Chapter 7 COMPRESSIBLE FLOW

In this chapter on compressible flow, one new variable enters, the density, and one extra equation is available, the equation of state, which relates pressure and density. The other equations-continuity, momentum, and the first and second laws of thermodynamics - are also needed in the analysis of compressible fluid-flow situations. When density changes are gradual and do not change by more than a few percent, the flow may be treated as incompressible with the use of an average density. The following topics are treated in this chapter: prefect-gas relations, speed of a sound wave, Mach number, isentropic flow, shock waves, Fanno and Rayleigh lines, adiabatic flow, flow with heat transfer, isothermal flow, and the analogy between shock waves and open-channel waves.

7.1 PERFECT-GAS RELATIONS Perfect gas is defined as a fluid that has constant specific heats and follows the law (7.1.1) in which p and T are the absolute pressure and absolute temperature, respectively, ρ is the density, and R the gas constant. In general, the specific heat cv is defined by (7.1.2) The specific heat cp at constant pressure is defined by (7.1.3)

For perfect gases Eq. (7.1.2) becomes (7.1.4) and Eq. (7.1.3) becomes (7.1.5) Then, from differentiating gives and substitution of Eqs. (7.1.4) and (7.1.5) leads to (7.1.6)

The specific heat ratio k is defined as the ratio                                                                       (7.1.7) Solving with Eq. (7.1.6) gives (7.1.8)

Entropy Relations The first law of thermodynamics for a system states that the heat added to a system is equal to the work done by the system plus its increase in internal energy [Eq. (3.8.4)]. In terms of the entropy s the equation takes the form (3.8.6) The internal-energy change for a perfect gas is (7.1.9) and the enthalpy change is (7.1.10) The change in entropy (7.1.11)

may be obtained from Eqs. (7.1.4) and (7.1.1). Alter integrating, (7.1.12) By use of Eqs. (7.1.8) and (7.1.1), Eq. (7.1.12) becomes (7.1.13) or (7.1.14) And (7.1.15) These equations are forms of the second law of thermodynamics.

From Eq. (7.1.14) for s2=s1, (7.1.16) Equation (7.1.16) combined with the general gas law yields (7.1.17) The enthalpy change for an isentropic process is (7.1.18) The polytropic process is defined by (7.1.19) and is an approximation to certain actual processes in which p would plot substantially as a straight line against ρ on log-log paper.

Example 7.1 Helium has R=2.077 kJ/kg K. Find cv and k and check against Table C.3. Solution

Example 7.2 Compute the value of R from the values of k and cp for air and check in Table C.3. Solution From Eq. (7.1.8)

Example 7.3 Determine the entropy change in 4.0 kg of water vapor when the initial conditions are p1=42 kPa, t1=430C and the final conditions are p2=280 kPa, t2=30C. Solution From Eq.(7.1.15) and Table C.3

7.2 SPEED OF A SOUND WAVE; MACH NUMBER The speed of a small disturbance in a conduit can be determined by application of the momentum equation and the continuity equation. The question is first raised whether a stationary small change in velocity, pressure, and density can occur in a channel. By referring to Fig. 7.1 the continuity equation can be written in which A is the cross-sectional area of channel. The equation can be reduced to When the momentum equation (3.11.2) is applied to the control volume within the dotted lines,

Figure 7.1 Steady flow in prismatic channel with sudden small change in velocity, pressure, and density.

If ρdV is eliminated between the two equations, (7.2.1) The equation for speed of sound (7.2.2) may be expressed in several useful forms. The bulk modulus of elasticity can be introduced: in which V is the volume of fluid subjected to the pressure change dp. Since

K may be expressed as Then, from Eq.(7.2.2), (7.2.3) This equation applies to liquids as well as gases. Since the pressure and temperature changes due to passage of a sound wave and extremely small, the process is almost reversible. In the limit, the process may be considered to be isentropic, (7.2.4)

or, from the perfect-gas law p=ρRT, (7.2.5) which shows that the speed of sound in a perfect gas is a function of absolute temperature only. The Mach number has been defined as the ratio of velocity of a fluid to the local velocity of sound in the medium, (7.2.6) The Mach number is a measure of the importance of compressibility. In an incompressible fluid K is infinite and M=0. For perfect gasses. (7.2.7) when the compression is isentropic.

