ftcftWVttJN RM No. L8J14 NACA RESEARCH MEMORANDUM NOTE ON THE IMPORTANCE OF IMPERFECT -GAS EFFECTS AND VARIATION OF HEAT CAPACITIES ON THE ISENTROPIC FLOW OF GASES By Coleman duP. Donaldson Langley Aeronautical Laboratory Langley Field, Va. UNlVERSnY OF FLORIDA DOCUMENTS DEPARTMENT 1 20 MARSTON SCIENCE LIBRARY P.O. BOX 117011 GAINESVILLE, a 3261 1-701 11 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON December 10, 1948 NACA EM No. L8JlA NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS RESEARCH MEMORANDUM NOTE ON THE IMPOETANCE OF IMPERFECT-GAS EFFECTS AND VARIATION OF HEAT CAPACITIES ON THE ISENTEOPIC FLOW OF GASES By Coleman duP. Donaldson SUMMAEY The errors involved in using the perfeci>-gaB law pv = ET and the assumption of constant heat capacities are evaluated. The "basic equations of gas flows taking into account these phenomena separately and at the same time are presented. INTRODUCTION The conventional method of obtaining high Mach numbers for aerodynamic tests is to accelerate the air by means of a pressure difference so that the random kinetic energy of the molecules of air at rest is converted into kinetic energy in the test section. For very high Mach numbers this may occasion high stagnation temperatures and pressures which introduce effects due to the vibrational heat capacity and molecular forces and size such that the perfect— gas law pv = ET and the assumption of constant heat capacities may be no longer sufficiently accurate to evaluate gas flows. It is the purpose of this paper to present formulas which are suitable for handling such problems and to point out the magnitude of the errors that may be involved in using the perfect— gas law and the assumption of constant heat capacities. Tsien (reference l) has published a theoretical discussion of this problem in which certain approximations were introduced in order to obtain solutions that were in a very neat form when the imperfect- gas effects were moderate. A comparison of Tsien' s results with this work is presented to show the magnitude of these approximations. In England Goldstein has previously investigated this problem at moderate temperatures and pressures in order to prove the small magnitude of imperfect-gas and vibrational— heat— capacity effects in most supersonic wind tunnels. This report indicates, in general, the range in which these effects are small but does not present formulas for handling problems in gas dynamics when these effects are large. NACA EM No. L8J11+ The present paper la arranged in the following three parts: temperature effects on perfect— gas flows due to variation of heat capacities; imperfect— gas effects on gases without variation of heat capacities; and gas flows in which both effects are present. This is done since the formulas in the first part may prove use- ful to those dealing with the flow of hot exhaust gases and since it may bring out more clearly the differences between the two effects. SYMBOLS a term in Van der Waals' equation correcting for the effect of molecular forces b term in Van der Waals' equation correcting for the effect of molecular size c speed of sound, feet per second Cp heat capacity at constant pressure C v heat capacity at constant volume E . energy, foot-pounds M Mach number (w/c) p pressure, pounds per square foot B gas constant T absolute temperature, degrees Fahrenheit v specific volume, cubic feet .per slug w velocity, feet per second y ratio of heat capacities (Cp/C T ) p density, slugs per cubic foot 6 characteristic temperature of molecular vibration Subscripts: o stagnation conditions c critical conditions NACA EM Wo. I&JlA Errors Involved in Assuming Constant Specific Heats in the Presence of High Temperatures in a Perfect Gas For a perfect gas with constant heat capacities the equation for conservation of energy of a steady isentropic process may "be written as C p T + ^2 = c?To If this equation is combined with the equation for the isentropic speed of sound A - 1)' (3) where 6 is a constant depending on the gas. The formula may be used for the mixture air if the value of 9 is placed equal