RETURN TO A-Z^A / , Utr "t mmi, "^^P*- °^ A£i-6n£.ulical Engineering // f *t-n U ' ^ ♦ nraB.1 ROOM 5 - ARMORY UNIVERSITY OF MINNESOTA ^^ ^°- ^^^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED January 19^^ as Advance Restricted Report Ua22 EFFECTS OF HEAT-CAPACITT LAG IN GAS DTCNAl^CS By Arthur Kantrovrltz Langley Memorial Aeronautical Laboratory Langley Field, Va. ^ NACA ^ 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. U57 DOCUMENTS DEPARTMENT Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/effectsofheatcapOOIang 1/1. 5-7'/ i'S KATIOrAL ADVISORY COMIIITTSE FOR AEROITAUTICS T"D REPORT EP'FECTS OF IlEAT-CAP^iCITY LaG IN GAS DYNAMICS B7 Arthur Kantrowltz The existence of enerry diss lij at ions In gas d;;manics, which must be attributed to a Js.p: in the vibrational hp.nt canacity of t2 ie__gjifL, has been established both theoreti- cally and experimentally. The flcv about a very sinall ir.pact tr.be is discussed. It Is shovn that total-head defects due to hsat -capacity lag during and after the cojr.pre.Tsloh of the ga3~aT~E7Te" nose of an ii.apact tube are to be anticipated. Experi- ments quantit.'.M,lvely verifying these anticipations in carbon dioxide are discussed. A general theory of the dissipations in a nore general flov problem "is developed and applied to scr-ie special cases. It is pointed out that energy di 3_si]_^ajbion3_due^ to this _eff ect_..ar_g .^t.9.J^^-. antj cipr..t& d__I^Turbir;e.a'. Dissipations of this kind might also 1 n 1 1- ■ odu c e errors in cases in which the flow of one gas is used in an a-<:terapt to simulate the flo\'.' of another gas. Niif ortunatel;', the relaxation times of nost of the"" gases of engineering importance have not been studied. k new method of measuring the relaxation tine of _ gases~~i3 __irvtro^xiajejd- in which the total-head defects ob- served with a spec j ally shaped Impact tvhe are compared with theoretical considerations. A parameter is thus evaluated in v.'hich the only unl-oiov.m quantity is the re- laxation time of the gas. This method has been ariplied to carbon dioxide ''-nd ..las given consistent results for tv/o impact tubes at a variety of gas veloclbles. INTRODITCTION The heat content of gases is primarily three forms of m.olecular mechanical energy. JT'lrst^ there is the trans lational kinetic energy vhlch is n^.T , where R \ is the gas constant and T is the abnoliate teriperatijre, Secondlx, thero Is the rotat ional kineti_c__energ:gj.. For "all gases near or above rooiri tpraperatv.re, the n rota- tional defjrees of freedom involvin.n; morcents of inertia due to the separation of atomic neuclei have energy states close enough together that the rotational internal energy is closo to the classical value wRT. The third prin- cipal form of internal energy is the v ihro.tional energy of the rnolecul ££«_ If the frequencies of the normal modes of vibration of the molecule are Icnown (say, from spectra), the vibrational heat capacity car be computed by the methods of statistical mechanics, (See, for exam.ple, reference 1.) The possibility of dispersion an d absorption_aC sound d ue to parts of the heat; capacity lagging' behind the rapid temperature changes acco;:.panying the propaga- tion of a so^'ond v;ave in a gas vas first discussed theo- retically by Jeans and Einstein, Dispersion and ab- sorption in carbon dioxide observed by Pierce v.-erc sho^vn by Herzfeld and Rice to be attributable to lagging of the vibrational heat capacit:/ of the gas, looser vas able to account quantitatively for dispersion and absorption in COo and oxygon on the assumption that the vibrational heat capacity lagged. The dispersion and absorption of sound In several gases have been investigated and o. fairly comipletc bibliography is available in reference 2, It is fo^und, in general, that disporsion and absorption many timics larger than those attributable to vicosity and heat conduction are to be ejrpected in gases Vi,'ith vibrational heat capacity. These effects can be described 'oj rela- tions such as t?iose given by Kneser and can be attributed to the vibrational heat capacity of the gas. All the measurements of dispersion and absorption have demonstrated that most im.purities miarkedlv reduce the relaxation time of a ^j 3^; for examiple, Kneser and lOiudscn (rel'erences 5 and 4) concluded that the adjust- m^ent of the vibrational heat capacity of oxygen was dependent cntirel;/ on the action of Impurities. V.arious er-coeriments -.vith COo have shown that, at room tem.peraturc, collisions with w.itcr molecules are 500 timics as effective as collisions with COo-molecules In exciting vibration in COo-molecules, This strong o .3 dependence on lourity lias produced rrsat discrepancies among the relaxation tlvae^^ riea^ured by the various v/orl:ers In this field. There has been much better agreenent amonr, the neasurements of the effectiveness of 'impurities. onic measurements in CO^ are discussed in appendix A. n quant uin-mechanlcal theory of relaxation times developed by Landau and Teller is discussed in appendix 3. Tv/o conclusions, which are verified by the sonic v;ork in CO2, are important to the present paper: (1) All the vibra- tional states of a single normal mode adjust vrith the same relaxation tlm.e and (2) the logaritlim of the relaxation time (expressed in molecular collisions) is proportional _1 " to T "^ . Verification of conclusion (2) is presented in figure 1. Dr. Yannevar Bush helped to initiate this work by asking the v/riter a stimulating question. The author also is very grateful to Professor 7!.. Teller for heli:'ful discussions . EFFECTS IN GAT DYI'TaMICS In the flov-' of gases about obstacle.