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Chapter 9

Gas Power Systems

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Learning Outcomes

Performair-standard analysesof gas

turbine power plants based on the Brayton

cycle and its modifications, including:

sketchingT-sdiagrams and evaluating

property data at principal states.

applyingmass, energy, entropy, and exergy

balances.determiningnet power output, thermal

efficiency, back work ratio, and the effects of

compressor pressure ratio.

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Learning Outcomes

For subsonicand supersonic flowsthrough

nozzlesand diffusers:

demonstrateunderstanding of the effects of

area change, the effects of back pressure on

mass flow rate, and the occurrence of choking

and normal shocks.

analyzethe flow of ideal gases with constantspecific heats.

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Considering Compressible FlowIn many applications of engineering interest, gases

move at relatively high speedsand exhibit significantchanges in specific volume(density). They include

Flows through the nozzles and diffusers of jet

engines.

Flows through wind tunnels, shock tubes, and steamejectors.

These flows are known as com press ib le flows.

Next, we consider some important preliminaries, includingthe

momentum equationfor steady one-dimensional flow

velocity of soundand Mach number

stagnation state

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net rate at which momentum is

transferred intothe control

volume accompanying mass flow

Momentum Equation for

Steady One-Dimensional Flow

In words, Newtons second lawfor a control volume is:

time rate of change

of momentum contained

within the control volume

resultant force

acting onthe

control volume

= +

(= 0 at steady state)

The form of the momentum

equation for a one-inlet, one-exit

control volume at steady stateis:

(Eq. 9.31)

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Velocity of Sound

Analyses using mass and momentum equations

supported by experimental data reveal that therelation between pressure and specific volume

across a sound wave is nearly isentropic, and that its

velocity ccalled the velocity of soundis given by

(Eq. 9.36b)

A sound waveis a small pressure disturbance that

propagates through a gas, liquid, or solid at a velocitycthat depends on the properties of the medium.

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Velocity of Sound

The special case of an ideal gas with constant

specific heatsis used extensively in Chapter 9. Forthis case, the relationship between pressure and

specific volume for fixed entropy is pvk= constant

where kis the specific heat ratio. Using this

relationship, Eq. 9.36bbecomes

(Eq. 9.37)

The velocity of sound is an intensive propertywhose value depends on the state of the medium

through which sound propagates. While sound

propagates nearly isentropically, the medium itself

may be undergoing any process.

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Velocity of Sound

Example: Calculate the velocity of sound in air at

300 Kand 650 K.

From Table A-20at 300 K, k= 1.4. Thus from Eq.

9.37

N1

m/skg1K)300(

Kkg

mN

97.28

83144.1

2

c = 347 m/s(1138 ft/s)

From Table A-20at 650 K, k= 1.37. Thus from Eq.

9.37

N1

m/skg1K)650(

Kkg

mN

97.28

831437.1

2

c = 506 m/s(1660 ft/s)

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Mach Number

In subsequent discussions, the ratio ofvelocity Vat

a state in a flowing fluid to the value of sonic velocitycat the same state plays an important role. This ratio is

called the Mach number, M.

(Eq. 9.38)

Several important

termsassociated withMach number are

shown in the table.

Mac h Number Term

M < 1 Subsonic

M = 1 S onic

M > 1 S upers onic

M >> 1 Hypers onic

M near 1 Transonic

Mac h Number Term

M < 1 Subsonic

M = 1 S onic

M > 1 S upers onic

M >> 1 Hypers onic

M near 1 Transonic

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Actual compressible

flow process

h

h

s

p

V

ho

po

Vo

= 0

Actual compressible

flow process

h

h

s

p

V

ho

po

Vo

= 0

Stagnation State Properties

The h-sdiagram shows a

compressible flow process.Associated with each state of the

flow is a reference stateknown

as the stagnation state.

Thestagnat ion stateis the

state the flowing gas would

attain if it were decelerated to

zero velocity isentropically.

