of each component in the dependent variable vector and returns it

to the main program. A mass fraction less than 1E-20 is treated as error and the value is

set to zero. Solutions for the mini-mesh are spliced to their corresponding variables for

the entire stage. The program moves one differential step downstream and loops back to

the momentum equation. Recall that before solving the equations for a given mini-mesh

the temperature and composition must be defined at all three axial locations. The

program linearly extrapolates these values from the previous two x-locations. The same

process solves the governing equations for the next mini-mesh. Its solution is spliced to

the stage's solution variable and the program moves on. This continues until the end of

the stage is reached or the differential x-step is not small enough to resolve the solution.














CHAPTER 4
TESTING

A process of running the code for several cases and examining the solutions is

performed in order to test the program. Four cases are used to test the software.

Beginning with no chemistry in the first case, chemistry is slowly introduced to the other

cases. The second case involves only gas chemistry while the last two tests include both

gas and surface chemistry. Gas and surface chemical reactions are modeled at five

hundred and seven hundred degrees Kelvin. Slowly introducing chemistry to the model

will aid in locating errors during the debugging process should any problem arise. Each

case uses the input file named ptcombust.cti, which is provided by Cantera as part of the

software package. This file contains data for the methane/oxygen surface mechanism on

platinum developed by O. Deutschmann. The input file ptcombust.cti calls on the file

gri30.cti, also part of the Cantera package, to manage the gas reactions. The file gri30.cti

contains data for the optimized GRI-Mech mechanism and for this program calculates

transport properties based on a mixture-averaged transport model. Once the program

finds the solution for a given case, the results are examined. No experimental data is

available at this time to compare to the program's solutions. However the different tests

can confirm that the software produces reasonable results and is operational.

Several parameters and conditions that characterize the reactor and incoming flow

are similar for the four cases. The reactor has a radius or thickness of two centimeters

and a length of thirty centimeters. Therefore, the height of the mesh is two centimeters

and the sum of stage lengths equals thirty centimeters. The distance of the non-reactive









surface at the entrance of the reactor is given the variable name Lnocat and is equal to

one centimeter. In case one, a catalytic surface is not present and the one-centimeter

value only determines the location of the initial velocity condition found with Blasius

solution. The differential step size in the vertical direction is set at four-hundredths of a

centimeter. This mesh spacing in they-direction is not linear. In order to place more

points near the surface, the power discussed in the section Discretization is set to four.

The PC variable is equal to one so no iteration occurs. Temperature of the surface is set

to four hundred degrees Kelvin for case one and two. The initial temperature of the flow

is also four hundred degrees Kelvin for the first two cases. A mixture of air and methane

at one atmosphere of pressure comprise the fluid entering the reactor of every test.

Case One

Case one models a chemically inactive gas passing through a reactor with no

catalytic surface. The flow is essentially a non-reactive flow through a pipe or channel

with a pressure gradient. With no chemical reactions taking place, density remains the

same and the entire system of governing equations is altered. All of the equations could

be simplified for an incompressible flow and the set of species equations could be

removed all together. Although modifying the code in this way would defeat the purpose

of the test. To test the program only fluid properties are changed, while the code remains

unaltered. Turning the gas chemistry off is achieved by equating the species mass

production rate to zero. This only affects the source term, s3 in Figure 3-3, in the species

continuity equations. Creating a surface with no catalyst also exclusively affects the

species continuity equations. Production rates at the surface are forced to zero changing

the lower boundary conditions in Figure 3-4.









One stage is used to model the flow of the first case. Recall that the computational

space of the reactor can be broken up into stages. Being able to change the axial

differential step size of each stage allows the program to resolve the changing

composition. However there is no varying composition because the chemistry is

removed in this test. The reactor does not need to be split into stages for this reason. The

fluid mixture, given in mass fractions in Table 4-1, enters with a velocity of one meter

per second. Other parameters and conditions of this test run are listed in Table 4-1.

