Chalmers University of Technology

Lecture 4

• Ideal cycles III– Reheat cycle– Intercool cycle

• The WR21 engine

• Polytropic efficiencies

• Exercise– Problem 2.1, 2.3 and 2.9

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Reheat cycle/Reheat with heat exchanger

• Split expansion into a high pressure and a low pressure step and reheat in between

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Selection of pressure ratio –

reheat cycle

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Theory 4.1 - Selection of optimal pressure ratio – reheat cycle

6543 TTcTTcW ppturbines

Introduce auxiliary variable β according to:

get tocombine

r ][

,

1

1

56

1

6

5

6

354

6

5

4

3

34

1

4

3

4

3

r

TTrP

P

P

PPP

P

P

P

P

TT

P

P

T

T

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Theory 4.1 - Selection of optimal pressure ratio – reheat cycle

cwith

accordanceinr

rTc

W

torespectwithatedifferenti

r

TT

TT

c

W

p

t

p

t

C.R.S.

c introduce 0

11

:

1

123

TT

15

53

3

53

Insert result into power formula:

1

221

1

2

111

c

c

tt

T

T

c

ctt

c

tt

Tc

W

Tc

W

Tc

W

c

p

c

p

t

p

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Efficiency for reheat cycle (at pressure division for max. power output)

c

tct

cc

tt

c

ttct

cc

tt

c

TT

TTT

T

cc

tt

c

TTTT

Tc

TTcTTc

TcW

TcQ

TcW

p

pp

p

p

p

2

12

212

2

11

12

2

] , [

33

123

1

3435

1

4523

1

1

1

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Cycle changes due to reheat

You introduce an “additional cycle” operating at lower pressure ratio. We have already derived what we want to know!!! Decreasing pressure ratio in simple cycle => efficiency decreases.

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Reheat/reheat with heat exchange compared to single cycle

• Simple reheat– Power output increases– Decrease in efficiency (added cycle is

worse than underlying cycle, since simple cycle efficiency decreases with pressure ratio)

• Reheat with heat exchange

– Power output increases– Increase in efficiency. Heat is added

at a higher average temperature and removed at a lower temperature than in simple cycle. See figure to the right.

Simple cycle

Reheat with heatexchange

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Intercooling• Bulky and requires

large amounts of cooling water– Compactness and self-

containedness of gas turbine is lost

• What about efficiency and power output of cycle ?....– Try to draw a T-S

diagram and make some arguments. Check with CRS.

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The WR 21• ICR cycle - Intercooled Recuperated

Cycle• Improved part load performance =>

30% reduction in fuel burn for a typical operating profile

• 25 MW output • Fits in footprint of current naval

engines of similar power. • LM2500 ηth=37. ICR ηth=43.• Starts in two minutes instead of 4

hours for comparable steam engine.• Greater power for given space when

compared with steam/diesel.

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The WR 21

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Polytropic efficiencies - motivation• If we study multistage designs the isentropic

efficiency for high pressure compressors tend to be lower than for low pressure compressors. Why?

sss

s TT

T1

• Assume ηs (stage efficiency) constant, the overall temperature rise ΔT is obtained by:

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Polytropic efficiency - motivation

But: ss

s

sc T

TTT

T

T

Preheat effect: as you go through the stages you move to the right in the T-s diagram. Isobars diverge in that direction!

Thus, the total efficiency is always

less than the stage efficiency.

sTT

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Polytropic efficiency – “preheat independence”

• Define the polytropic efficiency (differential stage efficiency) as:

P

dP

T

dT

T

Tdc

1,

(1) , dT

Tdc

11

1

2

1

2,

c

P

P

T

T

We have (second revision question – lecture 1 – before integrating):

(1) + (2) produces:

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Polytropic efficiency – “preheat independence”

1

4

3

4

3

,

t

P

P

T

T

1

4

3

4

3

,

t

P

P

T

TSimilarly for a turbine:

Polytropic efficiencies are useful for preliminary design, when many compressor concepts with different pressure ratios may be evaluated for a given application.

90.0

90.0

,

,

t

c

First guess for preliminary design work

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Recommendation to get started with the course

• Work through Example 2.1 and 2.2 at home (page 74-78).– Derive the optimal pressure ratio (for maximum power) for the

simple cycle gas turbine (Theory 2.1)– Derive the efficiency for the heat-exchange cycle (Theory 3.1)

• Read ”very important” sections as stated in course PM (so far section 2.1, 2.2 and 2.3)

• Start with Design Task 1 !!!

If you have time:– Read all ”important” sections as well. – Work through example 2.3 and attempt to solve problem 2.5.

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Learning goals• Know how to show (by arguments or T-S

diagrams) how the efficiency of the reheat cycle with and without heat exchange changes in comparison with the simple cycle (ideal case)

• Be able to derive the optimal pressure division in ideal reheat cycles

• Be familiar with the polytropic efficiency concept and state reasonable loss levels for turbine and compressors