| DEPARTMENT OF ENGINEERING RESEARCH.-UNIVERSITY OF MICHIGAN. UMM-6 Copy No. /14 UNIVERSITY OF MICHIGAN Ann Arbor EXTERNAL MEiMRANDUM NO. 6 Project MX-794 (AAF Contract W33-038 ac-14222) Project "Wizard" "Aerodynamic Relations with Variable Specific Heats" Prepared by 9 &. D. Schetzer C. H. Lauritsen April 15, 1947

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Pageii CONTENTS Page Summary and Conclusions............... 1 Symbols...................* 3 Introduction. 0........*. ~. ~ 4 Discussion............... 6 References.....................11 Appendix......... ~ ~ ~ ~ ~ ~ ~ ~ ~12

DEPARTMENT OF ENGINEERING RESEARCH Page iii UNIVERSITY OF MICHIGAN Report No. UMM-6 LIST OF ILLUSTRATIONS Figure Title Page 1 Variation of ( 1- l) with Temperature.... 16 2 Variation of p with Mach Number....... 17 PO 3 Variation of/- with Mach Number....... 18 /o0 4 Variation of T with Mach Number...... 19 TO 5 Errors; Isentropic Flow, S Constant..... 20 6 P2 6 Variation of P2 with........... 21 P1 7 Variation ofQ2 with............. 22 Variation of T1 P2~ 2 T 8 9 Variation of.2. with il............ 24 0P1 Pl~ 10 Variation of M2 with M........... 25 11 Errors; Shock Flow, 6 Constant....... 26 12 Variation of / with T2.......... 27

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 1 SUMMARY AND CONCLUSIONS It has been common practice to base thermodynamic calculations concerning the flow through ducts and across shock waves on formulae which have been derived under the assumption that the specific heats of air are constant. For Mach Numbers greater than two, the range between the static and stagnation values of the temperature is large, and the variation in the specific heats of air between these temperatures cannot be neglected. In this memorandum, the influence of variable specific heats on certain flow relations has been investigated. For isentropic flow, it has been found that the error in the ratio of the static value to the stagnation value of the pressure, density, and temperature calculated on the basis of a specific heat ratio of 1.4, is less than 1% for flow Mach Numbers that do not exceed two. For Mach Numbers greater than two, the errors rise rapidly. For a Mach Number of six, the errors in the pressure, density, and temperature ratios are 26%, 34% and 13% respectively. Values of the ratios versus Mach Number are plotted on Figures 2, 3 and 4. Errors versus Mach Number are plotted on Figure 5. For flow through a normal shock wave, the ratios of the upstream to the downstream values of the pressure, density, temperature,and stagnation pressure have been plotted versus Mach Number on Figures 6, 7, 8, and 9. On Figure 10, the Mach Number downstream of a normal shock has been plotted l 1I llll1 1I,.....', l

DEPARTMENT OF ENGINEERING RESEARCH Page 2 UNIVERSITY OF MICHIGAN Report No. UMM-6 versus the Mach Number upstream. On Figure 11, the error arising as a consequence of holding the specific heat ratio constant at the value 1.4 is plotted versus Mach Number. For Mach Numbers less than two, the use of a constant value for the specific heat ratio leads to an error of less than 1% in the quantities considered. At an upstream Mach Number of six, the errors in the downstream values of the pressure, density, temperature, and stagnation pressure ratios are 3%, 16%, 11%, and 24% respectively. The error in the downstream value of the Mach Number is 5%. The calculations for flow through a shock wave have been made for an upstream static temperature of 520~R. At lower temperatures, the errors resulting from the use of a constant specific heat ratio are smaller. Calculations at a Mach Number of six using an upstream static temperature of 393~R (stratosphere temperature) have been made and this one point is plotted on Figures 6 through 10.

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 3 SYMBOLS a = Local speed of sound, ft per sec c = Specific heat at constant pressure, Btu per lb per degree p c = Specific heat at constant volume, Btu per lb per degree h = Enthalpy, Btu per lb mass; OR, ft lb per slug p = Pressure, lb per sq ft u = Velocity, ft per sec v = Specific volume, cu ft per slug M= 1Mach Number R = The gas constant (1715 ft lb per slug per degree Rankine) T = Absolute temperature, degrees Rankine RT 3 u2 + 2, ft per sec U2 = cE t atio of specific heats cv = u + R, ft per sec u1 p = Density, slug per cu ft Subscripts: O = Stagnation value 1 = Position immediately upstream of shock 2 = Position immediately downstream of shock s = Arbitrary standard value Superscripts: * = Critical value (M = 1) 0 = Stagnation value

