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Chapter 9 Compressible Flow Motivation. All eight of our previous chapters have been concerned with “low-speed’’ or “incompressible’’ flow, i.e., where the fluid velocity is much less than its speed of sound. In fact, we did not even develop an expression for the speed of sound of a fluid. That is done in this chapter. When a fluid moves at speeds comparable to its speed of sound, density changes become significant and the flow is termed compressible. Such flows are difficult to obtain in liquids, since high pressures of order 1000 atm are needed to generate sonic velocities. In gases, however, a pressure ratio of only 21 will likely cause sonic flow. Thus compressible gas flow is quite common, and this subject is often called gas dynamics. Probably the two most important and distinctive effects of compressibility on flow are (1) choking, wherein the duct flow rate is sharply limited by the sonic condition, and (2) shock waves, which are nearly discontinuous property changes in a supersonic flow. The purpose of this chapter is to explain such striking phenomena and to familiarize the reader with engineering calculations of compressible flow. Speaking of calculations, the present chapter is made to order for the Engineering Equation Solver (EES) in App. E. Compressible-flow analysis is filled with scores of complicated algebraic equations, most of which are very difficult to manipulate or invert. Consequently, for nearly a century, compressible-flow textbooks have relied upon extensive tables of Mach number relations (see App. B) for numerical work. With EES, however, any set of equations in this chapter can be typed out and solved for any variable—see part (b) of Example 9.13 for an especially intricate example. With such a tool, App. B serves only as a backup and indeed may soon vanish from textbooks. 9.1 Introduction We took a brief look in Chap. 4 [Eqs. (4.13) to (4.17)] to see when we might safely neglect the compressibility inherent in every real fluid. We found that the proper criterion for a nearly incompressible flow was a small Mach number V Ma    1 a where V is the flow velocity and a is the speed of sound of the fluid. Under small-Machnumber conditions, changes in fluid density are everywhere small in the flow field. The energy equation becomes uncoupled, and temperature effects can be either ignored or 571 572 Chapter 9 Compressible Flow put aside for later study. The equation of state degenerates into the simple statement that density is nearly constant. This means that an incompressible flow requires only a momentum and continuity analysis, as we showed with many examples in Chaps. 7 and 8. This chapter treats compressible flows, which have Mach numbers greater than about 0.3 and thus exhibit nonnegligible density changes. If the density change is significant, it follows from the equation of state that the temperature and pressure changes are also substantial. Large temperature changes imply that the energy equation can no longer be neglected. Therefore the work is doubled from two basic equations to four 1. 2. 3. 4. Continuity equation Momentum equation Energy equation Equation of state to be solved simultaneously for four unknowns: pressure, density, temperature, and flow velocity (p, , T, V). Thus the general theory of compressible flow is quite complicated, and we try here to make further simplifications, especially by assuming a reversible adiabatic or isentropic flow. The Mach Number The Mach number is the dominant parameter in compressible-flow analysis, with different effects depending upon its magnitude. Aerodynamicists especially make a distinction between the various ranges of Mach number, and the following rough classifications are commonly used: Ma  0.3: incompressible flow, where density effects are negligible. 0.3  Ma  0.8: subsonic flow, where density effects are important but no shock waves appear. 0.8  Ma  1.2: transonic flow, where shock waves first appear, dividing subsonic and supersonic regions of the flow. Powered flight in the transonic region is difficult because of the mixed character of the flow field. 1.2  Ma  3.0: supersonic flow, where shock waves are present but there are no subsonic regions. 