^ , compressions and rarefactions accom.panied by temperature changes occur. The timie in which these temperature changes take place is controlled by the dimensions of the ob- stacles and the velocity of flov?. If these tir^e inter- vals are com.parable vith or sliorter than the time re- quired for the gas to absorb its full heat capacity, the gas v;ill depart from its equilibrium partition of energy, Jn this case, the transfer of energy from parts of ti^e heat capacity that have more than their share to parts th3.t have less than their share will be an irre- versible process and will increase the entropy of the gas, If the tim.e intervals involved are comparable with the rela:-;ation tim.e of the gas, this increase in entropy can be used to m.easure the relaxation time of the gas (refer- ence 5) . Turbine-working fluids such as steam, air, and ex- haust gas have appreciable vibrational heat capacity at high temperatures. If tjiese gases have relaxation tim.es comparable with o r ; sho rt er^ than the intervals during which temperature changes\ occur in the gas, losses \ attributable to heat-car'aclty lap mn.st be anticipated. A rouph estiriiate has indicated that, unless the rela;:a- tion tliTiss of the v/orkinr fluids a-po very short, the losses at high teTiperatui^es due to heat-capacity laj: can be comparable v^ith the losses due to skin friction. Unfortunately, no measurenents of the relaxation times of the usual turbine-v-orkinp" f].uids erist. Various persons have proposed, in v;ind-tunnel tests and in tests of rotating- riac-iinery, tVie substitution of gases that have properties enabling tests to be made riore conveniently at a given I'ach nuj'^oer or Reynolds number than v.'ith the actual v/orhing fluid. In such cases, care must be taken to ensure that an error due to dif- ferences in heat-cajiecity-la^-; behavior of the fluid used and the v:ork"ing fluid 2.3 not ii^troduced. For example, according to a rough calculation, a wing in pure COg might have a drag coefficient tv;ice as large as t;ie sane wing in air at the sane "lach nrrr.oor and Reynolds nui.iber. In the follov/lng discussion, ti^e existence of these dissipations in gas dynamics is demonstrated and a gas- dynamics m.ethod of r^easuring the relaxation tines is developed. The application of this method to the meas- urement of the relaxation tim.es of gases of engineering importance is proposed. r'LOVi ABOUT A VJ:RY ^IIALL IrlFACT TUBi: AS a first e:'.am.ple of thxe energy dissipations to be expected from heat-capacity lag, consider the tota.l head measured by an im.p.act tube in a perfect gas. For defl- niteness, consider the apparatus illustrated schematically in figure 2. The gas enters the chamber and settles at the pressure Pq and the temperature Tq. It then ex- pands to a pressure p-, and a temperature T-, ou.t of the faired orifice, Vv'hich :" s designed to give a tem.perature drop gradual enough that the expansion through the orifice is isentropic. The gas that flovs along the axial stream- line of the impact tube is then brought to rest at the nose of the tube and, during this process, its pressure rises to po a.nd its temperature rises to To. If this second process is slow enough to be isentropic also, the entropy and the energy of the gas that has reached equi- librium, at the nose of the impact tube are equal to the values- in the, nhamber and hcnco the pressure Pp equals p,-, and tlic roadin,^ on the alcohol manometer is zero. Consider, hov;evcr, the other extreme case in which the compression time - that is, the timie required for the gas to undergo the greater part of its temperature rise - at the nose of the Impact tune is sm.all compared with the t5-rae required for the gas to absorb its full heat capacity. The orifice is conaldercd large enough that, during the expansion through It, the gas maintains equilibrium, A part of the heat capacity of the gas O-^y:^ does not follov; the rise in temperature du.ring the com.prosslon as the gas is brought to rest at the nose of the impact tube and adjusts irreversibly after the compression is over. The resultant increase of entropy in this case means that the pressure Pp is lower than Pq« This increase in entropy is now calculated. All temperature changes are assum.ed small enough that the heat capacities of the gas can be taken as constants 9 At the beginning of the adjustment, . the lagging part o'f^ the heat capacity c^.ti^ is still In equiibrium with a thermometer at the .tenperat\^-rc T-| wriile the transla- tlonal and other dcr-'rees of freedbm"v/ith heat capacities totaling c- ' , tlie relaration time of T.