download the script: Isentropic Relations

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

Δs=s_{1}-s_{2}=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:

- reversible
- 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:

- adiabatic →δq=0
- 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