Isentropic relations: Difference between revisions
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=== First law of thermodynamics === | === First law of thermodynamics === | ||
First law of thermodynamics: | First law of thermodynamics: | ||
<math | {{NumEqn|<math>de=\delta q - \delta w</math>}} | ||
de=\delta q - \delta w | |||
</math> | |||
For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math> | For a reversible process: <math>\delta w=pd(1/\rho)</math> and <math>\delta q=Tds</math> | ||
<math | {{NumEqn|<math> | ||
de=Tds-pd\left(\frac{1}{\rho}\right) | de=Tds-pd\left(\frac{1}{\rho}\right) | ||
</math> | </math>|label=eq-first-law-b}} | ||
Enthalpy is defined as: <math>h=e+p/\rho</math> and thus | Enthalpy is defined as: <math>h=e+p/\rho</math> and thus | ||
<math | {{NumEqn|<math> | ||
dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp | dh=de+pd\left(\frac{1}{\rho}\right)+\left(\frac{1}{\rho}\right)dp | ||
</math> | </math>|label=eq-dh}} | ||
Eliminate | Eliminate <math>de</math> in {{EquationNote|label=eq-first-law-b}} using {{EquationNote|label=eq-dh}} | ||
<math | {{NumEqn|<math> | ||
Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)} | Tds=dh-\cancel{pd\left(\frac{1}{\rho}\right)}-\left(\frac{1}{\rho}\right)dp+\cancel{pd\left(\frac{1}{\rho}\right)} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
ds=\frac{dh}{T}-\frac{dp}{\rho T} | ds=\frac{dh}{T}-\frac{dp}{\rho T} | ||
</math> | </math>}} | ||
Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get | Using <math>dh=C_p T</math> and the equation of state <math>p=\rho RT</math>, we get | ||
<math | {{NumEqn|<math> | ||
ds=C_p\frac{dT}{T}-R\frac{dp}{p} | ds=C_p\frac{dT}{T}-R\frac{dp}{p} | ||
</math> | </math>|label=eq-ds}} | ||
Integrating | Integrating {{EquationNote|eq-ds}} gives | ||
<math | {{NumEqn|<math> | ||
s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right) | s_2-s_1=\int_1^2 C_p\frac{dT}{T}-R\ln\left(\frac{p_2}{p_1}\right) | ||
</math> | </math>|label=eq-ds-b}} | ||
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 | 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 | ||
<math | {{NumEqn|<math> | ||
s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right) | s_2-s_1=C_p\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{p_2}{p_1}\right) | ||
</math> | </math>|label=eq-ds-c}} | ||
An alternative form of | An alternative form of {{EquationNote|label=eq-ds-c}} is obtained by using <math>de=C_v dT</math> in {{EquationNote|label=eq-first-law-b}}, which gives | ||
<math | {{NumEqn|<math> | ||
s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right) | s_2-s_1=\int_1^2 C_v\frac{dT}{T}-R\ln\left(\frac{\rho_2}{\rho_1}\right) | ||
</math> | </math>|label=eq-ds-d}} | ||
Again, for a calorically perfect gas, we get | Again, for a calorically perfect gas, we get | ||
<math | {{NumEqn|<math> | ||
s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right) | s_2-s_1=C_v\ln\left(\frac{T_2}{T_1}\right)-R\ln\left(\frac{\rho_2}{\rho_1}\right) | ||
</math> | </math>|label=eq-ds-e}} | ||
=== Isentropic Relations === | === Isentropic Relations === | ||
Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus | Adiabatic and reversible processes, i.e., isentropic processes implies <math>ds=0</math> and thus {{EquationNote|label=eq-ds-c}} reduces to | ||
<math | {{NumEqn|<math> | ||
\frac{C_p}{R}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right) | \frac{C_p}{R}\ln\left(\frac{T_2}{T_1}\right)=\ln\left(\frac{p_2}{p_1}\right) | ||
</math> | </math>}} | ||
<math display="block"> | <math display="block"> | ||
| Line 81: | Line 89: | ||
</math> | </math> | ||
<math | {{NumEqn|<math> | ||
\frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)} | \frac{p_2}{p_1}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)} | ||
</math> | </math>|label=eq-isentropic-a}} | ||
In the same way, | In the same way, {{EquationNote|label=eq-ds-e}} gives | ||
<math | {{NumEqn|<math> | ||
\frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)} | \frac{\rho_2}{\rho_1}=\left(\frac{T_2}{T_1}\right)^{1/(\gamma-1)} | ||
</math> | </math>|label=eq-isentropic-b}} | ||
Eqn. | Eqn. {{EquationNote|eq-isentropic-a}} and Eqn. {{EquationNote|eq-isentropic-b}} constitutes the isentropic relations | ||
<math | {{NumEqn|<math> | ||
\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)} | \frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^{\gamma}=\left(\frac{T_2}{T_1}\right)^{\gamma/(\gamma-1)} | ||
</math> | </math>}} | ||