-hich can be neg- lected^ are' in cquilibritim with a thermom.eter at some higher temperature T. I'nergy then flows from the heat capacity Cp ' to the hcr.t capacity Cy-ij^ increasing the temperature TT^n't associated v:n.th ^^ih '^^^'^^ ^'l to the final cquil:'brium temperature, whicli is Tq . Conservation of onorgy gives the follov/ing relation between T and '^yVo ' ^vib^vib + ^p'T - CpTo: (1) vdiere Cp is the total heat capacity at constant pres- sure. The entropy increase v.hen an eler.ent of energy dq flovrs from T to Ty3_^ is ds - - - TT - °vib ^-vib I rrr T ■^Vib , " \^vib ^ / \ Eliminating T in equation (2) from equation (1) and in- tegrating over the v;hole process gives -^T^ ^ 1 ^vib'^vib 1'vib^ (3) Equation (3) gives the entropy difference betv/een 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 Po P2 and Po P2 (4) It may be instructive to derive this relation by considering the isentropic parts of the process. During the slov)^ expansion, the enthalpy theorem (see append!:?: C) 1 2 gives CpT + 75U = Constant (u is flow velocity) and, '^ 1 2 during the instantaneous compression, Cp'T + tjU = Constant, *^yib being onitted 'becau.'^e it takep no part in the com- pression. Combining th^se equations gives CpCTo - Ti) - Cp'(^T2 - Ti) (5) where T^ is the ternpei-^ature reached oy the translational degrees of freedom before the adjustment period starts. The adiabatic-conpression relation can be applied to both the expun^ion and the com.pression v/lth the appropriate heat capacities to calculate P2 I thus, Combining eqn.rtions (5) and (6) gives, after several manip Vila b i ons . y '^p ^vib 1 \-i. \;P - -vih >:-^/ which is of cour3e identical v/ith equai:ior. \^l) . The pcrcentare total-head defect ■ 100 is PO - Pi plotted against chamber pressure Pq/pt in figure 5 for Cp ' = 3.5R. The apparatus scheriati'sed in fin-ure 2 v/as used to check equation (4) for CO2 Vi'here the vibra- tional heat canacity v/ould be expected to lag. The orifice was a hole in a —-inch plate v/ith its diameter variation deslgried for constant tlmLe rate of tem.perati.ire drop. The last I/I6 inch of the flov/ passage v/as straight in order that the strear.lines in the jet v;ould be straight arid axial and hence the static pressure at the orifice exit would equal atmospheric, pressure. The glass impact tube was 0.005 inch In diar:eter and its end \ 8 were between 300 and 600 foct per second. The expansion therefore took place in times ranging between 1.4 x 10"'^ end. 2,8 X 10" second, '.Che compression at the nose of an inpact tube takes place while the gas flows a distance of the order of 1 tube radius. (See fig. E.) The com- pression times then ranged between 7 x 10""''' and 14 X 10**' second, Comm.orcial COq w-as used and, be- cause it v/as fairly dry, a relaxation time of the order of 10"^ second was expected. It seemed likely, therefore, that this setup would approach the case of an nsentropic e:cpansion and an instantoneous compression closely enough for the result? to bear at least a qualitative resemblance to equation (4) . Preliminary to the Investigation of heat-capacity lagr, it was necessary to make sure that hydrodjiiam-ic effects other than heat-capacity lag woxxld not produce a reading on the alcohol manometer. Air and later nitrogen at 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 v/as properly alined, the difference in pressure measured by the alcohol manometer xvas very small and co\ild be accounted for entirely by lags in the small vibrational heat capacity of sir (about 0.02P.) . Carbon dioxide was then introdu.ced into the apparatus and the observations shown in fi;:ure 4 were made. The gas was heated before entering the chamber, and its temperature was measured by a small ' thermocouple inserted in the jet close to the impact tube. In accordance with aerod;'/namic experience, the temperature measured by the thermocouple was asswaed to be O.QTq + 0.n\. The dif- ference betv/ren Tq and T^ was alv/ays less than 30° P, corrr?3ponding to a difference in c^j_i-| of less than 8 percent, and v/as thus considered accurate enough to assuiae a constant <^-^ii, ^nd to compute this value at a temperatui'-e t = — 1 , o The pressure p„ - p^ was read by the mercury manometer, p-,^ by a barometer, and p^ - p^ by the alcohol manometer, v/hich vras fitted v;ith a microscope to make pos3iblc readings to 0.001 inch. In figure 4 the reading of the alcohol manometer is plotted a;--:ainst the chamber pressure Pq/pi* The ex- perimental values at both temperatures agree v/ith the theoretical values more closely than could have been anticipated. It will becom.e clear later that the theo- retical and experimental values agreed so closely because sma] 1 entropy increases in the orifice, attributable to too-rapid expsjision, just about compensated for the fact that the compression was not quite instantaneous compared \\"ith the relaxation time of the go.c. It should be pointed out that ordinary hydrodynamic effects such as misalinoFient of the impact tube vould be expected to pro- duce a total-head defect \;hich would vary directly as PO p. - 1 GENERAL THEORY OE EIlERiT-: DISSIPATIONS IN GASES EXHIBITIIIG HEAT -CAPACITY LAG In the general case in which the temperature changes may be neither very fn=:t nor ver^-'y slov; compared v;lth the relaxation time of the gas, the temperature history of a gas particle as it flov/s along a streamline must be con- sidered. The problem, is grcatl;/- simplified if tlie effect of heat-capacit;,'" lag on velocity distribution is neglected in order to get the effect of the lag on energy dissipation. This procedure can be regarded as the first step in an iteration process and is probably ade- quate for the applications novr contemplated. The re- striction that the temperature changes involved in the flow arc sm.all enough for the heat capacities to be con- sidered constant is also retained. Assume, therefore, that the velocity distribution in the field of flov; is doterm.ined by standard gas-dynamics methods. The velocity distribution is usually given as a function of space coordinates u(x,y,z) or along the c^- stream-line as u^Cs), v/bero s is the distance along the streamline. This expression can be converted bo a function of time u.