Fluid Mechanics: Fundamentals and Applications, 4Th Edition Yunus A

Fluid Mechanics: Fundamentals and Applications, 4th edition Yunus A. Cengel, John M. Cimbala Lecture slides by Mehmet Kanoglu

©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Chapter 10

COMPRESSIBLE FLOW

. © © Settles, Lab, Dynamics G.S. PennGas State with permission Used University.

High-speed color schlieren image of the bursting of a toy balloon overfilled with compressed air. This 1-microsecond exposure captures the shattered balloon skin and reveals the bubble of compressed air inside beginning to expand. The balloon burst also drives a weak spherical shock wave, visible here as a circle surrounding the balloon. The silhouette of the photographer's hand on the air valve can be seen at center right.

©McGraw-Hill Education. Objectives

• Appreciate the consequences of compressibility in gas flow • Understand why a nozzle must have a diverging section to accelerate a gas to supersonic speeds • Predict the occurrence of shocks and calculate property changes across a shock wave • Understand the effects of friction and heat transfer on compressible flows

©McGraw-Hill Education. 12–1 ■ STAGNATION PROPERTIES

Stagnation (or total) enthalpy V 2 hh  kJ/kg 0 2 Static enthalpy: the ordinary enthalpy h

Energy balance (with no heat or work interaction, no change in potential energy)

22 VV12 hh   hh01 02 1222

(a) © Corbis RF; (b) Photo courtesy of United Technologies Corporation/Pratt & Whitney. Used by permission. All rights reserved. Aircraft and jet engines involve high speeds, Steady flow of a fluid through an and thus the kinetic energy term should adiabatic duct. always be considered when analyzing them.

©McGraw-Hill Education. If the fluid were brought to a complete stop, the energy balance becomes V 2 h1 h h Stagnation enthalpy: The enthalpy of a 12 2 02 fluid when it is brought to rest adiabatically.

During a stagnation process, the kinetic energy of a fluid is converted to enthalpy, which results in an increase in the fluid temperature and pressure. The properties of a fluid at the stagnation state are called stagnation properties (stagnation temperature, stagnation pressure, stagnation density, etc.). The stagnation state is indicated by the subscript 0.

©McGraw-Hill Education. Isentropic stagnation state: When the stagnation process is reversible as well as adiabatic (i.e., isentropic). The stagnation processes are often approximated to be isentropic, and the isentropic stagnation properties are simply referred to as stagnation properties. When the fluid is approximated as an ideal gas with constant specific heats V 2 V 2 cpp T0  c T TT0  2 2cp

Stagnation (or total) temperature T0 : It represents the temperature an ideal gas attains when it is brought to rest adiabatically. The actual state, actual stagnation state, and isentropic stagnation state 2 Dynamic temperature V /2cp : of a fluid on an h-s diagram. corresponds to the temperature rise during such a process.

©McGraw-Hill Education. Stagnation pressure P0 :The pressure a fluid attains when brought to rest isentropically. kk/1   PT00   PT

Stagnation density 0 1/k  1   1/V 00T   kk  T PPvv 00  When stagnation enthalpies are used, the energy balance for a single-stream, steady-flow device The temperature of an ideal

ĖĖin out gas flowing at a velocity V 2 rises by V /2cp when it is qin w in  h 01  gz 1  q out  w out  h 02  gz 2  brought to a complete stop. When the fluid is approximated as an ideal gas with constant specific heats

qin q out  w in  w out  cp  T 02  T 01  g z 2  z 1 

©McGraw-Hill Education. 12–2 ■ ONE-DIMENSIONAL ISENTROPIC FLOW

cP /  speed of   s sound

c k  P /   T

c kRT for an k the specific heat ratio of the gas ideal gas R the specific gas constant

V Ma  Mach c number

During fluid flow through many devices such as nozzles, diffusers, and turbine blade passages, flow quantities vary primarily in the flow direction only, and the flow can be approximated as one- dimensional isentropic flow with good accuracy.

©McGraw-Hill Education. EXAMPLE: Gas Flow through a Converging– Diverging Duct

Variation of normalized fluid properties and cross-sectional area along a duct as the pressure drops from 1400 to 200 kPa.

