6.12 Key equations

Claire Yu Yan

6. Entropy and the Second Law of Thermodynamics

6.12 Key equations

Heat engine

Net work output

{˙ W}_{n e t, o u t} = {˙ Q}_{H} - {˙ Q}_{L}

${\dot{W}}_{net,\ out}={\dot{Q}}_H-{\dot{Q}}_L$

Thermal efficiency of any heat engine

η_{t h} = \frac{d e s i r e d o u t p u t}{r e q u i r e d i n p u t} = \frac{{˙ W}_{n e t, o u t}}{{˙ Q}_{H}} = 1 - \frac{{˙ Q}_{L}}{{˙ Q}_{H}}

$\eta_{th}=\displaystyle\frac{desired\ output}{required\ input}=\frac{{\dot{W}}_{net,\ out}}{{\dot{Q}}_H}=1-\frac{{\dot{Q}}_L}{{\dot{Q}}_H}$

Thermal efficiency of Carnot heat engine

η_{t h, r e v} = 1 - \frac{T_{L}}{T_{H}}

$\eta_{th,\ rev}=1-\displaystyle\frac{T_L}{T_H}$

Refrigerator and heat pump

Net work input

{˙ W}_{n e t, i n} = {˙ Q}_{H} - {˙ Q}_{L}

${\dot{W}}_{net,\ in}={\dot{Q}}_H-{\dot{Q}}_L$

COP of any refrigerator

\begin{matrix} {C O P}_{R} & = \frac{d e s i r e d o u t p u t}{r e q u i r e d i n p u t} = \frac{{˙ Q}_{L}}{{˙ W}_{n e t, i n}} = \frac{{˙ Q}_{L}}{{˙ Q}_{H} - {˙ Q}_{L}} = \frac{1}{{˙ Q}_{H} / {˙ Q}_{L} - 1} \end{matrix}

$\begin{align*} {COP}_R &=\displaystyle\frac{desired\ output}{required\ input} \\&=\frac{{\dot{Q}}_L}{{\dot{W}}_{net,\ in}} =\frac{{\dot{Q}}_L}{{\dot{Q}}_H-{\dot{Q}}_L} =\frac{1}{{\dot{Q}}_H/{\dot{Q}}_L-1} \end{align*}$

COP of Carnot refrigerator

{C O P}_{R, r e v} = \frac{T_{L}}{T_{H} - T_{L}} = \frac{1}{T_{H} / T_{L} - 1}

${COP}_{R,\ rev}=\displaystyle\frac{T_L}{T_H-T_L}=\frac{1}{T_H/T_L-1}$

COP of any heat pump

\begin{matrix} {C O P}_{H P} & = \frac{d e s i r e d o u t p u t}{r e q u i r e d i n p u t} = \frac{{˙ Q}_{H}}{{˙ W}_{n e t, i n}} = \frac{{˙ Q}_{H}}{{˙ Q}_{H} - {˙ Q}_{L}} = \frac{1}{{1 - ˙ Q}_{L} / {˙ Q}_{H}} \end{matrix}

$\begin{align*} {COP}_{HP} &=\displaystyle\frac{desired\ output}{required\ input} \\&=\frac{{\dot{Q}}_H}{{\dot{W}}_{net,\ in}}=\frac{{\dot{Q}}_H}{{\dot{Q}}_H-{\dot{Q}}_L}=\frac{1}{{1-\dot{Q}}_L/{\dot{Q}}_H} \end{align*}$

COP of Carnot heat pump

{C O P}_{H P, r e v} = \frac{T_{H}}{T_{H} - T_{L}} = \frac{1}{{1 - T}_{L} / T_{H}}

${COP}_{HP,\ rev}=\displaystyle\frac{T_H}{T_H-T_L}=\displaystyle\frac{1}{{1-T}_L/T_H}$

Entropy and entropy generation

The inequality of Clausius

\oint \frac{δ Q}{T} \leq 0 (= f o r r e v e r s i b l e c y c l e s; < f o r i r r e v e r s i b l e c y c l e s)

$\displaystyle\oint\displaystyle\frac{\delta Q}{T}\le0 \ \rm{(= for \ reversible \ cycles; \ < for \ irreversible \ cycles)}$

Definition of entropy

\begin{matrix} I n f i n i t e s i m a l f o r m : & d S = {(\frac{δ Q}{T})}_{r e v} I n t e g r a l f o r m : & Δ S = S_{2} - S_{1} = \int_{1}^{2} {(\frac{δ Q}{T})}_{r e v} \end{matrix}

$\begin{align*} \rm{Infinitesimal \ \ form:} & \ dS =\left(\displaystyle\frac{\delta Q}{T}\right)_{rev} \\ \rm{Integral \ \ form:} & \ \Delta S = S_2-S_1=\displaystyle\int_{1}^{2}\left(\displaystyle\frac{\delta Q}{T}\right)_{rev} \end{align*}$

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

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Introduction to Engineering Thermodynamics Copyright © 2022 by Claire Yu Yan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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