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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