Isentropic flow relations

The first law of thermodynamics

$$dU=\delta Q-\delta W$$

 

The second law of thermodynamics

$$\delta S\ge \frac{\delta Q}{T}$$

 

For the second law of thermodynamics, when we assume it is isentropic

$$\delta Q=TdS$$

 

Combine the two equation...

$$dU=\delta Q-\delta W$$

$$dU=TdS-PdV\cdots (1)$$

 

And bring the concept/definition of enthalpy and differential form

$$u=h-Pv$$

$$du=dh-vdP-Pdv\cdots (2)$$

 

Combine the equation (1) and (2)

$$du=Tds-Pdv\cdots (1)$$

$$du=dh-vdP-Pdv\cdots (2)$$

 

$$Tds-Pdv=dh-vdP-Pdv$$

$$Tds=dh-vdP$$

$$dh=Tds+vdP\cdots (3)$$

 

And bring the concept of specific volume$v$

$$m=\rho V$$

$$\frac{m}{m}=\rho \frac{V}{m}=\rho v$$

$$1=\rho v$$

$$v=\frac{1}{\rho }$$

 

Apply the specific volume in the equation (3)

$$dh=Tds+\frac{dP}{\rho }\cdots Gibbs$$

 

Again, bring the concept of specific heat at constant pressure $C_P$

$$dh=C_{P}dT$$

$$dh=Tds+\frac{dP}{\rho }=C_{P}dT\cdots (4)$$

 

And for ideal gas we can apply the equation below

$$P=\rho RT$$

$$\frac{1}{\rho }=\frac{RT}{P}$$

 

Then, the equation (4) can be rearrange into

$$C_P=Tds+\frac{RT}{P}dP$$

 

And for isentropic process, $ds=0$

$$C_P=\frac{RT}{P}dP\cdots (5)$$

 

Now, Integrate the equation (5)

$${\int }_{T_1}^{T_2}\frac{C_P}{R}\frac{dT}{T}={\int }_{P_1}^{P_2}\frac{dP}{P}$$

 

$$\frac{C_P}{R}[\ln T_2-\ln T_1]=\ln P_2-\ln P_1$$

$$=\frac{C_P}{R}\ln \frac{T_2}{T_1}=\ln \frac{P_2}{P_1}$$

$$=\ln {\left(\frac{T_2}{T_1}\right)}^{C_P/R}=\ln \left(\frac{P_2}{P_1}\right)\cdots (6)$$

 

And we can rearrange the exponent $C_P/R$

$$\frac{C_P}{R}=\frac{C_P}{C_P-C_v}=\frac{C_P/C_v}{C_P/C_v-C_v/C_v}=\frac{\gamma }{\gamma -1}$$

 

Applying the rearranged exponent in equation (6)

$$\frac{P_2}{P_1}={\left(\frac{T_2}{T_1}\right)}^{\gamma /\gamma -1}\cdots (7)$$

 

We can also derive the eqaution between pressure and density relation.

$$P=\rho RT$$

$$T=\frac{P}{\rho R}$$

$$\frac{T_2}{T_1}=\frac{P_2/({\rho }_{2}R)}{P_1/({\rho }_{1}R)}$$

$$=\frac{P_2}{P_1}\frac{{\rho }_1}{{\rho }_2}\cdots (8)$$

 

Insert the eqaution (8) on equation (7)

$$\frac{P_2}{P_1}={\left(\frac{P_2}{P_1}\frac{{\rho }_1}{{\rho }_2}\right)}^{\gamma /\gamma -1}$$

$$\frac{P_2}{P_1}={\left(\frac{P_2}{P_1}\right)}^{\gamma /\gamma -1}{\left(\frac{{\rho }_1}{{\rho }_2}\right)}^{\gamma /\gamma -1}$$

$${\left(\frac{P_2}{P_1}\right)}^{1-\frac{\gamma }{\gamma -1}}={\left(\frac{P_2}{P_1}\right)}^{-\frac{1}{\gamma -1}}={\left(\frac{{\rho }_1}{{\rho }_2}\right)}^{\frac{\gamma }{\gamma -1}}$$

$$\frac{P_2}{P_1}={\left(\frac{{\rho }_1}{{\rho }_2}\right)}^{-\gamma }={\left(\frac{{\rho }_2}{{\rho }_1}\right)}^{\gamma }$$

 

$$\frac{P_2}{P_1}={\left(\frac{{\rho }_2}{{\rho }_1}\right)}^{\gamma }$$

 

Reference

Hypersonics - from Shock Waves to Scramjets Section 2 - Isentropic Flow