-(t) by integration of dt = -4^ ^' ' •' ^- u(s) \ 10 along a streamline. The funct.lon Hg(t) is taken for granted and the entr-opy increar.e in the flow along a strear'iline is determined, Ey Introducing the variable e, which represents the excess ener.^ry par unit m.ass in the lagging heat capacity over the energ3'- at equllibriixm partition at the translatlonal temperature T, it is seen that T + iu^ + e= Constant (7) The assumption is no": Introduced that there is only one type of heat energy in the gas E„«v, which lags appreciably behind the translation temperature and that its time rate of adjustment is proportional to its de- parture fro:n equillbriTim* that is, dt This assxmption is in agreement vr'.th the sonic theories previov.sly discussed. From the definition of c , = E - f" '' ^vxb "vib' because c^^-j^T^.^ is the equilibrium value of S-y-.-v, (measured from an arbiti-ary zero). By combining these equations, ^vib "^'^^^ '''^^ eliminated to yield OlI. .- „ d' dt IF " -'^vib^ -^'' ^s) The meaning of 1: can be made clear if the variation of € with time is exam.ined for the case in which the total heat energy of the gas remains constant. In this case, c T + c = Constant P 11 Equation (8) then becor-'.es or dt S-e from 'A'hich k ^P' is the reclDrocal of the relaxation time T of the gas. It will be seen tiiat these equa- tions are restricted to £;ases with only one relaxation time. In order to simplify later expressionf' and to clarify their ph^/sical m.eanlngj there are introduced the dimiens i onles s v&.riab les £l Cr-ib u' = ~ t' = K _t_ h/U h TU r (9) h and U are a typ^'.cal lenp'th and a t7,^nical city in the flov/ and L. is a dimensionless parameter is a measure of the ratio of the times in which ^erature of the cna: eas wher velo that tern time fine pans part betw dlmenslorial quantities gives pes occur in the gas to the relaxation It v.ill be seen later that <-' ' is de- d to m.ake it becor.e unity after an instantaneous ex- ion which starts from rest vrith equilibriran enerf';y ition and ends ^rXYi the velocity Lh Eliminating T 3cn equations (7) and (8) and introducing the non- dC dt' + Ke^ du' _ of (10) If u'(t') is known, the intet^.ral of equation (10) can be v/ritten as \ 12 e' = e t/ f j -^-e dt» + Constant (11) The rate of entropy increase in the flov; can nov; be calcu- lated from equation (11), The rate of heat flow from the temperature Ty-i_t) '^^ 1" ^^ ^^-'i hence, vih/ Nnw e = '^vT'h Fvib ~ '^'^' ^'"^ equation (12) can he v/ritten as "^ ^ I d8 _ ^ c A 1 ;n dt ( T " rp a. _ll._ ) ^ T + -r— - ; The entropy Increase along the strearaline in question betreen the starting; time t^ and the tline t is 1 \ AS := / ke/i 5: \ dt \ ' T + — '^- V ^vib/ (13) uto In order to obtain the total entropy increase, equa- tion (13) v/ould have to be integrated over a3.1 the streariilines in the flov; v.ith the use of equation (7) Similarity La\u for Lov:-Velocity Flows The calculation of energy dissipations can be simpli- fied if the restriction to flows involving pressure and temperature changes that are sriall compared with ambient pressure and temperature is adopted. The greatest ad- vantage of this procedure is that the flow pattern ob- tained in an incompressible fluid can be used as an ap- proximation. This fact is importarit because few com.pres- sible fluid flows are known accurately. If this restriction is accepted, k and hence K can be assumed constant for the flow. Equation (11) then becomes C = e-^^' rfl^eKt' at' + Constant (14) vJ dt ' / 13 Nov; both — and variations of T are snail comi^ared cvlb v t'l. If K is snail enough,, tli.o chan,^e in e' due to the t"'-term in equation (10) is small compered v/ith'the change due to the •%x--,- -tern and can be neglected in d t ' *- the calculation of the entropy increase AS ' ]_ during the first Interval; thus, where u.'q'- is the velocity squared at t' = 0. Hence, ■■^.'^ = 2K / €'^ dti = 2K I ^u'*" - u'o") dt' ■ ^'0 In order to ccipute the valu.e of e' at t']_.. tl-ie total contribution of tho ti-teiT: in equation (10) is added to the toti'l change in the squ.are of the velocity during the first intorval Au'"". Thus^ e 1 ^ = Au' ^ - i: / e ' dt ' = Aui ^- - K [ (u' ^ _ ^^i ^2 ^ ^^ In t}ie period after t'^.,- ^'o ^0 e' ---. c'le -K(tf-t'i) and the entropy increase in this second period ^S'g is A 00 Q ( -PT'f ti-t. ' -1 ] O 1 * ^J-J 1 1 The total ertrop" increase in the flo'.v is therefore, for K « 1, 16 \ AS I — 2K u ,2f dt' + ^0 Calculation of Total-Head Defect In Plow about a "Source-Shaped" Impact Tube The total-head defect to be anticipated in a com- pression at the nose of an impact tube of a special shape is calculated to be used in the m.easurertient of the re- laxation time of gases. The restriction to lov veloci- ties adopted previously is retained, chiefly