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Tx NACA RM No. LJal4 NATIONAL ADVISORY COMIK'EE FOR AERONAUTICS RESEARCH MEMORAWNUM NOTE ON THE IMPORTANCE OF IMPERFECTGAS EFFECTS AND VARIATION OF HEAT CAPACITIES ON THE ISENTROPIC FLOW OF GASES By Coleman duP. Donaldson SUMMARY The errors involved in using the perfectgas law pv = RT 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 perfectgas law pv = RT 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 perfectgas law and the assumption of constant heat capacities. Tsien (reference 1) 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 imperfectgas and vibrationalheatcapacity 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 RM No. IBJ14 The present paper is arranged in the following three parts: temperature effects on perfectgas flows due to variation of heat capacities; imperfectgas effects cn 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 b c Cp Cv E .. M p P R T v V 7 P Subscripts o term in Van der Waals' equation correcting for the effect of molecular forces term in Van der Waals' equation correcting for the effect of molecular size speed of souni, feet per second heat capacity at constant pressure heat capacity at constant volume energy, footpounds Mach number (v/c) pressure, pounds per square foot gas constant absolute temperature, degrees Fahrenheit specific volume, cubic feet.per slug velocity, feet per second ratio of heat capacities (Cp/CV) density, slugs per cubic foot characteristic temperature of molecular vibration stagnation conditions critical conditions NACA PI No.. L814 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 CpT + j2 = CpTo P 2 p 0 If this equation is combined with the equation for the isentropic speed of sound c2 = 7RT the resulting equation is =2 2 T 1i) (1) If the expansion is Isentropic, the pressure and density ratios corresponding to the Mhch number are and (2) 0 (T /R However, if the temperature of the gas is high enough the heat capacities may not be assumed constant because the vibrational degrees of freedom of polyatomic molecules are excited. The variation of the equilibrium value of the heat capacity at constant volume of a perfect diatomic gas is found from quantum mechanical considerations to be of the form C v 5 ) 2 e/T E 2 T (e/T 1)2 where e is a constant depending on the gas. The formula may be used for the mixture air if the value of 0 is placed equal to 5526 when absolute temperature is measured in degrees Fahrenheit. (See the 4 NACA RM No. I&T14 appendix.) The value of the heat capacity at constant pressure far a perfect gas is then p = 7 9 2 R 2 T2 e /T (e/T 1)2 Figure 1 is a plot of equation (4) and shows that may not be considered constant above 6000 F absolute. the heat capacity When the heat capacity at constant pressure varies according to equation (4) the energy equation must be written J/To 2 To Substituting equation (4) into equation (5) and integrating yields 7RT + 2R w2 = 7RT + 06/T 1 The Mach number is obtained from equation (6) by 7RT = c2, which gives 1) 29 1 /1To  1 2RO ee o 1 by dividing through 1 i   e9/T  1 Cp/ 7 (ee/T CV/P 5 (e/T  2 + 2) 2e/T  12 + 'Tle pressure ratio corresponding to this Mach number is obtained from tile luentropic equation where M2 7 I f M (T S2 (e/T NACA RM No. L8J14 log = R T POii by substituting equation (4) into equation (9) and integrating to give p 7/2 1 e/To Po :o 1 1 ee/T eO/T e0 /T1 e ee/To To ee/TO_ (10) Similarly, the density ratio is found to be P /T 5/2 P o 1 e/To 1  e/T ee/T e0/T1 e To e0/To e /To_1/ (11) The differences involved in the use of equations (1) and (2) to predict the temperature, density, and pressure ratios corresponding to a given Mach number are given in figures 2(a) and 2(b), n terms of the percentage differences from the value given by equations (7), (10), and (1) for stagnation temperatures of 10000 and 20000 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 10000 F absolute are involved. Errors Involved in the Adsumption of the PerfectGas Law pv = RT for a Gas with Constant Heat Capacities In order to.evaluate flows in which imperfectgas 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 Waals (p (v b = RT (12) \ v2i NACA RM No. IBJ14 where b is a term correcting