GAS POWER CYCLES 631

dharm

\M-therm\Th13-3.pm5

=

r

ρ

F γ

HG

I

KJ

− 1

Q v

v

v

v

v

v

v

v

4 r

3

1

3

1

2

2

3

==×=

F

HG

I

ρKJ

∴ T 4 =

T

r

Tr

r

(^311) T
1
1 1
ρ
ρ
ρ
γ ρ
γ
γ
γ
F
HG
I
KJ

F
HG
I
KJ
− =
−
−
.().
By inserting values of T 2 , T 3 and T 4 in eqn. (i), we get
ηdiesel = 1 –
(. )
(. .() .() )
TT
Tr Tr
11
1
1
1
1
ρ
γρ
γ
γγ
−
−−−
= 1 –
()
() ( )
ρ
γρ
γ
γ
−
− −
1
r^11
or ηdiesel = 1 –^11
γ^11
ρ
γ ρ
γ
()r −
−
−
L
N
M
M
O
Q
P
P
...(13.7)
It may be observed that eqn. (13.7) for efficiency of diesel cycle is different from that of the
Otto cycle only in bracketed factor. This factor is always greater than unity, because ρ > 1. Hence
for a given compression ratio, the Otto cycle is more efficient.
The net work for diesel cycle can be expressed in terms of pv as follows :
W = p 2 (v 3 – v 2 ) +
pv 33 pv 44 pv 22 pv 11
11
−
−
−
−
γγ−
= p 2 (ρv 2 – v 2 ) +
pv prv pv prv 32 42 22 12
11
ρ
γγ
−
−
−
−
−
Q v
v
vv v
v
rvrv
vv vrv
3
2
32 1
2
12
41 4 2
=∴ = =∴=
=∴ =
L
N
M
M
M
O
Q
P
P
P
ρρand
But
= p 2 v 2 (ρ – 1) + pv prv pv prv32 42 22 12
11
ρ
γγ
−
−
− −
−

vp 22 11 p pr p pr 3 4 2 1
1
[( )( )ργ ρ ( )]
γ
−−+−−−
−

vp p pr
p
p pr
(^223) p
4
3
2 1
2
11 1
1
()()ργ ρ
γ
−−+ −
F
HG
I
KJ
−−
F
HG
I
KJ
L
N
M
M
O
Q
P
P
−
= pv^22 r r
11 111
1
[(ργ ρρ)( ). ( )
γ
−−+−γγ−−γ
−
−−
Q p
p
v
vr
(^4) r
3
3
4

F
HG
I
KJ
=F
HG
I
KJ

L
N
M
M
O
Q
P
P
−
γ γ
ρ ργγ

pvr 11 1111 r 1 r^1
1
γγργ ρργγ
γ
−−−−+− −−−
−
[( )( ) ( )
Q p
p
v
v
ppr v
v
(^2) rvvr
1
1
2
21 1
2
= 211
F
HG
I
KJ

L
N
M
M
O
Q
P
P
−
γ
or. γand or

pvr 11 1111 r
1
γγγρ ργ
γ
−−−− −
−
[( ) ( )]
()
...(13.8)