download the script: Isentropic Relations

An isentropic process is a process during which the entropy of a system remains constant:

Δs=s₁-s₂=0

This can also be described in the T-s diagram:

According to the definition of entropy which is expressed as:

for an isentropic process (dS=0), we obtain:

δQ_rev=0

We can now conclude from the above equation that no reversible heat transfer with surrounding occurs during an isentropic process. So if a process is carried out in an isentropic manner, the following two conditions must be satisfied:

1. reversible
2. adiabatic

Since a reversible adiabatic process is necessary for an isentropic process, let’s see what kind of relationship between properties of state will be obtained from the 1.law of thermodynamics:

du=δq+δw

where:

1. adiabatic →δq=0
2. reversible: →δw=-p·dν

The 1.law of thermodynamics for an isentropic process is now:

du=-p·dν    (1)

For ideal gas, we have additionally:

du=c_v·dT    (2)

d(p·ν)=d(R·T) → p·dν+ν·dp=R·dT   (3)

c_p=c_ν+R     (4)

We combine above three equations (1), (2), (3) and (4) and simplify it, then we obtain a differential equation:

c_p·p·dν + ν·c_ν·dp=0

We define here the ratio of heat capacity as isentropic exponent κ (kappa):

κ=c_p/c_ν

So after integrating the above differential equation and by using the ideal gas law, we obtain the following important isentropic relations for ideal gas:

p·ν^κ=constant

p^1-κ·T^κ=constant

T·ν^κ-1=constant