6. Entropy and the Second Law of Thermodynamics

# 6.12 Key equations

Heat engine

 Net work output ${\dot{W}}_{net,\ out}={\dot{Q}}_H-{\dot{Q}}_L$ Thermal efficiency of any heat engine $\eta_{th}=\displaystyle\frac{desired\ output}{required\ input}=\frac{{\dot{W}}_{net,\ out}}{{\dot{Q}}_H}=1-\frac{{\dot{Q}}_L}{{\dot{Q}}_H}$ Thermal efficiency of Carnot heat engine $\eta_{th,\ rev}=1-\displaystyle\frac{T_L}{T_H}$

Refrigerator and heat pump

 Net work input ${\dot{W}}_{net,\ in}={\dot{Q}}_H-{\dot{Q}}_L$ COP of any refrigerator \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*} COP of Carnot refrigerator ${COP}_{R,\ rev}=\displaystyle\frac{T_L}{T_H-T_L}=\frac{1}{T_H/T_L-1}$ COP of any heat pump \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*} COP of Carnot heat pump ${COP}_{HP,\ rev}=\displaystyle\frac{T_H}{T_H-T_L}=\displaystyle\frac{1}{{1-T}_L/T_H}$

Entropy and entropy generation

 The inequality of Clausius $\displaystyle\oint\displaystyle\frac{\delta Q}{T}\le0 \ \rm{(= for \ reversible \ cycles; \ < for \ irreversible \ cycles)}$ Definition of entropy \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*} Definition of entropy generation ${\rm{Infinitesimal \ \ form:}} \ dS =\displaystyle\frac{\delta Q}{T}+\delta S_{gen} \\ {\rm{where}} \ \delta S_{gen}\geq0 \\ \rm{(= for \ reversible \ processes; \ > for \ irreversible \ process)}$

The second law of thermodynamics

 For closed systems (control mass) \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*} where $T_k$ is the absolute temperature of the system boundary, in Kelvin. For steady-state, steady flow in a control volume (open systems) $\sum{{\dot{m}}_es_e}-\sum{{\dot{m}}_is_i}=\sum\displaystyle\frac{{\dot{Q}}_{c.v.}}{T}+{\dot{S}}_{gen}\ \ \ \ \ \ \left({\dot{S}}_{gen}\geq0\right)$ For steady and isentropic flow $\sum{{\dot{m}}_es_e}=\sum{{\dot{m}}_is_i}$ Change of specific entropy between two states of a solid or liquid $s_2-s_1=C_pln\displaystyle\frac{T_2}{T_1}$ Change of specific entropy between two states of an ideal gas Assume constant  $C_p$ and $C_v$ in the temperature range,  $s_2-s_1=C_pln\displaystyle\frac{T_2}{T_1}-Rln\frac{P_2}{P_1}$ $s_2-s_1=C_vln\displaystyle\frac{T_2}{T_1}+Rln\frac{v_2}{v_1}$ Isentropic relations for ideal gases $Pv^k= \rm {constant}$ $\displaystyle\frac{P_2}{P_1}=\displaystyle\left(\displaystyle\frac{v_1}{v_2}\right)^k=\left(\frac{T_2}{T_1}\right)^{k/(k-1)}$ where  $k=\displaystyle\frac{C_p}{C_v}$  and $T$ is in Kelvin