to 5526 when absolute temperature is measured in degrees Fahrenheit. (See the WACA RM Wo. L8J14 appendix.) The value of the heat capacity at constant pressure for a perfect gas is then * 2 W ( e e/T _ l} 2 M Figure 1 is a plot of equation (k) and shows that the heat capacity may not he considered constant above 600° F absolute. When the heat capacity at constant pressure varies according to equation (k) the energy equation must he written flT c B / IE dT + 1*2 = (5) U T o Substituting equation (k) into equation (5) and integrating yields ^ T + e eW- +w2 = ' mo+ e/f° (6) The Mach numher is obtained from equation (6) by dividing through by 7FT = c^ } which gives e-lfi?-l) + ± I ^ -. 1 ] + f| (_1 i ) (7) where The pressure ratio corresponding to this Mach number is obtained from the iuentropic equation NACA EM No. I&JlU 5 lo S X = / T £P AT , 9) Po j To E T ^ "by substituting equation (Ij-) into equation (9) and integrating to give e e / T e e / T c X = fX\ 7/2 1 - e / T o e V T e0/T - 1_ T ° e Q ^-l J Po vW x _ e e/T Similarly, the density ratio is found to "be e e/T e e e/T Q (10) p / T\5/2 x _ e e Ao \ T eS/T-1 T e e/T _^ (11) The differences involved in the use of equations (l) and (2) to predict the temperature, density, and pressure ratios corresponding to a given Mach number are given in figures 2(a) and 2(h), in terms of the percentage differences from the value given "by equations (7), (10), and (ll) for stagnation temperatures of 1000° and 2000° F absolute.. It is seen that the assumption of constant heat capacity leads to appreciable differences in applying the isentropic law for a perfect gas if stagnation temperatures above 1000° F absolute are involved. Errors Involved in the Assumption of the Perfect-Gas Law pv = ET for a Gas with Constant Heat Capacities In order to. evaluate flows in which imperfect— gas effects are present, an equation of state that takes into account these effects must be chosen. For the purposes of this paper an equation which takes into account the effects of molecular forces and size should be sufficient. A suitable equation is that of Van der WaalB P + -\\(r - b) = ET (12) NACA EM No. L8J1U where b is a term correcting for the volume occupied ty the molecules and a is a term correcting for the effect of molecular forces. Figure 3 is a graph of Van der Waals' equation in which the quanti- ties Pj v , and T have been made nondimensional by dividing by the values of these quantities at the critical point p Cj v c , and T Cf thus making the graph suitable for any gas. (See reference 2.) The graph may be used for air if an empirical critical point (p c = 37.2 atm, T c = 238. 5° F abs., v c = 0.6^38 slugs /ft 3) is assigned to that mixture of oxygen and nitrogen. To give this critical point the values of a, b, and E for air when the pressure is measured in pounds per square foot, the specific volume in cubic feet per slug and the absolute temperature in degrees Fahrenheit are a = 8.78 x 10-5, b = 0.65^, and E = 1716. The proper equation for an isentropic expansion of a real gas is (see reference 3) dE = C v dT + \dT/ v= Constant dv = (13) which for Van der Waals 1 equation becomes dE = C v dT + p dv + -% dv = •7 d (Ik) Equation (lU) may be written as dE = C v dT + d(pv) - v dp + -% dv = v^ and since — v dp = w dw dE = C v dT + d(pv) + -% dv + w dw = (15) Assuming constant heat capacity at constant volume and integrating equation (15) gives w 2 E = C V T + v (v - -%A + £■ = Constant = E Q (16) NACA RM Wo. L8J1^ This ie then the energy equation for a Tan der Waals gas. Dividing through by the isentropic speed of sound c 2 = & = (l + £-\ t2bT ^ (17) dp ^ C Y ) (v _ t) 2 v ^ and since p _ _&_ - RT _ 2a t^ v — b v 2 then VTo ♦ ^o ( ^ - %) - 2C T T - Br (j^ - £) 2C M2 The value of v for an isentropic expansion to he placed in equation (l8) can he formed from equation (l4) as follows: (V dT + p dv + -i dv = (^ dT + R ^ . dv = v then ii! = _jr_ (19 , T v - h and if C v is constant /T \°v^ v=(v -h)(4pj + b (20) From equations (l8) and (20) f knowing the stagnation conditions for an expansion from T to T^ the Mach number