By reducing an energy

balance for a hypotheticaldiffuser thatin principle only

decelerates the gas, we get

(Eq. 9.39)

Stagnation state:ho= stagnation enthalpy,

po = stagnation pressure,

To= stagnation temperature

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One-Dimensional Steady Flow

in Nozzles and Diffusers

One of these equations relates velocity andpressure changes in the direction of flow:

(Eq. 9.44)

When velocity increases: dV > 0, then pressure

decreases: dp< 0.

When velocity decreases: dV < 0, then pressureincreases: dp> 0.

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Another equation relates velocity and area changes inthe direction of flow:

(Eq. 9.45)

There are four cases, each of which depends on the

local Mach numberM:

Supersonic nozzle: dV > 0and M> 1 dA > 0.

The ductdiverges.

Supersonic Nozzle

Velocity increases

Area increases

Pressure decreases

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(Eq. 9.45)


local Mach numberM:

Supersonic diffuser: dV < 0and M> 1 dA < 0.

The ductconverges.

Supersonic Diffuser

Velocity decreases

Area decreases

Pressure increases

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(Eq. 9.45)


local Mach numberM:

Subsonic diffuser: dV < 0and M< 1 dA > 0.

The ductdiverges.

Subsonic Diffuser

Velocity decreases

Area increases

Pressure increases

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Exploring the Effects of Area Change

in Subsonic and Supersonic Flows

Consider a converging section with subson icflow connectedto a diverging section to form a converging-diverging duct.

If the Mach number is unity at the end of the converging

section, and the flow continues to accelerate, the flow will

become supersonic in the diverging section.

M= 1at the throat.

This is called a converging-diverging nozzle.

Velocity increases, pressure decreases

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in Subsonic and Supersonic FlowsConsider a converging section with superson icflow

connected to a diverging section to form a converging-divergingduct.

If the Mach number is unity at the end of the converging

section, and the flow continues to decelerate, the flow will

become subsonic in the diverging section.

M= 1at the throat.

This is called a converging-diverging diffuser.

Velocity decreases, pressure increases

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in Subsonic and Supersonic Flows

These findings indicate that a Mach number ofunity can occur only at the location in a nozzle or

diffuser where the flow area is a minimum. This

location of minimum areais called the throat.

As shown in the following discussion, a Machnumber of unity does not necessarily occur at the

location where flow area is a minimum.

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Effects of Back Pressure on

Mass Flow Rate Converging Nozzle

Consider a convergingduct with stagnation conditionsat the inlet, discharging into a region outside the nozzle

where the pressure pBcalled the back pressurecan

be varied.

We explore how the mass flow rate through the nozzle

and the pressure at the nozzle exit vary as the back

pressure is decreased while keeping the nozzle inlet

conditions fixed.

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Case a. WhenpB=pE=po, there is no

flow: .0m

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Case a. WhenpB=pE=po, there is noflow: .0m

Cases b and c. AspBis decreased, the

mass flow rate increases, flow is

subsonic throughout. The pressure at the

nozzle exit equals the back pressure.Case d. Eventually, aspBis decreased, a

Mach number of unityis attained at the

nozzle exit. The corresponding exit

pressure is called the critical pressure,denoted byp*.The mass flow rate is the

maximum possible and the nozzle is saidto be choked.

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Case a. WhenpB=pE=po, there is noflow: .0m

Cases b and c. AspBis decreased, the

mass flow rate increases, flow is

subsonic throughout. The pressure at the

nozzle exit equals the back pressure.Case d. Eventually, aspBis decreased, a

Mach number of unityis attained at the

nozzle exit. The corresponding exit

pressure is called the critical pressure,denoted byp*.The mass flow rate is the

maximum possible and the nozzle is saidto be choked.

Case e: Further reductions inpBbelowp*

have no effect on the flow conditions

within the nozzle. The mass flow rate

does not change.

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Effects of Back Pressure

Converging-Diverging Nozzle

Consider a converging-divergingduct with stagnationconditions at the inlet, discharging into a region outside

the nozzle where the pressure pBcalled the back

pressurecan be varied.

We explore how the mass flow rate through the nozzle

and the pressure at the nozzle exit vary as the back

pressure is decreased while keeping the nozzle inlet

conditions fixed.

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Mass flow rate increasesas pBis reduced in steps

from poto pb, pc, and pd.