to permit the use of incompressible-fluid theory and of the simi- larity theorem. The flov/ about bodies of revolution in a uniform stream is usually calculated by considerinr the flov; about sources in the fluid. (Compare reference 6, p. 146.) It is possible to find a surface in the flow across which no fluid flov/s. If a solid body shaped like this surface is substituted for the sources, no alteration outside the surface occurs; the flov; about the solid body is thiis identical with that about the sources. The flov; about a single source in a uniform flov; 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) = TJ 1 - d^ 3 V^ 16x 17 X distance along central streainline from source U velocity far from body d diameter of Impact tube This expression can be converted to the following non- dimensional form by using U as the typical velocity and d as the typical df-mension: u' (>:') = 1 1 15x'2 (19) The next ster) is to find u'(t'). 1^-"-^e quantity t' be found as a function of u' b^'' Integrating can di 1 '8 du' u' (1 - u') , ^ 3/2 The choice of the zero of t' is arbitrary, venience, if t' =: when u' = 0.99, then du' t' 1 •8 / a ' 7^ v^O.99 u' (1 - u')'^' ^ For con- 2 8 V 1 + V 1 - u7 -,/i - u' 20 (20) The next step is to determine e'(t') from, equation (14). Then, by use of equation (19).. du'' dt' O nil' 2u''^ -rV -16u'^(l - u')^/~' (21) Eeca.ur>e e' is sero initially (f = -co) and remains zero ijintil u' begins to vary rapidly with time, if K is not 18 \ too small, the lagging heat capacity can be asstiriied to follow the temperature changes in the gas l,^p to the point u' = 0.995 that is, e' = can "be used for t' = o'. Combining this fact with equations (14) and (21) yields Kb' -Kt' (f) = 16u'2(i _ u')-2eKt' dt' (22) In view of the partly transcendental nature of equa- tion (20), it v/as necessary to integrate equation (22) nu:nerically. Equation (20) was plotted (fig. 6) in such a v/ay that the values of u' corresponding to regularly spaced values of t' could be found easily. By Simpson's rule, e'(t') Vv-as then found for a series of values of K. An example of the result of such a calculation is given in figure 6 for K ?he entropy increase along tlie central streamline v/as then found from equation (16), Values of A3' found frora integrating equation (22) by oimpson's rule and equation (16) with a planim.eter are plotted in figure 7 and are given i!i the following table: RESULTS OP NUMERICAL CALCULATIONS OP AS' FOR SOURCE-SHAPED IMPACT TUBE K AS' 10 3 o .3 .1 0.1685 .405 .516 .676 .868 .952 For large and small values of K, the approximations developed earlier were used to reduce the labor of cal- culations and yielded 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 . 19 Ca].culatlon of Entropy mcreace In Flow through a Nozzle of Special Do sign : For the mc a sur omenta of the rolayation time' in COo, a nozzle is e-mployoc'L in v/hich the gases expand and ac- celerate before iri-eoting the impact tube. This expansion cannot alv^'ays be made slow enough - that is, the nozzle large enough - that tho expansion through the nozzle in- volves a really nogligihle entropy increase; hence, the results of figure 7 must be corrected for the entropy increaser; in the n.:::3le. In order to slKplify the cal- culations, the nozzle was so designed that the time rate of temperature drop v/as constant. It can be shov.ai that the entropy increase i.u a nozzle of this design is 31.,, 9---^ K " y c- 2 I" where Iw = - — anc-. "J is the ftljial voj.ocity attained ■'•" li 'i "■ ■ by tho flow :.,n the nc^izle* It must be remembered that the calculations for tlie Impact tube presumed e* to be zero initially;. 0:his condition is tho case only if Kj,.- » 1 and hsnce"tho calculation given hoi'e is valid only for this case* From the dofiritions of K nnd Ztt, it is seen that Ktvj ~ -rK and bonce the tota3. entropy increase can bo OOTressed as a function of K alone for a given l/d. This total entropy Increase is plotted in fig- ure B against K for the two values of l/d used in these experiments and for l/d = c» , MSASTJREMEIIT OF RI^L/lXATION TIMS OF CO^ . The theory vill row be applied to the measurem.ont of the relaxation time of CO2. This v/ork v;as undertaken both to test bhe theory and to develop a 20 technique that would supolement the sonic methods pre- viouoly used for mear-uriii^, rolr.xation tiiaes. The method e3sentia?-ly conslatf^ in expanding the gas through a knov;n 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 3 and the value of K appro- priate to the flow is found. From this value of K, the relaxation time of the gas can -bo easily computed if the velocity before compression and the diameter of the impact tube are know. During the coiiipression of the gas, the tcm.perature and pressure rise fi"-om T.. and p.. to T_ and Pp, respectively. The relaxation time and the heat capacity of the gas thus change along a streamline. The pro- cedure previously outlined then gives an average relaxa- tion time for the flow. It is assumed that this average relaxation tim.e is the relaxation time appropriate to conditions halfway between compressed and expanded con- ditions. Because p is__close to p^ and Tg = Tq, these conditions p and T ca:a be fomid from. md T = 2 The errors introduced in this way certainly are no J greater than those duo to the lov'-velocity assumption introduced in the theory upon which figui^o 8 is based. Gas The gas used in these exxjerim,:nts was coraiaercial "bone-dry" COg. This gas was dried by passing it through calcium chloride and then deh^'drite 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 v.