DEPARTMENT OF ENGINEERING RESEARCH Page 4 UNIVERSITY OF MICHIGAN Report No. U-6 INTRODUCTION In aerodynamic calculations at low hLoch Numbers the temperature variationsare sufficiently small that the specific heat ratio b = E can be considered constant. This permits cv the quantities, - and T for isentropic flow and P fPO' TO the quantities P2 P2, T2 and 2 for shock flow to be 1 P 1 T1 P10 expressed conveniently in terms of the flow Mach Number. At high Mach Numbers V cannot be considered constant and the validity of formulae based on this assumption must be investigated. In Reference 1, values of the specific heats of air at vanishing pressure, computed from experimentally determined data on nitrogen and oxygen, are listed for temperatures between 200~R and 6600~R. Values of the enthalpy computed from these specific heats are shown to be accurate within close limits, even when the pressures are very high. Values of V have also been computed from these data and are listed in Table 3 of Reference 1. On Figure 1 of this report, the important parameter ( 6 - 1) is plotted versus temperature. The variation of this factor in the temperature range between 200~R and 6600~R is 42%. It is interesting to note that at 6400~R, the value of V computed in Reference 1 is 1.280, which is close to the value 1.286 predicted by Kinetic Theory for gas molecules with seven degrees of freedom. The subject of variation

DEPARTMENT OF ENGINEERING RESEARCH Report No. UhM-6 UNIVERSITY OF MICHIGAN Page 5 in specific heats of gases with temperature is discussed in References 2 and 3. From the enthalpy tables of Reference 1, it is a simple matter to compute the isentropic flow relations versus Lach Number. The flow through a normal shock is more involved and a graphical solution to the equations is employed. At present, the very high stagnation temperatures resulting from supersonic flow at Mach Numbers between three and six are encountered only in free flight. Therefore calculations have been based on a temperature of the undisturbed flow of 520~R at all Mach Numbers. It should be noted that in flight at high altitudes the undisturbed flow temperature is lower and errors in the formulae based on constant r may be expected to be smaller. Cne calculation of the parameters has been made at a Mach Number of six and an undisturbed flow temperature of 393~R to illustrate this point. It is also pointed out that the temperature rise through a plane, or conical, oblique shock is smaller than the rise through a normal shock under the same conditions. Therefore, the errors indicated in this report serve as an upper limit on the errors one may expect if V is treated as a constant in calculating the parameters downstream of any shock wave.

DEPARTMENT OF ENGINEERING RESEARCH Page 6 UNIVERSITY OF MICHIGAN Report No. UbM-6 DISCUSSION I. Isentropic Flow The energy equation for a gas can be written: (1) -2 - c + dT = c dT 2 + P P If the specific heats are considered constant, the energy equation reduces to: u2, ~ p.. ~ PO 2' f1 Po Assuming further that the flow is isentropic, and employing the relation iy equals a constant, the static to stagnation ratios re easily derived. For 6= 1.4 (2) 2 = (1 + 0.2M2) 3'5 PO (3) P- = (1 + 0.22)-2'5 Po (4) T- (1 + 0.22)TO For high Mach Numbers, Equations 2, 3, and 4 are invalid because of the large variation in', aid air tables similar to those of Reference 1 must be employed. In the air tables, the values of enthalpy f c dT, pressure ratio 2-, and specific 5 PPs volume ratio., are tabulated for values of the temperature between 300~R and 6500~R. From these figures it is a simple matter to determine the ratios E,mP, and T, versus stream PO pO T

Re t DEPARTMENT OF ENGINEERING RESEARCH P Report No. UNM-6 UNIVERSITY OF MICHIGAN Page 7 Mach Number for any given value of stream temperature. This has been done for a Stream temperature of 520~R and the results have been plotted on Figures 2, 3, and 4. A sample calculation is given in the Appendix. For comparison, the relations for a constant' given by Equations 2, 3, and 4 are plotted on the same figures. The correction factors which must be applied to Equations 2, 3, and 4 are plotted versus Mach Number on Figure 5. II. Flow Through a Normal Shock \J ~ The relations across a normal shock can be found by combining the three equations expressing the conservation of mass, momentum,:i and energy and the equation of state, in the proper manner* These equations are listed below. The subscripts 1 and 2 refer to positions immediately upstream and downstream of the shock, respectively. (5) fu1l = 2U2 conservation of mass (6) P1- P2 =p 2u22 -p1ul2 conservation of momentum 2 u 2 (7) l + hl 2 + h2 = ho conservation of energy (8) p =RT equation of state If 5 is considered constant, conservation of energy can be reduced to: (7A) u12 al2 u2 a22 + 1 = 2 e 2 + o-e r 2a nm i s2(o- u) a The relations across a normal shock, using Equations 5, 6, 7h,.... 1,,,,1 1,,