©McGraw-Hill Education. Variation of fluid properties in flow direction in the duct described in Example 12–2 for m = 3 kg/s = constant

P, kPa T, K V, m/s ρ, kg/m3 c, m/s A, cm2 Ma 1400 473 0 15.7 339.4 ∞ 0 1200 457 164.5 13.9 333.6 13.1 0.493 1000 439 240.7 12.1 326.9 10.3 0.736 800 417 306.6 10.1 318.8 9.64 0.962 767* 413 317.2 9.82 317.2 9.63 1.000 600 391 371.4 8.12 308.7 10.0 1.203 400 357 441.9 5.93 295.0 11.5 1.498 200 306 530.9 3.46 272.9 16.3 1.946

* 767 kPa is the critical pressure where the local Mach number is unity.

Note that as the pressure decreases, the temperature and speed of sound decrease while the fluid velocity and Mach number increase in the flow direction. The density decreases slowly at first and rapidly later as the fluid velocity increases.

©McGraw-Hill Education. We note from Example that the flow area decreases with decreasing pressure up to a critical-pressure value (Ma = 1), and then it begins to increase with further reductions in pressure. The Mach number is unity at the location of smallest flow area, called the throat. The velocity of the fluid keeps increasing after passing the throat although the flow area increases rapidly in that region. This increase in velocity past the throat is due to the rapid decrease in the fluid density. The flow area of the duct considered in this example first decreases and then increases. The cross section of a nozzle at Such ducts are called converging– the smallest flow area is called diverging nozzles. the throat. These nozzles are used to accelerate gases to supersonic speeds and should not be confused with Venturi nozzles, which are used strictly for incompressible flow.

©McGraw-Hill Education. Variation of Fluid Velocity with Flow Area In this section, the relations for the variation of static-to-stagnation property ratios with the Mach number for pressure, temperature, and density are provided. m AV constant d dA dV    0  AV dP V dV 0  dA dP1 d  2 A  V dP

dA dP 2 2  1 Ma  AV Derivation of the differential form This relation describes the variation of of the energy equation for steady pressure with flow area. isentropic flow.

©McGraw-Hill Education. At subsonic velocities, the pressure decreases in converging ducts (subsonic nozzles) and increases in diverging ducts (subsonic diffusers). At supersonic velocities, the pressure decreases in diverging ducts (supersonic nozzles) and increases in converging ducts (supersonic diffusers).

dA dV This equation governs the shape of a    1  Ma 2  nozzle or a diffuser in subsonic or AV supersonic isentropic flow. dA For subsonic flow  Ma 1 , 0 dV dA For supersonic flow  Ma 1 , 0 dV dA For sonic flow  Ma 1 , 0 dV The proper shape of a nozzle depends on the highest velocity desired relative to the sonic velocity. To accelerate a fluid, we must use a converging nozzle at subsonic velocities and a diverging nozzle at supersonic velocities. To accelerate a fluid to supersonic velocities, we must use a converging–diverging nozzle.

©McGraw-Hill Education. We cannot obtain supersonic velocities by attaching a converging section to a converging nozzle. Doing so will only move the sonic cross section farther downstream and decrease the mass flow rate.

©McGraw-Hill Education. Variation of flow properties in subsonic and supersonic nozzles and diffusers.

©McGraw-Hill Education. Property Relations for Isentropic Flow of Ideal Gases The relations between the static properties and stagnation properties of an ideal gas with constant specific heats

T0 k 1 2 1 Ma T 2 kk/1   P0 k 1 2  1 Ma P 2 1/k  1 0 k 1 2  1 Ma  2

T *2  Tk1 Critical kk/1   ratios P*2   (Ma = 1) Pk0 1 1/k  1 When Ma = 1, the properties at the nozzle  *2 t   throat become the critical properties. 0 k 1

©McGraw-Hill Education. The critical-pressure, critical-temperature, and critical-density ratios for isentropic flow of some ideal gases

Superheated Hot products of Monatomic steam, combustion, Air, gases, k = 1.3 k = 1.33 k = 1.4 k = 1.667 푃∗ 0.5457 0.5404 0.5283 0.4871 푃0 푇∗ 0.8696 0.8584 0.8333 0.7499 푇0 휌∗ 0.6276 0.6295 0.6340 0.6495 휌0