Table 4-1. Parameters and conditions of case one.
Reactor Parameters Initial Flow Conditions
Radius 0.02 m Velocity 1 m/s
Lnocat 0.01 m Temperature 400 K
Stage length 0.30 m Pressure 101325 Pa
Surface temp 400 K CH4:0.004,
dy 0.0004 m Composition 02:0.23,
power 4 (mass fractions) N2:0.752,
AR:0.014
No. of Stages 1 PC 1
dx (Stagel) 0.01 m Input file ptcombust.cti

Results of Case One

The solution should mirror that of a viscid two-dimensional laminar flow through

two flat plates with the pressure slowly decreasing. Figure 4-1 illustrates the axial

velocity profile at four different locations. As expected the axial velocity solution

resembles a boundary layer flow increasing from zero at the surface to the centerline

velocity. The centerline velocity increases to compensate for the loss of mass flux near

the surface. This is shown in Figure 4-1, where the centerline velocity increasing

downstream as the boundary layer grows. Note the overshoot in the velocity profile at x

equal zero. This is due to the program creating a function representation of the initial










velocity condition. Other than the overshoot the velocity solution is a smooth continuous

model of what is expected for flow over a flat plate with changing pressure.


0.020 i

0.018
SI
0.016

0.014 x=0

0.012 x=.1
- x=0.2 '
y (m) 0.010 -x=0.3 --
0.008

0.006

0.004

0.002

0.000
0.0 0.2 0.4 0.6 0.8 1.0 1.2
x-velocity (m/s)



Figure 4-1. Axial velocity profiles of case one.

Increase in the centerline velocity should lead to a decrease in pressure. The

pressure change of Figure 4-2 shows this to be the condition. The pressure slowly

decreases downstream from its initial value of one atmosphere. The incompressibility of

case one allows the use of Bernoulli's Equation to calculate the change in pressure. This

provides an alternate means of finding the pressure with the axial velocity and ensures

that the solutions of the program are consistent. The velocity at the streamline or

centerline, where viscous effects are not present, is used in Bernoulli's Equation. The

pressure difference calculated from both the program and Bernoulli's Equation is graphed

in Figure 4-2. The change in pressure predicted by the program and Bernoulli's Equation

is very similar and the behavior is typically found in the beginning stages of a pipe or

channel flow.










0.00

-0.02

-0.04

-0.06
p- program
P- -0.08 Bernoulli
(Pa) -0.10

-0.12

-0.14

-0.16

-0.18
0 0.05 0.1 0.15 0.2 0.25 0.3
x(m)


Figure 4-2. Pressure plot of case one.

Variations in the species mass fractions should not exist because all chemistry is

neglected. Temperature should also have minor changes for the same reason in addition

to the low Mach number of the flow. This is the case for the first test of the program. As

expected, the calculated composition and temperature remain constant. The software

produces the expected solutions for all variables in case one.

Case Two

A flow characterized by gas reactions and no surface reactions is modeled in case

two. The flow is that of a chemically reacting fluid passing over a non-catalytic surface.

The only difference between case two and case one is the presence of gas chemistry in

the flow. Cantera determines the value of the species mass production rates. Unlike case

one, these values are not forced to equal zero. Removing the surface chemistry is

achieved by altering the lower boundary conditions of the species continuity equations.

Surface production rates are set to zero just as they are in case one.









Three stages are used to model the flow in case two. Because the temperature of

the flow is relatively low, little change in the composition is expected. However, using

more than one stage will test the process involved with multiple stages. This includes the

saving and loading of variables and the smooth connection of the stages. The length of

each stage is one centimeter. The same fluid composition enters the reactor, but the

initial velocity is now half a meter per second. Parameters and conditions of this test run

are listed in Table 4-2.

Table 4-2. Parameters and conditions of case two.
Reactor Parameters Initial Flow Conditions
Radius 0.02 m Velocity 0.5 m/s
Lnocat 0.01 m Temperature 400 K
Stage length 0.30 m Pressure 101325 Pa
Surface temp 400 K CH4:0.004,
dy 0.0004 m H
dy 0.0004 Composition 02:0.23,
Power 4
power (mass fractions) N2:0.752,
No. of Stages 3 AR0.014
AR:0.014
dx (Stagel) 0.01 m
dx (Stage2) 0.01 m PC 1
dx (Stage3) 0.01 m Input file ptcombust.cti

Results of Case Two

The axial velocity calculated in the second test is graphed in Figure 4-3 for three x-

locations. Similar to the first test, the velocity is recognized as a typical pipe or channel

flow solution. The initial velocity is half a meter per second and the centerline velocity

increases from this value as the flow becomes fully developed. The presence of gas

chemistry does not appear to affect the solution of the momentum equation. Variation in

the composition is not anticipated and the velocity profile is comparable to that in case

one. The reduction in the initial velocity does remove the overshoot found in Figure 4-1.