-orlc is to establish the self- consistency of this test method rather than to obtain an accurate relaxation time for pure COg. Apparatus The apparatus used is essentially the same as that schematized in figure 2. A lonrrltudinal section through a chamber of the most recent design is shov/n 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 sm.all to stabilize the gas flow into the chamber. The glass wool is necessary to remove turbu- lence from the gas in the cham^ber and contributes materi- ally tov/ard reducing the total-head defects obtained in gases without heat-capacity lag. It v/as found that total-head defects traceable to nonuniformities in tem- perature existed and could be reduced by the use of the lined chamber shown. The fact that the gas flov;s around the inner chamber before entering helps to keep the gas in the inner chamber at uniform temperature. The tem.perature nonuniformities can be almost elimi- nated if the 'gas entering the outer chamber is at the samie temperature as the chamber, A mechanism was used to adjust the alinement of the im.pact tube without m.ovlng the tip from, the center of the nozzle. The impact tube must be adjustable in order that sm.all errors in shape near the hole will not give spurious total-head defects (in helium, for example). The gas and the cham^ber v\/ere heated electrically and a thermiocouple inside the chamber was used to measure the gas temperature. The nozzle used had a circular cross section, v/as 1.6 inches long, and \"as designed according to the methods previously described to give a constant tine rate of du temperature drop; that is, — — = Constant for th.e first 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. \ 22 Two Impact tu.be s with diameters 0.0299 inch and 0.0177 inch were used in these experiiucnts . They were made by drawing out rlasp tubing until a piece of ap- propriate diameter and hole v-as obtained. The hole v;as kept larger than about 0.004 Inch and the fine section not too long (=il/4 in.) to prevent the response of the alcohol Tianoraeter fron beinf^ too sluggish. The ends of the tubes were ground to a source shape (fig. 5) on a fine stone. During the prindine; process, a silhouette of the tube Vi^as cast on the screen of a projecting micro- scope and the contour superimposed on a .'^ource-shaped curve. By this technique the contour could be gro-ond 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 v/as measured v;lth the two impact tubes over a range of cliamber pressures. The consistency of relaxation times obtained at various pres- sure ratios and v/ith various impact tubes serves as a check on this method of measuring I'elaxation time and on the theory on v/hlch the method is based. Before each series of measurements nitrogen, which has only a negligible vibrational heat content at room temperature, v/as 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 kept to less than 0.01 percent of the chamber pres- sure; therefore, the resultant error in rela:cation tim.e due to these causes v «— u^ = 7.050 X 10^^ 26 APPEIIDIX A SONIC MEASTTRE^ENTS ITT CARBON mOXTDT) Much careful work has been done on the lag in the vibratlonsl 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 norr.ial modes in vibration that are dla^'rainried v/ith their frequencies as follov'/s (data from reference 8): 2 modes The heat capacity of COo is somewhat complicated by the fact that the second excited state of the oscil- lation Ug has almost the seme energy as the first excited state of U]_ . The near resonance results in a strong interaction through the first-order perturbation (the first-order departure of the potential energy from the square lav/) between the tvv'o states involved, as was pointed out by Permi (reference 9) . This perturbation produces significant disturbances i^dO cm"-^) of the levels Involved but does not have a large effect on the heat capacity of the gas. The heat capacity of CO2 v;as computed by Kassel (reference 10) and his results are used in the present calculations, Eucken and his coworkers have carefully studied over a period of years the dispersion of sound in COg (references 11 to 15) . One conclusion of this v'ork - that the vibrational energy levels in COg adjust with the same relaxation time - is demionstrated by showing that the dispersion curves obtained fit a simple dispersion formula such as Ilneser's. Kuchlfer ,- . for example, obtained a simple dispersion curve at 410° C, at which appreciable heat capacity due to all three normal modes would be expected. Richards and Reid (reference 10) a?id others (see bibliography of 27 reference 2) have maintained that the symiTietrlcal valence vibration v-\ of COo doe? not adjust at 9 kilocyclerj in oOno dir.!persion neasureraents r/ade near rooin teirpera- trrs. As they point out, this fact l.^^ rei,;arkablo be- cause the second excited state of Uo strongly perturbs the first excited state of u,. In any case, the con- tribution of this norniPl :;oce to the heat capacity is ver'y r-rall at room tcraperarure and the ef '"ects found are near the llr.