Effects of heat-capacity lag in gas dynamics

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Title:
Effects of heat-capacity lag in gas dynamics
Alternate Title:
NACA wartime reports
Physical Description:
34, 11 p. : ill. ; 28 cm.
Language:
English
Creator:
Kantrowitz, Arthur
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Carbon dioxide   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: The existence of energy dissipations in gas dynamics, which must be attributed to a lag in the vibrational heat capacity of the gas, has been established both theoretically and experimentally. The flow about a very small impact tube is discussed. It is shown that total-head defects due to heat-capacity lag during and after the compression of the gas at the nose of an impact tube are to be anticipated. Experiments quantitatively verifying these anticipations in carbon dioxide are discussed. A general theory of the dissipations in a more general flow problem is developed and applied to some special cases. It is pointed out that energy dissipations due to this effect are to be anticipated in turbines. Dissipations of this kind might also introduce errors in cases in which the flow of one gas is used in an attempt to simulate the flow of another gas. Unfortunately, the relaxation times of most of the gases of engineering importance have not been studied. A new method of measuring the relaxation time of gases is introduced in which the total-head defects observed with a specially shaped impact tube are compared with theoretical considerations. A parameter is thus evaluated in which the only unknown quantity is the relaxation time of the gas. This method has been applied to carbon dioxide and has given consistent results for two impact tubes at a variety of gas velocities.
Bibliography:
Includes bibliographic references (p. 31-32).
Statement of Responsibility:
by Arthur Kantrowitz.
General Note:
"Report no. L-457."
General Note:
"Originally issued January 1944 as Advance Restricted Report 4A22."
General Note:
"Report date January 1944."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

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University of Florida
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oclc - 71257458
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RETURN TO
Dept of Aeronautical Engineering
ROOM 5 ARMORY
UNIVERSITY OF MINNESOTA


ARR No. 4A22


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS






WARTIME REPORT
ORIGINALLY ISSUED
January 1944 as
Advance Restricted Report iA22


EFFECTS OF HEAT-CAPACITY LAG IN GAS DYNAMICS

By Arthur Kantrowitz


Langley Memorial Aeronautical
Langley Field, Va.


Laboratory


WASHINGTON


NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


. L 457


DOCUMENTS DEPARTMENT


_


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Digitized by Ihe Inlernel Archive
in 2011 with funding from
University ol Florida, George A. Smathers Libraries wiLh support from LYRASIS and the Sloan Foundation


hltp: www.archive.org details effectsolhealcapOOlang









iATIOI'AL ADVISORY COi;TTTrTTE FOR AERO.AUTICS


ADVAIUCE R3STRLICTD R:?PORT


C'FE'CTS OFP HEAT-CAPCITY L".C Ili CAS DYIITATCS

By ,rthur Ksntrowitz





The existence of energy dissipations in gas dynanics,
which must be attributed to a q'r in the vibratjonsal _henn
capacity of the- ras has been establis-hed both theoreti-
cally and experimentally.

The fl,',i about a very s.-'all i-.:,act t, be is discussed.
It is shown that total-;-ead defect- due to ,ieat-capacity
lag during and after the cc'c-.s ion )f tle g'as at theV
nose of an impact tube are to be a.-tic ipted. Lxperi-
ments quanti t'.Lively verifyin,, these ant.cipations in
carbon dioxide : are d'.sci;use A C 2,e.;-l tleo of the
dissipations in a inre eiier'il lo'.' r.lrm l developed
and applied to s-ie special c.*.ses. It is rjoirntcd out
tjiat energy ,lssliat 'C 3 dcle to this e.-'fct jare to be
ant- ci :.te, in turt', e. Diipslr -ions _f ti C ;:ind night
also inc~ odrce errr- in cas'.s in Jhich the flo\: of rne
gas is used in an al.te-p)t t-o ir'ulate th { fli .: of another
gas. UTifortura-tc-l-'i, tue re-laxa.ioi t s of t
gases of EnginceriLcg i:.i.- r'tance have not been studied.

A ne, method of re-tss.rinr t-he- relaxation tine of
gas3s is intr .- n ':.-ch the total-.radc defects cb-
served with a ?sec.'orili"' ha'icd .ir.pact ti'Le are co.napred
with theoreLical 2'oi. ilerations. a parar.ster is t.ius
evaluated in hic.i the only ,ir.'r-n'\n qua:.ltit is3 the re-
laxatini tine of thec -a?. T'.1 msthrld :.as bcee applied
to carbon dio.:l de 'nd .,as ,iven c';nsist r.nt re..ults 'cr
two impact tubes at a variety of -gas velocities.


I'TT GDJ-CTIOIl


The heat content of gases 13 nri'-arily three fcrms
of molecular riechmnical enerfy. First, there is the
translational kineti.c encer-'y 0"' ich is tT, where I








2


is the gas constant and T is the absolute temperature.
Secondld there is the rotational kinetic energy<, For
all gases near or above room temperature, the n1 rota-
tional degrees of freed.on involving moments of inertia
due to the separation of atomic neuclei have energy states
cloce enough together that the rotational internal energy
is close to the classical value nRT. The third prin-
cipal form of internal energy is the vibrvtional energy
of the molecules. If the frequencies of the normal
mr'..co of vibration of the rmolecule are known (say, from
spectra), the vibrational heat capacity car be computed
bte the Pethods of statistical t..echanics. (See, for
e'arrple, reference 1.)

The possibilit- of dispersion and absorption of
sound diau to parts of the heat capacity lagfing behind
the raid temperature changes accoI.panying the propaga-
tion of a sound wave in a gas was first r'iscucsed theo-
r.tscally by Jeanr and Einsteln. Dispersion and ab-
sorftion in carbon dioxide ob-crved by Pierce v'erc shown
by Herzfeld and Rice to be attributable to laFring of
the vibrational heat capacit-y of t.he gas. Kescer was
able to account quantib t.tivl:l- for dispersion and
absorption in Co e.nd oxy-oen on the assumption that the
vibrati-,nal cat capacity la-ged.

The disper.-icn and absorption of sound in several
gares have been inve ti-atcd and r. fairly complete
bibliograp,.n is availabic in reference 2. It is found,
in generral, that disp:r.ion and an:-orption many times
larger than thse attributable to vicozity and heart
conduction are to be c:.rpected in gases with vibrational
heat capacity. These cffecs t can et described by rela-
tions such as those rgivn biy Tnesc'r and can be attributed
to the vibra.tional heat cap .rity of the gas.