0.020

0.018

0.016
-x=0
0.014 x=0.1

0.012- --x=0.2
x=0.3
y (m) 0.010-

0.008 .0

0.006 -_,_

0.004

0.002

0.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6

x-velocity (m/s)


Figure 4-3. Axial velocity profiles of case two.

Minor variations in the density do not change the velocity, meaning the pressure

should also behave the same. Figure 4-4 illustrates that the pressure decreases

downstream much like the pressure in case one. A difference in the program's solution

and Bernoulli's solution is noticeable and there is almost a twenty-two percent difference

between the two. The general behavior of the pressure is consistent with expectations;

however, the software produces values that are not validated by Bernoulli's Equation.

As expected, the temperature remains constant at four hundred degrees Kelvin.

Some changes in the temperature do occur but are very small and can be considered

numerical error. Some of the mass fractions also contain small fluctuations. Figure 4-5

shows the change in the mass fraction of the species methane. While some of the species

mass fractions behave oddly, it is most likely a product of numerical error. The program

has a second-order accuracy and the largest step size equals one centimeter. The







78



resulting error has the size of one ten-thousandths, which is greater than the error seen in


Figure 4-5.


0.00

-0.01

-0.02

-0.03

-0.04

-0.05

-0.06

-0.07

-0.08


0 0.05 0.1 0.15 0.2 0.25

x (m)


Figure 4-4. Pressure plot of case two.


0.02

0.018

0.016

0.014

0.012

y (m) 0.01

0.008

0.006

0.004

0.002

0
-3E-12


-2E-12 -1E-12 0 1E-12

reduction of CH4 mass fraction


Figure 4-5. Reduction in methane concentrations of case two.


2E-12









The software produces reasonable solutions for a chemically reacting flow without

a catalyst. The low temperature in this test results in little gas reactions and the mass

fractions remain nearly constant. A large incoming temperature will lead to combustion

of the fuel/air mixture. Care is taken to avoid combustion because the governing

equations are reduced based on the expectation that characteristic length scales in the

axial direction are large. The smooth connection of multiple stages is also confirmed by

the second test. This can be seen in Figure 4-4 where the pressure is a continuous

function ofx. At this point the code calculates expected values for a flow with and

without chemical gas reactions.

Case Three

A complete test of the software is performed in case three where gas and surface

chemistry both exist. As it is originally intended, the model is that of a chemically

reacting fluid flow over a catalytic surface. Cantera finds the gas and surface production

rates used by the set of species continuity equations. Unlike the previous two tests, these

values are not forced to equal zero.

Table 4-3. Parameters and conditions of case three.
Reactor Parameters Initial Flow Conditions
Radius 0.02 m Velocity 0.5 m/s
Lnocat 0.01 m Temperature 500 K
Stage length 0.30 m Pressure 101325 Pa
Surface temp 500 K CH4:0.004,
dy 0.0004 m Composition 02:0.23,
power 4 (mass fractions) N2:0.752,
No. of Stages 2 AR:0.014
dx (Stagel) 0.001 m PC 1
dx (Stage2) 0.001 m Input file ptcombust.cti

Minimal change in the composition is encountered due to the low temperature and

only two stages are applied. The temperature of the gas and catalytic surface is increased









to five hundred degrees Kelvin and the differential step size in the axial direction

decreases to one millimeter. The other parameters and conditions are similar to the

second test and all are found in Table 4-3.

Results of Case Three

At first the original software does not obtain a solution for the entire flow in case

three and changes are made accordingly. Two centimeters into the reactor the program is

unable to resolve the changing composition and the code prematurely terminates. This

problem is found to be associated with the application of the dot products in the

subprogram SpeciesEnergy. Once these dot products are removed, the program is able

to solve the entire computational space. The dot product of Equation 3-20 is replaced

with the mixture-averaged diffusion coefficient. The Nh species equation is now similar

to the rest of the species equations. The other sum, Equation 3-21, is no longer

performed by the dot product but is found by adding the function representations of each

species. These two dot products create the sums involving the gradients of the species

mass fractions. The problem is not noticeable in case one because the change in the mass

fractions is zero. This problem might be the source of the odd behavior seen in some of

the mass fractions and pressure difference of the second test.

