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30JUn'48 MDDC990 ..4. H. I. ,. .. I *.. qf r UNITED STATES ATOMIC ENERGY COMMISSION OAK RIDGE TENNESSEE HEAT TRANSFER IN SPHERE BEDS by Frederick Johnson Robert Bentley Robert Maurer ae J : ,, ,, ,,, , H .A :. F *%.... R"~~.:... ,i *" ,, i ;'W :!: .*SA .A S ..... I r*.: ": **4 "** S:, ** ** i *.flr * : i. i".* . .r. ::. .. ,.. .. ,: **:': " .. .i ."i: ": ..iF E '. . i .". ** x. :: :"i' *,  ., ,. ; . . H .fl*:: ": ., *; ", p,,,, * ..." .. "...... .". . ,,A S i ," :fl ,^ " ,,S .*"'A .. ** * .* . f' ,z "":: x. il, ifj . ,, ^;K~~f,, , .. . SLH r a. ,~.... a: :: . ; .^ :. .:: .*.1 . .. ". ... . Ks i ,* . ,,' ,,, ,,, published for use within the Atomic Energy Commission. Inquiries for additional copies and any questions regarding reproduction by recipients of this document may be referred to the Technical Information Division, Atomic Energy Commission, P. O. Box E, Oak Ridge, Tennessee. Inasmuch as a declassified document may differ materially from the original classified document by reason of deletions necessary to accomplish declassification, this copy does not constitute authority for declassification of classified copies *may bear the same title and authors. Date of Manuscript: July 4 1945 : Document Declassified: May 19, 1947 .. .... .II flYnrr LIr~~ a ai al Inr. S 0 of a similar document which . a 4< Digitized by the Internet Archive Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Fou t t t~t i t^.' .*s ' ^f *^>0 ndti* ^:i3 ^. F. ' >4> i~lll' '* A^.K KKKKKKK I I MDDC990 :.. : * " I .* I *: : . :"".. "i ." . mIx:. " i : . e. .. . .... : .4 . ki.. ". .*. i .. ..h * *9:: I.. 4. ...... I .. y"i ** *"y *;.:4." . i .. . .: .. . ....... . i.....i " iiiJ ii*.". . * 4:* ..." *:" : H..i::. .. . iE" "h *FK ME :* " .ii" .' .1.: .* . .* .: .S fi .: . * 1 a mm : ABSTRACT The rate of solution of benkoic acid in water has been determined for a system consist ing & a bed of acid spheres with water flowing through the interstices of the bed. It is pro posed to use the results to estimate the heat transfer between a sphere bed and a liquid coolant. The equation proposed for this purpose is = 2.36 x 101 Re0"3 Sc"06 where St, Re and Sc are the Stanton, Reynolds and Schmidt numbers. 4r I. ~. V S..  %* 'I ilL' EL . .5..... V ~ ~< ~<> MDDC990 HEAT TRANSFER IN SPHERE BEDS By Robert Bentley, Frederick Johnssn, Pd Roert Maurer INTRODUCTION 4:" ~ This report represents an attem to estimatethe heat transfer properties ed More specifically the problem is as follows. A bed of metal or oxide sphereswith which bt is being generated, is to be cooled by water'movimgthrough the interstices the bed. For design purposes it is necessary to know the flux of heat from the s] the water as a function of the temperature difference between spheres and water a function Of the water velocity and bed geometry (See MUC GY31). The term, etry, includes such factors as sphere diameter, bed porosity, and overall bed da tmetew.i The direct measurement of heat transfer coefficients forsach a system is a dif and tedious task. Mr. Young suggested that preliminary estimates might be made by the measurement of material transfer coefficients and the use of this data to calculate heat transfer. THEORY The heat tranter coefficient his defined (see MUCGY28) by the equation U where F. is the flux of heat across the solidliquid interface, expressed in calories per square centimetersend. AT is the difference between the temperature of the interface and the average temperature of the liquid. A transfer velocity Hn may also be defined by the equation = HrAC A where A C is the dierenee in heat ecmaentrati corresponding to the temperature dif ~Th~ 0 0 0 0 0 0 4o~ 0 0 0 0 7 flr~ A. ference AT. The heat ccentrati C is defined by the *0 eqcratic c PZT rnd the relation between hI, and H1 is = e/Pc fhe:specfie heatof the liquid is c, and Pis its density. For the analogous situation of a material dssolvingrin a liquid stream from a solid surface, the e quafl 0x8 ,! . ****: = B1~F 6T H.: I *~I  Si.  