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9.12 Compressible Flow Preliminaries 553

moving away on the left with velocity c 2 DV, pressure p 1 Dp, density r 1 Dr, and

temperature T 1 DT.

At steady state, the conservation of mass principle for the control volume reduces

to m

5 m

, or

rAc 5

r 1 ¢r

c 2 ¢V

On rearrangement

0 5 c¢r 2 r¢V 2 ¢r¢V

(9.32)

If the disturbance is weak, the third term on the right of Eq. 9.32 can be neglected,

leaving

¢V 5

¢r (9.33)

Next, the momentum equation, Eq. 9.31, is applied to the control volume under

consideration. Since the thickness of the wave is small, shear forces at the wall are

negligible. The effect of gravity is also ignored. Hence, the only significant forces act-

ing on the control volume in the direction of flow are the forces due to pressure at

the inlet and exit. With these idealizations, the component of the momentum equation

in the direction of flow reduces to

A 2

p 1 ¢p

A 5 m

c 2 ¢V

2 m

5 m

c 2 ¢V 2 c

rAc

2¢V

¢p 5 rc ¢V (9.34)

Combining Eqs. 9.33 and 9.34 and solving for c

c 5

¢p

¢r (9.35)

SOUND WAVES. For sound waves the differences in pressure, density, and

temperature across the wave are quite small. In particular, Dr ,, r, justifying the

neglect of the third term of Eq. 9.32. Furthermore, the ratio Dp/Dr in Eq. 9.35 can

be interpreted as the derivative of pressure with respect to density across the wave.

Experiments also indicate that the relation between pressure and density across a

sound wave is nearly isentropic. The expression for the velocity of sound then

becomes

c 5

(9.36a)

or in terms of specific volume

c 5

(9.36b)

The velocity of sound is an intensive property whose value depends on the state of

the medium through which sound propagates. Although we have assumed that sound

propagates isentropically, the medium itself may be undergoing any process.

Means for evaluating the velocity of sound c for gases, liquids, and solids are intro-

duced in Sec. 11.5. The special case of an ideal gas will be considered here because it

is used extensively later in the chapter. For this case, the relationship between pressure

and specific volume of an ideal gas at fixed entropy is py

5 constant, where k is the

velocity of sound

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554 Chapter 9

Gas Power Systems

specific heat ratio (Sec. 6.11.2). Thus,

52k

y, and Eq. 9.36b gives c 5 1k

Or, with the ideal gas equation of state

c 5 1kRT





ideal gas

(9.37)

to illustrate the use of Eq. 9.37, let us calculate the velocity of

sound in air at 300 K (5408R) and 650 K (11708R) From Table A-20 at 300 K,

k 5 1.4. Thus

c 5

1.4a

8314

28.97

N ? m

? K

b1300 K2`

1 k

g ? m

1 N

`5 347

a1138

At 650 K, k 5 1.37, and c 5 506 m/s (1660 ft/s), as can be verified. As examples in

English units, consider next helium at 4958R (275 K) and 10808R (600 K). For a

monatomic gas, the specific heat ratio is essentially independent of temperature and

has the value k 5 1.67. Thus, at 4958R

c 5

1.67

1545

ft ? lbf

? 8R

14958R2

32.2 lb ? ft

1 l

5 3206

977

At 10808R, c 5 4736 ft/s (1444 m/s), as can be verified. b b b b b

Mach Number

In subsequent discussions, the ratio of the velocity V at a state in a flowing fluid to

the value of the sonic velocity c at the same state plays an important role. This ratio

is called the Mach number M

M 5

(9.38)

When M . 1, the flow is said to be supersonic; when M , 1, the flow is subsonic;

and when M 5 1, the flow is sonic. The term hypersonic is used for flows with Mach

numbers much greater than one, and the term transonic refers to flows where the

Mach number is close to unity.