©McGraw-Hill Education. 12–3 ■ ISENTROPIC FLOW THROUGH NOZZLES Converging or converging–diverging nozzles are found in steam and gas turbines and aircraft and spacecraft propulsion systems. In this section we consider the effects of back pressure (i.e., the pressure applied at the nozzle discharge region) on the exit velocity, the mass flow rate, and the pressure distribution along the nozzle. Converging Nozzles Mass flow rate through a nozzle

AMa P00 k /  RT  m  2 kk1 / 2 1 1k 1 Ma /2 Maximum mass flow rate kk1 / 2 1 The effect of back pressure on the k 2  m A* P pressure distribution along a max 0  converging nozzle. RT0  k 1

©McGraw-Hill Education. PPPbbfor  * Pe   P* forPPb  *

The variation of the mass flow rate through a nozzle with inlet stagnation properties.

The effect of back pressure Pb on the mass flow rate and the exit pressure Pe of a converging nozzle.

©McGraw-Hill Education. kk1 / 2 1 Ma* is the local Ak1 2   1 2   velocity  1 Ma  Ak* Ma 1  2  nondimensionalized with respect to the V V cMa c Ma kRT T sonic velocity at the Ma*  Ma*    Ma c* c c*** ckRT * T throat. Ma is the local k 1 velocity Ma* Ma 2k 1 Ma 2 nondimensionalized with respect to the local sonic velocity.

Various property ratios for isentropic flow through nozzles and diffusers are listed in Table A–13 for k = 1.4 for convenience.

©McGraw-Hill Education. Converging–Diverging Nozzles The highest velocity in a converging nozzle is limited to the sonic velocity (Ma = 1), which occurs at the exit plane (throat) of the nozzle. Accelerating a fluid to supersonic velocities (Ma > 1) can be accomplished only by attaching a diverging flow section to the subsonic nozzle at the throat (a converging–diverging nozzle), which is standard equipment in supersonic

aircraft and rocket propulsion.

Courtesy NASA Courtesy Converging–diverging nozzles are commonly used in rocket engines to provide high thrust.

©McGraw-Hill Education. When Pb = P0 (case A), there will be no flow through the nozzle.

1. When P0 > Pb > PC, the flow remains subsonic throughout the nozzle, and the mass flow is less than that for choked flow. The fluid velocity increases in the first (converging) section and reaches a maximum at the throat (but Ma < 1). However, most of the gain in velocity is lost in the second (diverging) section of the nozzle, which acts as a diffuser. The pressure decreases in the converging section, reaches a minimum at the throat, and increases at the expense of velocity in the diverging section.

The effects of back pressure on the flow through a converging– diverging nozzle.

©McGraw-Hill Education. 2. When Pb = PC, the throat pressure becomes P* and the fluid achieves sonic velocity at the throat. But the diverging section of the nozzle still acts as a diffuser, slowing the fluid to subsonic velocities. The mass flow rate that was

increasing with decreasing Pb also reaches its maximum value.

3. When PC > Pb > PE, the fluid that achieved a sonic velocity at the throat continues accelerating to supersonic velocities in the diverging section as the pressure decreases. This acceleration comes to a sudden stop, however, as a normal shock develops at a section between the throat and the exit plane, which causes a sudden drop in velocity to subsonic levels and a sudden increase in pressure. The fluid then continues to decelerate further in the remaining part of the converging–diverging nozzle.

©McGraw-Hill Education. 4. When PE > Pb > 0, the flow in the diverging section is supersonic, and the

fluid expands to PF at the nozzle exit with no normal shock forming within the nozzle. Thus, the flow through the nozzle can be approximated as isentropic.

When Pb = PF, no shocks occur within or outside the nozzle.

When Pb < PF, irreversible mixing and expansion waves occur downstream of the exit plane of the nozzle.

When Pb > PF, however, the pressure of the fluid increases from PF to Pb irreversibly in the wake of the nozzle exit, creating what are called oblique shocks.