H I.. H * A It': * ic 4. I. 3 MDDC990 Fm = ELmAC may be written. Here Fm is the flux of material across the solidliquid interface, expressed in grams per square centimetersecond. The difference between the concentration of mater ial in the liquid at the interface and the average concentration of material in the liquid stream is AC. It is assumed that the film of liquid in contact with the interface is saturated with the material in question. ma .... The transfer velocities for heat and material defined by equations (2) and (5) will be iP. functions of the properties of the system. If it is assumed that the pertinent variables are i:::'.the viscosity, density, and velocity of flow of the liquid, the diameter of the spheres, and .i!.:i the diffusion coefficient for heat (or material) in the liquid, then a dimensional analysis i':JIiadicates that the relationship between these variables and the transfer velocity may be ::.:~*: written in the following form. *:.: il "M"* I. ...**....* H:: S. t'S Ii The Stanton I 1 the Schmidt .. i*,v : ." * " : .. . .. : I. ::. * B ij=="= gp = *p* ru* ..:, * i3 .: 11 .:* 11: '. *v .' **,'" .. ZiiSl: ..  =====i= ,F ' ,".iH. number, St, is (H/V). Re is the modified Reynolds number ( V P d/f). Sc is number (p /PD). In the above definitions V = velocity of flow of the liquid d = diameter of the spheres p = viscosity of the liquid D = diffusion coefficient of heat (or material) in the liquid 9 If one works with a given fluid and dissolved material at a fixed temperature the Schmidt number is a constant. The diffusion coefficient for heat in a liquid may be related to the  thermal conductivity of the liquid by the equation. D = k/Pc (7) . The experimental problem is to determine explicitly equation (6). The assumption is Then made that the relationship between the Stanton, Reynolds, and Schmidt numbers which is obtained experimentally for material transfer may be used to calculate the heat transfer velocity if the diffusion coefficient for material is replaced by the diffusion coefficient for heat in the Schmidt number. EXPERIMENTAL Benzoic acid and water were chosen as the materials to be used in the experiment. llBI ,... ^s. 'T llMDDC T^:** wG90;. Benzoic acid is slightly soluble in water and the> concentration is easily determined b titration. The solubility6 of benzoic acid in water has been determined as a function of temperature near room temperature. The diffusion coefficient of benzoic aid has been determined at Spheres of benzoic acid were cast from C.P. material in a brass mold. TI were obtained by having the acid at 140C and the mold at room temperature. Linwa~r >0 beWre The cast spheres had a smooth, white surface. The average diameter of the spheres was 0.635 inches. ... ,, .. ^ _ The form of the sphere bed is shown in Figure 1. The spheres were held in a cylinder f galvanized iron closed at top and bottom by wire screen. It was found that the porosity of the bed, which is the fraction of the bed volume which is void space, could not be varied but was fixed by the diameter of the ed. legt wer exmnd n" nhdimtrbd 8Ice I egh a sd length were examined. One 8 inch diameter bed, 18 inches in length, was used. I cerning the beds are given in Table 1. The velocity of flow of the water through the bed was regulated by the head of the area of the orifice, 0, at the bottom of the bed. The watervelocity, Two 4.5 inch diameter beds 14 and 21 inches in 5 inh eer bds 1 )ata con tat Tfn^^ilBy1' N "' N~ JBg ^ kk f ^^ ^^^ ^H ^^^ H^f^ V, is