Thermodynamics

Equations

Contents:

Fluid Properties
Pressure
Pure Substances
First Law
Work
Internal Energy
Enthalpy
Power
Conservation of Mass
Transient Flow Process
Thermal Efficiency
Heat Engines
Entropy
Entropy Analysis
Device Efficiency
Mixtures
Air Mixtures
Combustion

Fluid Properties

$\nu=\dfrac{V}{m}$

$P=\dfrac{F}{A}$

\rho=\dfrac{m}{V}=\dfrac{1}{\nu}

Pressure

$P_fA=P_0A+mg$

$P_f=P_0+\sum_0^n\rho_iz_ig$

P_f=P_0+\rho_fzg

Pure Substances

$\nu_{fg}=\nu_g-\nu_f$

$\nu(T)\approx\nu_f(T)$

$\nu=x\nu_{fg}+\nu_f$

$R=\dfrac{\overline{R}}{M}$

$\displaystyle x=\dfrac{m_g}{m}$

$\displaystyle \nu=(1-x)\nu_f+x\nu_g$

$\displaystyle PV=n\overline{R}T$

$\displaystyle P\nu=RT$

Phase	$P$	$T$	$\nu$
Compressed Liquid	$P>P_{sat}$	$T < T_{sat}$	$\nu < \nu_f$
Saturated Mixture	$P=P_{sat}$	$T=T_{sat}$	$\nu_f < \nu < \nu_g$
Superheated Vapour	$P < P_{sat}$	$T>T_{sat}$	$\nu>\nu_g$

First Law

$Q=\Delta U+\Delta KE+\Delta PE+W$

$\Delta KE=\frac{1}{2}\sum m_ev^2_e-\frac{1}{2}\sum m_iv^2_i$

$Q=\Delta H+\Delta KE+\Delta PE+W$

$\Delta PE=\sum m_egz_e-\sum m_igz_i$

Work

$\displaystyle W=\int_{V_1}^{V_2}PdV$

$PV^n=C$

\displaystyle w=\dfrac{W}{m}

\displaystyle

\displaystyle

Internal Energy

$\displaystyle U=mu$

$\Delta u=C_{v 0}\Delta T$

\displaystyle \Delta U=\sum m_eu_e-\sum m_iu_i

Enthalpy

$H=mh$

$h=u+P\nu$

$\Delta h=C_{p0}\Delta T$

$\displaystyle H=U+PV$

$\displaystyle \Delta H=\sum m_eh_e-\sum m_ih_i$

Power

$\dot{W}=\dfrac{dW}{dt}$

$\displaystyle\dot{Q}-\dot{W}=\sum\dot{m}_e\left(h_e+\dfrac{v^2_e}{2}+gz_e\right)-\sum\dot{m}_i\left(h_i+\dfrac{v^2_i}{2}+gz_i\right)$

\dot{Q}=\dfrac{dQ}{dt}

Conservation of Mass

\displaystyle\sum\dot{m}_i=\sum\dot{m}_e

Transient Flow Process

\displaystyle Q_{cv}-W_{cv}=\sum m_e\left(h_e+\dfrac{v^2_e}{2}+gz_e\right)-\sum m_i\left(h_i+\dfrac{v^2_i}{2}+gz_i\right)+m_2\left(u_2+\dfrac{v^2_2}{2}+gz_2\right)-m_1\left(u_1+\dfrac{v^2_1}{2}+gz_1\right)

Thermal Efficiency

$\displaystyle\eta=\dfrac{\dot{W}_{net}}{\dot{Q}_{in}}$

$\displaystyle\beta^\prime=\dfrac{\dot{Q}_{heating}}{\dot{W}_{compressor}}$

\displaystyle\beta=\dfrac{\dot{Q}_{cooling}}{\dot{W}_{compressor}}

Heat Engines

$W=Q_H-Q_L$

$\eta=\dfrac{W}{Q_H}=\dfrac{T_H-T_L}{T_H}$

$\beta^\prime=\dfrac{Q_H}{W}=\dfrac{T_H}{T_H-T_L}$

$\left(\dfrac{Q_H}{Q_L}\right)_{rev}=\dfrac{T_H}{T_L}$

$\beta=\dfrac{Q_L}{W}=\dfrac{T_L}{T_H-T_L}$

Entropy

$\displaystyle\oint\dfrac{\delta Q}{T}=\dfrac{Q_H}{T_H}-\dfrac{Q_L,irr}{T_L}<0$

$s=\dfrac{S}{m}$

$TdS=dU+PdV$

$s_2-s_1=C\ln\left(\dfrac{T_2}{T_1}\right)$

$s_2-s_1=C_{v 0}\ln\left(\dfrac{T_2}{T_1}\right)+R\ln\left(\dfrac{\nu_2}{\nu_1}\right)$