3.0  Ma: hypersonic flow [13], where shock waves and other flow changes are especially strong. The numerical values listed above are only rough guides. These five categories of flow are appropriate to external high-speed aerodynamics. For internal (duct) flows, the most important question is simply whether the flow is subsonic (Ma  1) or supersonic (Ma  1), because the effect of area changes reverses, as we show in Sec. 9.4. Since supersonic-flow effects may go against intuition, you should study these differences carefully. The Specific-Heat Ratio In addition to geometry and Mach number, compressible-flow calculations also depend upon a second dimensionless parameter, the specific-heat ratio of the gas: cp k   c (9.1) 9.1 Introduction 573 Earlier, in Chaps. 1 and 4, we used the same symbol k to denote the thermal conductivity of a fluid. We apologize for the duplication; thermal conductivity does not appear in these later chapters of the text. Recall from Fig. 1.3 that k for the common gases decreases slowly with temperature and lies between 1.0 and 1.7. Variations in k have only a slight effect upon compressibleflow computations, and air, k  1.40, is the dominant fluid of interest. Therefore, although we assign some problems involving, e.g., steam and CO2 and helium, the compressibleflow tables in App. B are based solely upon the single value k  1.40 for air. This text contains only a single chapter on compressible flow, but, as usual, whole books have been written on the subject. References 1 to 6, 26, 29, and 33 are introductory, fairly elementary treatments, while Refs. 7 to 14, 27 to 28, 31 to 32, and 35 are advanced. From time to time we shall defer some specialized topic to these texts. We note in passing that there are at least two flow patterns which depend strongly upon very small density differences, acoustics, and natural convection. Acoustics [9, 14] is the study of sound-wave propagation, which is accompanied by extremely small changes in density, pressure, and temperature. Natural convection is the gentle circulating pattern set up by buoyancy forces in a fluid stratified by uneven heating or uneven concentration of dissolved materials. Here we are concerned only with steady compressible flow where the fluid velocity is of magnitude comparable to that of the speed of sound. The Perfect Gas In principle, compressible-flow calculations can be made for any fluid equation of state, and we shall assign problems involving the steam tables [15], the gas tables [16], and liquids [Eq. (1.19)]. But in fact most elementary treatments are confined to the perfect gas with constant specific heats cp p  RT R  cp c  const k    const (9.2) c For all real gases, cp, c , and k vary with temperature but only moderately; for example, cp of air increases 30 percent as temperature increases from 0 to 5000°F. Since we rarely deal with such large temperature changes, it is quite reasonable to assume constant specific heats. Recall from Sec. 1.6 that the gas constant is related to a universal constant divided by the gas molecular weight Rgas   Mgas where (9.3)  49,720 ft2/(s2 °R)  8314 m2/(s2 K) For air, M  28.97, and we shall adopt the following property values for air throughout this chapter: R  1717 ft2/(s2 °R)  287 m2/(s2 K) k  1.400 R c    4293 ft2/(s2 °R)  718 m2/(s2 K) k 1 kR cp    6010 ft2/(s2 °R)  1005 m2/(s2 K) k 1 (9.4) 574 Chapter 9 Compressible Flow Experimental values of k for eight common gases were shown in Fig. 1.3. From this figure and the molecular weight, the other properties can be computed, as in Eqs. (9.4). The changes in the internal energy û and enthalpy h of a perfect gas are computed for constant specific heats as û2 û1  c (T2 T1) h1  cp(T2 h2 T1) (9.5) For variable specific heats one must integrate û   c dT and h   cp dT or use the gas tables [16]. Most modern thermodynamics texts now contain software for evaluating properties of nonideal gases [17]. Isentropic Process The isentropic approximation is common in compressible-flow theory. We compute the entropy change from the first and second laws of thermodynamics for a pure substance [17 or 18] T ds  dh dp   (9.6) Introducing dh  cp dT for a perfect gas and solving for ds, we substitute T  p/R from the perfect-gas law and obtain  2 1 ds   2 1 dT cp  T  2 R 1 dp  p (9.7) If cp is variable, the gas tables will be needed, but for