Isothermal Flow Gas ﬂow in Long Constant-Area Ducts, Such As Natural Gas Pipelines, Is Essentially Iso- Thermal

80812 c13a.3d GGS 6/6/08 19:3

13-3 FLOW IN A CONSTANT-AREA DUCT WITH FRICTION (continued)

Isothermal Flow Gas flow in long constant-area ducts, such as natural gas pipelines, is essentially isothermal. Mach numbers in such flows are generally low, but significant pressure changes can occur as a result of frictional effects acting over long duct lengths. Hence, such flows cannot be treated as incompressible. The assumption of isothermal flow is much more appropriate. For isothermal flow with friction (as opposed to the adiabatic flow with friction we previously discussed), the heat transfer dQ/dm is not zero. On the other hand, we have the simplification that the temperature is constant everywhere. As for adiabatic flow, we can start with our set of basic equations (Eqs. 13.1), describing one- dimensional flow that is affected by area change, friction, heat transfer, and normal shocks, ϭ ϭ ϭ ϭ r1V1A1 r2V2A2 rVA m constant ð13:1aÞ Rx ϩ p1A1 Ϫ p2A2 ϭ mV2 Ϫ mV1 ð13:1bÞ

dQ V2 V2 ϩ h ϩ 1 ϭ h ϩ 2 ð13:1cÞ dm 1 2 2 2 Z : 1 Q mðs2 Ϫ s1Þ Ն dA ð13:1dÞ CS T A p ϭ rRT ð13:1eÞ

Dh ϭ h2 Ϫ h1 ϭ cpDT ϭ cpðT2 Ϫ T1Þð13:1fÞ

T2 p2 Ds ϭ s2 Ϫ s1 ϭ cp ln Ϫ R ln ð13:1gÞ T1 p1

We can simplify these equations by setting DT ϭ 0, so T1 ϭ T2, and A1 ϭ A2 ϭ A.In addition we recall from Section 13-1 that the combination, h ϩ V 2/2 is the stagnation enthalpy, h0. Using these, our ﬁnal set of equations (renumbered for convenience) is m r V ϭ r V ϭ rV ϭ G ϭ ϭ constant ð13:22aÞ 1 1 2 2 A Rx ϩ p1A Ϫ p2A ϭ mV2 Ϫ mV1 ð13:22bÞ

dQ V2 Ϫ V2 q ϭ ϭ h Ϫ h ϭ 2 1 ð13:22cÞ dm 02 01 2 Z : 1 Q mðs2 Ϫ s1Þ Ն dA ð13:22dÞ CS T A p ϭ rRT ð13:22eÞ

p2 Ds ϭ s2 Ϫ s1 ϭϪR ln ð13:22fÞ p1 Equations 13.22 can be used to analyze frictional isothermal flow in a channel of constant area. For example, if we know conditions at section 1 (i.e., p1, r1, T1, s1, h1, and V1) we can use these equations to find conditions at some new section 2 after the fluid has experienced a total friction force Rx. We have five equations (not includ- ing the constraint of Eq. 13.22d) and five unknowns (p2, r2, s2, V2, and the heat 80812 c13a.3d GGS 6/6/08 19:3

13-3 FLOW IN A CONSTANT-AREA DUCT WITH FRICTION (continued) W-25

transfer q that was necessary to maintain isothermal conditions). As we have seen before, in practice this procedure is unwieldy—we once again have a set of nonlinear, coupled algebraic equations to solve. Before doing any calculations, we can see that the Ts diagram for this process will be simply a horizontal line passing through state 1 . To see in detail what happens to the ﬂow, in addition to Eqs. 13.22, we can develop property relations as func- tions of the Mach number. For isothermal ﬂow, c ϭ constant, so V2/V1 ϭ M2/M1, and from Eq. 13.22a we have r 2 ϭ V1 ϭ M1 r1 V2 M2 Combining with the ideal gas equation, Eq. 13.22e, we obtain p r V M 2 ϭ 2 ϭ 1 ϭ 1 ð13:23Þ p1 r1 V2 M2 At each state we can relate the local temperature to its stagnation temperature using Eq. 12.21b, T k Ϫ 1 0 ϭ 1 ϩ M2 ð12:21bÞ T 2