defined as the quotient of the volume of water passing through the bed per unit~ime and the cro area of the cylinder containing the bed. The volume of water passing through the unit time was measured by weighing the water issuing from the bed in a known in water was used in all the experiments. S i c t h w a e v e o i y de o a e h cm d ub r e emn t o SSince the water velocity, V, does not e dependence of the transfer velocity, Hm, Stantop number and the Reynolds number. ceded in the following fashion. Water, pa collected and weighed. From this data the SS enter the Schmidt number, a determination upon V, determines the relationship betwe The determination of Hm as a function oi missed through the bed for a timed interval, e water velocity, V, was calculated. sectional Swas The concen tration of benzoic acid in the water after passage through the bed was determined by titra tion with 0.01 molar sodium hydroxide, using phenolpthalein as an indicator. were made on the tap water used in the.experiments. Blank titrations The total weight of acid which dissolved into the water stream was calculated from the concentration of acid and the total weight of water. The flux, F, may then be calculated from the above data, the known total area of sphere surface, and the time of contact of the water with the bed. The concentration of dis solved acid was in all cases negligible by comparison with the solubility of the acid so that the concentration difference A C was taken as the solubility, ot the acid in water. , A sample of the data from a typical experiment is given i at the temperature of the water, STable 2. In Table 3o ledted a summary of all the experimental data. A lgtog plot o the transfer velocity the water velocity is given In Figure 2. rom this data one qbts the equation r suif i i F ;1 ';:E ",; ii """I,; i " """ i :. * 0.1 5 MDDC990 Hrf7.1 x 104 V0.70 for Hm and V. In experiments of this type it is difficult to determine the functional dependence of the Stanton number upon the Schmidt number. This is so because none of the quantities which enter the Schmidt number may be readily varied. Heat transfer experiments upon liquids flowing in pipes indicate that the Stanton number is inversely proportional to the Schmidt number raised to the sixtenths power1 I .Hii~~ Li. Er.:!. I:,, L CA'. I. . I... ~r .i~, in.. ... :~H. CU .iH~ ~ * :1::. . *bt!s ~.* 1~ I.. S H4.~. I!" *. IA K. L 1* '~1E I' . Experiments upon heat and material transfer from solids to agitated liquids indicate an inverse square root dependence of the Stanton number upon the Schmidt number2 . For flow of liquids normal to cylinders the Stanton num her is, inversely proportional to the seventenths power of the Schmidt number1 .We have :assumed that the Stanton number is inversely proportional to the Schmidt number raised to the sixtefiths power and have rewritten equation (8) in the form St = 2.36 x 101 Re 0.3 Sc0.6 Since the data for the three beds, of varying height, diameter, and porosity are fairly well represented by the single equation (8), there is reason for belief that these variables are not of primary importance. I I DISCUSSION The assumptions involved the use of equation (9) for the calculation of heat transfer The assumptions involved in the use of equation (9) for the calculation of heat transfer velocities must be kept in mind. The first assumption concerns the manner in which the Schmidt number enters the equation. There is little doubt that the inverse sixtenths po is approximately correct. A better representation of the behavior of the system might be obtained, however, by a change in the exponent of the Schmidt number of +20%. Such a change would be of some importance since the diffusion coefficients for heat and benzoic Said in water differ by a factor