DEPARTMENT OF ENGINEERING RESEARCH Page 8 UNIVERSITY OF MICHIGAN j Report No. U13i-6 and 8 and the value b = 1.4, are: (9) p = 1.165 M2.167 P1 ~20 2 (I2 =) L2 2, 20 (12) 1 Pi 0 M2 o _, 12 + 0.4 Li2i' (12) 2[:e 5+E1 7 M2 1 When dy is considered to be a function of temperature, Equation 7A is not available. Air tables must be used in its place. Equations 5, 6, and 8 do not involve the specific heats. The four anknowns P2,,p2, T2, and u2 occur in these three equations and from them u2 may be solved as a function of T2 in the following manner: Substituting (8) in (6): p2RT2,'01u..f 2 R2 which becomes, after making use of (5) and simplifying: (13) RT2 Ru+a 2 1 u Another value of u2 is found from Equation 7 (14) u2= (h0 -h) Substituting this in Equation 13 yields: (15) ^(!h,-i^.^.uRT RT1 (1) (h -h2~ =u 2 U1

.DEPARTMENT OF ENGINEERING ESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 9 From Equation 15 and the air tables (which give h as a function of T), the value of the downstream static temperature T2 may be solved for any set of upstream conditions ul and T1. A graphi-. cal solution shows two values of T that satisfy Equation 15. One solution indicates that T2 = T1 and corresponds to the trivial case of a discontinuity of zero intensity. After T has been determined for the specific upstream conditions u1 and T1, h2 can be found from the air tables and u2 can be computed from Equation 14. Then T2 and 2 are u2 T1 Il known. 2 can be computed from the continuity relaion:.~2 = l,?ip1 u2 and P2 from the equation of state: XI p1 P2 _ 2 T2 _ A _. _T2. /P1 T1 2 can be determined for any value of T from the air tables, and I2 computed from the definition M = 2. This leaves only the ~~~~2 2 a2 ratio of the stagnation pressures, which is computed from the following expression: p 0 P2 P2 P2 Pl Ip -P2 Pl Pl0 P20 Pl is read from the curves of Part I of this discussion, p2 P10 P2 can be computed from Equation 2 because the Mach Number downstream of the shock is less than unity. However,in this equation, care must be taken to use a value of y corresponding to

DEPARTMENT OF ENGINEERING RESEARCH Page 10 |UNIVERSITY OF MICHIGAN Report No. UM-6 the stagnation temperature of the flow. The ratios P2,P-, T2 and P20,computed in the manner P1 cJ~ T1 Pl described above, have been plotted versus Mach Number on Figures 6, 7, 8, and 9 respectively. On Figure 10, M2 has been plotted versus 1. On the same figures, these quantities as computed from Equations 9 through 12 are also plotted for purpose of comparison. The error which arises as a result of holding 5 equal to 1.4 is plotted versus Mach Number on Figure 11.

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 11 REFERENCES (1) Keenan and Kaye - Thermodynamic Properties of Air, John Wiley & Sons, Inc., New York, 1945, First Edition (2) Loeb - Kinetic Theory of Gases, McGraw-Hill Book Company, Inc., New York, 1927, First Edition (3) Kennard - Kinetic Theory of Gases, McGraw-Hill Book Company, Inc., New York, 1938, First Edition

DEPARTMENT OF ENGINEERING RESEARCH Page 12 UNIVERSIrY OF MICHIGAN Report No. UMM-6 APPENDIX I. Isentropic Flow Calculations As an example, the flow properties are calculated for a Mach Number of six and a static temperature of 520~R. From the air tables (Ref. 1), corresponding to T = 520~R: (1) a = 1118 ft per sec (2) (h - h ) < 28.77 Btu per lb mass Ps (4) = 1 5192 Vs jo /Os Then u = Ma = (6)'(1118) = 6708 ft per sec, and from the energy equation: (h - h) =(h-h ) + 2 o s s 2 28.77 + (608)2. 2 106 (40 x 10-6 converts ft lb per slug to Btu per lb mass) = 928.715 Btu per lb mass Again from the air tables (Ref. 1), corresponding to h - h = 928.715 Btu per lb mass o S To = 3785~R Po = 5345.0 Ps o~ = 1 _ 17.70 Vs Po

