1、TECHNICAL NOTE 4230 PRANDTL-MEYER EXPANSION OF CHEMICALLY REACTING GASES IN LOCAL CHEMICAL AND THERMODYNAMIC EQUZIBIZUM By Steve P. Eeims Ames Aeronautical Laboratory Moffett Field, Calif. Washington March 1958 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from
2、 IHS-,-,-TECH LIBRARY KAFB, NM 1T NATIONALADVISORYCOMMITTEE FORAERONAUTICS TECHNICAL NOTE 4230 PRANDTL-MEXEREX3NSIONOFCHFMICYRRACTINGGASES INLOCALCHESIICALANDTHNRMODCEQUILIRRIUM By Steve P. Heims It is found that Prandtl-Meyer flow, in which chemical reactions = 7 are occurring and are in equilibriu
3、m, can be simply and exactly cslcu- lated. The property of air which governs the flow is found to be a quantity which depends only on the ratio of enthalpy to the square of the speed of sound; the analogous quantity for an inert gas depends only on the ratio of specific heats. The maximum angle thro
4、ugh which the flow may turn is generally larger when chemical reactions are occurring than it is in nonreacting air. A numerical example shows that the pres- sure variation with angle, as well as temperature and Mach number varia- tions,may be considerably affected by the presence of the chemical re
5、action. INTRODUCTION At the high temperatures encountered in hypersonic flight, the air may no longer be regarded as an inert gas. It does not have a constant ratio of specific heats, 7, nor does it generally obey the simple equa- tion of state, p/p = RT. These thermodynamic features reflect the fac
6、t that at high temperatures moleculsx vibrations are excited and chemical reactions are taking place in air. Because of this, any flow solutions depending on constancy of 7 and the perfect gas law are not valid. One elementary supersonic flow solution is the Prandtl-Meyer expan- sion around a corner
7、. In this paper the theory of the Prandtl-Meyer expansion is extended to include high temperature flow in chemical and thermal equilibrium. When only molecular vibrations are active, and no chemical reactions occur, one can still use the usual flow equations in terms of 7, if the appropriate functio
8、n of temperature is inserted for 7 (see ref. 1). However, when chemical reactions are occurring, then 7 is no longer a useful concept in the Prandtl-Meyer flow. Instead of working with 7, we shall employ a quantity 9 to des- cribe the thermodynamics of the gas, because it is q and not 7 which enters
9、 into the flow equations at high temperatures. When no reactions are taking place, 9 reduces to (7+1)/(7-l). In the present anslysis, it will be shown that by introducing also an auxili.ary variable q, the Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-
10、,-,-2 NACA TN 4230 Prsndtl-Meyer expansion can be sFmply and exactly calculatec for equi- librium flow without employing any iterative procedures or e.utensive numerical. integrations. However, a table orla Moldier diagram for the thermodynamic properties of air at high temperatures is required. At
11、I b- - high temperatures, the table of reference 2 or the Mollier chart of reference 3 is suitable. At temperatures below 3000 K, reference 4 is useful. - a speed of sound C a constant ofmotion with dimensions of velocity, defined by equation (3) enthalpy per unit mass Plsncks constant divided by 2n
12、 Bolt- constant Mach number P r R S T V vr % 7 9 8 V vm P pressure radial coordinate gas constant for air entropy absolute temperature speed, ,/m component of velocity along the radius vector component of velocity perpendicular to the radius vectqr ratio of specific heats, cv parameter defined by l+
13、 9 angular coordinate angle through which the flow has turned theoretical maximum value of v density SYMBOLS c - cP Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 4230 3 . Ik an angle defined by equation (7) Jr an integration variable Subscr
14、ipt 0 value along the radisl line on which the Prandtl-Meyer expansion begins In the usual derivation of Prandtl-Meyer flow (ref. 5) one seeks a solution for supersonic flow for which the pressure, density, and velocity ANAIJYsrs are constsnt along radiel lines emanating from the corner around which
15、 the gas is expanding. Such a solution can exist only when no character- istic length enters into the problem. Thus when chemicsl relaxation occurs in the air, one cannot have such a solution. However, when the air is in equilibrium everywhere, there is no length and one may expect to find such a so
16、lution. More generally, if each of the various reac- tions and internal degrees of freedom of the molecules are either frozen or in chemical equilibrium, a Prandtl-Meyer type solution is expected to exist. We Seek a formal solution of the Prandtl-Meyer type which is valid for any of these isentropic
17、 flows. If we define a2 = 22 ap s the variation of pressure with density at constant entropy, we obtain from the requirement of continuity and from momentum conservation the usual equations: I=v de 8 (1) (2) Equation (2) requires that the flow be supersonic. The energy equation iS 1 c2 vg + vr2 2 =h
18、+ 2 (3) Combining equations (1), (2), and (3) gives the differential equation for Vy: Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACA TN 4230 1 1 + (2h/a2) b“ - vr2) (4) Let 8, be the angle at which the Prandtl-Meyer expansion begins. Then in