for the volume occupied by 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 p, v, and T have been made nondimenslonal by dividing by the values of these quantities at the critical point pc, cs, and Tc, thus making the graph suitable for any gas. (See reference 2.) The graph may be used for air if an empirical critical point (pe = 37.2 atm, Tc = 238.50 F abs., vc = 0.6438 slugs/ft3) is assigned to that mixture of oxygen and nitrogen. To give this critical point the values of a, b, and R 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 105, b = 0.654, and R = 1716. The proper equation for an isentropic expansion of a real gas is (see reference 3) dE = Cv dT + T c dv = 0 (13) [ TTv=Conetant] which for Van der Waals' equation becomes dE = Cv dT + p dv + 1 dv = 0 (14) v2 Equation (14) may be written as dE = Cv dT + d(pv) v dp + dv = 0 and since v dp = w dw dE = Cv dT + d(pv) + L dv + w dw = 0 (15) v2 Assuming constant heat capacity at constant volume and integrating equation (15) gives E = CT + vp + Constant = Eo (16) v2L + NACA RM No. L8J14 This is then the energy equation for a Van der Waals gas. Dividing through by the Isentropic speed of sound c2 = d + T RT dp CV (v b)2 v and since p . = RT 2a Sv2 v b v2 then 2 RTo 2CvTo + 2vo v b (vo b (1+  2a 2 2CvT 2v vo2/ ( PT _a v b v2) S\) v2RT g (v b)2 v The value of v for an isentropic expansion to be placed in equation (18) can be formed from equation (14) as follows: Cv dT + p dv + dv = CT dT+  dv= 2 v b then C dT  T vb and if C, is constant Tv = (v/ R v = (vo b)(0r) Sb (20) From equations (18) and (20), mkowing the stagnation conditions for an expansion from To to T, the Mach number may be calculated. The pressure ratio is then found to be (17) (18) (19) 8 NACA RM No. IBJ11 RT a P = v v (21) Po 0o _a vo b Vo2 and substituting the value of v from equation (20) we obtain RT a (Vo b) vo b)(2 + b Po RTo a ^b Vo b Vo2 Figure 4 shows the conventional pressure ratio and area ratio pw/(pw)F. l plotted against Mach number for air starting from stagnation conditions of 5200 F absolute and various pressures compared with the value obtained using constant ratio of heat capacities and the perfectgas law. Also shown in figure 4 are the values of pw/(pw)?W1 computed by Tslen's method. It is seen that as the imperfectgas effects become large it is no longer possible to simplify the analysis by neglecting terms containing the squares of and i although Tsien's results v pv2' pV2 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 Van der Waals gas c2 = l + 2 v2PT (17) dp C.1 (v b)2 v is not equal to 7PT. The expression for the ratio of specific heats in a Van der Waals gas is 7 pv2 a (23) v pv2 a + ab NACA RM No. L8J14 9 Srcu CMj S p ' * H 1 0 rMl o o. P 0 l 00 0 0 I &H 4w' E] rlcP ca S ) a + IC o 0 0 w o + ci H p 0 0 4 + a 1' ao 44 0 0 c 6 I Ep ' 0 000 0 Hi i s .4a + 0 0 o ) + 1 H H H +2+' CL p H I ( 0 C1 m h' 0 0 V + 0 H M + 0 .4H 11H . 4n 4 0 a) E4 r0 +i + 41 E + P O6 e8 0 _I ( +0m 0 C 0 0 m 4+2 ro tio H ( ^ Fd0 0 H i 0 I 4q0 +) e' FP 02 0 nCa r.i (Di i H rj r0Pi HD 4) 0 , i )g 0 q cr < u ul c (d H *g 1 0 5 0 4) ID 0 a ) )C 0 rI 7 F4 4) M 4 tI CL( p NACA RM No. IBJ14 r1 ccu (D 0 0 01 V 0 0as d 0 0 o o E 0 0 a I I ,I oE H H I 0 0 iO 0 D c(D o 0 la 04 0 H > 0 0 m r ? o ,o + 0 0 t oS H 0 0 0 0 P P E0 H0 I O 0 I O0 0I 0 0 0 H ) o i m A , oU 0 0 $ ii { <0 0 ^ S o NACA RM No. I&14 The value of the ratio of heat capacities 7 in this case is (er2 T}) 7 = ee/T (e e/ 1)2 5 \2 + (+ pv2 + a p2 a b v ee/ (e 1) 2 but the speed of sound is found from equation (17) by substituting the value of Cv/R from equation (3) to be c2= 1 5 fr  + ' 2 T) ee/T (e/T ) vT 2a (v b)2 v Figure 6 shows the conventional pressure ratio and area ratio plotted against Mach number for air starting from stagnation conditions of 20000 F absolute and various pressures compared with the value obtained using constant ratio of heat capacities and the perfectgas law. DISCUSSION The foregoing analyses show that the effects of variation of heat capacities with temperature do not became important in isentropic expansions of air until stagnation temperatures of the order of 10000 F absolute are encountered. Above 10000 F absolute, however, for accurate analysis this variation must