Expansion Process

7/28/2019 Expansion Process

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Unit-2, Chapter-4

Expansion Process


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Expansion process

Work done expressions for a turbine

Ideal and actual expansion process in a turbine is shown in the figure.

01-02 is the ideal expansion (isentropic)on stagnation basis

01-02 is the actual expansion (adiabatic) on stagnation basis

1-2 is the ideal expansion (isentropic)on static basis

1

Wa

2

2

T2

Wisen

ENTROPY

TEMP

ERATURE

T1

01

02

02

T2

Isentropic

1-2 is the actual expansion

(adiabatic) on static basis

In a compressor the air enters from

atmosphere. But in a turbine, the

fluid enters from nozzles or a

previous stage. Hence stagnation

properties should be considered.

The actual shaft work output is

calculated from the difference in

stagnation enthalpies.

0201 hhWa


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Expansion process

If the working fluid is air,

The shaft power output from the turbine is,

The isentropic work done can be defined in two ways:

Based on stagnation states : Valid if the kinetic energy of a stage is not

wasted as in the case of an aircraft gas turbine engine where the

exhaust goes to a propulsion nozzle.

Based on stagnation-static states: Valid if the kinetic energy is lost as in

the case of a single stage gas turbine where exhaust goes to

atmosphere

0201TTcW

pa

aa WmP

(in kW when mass flow rate of air is in kg/s)

'

0201

'

0201 TTcWorhhW pss

'

201

'

201 TTcWorhhW pss


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Expansion process

Efficiency expressions for a turbine

The efficiency of the turbine is defined as the ratio of actual work output to

the ideal work output for the same pressure ratio.

1

01

0201

1

1

02

01

'

02

01

'

0201

0201

1 ro

tt

ro

tt

tt

pT

TT

p

p

p

T

TSince

hh

hh

endsstagnationforoutputworkIdeal

outputworktotalActual


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Expansion process

The actual total power output is,

If the mechanical efficiency is known,

Similarly,

1

01 1 ropttaa pTcmWmP

1

011 ropttmechamechSP pTcmPP

'

201

0201

'

201

0201

TT

TT

hh

hh

outletstaticandinletstagnationforoutputworkIdealoutputworktotalActual

ts

ts


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1

01'1 rptsaa pTcmWmP

Expansion process

For the isentropic process 01-2, with pr = (p01/p2),

The total power output is,

If the mechanical efficiency is known, then

1

01

201

1'

1

2

01

'

2

01

'1 r

ts

r

pT

TT

pp

p

T

TSince

1

01'1 rptsmechamechSP pTcmPP


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Expansion process

Finite stage efficiency

A stage with finite pressure drop is a finite stage. All the work and efficiency

equations derived earlier hold good for the finite stage. Due to largepressure drop in the beginning stages and the thermodynamic effects, more

work will be done in the last stages (LP or Low Pressure stages) of

multistage turbines for the same finite pressure drop per stage.

p1

pB

pA

1

22

ENTROPY

TEMPERAT

URE

p2

X

Y

Z

A

BWS

WS1

WS2

WS3

Wa

O

Reheat effect

Consider a 3-stage turbine workingbetween pressures p1 and p2. The

intermediate pressures are pA and

pB.

Assuming that the pressure ratio

and efficiency are the same for allstages,

Cp

p

p

p

p

p B

B

A

A

2

1


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Expansion process

IfO is the overall efficiency, then the actual work done (process 1-2), Wa isgiven by,

The total actual work done can also be written as the sum of actual work

done in each stage.

where,

Combining, we can write,

sa WW 0

sststageaa WWW ,

321 ssss WWWW

s

sstssts

W

WorWW

00 ,


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Expansion process

As the constant pressure lines diverge towards the right hand side, the

isentropic work per stage increases as the temperature difference increasesfor the same pressure ratio and stage efficiency (e.g., (A-Y) > (X-O) in the T-S

diagram).

Therefore, , and the term is called the Reheat Factor

This implies that 0 > st due to effect of reheating where the gas is

unintentionally heated at the end of each expansion stage but this appears

as losses in subsequent stages.

Infinitesimal Stage Efficiency or Polytropic Efficiency:

A finite turbine stage can be assumed to be made up of infinite number of

small stages.

Each of the small stages has an efficiency p called polytropic or small stage

efficiency.

1

s

s

W

W

s

s

W

W


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Expansion process

Considering a single stage turbine with a stage efficiency st operating

between pressures p1 and p2, and an infinitesimal stage with efficiency pworking between pressures p and (p-dp), and considering working fluid as

perfect gas, we have

'' dT

dT

dh

dh

dropenthalpyIsentropic

dropenthalpyActualp

p2

p

P-dp

p1

1

3

22

3

dT

ENTROPY

TEMPERAT

URE

dT

XT

1

'

1'

11

,

p

dp

T

dT

orp

dpp

T

dTT


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Expansion process

Expanding RHS and dropping higher order terms,

On substituting for dT and integrating between limits 1 and 2, we get

p

dp

T

dT

p

dp

T

dT

1

111

'

'

1

2

1

2

ln1

ln

p

pT

T

p


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Expansion process

The irreversible adiabatic (actual) expansion process can be considered asa polytropic process with index n.

Equating the indices,

n

n

p

p

p

p

T

Tp

1

1

2

1

1

2

1

2

pp

pn

n

1

11


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Expansion process

We also know,

Applying this equation to one stage (p1 to p2) and denoting its efficiency by

s,

pp

p

pT

p

pTTTT

1

1

2

1

1

1

2

1121 1

1

2

1

121

1

2

11

21

1

1

p

pTTT

p

pT

TT

s

s


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Expansion process

Equating the two values so derived for (T1-T2), we get

For multistage expansion, s is replaced by the overall efficiency o and the

stage pressure ratio by the overall pressure ratio pro.

1

1

1

2

1

1

2

1

1

2

1

1

1

2

1

1

1

1

1

1

r

rs

s

p

p

p

p

pp

p

p

pp

p

pp

1

1

1

1

p

ro

roo

p

p

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Expansion Process