Adiabatic Reversible Expansion and Compression: Screencasts
The relation between temperature and pressure for an adiabatic reversible process of an ideal gas is derived for a constant heat capacity.
We suggest that after watching this screencast, you list the important points as a way to increase retention.
The relation between temperature and pressure for an adiabatic reversible process of an ideal gas is derived for a temperature-dependent heat capacity.
We suggest that after watching this screencast, you list the important points as a way to increase retention.
Important Equations:
These equations are for both closed systems and steady-state flow systems.
\[\Delta S = 0 = \int_{T_1}^{T_2} C_P\,dT – R\,ln\left( \frac{P_2}{P_1} \right) = \int_{T_1}^{T_2} C_V\,dT + R\,ln\left( \frac{V_2}{V_1} \right) \]
Adiabatic reversible compression or expansion for ideal gas with constant heat capacity
\[\Delta S = 0 = C_P\,ln\left(\frac{T_2}{T_1}\right) – R\,ln\left(\frac{P_2}{P_1}\right) = C_V\,\left(\frac{T_2}{T_1}\right) + R\,ln\left(\frac{V_2}{V_1}\right)\]
\[\frac{T_2}{T_1} = \left( \frac{P_2}{P_1}\right)^{\frac{R}{C_P}}\]
\[\frac{T_2}{T_1} = \left( \frac{V_2}{V_1}\right)^{\frac{R}{C_V}}\]
\(W_{EC} = \Delta U = C_V(T_2 – T_1)\)
\(W_S = \Delta H = C_P(T_2 – T_1)\)
where \(T_2\) = final or outlet temperature (K)
\(T_1\) = initial or feed temperature (K)
\(P_2\) = final or outlet pressure (MPa)
\(P_1\) = initial or feed pressure (MPa)
\(V_2\) = final or outlet molar volume (m3/mol)
\(V_1\) = initial or inlet molar volume (m3/mol)
\(R\) = ideal gas constant (J/(mol K))
\(C_P\) = constant-pressure heat capacity (J/(mol K))
\(C_V\) = constant-volume heat capacity (J/(mol K))
\(W_{EC}\) = expansion/compression work (J/mol)
\(W_S\) = shaft work (J/mol)
\(\Delta U\) = change in molar internal energy (J/mol)
\(\Delta H\) = change in molar enthalpy (J/mol)
Adiabatic reversible compression or expansion for ideal gas with temperature-dependent heat capacity
\[C_P = A + BT + CT^2 + DT^3\] where \(A\), \(B\), \(C\), and \(D\) are constants and \(T\) is absolute temperature
\[R\,ln\left(\frac{P_2}{P_1}\right) = A\,ln\left(\frac{T_2}{T_1}\right) + B(T_2 – T_1) + \frac{C}{2} (T_2^2 – T_1^2) + \frac{D}{3} (T_2^3 – T_1^3)\]
\[W_{EC} = \Delta U = \int_{T_1}^{T_2} C_V \,dT\]
\[W_S = \Delta H = \int_{T_1}^{T_2} C_P \,dT\]
\[W_{EC} = A(T_2 -T_1) +\frac{B}{2} (T_2^2 -T_1^2) + \frac{C}{3} (T_2^3 – T_1^3) + \frac{D}{4} (T_2^4 – T_1^4) – R\,(T_2 – T_1)\]
\[W_S = A(T_2 -T_1) +\frac{B}{2} (T_2^2 -T_1^2) + \frac{C}{3} (T_2^3 – T_1^3) + \frac{D}{4} (T_2^4 – T_1^4)\]