Mach number

supersonic

subsonic

BIOCONNECTIONS For centuries, physicians have used sounds emanating

from the human body to aid diagnosis. Since the early nineteenth century, stetho-

scopes have enabled more effective sound detection. Sounds within the human

body have also interested researchers. For instance, in the 1600s an observer described a

phenomenon that now mainly serves as a diversion: Place your thumbs over your ear open-

ings so the ear canals are completely covered. Then, with elbows raised, tighten your hands

into fists and hear muscular noise as a soft distant rumbling.

Today’s medical researchers and practitioners use sound in various ways. Researchers

have found that during strong and repeated activity, the wrist muscles of people with

untreated Parkinson’s disease produce sound at much lower frequencies than those of

healthy individuals. This knowledge may prove useful in monitoring muscle degeneration

in Parkinson’s patients or aid in early diagnosis of this debilitating disease.

A more commonly encountered sound-related method in medical practice is the use of

sound at frequencies above that audible by the human ear, known as ultrasound. Ultrasound

imaging allows physicians to peer inside the body and evaluate solid structures in the

abdominal cavity. Ultrasound devices beam sound waves into the body and collect return

echoes as the beam encounters regions of differing density. The reflected sound waves

produce shadow pictures of structures below the skin on a monitor screen. Pictures show

the shape, size, and movement of target objects in the path of the beam.

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9.13 Analyzing One-Dimensional Steady Flow in Nozzles and Diffusers 555

9.12.3

Determining Stagnation State Properties

When dealing with compressible flows, it is often convenient to work with properties

evaluated at a reference state known as the stagnation state. The stagnation state is

the state a flowing fluid would attain if it were decelerated to zero velocity isentropi-

cally. We might imagine this as taking place in a diffuser operating at steady state. By

reducing an energy balance for such a diffuser, it can be concluded that the enthalpy

at the stagnation state associated with an actual state in the flow where the specific

enthalpy is h and the velocity is V is given by

5 h 1

(9.39)

The enthalpy designated here as h

is called the stagnation enthalpy. The pressure

and temperature T

at a stagnation state are called the stagnation pressure and

stagnation temperature, respectively.

Obstetricians commonly use ultrasound to assess the fetus during pregnancy. Emer-

gency physicians use ultrasound to assess abdominal pain or other concerns. Ultrasound

is also used to break up small kidney stones.

Cardiologists use an ultrasound application known as echocardiography to evaluate the

heart and its valve function, measure the amount of blood pumped with each stroke, and

detect blood clots in veins and artery blockage. Among several uses of echocardiography

are the stress test, where the echocardiogram is done both before and after exercise, and

the transesophageal test, where the probe is passed down the esophagus to locate it closer

to the heart, thereby enabling clearer pictures of the heart.

stagnation state

stagnation pressure

and temperature

9.13 Analyzing One-Dimensional Steady

Flow in Nozzles and Diffusers

Although the subject of compressible flow arises in a great many important areas of

engineering application, the remainder of this presentation is concerned only with

flow through nozzles and diffusers. Texts dealing with compressible flow should be

consulted for discussion of other areas of application.

In the present section we determine the shapes required by nozzles and diffusers

for subsonic and supersonic flow. This is accomplished using mass, energy, entropy,

and momentum principles, together with property relationships. In addition, we study

how the flow through nozzles is affected as conditions at the nozzle exit are changed.

The presentation concludes with an analysis of normal shocks, which can exist in

supersonic flows.

9.13.1

Exploring the Effects of Area Change in Subsonic

and Supersonic Flows

The objective of the present discussion is to establish criteria for determining whether

a nozzle or diffuser should have a converging, diverging, or converging–diverging

shape. This is accomplished using differential equations relating the principal vari-

ables that are obtained using mass and energy balances together with property rela-

tions, as considered next.

stagnation enthalpy

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556 Chapter 9

Gas Power Systems

GOVERNING DIFFERENTIAL EQUATIONS. Let us begin by considering a

control volume enclosing a nozzle or diffuser. At steady state, the mass flow rate is

constant, so

V 5 constant

In differential form

rAV

5 0

AV dr 1 rA dV 1 rV dA 5 0

or on dividing each term by rAV

5 0

(9.40)

Assuming

5 W

5 0 and negligible potential energy effects, an energy rate

balance reduces to give

5 h

Introducing Eq. 9.39, it follows that the stagnation enthalpies at states 1 and 2 are

equal: h

5 h

. Since any state downstream of the inlet can be regarded as state 2,

the following relationship between the specific enthalpy and kinetic energy must be

satisfied at each state

h 1

5 h





1constant2

In differential form this becomes

(9.41)

This equation shows that if the velocity increases (decreases) in the direction of flow,

the specific enthalpy must decrease (increase) in the direction of flow.