Applying this to states 1 and 2 , with the fact that T1 ϭ T2, we obtain Ϫ ϩ k 1 2 1 M2 T02 2 ϭ ð13:24Þ T k Ϫ 1 01 1 ϩ M2 2 1 To determine the variation in Mach number along the duct length, it is necessary to consider the differential momentum equation for flow with friction. The analysis leading to Eq. 13.18 is valid for isothermal flow. Since T ϭ constant for isothermal flow, then from Eq. 13.18, with dT ϭ 0, 2 Ϫ 2 d M2 f kM ϭ 1 kM ð Þ dx 2 Dh 2 2 M and 1 Ϫ kM2 d M2 f ϭ ð Þ ð Þ dx 4 ð13:25Þ Dh kM ϭ Equation 13.25 shows (setpdxffiffiffi 0) that the Mach number at which maximum length Lmax is reached is M ϭ 1/ k. Since T is constant, then the friction factor, f ϭ f(Re), ϭ ϭ is alsop constant.ffiffiffi Integration of Eq. 13.25 between the limits of M M at x 0 and M ϭ 1/ k at x ϭ Lmax, where Lmax is the distance beyond which the isothermal flow may not proceed, gives Ϫ 2 fLmax ϭ 1 kM ϩ 2 2 ln kM ð13:26Þ Dh kM

The duct length, L, required for the ﬂow Mach number to change from M1 to M2 can be obtained from L L Ϫ L f ϭ f max1 max2 Dh Dh 1 Ϫ kM2 1 Ϫ kM2 M2 L ϭ 1 Ϫ 2 ϩ 1 f 2 2 ln 2 ðÞ13:27 Dh kM1 kM2 M2 80812 c13a.3d GGS 6/6/08 19:3

W-26 CHAPTER 13 / COMPRESSIBLE FLOW

The distribution of heat exchange along the duct required to maintain isothermal flow can be determined from the differential form of Eq. 13.22c as k Ϫ 1 dq ϭ dh ϭ c dT ϭ c d T 1 ϩ M2 0 p 0 p 2 or, since T ϭ constant, k Ϫ 1 cpT0ðk Ϫ 1Þ dq ϭ c T dM2 ϭ dM2 p 2 k Ϫ 1 21ϩ M2 2 Substituting for dM 2 from Eq. 13.25, 4 cpT0ðk Ϫ 1ÞkM f dq ϭ dx ð13:28Þ k Ϫ 1 D 21ϩ M2 ð1 Ϫ kM2Þ h 2 pffiffiffi From Eq. 13.28 we note that as M ! 1/ k, then dq/dx ! ϱ. Thus, an infinite rate of heat exchange is required to maintain isothermal flow as the Mach number ap- proaches the limiting value. Hence, we conclude that isothermal acceleration of flow in a constant-area duct is only physically possible for flow at low Mach number. We summarize the set of Mach number–based equations (Eqs. 13.23, 13.24, and 13.27, respectively, renumbered) we can use for analysis of isothermal flow of an ideal gas in a duct with friction:

p r V M 2 ϭ 2 ϭ 1 ϭ 1 ð13:29aÞ p1 r1 V2 M2 Ϫ ϩ k 1 2 1 M2 T02 2 ϭ ð13:29bÞ T k Ϫ 1 01 1 ϩ M2 2 1 1 Ϫ kM2 1 Ϫ kM2 M2 fL ϭ 1 Ϫ 2 ϩ 1 2 2 ln 2 ð13:29cÞ Dh kM1 kM2 M2

13-6 SUPERSONIC CHANNEL FLOW WITH SHOCKS (continued)

Supersonic Diffuser Analysis of the effects of area change in isentropic flow (Section 13-2) showed that a converging channel reduces the speed of a supersonic stream; a converging channel is a supersonic diffuser. Because flow speed decreases, pressure rises in the flow di- rection, creating an adverse pressure gradient. Isentropic flow is not a completely ac- curate model for flow with an adverse pressure gradient,2 but the isentropic flow model with a normal shock may be used to demonstrate the basic features of supersonic diffusion. For isentropic flow, a shock cannot stand in a stable position in a converging passage; a shock may stand stably only in a diverging passage. Real flow near M ϭ 1

