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Isentropic flow relations 본문
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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