Example 7.6 Carbon tetrachloride has a bulk modulus of 1.124 GPa and  a density of 1593 kg/m3. What is the speed of sound in the medium? Solution

Example 7.7 What is the speed of sound in dry air sea level when t=200C and in the stratosphere when t=-200C ? Solution At sea level, from Eq.(7.2.5)               and in the stratosphere

7.3. ISENTROPIC FLOW Frictionless adiabatic, or isentropic, flow is an ideal that cannot be reached in the flow of real gases. Some very general results can be obtained by use of Euler's equation (3.5.8), neglecting elevation changes, (7.3.1) and the continuity equation (7.3.2)

Differentiating ρAV and then dividing through by ρAV gives                                                               (7.3.3) From Eq.(7.2.2) dρ is odtained and sudstituted into Eq.(7.3.1) yielding                                                                   (7.3.4) Eliminating dρ/ρ in the last two equations and rearranging give (7.3.5) The assumptions underlying this equation are that the flow is steady and frictionless. No restrictions as to heat transfer have been imposed. Equation (7.3.5) show that, for subsonic flow (M<1), dA/dV is always negative; i.e., the channel area must decrease for increasing velocity.

When the analysis is restricted to isentropic flow, Eq. (7. 1 When the analysis is restricted to isentropic flow, Eq. (7.1.16) may be written (7.3.6) Differentiating and substituting for dp in Eq.(7.3.1) give Integration yields or (7.3.7)

From p=ρRT (7.3.8) For adiabatic flow from a reservoir where conditions are given by p0, ρ0, T0 at any other section (7.3.9) In terms of the local number V/c, with c2=kRT (7.3.10)

From Eqs. (7.3.10) and (7.1.17), which now restrict the following equations to isentropic flow, (7.3.11) (7.3.12) Flow conditions are termed critical at the throat section when the velocity there is sonic. Sonic conditions are marked with an asterisk. By applying Eqs. (7.3.10) to (7.3.12) to the throat section for critical conditions (for k=1.4 in the numerical portion), (7.3.13) (7.3.14) (7.3.15)

The variation of area with the Mach number for the critical case is obtained by use of the continuity equation Eqs. (7.3.10) to (7.3.15). First (7.3.16) in which A* is the minimum, or throat, area. Then (7.3.17) Now, and , so that (7.3.18) by use of Eqs.(7.3.13) and (7.3.10). In a similar manner (7.3.19)

By substituting the last two equations into Eq. (7.3.17) gives (7.3.20) For gasses with k=1.40, Eq.(7.3.20) reduces to (7.3.21) The maximum mass flow rate mmaxcan be expressed in terms of the throat area and reservoir conditions: Replacing ρ0 by p0√(R0T0) gives (7.3.22)

For k=1.40 this reduces to (7.3.23) The mass rate of flow m is obtained from (7.3.24) This equation holds for any section and is applicable as long as the velocity at the throat is subsonic. It may be applied to the throat section, and for this section, from Eq. (7.3.14), When the equals sign is used in the expression, Eq. (7.3.24) reduces to Eq. (7.3.22).

For maximum mass flow rate, the flow downstream from the throat may be either supersonic or subsonic, depending upon the downstream pressure. Substituting Eq. (7.3.22) for m in Eq. (7.3.24) and simplifying gives (7.3.25) A may be taken as the outlet area and P as outlet pressure. For a given A*/A (less than unity) there will be two values of p/p0 between zero and unity, the upper value for subsonic flow through the diverging duct and the lower value for supersonic flow through the diverging duct. For all other pressure ratios less than the upper value complete isentropic flow is impossible and shock waves form in or just downstream from the diverging duct. They are briefly discussed in the following section.

Example 7.9 A preliminary design of a wind tunnel to produce Mach number 3.0 at the exit is desired. The mass flow rate is 1 kg/s for p0=90 kPa abs, t0=250C. Determine (a) the throat area, (b) the outlet area, and (c) the velocity, pressure, temperature, and density at the outlet. Solution (a) The throat area con be determine form Eq. (7.3.23);

(b) The area of outlet is determined from Table C.4:                     (c) From Table C.4                           From the gas law

hence, at the exit          From the continuity equation

Example 7.10 A converging-diverging air duct has a throat cross section of 372 cm2 and an exit cross section of 929 cm2. Reservoir pressure is 210 kPa abs, and temperature is 16 0C. Determine the range of Mach numbers and the pressure range at the exit for isentropic flow. Find the maximum flow rate. Solution From Table C.4 [Eq.(7.3.21)] M=2.44 and 0.24. Each of these values of Mach numbers at the exit is for critical conditions; hence, the Mach number range for isentropic flow is 0 to 0.24 and them one value 2.44.