constant cp we obtain the analytic results s2 T s1  cp ln 2 T1 p T R ln 2  c ln 2 p1 T1  R ln 2 1 (9.8) Equations (9.8) are used to compute the entropy change across a shock wave (Sec. 9.5), which is an irreversible process. For isentropic flow, we set s2  s1 and obtain the interesting power-law relations for an isentropic perfect gas   p T 2  2 p1 T1 k/(k 1)   2 1   k (9.9) These relations are used in Sec. 9.3. EXAMPLE 9.1 Argon flows through a tube such that its initial condition is p1  1.7 MPa and 1  18 kg/m3 and its final condition is p2  248 kPa and T2  400 K. Estimate (a) the initial temperature, (b) the final density, (c) the change in enthalpy, and (d) the change in entropy of the gas. Solution From Table A.4 for argon, R  208 m2/(s2 K) and k  1.67. Therefore estimate its specific heat at constant pressure from Eq. (9.4): 9.2 The Speed of Sound 575 1.67(208) kR cp      519 m2/(s2 K) 1.67 1 k 1 The initial temperature and final density are estimated from the ideal gas law, Eq. (9.2): 1.7 E6 N/m2 p1 T1      454 K (18 kg/m3)[208 m2/(s2 K)] 1R Ans. (a) 248 E3 N/m2 p2 2      2.98 kg/m3 (400 K)[208 m2/(s2 K)] T2R Ans. (b) From Eq. (9.5) the enthalpy change is h2 h1  cp(T2 T1)  519(400 454)  28,000 J/kg (or m2/s2) Ans. (c) The argon temperature and enthalpy decrease as we move down the tube. Actually, there may not be any external cooling; i.e., the fluid enthalpy may be converted by friction to increased kinetic energy (Sec. 9.7). Finally, the entropy change is computed from Eq. (9.8): s2 T s1  cp ln 2 T1 400  519 ln  454  p R ln 2 p1 0.248 E6 208 ln  1.7 E6 400  334 m2/(s2 K) 66 Ans. (d) The fluid entropy has increased. If there is no heat transfer, this indicates an irreversible process. Note that entropy has the same units as the gas constant and specific heat. This problem is not just arbitrary numbers. It correctly simulates the behavior of argon moving subsonically through a tube with large frictional effects (Sec. 9.7). 9.2 The Speed of Sound The so-called speed of sound is the rate of propagation of a pressure pulse of infinitesimal strength through a still fluid. It is a thermodynamic property of a fluid. Let us analyze it by first considering a pulse of finite strength, as in Fig. 9.1. In Fig. 9.1a the pulse, or pressure wave, moves at speed C toward the still fluid (p, , T, V  0) at the left, leaving behind at the right a fluid of increased properties (p p,  , T T) and a fluid velocity V toward the left following the wave but much slower. We can determine these effects by making a control-volume analysis across the wave. To avoid the unsteady terms which would be necessary in Fig. 9.1a, we adopt instead the control volume of Fig. 9.1b, which moves at wave speed C to the left. The wave appears fixed from this viewpoint, and the fluid appears to have velocity C on the left and C V on the right. The thermodynamic properties p, , and T are not affected by this change of viewpoint. The flow in Fig. 9.1b is steady and one-dimensional across the wave. The continuity equation is thus, from Eq. (3.24), AC  ( or )(A)(C  V  C    V) (9.10) 576 Chapter 9 Compressible Flow C p + ∆p ρ + ∆ρ T + ∆T p ρ T V=0 ∆V Moving wave of frontal area A (a) Friction and heat transfer effects are confined to wave interior Fig. 9.1 Control-volume analysis of a finite-strength pressure wave: (a) control volume fixed to still fluid at left; (b) control volume moving left at wave speed C. p ρ T p + ∆p ρ + ∆ρ T + ∆T V=C V = C – ∆V Fixed wave (b) This proves our contention that the induced fluid velocity on the right is much smaller than the wave speed C. In the limit of infinitesimal wave strength (sound wave) this speed is itself infinitesimal. Notice that there are no velocity gradients on either side of the wave. Therefore, even if fluid viscosity is large, frictional effects are confined to the interior of the wave. Advanced texts [for example, 14] show that the thickness of pressure waves in gases is of order 10 6 ft at atmospheric pressure. Thus we can safely neglect friction and apply the one-dimensional momentum equation (3.40) across the wave  Fright  ṁ(Vout or pA (p Vin) p)A  (AC)(C V C) (9.11) Again the area cancels, and we can solve for the pressure change p  C V (9.12) If the wave strength is very small, the pressure change is small. Finally combine