In addition to Eqs. 9.40 and 9.41 expressing conservation of mass and energy,

relationships among properties must be taken into consideration. Assuming the flow

occurs isentropically, the property relation (Eq. 6.10b)

T ds 5 dh 2

reduces to give

dh 5

(9.42)

This equation shows that when pressure increases or decreases in the direction of

flow, the specific enthalpy changes in the same way.

Forming the differential of the property relation p 5 p(r, s)

dp 5

dr 1

The second term vanishes in isentropic flow. Introducing Eq. 9.36a, we have

dp 5 c

dr (9.43)

which shows that when pressure increases or decreases in the direction of flow, den-

sity changes in the same way.

Additional conclusions can be drawn by combining the above differential equa-

tions. Combining Eqs. 9.41 and 9.42 results in

dp 52V dV

(9.44)

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9.13 Analyzing One-Dimensional Steady Flow in Nozzles and Diffusers 557

which shows that if the velocity increases (decreases) in the direction of flow, the

pressure must decrease (increase) in the direction of flow.

Eliminating dp between Eqs. 9.43 and 9.44 and combining the result with Eq. 9.40

gives

1 2

or with the Mach number M

11 2 M

(9.45)

VARIATION OF AREA WITH VELOCITY. Equation 9.45 shows how area

must vary with velocity. The following four cases can be identified:

Case 1: Subsonic nozzle. dV . 0, M , 1 1 dA , 0: The duct converges in the

direction of flow.

Case 2: Supersonic nozzle. dV . 0, M . 1 1 dA . 0: The duct diverges in the

direction of flow.

Case 3: Supersonic diffuser. dV , 0, M . 1 1 dA , 0: The duct converges in the

direction of flow.

Case 4: Subsonic diffuser. dV , 0, M , 1 1 dA . 0: The duct diverges in the direc-

tion of flow.

The conclusions reached above concerning the nature of the flow in subsonic

and supersonic nozzles and diffusers are summarized in Fig. 9.30. From Fig. 9.30a,

we see that to accelerate a fluid flowing subsonically, a converging nozzle must be

used, but once M 5 1 is achieved, further acceleration can occur only in a diverg-

ing nozzle. From Fig. 9.30b, we see that a converging diffuser is required to decel-

erate a fluid flowing supersonically, but once M 5 1 is achieved, further decelera-

tion can occur only in a diverging diffuser. These findings suggest that a Mach

number of unity can occur only at the location in a nozzle or diffuser where the

cross-sectional area is a minimum. This location of minimum area is called the

throat.

The developments of this section have not required the specification of an equa-

tion of state; thus, the conclusions hold for all gases. Moreover, although the conclu-

sions have been drawn under the restriction of isentropic flow through nozzles and

diffusers, they are at least qualitatively valid for actual flows because the flow through

well-designed nozzles and diffusers is nearly isentropic. Isentropic nozzle efficiencies

(Sec. 6.12) in excess of 95% can be attained in practice.

throat

Fig. 9.30 Effects of area change in subsonic and supersonic flows. (a) Nozzles: V increases;

h, p, and r decrease. (b) Diffusers: V decreases; h, p, and r increase.

M < 1

Subsonic

M > 1

Supersonic

Case 4

(b)

Case 3

M > 1

Supersonic

M < 1

Subsonic

Case 2

(a)

Case 1

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558 Chapter 9

Gas Power Systems

9.13.2

Effects of Back Pressure on Mass Flow Rate

In the present discussion we consider the effect of varying the back pressure on the

rate of mass flow through nozzles. The

back pressure is the pressure in the exhaust

region outside the nozzle. The case of converging nozzles is taken up first and then

converging–diverging nozzles are considered.

