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| __TOC__ | | __TOC__ |
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| === Specific Heat Relations ===
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| For thermally perfect and calorically perfect gases
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| <math display="block">
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| \begin{aligned}
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| &C_p=\frac{dh}{dT}\\
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| &C_v=\frac{de}{dT}
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| \end{aligned}
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| </math>
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| From the definition of enthalpy and the equation of state <math>p=\rho RT</math>
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| <math display="block">
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| h=e+\frac{p}{\rho}=e+RT
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| </math>
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| Differentiate Eqn. \ref{eq:enthalpy} with respect to temperature gives
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| <math display="block">
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| \frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT}
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| </math>
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| Inserting the specific heats gives
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| <math display="block">
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| C_p=C_v+R
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| </math>
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| Dividing Eqn. \ref{eq:specificheat:b} by <math>C_v</math> gives
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| <math display="block">
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| \frac{C_p}{C_v}=1+\frac{R}{C_v}
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| </math>
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| Introducing the ratio of specific heats defined as
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| <math display="block">
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| \gamma=\frac{C_p}{C_v}
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| </math>
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| Now, inserting Eqn. \ref{eq:gamma} in Eqn. \ref{eq:specificheat:c} gives
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| <math display="block">
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| C_v=\frac{R}{\gamma-1}
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| </math>
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| In the same way, dividing Eqn. \ref{eq:specificheat:b} with <math>C_p</math> gives
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| <math display="block">
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| 1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma}+\frac{R}{C_p}
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| </math>
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| and thus
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| <math display="block">
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| C_p=\frac{\gamma R}{\gamma-1}
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| </math>
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| === Isentropic Relations ===
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| First law of thermodynamics:
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| <math display="block">
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| de=\delta q - \delta w
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| </math>
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| For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math>
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| <math display="block">
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| de=Tds-pd\left(\frac{1}{\rho}\right)
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| </math>
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| Enthalpy is defined as: <math>h=e+p/\rho</math> and thus
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| <math display="block">
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| dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp
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| </math>
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| Eliminate $de$ in Eqn. \ref{eq:firstlaw:b} using Eqn. \ref{eq:dh}
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| <math display="block">
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| Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)}
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| </math>
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| <math display="block">
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| ds=\frac{dh}{T}-\frac{dp}{\rho T}
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| </math>
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| Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get
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| <math display="block">
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| ds=C_p\frac{dT}{T}-R\frac{dp}{p}
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| </math>
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| Integrating Eqn. \ref{eq:ds} gives
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| <math display="block">
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| s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right)
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| </math>
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| For a calorically perfect gas, <math>C_p</math> is constant (not a function of temperature) and can be moved out from the integral and thus
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| <math display="block">
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| s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right)
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| </math>
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| An alternative form of Eqn. \ref{eq:ds:c} is obtained by using <math>de=C_v dT</math> Eqn. \ref{eq:firstlaw:b}, which gives
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| <math display="block">
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| s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right)
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| </math>
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| Again, for a calorically perfect gas, we get
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| <math display="block">
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| s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right)
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| </math>
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|
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| === Isentropic Relations ===
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| Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus Eqn. \ref{eq:ds:c} reduces to
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| <math display="block">
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| \frac{C_p}{R}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right)
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| </math>
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| <math display="block">
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| \frac{C_p}{R}=\frac{\gamma}{\gamma-1}
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| </math>
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| <math display="block">
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| \frac{\gamma}{\gamma-1}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right)\Rightarrow
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| </math>
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| <math display="block">
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| \frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
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| </math>
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| In the same way, Eqn. \ref{eq:ds:e} gives
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| <math display="block">
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| \frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)}
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| </math>
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| Eqn. \ref{eq:isentropic:a} and Eqn. \ref{eq:isentropic:b} constitutes the isentropic relations
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| <math display="block">
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| \frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)}
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| </math>
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| === Flow Processes ===
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| <math display="block"> | | <math display="block"> |