From Table C. 4 [Eq. (7. 3. 11)] M=2. 44, p=210×0. 064=13 From Table C.4 [Eq.(7.3.11)] M=2.44, p=210×0.064=13.44 kPa, and M=0.24, p=210×0.961=201.8 kPa. The downstream pressure range is then from 201.8 to 210 kPa abs, and the isolated point is 13.44 kPa. The maximum mass flow rate is determined form Eq.(7.3.23):

Example 7.11 A converging-diverging duct in an air line downstream from a reservoir has a 50-mm-diameter throat. Determine the mass rate of flow when p0=0.8 MPa abs, t0=330C, and p=0.5 MPa abs at the throat. Solution         From Eq.(7.3.24)

7.4 SHOCK　WAVES In one-dimensional flow the only type of shock wave that can occur is a normal compression shock wave, as illustrated in Fig. 7.3. In this section the normal shock wave in a diffuser is studied, with isentropic flow throughout the tube, except for the shock wave surface. The shock wave occurs in supersonic flow and reduces the flow to subsonic flow, as proved in the following section. It has very little thickness, of the order of the molecular mean free path of the gas. The controlling equations for adiabatic flow are (Fig.7.3) Continuity:        (7.4.1) Energy: (7.4.2)

Figure 7.3 Normal compression shock wave.

The momentum equation (3. 11 The momentum equation (3.11.2) for a control volume between sections 1 and 2 becomes (7.4.3) The value of p2 is (7.4.4) By combination of the continuity and momentum equations (7.4.5) The Rankine-Hugoniot equations are obtained: (7.4.6) (7.4.7)

From Eq. (7.4.2), the energy equation, (7.4.8) Dividing Eq.(7.4.3) by Eq. (7.4.3) gives by use of Eq.(7.4.8), leads to (7.4.9) which is satisfied by V1=V2 (no shock wave) or by (7.4.10) It may be written (7.4.11)

An expression for change of entropy across a normal shock wave may be obtained in terms of M1 and k. From Eq.(7.4.4)  (7.4.14) From Eq. (7.4.12), (7.4.13) Placing this value of p2/p1 in Eq. (7.4.7) yields Substituting these pressure and density rations into Eq. (7.1.14) gives

Example 7.12 If a normal shock wave occurs in the flow of helium, p1=7 kPa, t1=5°C, V1= 1372 m/s, find p2, ρ2, V2 and t2. Solution From Table C.3, R = 2077, k = 1.66. and From Eq. (7.4.4) From Eq. (7.4.5)

From Eq. (7.4.1)                and

7.5 FANNO AND RAYLEIGH LINES To examine more closely the nature of the flow change in the short distance across a shock wave, where the area may be considered constant, the continuity and energy equations are combined for steady, frictional, adiabatic flow. The lines on such a plot for constant mass flow G are called Fanno lines (Fig. 7.3). The most revealing plot is that of enthalpy against entropy, i.e., an hs diagram. The entropy equation for a perfect gas, Eq. (7.1.14), is (7.5.1) The energy equation for adiabatic flow with no change in elevation, from Eq. (7.4.2), is (7.5.2)

And the continuity equation for no change in area, from Eq. (7. 4 And the continuity equation for no change in area, from Eq. (7.4.1), is (7.5.3) The equation of state, linking h, p, and ρ, is (7.5.4) By elimination p, ρ, and V from the four equations (see Fig. 7.4), (7.5.5) By indicating by subscript a values at the maximum entropy point,

Figure 7.4 Fanno and Rayleigh lines.

After substituting this into Eq. (7.5.2) to find Va, and (7.5.6) Hence, the maximum entropy at point a is for M = 1, or sonic condition. For h> ha the flow is subsonic, and for h< ha the flow is supersonic. The two conditions, before and after the shock, must lie on the proper Fanno line for the area at which the shock wave occurs. The momentum equation was not used to determine the Fanno line, and so the complete solution is not determined yet.

Rayleigh Line Conditions before and after the shock must also satisfy the momentum and continuity equations. Assuming constant upstream conditions and constant area, Eqs. (7.5.1), (7.5.3), (7.5.4), and (7.4.3) are used to determine the Rayleigh line. Eliminating V in the continuity and momentum equations gives (7.5.7) Next, eliminating p from thus equation and the entropy equation gives (7.5.8) Enthalpy may be expressed as a function of and upsteam condition, from Eq. (7.5.7): (7.5.9)

The value of maximum entropy is found by taking ds/dρ and dh/dρ from the equations; then by division and equating to zero, using subscript b for maximum point, To satisfy this equation, the numerator must be zero and the denominator not zero. The numerator set equal to zero yields For this value the denominator is not zero. Again, as with the Fanno line, sonic conditions occur at the point of maximum entropy. Since the flow conditions must be on both curves, just before and just after the shock wave, it must suddenly change from one point of intersection to the other.