All the rc:asurrm,:-nt.r c. dI-pcrsion and abs-)rption
h've demonstrated that most impuritic markedly reduce
thc rcla:;ati.cn timn of a 07 c; for example, Kneer and
7.nudscen (-fterE.ncev 5 and 4) concluded that the adjust-
ment of the vibr.-.tional heat capacity of oxygen vas
depcnd-ent entir.-l on the action of impurities.
Various e:-.mcrime-uts -vith CO4 hv-.e r.hown that, at room
temp-raturr', collisions with \ltcr molecule : are 500
times as effective as collisions with CO-molecules in
exciting; vibration in C02-molecules. This strong












dependence on purity has :ro.Duccd rr-at discre.panlcies
among the relaxation tinev neas.rreci by the various vwor:ers
in this field. There ,has been much better agreement
aronr the measurements of te effe': ienless." f 'im[.urities.
Sonic reasurenenits in CO.G are discussed in appendix 41.
I quanttun-rechanical tlheory of relax.:.tion tinres developed
by Landau and Teller is discus--e, in appendix 3. Tvw.
conclusions, which are verified b; t;.e so.i,'c work in C02,
are important to the present -rper: (1) All the vibra-
tional states 7f a single normal .,ide adjust v.'ith the same
rela:atio; tile and (2) the loo.rith-:n f the relaxation
time (expressed in m.,lecu.ar collisions) is proportional
I
to 7 Verification of conclusion (i) is presented
in figure 1.

Dr. Vannevar 3Bu'h helped to initiate tlis -cork by
asking the v.riter a etimulatin.r que-tiocn. The auth-or
als is very .-r.ateful to Prcfes~o.r 7. Teller for helpful
discussions.


7:F".C'7CS T" r2J .""..:':SC


In the flw.':: f gases ato'lt ob-te.cl 7., conpressio.rns
and rarefact iZoS. acor:panied b ter -:,c.rature cha ntes
occur. T7e- ti.re .i '"ilc': t"esc te,!perat'ire chaines
ta'e place is contrilledl b- th': di,-ensi ns cof the ob-
stacles and the velocit" of flo'w. If these tire inter-
vals are comparable w-it' cr sn'-ter a t-.an tlhe time re-
quiredc for the Cas to ab'-.orb its full heat c-pcity,
thie as will depart frI. its equilibri;um .iZt'...'n of
ener-gy. In this csE the transfer of ener:rv ^or-.:1 parts
of tl.e heat carp.acit.y; tiiat ha.v mor'e than th .eir rp.are to
parts that ha"e less tan t:a -.eir share :.il be an irre-
versible process and ,'ill in-.crea e tre Ei.tr-op, f the ras.
If the time intervals involve are comnrou.ble with the
rela::ation time of the pas, this increase in entropy can
be used to measure the rela::ation time of the ;-s (refer-
ence 5).

Turbine-w.orkinp fluids such a- stean, air, and ex-
haust gas have aprreciablc vi'-rati-nal heat capacity at
hirh temperatures. If these gases have relaxation tim-es
comparable with or bshnrter.'than the intervals during
which temperature chianes,. occur in th- ,gas, losses





. -


4


sttributable to heat-car:acitr 1 v -,l.st be anticipated.
A r-urFh esti.iLoP '-as indic-.tr:. tPat, unless the rela:a-
tion times of tle .'orl-inr fluids acre very rhort, the
losses at hi.i te-.peis.tures d-l to .eat-capacity lag can
be conpaseble vwth the losses due to skin friction.
U',fortunately, no measurement: of the relaxation tines
of the u?,sial turbine-v ork :. nr 'luid rs e:ist.

Various persons have prco-cd, in vrnd-tunnel tests
a:-l in tests of rotat:.in r,'c ..n:Per,-, the substitution of
,gaes that ihave Frol'ertirs en"bliln tests to be made more
cn-vcn ienfltl at a riven l'c-h nri..ber_ or Reynolds number
tha.n with the actual vorl,:r.r -luid. TI s.ich cases,
care nmst be takein to er.svre t':at ar error du'. to dif-
ferences in heat-cj ac it-laej heh-avior of t:he fluid used
and the vor:ir.inp fluid is rnot 'iLtro,.uced. For example,
.according to a roubh calcui.at-.or., a win, in pure C02
mi.'ht have a drag coefficie-nt t"'ice as large as the same
winr in air at th-e seae ":ach n'.:ocr and e:'nolds nu..ber.

In the follow;lnr: discur'..on, the e:ristence of these
dir.sinaticons in i-as d:'n!'.cs '.s Jeronr-trated and a gas-
d :r!i-a-ics ret-Iho of reacirin,- t-e rc.la-ation tines is
develocpd. 'he appllcatio. of th.is mret-.od to the meas-
urL.'en-t of the relsa:xation tin %. of J ss of engineering
importance is pr;p.:se'..


P:,O'. ABOUT A. VT'.R'' T:LL I?;:.& TJR..


As a first e::a.nle of the energy dissipations to be
expected from heat-cannc'.ty lag, consider the total head
measured by an impact tube In a perfect gas. For defi-
niteness, consider the naparatus Illustrated schenatically
in firs:re 2. The raQ enters t'.e c'L-nt-er and settles at
the pressure p.1 and t!.e te-r'perature T,. It then ex-
pands to a pressure pl and a ',prn .rrature _1 o'i.t of the
faired orifice, which h .: s dez.r-ned t? give a temperature
drop praldual enoJii.-] that the ey.ansion thorough the orifice
is ise -tropic. Ti-e ,~s that flov"s alonr the axial stream-
line of the impact tube is then brought to rest at the
nose of the tube -nd, during: thi". process, its pressure
rises to p2 and its tcmpc-:'ature rises to T2. If this
second process is sl:vr enou'-r1 to be isentropic also, the
en*4rory and th'-e f.ervy of c'-e r s that has reached equi-
librium at the nrose of the '-r,.'ct tube are equal to the











values in the chamber and hence the pressure p, equals
pO and the reading on the alcohol nmanometer is zero.

Consider, however, the other extreme case in which
the compression time that is, the time required for the
gas to undergo the greater part of its temperature rise -
at the nose of the impact tube is small compared with the
time required for the gas to absorb its full heat capacity.
The orif'.ce is concidercd larrc enough that, during the
expansion through it, the gas maintains equilibrium. A
part of the heat capacity of the gas Cvib does not follow
the rise in temperature during the compression as the
gas is brouji':t to rest at the nose of the inopact tube and
adjusts .rrlvcrsibly after the compression is over. The
resultant increase f entropy in this case r'cans that the
pressure P2 is lower than pp. This increase in entropy
is now crlculatcd. All temper'turo changes arc assumed
small .'nc.ugh that the heat capacities of the gas can be
taken ac constant.,

At tie bcrinning of the adjurtm-ent, the lagilng part
of the heat capacltTV cvi It still in rquilbri.un with
a th-ermoicter at t'.h tcnperati.re n;1 '.:h1ile the transla-
tional and other dcl-rce, olf freed ;:.ri7h heat capn-citics
totLling c-', thl;c rcla:-ation timc of which h can be neg-
lected, are' in c.qullib'iun1 -vith a t-he -;,rnc ter al some
highe1 r impc?- .ture T. 1.rcr,3- thcn flovws from the heat
cap?.ci t ce o the heat capacity c,.,,I, increasin
the temiperatu-re Tvb assocf.tcd "-th ev1ib fr-.omI T1
to the f'.nal cquil2'rilu temecrature, "i'ich i" TO.
Con.-,crvation of ,.-c.r'.T7 jgves the follov' ilr rolPtion
between T and Tvib