may he calculated. The pressure ratio is then found to he 8 NACA RM No. L8J1^ p RT a t-Tj t 2 Po ET o (21) v„ - to and substituting the value of v from equation (20) we obtain Po (22) Figure h shows the conventional pressure ratio and area ratio pw/(pw)j4_;L plotted against Mach number for air starting from stagnation conditions of 520° F absolute and various pressures compared with the value obtained using constant ratio of heat capacities and the perfect— gas law. Also shown in figure k are the values of pw/(pw)j^-j_ computed by Tsien's method. It is seen that as the imperfect— gas effects become large it is no longer possible to simplify the analysis by neglecting terms containing the squares of — and — ^—, although Tsien's results v pv are in good agreement at 50 atmospheres when the Van der Waals effect is moderate. It is interesting to note that the speed of sound in a Tan der Waals gas dp I 1 Cj (v _ t) 2 v (17) is not equal to 7FT. The expression for the ratio of specific heats in a Van der Waals gas is r-i + ff r 2 + %ab ^ Ce3) W \ pT 2 _ a + £M NACA EM No. LBJlU -3- OJ CJ vo OJ p tQ © ^ m © p •h o o o © OJ Pmr\ ©^- o i © -sa © P .3 © © P cd © ft h En th a) cm CO ID © O o © © p to p pi p •H p in a o ■H P cd & © o ii + -1% CDlEH © Ch CQ © O © § •H P © © M p ■a © OJ ©I ojI Eh OJ OJ I 1 .EH t OJ H H CD © 1 CD © OJ ,CD|fH, + lfA|OJ CD P 10 NACA BM No. L8«Jl4 CO OJ H CJ On S •H -P cd Ih Cm «M © O ■H P aj o< > ■H -P OJ (D a •H OJ o •H P 0} © o p n o •H P a) & CD ID CD CD|Eh o CD ©I o EH CD 4h° + CD (D I O Eh CD © I 60 O H 00 O lOJ OJ cdl o t> i 1 o 1 t> « o > *l ft P © CD © o p © Eh £ cTI Ph © ffj © P bO •H P •H P CD CO LfAlOJ II I o 60 o H o 03 CO © © NACA RM Wo. L8JTJ+ 11 The value of the ratio of heat capacities 7 in this case is 2 + ( T 7 = 6/T pv^ + a (e^-lf P- 2 ~a + 2ab T l + M 3 e/T T ^ (e*A-l)' (29) "but the speed of sound is found from equation (17) by substituting the value of C v /R from equation (3) to he C 2 = 1 + 2 + ( T .e/T (e*/ T - l)' v^T (v - h) 2 2a v Figure 6 shows the conventional pressure ratio and area ratio plotted against Maeh number for air starting from stagnation conditions of 2000° F ahsolute and various pressures compared with the value obtained using constant ratio of heat capacities and the perfect— gas law. DISCUSSION The foregoing analyses show that the effects of variation of heat capacities with temperature do not "become important in isentropic expansions of air until stagnation temperatures of the order of 1000° F ahsolute are encountered. Above 1000° F ahsolute, however, for accurate analysis this variation must he taken into account. In general, it may "be stated that for diatomic gases these effects are important when (— becomes appreciable compared to the number 2.5. e/T 'e/T - 1 The effects of "Van der Waals 1 forces become important when either the temperature is extremely low for near atmospheric pressures or the pressure very high for moderate temperatures. These forces must be taken into account when the value of a/v 2 becomes appreciable compared to the pressure p, or b becomes appreciable compared to v. For air these effects are unimportant until stagnation pressures of the order of 50 atmospheres at stagnation temperature of 520° F absolute are encountered. 12 NACA EM Wo. LSJlU Tsien's method agrees well with the results of this investigation up to 50 atmospheres in this case, hut it appears that it is not possible to neglect the squared terms of — and — j|— when the effects of Van der Waals' forces become appreciable. CONCLUSIONS v^p In many cases found in very high Mach number wind tunnels and in flows of high stagnation temperature or pressure, imperfect— gas effects and the effects of variation of heat capacities may be present. For diatomic gases the effect of variation of heat capacities becomes /fl\ 2 p / t / important when ( — ) s ^ becomes appreciable compared to 5/2. For air these effects become appreciable when stagnation conditions of 1000° F absolute or larger are encountered. Imperfect— gas effects become important in gas dynamics when a/v 2 becomes appreciable compared to