TITLE.PM5

116 ENGINEERING THERMODYNAMICS

dharm

M-therm/th4-1.pm5

i.e., Work done, W pv p v

= n

−

−

11 2 2

1

...(4.39)

or W
RT T
= n
−
−

() 12
1 ...(4.40)
Eqn. (4.39) is true for any working substance undergoing a reversible polytropic process. It
follows also that for any polytropic process, we can write

p

p

v

v

n

2

1

1

2

=FHG IKJ ...(4.41)
The following relations can be derived (following the same procedure as was done under
reversible adiabatic process)

T

T

v

v

n

2

1

1

2

1

=FHG IKJ

−

...(4.42)

T

T

p

p

n

2 n

1

2

1

1

=FHG IKJ

−
...(4.43)
Heat transfer during polytropic process (for perfect gas pv = RT) :
Using non-flow energy equation, the heat flow/transfer during the process can be found,
i.e., Q = (u 2 – u 1 ) + W

= cv(T 2 – T 1 ) +

RT T

n

() 12

1

−

−

i.e., Q =

RT T

n

() 12

1

−

− – cv (T^1 – T^2 )

Also cv =

R

()γ− 1

On substituting,

Q = nR− 11 ()TT 12 −−()γR− (T 1 – T 2 )

i.e., Q = R(T 1 – T 2 )

1

1

1

n− − − 1

F

H

I

γ K

=RT T()(^12 ()()−−−+γ−− 11 γn^11 n )=RT T()()()()γ^12 −−−− 11 nγ n

∴ Q

n RT T

= n

−

−

−

−

()

()

()

()

γ

γ 11

12

or Q

= −n W

−

F

HG

I

KJ

γ

γ 1

LQW=RT Tn−−

NM

O

QP

()

()

12
1 ...(4.44)
In a polytropic process, the index n depends only on the heat and work quantities during
the process. The various processes considered earlier are special cases of polytropic process for a
perfect gas. For example,
When n = 0 pv° = constant i.e., p = constant
When n = ∞ pv∞ = constant
or p1/∞ v = constant, i.e., v = constant
When n = 1 pv = constant, i.e., T = constant
[since (pv)/T = constant for a perfect gas]
When n = γ pvγ = constant, i.e., reversible adiabatic