6.12 Key equations

6. Entropy and the Second Law of Thermodynamics

Heat engine

Net work output

{\dot{W}}_{n e t, o u t} = {\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{{\dot{W}}_{n e t, o u t}}{{\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}}

Refrigerator and heat pump

Net work input

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

COP of any refrigerator

\begin{aligned} {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{{\dot{Q}}_{L}}{{\dot{W}}_{n e t, i n}} = \frac{{\dot{Q}}_{L}}{{\dot{Q}}_{H} - {\dot{Q}}_{L}} = \frac{1}{{\dot{Q}}_{H} / {\dot{Q}}_{L} - 1} \end{aligned}

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 of any heat pump

\begin{aligned} {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{{\dot{Q}}_{H}}{{\dot{W}}_{n e t, i n}} = \frac{{\dot{Q}}_{H}}{{\dot{Q}}_{H} - {\dot{Q}}_{L}} = \frac{1}{{1 - \dot{Q}}_{L} / {\dot{Q}}_{H}} \end{aligned}

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}}

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)

Definition of entropy

\begin{aligned} 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{aligned}

Definition of entropy generation

I n f i n i t e s i m a l f o r m : d S = \frac{δ Q}{T} + δ S_{g e n} w h e r e δ S_{g e n} \geq 0 (= f o r r e v e r s i b l e p r o c e s s e s; > f o r i r r e v e r s i b l e p r o c e s s)

The second law of thermodynamics

For closed systems (control mass)

\begin{aligned} S_{2} - S_{1} & = \int_{1}^{2} \frac{δ Q}{T} + S_{g e n} \\ ≅ \sum \frac{Q_{k}}{T_{k}} + S_{g e n} (S_{g e n} \geq 0) \end{aligned}

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}}_{e} s_{e} - \sum {\dot{m}}_{i} s_{i} = \sum \frac{{\dot{Q}}_{c . v .}}{T} + {\dot{S}}_{g e n} ({\dot{S}}_{g e n} \geq 0)

For steady and isentropic flow

\sum {\dot{m}}_{e} s_{e} = \sum {\dot{m}}_{i} s_{i}

Change of specific entropy between two states of a solid or liquid

s_{2} - s_{1} = C_{p} l n \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_{p} l n \frac{T_{2}}{T_{1}} - R l n \frac{P_{2}}{P_{1}}

$s_{2} - s_{1} = C_{v} l n \frac{T_{2}}{T_{1}} + R l n \frac{v_{2}}{v_{1}}$

Isentropic relations for ideal gases

P v^{k} = c o n s t a n t

$\frac{P_{2}}{P_{1}} = {(\frac{v_{1}}{v_{2}})}^{k} = {(\frac{T_{2}}{T_{1}})}^{k / (k - 1)}$
where $k = \frac{C_{p}}{C_{v}}$ and $T$ is in Kelvin

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