Cvib' vlb + cp'T = cpT3 (1)

where Cp is the total heat c:.pacity at con-tant pres-
sure. The entropy increase v.'~tn an lecr.ent of energy
dq flows from T to Tvib is

dS dq dq d / 2
Tb ." -- cvib vib ITi
Tvib vLb T





-


6


Eliminating T in equation (2) from equation (1) and in-
tegrating over the whole process gives


AS = vib dTvib -co T +
SpTo CvibTvib T


og vib P v/ (3)

T1 1
S vilb TO-


Equation (3) gives the entropy difference between
the gas in the chamber and the gas at equilibrium at the
nose of the impact tube. Because the energy and hence
the temperature is the same at the beginning and at the
end of the process, the ratio of chamber pressure to
impact-tube pressure can readily be computed from the
perfect gas relation

S = Cp log T R log p + Constant

which gives

AS = R log -
P2

and
c :- Cvib



Cvib To

It may be instructive to derive this relation by
considering the isentropic parts of the process. During
the slow expansion, the enthalpy theorem (see appendix C)
1 2
gives cpT + fu = Constant (u is flow velocity) and,
during the instantaneous compression, Cp'T + 1u = Constant,











cvib being omitted because it tak:es no part in the com-
rression. Combining thl-e equation? gives

cpTo0 TI.) Cp'(T2 T1) (5)


where T2 i' the tFmpeC t,..'re re',ch1ed L"' the translational
degrees of frecdorn before the adjustLr.ent period starts.
The adcin tb ctic-coriirssion relc.tion can be applied to both
the exr"..:.-ion dr.:l the comr.re:s."ion vith the appropriate
heat cS.acltie; to calculate p2; thus,

E FR

1) \1/ lT



Comlbining eqigtinotl (E) and (f') gI es, after several
mani;:ulci l io.ns


c. i,- 7-





which 1s of cour-'e ie-ntical with cu'J1: ).
S.- ,P.2
The percentage 'otsl-hesad defect 10'----- is
''0 "1
plotted against ceimner pressure c,'. in f ureu 5 for
Cp' = 3.5R. The p- arEatus scher;ati Led in f.rure 2 v",
used to chtmc. equation (4) for CC verc. tie- vibra-
tional Ieat car:.acit w-jould be exp-cted to Ir rhe
orifice was a hole in a -inch plate with its dia-;meter
variation desir:;ed f.r constant tir,e ;-ate of t e.;pcrat'ire
drop. The last 1/16 ir:ch of' the flot passage w.s
straight in order that the strea.lirLes in the jet would
be straight and axial and I.ence the static pressure at
the orifice exit would equal at:-o.spheric. pressure. The
glass impact tube wa3? O.C00 inch in diareter and its end





N
-~


8


were between 300 and 600 fe t per second. The expansion
therefore took place in times ranging between 1.4 x 10-4
and 2.8 x 10- second. The compression at the nose of
an impact tube takes place while the gas flows a distance
of the order of 1 tube radius. (See fig. 5.) The com-
pression times then ranged between 7 x 10-7 and
14 x 10" second. Commercial CO2 was used and, be-
cause it was fairly dry, a relaxation time of the order
of C0-5 second was expected.. It seomed likely, therefore,
that thick setup would approach the case of an isentropic
expansion and an instantaneous compression closely enough
for the results to bear at least a qualitative resemblance
to equation (4).

Preliminary to the investigation of heat-capacity
las., it was necessary to iake sure that hydrodynamic
effects other than heat-capacity lag would not produce a
reading on the alcohol mano-meter. Air and later nitrogen
ail room temperature were therefore substituted for CO2
at the beginning of each run. It was always found in
these preliminary tests that, v.hen the tube was properly
alined, the difference in pressure measured by the alcohol
manometer ,7aS vcry small and could be accounted for
entirely ty lags in the small vibrational heat capacity
of eir (at'rit 0.02P).

Carbon dio .ide w.prs then introduced into the apparatus
and t.he observations chown in fiure 4 vcrc. made. The
ras. was heated before cr.toring the chamber, and its
ten~pc.raturo vas measured by n small thermocouple inserted
in the jet close to the impact tube. In accordance with
acrodynamic experience, the temperature measured by the
thermoccuple was cr.s!sicd to be 0.9TO + O.1T1. The dif-
ference betvw'en Tr and T1 was" always less than 300 F,
corresponding to a difference in Cvib of less than
8 percent, and v;wc. thus considered accurate enough to
assu:ie a constant cvi and to compute this value at a
T +
temperature T --L


The pressure pO p, was read by the mercury
manoneter, pl, by a barom;cter, and pO P2 by the










alcohol manometer, which was fitted with a microscope to
make possible readings to 0.001 inch,

In figure 4 the reading of the alcohol manometer is
plotted against the chamber pressure PO/p,. The ex-
perimental values at both temperatures agree with the
theoretical values more closely than could have been
anticipated. It will become clear later that the theo-
retical and experiri:nrtal values agreed soo clo.-cly because
smrn]l entropy increases in the oriffice, attributahle to
too-rap'id e:p.nrsion, just about compensated for tbc fact
t'at the ccToprecssionr was not anite Instantaneous compared
w'th the relao.:ati-n time of thbo pac. It should be
pointed out that ordinary h;rdrodyncamic c 'ffcts such as
inldalinemnent of the impact tube v'r.nld bc expected to pro-
duce a total-head defect ,.hich would vary directly as
Pn
-- 1 .
-1


G7i;ER.AL TI71,T7 o w7 i!C- DISCFTTIOIIS. IN GASES

-EhIFITTI-G HEr -C-APAC. LAG


In the general case in ',nhich the temperature changes
nay be neither t verve, i. sl. ::' compared with the
relaxation ir'ic of th? gas, the tei-r.pcrature history of a
gas particle a: it floeves alon] strea.mline n'ust be con-
sidered. The problc,! ~ rc greatly siinlified if the effect
of hcat-capacity lag on vclocity dilsributi'on is
neglected in order to .*:et the cffcct of the 1-,. on energy
dissipation. This *;rocicLurc cin be recardo.1 aj the
first step in an iteration procers "nd is .robah-ly aice-
quate for tlie applications no'v c .nt !nrlatc-. 'Th re-
striction t1lrt the tc.r;c-rat,.re changes involv:rd :'.n the
flo" are sr'all Lemu1.: for ti.c '.acnt capacitiec to bc con-
sidered constant is also retaiicd.