The software is able to model the entire reactor after the corrections are made. The

velocity profiles graphed in Figure 4-6 are typical of boundary-layer growth in the

presence of a pressure gradient and are consistent with the models of the first two tests.

Again, the velocity increases at the centerline and the boundary layer grows as the fluid

moves downstream.











0.02

0.018
--x=O ,
0.016 x=0.05 -

0.014 ------.- x=0.1 : _
0.0 -x=0.2
0.012 ,-
x=0.3 "
y (m) 0.01
r*
0.008

0.006 1

0.004 -- ''"

0.002 0


0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

x-velocity (rms)


Figure 4-6. Axial velocity profiles of case three.

Figure 4-7 shows the pressure change calculated from the program and Bernoulli's

Equation. Little difference is seen between the two solutions and both agree favorably.

It is evident that after the corrections are made, the program produces reasonable values

for the pressure in case three. The general behavior is also consistent with that of the

other two tests.

Temperature of the computational space remains constant at five hundred degrees

Kelvin. Like case two the low temperature means gas reactions are at a minimum, but

the presence of the catalytic surface generates chemical reactions. The reactions produce

a slight increase in the temperature just above the surface but the change is minimal.

The chemical decomposition of methane illustrated in Figure 4-8 also seems

logical, but the values are small enough to be considered numerical error. The species

mass fraction decreases from its initial value at the catalyst and the effect diffuses to the








82



centerline as the flow moves downstream. The decomposition is not sufficient to


generate any significant chemical activity.


0.00 -


-0.01


-0.02


-0.03


S-0.04
(Pa)

-0.05


-0.06


-0.07
0.00


0.05 0.10 0.15 0.20 0.25

x(m)


Figure 4-7. Pressure plot of case three.


0.02

0.018

0.016 x=O
-x=0.1
0.014 x=0
.-.-.-. x=0.2
0.012 -x=0.3

y (m) 0.01 -

0.008 -

0.006 -

0.004 -t """

0.002

0
-3.5E-11 -3E-11 -2.5E-11 -2E-11 -1.5E-11 -1E-11 -5E-12

reduction of CH4 mass fraction



Figure 4-8. Reduction in methane concentrations of case three.


0 5E-12









The third case test reveals that the use of dot products in the subprogram,

SpeciesEnergy, leads to resolution problems in the code. After removing the dot

products, the software produces good results. However, case three does not produce a

considerable amount of chemical activity and a higher temperature is use in case four.

Case Four

Similar to the third test, case four is another complete test of the software where

gas and surface chemistry both exist. The incoming gas temperature and surface

temperature is increased to seven hundred degrees Kelvin in an attempt to generate

chemical reactions. Three stages are applied in an attempt to resolve the changing gas

composition. Parameters and conditions are listed in Table 4-4.

Table 4-4. Parameters and conditions of case four.
Reactor Parameters Initial Flow Conditions
Radius 0.02 m Velocity 0.5 m/s
Lnocat 0.01 m Temperature 700 K
Stage length 0.30 m Pressure 101325 Pa
Surface temp 700 K CH4:0.004,
CH4:0.004,
dy 0.0004 m
y 0.0004n- Composition 02:0.23,
Power 4
power (mass fractions) N2:0.752,
No. of Stages 3 AR0.014
AR:0.014
dx (Stagel) 0.001 m
dx (Stage2) 0.00001 m PC 1
dx (Stage3) 0.000001 m Input file ptcombust.cti

Results of Case Four

The program is not able to obtain a solution for the entire flow in case four. Nearly

five centimeters into the catalytic reactor, rapid change in the fluid's composition is

followed by a large increase in temperature. It appears that an initial temperature of

seven hundred degrees Kelvin is sufficient to cause ignition of the air/fuel mixture over

the catalytic surface. The software is unable to resolve the rapidly changing flow

variables after this point. This is due to the fact that the code being tested is not designed










to model a combustion process. Governing equations are reduced based on the

assumption of a relatively large characteristic length. Large axial gradients involved with

the ignition of the fuel will cause the code to terminate at the point of ignition.

