- #1
Chetty
- 32
- 3
- Homework Statement
- In the text "Elements of Gasdynamics" by Liepmann & Roshko, the first chapter exercise asks for confirmation of the following relation.
- Relevant Equations
- Equation is below
##(\frac {∂p} {∂ρ})_s=ϒ(\frac {∂p} {∂ρ})_T##
The variables are p for pressure, ρ for specific mass density and γ is ratio of specific heats. I am able to show that the relation is valid for a perfect gas but cannot show its validity in general.
The closest I get is ##dp=(\frac {∂p} {∂ρ})_s dρ+(\frac {∂p} {∂s})_ρ ds=(\frac{∂p} {∂ρ})_T dρ+(\frac {∂p} {∂T})_ρ dT##
For the perfect gas, the first part with constant 's' implies an isentropic relation for a partial derivative of ##\frac p {ρ^γ} =const## and the second part is the partial derivative of the perfect gas expression ##p=ρRT##.
The variables are p for pressure, ρ for specific mass density and γ is ratio of specific heats. I am able to show that the relation is valid for a perfect gas but cannot show its validity in general.
The closest I get is ##dp=(\frac {∂p} {∂ρ})_s dρ+(\frac {∂p} {∂s})_ρ ds=(\frac{∂p} {∂ρ})_T dρ+(\frac {∂p} {∂T})_ρ dT##
For the perfect gas, the first part with constant 's' implies an isentropic relation for a partial derivative of ##\frac p {ρ^γ} =const## and the second part is the partial derivative of the perfect gas expression ##p=ρRT##.