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Formula Units
Total Work

    \[W_{net,out}=w_{out}-w_{in}\]

    \[kJ\]
Thermal Efficiency

    \[n_{th}=\frac{W_{net,out}}{Q_{in}}=1-\frac{Q_{out}}{Q_{in}}\]
Carnot Thermal Efficiency

    \[n_{th}=1-\frac{Q_L}{Q_H}=1-\frac{T_L}{T_H}\]
The Net Power of Heat Engine

    \[W_{net,out}=\dot{Q}_{out}-\dot{Q}_{in}\]

    \[kJ\]
Coefficient of Performance

    \[COP_R=\frac{Q_{out}}{W_{net,in}}=\frac{Q_L}{Q_H-Q_L}=\frac{1}{(\frac{Q_H}{Q_L})-1}\]
Real Heat Pump

    \[COP_{HP}=\frac{Q_{out}}{W_{net,in}}=\frac{Q_H}{Q_L-Q_H}=\frac{1}{1-(\frac{Q_L}{Q_H})}\]

    \[COP_{HP}=COP_R+1\]
Energy Efficiency Rating

    \[EER=3.412 \ COP_R\]
Clausius Inequality

    \[\oint \frac{\delta Q}{T} \leq 0\]
Entropy

    \[dS=(\frac{\delta Q}{T})_{int \ rev}\]

 kJ/K
Change of Entropy

    \[\Delta S=S_2-S_1= \int^2_1 \frac{\delta Q}{T}_{int \ rev}\]

 kJ/K
Specific Entropy

    \[s=s_f + xs_{fg}\]

    \[s_{@T,P}=S_{f@T}\]

 kJ/K
Entropy Change

    \[\Delta S=m \Delta S = m(s_2-s_1)\]

 kJ/K
  • Carnot Cycle
 Isothermal Heat Transfer: S_2-S_1=\frac{1}{T_H} \int^2_1 \delta Q = \frac{_1Q_2}{T_H}
Reversible Adiabatic (Isentropic Process): dS=(\frac{\delta Q}{T})_{rev}
Reversible Isothermal Process: S_4-S_3 = \int^4_3(\frac{\delta Q}{T})_{rev}=\frac{_3Q_4}{T_L}
Reversible Adiabatic (Isentropic Process): Entropy decrease in process 3-4 = the entropy increase in process 1-2.
  • Reversible Heat-Transfer Process

    \[s_2-s_1=s_{fg}=\frac{1}{m} \int^2_1 (\frac{\delta Q}{T})_{rev}=\frac{1}{mT} \int^2_1 \delta Q = \frac{_1q_2}{T}=\frac{h_{fg}}{T}\]
Entropy Generation

    \[dS=\frac{\delta Q}{T}+\delta S_{gen}\]

    \[\delta W_{irr}=PdV-T \delta S_{gen}\]

    \[S_2-S_1= \int^2_1 dS = \int^2_1 \frac{\delta Q}{T}+_1S_{2 gen}\]
Principle of the Increase of Entropy

    \[dS_{net}=dS_{c.m.}+dS_{surr}= \Sigma \delta S_{gen} \geq 0\]
Entropy Change
  • Solids & Liquids

    \[s_2-s_1=c \ln(\frac{T_2}{T_1})\]

    \[\text{Reversible Process: }ds_{gen}=0\]

    \[\text{Adiabatic Process: }dq=0\]
  • Ideal Gas
 Constant Volume:  s_2-s_1=\int ^2_1 C v_0 \frac{dT}{T} + R \ln(\frac{v_2}{v_1})
Constant Pressure: s_2-s_1= \int ^2_1 C p_0 \frac{dT}{T} - R \ln(\frac{P_2}{P_1})

Constant Specific Heat: s_2-S_1=Cv_0 \ln(\frac{T_2}{T_1})+R \ln(\frac{v_2}{v_1})                              s_2-s_1=Cp_0 \ln(\frac{T_2}{T_1}) - R \ln(frac{P_2}{P_1})
Standard Entropy

    \[s^0_T=\int^T_{T_0}\frac{C_o0}{T}dT\]

    \[kJ/kgK\]
Change in Standard Entropy

    \[s_2-s_1=(s^0_{T2}-s^0_{T1})-R \ ln(\frac{P_2}{P_1})\]

    \[kJ/kgK\]
Ideal Gas Undergoing an Isentropic Process

    \[s_2-s_1=0=C_p \ln(\frac{T_2}{T_1})-R \ln(\frac{P_2}{P_1}) \rightarrow \frac{T_2}{T_1}=(\frac{P_2}{P_1})^{\frac{R}{C_p0}}\]

    \[\text{but }\frac{R}{C_{p0}} = \frac{C_p0-C_v0}{C_{p0}}=\frac{k-1}{k}, k=\frac{C_p0}{C_v0}=\text{ratio of specific heats}\]

    \[\Rightarrow \frac{T_2}{T_1}=(\frac{v_1}{v_2})^{k-1}, \frac{P_2}{P_1}=(\frac{v_1}{v_2})^k\]

    \[\text{Special case of polytropic process where }k=n: Pv^k=const\]
Reversible Polytropic Process for Ideal Gas

    \[PV^n=const=P_1V_1^n=P_2V_2^n\]

    \[\rightarrow \frac{P_2}{P_1}=(\frac{V_1}{V_2})^n, \frac{T_2}{T_1}=(\frac{P_2}{P_1})^{n-\frac{1}{n}} = (\frac{V_1}{V_2})^{n-1}\]
  • Work

    \[W_{1-2}=\int^2_1PdV = const\int^2_1\frac{dv}{v^n}=\frac{P_2V_2-P_1V_1}{1-n}=\frac{mR(T_2-T_1)}{1-n}\]
  • Values for n

    \[\text{Isobaric Process}: n=0, P=const\]

    \[\text{Isothermal Process}: n=1, T=const\]

    \[\text{Isentropic Process}: n=k, s=const\]

    \[\text{Isochronic Process}: n= \infty, v=const\]

License

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Thermodynamics Copyright © by Diana Bairaktarova (Adapted from Engineering Thermodynamics - A Graphical Approach by Israel Urieli and Licensed CC BY NC-SA 3.0) is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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