2 Boundary layers develop rapidly in adverse pressure gradients, so viscous effects may be important or even dominant. In the presence of thick boundary layers, supersonic ﬂows in diffusers may form complicated systems of oblique and normal shocks. 80812 c13a.3d GGS 6/6/08 19:3

13-6 SUPERSONIC CHANNEL FLOW WITH SHOCKS (continued) W-27

Flow

1 2 3 4

T p 04 5 T0 = constant p01 4 p1* T* = constant

2 3

p1 T1

1 s Fig. 13.22 Schematic Ts diagram for ﬂow in a supersonic diffuser with a normal shock.

is unstable, so it is not possible to reduce a supersonic flow exactly to sonic speed. The minimum Mach number that can be reached at a throat is 1.2 to 1.3. Thus in real supersonic diffusers, flow is decelerated to M & 1.3 in a converging passage. Downstream from the throat section of minimum area, the flow is allowed to accelerate to M & 1.4, where a normal shock takes place. At this Mach number, the stagnation pressure loss (from Eq. 13.41b) is only about 4 percent. This small loss is an acceptable compromise in exchange for flow stability. Figure 13.22 shows the idealized process of supersonic diffusion, in which flow is isentropic except across a normal shock. The slight reduction in stagnation pressure all takes place across the shock. In the actual flow, additional losses in stagnation pressure occur during the supersonic and subsonic diffusion processes before and after the shock. Experimental data must be used to predict the actual losses in supersonic and subsonic diffusers [3, 4]. Supersonic diffusion also is important for high-speed aircraft, where a supersonic external freestream flow must be decelerated efficiently to subsonic speed. Some diffusion can occur outside the inlet by means of a weak oblique shock system [5]. Vari- able geometry may be needed to accomplish efficient supersonic diffusion within the inlet as the flight Mach number varies. Multi-dimensional compressible flows are discussed in Section 13-7, and are treated in detail elsewhere [6, 7].

Supersonic Wind Tunnel Operation To build an efficient supersonic wind tunnel, it is necessary to understand shock behavior and to control shock location. The basic physical phenomena are described by Coles in the NCFMF video Channel Flow of a Compressible Fluid. (See http:// web.mit.edu/fluids/www/Shapiro/ncfmf.html for free online viewing of this film; it is old but good!) In addition to choking—sonic flow at a throat, with upstream flow independent of downstream conditions—Coles discusses blocking and starting conditions for supersonic wind tunnels. A closed-circuit supersonic wind tunnel must have a converging-diverging nozzle to accelerate flow to supersonic speed, followed by a test section of nearly constant area, and then a supersonic diffuser with a second throat. The circuit must be completed by compression machinery, coolers, and flow-control devices, as shown in Fig. 13.23 [9]. 80812 c13a.3d GGS 6/6/08 19:3

W-28 CHAPTER 13 / COMPRESSIBLE FLOW

A. Dry Air Storage Spheres G. Cooling Tower B. Aftercooler H. Flow Diversion Valve C. 3-Stage Axial Flow Fan I. Aftercooler D. Drive Motors J. 11-Stage Axial Flow Compressor E. Flow Diversion Valve K. 9- by 7-Foot Supersonic Test Section F. 8- by 7-Foot Supersonic Test Section L. 11- by 11-Foot Transonic Test Section Fig. 13.23 Schematic view of NASA-Ames closed-circut, high-speed wind tunnel with supporting facilities [9]. (Photo courtesy of NASA.)

Consider the process of accelerating flow from rest to supersonic speed in the test section. Soon after flow at the nozzle throat becomes sonic, a shock wave forms in the divergence. The shock attains its maximum strength when it reaches the nozzle exit plane. Consequently, to start the tunnel and achieve steady supersonic flow in the test section, the shock must move through the second throat and into the subsonic diffuser. When this occurs, we say the shock has been swallowed by the second throat. Consequently, to start the tunnel, the supersonic diffuser throat must be larger than the nozzle throat. The second throat must be large enough to exceed the critical area for flow downstream from the strongest possible shock. Blocking occurs when the second throat is not large enough to swallow the shock. When the channel is blocked, flow is sonic at both throats and flow in the test section is subsonic; flow in the test section cannot be controlled by varying conditions downstream from the supersonic diffuser. When the tunnel is running there is no shock in the nozzle or test section, so the energy dissipation is much reduced. The second throat area may be reduced slightly during running to improve the diffuser efficiency. The compressor pressure ratio may be adjusted to move the shock in the subsonic diffuser to a lower Mach number. A combination of adjustable second throat and pressure ratio control may be used to achieve optimum running conditions for the tunnel. Small differences in efficiency are important when the tunnel drive system may consume more than half a million kilowatts [10]!