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 13 Then: T = 520.1374 TO 3785. E, Ps = (2.504). =.004685 P0 Ps P0 5345.0 V0 s = _v.&v 0 - 1 (17.70) =.003409 /o0 v v V 5192 T R' Assuming y to be constant, the expressions for T, and 0 p0 ~- may be obtained analytically. Thus:,/~o P0 L 2 =.000630 Tor.= 1.40 The error due to assuming y constant is therefore: A ( o ) =.000630 -.000468 =.000162 P0 The per cent error being: A (P- x 100 (po x 2o1 Based on constant specific.= -t - o2 = 25.7 heat ratio determination'.000630 of. PO Po Based on variable specific. 0162 = 34.6 heat ratio determination.0004685 of E PO II. Normal Shock Wave Calculations As an example of these calculations the flow parameters are computed for an upstream Mach Number of six and an upstream static

DEPARTMENT OF ENGINEERING RESEARCHeport No. UMM-6 Page 14 UNIVERSITY OF MICIGAN _ Report No. UMM-6 temperature of 520~R. Equation 15 is repeated below. ERT RT (15) 2(h -h2). h2 ) u RT or by defining/3 =2-i + RT1 and Y = ul + RTl 1 U1 (15a) 3t a This gives T2 and h2, (h2 = f(T2)), expressed only in terms of the initial conditions upstream of the shock wave. This equation is solved by plotting f against T2 for a specific u, and T1, and observing the values of T2 for which 3 = Y. See Figure 12. Intersections may be noted for values of T of 520~R and 3703~R. The first intersection represents the trivial case of a discontinuity of zero intensity. The remainder of the quantities may be obtained as follows: where h = 904.43 Btu per (1) u2 - V2(ho - h2) lb mass2 from Reference 1, 6(928 -904) --- corresponding to T2 " 3703.5 R. =V/2(928.8 - 904.43) 10 = 1103.8 ft per sec (2) /~_ =.67o08 6.0772,P u2 1103.8 (3) T2 223T039 7 7.1221 T1 520 7.122 (4) PP2 2 T2 - (6.0772)(7.1221) P1 1 T1 = 43.282

. 5 DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 15 (),,, 1103.8 ( u_ 2 =11ll 8 2 a 2872 =.3843 (6) P2 P. Pp22. P._ = (1.0997)(43.28-)(.004685) Pi0 P2 PI Pi0 =.0223 I

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DEPARTMENT OF ENGINEERING RESEARCH Page 18 |UNIVERSITY OF MICHIGAN Report No. UM-6 VARIATION OFIG. WITH MACH NUiMBER L,.... Ll~~~~~~~~~~~~~........m ~ ~ FiXgur No*ta

...I DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF MICHIGAN Page 19 Figure No. 4 VARIATION OF T WITH MACH NUMBER T 0 | k$ W @@ @ XX X XXX Xm X X X t~~~~~~~~~rT Xt Xs gg 4iS*8i| *S||*|ii~~~~4t SS % gW W i WAX~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~gREEE~~~~~~~I T EXX mm mmAV XX XX X gX fflX XmH S A~~~~~~~~~~~~~~~~~~~~~~m 2It4II!+I SUI+ItW e~~~~~~~~-L 7 S S L SS SiX S LX S S X XW x Wmm X XXX S~~~~~~~~~~~~~~~~~~~~~~~~I T L LX |tW4W X AT X A S FiXgur NoI+l~~lllll~lt..X i X S >: 1t1; 1t M

DEPARTMENT OF ENGINEERING RESEARCH Page u ____20 lUNIVERSITY OF MICHIGAN Ieport No. UMM-6 Figure No. 5 ERRORS; ISiNTROPIC FLOW, I CONSTANT H iT- -Y ------- H~~~~~~