CONVERGING NOZZLES. Figure 9.31 shows a converging duct with stagnation

conditions at the inlet, discharging into a region in which the back pressure p

can

be varied. For the series of cases labeled a through e, let us consider how the mass

flow rate m

and nozzle exit pressure p

vary as the back pressure is decreased while

keeping the inlet conditions fixed.

When p

5 p

, there is no flow, so m

. This corresponds to case a of Fig. 9.31.

If the back pressure p

is decreased, as in cases b and c, there will be flow through the

nozzle. As long as the flow is subsonic at the exit, information about changing condi-

tions in the exhaust region can be transmitted upstream. Decreases in back pressure

thus result in greater mass flow rates and new pressure variations within the nozzle. In

each instance, the velocity is subsonic throughout the nozzle and the exit pressure

equals the back pressure. The exit Mach number increases as p

decreases, however,

and eventually a Mach number of unity will be attained at the nozzle exit. The corre-

sponding pressure is denoted by p*, called the critical pressure. This case is represented

by d on Fig. 9.31.

Recalling that the Mach number cannot increase beyond unity in a converging

section, let us consider next what happens when the back pressure is reduced further

to a value less than p*, such as represented by case e. Since the velocity at the exit

equals the velocity of sound, information about changing conditions in the exhaust

region no longer can be transmitted upstream past the exit plane. Accordingly, reduc-

tions in p

below p* have no effect on flow conditions in the nozzle. Neither the

pressure variation within the nozzle nor the mass flow rate is affected. Under these

back pressure

Fig. 9.31 Effect of back

pressure on the operation

of a converging nozzle.

, T

V ≈ 0

1.0

Exhaust region

Valve to adjust

back pressure

)

1.0

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conditions, the nozzle is said to be choked. When a nozzle is choked, the mass flow

rate is the maximum possible for the given stagnation conditions. For p

less than p*,

the flow expands outside the nozzle to match the lower back pressure, as shown by

case e of Fig. 9.31. The pressure variation outside the nozzle cannot be predicted using

the one-dimensional flow model.

CONVERGING–DIVERGING NOZZLES. Figure 9.32 illustrates the effects of

varying back pressure on a converging–diverging nozzle. The series of cases labeled

a through j is considered next.

c Let us first discuss the cases designated a, b, c, and d. Case a corresponds to

5 p

for which there is no flow. When the back pressure is slightly less than

(case b), there is some flow, and the flow is subsonic throughout the

nozzle. In accordance with the discussion of Fig. 9.30, the greatest velocity and low-

est pressure occur at the throat, and the diverging portion acts as a diffuser in which

pressure increases and velocity decreases in the direction of flow. If the back pres-

sure is reduced further, corresponding to case c, the mass flow rate and velocity at

the throat are greater than before. Still, the flow remains subsonic throughout and

qualitatively the same as case b. As the back pressure is reduced, the Mach number

at the throat increases, and eventually a Mach number of unity is attained there

(case d). As before, the greatest velocity and lowest pressure occur at the throat, and

the diverging portion remains a subsonic diffuser. However, because the throat

velocity is sonic, the nozzle is now choked: The maximum mass flow rate has been

attained for the given stagnation conditions. Further reductions in back pressure

cannot result in an increase in the mass flow rate.

c When the back pressure is reduced below that corresponding to case d, the flow

through the converging portion and at the throat remains unchanged. Conditions

within the diverging portion can be altered, however, as illustrated by cases e, f, and

g. In case e, the fluid passing the throat continues to expand and becomes supersonic

in the diverging portion just downstream of the throat; but at a certain location an

abrupt change in properties occurs. This is called a normal shock. Across the shock,

there is a rapid and irreversible increase in pressure, accompanied by a rapid decrease

from supersonic to subsonic flow. Downstream of the shock, the diverging duct acts

as a subsonic diffuser in which the fluid continues to decelerate and the pressure

choked flow:

converging nozzle

choked flow:

converging–diverging

nozzle

normal shock

9.13 Analyzing One-Dimensional Steady Flow in Nozzles and Diffusers 559

Fig. 9.32

Effect of back pressure on the operation of a converging–diverging nozzle.