Eqs. (9.10) and (9.12) to give an expression for the wave speed  p C2   1      (9.13) The larger the strength / of the wave, the faster the wave speed; i.e., powerful explosion waves move much more quickly than sound waves. In the limit of infinitesimal strength  → 0, we have what is defined to be the speed of sound a of a fluid: p a2    (9.14) 9.2 The Speed of Sound Table 9.1 Sound Speed of Various Materials at 60°F (15.5°C) and 1 atm Material Gases: H2 He Air Ar CO2 CH4 238 UF6 Liquids: Glycerin Water Mercury Ethyl alcohol Solids:* Aluminum Steel Hickory Ice a, ft/s a, m/s 4,246 3,281 1,117 1,040 873 607 297 1,294 1,000 340 317 266 185 91 6,100 4,890 4,760 3,940 1,860 1,490 1,450 1,200 16,900 16,600 13,200 10,500 5,150 5,060 4,020 3,200 *Plane waves. Solids also have a shear-wave speed. 577 But the evaluation of the derivative requires knowledge of the thermodynamic process undergone by the fluid as the wave passes. Sir Isaac Newton in 1686 made a famous error by deriving a formula for sound speed which was equivalent to assuming an isothermal process, the result being 20 percent low for air, for example. He rationalized the discrepancy as being due to the “crassitude’’ (dust particles, etc.) in the air; the error is certainly understandable when we reflect that it was made 180 years before the proper basis was laid for the second law of thermodynamics. We now see that the correct process must be adiabatic because there are no temperature gradients except inside the wave itself. For vanishing-strength sound waves we therefore have an infinitesimal adiabatic or isentropic process. The correct expression for the sound speed is p a      1/2 s p  k     1/2 (9.15) T for any fluid, gas or liquid. Even a solid has a sound speed. For a perfect gas, From Eq. (9.2) or (9.9), we deduce that the speed of sound is   kp a    1/2  (kRT)1/2 (9.16) The speed of sound increases as the square root of the absolute temperature. For air, with k  1.4 and R  1717, an easily memorized dimensional formula is a (ft/s)  49[T (°R)]1/2 (9.17) a (m/s)  20[T (K)]1/2 At sea-level standard temperature, 60°F  520°R, a  1117 ft/s. This decreases in the upper atmosphere, which is cooler; at 50,000-ft standard altitude, T  69.7°F  389.9°R and a  49(389.9)1/2  968 ft/s, or 13 percent less. Some representative values of sound speed in various materials are given in Table 9.1. For liquids and solids it is common to define the bulk modulus K of the material K p    p      s (9.18) s For example, at standard conditions, the bulk modulus of carbon tetrachloride is 163,000 lbf/in2 absolute, and its density is 3.09 slugs/ft3. Its speed of sound is therefore [163,000(144)/3.09]1/2  2756 ft/s, or 840 m/s. Steel has a bulk modulus of about 29  106 lbf/in2 absolute and water about 320  103 lbf/in2 absolute, or 90 times less. For solids, it is sometimes assumed that the bulk modulus is approximately equivalent to Young’s modulus of elasticity E, but in fact their ratio depends upon Poisson’s ratio  E   3(1 K 2) (9.19) The two are equal for   13, which is approximately the case for many common metals such as steel and aluminum. 578 Chapter 9 Compressible Flow EXAMPLE 9.2 Estimate the speed of sound of carbon monoxide at 200-kPa pressure and 300°C in m/s. Solution From Table A.4, for CO, the molecular weight is 28.01 and k  1.40. Thus from Eq. (9.3) RCO  8314/28.01  297 m2/(s2 K), and the given temperature is 300°C 273  573 K. Thus from Eq. (9.16) we estimate aCO  (kRT)1/2  [1.40(297)(573)]1/2  488 m/s 9.3 Adiabatic and Isentropic Steady Flow Ans. As mentioned in Sec. 9.1, the isentropic approximation greatly simplifies a compressible-flow calculation. So does the assumption of adiabatic flow, even if nonisentropic. Consider high-speed flow of a gas past an insulated wall, as in Fig. 9.2. There is no shaft work delivered to any part of the fluid. Therefore every streamtube in the flow satisfies the steady-flow energy equation in the form of Eq. (3.66) h1 gz1  h2 V21 1 2 1 2 V22 gz2 q w (9.20) where point 1 is upstream of point 2. You may wish to review the details of Eq. (3.66) and its development. We saw in Example 3.16 that potential-energy changes of a gas are extremely small compared with kinetic-energy and enthalpy terms. We shall neglect the terms gz1 and gz2 in all gas-dynamic analyses. Inside