©McGraw-Hill Education. 12–4 ■ SHOCK WAVES AND EXPANSION WAVES For some back pressure values, abrupt changes in fluid properties occur in a very thin section of a converging–diverging nozzle under supersonic flow conditions, creating a shock wave. We study the conditions under which shock waves develop and how they affect the flow. Normal Shocks Normal shock waves: The shock waves that occur in a plane normal to the direction of flow. The flow process through the shock wave is highly irreversible.

Schlieren image of a normal shock in a

Laval nozzle. The Mach number in the

nozzle just upstream (to the left) of the . shock wave is about 1.3. Boundary layers distort the shape of the normal shock near

the walls and lead to flow separation

G.S. Settles, Gas Dynamics Lab, Penn State University. University. State Penn Dynamics Lab, Gas Settles, G.S. Used with permissionUsedwith beneath the shock. ©

©McGraw-Hill Education. Conservation of mass Control volume for flow across a normal shock wave. 1AV 1 2 AV 2 1VV 1 2 2 Conservation of energy 22 VV12 hh   hh01 02 1222 Conservation of momentum

A P1 P 2  m V 2  V 1 

ss210 Increase of entropy Fanno line: Combining the conservation of mass and energy relations into a single equation and plotting it on an h-s diagram yield a curve. It is the locus of states that have the same value of stagnation enthalpy and mass flux. Rayleigh line: Combining the conservation of mass and momentum equations into a single equation and plotting it on the h-s diagram yield a curve. The h-s diagram for flow across a normal shock.

©McGraw-Hill Education. The relations between various properties before and after the shock for an ideal gas with constant specific heats. 22 T PV PMa cPT2Ma 2 2  P   Ma  2 2 2  2 2 2    2   2  T1 PV 1 1 P 1Ma 1 c 1PT1Ma 1 1  P 1   Ma 1  Various flow property ratios 2 P Ma11 1 Mak 1 /2 across the shock Fanno line: 2  2 are listed in Table P1 Ma 1 Ma/k 1 2 22  A–14. 2 Pk211 Ma Rayleigh lin e:  2 P121 k Ma 2 2 Ma1  2/k 1 Ma 2  2 2Ma1 kk /  1 1 This represents the intersections of the Fanno and Rayleigh lines, and relates the Mach number upstream of the shock to that downstream of the shock.

Variation of flow properties across a normal shock in an ideal gas.

©McGraw-Hill Education. © StockTrek/Getty Images RF The air inlet of a supersonic fighter jet is designed such that a shock wave at the inlet © G.S. Settles, Gas Dynamics Lab, Penn State University. Used with permission. decelerates the air to subsonic Schlieren image of the blast wave (expanding velocities, increasing the spherical normal shock) produced by the explosion pressure and temperature of of a firecracker detonated inside a metal can that sat on a stool. The shock expanded radially the air before it enters the outward in all directions at a supersonic speed that engine. decreased with radius from the center of the explosion. The microphone at the lower right sensed the sudden change in pressure of the passing shock wave and triggered the microsecond flashlamp that exposed the photograph.

©McGraw-Hill Education. TP22 s2 s 1  cP ln  R ln TP11

Since the flow across the shock is adiabatic and irreversible, the second law of thermodynamics requires that the entropy increase across the shock wave. Thus, a shock wave cannot exist for values of Ma1 less than unity where the entropy change would be negative. For adiabatic flows, shock waves can exist only for Entropy change across the normal shock. supersonic flows, Ma1 > 1.

©McGraw-Hill Education. Example 12–8 illustrates that the stagnation pressure and velocity decrease while the static pressure, temperature, density, and entropy increase across the shock.

The rise in the temperature of the fluid downstream of a shock wave is of major concern to the aerospace engineer because it creates heat transfer problems on the leading edges of wings and nose cones of space reentry vehicles and the recently proposed hypersonic space planes. Overheating, in fact, led to the tragic loss of the space shuttle Columbia in February of 2003 as it was reentering earth’s atmosphere.

When a lion tamer cracks his whip, a weak spherical shock wave forms near the tip and spreads out radially; the pressure inside the expanding shock wave is higher than ambient air pressure, and this is what causes the crack when the shock wave

reaches the lion's ear. © Joshua Ets-Hokin/Getty Images RF

©McGraw-Hill Education. Oblique Shocks When the space shuttle travels at supersonic speeds through the atmosphere, it produces a complicated shock pattern consisting of inclined shock waves called oblique shocks. Some portions of an oblique shock are curved, while other portions are straight.