Assume, therefore, that t;,. velocity distribution in
the field of floi is dctcrmincd by standard gas-dynam-ics
nrethods. Tlio velocity- distribution is usually given as
a function of space cocrdinrtcs u(x,:y,z) or along the
'-:trc '.r'.lire s s(i) '. -re s is the distance along
the r:trcanlinrc. Ths cx-prc ssion can be converted to a
function of ti;. v n ,(t) b: integration of dt = -d-
uT 7T








10

along a streamline. The function ua(t) is taken for
granted and the entropy increase in the flow along a
streamline is determined,

Ly introducing the variable c, which represents
the excess energy penr unit mass in the lagging heat
capacity over the energy at equilibrium partition at the
translational temperature T, it is seen that

1 o
c T + -u + E= Constant (7)
P 2

The assaunption is now introduced that there is only
one type of heat energy in the gas Evib which lags
appreciably behind the translation temperature and that
its time rate of adjustment is proportional to its de-
parture from equ.iibri'u.i; that is,


Svib=
-I, E
dt

This a-si'mptlon is in a-reement w-.th the sonic theories
previor.sly dircuss~d, From the definition of ,

C E~ c T
vib vib


because cvibTvib is t.e eCvil.l.bViun valve of Evib
measurede d from an arbitrary zero). By combining these
eqi.ations, Evib can be eliminated to yield

dc -c dT
dt v-ib dt k

The meaning of 1: carn be made clear if the variation of
e ,with time is e::ar.i--cd for the case in which the total
heat energy of the gas remains constant. In this case,


c T + C = Constant
P











Equation (8) then becomes


at c P

or


L.t T '

from '.'.ich I--,- is the reciprocal of the relaxation
P
ti-e T of the gas. It will 1t' seen that these equa-
ti:,.s are 'restricted to gases with on.r-' onne relaxation
time.

In orner to sirmlif7 later 'r:'re3s-1n. and to
clarify their phTvical meiening, there are introduced the
dimrensionless variables


F r~t
l c =- ti "-- ,-an
-I
c--i "' / '--
Cp'






where h Rnrd L arc a tr'r-ctl len, th and a t'-nical
velocity- in tha f'lo-l ard I is a dil..ensi less araneter
that is a rieasiure. of t,-e r-at io of the t1r.ie. n r v'hi,-li
temperature c.:afre3 c'cur .:. the .cas to the rela:"ation
ti'ie of the !as It vill be seen later- that C' is de-
fined to r.al-e It becor.e .ul'.ty .aftc E:- inst.,ttaneous ex-
pansion "hich starts fro', rsc-t 'it- equ .ltlbri 6 '.: er
partition and ~cns %'th the ?elociY T" Lli:'.nat ig T
betvecn equ-ttioris (7) and (0) and introdicinc the non-
dimen-ioinal quanti -:ies rives


S+ E-c (10)
,:t' t'

If u'(t') is Ikno\rn, the inte,'ral -of equation (10) can
be written as





K I


12

-IKdt' du'2 ,/Kdt' t
= e t ---'I -e dt' + Constant (11)


Th- rate of entropy increase in the flow can now be calcu-
lated from equation (11). The rate of heat flow from the
temperature Tvib to T is k-.; hence,

d.s = -r I (12)
dt \, Tvib/

T-w C = Cevib ( ib T\ and equation (12) can be written
as

dS /1 1 1
dt T + --
\ c-vib/

The entvn-py increase along the strea rlrine in question
bet-:-ecn the startin-. time t and the tirme t is

;it
1 1
AS \ d t (13)
C. T +- i
SvO ib,

In order to obtain the total entropy increase, equa-
tion (13) v:ould have- to be ir:.regrrated over all the
strea:mlincs in the flo', '~it:- the uce of equation (7).

Similari.t:- Law for Lov:-Velocity Flows

The calculation of energy dissipstions can be simpli-
fied if the restriction to flows involving pressure and
te;riprature clharnes that are small compared. w3th ambient
pressure -nd tcmncuratuL-e is adopted. The greatest ad-
vantag: of third prre- i :rei i that the flow pna ttern ob-
tained in an incofiirsc.tle fluid can b' used as ani ap-
proximnation. Thi_ fa.ct is important because few compres-
sible fluid flows are kcn:w-n accurately. If this
restriction is accepted, k and hence K can be assumed
co'-,rtant for the flow. Equation (11) then becomes

t duy Kt'
E1 e --(-- dt' + Constan (14)
.dt '











Now both --- and variations of T are small compared
cvib
with T and equation (13) becomes


AS = k- e2 dt (15)
Cvib.T2- t (15)
to

It is now shown that there is a simple relation
among the dependencies of the energy dissipation in a
low-velocity flow on the scale of the flow, on the typical
velocity, and on the rela:.-tion ti-ie of -.he rar. This
relation i.s that the entropy increase, reduced to non-
dimensional form, depend, in ecmi-etrically sinmlar flovs
cn a single parameter K.

Equation (15) can be rewritten as

c E U2 cvib 2 .. 'v
vi2 \ 2 ct ''
CpCvib- \2t


Introducing the nondimensional entropy increase A3' by
dividing AS by the entropy increase followirr an
"instantaneous" conp;ressicn .ilves


AS' = 2K ,2 dr' (16)



From equation (14), it is 1-LA':'n thst e and hen-s AS'
depends only on K for similar flo. s.


Appro::irations for Large an'l :r.all Values of .

The integration of equations (14) and (1.) are
sometimes difficult to perfl rn anal-tically and laborious
to evaluate nurerlcall-. For the special cases in which
the relaxation tine is either lonr or short compared with
the times in w,:hich temperature changes ta!:e place in the
gas, it is possible to use approxirimations that consider-
ably reduce the numerical labor. In these cases, it is
possible to express AS' in terms of integral in which





K


14


K does not appear under the integral sign; thus, these
integral need be evaluated only once to determine AS'
for all values of K for which the approximation is
valid.

The case of short relaxation time, when K is large,
will be treated first. In order to avoid confusion, the
symbol tVa is introduced into equation (14), which
becomes


'a
C'(t'a) = f


du'2 -K(t'a-t')
--e dt'
dt'


(17)


For a large value of K, most of the contribution to
this integral cones from values of t' so close to t'a
that the following approximations can be made:


du'2 du'2
dt' \dt'/a


u12 ,a2
S- a
t' t'a (du,2
dut'
d dt' /a


and the lower limit of the integral in equation (17)
be replaced by c Equation (17) then becomes


u'a2
e'(t'a) = f/
1:'+,3


exp d, u'a2) du'2
dt' )a


where the sign of the lover limit is opposite that
of (du'a. Hence, for K >> 1,


' a)dt 'a


2_ 2 f 2du2
AS' =- Kjdt' dt
Kf -it-, )


The case of long relaxation time, when K is small
compared with 1, is now considered. In the usual flow
problem, the gas velocity changes appreciably during a
certain time interval say, from 0 to t'1 and then


can


(18)












settles to e new steady. value. T-e problem can be
divided into two parts: 0 < t' < tI1 and t' > t'l.
If I is small enough., the ch'-aive in Er due to the
c'-terr in equation (10) is small co'.pered with'the
change due to the dt--terl anc can be neglected in
dt'
the calculation of the entrpy increase AS' during
the first interval; tlhus,

c' = u ,.


where u'IO" is the velccity o'l.r-d. at t' = .


rt'
AS'I = 21:
% 0


r tI
' dtat = 2 :


'. u ,? )2 dt'


In trCder to c: -r e the value : c' t t'l, the
total contribution onf th- --. .- in eq.ration (10) is
added to the t-ot~ 1 lanhc In t:I sq:;r' cf the velocity


during the first i"t.rval ui".