of approximately one hundred (See MUCGY28, p.3). In passing by the use of equation (9) from material to heat transfer, a 20% change in the ex ponent of the Schmidt number changes the calculated heat transfer velocity by a factor of approximately two. If the results of the present work are compared with the results of other workers, it wer is found that equation (9) predicts smaller values for the heat transfer velocities. FurasS  S MDDC990 have arrived at tIh following e4ations for heat transfer from :+ : ' F ur Ha a Saunders and Ford O.8 Rp~'O.5 SC'06 .p"~0. i each case, the exponent of the Schmidt number has been assumed. In this conr is worth noting that the Schmidt number for heat in gases is of the order of unity iect it ai or benzoic acid in water is of the order of 1000. This large difference in the magni Schmidt number may make a comparison of the above equations with our results The Schmidt number for heat in water is approximately seven. In addition the da have been taken for Reynolds numbers of the order of 200. The work of Gamson tUde ofltk. Lta o nss and hi coworkers indicates that this is in the transition region between streamline and turbulent flow. Extrapolation of equations (10) and (11) into the region of turbulent flow not permissable. In the present work, the range of Reynolds number which w extended from 200 to 3000. Gamson, working with heat transfer from a sphere bed to a gas, found the = 1.064 Re 0.41 g0.67 for Reynolds numbers between 350 and 4000. The exponent of the Schmidt nun is there.fore Is covered (12) ber was @XAK K^ K*vv ^^^^^K X K K ^ K ::..f .: ^ iKKKIK KX KK KK ___ M ^ ^ Kt *_^K y^x ^ xxx^xx j~~~k ~ N B!k^i ^U^Hk^B iUy^UB^^U^vXv^ (12) < .:;: : K1 KK KK^K"KKKN KK S *' ".  be w as ^ *'. 1 assumed. Equation (12) leads also to larger heat transfer velocities than our results do. The work of Gamson was performed upon very shallow beds which may account for the discrepancies. Tht~ V ~ Th ~< ~vv~ Sand Saunders and Ford4 gases to beds of broken solids (10) ~~21 ~irl :',r,:: MDDC990 TABLE 1 ..H!*H ; I' W r.H*r~ .1 * I ~A ~HI r'" 4. H H I ~1j*~~~ H. ~ 'hi!.. 1=.~ I. 'I.. *H~~HH .iic.. Ft I!.. un'.". 'iW~'. 1I~ S. Sphere Bed Data Bed 1 Diameter of Sphere Bed Crosssectional.Area of Bed Number of Spheres Area of Sphere Surface Height of Sphere Bed Volume of Spheres Volume of Sphere Bed Porosity 1L4 102.6 1,002. 8,010. Bed 2 Bed 3 20.3 cm 102.6 1,416. 11,580. 322.1 cm2 4,330. 35,400. 45.7 cm 2,125. 3,647. 3,105. 5,460. 95,00. 14.720. 43.% ; jj j" MDBC 90 r* arm C ' I^o S. f* C Pt bar* 0~~ w a baa a. Is 4* Os. *~0 a. $~ CobB ~ $4%*~* g o  3 Ac 5.~ ~ .a. a * .o~C* a  a g~ It e~00 000 1' IC cbs a *~ * 00< 4 ~Th >1>0K r>i'4 :m A> @4<> Th. 0> 0 ~ t@ ~<>< 4<0<: 0 0 0 0 0 0 InM C 0 '4 A * 'V 9  MDDC900 TABLE 3 Summary of Data Fx 106 AC x 103 H xl04 Bed 1 A'.!' A hi' * I ~ " 'H.. ..h. A * ...H~ tL>H U a.... rye,. V btc.. 1.69 2.49 6.86 8.53 8.79 11.97 16.44 1.07 1.95 4.50 6.75 7.95 14.2 21.1 Bed 2 1.25 4.60 6.71 13.6 15.4 Bed 3 1.07 2.32 4.44 8.44 14.7 Diffusion coefficient of benzoic acid in water D= 1.lx 10o5 cm2 at 250C (6) see = 8.3 x 106 cm2 at 15C (Calculated; see CP2883) 1.14 x 10 poise at 15C Calculation of Reynolds and Schmidt numbers performed for a temperature of 15C. 4. V r ~~III~I 40  WIRE SCREEN S*E b REEN wmrK :sE2REEN S*:k. *f J. *C *. < Cf aS  *. SMDDC 900 S^M~s <^l 'Q~l '*'V .RESERTOIR VALVE GALVANIZED IRON. CYLINDER . BENZOIC ACID SPHERE BED RECEIVER **** * . a* . * 4 iR^I? ~ ~ 1 /W 4 4?