~DEPARTMENT OF ENGINEERING RESEARCH Report No, UMM-6 UNIVERSITY OF MICHIGAN Page 21 Figure No. 6 VARIATION OF 2 WITH P1

DEPARTMENT OF ENGINEERING RESEARCH Page 22 UNIVERSITY OF MICHIGAN Report No. UMh-6 Figure No. 7 VARIATION OF72 WITH M1?l 2 2S g 2 2 22 XE i i 2 gH Sg8 - - - i X m gE e ilelililil illil-lilililililil~~~~~~~~~~~~~~~~~~ge a-tilttil!; I~~~~lit tl~ 444 EEt W -!XXX HWllllii lpt-~ftl$i!tlitilli$ttl i g g R R X i g g t 21 X g ffi ffi g E E X A 1 11 1Ft] 1 111 4 i 1' 1 + t~~~~~~~~~~~~~lj 01+ W~~g~lililt~~ll jl'll llsl~t ii l i i Si int >;}* H*~ ~~~~ +i-t - - +HI Il -4T P IT 1 ii ii tt= 1 = i:+4 X; j 11!1 m mi~t m m m i #t lltttt # m m m t # ett m m llllil!I!I!II!!I~tll jl~ll!!j~il~!lllll!lllT ~ttltt'+,H7-l mm m m m m m m 111!11111 mm t m m mm $M tt t m Ill~~~~~ll!!ilitl ll i~~ll~lll!!!ll~~tlitl!;!lllitl!0ll1Lll tit~~~tI~titt~tI jm gT 2 X g ~ gt t 1flll:lulll|l-fis{ i m 1 l ill:1 1it tH e ili lii l 1 1il l!! 1!1!+ett 1!!!1:!!1 Et T!!!11!1!!!!!l!!!lilit!!il!!ll~l~i~liliilil!lilillilI+iII!ItqlF!T-t IT - X SE - -- -- -- -- -- t it i ffffiffi g X EE W EE iiiW i Xll#!|||l~l|l 4l|1l ik||1 |-tltil ll~jl 1ti:|4ll!j11IT_~ * EE ESX m E WXEE ESSS S L. ntx~~~~~~~~~~~~~~~~~~~~- XEE EEW X~~ii E EE EE t~~t W iX Itililt~~~il;!lit;Lliiili, l S 1 SSE EWW WNWW W XW Wg kWW W X< t||;t |111ll~~zi |:11. til|.;W

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-6 UNIVERSITY OF PaICHIGAN gage 23 Figure No. 8 VARIATION OF 2 WITH M T4 1

DEPARTMENT OF ENGINEERING RESEARCH o... Page ^24 UNIVERSITY OF MICHIGAN Report No, UMM-6 Figure No. 9 VARIATION OF WITH M....

DEPARTMENT OF ENGINEERING RESEARCH Report No. UhM-6 UNIVERSITY OF MICHIGAN Page 25:, |:':.'.......;:::..... i.:.........:;:;::: ^ ^:;;: ^ ^: ^ ^;:::: IN N ^ II I ~~~I II IN II1 IIII 11 1 1 1 1 1 i IIIi ^ 11^ IN 1 11 l lllh ^ 111 11 I 1 1111l^ l Figure No. 10 VARIATION OF M8 ~^ITH M1

DEPARTMENT OF ENGINEERING RESEARCH Page 26 UNIVERSITY OF MICHIGAN Report No. VMM-6 i -^: ^ l~ l^ ^^ n |i|^ i i ii^ ^ ^:t~~~~~~~~~~4 - -------- --!r ~+i~ i| ^ -!:; -- - i: | ^^: * i; - -1 n -^ ^l ill IP? mt 6 "t tt i F~-;';:;i':; 1! ^;1"t;;;;;;;; Nl:; i- +~~~~~~~4 4- -tttttttt -t-rt —t- t1 1 m1 1 tOtttff + + H ir Figare No. 1 ERRORS; SHOCK FLOW, ^ CONSTANT t4I 11 t i # i ttl i-;-, t.I - f T -' i al Li Li-~~~~~~~~~~~~i~rtti-tt MRORS; SHOCK FOW,, CONST 11

DEPARTMENT OF ENGINEERING RESEARCH Report No. UI"YM-6 UNIVERSITY OF MICHIGAN Page 27 - - - ------ --:::: N;; - -- - — ^ S t j ^ ^ j;;::;;:;:;;::::::;::;:: Figure No. 12 VA~.IRIATION OF fi A'lTH T2

DEPARTMENT OF ENGINEERING RESEARCH _UNIVERSITY OF MICHIGAN UNIVERSITY OF MICHIGAN IIIIIiilt llllllll11 3 9015 03524 4469 DISTRIBUTION Distribution of this report is made in accordance with AN-GM Mailing List No. 2, dated February 1947, also Amendment No. 1, dated 24 February 1947, to include Part A, Part C and Part DA.