Supersonic Flow with Friction in a Constant-Area Channel Flow in a constant-area channel with friction is dominated by viscous effects. Even when the main flow is supersonic, the no-slip condition at the channel wall guarantees subsonic flow near the wall. Consequently, supersonic flow in constant-area channels may form complicated systems of oblique and normal shocks. However, the basic 80812 c13a.3d GGS 6/6/08 19:3

13-6 SUPERSONIC CHANNEL FLOW WITH SHOCKS (continued) W-29

behavior of adiabatic supersonic flow with friction in a constant-area channel is revealed by considering the simpler case of normal-shock formation in Fanno-line flow. Supersonic flow along the Fanno line becomes choked after only a short length of duct, because at high speed the effects of friction are pronounced. Figure E.2 (Appendix E) shows that the limiting value of fLmax/Dh is less than one; subsonic flows can have much longer runs. Thus when choking results from friction and duct length is increased further, the supersonic flow shocks down to subsonic to match downstream conditions. The Ts diagrams in Figs. 13.24a through 13.24d illustrate what happens when the length of constant-area duct, fed by a converging-diverging nozzle supplied from

La La

T0 T0 Shock p0 p0

1 * 1 *

T p * p * p01 0 T p01 0 T0 = constant T0 = constant M < 1 p* Shock p* T* = constant T* = constant

M = 1 M > 1

Process path 1 p1 1 p1 T1 T1 s s (a) Choked supersonic flow in channel. (b) Choked flow in channel with shock.

Ld Lc L La a T T0 0 Shock p p0 0

x y 1 * 1 * T p0 p * p * p0 0 T p0 0 1 1 T0 = constant T0 = constant p1 T1 M < 1 1 y p* p* T* = constant T* = constant x M = 1

1 p1

T1 s s (c) Choked flow with shock in nozzle exit (d) Choked flow with shock in nozzle; subsonic plane. flow in channel. Fig. 13.24 Schematic Ts diagrams for supersonic Fanno-line ﬂows with normal shocks. 80812 c13a.3d GGS 6/6/08 19:3

W-30 CHAPTER 13 / COMPRESSIBLE FLOW

a reservoir with constant stagnation conditions, is increased. Supersonic flow on the Fanno line of Fig. 13.24a is choked by friction when the duct length is La. When additional duct is added to produce Lb Ͼ La, Fig. 13.24b, a normal shock appears. Flow upstream from the shock does not change, because it is supersonic (no change in downstream condition can affect the supersonic flow before the shock). In Fig. 13.24b the shock is shown in an arbitrary position. The shock moves toward the entrance of the constant-area channel (toward higher initial Mach number) as more duct is added. Flow remains on the same Fanno line as the shock is driven upstream to state 1 by adding duct length; thus the mass flow rate remains unchanged. The duct length, Lc, which moves the shock into the channel entrance plane, Fig. 13.24c, may be calculated directly using the methods of Section 13-3. When duct length Lc is exceeded, the shock is driven back into the C-D nozzle, Fig. 13.24d. The mass flow rate remains constant until the shock reaches the nozzle throat. Only when more duct is added after the shock reaches the throat does the mass flow rate decrease, and the flow move to a new Fanno line. If the shock position is known, flow properties at each section and the duct length can be calculated directly. When length is specified and shock location is to be determined, iteration is necessary.