, T

V ≈ 0

M = 1

Exhaust region

Throat

Valve to adjust

back pressure

Normal

shock

Normal

shock

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560 Chapter 9

Gas Power Systems

increases to match the back pressure imposed at the exit. If the back pressure is

reduced further (case f), the location of the shock moves downstream, but the flow

remains qualitatively the same as in case e. With further reductions in back pressure,

the shock location moves farther downstream of the throat until it stands at the exit

(case g). In this case, the flow throughout the nozzle is isentropic, with subsonic flow

in the converging portion, M 5 1 at the throat, and supersonic flow in the diverging

portion. Since the fluid leaving the nozzle passes through a shock, it is subsonic just

downstream of the exit plane.

c Finally, let us consider cases h, i, and j where the back pressure is less than that cor-

responding to case g. In each of these cases, the flow through the nozzle is not affected.

The adjustment to changing back pressure occurs outside the nozzle. In case h, the

pressure decreases continuously as the fluid expands isentropically through the nozzle

and then increases to the back pressure outside the nozzle. The compression that

occurs outside the nozzle involves oblique shock waves. In case i, the fluid expands

isentropically to the back pressure and no shocks occur within or outside the nozzle.

In case j, the fluid expands isentropically through the nozzle and then expands outside

the nozzle to the back pressure through oblique expansion waves. Once M 5 1 is

achieved at the throat, the mass flow rate is fixed at the maximum value for the given

stagnation conditions, so the mass flow rate is the same for back pressures correspond-

ing to cases d through j. The pressure variations outside the nozzle involving oblique

waves cannot be predicted using the one-dimensional flow model.

9.13.3

Flow Across a Normal Shock

We have seen that under certain conditions a rapid and abrupt change of state called

a shock takes place in the diverging portion of a supersonic nozzle. In a normal shock,

this change of state occurs across a plane normal to the direction of flow. The object

of the present discussion is to develop means for determining the change of state

across a normal shock.

MODELING NORMAL SHOCKS. A control volume enclosing a normal shock

is shown in Fig. 9.33. The control volume is assumed to be at steady state with W

5 0,

5 0 and negligible effects of potential energy. The thickness of the shock is very

small (on the order of 10

cm). Thus, there is no significant change in flow area

across the shock, even though it may occur in a diverging passage, and the forces

acting at the wall can be neglected relative to the pressure forces acting at the

upstream and downstream locations denoted by x and y, respectively.

The upstream and downstream states are related by the following equations:

Mass:

5 r

(9.46)

Energy:

5 h

(9.47a)

5 h

(9.47b)

Momentum:

2 p

5 r

2 r

(9.48)

Entropy:

2 s

5 s

(9.49)

Fig. 9.33 Control volume

enclosing a normal shock.

, M

, T

, p

, s

Normal shock

, M

, T

, p

, s

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When combined with property relations for the particular fluid under consideration,

Eqs. 9.46, 9.47, and 9.48 allow the downstream conditions to be determined for spec-

ified upstream conditions. Equation 9.49, which corresponds to Eq. 6.39, leads to the

important conclusion that the downstream state must have greater specific entropy

than the upstream state, or s

. s

FANNO AND RAYLEIGH LINES. The mass and energy equations, Eqs. 9.46 and

9.47, can be combined with property relations for the particular fluid to give an equa-

tion that when plotted on an h–s diagram is called a Fanno line. Similarly, the mass

and momentum equations, Eqs. 9.46 and 9.48, can be combined to give an equation

that when plotted on an h–s diagram is called a Rayleigh line. Fanno and Rayleigh

lines are sketched on h–s coordinates in Fig. 9.34. It can be shown that the point of

maximum entropy on each line, points a and b, corresponds to M 5 1. It also can be

shown that the upper and lower branches of each line correspond, respectively, to

subsonic and supersonic velocities.

