664 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th13-4.pm5

In case of a given turbine the minimum temperature T 1 and the maximum temperature T 3

are prescribed, T 1 being the temperature of the atmosphere and T 3 the maximum temperature

which the metals of turbine would withstand. Consider the specific heat at constant pressure cp to

be constant. Then,

Since,

T

T

r T

p T

3

4

1

2

1

==

−

()

γ

γ

Using the constant ‘z’ =

γ

γ

− 1

,

we have, work output/cycle

W = K T

r

Tr

p

z p

z

311

F − (^11)
H
GG
I
K
JJ−−
L
N
M
M
O
Q
P
P
()
Differentiating with respect to rp
dW
dr
KT z
rz
Tzr
p p
p
=× z

• −
  L
  N
  M
  M
  O
  Q
  P
  P
  −
  31
  1
  () 1
  () = 0 for a maximum
  ∴
  zT
  r
  Tz r
  p
  z p
  3 z
  1 1
  1
  ()
  ()()


• 
  

  = −
  ∴ r
  T
  p T
  2 z 3
  1

  
  

  ∴ rTTp=(/) 3112 /z i.e., rp = (/)TT 3121 ()
  γ
  γ− ...(13.17)
  Thus, the pressure ratio for maximum work is a function of the limiting temperature
  ratio.

  
  

13.10.3.Work ratio

Work ratio is defined as the ratio of net work output to the work done by the turbine.

∴ Work ratio =

WW

W

TC

T

−

where, obtained from this turbine,

and supplied to the compressor.

W

W

T

C

=

=

L

NM

O

QP

Work

Work

=

mc T T mc T T

mc T T

p p

p

()()

()

34 21

34

−− −

−

= 1 –

TT

TT

21

34

−

−

= 1 – T

T

r

r

T

T

p r

p

(^1) p
3
1
1
1
3
1
1
1 1
1
()
()
()
γ
γ
γ
γ
γ
γ
−
−
−
−
−
L
N
M M M M M M M
O
Q
P P P P P P P
=− ...(13.18)