Schlieren image of a small model of the space shuttle Orbiter being tested at Mach 3 in the supersonic wind tunnel of the Penn State Gas Dynamics Lab. Several oblique shocks are seen in the air surrounding the spacecraft.

©McGraw-Hill Education. An oblique shock of shock angle  Velocity vectors through an oblique shock of formed by a slender, two- shock angle  and deflection angle . dimensional wedge of half-angle . The flow is turned by deflection Unlike normal shocks, in which the angle  downstream of the shock, downstream Mach number is always and the Mach number decreases. subsonic, Ma2 downstream of an oblique shock can be subsonic, sonic, or supersonic, depending on the upstream

Mach number Ma1 and the turning angle.

©McGraw-Hill Education. The same velocity vectors of Fig. 12–35, but rotated by angle /2–, Relationships across an oblique shock for an so that the oblique shock is vertical. ideal gas in terms of the normal component of Normal Mach numbers Ma1,n and upstream Mach number Ma1,n. Ma2,n are also defined. All the equations, shock tables, etc., for normal shocks apply to oblique shocks as well, provided that we use only the normal components of the Mach number.

©McGraw-Hill Education. For known shock angle 훽 and known upstream Mach number Ma1, we use the first part of Eq. 12–44 to calculate Ma1, n, and then use the normal shock tables (or their corresponding equations) to obtain Ma2, n. If we also knew the deflection angle 휃, we could calculate Ma2 from the second part of Eq. 12–44. But, in a typical application, we know either 훽 or 휃, but not both. Fortunately, a

bit more algebra provides us with a relationship between 휃, 훽, and Ma1. We begin by noting that tan 훽 = V1, n/V1, t and tan(훽 − 휃) = V2, n/V2, t (Fig. 12–35). But since V1, t = V2, t, we combine these two expressions to yield

2 22 Vk2, nntan  2 1 Ma 1, 2 k  1 Ma1 sin   2  2 2 (12 - 45) V1, nntan  k 1 Ma 1,  k 1 Ma 1 sin

where we have also used Eq. 12–44 and the fourth equation of Fig. 12–36. We apply trigonometric identities for cos 2훽 and tan(훽 − 휃), namely, tan  tan cos 2 cos22   sin  and tan      1 tan  tan After some algebra, Eq. 12–45 reduces to

2 2 2 cot  Ma1 sin 1 The--Ma relationship: tan   2 (12 - 46) Ma1 k  cos2  2

©McGraw-Hill Education. The dependence of straight oblique shock deflection angle  on shock angle  for

several values of upstream Mach number Ma1. Calculations are for an ideal gas with k = 1.4. The dashed black line connects points of maximum deflection angle

( =  max). Weak oblique shocks are to the left of this line, while strong oblique shocks are to the right of this line. The dashed gray line connects points where

the downstream Mach number is sonic (Ma2 = 1). Supersonic downstream flow (Ma2 > 1) is to the left of this line, while subsonic downstream flow (Ma2 < 1) is to the right of this line.

©McGraw-Hill Education. A detached oblique shock occurs upstream of a two-dimensional wedge of half-angle  when  is greater than the maximum possible deflection angle . A shock of this kind is called a bow wave because of its resemblance to the water wave that forms at the bow of a ship.

Still frames from schlieren videography illustrating the detachment of an oblique shock from a cone with increasing cone half-angle  in air at Mach 3. At (a)  =20 and (b)  =40, the oblique shock remains attached, but by Mach angle: 1 (c)  =60, the oblique shock has detached,   sin 1/ Ma1  forming a bow wave.

©McGraw-Hill Education. Color schlieren image of Mach 3.0 flow from left to right over a sphere. A curved shock wave called a bow shock forms in front of the sphere and curves downstream. © G.S. Settles, Gas Dynamics Lab, Penn State University. Used with permission.

