6.13: Key equations

Last updated
Save as PDF

Page ID: 88863

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

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	$\text{[math]}$
COP of Carnot refrigerator	$859311692f9638cd5f64bdf673fc9861.png$
COP of any heat pump	$\text{[math]}$
COP of Carnot heat pump	$784d2d74e35ccb362ce96648ef65d4c3.png$

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)}" class="latex mathjax" title="{\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)}" src="/@api/deki/files/59331/84a1c10887a25607359ac46762759083.png">

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	$Pv^k= \rm {constant}$ $8daec76a30ebc5b7db9d1ebaf6f6a94d.png$ where $k=\displaystyle\frac{C_p}{C_v}$ and $T$ is in Kelvin