630 ENGINEERING THERMODYNAMICS

dharm
\M-therm\Th13-3.pm5

s

T

p = Const.

3

4

2

1

v = Const.

v

p

4

1

2 3

Adiabatic

Adiabatic

Fig. 13.15

volume increases from V 2 to V 3 and temperature T 2 to T 3 , corresponding to point 3. This point (3)
is called the point of cut-off. The air then expands adiabatically to the conditions p 4 , V 4 and T 4
respectively corresponding to point 4. Finally, the air rejects the heat to the cold body at constant
volume till the point 1 where it returns to its original state.
Consider 1 kg of air.
Heat supplied at constant pressure = cp(T 3 – T 2 )
Heat rejected at constant volume = cv(T 4 – T 1 )
Work done = Heat supplied – heat rejected
= cp(T 3 – T 2 ) – cv(T 4 – T 1 )
∴ηdiesel = Work done
Heat supplied

=

cT T cT T

cT T

p v

p

()()

()

32 41

32

−− −

−

= 1 –

()

()

TT

TT

41

32

−

−γ ...(i)

Q

c

c

p

v

=

L

N

M

O

Q

γP

Let compression ratio, r = v

v

1

2

, and cut-off ratio, ρ = v

v

3

2

i.e., Volume at cut-off

Clearance volume

Now, during adiabatic compression 1-2,

T

T

2

1

= v

v

(^1) r
2
1
F 1
HG
I
KJ

−
−
γ
()γ or T
2 = T 1. ()r
γ−^1
During constant pressure process 2-3,
T
T
3
2

v
v
3
2
= ρ or T 3 = ρ. T 2 = ρ. T 1. ()rγ−^1
During adiabatic expansion 3-4
T
T
3
4

v
v
4
3
F^1
HG
I
KJ
−γ