be taken into account. In general, it may be stated that /0Y 9/T for diatomic gases these effects are important when ()2 ee / T (e/T 1)2 becomes appreciable compared to the number 2.5. The effects of Van der Waals' 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/v2 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 5200 F absolute are encountered. (29) NACA RM No. 18j14 Talen's method agrees well with the results of this investigation up to 50 atmospheres in this case, but it appears that it is not possible to neglect the squared terms of h and a when the effects v v2p of Van der Waals' forces become appreciable. CONCLUSIONS In many cases found in very high Mach number wind tunnels and in flows of high stagnation temperature or pressure, imperfectgas effects and the effects of variation of heat capacities may be present. For diatomic gases the effect of variation of heat capacities becomes important when V2 ee 2 becomes appreciable compared to 5/2. vT (es l (T 'e 1) For air these effects become appreciable when stagnation conditions of 10000 F absolute or larger are encountered. Imperfectgas effects become important in gas dynamics when a/v2 becomes appreciable compared to the pressure p or b becomes appreciable compared to v. When air is expanded from a stagnation temperature of 5200 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 Van der Waals' forces become appreciable. Langley Aeronautical Laboratory National Advisory Committee for Aeronautics Langley Field, Va. NACA RM No. IJ14 13 APPENDTI DERIVATION OF TBE 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 Shrbdinger'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 (see references 4 and 5) per particle found in this way is hV+ heV E =2 V + 2 ehvAkT 1 where Planck's constant characteristic frequency of molecular vibration absolute temperature Differentiating to obtain the contribution to the heat capacity of this energy yields Cvib 1 LE = hY 2 R k 3T \kT/ ( hv/kT (ehv/kT 2 hv For a particular gas = e is a constant and may be determined from spectroscopic data. The heat capacity at constant pressure is then R 2 R 2 R 2 (T2. \T/ ee/T (e /T )2 NACA RM No. I &14 The value of 0 for oxygen is 4010.4 and for nitrogen is 6044.4 far absolute temperatures measured in degrees Fahrenheit. The value 5526 may be used for air. REFERENCES 1. Tsien, HsueShen: OneDimensional Flows of a Gas Characterized by van der Waals' Equation of State. Jour. Math. and Phys., vol. XXV, no. I, Jan. 1947, pp. 301324. 2. Jeans, James: An Introduction to the Kinetic Theory of Cases. Cambridge Univ. Press, 1946, pp. 9698. 3. Epstein, Paul S.: Textbook of Thermodynamics. John Wiley & Sons, Inc., 1937, pp. 6465. 4. Frenkel, J.: Wave Mechanics. Elementary Theory. Second ed., Oxford Univ. Press, 1936, pp. 7780. 5. Lindsay, Robert Bruce: Introduction to Physical Statistics. John Wiley & Sons, Inc., 1941, pp. 5359. NACA RM No. L8Jl4 15 o Cc 8 S(U cm U 40 I t o \ W '4 4 O ID .0 o Cc 34 43 Cr L0 A 1Q NACA RM No. L8J14 P/Pa I.o .T/To 02 1 6 8 0 Mach number, M (a) To = 10000 F abs. Figure 2. Percent error involved in the use of constantheatcapacity formulas to obtain T/T p/po, and p/po for air. NACA RM No. L8Jl& 2 h 6 8 10 Mach number, M (b) To = 2000 F abs. Figure 2. Concluded. 16 12 8 8 S0 NACA RM No. L8J14 N. 1.20 1.00 4 21 .60 4 0 1 2 3 4 5 Specific volume ratio, L Figure 3. Van dev aals' equation in nondenional form. Figure 3. Van def Waals' equation in nondinensiona.l form. U. 0 4) C. S Q0 p. O 4** r  a 00 NACA RM No. LJ&T1 m CD 4l I U, a) 0 o 0 C. L1 (D 4,O rU Ud 0P 0 a) IACA RM No. L&SJI 2.0 1.8 1.6 1.4 1.2 1.0 0 Pressure p atm Figure 5. Variation of the ratio of specific heats 7 Waals gas. T = 5200 F abs. for a Van der 2.2 50 100 150 200 NACA PM No. L8J14 21 0 / // 0  f. /// I I I " * Id O N c 0 __ S00 4. 0 Uo \/A ,^ I1 _  __ ___ _ _ 0 4, 4d T Od ___ ___U   ^A^ ^^^ ' C mo 0 to4 4he p43 SA +1 o 0 t ri *+ pq0 S 0 H Z 'a 0e9 0 4 0 TII 4d (d W0 Fl 0 z7 0 0 rd z/ i odo o d , a ao4 O M ; o 1 0 D S 0 C il M *(T\ < ' M 41 ad 0 IDp 4D a 4 p p ( O m 43 p C DO C 0 w 0 * Id 0' 143 EsM E rD H0 0 Il g^ ^S )^ s &. P tj V r4 4j, 4 ic o I,t I I T, Vv, "' I 'I i4'j ?N fv T ".4 4r j 3 1262 081 
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