the pressure p or b becomes appreciable compared to v. When air is expanded from a stagnation temperature of 520° F absolute these effects become important if the stagnation pressures are of the order of 50 atmospheres or greater. Formulas are presented for handling isentropic expansions taking into account these phenomena both separately and at the same time. Tsien's method is found to be applicable for small departures from a perfect gas but is not accurate when the effects of Tan der Waals' forces become appreciable. Langley Aeronautical Laboratory National Advisory Com mi ttee for Aeronautics Langley Field, Va. NACA EM Wo. iBJlk 13 APPENDIX DERIVATION OF THE VIBRATIONAL HEAT CAPACITY OF A DIATOMIC GAS To arrive at the vibrational heat capacity of a diatomic gas, the individual molecules are treated as linear harmonic oscillators of a fundamental frequency and Shrodinger's equation is solved for the allowable energy states of such an oscillator. These allowable states are then substituted into the equation for the canonical energy distribu- tion and the average energy per particle as a function of the absolute temperature is found. This may be differentiated to obtain the contribution of the vibrational degrees of freedom of the molecule to the heat capacity of the gas at any temperature. The average vibrational energy per particle found in this way is (see references k and 5) ! = £¥- + hv 2 ehv/kT.-L where h Planck's constant v characteristic frequency of molecular vibration T absolute temperature Differentiating to obtain the contribution to the heat capacity of this energy yields Cylb 1 M /hvf e hV ^ E k ST {yiJ ^ hv/kT _ *2 hv For a particular gas -s— = 6 is a constant and may be determined from spectroscopic data. The heat capacity at constant pressure is then £e - 1 + Cyj - b _ 2 + /£\ 2 e e / T E 2 E 2 \TJ ( e/T ,\< ) I -1) It NACA EM No. L8«JlJ+ The value of for oxygen iB U010.U and for nitrogen is 60kh.h for absolute temperatures measured in degrees Fahrenheit. The value 5526 may "be used for air. REFERENCES 1. Tsien, Hsue— Shen: One— Dimensional Flows of a Gas Characterized by van der Waals' Equation of State. Jour. Math, and Phys., vol. XXV, no. h, Jan. 19^7, PP. 301-321+ . 2. Jeans, James: An Introduction to the Kinetic Theory of Gases. Cambridge Univ. Press, 19^6, pp. 96-98. 3. Epstein, Paul S.: Textbook of Thermodynamics. John Wiley & Sons, Inc., 1937, PP. 6^-65. k. Frenkel, J.: Wave Mechanics. Elementary Theory. Second ed., Oxford Univ. Press, 1936, pp. 77-80. 5. Lindsay, Robert Bruce: Introduction to Physical Statistics. John Wiley & Sons, Inc., 19^1, pp. 53-59. NACA RM No. L8J14 15 \ (<\ V •H bD CO CD & P> P> •H ra CQ (D ?H Ph -p CO O O -P •H O & o 13 O o •H P> «5 •H o in o lO o to o K fr 16 NACA KM No. L&TlU -2 -1 S P/Po P/P T/T "^NACA^ J* 6 Uach number, M (a) T Q = 1000° F abs. ID Figure 2.- Percent error involved in the use of constant -heat-capacity formulas to obtain T/T , p/p , and- p/p f° r air. NACA RM No. L8jlU 17 -16 -12 -8 -h | § P/p . T)/d trl fQ ■ T/T Q i U 6 Uach number, M (b) T = 2000° F abs. Figure 2.- Concluded. 8 10 18 NACA RM No. L&Jllj- *\£ as b o u n m 4) 1.0 I . 8 T .6 *» 1.20 ^ 1.00 .4 *■* .80 .60 .2 / -.2 K 4 t= \NACA^ Specific volume ratio, — Figure 3 >_ Van der Waals ' equation in nondimensional form. NACA EM No. L8J3A 19 20 NACA RM No. L8J1U 2.2 ^ 2.0 n •P 1 .fl CO Q) X o ■rt (4 1.4 1.2 1.0 50 100 150 200 Pressure atm Figure 5- _ Variation of the ratio of specific heats y for a Van der Waals gas. T Q = 520° F ats. NACA RM No. L8jlU 21 >n +5 •H S ft 01 o -p 01 83 Jq ' — >h •H 01 • OJ lH fl O 01 ft ft ^ (DO >Q O s o J3 ( ) Pi OJ A II o H ft -p m • a ^ri Rj m ttl u 01 O Tl Ti •H CD CQ -p CI -p C) o o H ft en -p O o •H -P ft 01 ft h CD Bl m (D 01 h rid 01 i -p Ti o pj m aj ft h C) id •H ft -P B 01 •H Ph m £ m 03 (1) h ft MD ft ■J °d CO ra o -P U O -P © p; sj tM ft Pi O tt) O O CO © O © £ U p >d 5 £§ H >> 1=4 PM ScO J* • ON O H a © s ^> a <; © q o -P -P Pen a n) O Dj fn 03 to a 08 aJ *t-< © H o o m o o UNIVERSITY OF FLORIDA 3 1262 08106 605 1 UNIVERSITY OF FLORIDA, UMENTS DEPARTMENT SCIENCE LIBRARY 17011