6. Entropy and the Second Law of Thermodynamics
6.12 Key equations
Heat engine
| Net work output | [latex]{\dot{W}}_{net,\ out}={\dot{Q}}_H-{\dot{Q}}_L[/latex] |
| Thermal efficiency of any heat engine | [latex]\eta_{th}=\displaystyle\frac{desired\ output}{required\ input}=\frac{{\dot{W}}_{net,\ out}}{{\dot{Q}}_H}=1-\frac{{\dot{Q}}_L}{{\dot{Q}}_H}[/latex] |
| Thermal efficiency of Carnot heat engine | [latex]\eta_{th,\ rev}=1-\displaystyle\frac{T_L}{T_H}[/latex]
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Refrigerator and heat pump
| Net work input | [latex]{\dot{W}}_{net,\ in}={\dot{Q}}_H-{\dot{Q}}_L[/latex] |
| COP of any refrigerator | [latex]\begin{align*} {COP}_R &=\displaystyle\frac{desired\ output}{required\ input} \\&=\frac{{\dot{Q}}_L}{{\dot{W}}_{net,\ in}} =\frac{{\dot{Q}}_L}{{\dot{Q}}_H-{\dot{Q}}_L} =\frac{1}{{\dot{Q}}_H/{\dot{Q}}_L-1} \end{align*}[/latex] |
| COP of Carnot refrigerator | [latex]{COP}_{R,\ rev}=\displaystyle\frac{T_L}{T_H-T_L}=\frac{1}{T_H/T_L-1}[/latex] |
| COP of any heat pump | [latex]\begin{align*} {COP}_{HP} &=\displaystyle\frac{desired\ output}{required\ input} \\&=\frac{{\dot{Q}}_H}{{\dot{W}}_{net,\ in}}=\frac{{\dot{Q}}_H}{{\dot{Q}}_H-{\dot{Q}}_L}=\frac{1}{{1-\dot{Q}}_L/{\dot{Q}}_H} \end{align*}[/latex] |
| COP of Carnot heat pump | [latex]{COP}_{HP,\ rev}=\displaystyle\frac{T_H}{T_H-T_L}=\displaystyle\frac{1}{{1-T}_L/T_H}[/latex] |
Entropy and entropy generation
| The inequality of Clausius | [latex]\displaystyle\oint\displaystyle\frac{\delta Q}{T}\le0 \ \rm{(= for \ reversible \ cycles; \ < for \ irreversible \ cycles)}[/latex] |
| Definition of entropy | [latex]\begin{align*} \rm{Infinitesimal \ \ form:} & \ dS =\left(\displaystyle\frac{\delta Q}{T}\right)_{rev} \\ \rm{Integral \ \ form:} & \ \Delta S = S_2-S_1=\displaystyle\int_{1}^{2}\left(\displaystyle\frac{\delta Q}{T}\right)_{rev} \end{align*}[/latex] |
| Definition of entropy generation | [latex]{\rm{Infinitesimal \ \ form:}} \ dS =\displaystyle\frac{\delta Q}{T}+\delta S_{gen} \\ {\rm{where}} \ \delta S_{gen}\geq0 \\ \rm{(= for \ reversible \ processes; \ > for \ irreversible \ process)}[/latex] |
The second law of thermodynamics
| For closed systems (control mass) | [latex]\begin {align*} S_2-S_1 &=\displaystyle\int_{1}^{2}\displaystyle\frac{\delta Q}{T}+S_{gen} \\& \cong\sum\frac{Q_k}{T_k}+S_{gen}\ \ \ \ \ (S_{gen}\geq0) \end {align*}[/latex] where [latex]T_k[/latex] is the absolute temperature of the system boundary, in Kelvin. |
| For steady-state, steady flow in a control volume (open systems) | [latex]\sum{{\dot{m}}_es_e}-\sum{{\dot{m}}_is_i}=\sum\displaystyle\frac{{\dot{Q}}_{c.v.}}{T}+{\dot{S}}_{gen}\ \ \ \ \ \ \left({\dot{S}}_{gen}\geq0\right)[/latex] |
| For steady and isentropic flow | [latex]\sum{{\dot{m}}_es_e}=\sum{{\dot{m}}_is_i}[/latex] |
| Change of specific entropy between two states of a solid or liquid | [latex]s_2-s_1=C_pln\displaystyle\frac{T_2}{T_1}[/latex] |
| Change of specific entropy between two states of an ideal gas | Assume constant [latex]C_p[/latex] and [latex]C_v[/latex] in the temperature range, [latex]s_2-s_1=C_pln\displaystyle\frac{T_2}{T_1}-Rln\frac{P_2}{P_1}[/latex]
[latex]s_2-s_1=C_vln\displaystyle\frac{T_2}{T_1}+Rln\frac{v_2}{v_1}[/latex] |
| Isentropic relations for ideal gases | [latex]Pv^k= \rm {constant}[/latex]
[latex]\displaystyle\frac{P_2}{P_1}=\displaystyle\left(\displaystyle\frac{v_1}{v_2}\right)^k=\left(\frac{T_2}{T_1}\right)^{k/(k-1)}[/latex] |