The velocity profiles, Figure 4-9, behave similarly to the other test and do not show

any error prior to ignition. Inaccuracy in the axial velocity at the point of combustion is

visible just above the surface in the boundary layer. The combustion of the fuel leads to a

temperature increase in this same region. The large temperature change causes the

density, found in the momentum equation, to change rapidly leading to error in the

velocity solution.


0.02

0.018

0.016 x=0

0.014 x=0.025
....... ignition
0.012

y (m) 0.01
0.008

0.006 "

0.004 .

0.002 -- .
0 _---- -.
0.00 0.10 0.20 0.30 0.40 0.50 0.60

x-velocity (m/s)


Figure 4-9. Axial velocity profiles of case four.

The change in pressure is graphed in Figure 4-10. This plot shows that the pressure

decreases from its initial value of one atmosphere and appears more linear than the other

pressure graphs. Although combustion occurs, significant change in density is not

present until ignition and Bernoulli's Equation calculates a pressure difference nearly










identical to the pressure change found by the program. Ignition is predicted just before

five centimeters into the reactor where the pressure gradient becomes very steep. The

pressure's behavior at this point is unexpected and is attributed to the resolution problems

associated with combustion.


0

-0.001

-0.002

-0.003
--program
P--0.004
,'_; Bernoulli
(Pa) -0.005 Bern
-0 .0 0 5----------------------^ ^ --

-0.006

-0.007

-0.008

-0.009
0 0.01 0.02 0.03 0.04 0.05
x (m)

Figure 4-10. Pressure plot of case four.

Temperature is graphed at four axial locations in Figure 4-11. The temperature

continues to increase just above the catalytic surface as the flow moves downstream. The

exothermic reactions induced by the catalyst lead to the temperature increase in the

boundary layer. This variable becomes large and unstable just before the code

terminates, which is visible in Figure 4-11. At the point of ignition, the temperature

increases to over eight thousand degrees Kelvin. This value cannot be viewed as an

accurate representation of the temperature. However, it appears that the ignition of the

fuel is occurring just above the catalytic surface.












0.02

0.018

0.016

0.014

0.012

y (m) 0.01

0.008

0.006

0.004

0.002

0
650


700 750 800 850 900

Temp (K)


950 1000 1050 1100


Figure 4-11.


0.02

0.018

0.016

0.014

0.012

y (m) 0.01

0.008

0.006

0.004

0.002

0-


Temperature profiles of case four.


0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

CH4 mass fraction


Figure 4-12. Methane concentrations of case four.


The chemical decomposition of methane in case four, before resolution problems


arise, behaves much like the mass fraction reduction graphed in Figure 4-8. The mass


--x=0
x=0.025
- - ignition


i







87


fraction slowly decreases at the catalyst and the reduction effect diffuses up, away from

the surface. Significant methane decomposition occurs right before combustion ends the

program. This can be seen in the plot of the methane mass fraction in Figure 4-12.

Many species are produced once significant amounts of methane are broken down.

Two such species are atomic and diatomic hydrogen and their mass fractions are graphed

in Figure 4-13. It seems that the greater initial temperature (700K) produces the desired

effect of chemical activity. However, the large temperature also produces other species

such as OH radicals, and temperature continually grows to the point of ignition. The

code is not designed to process combustion and consequently ends at this point. As

expected the catalyst aids in the production of hydrogen and the mass fraction of both the

hydrogen atom and molecule increase at the surface. The "wiggle" found in the graph of

Figure 4-13 at the last axial position is a result of the absolute value of an overshoot.

Cantera cannot process negative mass fractions and any overshoot into the negative must

be adjusted.

002 002
0018 0018
0016 0016
0 014 0014
0012 0012
y(m) 001 y(m) 001
0008 0008
0006- 0006
0 004 0 004
0002 -.- 0002

-001 0 001 002 003 004 005 006 007 008 -002 002 006 01 014 018 022
H mass fraction A H2 mass fraction B


Figure 4-13. Hydrogen concentrations of case four. A) Mass fractions of atomic
hydrogen. B) Mass fractions of diatomic hydrogen.