19、view of equation (2), the initial radial velocity is (see sketch): vr(Q0) = 43 R-=i (5) It is seen from equations (1) and (4) that, for given initial condi- tions, the flow is completely specified if the quantity q is known, where we define -. L Y (6) For air at moderate temperatures, the ratio of s
20、pecific heats is a constant, and v = (7+1)/(7-l) = 6. For a diatomic gas in which the molecular vibrations are in equilibrium but no reaction is occurring, q is a function of temperature (but not of pressure) and varies from 6 at moderate temperatures to 8 at temperatures that are large compared to
21、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-c * NACA TN 4230 5 Hw/k, where w is the characteristic frequency of molecular vibrations. In general, when chemical reaction is also occurring, the quantity v is a function In order convenient to or of
22、local temperature to integrate equation (4) in this general case, it is introduce a variable q(6) defined by with q. = sin -e sin eo)= sin- and the frozen flow is identified with a relatively smaller and a constsnt q. (see appendix). %ch e point here is not so much that T = 0 and p = 0 at vm; in fac
23、t the ideal gas model breaks down completely at temperatures where the air begins to liquefy. For example, near T = 0 the specific heats are zero, and do not have the constant values associated with ideal gases at room temperature. The point is, more precisely, that the v corresponding to the state
24、of the air in which the ideal gas laws break down is much larger in the large 7 case than in the small q flow. In the text the simpler statement is preferred; however, all of the equations in this paper are generally valid and do not depend on any specific gas model. - - Q c Provided by IHSNot for R
25、esaleNo reproduction or networking permitted without license from IHS-,-,-2T NACA TN 4230 . 9 Pressure Y First of a31 one sees from figure 3 that unlike the density curve the equilibrium values for pressure are always higher than the frozen ones. Moreover the equilibrium and frozen pressure curves a
26、re well sepa- rated, mch more so than the density curves. For example at v = 70 the equilibrium pressure is nFne times as large as the frozen pressure. The corresponding ratio of the densities is 2. The point of interest is that in a Prsndtl-Meyer expsnsion the pres- sure is quite sensitive to the c
27、hemical reaction, much more so than the density. This is just the opposite of what o.ccurs In the cconpression by a normsl shock, in which the pressure ratio is very insensitive to sny chemical reaction, but the density ratio is not. Mach Number snd Turning Angle . L The local Mach number variation
28、and its strong dependence on 7 in the example are shown in figure 4. The more general qualitative variation oz;y number snd turning sngle can be derived from equations (X2), (13), It is seen from these equations that for a given angle 8, the Mach QP de 60 s.nct g2 = r12 de will be different. We ask
29、the 60 question: Under what conditions will the density p, of flow 1 be smaller than that of flow 2 for small angles? - Expansion of equation (14) for small angles 6 yields: 0 k, + g2M2 P/P,= F” 8 1 (g, - g2) + g Cg”3+ o(P) From this expression it is seen that the condition that p, be less than p, f
30、or sufficiently small 8 is either Q, vo2. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NELL TN 4230 ll REFxmmcEs 1. Ames Research Stbles, and Charts for Ccmpressible Flow. NACA Rep. n35, 1953. 2. Hilsenrath, Joseph, and Beckett, Charles W.: Tables
31、 of Thermodynamic Properties of Argon-Free Air to 15,OOOO K. AJDC TN 56-12, NBS, Sept. 1956. 3. Feldman, Saul: Rypersonic Gas Dynamics Chart for Equilibrium Air. AK!0 Research Lab., Jan. 1957. 4. Hilsenrath, Joseph, et al.: Tables of Thermal Properties of Gases. NBS Circular 564, Nov. 1955. 5. Taylo
32、r, G. I., and Maccoll, J. W.: The Mechanics of Compressible Fluids. Vol. III of Aerodynsmfc Theory, div. H, ch. IV, sec. 5, W. F. Durand, ed., Julius Springer (Berlin), 1943, pp. 243-246. . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA l!N 4
33、230 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 4230 13 1 t - Equilibrium - Frozen 7 6 5 I O0 100 2cP 300 400 5o” 6o” 70* 800 Flow deflection, u 2h Figure l,- Comparison of q = l-t 2 audfi for frozen and equilibrium Prskdtl-Meyer flow; M.
34、 = 1.00, To = 614-0 K, p. = 1.2 atiosppleres. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 IW!A TN 4230 _-. O0 IO0 20* 4o“ 60 80 Flow deflection, Y Fi .gure 2.- Comparison of density for frozen and equilibrium Prandtl-Meyer flow; M, = 1.00, To
35、= 6140 K, p. = 1.2. atmospheres. .-. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 4230 O0 100 2o“ 40* 60 80* Flow deflection, v 3 c Figure 3.- Comparison of pressure for frozen and equilibrium Prmdtl-Meyer flow; M, = 1.00, To = 6140 K, p.
36、= 1.2 atmospheres. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 12 I I I j I I I I I I M 0 - Equilibrium - Frozen 0 lo” 28 30 40 50 SO” 70 80 90” Flow deflection, Y Figure 4.- Comparison of Msch number for frozen and eqtilibrium Prandtl-Meyer f
37、low; M, = 1.00, !s T, = 6140 K, p. = 1.2 atmospheres. ;G % c . I c Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-rl I P L 2 E 1.4 :D F I f I I.2 .8 .6 - Equilibrium - Frozen 0 lo” 20 30 40 50” 60 70” 80 9C Flow deflection,u )0 Figure 5.- Ccmrparj.son of tmperature for frozen and equilibrium F?rmdtl-Player flow; M. = 1.00, To = 6140 K, p. = 1.2 atispheres. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
