Skip to content

06 2nd Law

Terms

Term Meaning Formula
\(\eta\) Efficiency \(\frac{\text{Desired Output}}{\text{Input}}\)
COP Coefficient of Performance \(\frac{\text{Desired Output}}{\text{Input}}\)
\(q\) Calorific/Heating Value \(\frac Q m\)
Gravimetric mass terms

Devices

Device Purpose
Heat Engine - Heat \(\to\) Work
- cycle		 $\begin{aligned} \eta_{\small\text{HE}} &= \frac{W_\text{net, out}}{Q_\text{H}} \
&= 1 - \frac{Q_\text{L}}{Q_\text{H}} \end{aligned}$ \(\eta_\text{HE} < 1\)
Kelvin-Plank Statement		 \(\begin{aligned} \Delta U &= 0 \\ Q_\text{net} &= W_\text{net} \\ W_\text{net, out} &= Q_\text{in} - Q_\text{out} \\ &= Q_\text{H} - Q_\text{L} \end{aligned}\)
Refridgerator - maintain cool temp
- Reverse HE		 $\begin{aligned} \text{COP}R &= \frac{Q\text{L}}{Q_\text{net, in}} \
&= \frac{1}{ \frac{Q_\text{H}}{Q_\text{L}} - 1 } \end{aligned}$ \(\text{COP}_R\) can be > 1
Heat Pump - maintain warm temp
- Reverse HE		 $\begin{aligned} \text{COP}{HP} &= \frac{Q\text{H}}{W_\text{net, in}} \
&= \frac{1}{ 1 - \frac{Q_\text{L}}{Q_\text{H}} } \end{aligned}$ \(\begin{aligned} \text{COP}_{HP} &= \text{COP}_{R} + 1 \\ \text{COP}_{HP} &> \text{COP}_{R} \end{aligned}\) 
flowchart LR

subgraph Heat Engine
direction LR
a([Warm]) -->
|Q<sub>H</sub>| b[System] -->
|Q<sub>L</sub>| c([Cool])

b --> |W<sub>net</sub>| d[ ]
end

subgraph Refridgerator/Heat Pump
direction LR
r([Cool]) -->
|Q<sub>L</sub>| q[System] -->
|Q<sub>H</sub>| p([Warm])
s[ ] --> |W<sub>net</sub>| q
end

Carnot Cycle

For Heat Engine

Adiabatic means polytropic process with**out** heat transfer

Transition Characteristic Constant Signs Work
1 - 2 Isothermal Expansion
Heat Absorbed		 \(PV = c\) $W_{12} > 0 \
Q_\text{H} > 0$ \(P_1 V_1 \ln \vert \frac{V_2}{V_1} \vert\)
\(P_2 V_2 \ln \vert \frac{P_1}{P_2} \vert\)
2 - 3 Adiabatic Expansion \(PV^\gamma = c\) \(W_{23} > 0\) \(\frac{P_3 V_3 - P_2 V_2}{1-n}\)
3 - 4 Isothermal Compression
Heat Released		 \(PV = c\) $W_{34} < 0 \
Q_\text{L} < 0$ \(P_3 V_3 \ln \vert \frac{V_4}{V_3} \vert\)
\(P_4 V_4 \ln \vert \frac{P_3}{P_4} \vert\)
4 - 1 Adiabatic Compression \(PV^\gamma = c\) \(W_{41} < 0\) \(\frac{P_1 V_1 - P_4 V_4}{1-n}\)
\[ \begin{aligned} W_\text{net, out} &= W_{12} + W_{23} + W_{34} + W_{41} \\ \eta &= \frac{W_\text{net, out}}{Q_\text{H}} \\ &= 1 - \frac{Q_\text{L}}{Q_\text{H}} \\ &= 1 - \frac{T_L}{T_H} \end{aligned} \]

Make sure of the signs when calculating \(W_\text{net, out}\)

Reverse Carnot Cycle

For Refridgerator, Heat Pump

\(Q_\text{L} > 0, Q_\text{H} < 0\)

\[ \begin{aligned} W_\text{net, in} &= W_{12} + W_{23} + W_{34} + W_{41} \\ \text{COP}_R &= \frac{Q_\text{L}}{W_\text{net, in}} \\ \text{COP}_{HP} &= \frac{Q_\text{H}}{W_\text{net, in}} \end{aligned} \]
Last Updated: 2024-01-24 ; Contributors: AhmedThahir

Comments