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

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

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

    \[Adiabatic\ Process: dq=0\]

  • 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

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

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

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

    \[Special\ case\ of\ polytropic\ process\ where\ k=n: Pv^k=const\]

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

    \[Isobaric\ process: n=0, P=const\]

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

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

    \[Isochronic\ Process: n= infinity, v=const\]