'1 = Au'2 ': /t
I.-


1 T t1-
e' .tt' = -u' .
J,-i


1(u,'- u'o' )dt'


In the period after t'

n ntC-t' i) i
a' tC- 'C(tl

and the entrcpy incrcr.:,e in this seccn,' eriod z53'2 is


as

3' = 2

1t'l


S -,'" 1) "t 1
(t )


The total ertro:- i!:rease in th.E flo- i.s ter-:c.e, fcr
K << 1,


enice,


1'-us,





N
~-a'


16



AS' = 2K 1 u'O u2 2 dt'




+ I: (u'02 u'2) dt' + Au12



Calculation of Total-Head Defect in Flow

about a "Source-Shaped" Impact Tube

The total-head defect to be anticipated in a com-
pression at the nose of an inr'act tube of a special shape
is calculated to be used In the measurement of the re-
laxation time of gases. The restriction to lov veloci-
ties adopted previously is retained, chiefly to permit
the use of incotrpressible-fluid theory and of the simi-
larity theorem.

The flow about bodies of revolution in a uniform
stream is usually calculated b' considering` the flow
about sources in the fluid. (Comoare reference 6,
p. 146.) It is possible to find a surface in the flow
across which no fluid flo:s. If a solid body shaped
like this surface is suoztituted for the sources, no
alteration outside the surface occurs; the flov' about
the solid body is thus identical with that about the
sources. The flow about a single source in a uniform
flow is calculated in reference 6 and the corresponding
shape is plotted in figure 5. The total-head defect
to be anticipated for a tube of this shape is calcu-
lated as follows:

The velocity along the central streamline is re-
quired. This velocity is given on page 147 of refer-
ence 6 and is plotted in figure 5 as


u(x) = U1 2
16 26







17


where

x distance along central streamline from source

U velocity far from body

d diameter of impact tube

This expression can be converted to the following non-
dimensional forn by usinr U as the typical velocity
and d as the typical dimension:


u'(x') = 1 (19)


The next sten is to find u'(t'). The quantity t' can
be found as a function of u' tb- integrating

dx- 1 du'
dt' I = -
11' (x') o .l /2
Sxu' (1 u' )


The choice of the zero of t' is arbitrary. For con-
venience, if t' = 0 when u' = 0.99, then

1 du'




= 1eg -+ 2- 2 1 (20)
1 i u7' -i u'


The next step is to determine e '(c) froi e',uLation (14).
Then, by use of equation (19),

du'2 /u,2 =-16u'2x u')3 2 (21)
dt'. : (21

Because c is zero initially (t' = -c) and remains zero
until u' begins to vary rapidly with tire, if ; is not











too small, the lagging heat capacity can be assumed to
follow the temperature changes in the gas up to the point
u' = 0.99; that is, E' = 0 can be used for t' = 0.
Combining this fact with equations (14) and (21) yields


'(t') = -e-Kt' 16u'2(1 u)3/2eKt dt' (22)


In view of the partly transcendental nature of equa-
tion (20), it was necessary to integrate equation (22)
numerically. Equation (20) was plotted (fig. 6) in
such a way that the valued of u' corresponding to
regularly spaced values of t' could be found easily.
By Sirmpson's rule, E'(t') was then fcund for a series
of values of K. An example of the result of such a
calculation is Fiven in figure C for K = 3. The
entropy increase alonr the central streamline was then
found front: equation (16).

Values of AS' found frcr.i inteirrating equation (22)
by Simpson's rule and equation (16) with a planimeter
are ;lotted in figure 7 and are given in the following
table:

RESULTS OF ITUIERICAL CALCULATIOIIS OF AS'

FOR SOURlCE-SIHAPED IMPACT TUBL


SC AS'
nf i

10 0.1685
3 .405
2 .516
1 .676
.3 .868
.1 .952


For large and small values of K, the approximations
developed earlier were used to reduce the labor of cal-
culations and yieldedd the result AS' = 1.743/K when
K is large and AS' = 1.452K + (1 1.008K)2 when K
is small. These results are plotted in figure 7; this
figure thus indicates the range of applicability of these
approximations.










Calculation of Entropy Increase in Flow

through a Nczzle of Srecial DCsign

For the mnas'irc';ents of the relaxation time in COp,
a nozzle is employed in which the gases expand and ac-
colerate before ime.-tin the impact tube. This expansion
cannot always bLt 'made slow enough that is, the nozzle
lar.:e c louh! that t.he oxran-icn through the nozzle in-
volves a rc;..:ll-- n..lJgi'-e e'itropy incrsese; hcnce, the
results orf ficLroe rmist be corret.'d for the entropy
increasoc: in the r.,c.zzle. In order to simplify the cal-
culations,T the r.zz).e vwas s-o dc'.im.eCd that t '.ic tine rate
of temeni-ratirture daop v.3- constC.nt. It c..n be shovrn that
the entropy incre-.sc ini nozale of thi- design is



S- + 3 I!

7.
whro = iKs tv:-ir- -fi:-a .-!;locity attained
b, th f'.. : t'-c r'-,:1l Tt uv.t b. reen'e-bercd that
the ca7.L':'.L n. ?r .:: ~-.e t o"nc c"-. umcd to be
zero in t'a '..t. cn-, i tio.n th-? cae. only if
1. >> a,.' c ,- t;i cal uil....t'..i giv,. n ,re is val'd
only for tLd c0.

Prom the da'1in.-r ons of K "nd Dh, it is soon
that Kf K n h-nce the tot.l entrzrpy increase

AS' = 3'" + t3'


can bc ex>.ressed .-- a fun:-.tilcn of :K nalon fo- given
I/d. This total cntiop:-' increa.- is plotted in fig-
ure 8 against K fo r th- tv.'o valn.u.: of 1/C. us';d in
these cxpori..nents ai, for 7./d = .


MrP.ASUTPR..E-:T OF RTLAXAITICt' TI:.iE OF C02


The thcor- :-'ill rov ::,' e arlc l j. to n: t r aea,.reatent
of the rela;::st on btir of 00. This ''.or:k Tas
underlta. :n o :,i, t' test the theory and to .dc:vclop a







20


technique that would supplement the sonic methods pre-
v.ously used for measuring relaxation tines. The
method essentially consists in expanding the gas through
a knovn pressure ratio in a nozzle and compressing it
again at the nose of a source-shaped impact tube. The
resultant total-head loss is divided by the total-head
loss that would be obtained in a very slow expansion and
a fast compression (equation (4)). This nondimensional
total-head loss is compared with a theoretical result
such as is shown in figure and the value of K appro-
priate to the flo-. is found. From this value of K,
the relaxation time of the gas can-be easily computed if
the velocity before compression and the diameter of the
impact tube are know.