©McGraw-Hill Education. Prandtl–Meyer Expansion Waves We now address situations where supersonic flow is turned in the opposite direction, such as in the upper portion of a two-dimensional wedge at an angle of attack greater than its half-angle . We refer to this type of flow as an expanding flow, whereas a flow that produces an oblique shock may be called a compressing flow. As previously, the flow changes direction to conserve mass. However, unlike a compressing flow, an expanding flow does not result in a shock wave. Rather, a continuous expanding region called an expansion fan appears, composed of an infinite number of Mach waves called Prandtl–Meyer expansion waves.

An expansion fan in the upper portion of the flow formed by a two-dimensional wedge at the angle of attack in a supersonic flow. The flow is turned by angle u, and the Mach number increases across the expansion fan. Mach angles upstream and downstream of the expansion fan are indicated. Only three expansion waves are shown for simplicity, but in fact, there are an infinite number of them. (An oblique shock is present in the bottom portion of this flow.)

©McGraw-Hill Education. Turning angle across an expansion fan:  Ma21   Ma  Prandtl–Meyer function

kk111 2 1 2  Ma  tan  Ma  1  tan Ma  1 kk11

© G.S. Settles, Gas Dynamics Lab, Penn State University. Used with permission. (a) Mach 3 flow over an axisymmetric cone of 10-degree half-angle. The boundary layer becomes turbulent shortly downstream of the nose, generating Mach waves that are visible in the color schlieren image. (b) A similar pattern is seen in this color schlieren image for Mach 3 flow over an 11-degree 2-D wedge. Expansion waves are seen at the corners where the wedge flattens out.

©McGraw-Hill Education.

. The complex interactions

between shock waves and German Research Institute of Saint Louis, ISL. (b) © G.S. Settles, Settles, (b) © ISL. G.S. Louis, Saint of Institute Research German

- expansion waves in an “overexpanded” supersonic jet. (a) The flow is visualized by a schlieren- like differential interferogram. (b) Color shlieren image. (c) Tiger

tail shock pattern.

a) Reproduced courtesy French the of courtesy Reproduced a)

( Fighter Courtesy StrikeUniversity. Joint with permission.State (c) Used Photo Penn Dynamics Gas Lab, Whitney & Pratt and Defense ofProgram, Department

©McGraw-Hill Education. 12–5 ■ DUCT FLOW WITH HEAT TRANSFER AND NEGLIGIBLE FRICTION (RAYLEIGH FLOW) So far we have limited our Rayleigh flows: Steady one- consideration mostly to isentropic dimensional flow of an ideal gas with flow (no heat transfer and no constant specific heats through a irreversibilities such as friction). constant-area duct with heat transfer, Many compressible flow problems but with negligible friction. encountered in practice involve chemical reactions such as combustion, nuclear reactions, evaporation, and condensation as well as heat gain or heat loss through the duct wall. Such problems are difficult to analyze exactly since they may involve significant changes in chemical composition during flow, and the conversion of latent, chemical, and nuclear energies to Many practical compressible flow problems thermal energy. involve combustion, which may be modeled as heat gain through the duct wall. A simplified model is Rayleigh flow.

22x-Momentum PVPV1 1 1  2  2 2 equation VV22 Energy q c T  T   21equation p 21 2

q h02  h 01  cp  T 02  T 01 

TP22Entropy s2 s 1  cp ln  R ln change TP11 Control volume for flow in a constant- area duct with heat transfer and PP 12 Equation negligible friction. 1TT 1 2 2 of state

Consider a gas with known properties R, k, and cp. For a specified inlet state 1, the inlet properties P1, T1, 1, V1, and s1 are known. The five exit properties P2, T2, 2, V2, and s2 can be determined from the above equations for any specified value of heat transfer q.

©McGraw-Hill Education. From the Rayleigh line and the equations 1. All the states that satisfy the conservation of mass,momentum, and energy equations as well as the property relations are on the Rayleigh line. 2. Entropy increases with heat gain, and thus we proceed to the right on the Rayleigh line as heat is transferred to the fluid. 3. Heating increases the Mach number for subsonic flow, but decreases it for supersonic flow. T-s diagram for flow in a constant-area duct with heat transfer and negligible 4. Heating increases the stagnation friction (Rayleigh flow). temperature T0 for both subsonic and supersonic flows, and cooling decreases it. 7. The entropy change corresponding to 5. Velocity and static pressure have a specified temperature change (and opposite trends. thus a given amount of heat transfer) is larger in supersonic flow. 6. Density and velocity are inversely proportional.