Supersonic Flow with Heat Addition in a Constant-Area Channel Supersonic flow with heat addition in a frictionless channel of constant area is shown in Fig. 13.25a. Assume the channel is fed by a converging-diverging nozzle, supplied from a reservoir with constant stagnation conditions, and flow is supersonic at state 1 . Heat addition causes state points to move up and to the right along the Rayleigh line. Figure 13.25a illustrates the condition in which the heat addition is just sufficient to choke the flow.R Flow is sonic at the exit, so pe ϭ p* and Te ϭ T*; the heat addition s per unit mass, e Tds, is represented by the shaded area beneath the Rayleigh line. s1 A normal shock involves no heat addition, so T0 is constant across a shock. Con- sequently, a shock in the constant-area channel would not change the heat addition required to change the flow state from the inlet condition to choking. When the shock stands at the channel inlet, Fig. 13.25b, the heat addition needed to reach Mach one at the exit is the same as in Fig. 13.25a; the shaded areas also must be identical. If more thermal energy is added to flow at the conditions shown in Fig. 13.25b, the shock will be pushed from the entrance of the constant-area duct back into the diverging portion of the nozzle, where the Mach number is lower. With a shock in the nozzle, conditions at the duct entrance are changed, and heat addition occurs along a different Rayleigh line, as shown in Fig. 13.25c. There is no T T T ϭ T T* ϭ T* change in 0 or * across the shock (thus 03 04 and 3 4 ), but the Mach number downstream changes. Additional subsonic diffusion occurs from state 4 to the nozzle exit (state 5 ), thus moving the choked condition upward on the Ts plane, allowing for increased heat addition on the new Rayleigh line. All of these changes occur at the same mass flow rate, because nozzle throat conditions remain unchanged. The Mach number immediately upstream from the shock (state 3 ) is less than M1 of Fig. 13.25b; the corresponding temperature, T3, is higher than T1. Since the shock strength is reduced, the entropy rise across the shock is less, (s4 Ϫ s3) Ͻ (s2 Ϫ s1). The subsonic diffusion following the shock results in a lower Mach number and higher temperature at the duct entrance. Thus M5 Ͻ M2 and T5 Ͼ T2. When the heat addition rate is increased enough to drive the shock to the nozzle throat, a further increase in heat addition will result in a decrease in mass flow rate. The Mach number at the channel inlet is reduced, M7 Ͻ M5, and the channel flow shifts to another new Rayleigh line, as shown in Fig. 13.25d. Thus for specified mass flow rate, there is a maximum rate of heat addition for supersonic flow throughout. For higher rates of heat addition, a shock occurs in the 80812 c13a.3d GGS 6/6/08 19:3

13-6 SUPERSONIC CHANNEL FLOW WITH SHOCKS (continued) W-31

p01 p01 M > 1 M > 1 M > 1 M < 1 T01 T01

t 1 e t 1 2 e M = 1 M = 2 M = 1 Mt = 1 Me = 1 1200 t 1 e 1200 p0e T0 T p e 0e 0e p p e p0 p p 01 T 1 02 e T0 T 01 1 e Te 800 800 T2 p2

1 T T1 1

p1 Temperature, (K) Temperature, (K) 400 400 p1

0 0 0 123 0 123 Nondimensional entropy, (s* – s)/R Nondimensional entropy, (s* – s)/R (a) Choked supersonic flow. (b) Choked flow with shock at nozzle exit plane.

M > 1 M < 1

p p0 01 1 M < 1 M < 1 M < 1 T T01 01

t 3 45 6 t 7 8 Mt < 1 M7 = 0.35 M8 = 1 Mt = 1 M4 = 0.701 M6 = 1 2000 2000 p M3 = 1.5 M5 = 0.407 08

T08

p p06 8 1600 T 1600 06 T8

T6 1200 1200

p02 p 01 p01 Temperature, (K) Temperature, (K) T01 T T0 01 4 T 800 T5 800 7 T4 Tt Tt

p3 400 400

0 123 0 123 Nondimensional entropy, (s* – s)/R Nondimensional entropy, (s* – s)/R

(c) Choked flow with shock in nozzle; same (d) Subsonic flow throughout; decreased mass flow mass flow rate, but flow shifts to a new rate and flow shifted to another new Rayleigh Rayleigh line. line. Fig. 13.25 Schematic Ts diagrams for supersonic Rayleigh-line ﬂows with normal shocks. 80812 c13a.3d GGS 6/6/08 19:3

W-32 CHAPTER 13 / COMPRESSIBLE FLOW

nozzle and flow is subsonic in the constant-area channel, but the exit flow remains sonic. If the shock position is specified, the heat addition along the Rayleigh line can be calculated directly. If the heat addition is specified but the shock position or mass flow rate are unknown, iteration is required to obtain a solution. Additional consideration of flow with shock waves is given in [11].