During the compression of the gas, the temperature
and pressure rise from T1 .nd p. to T2 and P2,
ros-ectively. The relaxation time and the heat capacity
of the gas thus charge along a streamline. The pro-
cedure previously outlined then gives an average relaxa-
tion tine for the flow. It is assumed Lhat this average
relaxation time ic the relaxation time appropriate to
conditions halfway between compressed and expanded con-
ditions. Because po is clp.oe to p. and T2 = T
these conditions p and T can be found from


PO +Pl
p-
2

and
TO + T1
T = ---
2

The errors introduced in this '.tay certainly are no
greater than those due to the low-velocity assumption
introduced in the theory upon which figure 8 is based.

Gas

The qas used in these experiments was commercial
"bone-dry' C02. This gas was dried by passing it
through calcium chloride and then dehydrite while it









21


was at a pressure greater than 40 atmospheres. The puri-
fication procedure was not so thorough as methods used
in some previous investigations, and it is to be-expected
that somewhat shorter relaxation times would be obtained.
The primary object of this vrork is to establish the self-
consistency of this test method rather than to obtain an
accurate relaxation time for pure CO2.


Apparatus

The apparatus used is essentially the same as that
schematized in figure 2. A longitudinal section through
a chamber of the most recent design is shown in figure 9.
(The chamber used in the tests discussed in the next sec-
tion did not incorporate the liner and the gas entered
from the bottom.) The gas enters through three holes
that were made small to stabilize the ras flow Into the
chamber. The glass wool is necessary to remove turbu-
lence from the ras in the chamber and contributes materi-
ally toward reducing the total-head defects obtained in
gases without heat-capacity lag. It was found that
total-head defects traceable to nonuniformities in tem-
perature existed and could be reduced by the u.se of the
lined chamber shown. The fact that the gas flows around
the inner chamber before entering helps to keep the gas
in the inner chamber at uniform temperature,

The temperature nonuniformities can be almost elimi-
nated if the gas entering the outer chamber is at the
same temperature as the chamber. A mechanism was used
to adjust the alinement of the impact tube without moving
the tip from the center of the nozzle. The impact tube
must be adjustable in order that small errors in shape
near the hole will not give spurious total-head defects
(in helium, for example). The gas and the chamber were
heated electrically and a thermocouple inside the chamber
was used to measure the gas temperature.

The nozzle used had a circular cross section, was
1.6 inches long, and was designed according to the methods
previously described to give a constant tine rate of
du2
temperature drop; that is, = Constant for the first
dt
1.5 inches, the last 0.1 inch being straight. The
radius of the nozzle r is plotted against the distance
along the center line x in figure 10.





N


22


Two impact tubes with diameters 0.0299 inch and
0.0177 inch were used in these experiments. They were
made by drawing out glass tubing until a piece of ap-
propriate diameter and hole was obtained. The hole was
kept larger than about 0.004 inch and the fine section
not too long (=1/4 in.) to prevent the response of the
alcohol manometer from being too sluggish. The ends
of the tubes were ground to a source sb&pe (fig. 5) on
a fine stone. During the grinding process, a silhouette
of the tube was cast on the screen of a projecting micro-
scope and the contour superimposed on a source-shaped
curve. By this technique the contour could be ground
to the source shape within 0.0005 inch, except for the
hole, in a short time.


Tests and Computations

The total-head defect in CO2 was measured with the
two impact tubes over a range of chamber pressures. The
consistency of relaxation times obtained at various pres-
sure ratios and vith various impact tubes serves as a
check on this method of measuring relaxation tine and on
the theory on which the method is based.

Before each series of measurements nitrogen, which
has only a negligible vibrational heat content at room
temperature, was run through the chamber to be sure that
no spurious effects and leaks were present. In the re-
sults reported herein, the errors due to these effects
were l-ept to less than 0.01 percent of the chamber pres-
sure; therefore, the resultant error in relaxation time
due to these causes was less than 4 percent. In sub-
sequent work (not reported herein), it was found that
most of these total-head aberrations could be eliminated
by ensuring uniform temperature in the issuing gases.
If care is taken to eliminate ter- rature nonuniformities,
tube misalinements, and turbulence in the chamber, the
total-head aberrations can be reduced to 0.002 percent or
less.

The total-head defects obtained were divided by the
result of equation (4) to reduce them to nondimensional
form. The appropriate value of K was found by refer-
ring to the appropriate curve in figure 8. The gas
velocity was computed from the reading of the mercury
manometer by the enthalpy theorem with adiabatic expansion











assumed. The relaxation time was then computed from the
definition of K by equation (9). The relaxation times
thus obtained were expressed in collisions per molecule.
The number of molecular collisions per second in C02
was assumed to be 8.888 x 190 at 150 C and 1 atmosphere
by combining tables of pages 26 and 149 oV reference 7.
At all other temperatures and pressures, the number of
collisions was assumed to vary inversely with VT and
directly with pressure. The number of molecular colli-
sions per second and the heat capacity of the gas were
computed at temperature T and pressure p. The data
obtained are given in tables I and II for the 3.0299-
and 0.0177-inch tubes, respectively.

The result. are plotted in figure 11, which indi-
cates that the relaxation time in collisions is nearly
independent of pressure ratio and impact-tube size.
This consistency constitutes the desired verification
of this test method. It was expected that a variation
at high pressure ratios would appear in view of the
assumption of low velocity made at several points in the
theoretical development.

A large part of the scatter of the results in fig-
ure 11, in particular the apparent drop at low pressures,
is attributed to the fact that in the tests the average
temperature (halfway between chamber end expanded tem-
peratures) was not held constant during the run.

The average number of collisions obtained with the
0.0299-inch tube was 33,100; with the C.0177-inch tube,
32,000. The final result at 1050 F thus is 32,C00,
which is somewhat lower than the result of recent inves-
tigations in which the CO2 has been mum. more carefully
purified than in the present investigation. (Compare
with fig. 1.)


IMPACT-TUBE METHOD OF MNASURITNG RELAXATIONT TIr; OF GaSES


The impact-tube method of measuring the relaxation
time of gases rests essentially on the fact that the
total-head defect not traceable to heat-capacity lag can
be reduced to a very small value say, 0.002 percent.
Very small dissipations due to heat-capacity lag are
therefore measurable. For example, a gas having a





'


24


lagging heat capacity 0.1 with a relaxation time of 10-"
second could give a total-head defect of 0.05 percent.
If the gas had a lagging heat .capacity as large as R, a
relaxation time as short as 10-8 second would be meas-
urable.

This method seems to be easier to carry out than the
sonic methods previously discussed and can be used to
measure relaxation times vjjth comparable precision. The
quantity of gas required to make a measurement will be
larger than for the sonic methods (a standard tank of C02
lasts about 5 hr in this apparatus) and thus may make it
more difficult to attain high purity.

If the gas to be studied has a long relaxation time -
greater than 50 microseconds, for example it should be
possible to measure the relaxation time in an apparatus
similar to the one discussed by comparing the total-head
defects obtained with a calculation of the entropy in-
crease in the nozzle. In this case, the time taken for
the gas to flow through the nozzle is compared with the
relaxation time of the gas. The shape of the impact
tube would be uniiinportant in this case as long as it was
small enough that K << 1.


CONCLUSIONS


The existence of energy dissipations in gas dynamics,
which must be attributed to a lag in the vibrational heat
capacity of the gas, has been established both theoreti-
cally and experimentally.