the thermal and velocity boundary layers in Fig. 9.2 the heat-transfer and viscous-work terms q and w are not zero. But outside the boundary layer q and w are zero by definition, so that the outer flow satisfies the simple relation 1 2 h1 V21  h2 V22  const 1 2 (9.21) The constant in Eq. (9.21) is equal to the maximum enthalpy which the fluid would achieve if brought to rest adiabatically. We call this value h0, the stagnation enthalpy of the flow. Thus we rewrite Eq. (9.21) in the form h V2  h0  const 1 2 h0 V δ T > δV if Pr < 1 δV Fig. 9.2 Velocity and stagnationenthalpy distributions near an insulated wall in a typical high-speed gas flow. Insulated wall (9.22) 9.3 Adiabatic and Isentropic Steady Flow 579 This should hold for steady adiabatic flow of any compressible fluid outside the boundary layer. The wall in Fig. 9.2 could be either the surface of an immersed body or the wall of a duct. We have shown the details of Fig. 9.2; typically the thermal-layer thickness T is greater than the velocity-layer thickness V because most gases have a dimensionless Prandtl number Pr less than unity (see, e.g., Ref. 19, sec. 4-3.2). Note that the stagnation enthalpy varies inside the thermal boundary layer, but its average value is the same as that at the outer layer due to the insulated wall. For nonperfect gases we may have to use the steam tables [15] or the gas tables [16] to implement Eq. (9.22). But for a perfect gas h  cpT, and Eq. (9.22) becomes cpT V2  cpT0 1 2 (9.23) This establishes the stagnation temperature T0 of an adiabatic perfect-gas flow, i.e., the temperature it achieves when decelerated to rest adiabatically. An alternate interpretation of Eq. (9.22) occurs when the enthalpy and temperature drop to (absolute) zero, so that the velocity achieves a maximum value Vmax  (2h0)1/2  (2cpT0)1/2 (9.24) No higher flow velocity can occur unless additional energy is added to the fluid through shaft work or heat transfer (Sec. 9.8). Mach-Number Relations The dimensionless form of Eq. (9.23) brings in the Mach number Ma as a parameter, by using Eq. (9.16) for the speed of sound of a perfect gas. Divide through by cpT to obtain 1 V2 T   0 2cpT T But, from the perfect-gas law, cpT  [kR/(k comes 1)]T  a2/(k 1), so that Eq. (9.25) be- (k 1)V2 T   0 2a2 T 1 or (9.25) T 0  1 T k 1  Ma2 2 V Ma   a (9.26) This relation is plotted in Fig. 9.3 versus the Mach number for k  1.4. At Ma  5 the temperature has dropped to 16T0. Since a  T1/2, the ratio a0 /a is the square root of (9.26)   a T 0  0 a T 1/2   1 1  (k 2 1)Ma2 1/2 (9.27) Equation (9.27) is also plotted in Fig. 9.3. At Ma  5 the speed of sound has dropped to 41 percent of the stagnation value. Isentropic Pressure and Density Relations Note that Eqs. (9.26) and (9.27) require only adiabatic flow and hold even in the presence of irreversibilities such as friction losses or shock waves. 580 Chapter 9 Compressible Flow 1.0 a a0 T T0 ρ ρ 0 p p0 0.5 Fig. 9.3 Adiabatic (T/T0 and a/a0) and isentropic (p/p0 and /0) properties versus Mach number for k  1.4. 0 1 2 3 Mach number 4 5 If the flow is also isentropic, then for a perfect gas the pressure and density ratios can be computed from Eq. (9.9) as a power of the temperature ratio   p T 0  0 p T k/(k 1)  T 0  0  T 1/(k 1)    1  1  (k 2 1) Ma2  1  (k 2 1) Ma2  1 k/(k 1) (9.28a) 1/(k 1) (9.28b) These relations are also plotted in Fig. 9.3; at Ma  5 the density is 1.13 percent of its stagnation value, and the pressure is only 0.19 percent of stagnation pressure. The quantities p0 and 0 are the isentropic stagnation pressure and density, respectively, i.e., the pressure and density which the flow would achieve if brought isentropically to rest. In an adiabatic nonisentropic flow p0 and 0 retain their local meaning, but they vary throughout the flow as the entropy changes due to friction or shock waves. The quantities h0, T0, and a0 are constant in an adiabatic nonisentropic flow (see Sec. 9.7 for further details). Relationship to Bernoulli’s Equation The isentropic assumptions (9.28) are effective, but are they realistic? Yes. To see why, differentiate Eq. (9.22) Adiabatic: dh V dV  0 (9.29) Meanwhile, from Eq. (9.6), if ds  0 (isentropic process), dp dh    (9.30) Combining (9.29) and (9.30), we find that an isentropic streamtube flow must be dp   V dV  0 (9.31)