The downstream state y must satisfy the mass, energy, and momentum equations

simultaneously, so state y is fixed by the intersection of the Fanno and Rayleigh lines

passing through state x. Since s

. s

, it can be concluded that the flow across the

shock can only pass from x to y. Accordingly, the velocity changes from supersonic

before the shock (M

. 1) to subsonic after the shock (M

, 1). This conclusion is

consistent with the discussion of cases e, f, and g in Fig. 9.32. A significant increase

in pressure across the shock accompanies the decrease in velocity. Figure 9.34 also

locates the stagnation states corresponding to the states upstream and downstream

of the shock. The stagnation enthalpy does not change across the shock, but there is

a marked decrease in stagnation pressure associated with the irreversible process

occurring in the normal shock region.

Fanno line

Rayleigh line

9.14 Flow in Nozzles and Diffusers of Ideal

Gases with Constant Specific Heats

The discussion of flow in nozzles and diffusers presented in Sec. 9.13 requires no

assumption regarding the equation of state, and therefore the results obtained hold

generally. Attention is now restricted to ideal gases with constant specific heats. This

case is appropriate for many practical problems involving flow through nozzles and

diffusers. The assumption of constant specific heats also allows the derivation of rel-

atively simple closed-form equations.

9.14 Flow in Nozzles and Diffusers of Ideal Gases with Constant Specific Heats 561

Fig. 9.34

Intersection of Fanno and

Rayleigh lines as a solution to the normal

shock equations.

M = 1 at a and b

M < 1 on upper branches

M > 1 on lower branches

Rayleigh line

Fanno line

= h

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562 Chapter 9 Gas Power Systems

9.14.1

Isentropic Flow Functions

Let us begin by developing equations relating a state in a compressible flow to the

corresponding stagnation state. For the case of an ideal gas with constant c

, Eq. 9.39

becomes

5 T 1

where T

is the stagnation temperature. Using Eq. 3.47a, c

5 kR/(k 2 1), together

with Eqs. 9.37 and 9.38, the relation between the temperature T and the Mach number

M of the flowing gas and the corresponding stagnation temperature T

5 1 1

k 2 1

(9.50)

With Eq. 6.43, a relationship between the temperature T and pressure p of the

flowing gas and the corresponding stagnation temperature T

and the stagnation

pressure p

k21

Introducing Eq. 9.50 into this expression gives

1 1

k 2 1

k21

(9.51)

Although sonic conditions may not actually be attained in a particular flow, it is

convenient to have an expression relating the area A at a given section to the area

A* that would be required for sonic flow (M 5 1) at the same mass flow rate and

stagnation state. These areas are related through

rAV 5 r*A*V*

where r* and V* are the density and velocity, respectively, when M 5 1. Introducing the

ideal gas equation of state, together with Eqs. 9.37 and 9.38, and solving for A/A*

where T* and p* are the temperature and pressure, respectively, when M 5 1. Then

with Eqs. 9.50 and 9.51

k 1 1

1 1

k 2 1

k11

k21

(9.52)

The variation of A/A* with M is given in Fig. 9.35 for k 5 1.4. The figure

shows that a unique value of A/A* corresponds to any choice of M.

However, for a given value of A/A* other than unity, there are two pos-

sible values for the Mach number, one subsonic and one supersonic. This

is consistent with the discussion of Fig. 9.30, where it was found that a

converging–diverging passage with a section of minimum area is required

to accelerate a flow from subsonic to supersonic velocity.

Equations 9.50, 9.51, and 9.52 allow the ratios T/T

, p/p

, and A/A* to

be computed and tabulated with the Mach number as the single indepen-

dent variable for a specified value of k. Table 9.2 provides a tabulation

of this kind for k 5 1.4. Such a table facilitates the analysis of flow through

nozzles and diffusers. Equations 9.50, 9.51, and 9.52 also can be readily

evaluated using calculators and computer software such as Interactive

Thermodynamics: IT.

Fig. 9.35 Variation of A/A* with Mach

number in isentropic flow for k 5 1.4.

0 0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

2.0

2.5

3.0

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