An approximate method of calculating the entropy in-
crease in a general flov problem has been developed. The
special case in which a gas at rest expands out of a
specially shaped nozzle and is compressed at the nose of
a source-shaped impact tube near the mouth of the nozzle
has been treated, and the dependence of the resultant
total-head defect on the relaxation time of the gas has
been found.

The total-head defect in this flow has been applied
to measure the relaxation time of CO2. The results
obtained with two impact tubes were in agreement within








25


about 3 percent. The consistency of these results is
regarded as a check on the general theory developed and
on this measurement method.



Langley Memorial Aeronautical Laboratory,
National Advisory Committee for Aeronautics,
L~:.irley Field, Va.





N.


26


AP1PECDIX A

SOINIC ESTTRE.miTS I: R CARB~O DIO):ID


Much careful work has been done on the lag in the
vibrational heat capacity of CC2. Carbon dioxide is a
linear molecule and thus has a translational and rota-
ticnal heat capacity of -R. It has four norn.ial modes
in vibration that are dieararnTed :with their frequencies
as follows (data from reference 8):

0-9 C *--0 v1 = 4.164 x 1013

0 0 U2 = 2.003 x 1013 2 modes

-0 C--) --0 U = 7.050 x 1013

The heat capacity of CO2 iF sormewih1at complicated
by the fact that the second excited state of the oscil-
lation '2 has alrrost the sLrme energy as the first
excited state of ul The near resonance results in a
strong interaction through the first-ord.er perturbation
(the first-order departure of the potential energy from
the square law) between the two states involved, as was
pointed out ~',:y 7err.i (reference 9). This perturbation
produces significant disturbances (-=0 cm-1) of the levels
involved but does not have a large effect on the heat
capacity of the gas. The heat capacity of CO2 was
computed by Kassel (reference 10) and his results are
used in the present calculations.

F'ucken and his cov'orlkers h-ave carefully studied over
a period of years the dispersion of sound in C02
(references 11 to 15). One conclusion of this work -
that the vibrational energy levels in CO2 adjust with
the same relaxation tire is demonstrated by showing that
the dispersion curves obtained fit a simple dispersion
formula suc' as 1neser's.

KuiChler-. for e::arple, obtained a simple dispersion
curve at 410 C, at which appreciable heat capacity due
to all three normal -.iodes would be expected. Richards
and Reid (reference 16) and others (see bibliography of










reference 2) have maintained that the sym-metrical valence
vibration vu of CO- d-e? not adjust at 9 kilocycles
in some dirpersion nee.sure:;ens r3de near :ocm tewrnrcre-
tivre. As they pcint out, this fact I. re,.-.akable be-
cauc tihe second excitcted .ttee cf u2, strongly perturbs
the first crcitcc state -, U1. In any c.se, the con-
trihu.tion of thi!s nor:m-l n;oce to th- heat capacity is
vert r-c.l. at r:,oii. tLuirerarure and the eff'ects found are
neasi tihe lli.'. -.f tuie accu~a,:; of .Richards and :eid.

The relaxation tire of CO in rrolecular collisions,
as J.veri by E.l-:en -.n h.'. cc'vrke-r-s, is nobttcd against

T 3 (T In 0K) in fl.rrc 1 for co.-iarizon with the theory
dir-rursed in aupendi:. 1. Vani Itterleek, de Eruyn, arnd
Mari r.n (rcfereence IT) -'measurep' the a' bsnrptSon at 590 kilo-
cyclee in very car.: f t. T lpr'i 'led C02. heir measure-
ment., ..'ich are al-n l-.-en iln fi_.urc 1, -hov: a longer re-
laxation tim-- th.ai .1. .c s.'rc.c ta of :"'.clir n and hi-
coworkers. Fy al-- .. ,t t'j'. incres'-es. releaation time
to careful po'v.- fic.o n of i.'. a.:.

All the riac.--.-," ,' i h CO ; :ave indicated that
the rol.x-ati3r' :-" .: i .-.cr.ely ri'opc-& iortl to the
pr r.. e ..of : ,: 1.- -t F n' e-r :T.: the process
rosl lir: n-'.' i c :-r L:'. r'a.rcb -.;C.rc .o *o" c '.ner- t.tV:ecn vibra-
tional a.d ot.'.er d1.-r--. : forced. is bi-'lecular.











APFE:?DTX B

TTEC2Y OF EX:CITATTIO 0P O OLECUTLA2 VIEtATION EY COLLISIONS

LTarda:u and Teller (reference 1:) have riven an ap-
pro;xiaste calculation of the probability of the excita-
tion of a vibratior.al quantmini in a molecular collision.
Arsur:iinr that the interaction nr.-rr v.'hich induces the
vibration dr. rendu lirn.arl: en the l' ori.al -tcrdinate of
a ina:-r'o;nc vibration, tay:: n.sl:e a first-order perturba-
tin.... e-ultt ion. The iatri.L 'clreent for the transi-
tijon from the lth to the (Z + 1)th or from the
(Z + i)th tc te i Ith vibratli.Oal !..tte ic then pro-
portional to V/7- +1. The trarltil1rn robabilities
k "i cre pro.oprvttional to the a v.aro of matrix elements
anrd th: before


I1' 12- lO:k,?1'32 = 1.:2:3

and, when i j A 1, :, 0. nThs result is shown
-"J
i -i r-eferc..ice 1 t?. t .- to te cred-lietion that all the
&alloved tra,-i-ti :n- in a rive.. .--' .-1 mode have the same
ar- .a: io i l..rn a' a-'nd ellcr ne:-: examine the
c ol is- .n p r-r.c s r.I c.r calrI, a :, tr7.n7 the interaction
energy oet we' :-. ra:l.::':iLn ir3d vi-:,,ation to be prrpor-

L"ic'n. -l o i whcre tIhe .f stance between the
i" rl cul'-c ainw.1 a i" an r.nd:. tcr::.inE.'i c..-t ant. They
al2 o ti..'i.u.;? -,t t t'.- a:sl:: ti lonnl cner:-. of the mole-
cucs tl ;. is, t-he co lisle, i' 1 idiabatic. The
'moic.unt of cncr-"'' trans.'c.rd t.. -ibra'tor. In a collision
?. then calci ]atr'.. on .1 d to :-stir..a ..C che tians..tion
lr'babi' t- : anid I!.L '-lexatlon t-re oi thc- gas.

_11 r;
Thley concllide t'hat ti.:. ;c;rr:.:rt'-bw variation 0O thc rc-





Colli. ions = ex') 7T


where "I in molecular v.w-irht.

Tn :- .re 1., exp'ri:.intal results for the relaxation
tinre .:,1 collins o CC :1Vn? nitrous oxide 1I20 are










1
plotted against T T. The theoretical results are seen
to be straight lines, within e::perimental error. The
value of a can be found from the slope of the straight
line. For C02 with u = 2.003 x 1013 a=0.22 x 10 cm

and, for N20 with u = 1.773 x 1013, a= 0.36 x 10 cam.
These reasonable values for a are a further check on
this theory.