Flow accelerates in the

converging section and then

decelerates subsonically inthe diverging section.

Eventually, when pB= pd,

the pressure at the throat

reaches p*, corresponding to

a Mach number of unitythere. At this condition the

flow is choked, and mass

flow rate cannot increase

with further decrease in back

pressure.

Cases a, b, c, and d

Subsonic

Nozzle Subsonic Diffuser

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Back pressure is less than

that of Case g.

Flow in thenozzle is not

affected; adjustment occurs

outsidethe nozzle.

Case h: Pressure increase

outside the nozzle involvesan oblique compression

shock.

Cases h, i, and j

Subsonic

Nozzle Supersonic Nozzle

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that of Case g.



outsidethe nozzle.



shock.

Case i: Unique back

pressure for which no shocks

occur within or outsidenozzle.

Cases h, i, and j

Subsonic


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that of Case g.



outsidethe nozzle.



shock.

Case i: Unique back

pressure for which no shocks

occur within or outsidenozzle.

Case j: The gas expands

outside the nozzle through

an oblique expansion wave.

Cases h, i, and j

Subsonic


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Modeling Normal Shocks

Depending on the back pressure,

a normal shock can stand in the

diverging section of a supersonic

nozzle, as shown in the figure

wherethe subscripts xand ydenote, respectively, thestates just upstream and downstream of the shock.

Since the thickness of the shock is small, there is

no appreciable change in flow area across the shock

and the only significant forces acting on the control

volume in the direction of flow are the pressure

forces. Additionally, for the control volume .0cv Q

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At steady state, mass, energy,

momentum, and entropy balancesreduce to give the following

relations between the upstream, x,

and downstream, y, locations.

Mass: (Eq. 9.46)

Energy: (Eq. 9.47a)

Momentum: (Eq. 9.48)

Entropy: (Eq. 9.49)

Since the shock is an irreversibility, the entropy

balance requires that the downstream specific entropy

syis greater than the upstream specific entropy sx.

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The massand energyequationstogether with

property data for the particular gas combine to give a

curve on an h-sdiagram called a Fanno line.

Mass:

Energy:

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The massand momentum equationstogether with

property data for the particular gas combine to give a

curve on an h-sdiagram called a Rayleigh line.

Mass:

Momentum:

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I t i Fl F ti f Id l G

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Isentropic Flow Functions for Ideal Gases

with Constant Specific Heats

Next, for the case of ideal gases with constant specificheats, Eqs. 9.50 and9.51are introduced.

They relate

temperatureTand

pressurepat a state of acompressible flow to the

corresponding

stagnation temperature

Toand stagnationpressurepoin terms of

the specific heat ratio k

and Mach number M:

Compressible

flow

T

T

s

p

M

To

po

Mo

= 0

Stagnation state

Compressible

flow

T

T

s

p

M

To

po

Mo

= 0

Compressible

flow

T

T

s

p

M

To

po

Mo

= 0

Stagnation state


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(Eq. 9.51)

(Eq. 9.50)

These equations are developed in Sec. 9.14.1using massand energy balances together with isentropic property

relations.

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Using Eqs. 9.50-9.52, this tabulation

for k= 1.4Table

9.2can be

developed. Use of

Table 9.2for

problem solving is

illustrated in

Example 9.15.

Eqs. 9.50-9.52arereadily programmed

for use with hand-

held calculators.

N l Sh k F ti f Id l G

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Normal Shock Functions for Ideal Gases


Consider a normal shockstanding in a duct as shown in

the figure.

The ratio of temperature acrossthe shockis

The ratio of pressure across the shockis

(Eq. 9.54)

(Eq. 9.53)


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Consider a normal shockstanding in a duct as shown in

the figure.

The Mach numbers across the

shockare related by

The ratio of stagnation pressures across the shockis

(Eq. 9.55)

(Eq. 9.56)


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Using Eqs. 9.53-9.56, this tabulation

for k= 1.4Table

9.3can be

developed. Use of

Table 9.3for

problem solving is

illustrated in

Example 9.15.

Eqs. 9.53-9.56are readily

programmed for

use with hand-held

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ch09 pt3