Universal Gas Law

$PV=n \bar RT = mRT$

$Pv = RT$

The First Law

Always true:

$Q-W=(U_2-U_1)$

$Q-W=m(u_2-u_1)$

For cycles:

$\sum Q = \sum W$

For constant volume (no work done):

$Q = m(u_2-u_1)$

Work Done

Always true:

$W_{12}=\int^{V_2}_{V_1} P dV$

True for constant $P$ :

$W_{12}=P(V_2-V_1)$

True for constant $T$ :

(PV is constant)

$W_{12}=c \ln \frac{V_2}{V_1}$

True when adiabatic & reversible:

(Isentropic)

$W_{12}=\frac{P_2V_2-P_1V_1}{1-\gamma}$

Shaft Work

$\dot W_{sh}=\omega (Torque)$

$\omega = 2\pi f = \frac{2\pi}{T}$

Constant Pressure

$\frac{T_1}{T_2}=\frac{V_1}{V_2}$

Constant Volume

$\frac{T_1}{T_2}=\frac{P_1}{P_2}$

Perfect, Adiabatic & Reversible (Isentropic)

$Pv^\gamma$ is constant:

$\frac{P_1}{P_2}=(\frac{v_2}{v_1})^\gamma$

$Tv^{\gamma -1}$ is constant:

$\frac{T_1}{T_2}=(\frac{v_2}{v_1})^{\gamma -1}$

$\frac{T}{P^{\frac{\gamma -1}{\gamma}}}$ is constant:

$\frac{T_1}{T_2}=(\frac{P_1}{P_2})^{\frac{\gamma -1}{\gamma}}$

Enthalpy

$h=u+Pv$

$h = u + RT$

Perfect Gases & Temperature

Change in energy, $u$ :

$u_2-u_1=C_V(\Delta T)$

Change in enthalpy, $h$ :

$h_2-h_1=C_P(\Delta T)$

Perfect gas constant, $R$ :

$R=C_P-C_V$

Perfect gas ratio, $\gamma$ :

$\gamma =\frac{C_P}{C_V}$

$C_P =\frac{R\gamma}{\gamma -1}$

Steady Flow Energy Equation (SFEE)

$\dot Q-\dot W_{sh}=\sum\dot m[h+\frac{c^2}{2}+gZ]_{out}-\sum\dot m[h+\frac{c^2}{2}+gZ]_{in}$

Mass flow rate, $\dot m$ :

$\dot m = \rho cA$

$\dot m=\frac{cA}{v}$

Volume flow rate, $\dot V$ :

$\dot V=\frac{\dot m}{\rho}$

$\dot v = cA$

$\rho$ is the density
$c$ is the speed
$A$ is the cross-sectional area of the duct
$v$ is the specific volume

Wet Vapour

Dryness fraction, $x$ :

$x=\frac{X-X_f}{X_g-X_f}$

Wet Property, $X$ :

$X = x_f+x X_{fg}$

Clausius’ Inequality

For a reversible cycle or process:

$\oint\frac{dQ}{T}=0$

$S_2-S_1=\int^2_1\frac{dQ}{T}$

For an irreversible cycle of process:

$\oint\frac{dQ}{T}<0$

$S_2-S_1>\int^2_1\frac{dQ}{T}$

Combining 1st & 2nd Laws

$T ds = du + P dv$

$s_2-s_1 = \int^2_1 \frac{1}{T} du + \int^2_1 \frac{P}{T} dv$

$T ds = dh - v dP$

$s_2-s_1=\int^2_1\frac{1}{T} dh - \int^2_1\frac{v}{T}dP$

These equations are valid for both reversible and irreversible processes, as all quantities are properties.

Perfect Gas Processes

$s_2-s_1 = C_V \ln (\frac{T_2}{T_1}) + R \ln (\frac{v_2}{v_1})$

$s_2-s_1=C_P \ln (\frac{T_2}{T_1})-R \ln (\frac{P_2}{P_1})$

$s_2-s_1 = C_P \ln (\frac{v_2}{v_1})+C_V \ln (\frac{P_2}{P_1})$

Again, these are applicable to both reversible and irreversible processes so long as the gases are perfect.

Efficiencies

General efficiency, $\eta$ :

$\eta=\frac{output}{input}$

Thermodynamic Efficiencies, $\eta$ :

$\eta = \frac{W_{net}}{Q_H}$

$\eta=1-\frac{Q_C}{Q_H}$

Thermal (reversible) efficiency, $\eta_{th}$ :

$\eta_{th}=1-\frac{T_C}{T_H}$

Isentropic Efficiencies, $\eta_S$

Turbine/Engine

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$\eta_S-\frac{T_1-T_2}{T_1}{T_{2S}}$

$\eta_S=\frac{h_1-h_2}{h_1-h_{2S}}$

Compressor/Pump

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$\eta_S=\frac{T_{2S}-T_1}{T_2-T_1}$

$\eta_S=\frac{h_{2S}-h_1}{h_2-h_1}$

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