-its of the accuiacy of Richards and Reid. The relaxation tirie of COg in ir'.olocular colliriions, as ffivea by Eacken and hi.i: cc'vorkers, is plotted afrainst T 3 (•71 --J.-, OK) in figure 1 for coar.parison with the theory dir.cursed in appendix E. Van Itterbeek, de Eruyn, and Mariers (I'cference 17) ineusured the absorption at 599 kilo- cycles in very carofu-lly puri"'ied COg. Their measure- mentr, which are al,7.o 3:iven in figure 1, show a longer re- laxation time than the nieasureinents of Eucken and hie cov/oi'kern. They attribute this increased relaxation time to caroful pur-°fi cation of the gas. Al], the measu:."-cr'ent::. with CO2 'iave indicated that tho relaxation ti'.ie 1.r, inver.'^ely proportional to the pressire of the ^cio o Thir rosu.lt shows that the process responsible fcr the intcrchanp;c of cnerp;y bGtv;een vibra- tional aiid other dcrrces of freedom is birrolecular. \ o 3 APFEMDIX B THEORY OF EXCITATION OP I'OLECULA?. VIBRATIOII DY COLLISIOrS Landau and Teller (reference 1') have £;iven an ap- proximate calculation of the proba'oility of the excita- tion of a Vibrational quantiini in a molecular collision. Arsurdnp- that the interaction encrry which induce.':^ tVte vlhral;ion depends linearly on the norraal cjordinate of a hariri.onic vibration, they rriSke a first-order perturba- tion calculation. Th.e i;vitri;>c elerr.ent for the transi- tion from the ith to tho (I + l)th or from the (Z + 1) th to the Ith vibrational state ic- then pro- portional to \/Z- "+'"l . The transition probabilities kj_ ^' are proportional to the square of •:natrix eleinents and therefore 'J± Xd l^-O -^■^' i-"± O^ and, when 1 - .1 ,-^ ±1, h.- .• ~ 0. This result is shown in reference 18 to load to the prediction that all the allov.'ed transitions in a ^iven norivial mode have the same relaxation tiine. Landai?. and Teller next examii^e the collision process classically, a'jsurijing the interaction energy between translation and vibration to be propor- tional to s ®- , whore x is the ri stance between the molecules and a Is an ti.ndotermined constant. They also assuine that the translational ener,f;y of the mole- cules - that is, the collision - is adiabatic. The amount of energy transfei-red to vibration in a collision is then calculated and used to ostiriiate the transition probab-i'lT ty ]:^-. a?id the relaxation tire of the gas. They conclude that the tem.perature variation of the re- laxation time e^qoressed in molecular collisions is given by / i '' '2 \ Collisions =5 exp 13T W — i \ V ^H where '.I Is molecular v;oir;ht« In fif-are 1, experimental results for the relaxation tine :'n collisions of CO^ and nitrous oxide K2O are 29 plotted against T ^. The theoretical results are seen to be straight lines, v;lthin e:;:pei'imental error. The value of a can he found from the slope of the straight line. For COg with u = 2.003 x lO^-^', a = 0.22'xlO"^ cm and, for N2O with u = 1.775 x 10^"^, a=0.56xlo"^ cm. These reasonahle values for a are a further check on this theory. It should be pointed out that the temperature varia- tion of catalvtic effects is quite different from that of pure gases, the number of collisions required being nearly independent of temperature. (See Kuchler, reference 15.) Various attempts have been made to associate the effective- ness of catalyrsts with their physical or chemical proper- ties but no generally successful rule seems to have been proposed. Gases that have somie chemAcal affinity, gases with large dipole moments, and gases with sm.all moments of inertia are usually most effective. \ 30 AFPEIFDIX C THE ENTHALPY TKEOREIvI If no enerr*y is tran'^iitted across the walls of a stream tube, the total energy (internal enei'gy E ulus Kinetic energy per ■'..mit macs ~u ' pins the \A'ork done by pressures pV must he the sariie at any cross section of the tube; that is, E + 7:u' + pV = Constant (CI) In the case of a perfect gas with constant heat capacity and with equllibriura partition of energy, equation (CI) b e 002116 s c.,^T + -u'" = Constant (02) where c. is the heat capacity at constant pressure and T is the absolute temperature, Vv'henever equilibriurri partition exists, even though nonequilibriuin states have been passed through, equation (CI) is applicable in the absence of viscosity and heat conduction and equation (C2) can be applied to perfect gases, provided the heat capacity of the gas can be considered constant. 31 REFSRErCES 1. Povjler, R. H., and Guggenheim, F. A.: Statist.! cal Thernodynaialca. The TIacmlllsn Co., 1939, 2. Rlchardo, William T.: Supersonic Phenoi:iena, Rev. •■odsrn Ph;/-:., vol, 11, no, 1, .Jan. 1939, pp. 36-64, 3. Knocer, II. C. " Interpretation of Anonialcas Sound- Absorption in Air and Ox7/gen in Terms of Llolecular Collision^:, Joi.ir, Acous. 3oc, Arri,, vol. 5, no. 2, Oct. 1933, pp. 122-120, 4. Knudsen, Vern C: The Absorption of Sound in Air, in Cx-jrsn, and in IJitrogen - Effects of Ei-.Tnidlty and TeiTiperature, Jonr. Acous, Sec, An:., vol. 5, no, 2, Oct, l'~o3, "p. 112-121. 5. Kantrovvitz, Arthur; Effect"^, of Heat Capacity La.^ in Has Dyna;nic;ri= Let. to Ed., Jour. Chem. Phys,, vol. 10, no. 2, Feb, 1942, p. 145. 6. Tietjens, 0, G.: ^u?idanientals of Hydr.-i- and Aerone- chanics. '"c Gray; -Hi 11 Book Co., Inc., 1934. 7. Kenriard, Earle E.: Kinetic Theory of nazes. McGrav;- Eill Book Co., Inc., 1938. 