Polytropic Process in Gas In Thermodynamics

 An equation of the form pv^n = constant, where n is the constant can be used approximately to describe many processes which occur in practise. Such a process is called polytropic process. It is not adiabatic, but it can be reversible. It may be noted that y(gamma) is the property of the gas, whereas n is not. The value of n depends upon the process. It is possible to find the value of n which more or less fits the experimental results. For two states on the process, 

p1v1 ^n = p2v2^n 

or 

(v2/v1 ) ^ n  = p1/p2 

 n = (log p1 - log p2) / (log v2 - log v1) 

for known values of  p1,p2,v1 and v2, n can be estimated from the above relation. 

'Two other relations of a polytropic process, 

(T2 / T1 ) = ( v1 - v2 ) ^ (n-1) 

(T2 / T1)  = (p2 -p1) ^ (n - (1/n))

Entropy Change in a Polytropic Process 

In a reversible adiabatic process, the entropy remains constant. But in a polytropic process, the entropy changes. 

s2-s1 = Cv ln (T2 / T1)  + R ln (v2 / v1) 

         = (R /  (y -1)) ln ( T2 /  T1 )  + (R / (n -1)) ln ( T1 / T2 ) 

         =  ((n-y) / ( (y-1) (n-1)) R  ln  (T2 / T1) 

Relations in terms of pressure and specific volume can be similarly derived. These are ,

s2 - s1 = ((n -y) / n(y-1)) R ln (p2/p1) 

s2 - s1 = ((n-y) / (y-1)) R ln (v2 / v1) 

It can be noted that when n = y, the entropy change becomes zero. If p2 > p1, for n <_ y,  the entropy of the gas decreases, and for n > y , the entropy of the gas decreases, and for n > y, the entropy of the gas increases. The increase of entropy may result from reversible heat transfer to the system from the surroundings, Entropy decrease is also possible if  the gas is cooled. 

Heat and Work in Polytropic processs,

Q-W = u2 -u1 

for polytropic process, 

Q - Wx - delta ( (V*2 / 2) + gz) = (y (p1v1) / ( y-1 ) ) ( (v1-v2)^ (n-1) - 1)

Integral Property relations in polytropic process 

Qr = Cn delta T ,

Qr = Cn ( T2 - T1 )