The fourth test predicts ignition of the fuel just above the catalytic surface nearly

five centimeters into reactor. The greater initial temperature reveals that the program is

unable to model catalytic combustion, but can forecast the point of ignition. At this point

the software is unable to resolve the rapidly changing flow variables. However, solutions

past this point are no longer physically realistic because the assumptions made to

simplify the governing equations are not valid. Characteristic length scales in the axial

direction become much shorter in the combustion process, which result in very large

gradients. The program's resolution problems can be attributed to these large gradients

found in some of the variables being determined. The incoming temperature of five

hundred degrees Kelvin in case three is too low to produce any significant chemical

activity. While the initial temperature of case four is too great and causes combustion.

Two additional tests are preformed to better understand the temperature

dependence of chemical activity in the reactor. Both tests are similar to case three and

four, but use an initial temperature of five hundred fifty and six hundred degrees Kelvin

respectfully. The solution of the case using a temperature of five hundred fifty is very

similar to the solution of case three. The temperature and composition of the flow remain

nearly constant. The other solution, using an initial temperature of six hundred degrees

Kelvin, is similar to case four. The temperature continues to increases as the flow moves

downstream until ignition is reacted. Comparable to case four, the composition begins to

change at this point with the decomposition of methane and the production OH radicals

and other species. The point of ignition is further downstream than case four due to the

lower initial temperature. It is clear that the chemical activity is highly dependent on the

initial temperature. An initial temperature in the range of five hundred fifty to six






89


hundred degrees Kelvin is the temperature needed to cause ignition in the reactor being

modeled.














CHAPTER 5
PROGRAM LIMITATIONS AND IMPROVEMENTS

The program possesses several limiting characteristics when modeling a reacting

flow. Calculated solutions are second-order estimates due to the finite difference

equations. Error from these estimates could propagate into the governing equations

causing inaccuracies in the calculated solutions. The software uses a mixture-averaged

transport model in order to minimize the time needed to solve the system of equations.

The temperature at the catalytic surface is held constant. Initial conditions of a mini-

mesh and fluid properties embedded in the equations are transformed into smooth cubic

spline functions. Errors are undoubtedly produced in this process and rapid changes are

not converted to smooth functions very well. Consequently, this program cannot model

past the point of combustion and is only physically accurate for a relatively slow

reformation process.

To improve the program, the mini-mesh could be enlarged to include more than

three axial locations and the use of higher-order finite difference equations would be

possible. Increasing the size of the mini-mesh worsens the effect of the delay discussed

in the section Parameters and Conditions and iteration would probably be needed. The

code could also be modified to support a multi-component transport model. Both

changes would improve the accuracy of the solution but greatly increase the computation

time. The lower boundary condition of the temperature could also be modified to

represent a more realistic adiabatic surface or a surface with heat transfer.














CHAPTER 6
CONCLUSION

A program is created to validate new surface mechanisms of heterogeneous

catalysts. The adaptable program models a chemically reacting flow over a catalytic

surface. The catalytic reactor is represented in two-dimensional Cartesian coordinate

form with negligible body forces acting on the fluid. The flow is characterized as a

steady, low Mach number, boundary layer flow of a Newtonian fluid. The principles of

mass, species mass, momentum, and energy conservation are expressed mathematically

and simplified into the governing equations. The model is constructed by numerically

solving the system of coupled partial differential equations. The code, which consists of

a main program with three subprograms, is written in MATLAB and uses Cantera to

calculate chemical properties based on a mixture-averaged transport model. Allowing

Cantera to manage the chemistry independent of the main code allows the program to

remain flexible with the varying reaction pathways. Four different cases are utilized to

test the program. Calculated solutions from each case are examined to confirm that the

software produces reasonable results and is operational. The software is found to predict

the point of ignition in the fourth test where the initial temperature is great enough to

cause catalytic combustion.

Calculated values need to be compared to experimental data to truly determine the

accuracy of the program. If the comparison between experimental data and the model

reveals error in the program, improvements could be made to the code. Sacrificing

computation time for accuracy might be necessary. Once the solutions of the program