~ 4 Ji*r #9 j> C ~Th 0 t >0 V ~ThTh 21 #4 Figure 1 d ~ "" ;; ~;,;; MDDC990 Figure 2 7<  12~ ~ MDDC990 APPENDIX Hatrnfr"""" crs se.n a SHeat transfer to liquids flowing in smooth walled pipes of circular cross section has been extensively investigated and for the region of well developed turbulent flow jRe> 10, the following equation is well established. v:5:h 0.2 S0.6 Material transfer velocities were measured in a. circular pipe as a function of water velocity to determine if an equation similar to (13) existed. Figure 3 (a) is oi the benzoic acid pipe used in this experiment. A brass core was centered in a by means of end caps and benzoic acid cast in the space between the core and thi The brass core was then withdrawn. Two such sections were used The water was led from a reservoir to thu acid pipes by means of a brass pipe approximately six feet long and of the same transfer velocity were the same as in the sphere bed experiments. LI Table 4 is a IMro irass tube *^ i*. h i ' K KKKK KKJK/KK K..: aL SiGJ. =fi  )rass tUbe Souter " in the at&Z e benzoic internal aEthe summartt UA' i~C'/^lic'^^ ***^*V p = ^^^^ ^ ^ ^ ^ ^ f A A r ^^ H / A K ^ V : : ^A^ . K K KK^KKK.KK of the data. Solution of the benzoic acid in the water during the experiment increased ap preciably the diameter of the pipe section. Assuming that the change in diameter was constant for each run, a corrected diameter was used in computing the Reynolds number and surface area for each experiment. In Table 4 are given the initial and final areas the benzoic acid surface. Figure 4 is a loglog plot of the material transfer velocity versus the linear water velocity. The straight line of Figure 4 represents the equation = 1.23 x 104 V0"8 Making the usual assumption concerning the dependence of the Stanton number upon the Schmidt number, equation (14) may be rewritten as V 21 C St= 2:04 x 1Q2 Re02 Sc"0"6 Cbmparison of this result with equation (13) for heat transfer indicates satisfactory agree ment. This result lends considerable weight to the proposed use of material transfer data for the estimation of heat transfer performance. It is rather surprising that the data of Figure 4 fit a straight line plot of slope 0.8 as well as they do. For Reynolds numbers less than 10,000 the heat transfer data indicate a greater slope. A partial explanation may lie in the fact that in the course of solution of C V 4" 1< 9A1L 4<4,0< Ut acid the surface was appreciably roughened. The excellent agreement of equations (13) and (15) must therefore be considered somewhat fortuitous. Thi~a& = 2.30 x 102 Re brass shell. rangement illustrated in Figure 3 (b). diameter as the acid pipe units. The details of procedure for the determination c, >" *** *****^***^ "i.w:""i. 7 tt. Il H n.: I ~:! I'. MDDC990 TABLE 4 Summary of Results for Pipe Experiment 0.95 cm Cross sectional area of pipe: Total Length of acid sections: Initial area of acid: Final area of acid: I a. Mc' 3:: V .709 cm2 58.5 cm 174.2 cm2 237.1 cm2 s 106 AC x io3 Hz io4 Re (PVd) Re (  sec :1.90 .8.25 .1.00 2.53 3.98 15.55 26.20 31.01 53.20 73.10 110.0 113.4 86.60 72.10 43.10 29.10 15.41 123.3 188.9 2,390 3,350 5,260 11,600 17,900 25,700 28,000 18,800 14,100 7,410 4,930 3,500 147.9 Diameter of .pipe: I a MDDC990 TO RESERVOIR m.  .. :a END.CAP BRASS  BENZIC ACID *'4' RBEU HOSEK a .  jC!~;~ BRASS PIPE A B CLAMP FI~pne 3~ 4 Th< 4> >0 ~ >2> >~ >~0+ 4 >~> ~< 4 vs t~ >4~~ 0 ~t V 0~ >~0<~ 1< 4 ~0 9<  yrE~~T VENTI C:, F,* II : 15 MDDC990 p^.. i. Hi.. . 1*' a i . >i.: Materiul Transfer in Pinps Figure 4 Th V~" 16  MDDC990 REFERENCES W. McAdams, "Heat Transfer" , (1942) A. Hixon and S. Baum, Ind. Eng. Chem., 33, 478, 1433, (1941) C. Furnas, Ind, Eng. Ehcm., 22, 26, (1930) Saunders and Ford, Jour. Iron and Steel Inst., CXLI, No. 1, 297, (1940) 5. B. Gamson, G. Thodos, Chem. and 0. Hougen, Trans. Am. Inst. Eng., 39, 1, (1943) 6. A. Hixon and Wilkens , Ind. Eng. Chem., 25, 1196, (1933) uK rA a tv~m. A ~ :*" ^ ^ xxn x ." V E Th xM xj x a; :, , ,Ir I 1 I II? hi 1~ UNIVERSITY OF FLORIDA l il 111111111111 lll ll ll 28ll lU lll 3 1262 08909 6928 