Converging-Diverging Nozzle Flow Following the presentation of Liepmann and Roshko the various flow situations for converging-diverging nozzles are investigated. By use of Eq. (7.3.11) the area ratio is obtained as a function of pressure ratio (7.5.10) Figure 7.5 is a plot of area ratio vs. pressure ratio and M, good only for isentropic flow (k = 1.4). By use of the area ratios the distribution of pressure and Mach number along a given converging-diverging nozzle can now be plotted. Figure 7.6 illustrates the various flow conditions that may occur.

Figure 7.5 Isentropic relations for a converging-diverging nozzle (k=1.4).

Figure 7.6 Various pressure and Mach-number configurations for flow through a nozzle.

7.6 ADIABATIC FLOW WITH FRICTION IN CONDUITS Gas flow through a pipe or constant-area duct is analyzed in this section subject to the following assumptions: 1. Perfect gas (constant specific heats) 2. Steady, one-dimensional flow 3. Adiabatic flow (no heat transfer through walls) 4. Constant friction factor over length of conduit 5. Effective conduit diameter D equal to four times hydraulic radius (cross-sectioned area divided by perimeter) 6. Elevation changes unimportant compared with friction effects 7. No work added to, or extracted from, the flow.

For exactly one length of pipe, depending upon upstream conditions, the flow is just sonic (M=1) at the downstream end. For shorter lengths of pipe, the flow will not have reached sonic conditions at the outlet, but for longer lengths of pipe, there must be shock waves (and possibly choking) if supersonic and choking effects if subsonic. Choking means that the mass flow rate specified cannot take place in this situation and less flow will occur. Table 7.1 indicates the trends in properties of a gas in adiabatic flow through a constant-area duct, as can be shown from the equations in this section. The gas cannot change gradually from subsonic to supersonic or vice versa in a constant-area duct.

The momentum equation must now include the effects of wall shear stress and is conveniently written for a segment of duct of length δx (Fig. 7.7) which simplifies to (7.6.1) (7.6.2) Dividing Eq.(7.6.2) by p gives (7.6.3)

Figure 7.7 Notation for application of momentum equation. Table 7.1 Figure 7.7 Notation for application of momentum equation.

Each term is now developed in terms of M. By definition V/c = M, (7.6.4) (7.6.5) for the middle term of the momentum equation. Rearranging Eq. (7.6.4) gives (7.6.6) Now to express dV/V in terms of M, from the energy equation, (7.6.7) Differentiating gives (7.6.8)

Dividing through by V2 = M2kRT yields Since cp/R=k/(k-1) (7.6.9) Differentiating V2 = M2kRT and dividing by the equation give (7.6.10) Eliminating dT/T in Eqs. (7.6.9) and (7.6.10) and simplifying lead to (7.6.11)

Which permits elimination of dV/V from Eq. (7.6.6), yielding (7.6.12) And finally, from p=ρRT and G=ρV, (7.6.13) By differentiation Eqs. (7.6.9) and (7.6.11) are used to eliminate dT/T and dV/V: (7.6.14) After rearranging, (7.6.15)

By using the limits x=0, M=M0, x=l, M=M, (7.6.16) (7.6.17) For k=1.4, this reduces to (7.6.18) For the limiting condition M=1 and k=1.4, (7.6.19) There is some evidence to indicate that friction factors may be smaller in supersonic flow

The pressure, velocity, and temperature may also be expressed in integral form in terms of the Mach number. From Eq. (7.6.14) (7.6.20) From Eq. (7.6.11) (7.6.21) From Eqs. (7.6.9) and (7.6.11) Which, when integrated, yields (7.6.22)

Example 7.13 Determine the maximum length of 50-mm-ID pipe, f=0.02 for flow of air, when the Mach number at the entrance to the pipe is 0.30. Solution From Eq.(7.6.19) From which Lmax=13.25 m.

