Equations
Fluid Properties
\nu=\dfrac{V}{m}
P=\dfrac{F}{A}
Pressure
P_fA=P_0A+mg
P_f=P_0+\sum_0^n\rho_iz_ig
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
Internal Energy
\displaystyle U=mu
\Delta u=C_{v 0}\Delta T
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)
Conservation of Mass
Transient Flow Process
Thermal Efficiency
\displaystyle\eta=\dfrac{\dot{W}_{net}}{\dot{Q}_{in}}
\displaystyle\beta^\prime=\dfrac{\dot{Q}_{heating}}{\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
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}}
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