$S_{gen}=m_{c.m.}(s_2-s_1)_{c.m.}-\dfrac{Q_{1\to 2}}{T_{surr}}$

$\displaystyle S_2-S_1=\int_1^2\left(\dfrac{\delta Q}{T}\right)_{rev}$

$\displaystyle Q_{1\to 2}=\int_1^2 TdS$

$TdS=dH-VdP$

$s=s_f+xs_{fg}$

$s_2-s_1=C_{p 0}\ln\left(\dfrac{T_2}{T_1}\right)-R\ln\left(\dfrac{P_2}{P_1}\right)$

Entropy Analysis

\displaystyle S_{1\to 2}=(m_2s_2-m_1s_1)_{cv}+\sum m_es_e-\sum m_is_i-\dfrac{Q_{1\to 2}}{T_{surr}}

\displaystyle w_{ideal}=\dfrac{1}{2}(v^2_i-v^2_e)+g(z_i-z_e)-\int_i^e\nu dP

Device Efficiency

$\displaystyle \eta_T=\dfrac{w_T}{w_{T,s}}=\dfrac{h_i-h_e}{h_i-h_{e,s}}$

$\displaystyle \eta_N=\dfrac{v^2_e}{v^2_{e,s}}$

\displaystyle \eta_{P/C}=\dfrac{w_{P/C,s}}{w_{P/C}}=\dfrac{h_i-h_{e,s}}{h_i-h_e}

Mixtures

$\displaystyle c_i=\dfrac{m_i}{m_{tot}}$

$\displaystyle y_i=\dfrac{n_i}{n_{tot}}$

$R_{tot}=\sum_{i=1}^k c_iR_i$

$M_{tot}=\left[\sum_{i=1}^k\dfrac{c_i}{M_i}\right]^{-1}$

$u=\sum_{i=1}^k c_iu_i$

$C_{v 0,tot}=\sum_{i=1}^k c_iC_{v 0,i}$

$s_2-s_1=\ln\left(\dfrac{T_2}{T_1}\right)\sum_{i=1}^k c_iC_{p0,i}-\ln\left(\dfrac{P_2}{P_1}\right)\sum_{i=1}^k c_iR_i$

$\displaystyle \sum_{i=1}^k c_i=1$

$\displaystyle \sum_{i=1}^k y_i=1$

$M_{tot}=\dfrac{m_{tot}}{n_{tot}}$

$P_i=y_iP$

$h=\sum_{i=1}^k c_ih_i$

$C_{p0,tot}=\sum_{i=1}^k c_iC_{p0,i}$

Air Mixtures

$\displaystyle P=P_a+P_v$

$\displaystyle \phi=\dfrac{m_v}{m_g}=\dfrac{P_v}{P_g}$

$h=C_{p0,a}T+\omega h_v$

$\dot{V}_{tot}=\dfrac{\dot{m}_{tot}}{\rho_{tot}}$

$\displaystyle \omega=0.622\dfrac{P_v}{P_a}$

$\displaystyle u=C_{v 0,a}T+\omega u_v$

$\dot{m}_{tot}=\dot{m}_a(1+\omega)$

$\omega_1=\dfrac{C_{p0,a}(T_2-T_1)+\omega_2h_{fg,2}}{h_{g,1}-h_{f,2}}$

Combustion

$C_xH_y+a(O_2+3.76N_2)\to bCO_2+cH_2O+3.76aN_2$

$\displaystyle AF_{mass}=\dfrac{m_{air}}{m_{fuel}}$

$\displaystyle AF_{mass}=AF_{mol}\dfrac{M_{air}}{M_{fuel}}$

$H_R=\sum_Rn_i\overline{h}_i$

$\overline{h}_{T,P}=\overline{h}^0_f+\Delta\overline{h}_{T,P}$

$\displaystyle U_{RP}=\sum_Pn_e(\bar{h}_f^0+\Delta\bar{h}-P\bar{\nu})_e-\sum_Rn_i(\bar{h}_f^0+\Delta\bar{h}-P\bar{\nu})_i$

$\displaystyle n_{air}=4.76(x+0.25y)$

$\displaystyle AF_{mol}=\dfrac{n_{air}}{n_{fuel}}$

$Q+H_R=H_P$

$H_P=\sum_Pn_e\overline{h}_e$

$\displaystyle H_{RP}=\sum_Pn_e(\bar{h}_f^0+\Delta\bar{h})_e-\sum_Rn_i(\bar{h}_f^0+\Delta\bar{h})_i$