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Isentropic flow relations 본문

공학/열역학

Isentropic flow relations

lightbulb_4999 2022. 9. 19. 00:53

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)

$$\int_{T_{1}}^{T_{2}}{\frac{C_{P}}{R}\frac{dT}{T}}=\int_{P_{1}}^{P_{2}}{\frac{dP}{P}}$$

 

$$\frac{C_{P}}{R}\left[\ln T_{2}-\ln T_{1}\right]=\ln P_{2}-\ln P_{1}$$

$$=\frac{C_{P}}{R}\ln\frac{T_{2}}{T_{1}}=\ln\frac{P_{2}}{P_{1}}$$

$$=\ln\left(\frac{T_{2}}{T_{1}}\right)^{C_{P}/R}=\ln\left(\frac{P_{2}}{P_{1}}\right)\cdots(6)$$

 

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 

 

 

 

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