It should be pointed out that the temperature varia-
tion of catalytic effects is Quite different from that of
pure gases, the number of collisions required being nearly
independent of temperature. (See Ku"bler, reference 15.)
Various attempts have been made to associate the effective-
ness of catalysts with their Fhysical or chemical proper-
ties but no generally successful rule sefrs to have been
proposed. Gases that have or.e chemical affinity, gases
with large dipole moments, and gases with snall moments
of inertia are usually rost effect ive.





- '


30


APPEIIJTX C

THE EITHTALFY THEOREM


If no energy is transmitted across the walls of a
stream tube, the total energy (Internal energy E plus

Kinetic energy per unit rma"z -u 1 plue the work done by
prezs.sres pV mist be the same at any cross section of
the tube; that is,

2
E + + pV = Constant (C1)


In the case of a perfect ims vith constant heat capacity
and with 'cq1.illbriuml partition of ?nergy, equation (Cl)
Lec ...:es


c..T + -u = Constarnt (C2)


where cp is the heat capacity at constant pressure and
T is the absolute Ltc-.,,rrature. ,Whenever equilibrium
partition exists, even though ronequiblibrium states have
been passed through, equation- (Cl) is a-plicable in the
absence of vl c.-.:s. 3aj,.; heat c.-.ndIction and equation (C2)
can be at. lir;..i. t .c rif :'.-t gases., provided the heat
cap-acity of the gan ca.- 1e considered' constant.












REFERENCES


1. Fo.lrer, R. H., and Ouggenheim, F. A.: Statistical
Thernodynaric The T'acmillsn Co., 1939.

2. Richard..:, V.ll am T.: Super.rnic PeIFnn'ena. Rev.
'Todern Ph:-s., vol. 11, no. 1, .Jar. 1339, T-v. 36-14.

3. :Inerer, I. C." Interpretation cf Anomalous Tound-
Absorption 1.n -.ir and Oxygen in Terms cf ':olec-.lar
Collision". J.ur. Acous. 3oc. ri., vol. 5, no. 2,
Oct. 1933, p.r. 122-120.

4. Knudse.n, Vern C.: The Absorption o-f Sound In Air, in
Cyv' env, and in TU'trogern E'ffect: of I;u.mirlitt, and
'CmcreI:el'r.tute. Jorir. Acous. Soc. Ar., vDl. 5, ni. 2,
Oct. .-1v, -p. e,2-1,".

5. Kantro'vitz, Arthur: Fffects of Heat Canccit-: T,a in
Ca.s DyTn.a-ic Let. tm Td., Jou:. Cher. P)-r'-.,
vol. 1", n 2. 2, PIet. 142, i.. 1 .

6. Tietjens, 0. G.: ThGunda r!.ntl.s of ':yd i- and Aerone-
chan ic. 'T"coral,4-.ilol .'1. Co., ITc., 11954.

7. I"enrar'rd, Fanle I : Kin ti. The:.rc:, of a-c-e c.c-ravr-
11ill Fe'ok Co., Inc., 1238.

E. Deinnison, David '.. The 'ibr'.tionJ..l Levels of Linecr
i t .r trical 'rlo.to:ric 'ole o le Pr R.v., vil.
1'-I, ::.. .3, e s t ., t. ,. 1, 1 v '-', pp. 0 i'- 12o.

9. Fcrrni, T.: a nls'. Effect in CCr. Ze.Itchr. ?. Phys.,
vol. ',], -o-. arnd 4, Aur. 15, 1931, Fp. C', -;;C 9.

10. jassol, r.. '.? T. 'ermoi :ra;n-.~ .r Fruntion of :;itrous
Oxide :nd Carb-on Dio-ido. Jou... Am. Chet--. Soc.,
vol. ,C, no. 9, Sept. 19-,., ;p. 183C-184:2.

11. Eu:l-en, A., and Bc-l:er, P.: c-itcition of intra'.olec-
u]Lar l'ibrations in Gases al.d "-ar: Mixtures ihy
Coll].sioZ1 'oCed on :"ea.sre:!m. nt. of Sound Di-sper-
sion. Pe.li:.,'ra:; C'nLu..nhati,?.. 2c tschr. f.
p'." Ch'..., Abt. ', vrl. 20, no. 5/6, April 1933,
pc. 4: 7 ,-












12. Eucken, A., and Becker, R.: Excitation of Intra-
rmolecular Vibrations in Sases and Gas ?ii.;tures
by %oliioens, 7ased on :easlurements of Sound
Dislerision. Part TI. Ze.itccnr. f. phys. Chem.,
.:,-. F, :-ni. 27, njs. 3 and 4, Dec. 1934, pp.
5-:-"i:2.

13. Eucko-, '., and Jancks, H.: Exr.itati.n of Tntra-
r'e '.a 'ibral-ions in G-ases and GOs .1Mitures
bt- .'.1. lens, ,asod on :easurem-ents of sound
S:~~ si.-. 3t T7. ZLe-tschr. f. phva. Chen.,
..' 'l. 530, no". 2 aiid 3, Oct. 1935, pp.


14. r"ck--, ,/'.., .ar-d Ir-ann, E.: Excitation of Tntra-
in-Coiuila VioaL.'ions in jases and Gas Mixtures
'yy ".:. _.-n-. Part TV. Leitschr. f. phys.
Ch-, ,. t. vol. c, no. 3, July 1937, pp.
l..-,-] .:,5.

15. ::ichler, .. t of TI tr:-tLr'olocular Vibra-
tio:n-s :i '.:seze and Gas .i:.:turev: by Collisions.
F ,: V. z t. L.r. .f rc..v Cl'r-F At 3, vol.
41, no. Z, C.t. 193%, pp. 1.'-214.

16. P.ichardr, -,-1 Re.d, J. A.: Acoistical
St'.'ile-. Part :i. II Rat.e of' Txc -.tvtion of
Vi,'at .c:-al 'rer.Tr' in 1 C~, and SOs.Jour.
-:. .-- ., r.l. 2, no. -, ,nril i....4, pp. 193-
*.'.. ?:.'t l.1 .o.n E' Il' c. ll ie 3 of Vari-
oi "i --.e 'ci 'inr t'-e TL-er Vibrational
tctes .'.-. cn".. .::nI ati. n of Fotat onal
Encr ;,' 1 "-r', c -_ J.T '1 E-.'' P:;ys.., vol. 2,
uin 4, n.'--l 1 2-.T', pF-. 27,e-21?..

17. v:an -tter .r-c.', .., de :ru:n, 2., an,' Mari6ns, P.:
'".--s .,.-. .t t:. e ?.bs.:'r-ti.on of '.i.ud in COT
ara nr". Iso in 'ixt:.i: of C a .
Ph,-sia, voIl. -., I'-. 6, Juno 1.3 J, pF. &11-518.

18. Lan, dqn.: Tel 1.e- .: Thlcory of -ound Dis-
..i..Oi. y.:-. l.eitsjb.r. der Sowjetunion, vol.
1.0, no. 1, 19'23, pp. 34--4,








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