Specific heat: Difference between revisions
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Created page with "Category:Compressible flow Category:Thermodynamics __TOC__ For thermally perfect and calorically perfect gases <math display="block"> \begin{aligned} &C_p=\frac{dh}{dT}\\ &C_v=\frac{de}{dT} \end{aligned} </math> From the definition of enthalpy and the equation of state <math>p=\rho RT</math> <math display="block"> h=e+\frac{p}{\rho}=e+RT </math> Differentiate Eqn. \ref{eq:enthalpy} with respect to temperature gives <math display="block"> \frac{dh}{dT}=\fra..." |
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For thermally perfect and calorically perfect gases | For thermally perfect and calorically perfect gases | ||
<math | {{NumEqn|<math> | ||
\begin{aligned} | \begin{aligned} | ||
&C_p=\frac{dh}{dT}\\ | &C_p=\frac{dh}{dT}\\ | ||
&C_v=\frac{de}{dT} | &C_v=\frac{de}{dT} | ||
\end{aligned} | \end{aligned} | ||
</math> | </math>|label=eq-specific-heat}} | ||
From the definition of enthalpy and the equation of state <math>p=\rho RT</math> | From the definition of enthalpy and the equation of state <math>p=\rho RT</math> | ||
<math | {{NumEqn|<math> | ||
h=e+\frac{p}{\rho}=e+RT | h=e+\frac{p}{\rho}=e+RT | ||
</math> | </math>|label=eq-enthalpy}} | ||
Differentiate | Differentiate {{EquationNote|label=eq-enthalpy}} with respect to temperature gives | ||
<math | {{NumEqn|<math> | ||
\frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT} | \frac{dh}{dT}=\frac{de}{dT}+\frac{d(RT)}{dT} | ||
</math> | </math>}} | ||
Inserting the specific heats gives | Inserting the specific heats gives | ||
<math | {{NumEqn|<math> | ||
C_p=C_v+R | C_p=C_v+R | ||
</math> | </math>|label=eq-specific-heat-b}} | ||
Dividing | Dividing {{EquationNote|label=eq-specific-heat-b}} by <math>C_v</math> gives | ||
<math | {{NumEqn|<math> | ||
\frac{C_p}{C_v}=1+\frac{R}{C_v} | \frac{C_p}{C_v}=1+\frac{R}{C_v} | ||
</math> | </math>|label=eq-specific-heat-c}} | ||
Introducing the ratio of specific heats defined as | Introducing the ratio of specific heats defined as | ||
<math | {{NumEqn|<math> | ||
\gamma=\frac{C_p}{C_v} | \gamma=\frac{C_p}{C_v} | ||
</math> | </math>|label=eq-gamma}} | ||
Now, inserting | Now, inserting {{EquationNote|label=eq-gamma}} in Eqn. \ref{eq:specificheat:c} gives | ||
<math | {{NumEqn|<math> | ||
C_v=\frac{R}{\gamma-1} | C_v=\frac{R}{\gamma-1} | ||
</math> | </math>}} | ||
In the same way, dividing | In the same way, dividing {{EquationNote|label=eq-specific-heat-b}} with <math>C_p</math> gives | ||
<math | {{NumEqn|<math> | ||
1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma}+\frac{R}{C_p} | 1=\frac{C_v}{C_p}+\frac{R}{C_p}=\frac{1}{\gamma}+\frac{R}{C_p} | ||
</math> | </math>}} | ||
and thus | and thus | ||
<math | {{NumEqn|<math> | ||
C_p=\frac{\gamma R}{\gamma-1} | C_p=\frac{\gamma R}{\gamma-1} | ||
</math> | </math>}} | ||