6, Dennison, David ^".t The Vibrational Levels of Linear Syjin-ictrical Trlato;nic Molecules. Phyr. . Rev., vol, 41, no, 3, 2d ser,, Au^;. 1, 1932, pp. 304-312, 9, Ferrrd, E,; Raman Effect in CCg . Zeitschr, f. Phys., vol, V], noe, 3 and 4, Aug. 15, 1931, pp, £50-259. 10. '"assel, L. 3.; Thermo d^ma-fiic Functions of Nitrous Oxide and Carbon Dioxide. Jour. hxn. Che^;:. Soc, vol. 5G, no. 9, Sept. 1934, pp. 1838-1842. 11. Euckeh, A., and Becker, R.: Excitation of IntraiTiolec- ular Vibrations in Gases and Ga^s r.'ixtures by Collisions, Based on Eea^urements of Sound Disper- sion. Prelirriinary Coinmunicatlou. Zeitschr. f . rhys. Cher-;., Abt. P, vol. 20, no. 5/6, April 1933, pp. 467-474. \ 12, Eucken, A,, and Eeckerj R.; Excitation of Intra- Tnclecnlar Vibrations in Gases and G-as Mixtui-'es by Collisions,' BasoJ. on r-leasureraents of Sound Dispersion. Part IT. Zeitschr. f. phys. Chen., Abt'. E-, vol. 27, nos. 3 and 4, Dec. 1934, pp. 12, EuckoM, A,, and Japcks, K.: Excitation of Intra- rolecu.lar Vibrations in Gases and Gas Mixtures lij Colli.'i'lcns, EasGd on Iieasurements of Sound Disparsion. Tart III, Zejtschr, f. pliys, Chen,, " Aj';. i:^ vol, 30,- no''. 2 and 3, Oct. 1935, Dp, 85-'i3 2, 14. Eii.cken, A,, ard TJurann, E.s Excitation of Intra- molecular Vibrations in Gases and Gas Mi;<:tures by Oolliaicns. Part IV. Zeitscbr. f. ph^rs. Ches-i^, Abt, 3, vol. 36, no. 5, July 2 937, pp. 163-183. 15, Iluchler, E.; Excitation of IntraTTolocu.lar Vibra- tions in Gases and Gas ?.'Ii r-rture s by Collisions, P'vrt V. Zeitscbr. f, r;hvs. Ghe-i. Abt. B, vol, 41, no. 3, Oct. 193P, pp. 199-214. 16. Rlchardr, '"'. T., andRe5d, J. A.; Acoustical Studie-,. Part III.. Rates of Excitation, of Vibrational Energy in COg , CS3, and SO 2* Jour, Cher.. ?h:rr.., vol. 2, no, 4, April 1934, pp. 193- 205, Part IV. Collision E^Ticioncies of Vari- ous Folecule:"' in Excitinr: the Lov/er Vibrational States of Eth^'leno. Excitation of Rotational Encr'-y in TTyciT.'o,c'en. Jjur. Che!?... Phys., vol. 2, nn. 4, April ]93'i, pp. 206-214. 17, van ItterbPGV, A,, de Bruyn, ?., and Marions, P.: .Moas-arcrr'ent on the Absorption of 3ound in COg Gas . . . and also in 'fixtures of CO2 . , . Physica, vol. €, no, 6, June 1939, pp. 511-518. IQ. Land9u, L., and Teller, E,; Theory of -ounc Dis- persion. Phyc . Zeitscbr, der Sowjetunion, vol, 10, no. 1, 1956, pp. 34--i3, 33 (J> m C iHOcx)D-co(J>'^rH'vt r — o o rH rH O p; rH O ^ O y f-i Q) o '^ <3J a ■^ !>- to ^ U) uj U5 ^ ■* CO O to OJ to ^ '^ oj oj to to t-o to to to to to to to to c:) to to CO to • 03 Relaxa- tion time, r= ^ Uk (micro- sec) iHtO(X)C\JLr:)ir5cD'^LOcD£>tOiOCOlO(7i coa■Jto^^JCV!cvlOC7iOrHtocx!(^J(^^H CM (M to Kj tO f i to to C\i to -^ tO to lO fi to !> Gas velocity at end of nozzle, (in./sec) O'^OiHOrHOrHCDlOOtOrHrHtO^ '^ c£i CX) t^j to fX) CT' rH to lO iH CO Cl:) lO £> W CD ^ rH ^- to £> O f-0 cr. 05 Cj) -^i lO O LO CT> lOcD!>£>COa)CT>OiCTjCj5C£)J>J>COCOO c£l K (from fig. 8) to W CO to CO to to t- lO tH CO CJ 00 CO H fH ^ £> LO OLf;)c\jrHOOooocritoc\:cv!iHOO CVJrHrHHHrHrHrHiH rHrHrHiHrHrH iO Total AS' /Col. 2] • rH o O tOrHtOCC-^fOCDLOCD^tOtDCOrHOOCO CO O lO C- C3 C7i Ci Ci C^ O CJ Li"j lO L- CO CO Lf:)LOcX3tO!^cOcOCOCC£>-cDc£)tDcOCDCO O ■^ 13 » O r-i ca -p 'H .^ CD 4J to c: i>. I ^ O a £/:■ fn 1 CD .H CD O O C\2 rH 4H ?-l CD a ft cd cP';Jic£)'*rHCCCMCV!Cv3H rHrHC^lOiHOOtOD-OO'^t'CTilOHO^ cOCOOrH-^lOt-(7iOHtOlO«3 O iHrHtHiHHrHCaOJ rHHrHHH to Average tempera- ture, (Op) OCOUj£>tX)COOrHOO-)CD rHHHHH rHiHrHrH W Ti .H CD -P 1 ^ O 1 LOCDCOtO^U.)J>«3iHGOOcOO c\jcocDOcDcocn';j<'^iu3coojLOcocjto tO'shcDCOO>OHtO-^^£>COOH cd c D- CV! t<0 to ■* '^J* to O fn ft ft rHiHrHrHrHiHiHiHrHiHrHrHiHrHrHiH 54 < \ Ci Collisions (t X Collisions per molecule per sec) oooooooooooooooooo oooooooooooooooooo C-£>C~-'5iJJ>Oui(HC»b'J^i CO Relaxa- tion time, (micro- sec) fOC3■<;^^^r^lOOcncv!LO•^H(750^cv:Ot-f 05 O to iH fH 0-j 03 (H r~ GO CO C^ O iH iH H O fH C\3 fO C\3 C."> fi CJ to t<0 OJ 03 C\3 '03 to to f J to to to r- Gas velocity at end of nozzle, U (in. /sec) CO 03 05 f .) £> CJ O !> to to Lf3 CO £> CJ t- CO 03 CO '^f:)C0C}C0C0-^C-OC0C0C0t0iHO03C0rH rH 03 03 t- CD rH G5 t'O t- vOJ LO O Ci C7> CO £> '^ O COt>L>I>(DOLf;CX)CJ5Cr)CSC75COCOCOCOCOCO C£) CO k1 o . • ■ ^ to tOHC0'4tOr>J>!>LOt-'^'^^WC:5rH £>CDC5t>(DCr-05OcDCDCDOcDO o LO Total AS' /Col. 2\ t H . . . . O J , 1 ;=: O G K; ?H , 1 . CD. -H. OJ. O O CV! rH 'hh m o 3 ft H -— • CJr^^■cDcDoJtoc'5CJCoc:5co•^^-^'^coco '^'^u'.^cn£><C-CDCCOOrHlOU'iCDtOLC'DCJ50^C\3I> a>C\3OC5CHCJOCr5CJ5CT505Cr!C5OOOO rHfH iHfHrHrH' HfHHrH CVJ cd ^ fH -*fHCUOO O r- t> Cr> CD' lO -^i rH lO to •^ to C^ CJ 03 CJ rH (35 fH r-tiHr-ir-'. r-ir-ir-il-ir-ir-i fH ^ S^ft s 0JlO';)* CD CD "D CO iTj in CO lO •* ^ x: Hr-ir-ir-t fHj-j H MACA FIG. 1 LOOxlO- m C o o o i^ir- -jz: rl^ -iL-_--l_* -- " -:= =s4 ~-i=~-. - ^"T 1 1 1 i i 1 1- Cu2 --i3 /r-ir .rf3ii^ ^ ~:r" •;; ^ rVH-r -— ^-^- Mil -1 — + O Present in ves tige tlon -"S J -_i"_ L-£^zr^ ^i^i ~-~s p~^-4TTf4H D van Itterbeek, de Bruyri, and Marlens (reference 17 itz: ^-3^ = Gtr^^ ^^ cOo f M — A Richards and Reid (reference 16).^ _.-... . . f r = -^^T? i,==^ 1 ; ! 1 1 1 U Eucken and coworkers _ -J M-i— : 1 , - ' ' 1 1 ■ ---1 =^"=d^ -> 1 -pr 1 1 i H ( " ' ----■-■ ^ - - i-T 1 tr tl ces 1 Jasc 1 z x^ , ^ ! 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