7.7 FRICTIONLESS FLOW THROUGH DUCTS WITH HEAT TRANSFER The steady flow of a perfect gas (with constant specific heats) through a constant-area duct is considered in this section. Friction is neglected, and no work is done on or by the flow. The appropriate equations for analysis of this case are Continuity: (7.7.1) Momentum: (7.7.2) Energy: (7.7.3)

The Rayleigh line, obtained from the solution of momentum and continuity for a constant cross section by neglecting friction, is very in examining the flow. First, eliminating V in Eqs. (7.7.1) and (7.7.2) gives (7.7.4) Since by Eq.(3.8.4), for no losses, entropy can increase only when heat is added, the properties of the gas must change as indicated in Fig.7.8, moving toward the maximum entropy point as heat is added. From Eq. (7.7.4), by differentiation, From Eq. (7.7.3) it is noted that the increase in isentropic stagnation pressure is a measure of the heat added. From , and continuity,

Figure 7.8 Rayleigh line.

Now, from the momentum equation, (7.7.5) Writing this equation for the limiting case p2=p* when M2=1 gives (7.7.6) To develop the other pertinent relations, the energy equation (7.7.3) is used, Applying this to section 1, after dividing through by kRT1/(k-1), yields (7.7.7)

and for section 2 (7.7.8) Dividing Eq. (7.7.7) by Eq. (7.7.8) gives (7.7.9) From the perfect-gas law, (7.7.10) From continuity, and by definition so that and (7.7.11)

Now substituting Eqs. (7. 7. 5) and (7. 7. 11) into Eq. (7. 7 Now substituting Eqs. (7.7.5) and (7.7.11) into Eq. (7.7.10) and simplifying gives (7.7.12) This equation substituted into Eq.(7.7.9) leads to (7.7.13) When this equation is applied to the downstream section where T02=T0* and M2=1 and the subscripts for the upstream section are dropped, the result is (7.7.14) All the necessary equation for determination of frictionless flow with heat transfer in constant- area duct are now available.

7.8 STEADY ISOTHERMAL FLOW IN LONG PIPELINES In the analysis of isothermal flow of a perfect gas through long ducts, neither the Fanno nor the Rayleigh line is applicable, since the Fanno line applies to adiabatic flow and the Rayleigh to frictionless flow. The appropriate equations are Momentum [Eq. (7.6.3)]: (7.8.1) Equation of state: (7.8.2) Continuity:  (7.8.3) Energy [Eq. (7.7.7)]:   (7.8.4)

Stagnation pressure [Eq. (7.3.11)]: (7.8.5) From definitions and the above equations Substituting into the momentum equation (7.8.1) yields (7.8.6)

To determine the direction of heat transfer, differentiate Eq. (7. 8 To determine the direction of heat transfer, differentiate Eq.(7.8.4) and then divide by it, remembering that T is constant: (7.8.7) Eliminating dM2 in this equation and Eq. (7.8.6) gives (7.8.8) From Eqs. (7.8.5) and (7.8.6) (7.8.9) Table 7.3 shows the trends of fluid properties.

Table 7.3 Trends in fluid properties for isothermal flow

By integration of the various Eqs. (7. 8 By integration of the various Eqs. (7.8.6) in terms of M, the change with Mach number is found. the last two terms yield (7.8.10) To find the pressure change, (7.8.11) The superscript *t indicates conditions at M = 1/√k, and M and p represent values at any upstream section.

Example 7.16 Helium enters a 100-mm-ID pipe from converging-diverging nozzle at M=1.30, p =14 kPa, T=225 K. Determine for isothermal flow (a) the maximum length of pipe for no choking, (b) the downstream conditions, and (c) the length from the exit to the section where M=1.0, f=0.016. Solution (a) From Eq. (7.8.10) for k = 1.66 from which Lmax=2.425 m.

(b) From Eq. (7.8.11) From Eqs.(7.8.6) At the upstream section (c) From Eq.(7.8.10) for M = 1, or L’max = 0.683 m. M = 1 occurs 0.683 m from the exit.

7.9 ANALOGY OF SHOCK WAVES TO OPEN-CHANNEL WAVES Both the oblique and normal shock waves in a gas have their counterpart in open-channel flow. The continuity equation is an open channel of constant width is and the continuity equation for compressible flow in a tube of constant cross section is Compressible fluid density ρ and open-channel depth y are analogous. The same analogy is also present in the energy equation. The energy equation for a horizontal open channel of constant width, neglecting friction, is

After differentiating By substitution from Vc=√(gy) to eliminate g. which is to be compared with the energy equation for compressible flow [Eq.(7.3.4)] By applying the momentum equation to a small depth change in horizontal open-channel flow, and to a sudden density change in compressible flow, the density and the open-channel depth can again be shown to be analogous. In effect, the analogy is between the Froude number and the Mach number.