©McGraw-Hill Education. During heating, fluid temperature always increases if the Rayleigh flow is supersonic, but the temperature may actually drop if the flow is subsonic.

Heating or cooling has opposite effects on most properties. Also, the stagnation pressure decreases during heating and increases during cooling regardless of whether the flow is subsonic or supersonic.

The effects of heating and cooling on the properties of Rayleigh flow

Heating Heating Cooling Cooling Property Subsonic Supersonic Subsonic Supersonic Velocity, V Increase Decrease Decrease Increase Mach number, Ma Increase Decrease Decrease Increase

Stagnation temperature, T0 Increase Increase Decrease Decrease Temperature, T Increase for Ma < 1/k1/2 Increase Decrease for Ma < 1/k1/2 Decrease Decrease for Ma > 1/k1/2 Increase for Ma > 1/k1/2 Density, ρ Decrease Increase Increase Decrease

Stagnation pressure, P0 Decrease Decrease Increase Increase Pressure, P Decrease Increase Increase Decrease Entropy, s Increase Increase Decrease Decrease

2 2 T Ma21 1 k Ma  2   T Ma 1 k Ma 2 1 12 

22  V Ma12 1 k Ma  21 22 12V Ma21 1 k Ma  2 P1* k TMa 1kk V   1 Ma 2 2  2and   2 P*1Ma k T *1Ma  k V * 1Ma  k

kk1 Ma22 2   1 Ma T0      22 Tk0 * (1 Ma )

kk/1   P k 1 2k 1 Ma 2 Representative 0  results are given in 2  P0 * 1 k Ma k 1 Table A–15.

©McGraw-Hill Education. Choked Rayleigh Flow The fluid at the critical state of Ma =1 cannot be accelerated to supersonic velocities by heating. Therefore, the flow is choked. For a given inlet state, the corresponding critical state fixes the maximum possible heat transfer for steady flow:

qmaxhT00**  h 01  cp  T01 

For a given inlet state, the maximum possible heat transfer occurs when sonic conditions are reached at the exit state.

Wall friction associated with high-speed flow through short devices with large cross-sectional areas such as large nozzles is often negligible, and flow through such devices can be approximated as being frictionless. But wall friction is significant and should be considered when studying flows through long flow sections, such as long ducts, especially when the cross-sectional area is small. In this section we consider compressible flow with significant wall friction but negligible heat transfer in ducts of constant cross-sectional area. Consider steady, one-dimensional, adiabatic flow of an ideal gas with constant specific heats through a constant-area duct with significant frictional effects. Such flows are referred to as Fanno flows.

F x-Momentum P A P A  F mV  mV  P  P  friction 1 2 friction 2 1 1 2 A equation

 (2VVVV 2 ) 2 ( 1 1 ) 1 TP F 22Entropy 22friction s21 s  cp ln  R ln  0 PVPV1 1 1  2  2 2  TPchange A 11 VV22 V 2 h12  h   h  h  h  h   constant Energy 12 2 2 01 02 0 2 equation

h cp T For an ideal gas

22 VV12 TTTT1  2  01  02 22ccpp V 2 TT0    constant 2cp PP Control volume for adiabatic flow in 12 Equation a constant-area duct with friction. 1TT 1 2 2 of state

©McGraw-Hill Education. Consider a gas with known properties R, k, and cp. For a specified inlet state 1, the inlet properties P1, T1, 1, V1, and s1 are known. The five exit properties P2, T2, 2, V2, and s2 can be determined from the above equations for any specified value of the friction force Ffriction.

T-s diagram for adiabatic frictional flow in a constant- area duct (Fanno flow). Numerical values are for air with k = 1.4 and inlet

conditions of T1 = 500 K, P1 = 600 kPa, V1 = 80 m/s, and an assigned value of s1 = 0.

1. All the states that satisfy the conservation of mass, momentum, and energy equations as well as the property relations are on the Fanno line. 2. Friction causes entropy to increase, and thus a process always proceeds to the right along the Fanno line. At the point of maximum entropy, the Mach number is Ma = 1. All states on the upper part of the Fanno line are subsonic, and all states on the lower part are supersonic. 3. Friction increases the Mach number for subsonic Fanno flow, but decreases it for supersonic Fanno flow. 4. The energy balance requires that stagnation temperature remain constant during Fanno flow. But the actual temperature may change. Velocity increases and thus temperature decreases during subsonic flow, but the opposite occurs during supersonic flow. 5. The continuity equation indicates that density and velocity are inversely proportional. Therefore, the effect of friction is to decrease density in subsonic flow (since velocity and Mach number increase), but to increase it in supersonic flow (since velocity and Mach number decrease).

©McGraw-Hill Education. Friction causes the Mach number to increase and the temperature to decrease in subsonic Fanno flow, but it does the opposite in supersonic Fanno flow.

The effects of friction on the properties of Fanno flow Property Subsonic Supersonic Velocity, V Increase Decrease Mach number, Ma Increase Decrease

Stagnation temperature, T0 Constant Constant Temperature, T Decrease Increase Density, ρ Decrease Increase

Stagnation pressure, P0 Decrease Decrease Pressure, P Decrease Increase Entropy, s Increase Increase

Differential control volume for adiabatic flow in a constant- area duct with friction.

Continuity equation d dV  dV V d  0     V x-Momentum equation

PAc  P  dP A  Ffriction  m V  dV  mV  F dPA  F   AV dV or dP friction   V dV  0 friction A

©McGraw-Hill Education. 2 ffxx224A A dx Vfx Ffriction w dA s   w p dx   V dx   V dP dx  V dV  0 82DDhh 2Dh 1 dPfx dV 2  dx   0 kMa P2 Dh V Energy equation 2 k 1 2 dT2k  1 Ma dMa TT0   1  Ma  constant  2 Tk2 1 Ma2 Ma Mach number 2V dV 2Ma kRT d Ma  kR Ma2 dT  dV dMa 1 dT VV22  2V dV 2 d Ma dT VTMa 2 Ma T dV dMak 1 Ma 2 d Ma dV 2 d Ma   or  VMa2 k  1MaMa 22 V 2  k  1MaMa Ideal gas dP dT d dP dT dV dP2 2k 1 Ma 2 dMa dP R dT  RT d      PT  PTV Pk2 1 Ma2 Ma 2 f 4 1 Ma  x dx d Ma D 3  2 h kkMa 2 1 Ma

©McGraw-Hill Education. fL* 1 Ma2 k 1 k 1 Ma 2 22 ln Dh kMa 2 k 2 k 1 Ma fL fL**   fL   DDD    h h 12  h  1.11 1 6.9 /D  1.8 log    f Re 3.7

1/2 Pk11  Pk* Ma 2 1 Ma 2 Tk1  2 The length L* represents the distance Tk* 2 1 Ma between a given section where the 1/2 Mach number is Ma and a real or Vk*1 Ma imaginary section where Ma* = 1. Vk*  2 1 Ma 2

2 kk1 / 2 1 P00 1 2k 1 Ma **  Pk00 Ma 1

Friction causes subsonic Fanno flow in a constant-area duct to accelerate toward sonic velocity, and the Mach number becomes exactly unity at the exit for a certain duct length. This duct length is referred to as the maximum length, the sonic length, or the critical If duct length L is greater than L, length, and is denoted by L*. supersonic Fanno flow is always sonic at the duct exit. Extending the duct will only move the location of the normal shock further upstream.

©McGraw-Hill Education. • Stagnation properties Summary • One-dimensional isentropic flow • Variation of fluid velocity with flow area • Property relations for isentropic flow of ideal gases • Isentropic flow through nozzles • Converging nozzles • Converging–diverging nozzles • Shock waves and expansion waves • Normal shocks • Oblique shocks • Prandtl–Meyer expansion waves • Duct flow with heat transfer and negligible friction (Rayleigh flow) • Property relations for Rayleigh flow • Choked Rayleigh flow • Adiabatic duct flow with friction (Fanno flow) • Property relations for Fanno flow • Choked Fanno flow