5.6: Key equations

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Constant-pressure and constant-volume specific heats

Constant-pressure specific heat	$C_p=\left(\displaystyle\frac{\partial\ h}{\partial\ T}\right)_p$
Constant-volume specific heat	$C_v=\left(\displaystyle\frac{\partial\ u}{\partial\ T}\right)_v$
Relations between $C_p$ and $C_v$ for ideal gases	$k=\displaystyle\frac{C_p}{C_v} \qquad \qquad C_p=C_v+R$ \[C_v=\displaystyle\frac{R}{k-1} \qquad C_p=\displaystyle\frac{kR}{k-1}\]

Specific enthalpy

Change in specific enthalpy	$\Delta h = h_2-h_1$
Change in specific enthalpy for ideal gases	$\Delta h = h_2-h_1 = C_p(T_2-T_1)$ (assuming constant $C_p$ in the temperature range)
Relation between $\Delta h$ and $\Delta u$ for solids and liquids	$\Delta h \approx\Delta u\approx C_p(T_2-T_1)$

Mass conservation equations in a control volume

Volume flow rate	$\dot{\mathbb{V}}=\displaystyle\frac{d\mathbb{V}}{dt}=V_{avg,\ n}A=\dot{m}v$
Mass flow rate	$\dot{m}=\displaystyle\frac{dm}{dt}=\rho\ V_{avg,\ n}A=\rho\dot{\mathbb{V}}$
Transient flow	$\displaystyle\frac{dm_{CV}}{dt}=\sum{\dot{m}}_i-\sum{\dot{m}}_e\neq0$
Steady flow	$\displaystyle\frac{dm_{CV}}{dt}=\sum{\dot{m}}_i-\sum{\dot{m}}_e=0$

Energy conservation equations in a control volume

Transient flow	$\displaystyle\frac{dE_{CV}}{dt}\neq0$ \[\begin{align} \displaystyle\frac{dE_{CV}}{dt}={\dot{Q}}_{cv}-{\dot{W}}_{cv} &+\sum{{\dot{m}}_i(h_i+\frac{1}{2}V_i^2+gz_i)} \\&-\sum{{\dot{m}}_e(h_e+\frac{1}{2}V_e^2+gz_e)} \end{align}\]
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Transient flow

$\displaystyle\frac{dE_{CV}}{dt}\neq0$

\[\begin{align*} \displaystyle\frac{dE_{CV}}{dt}={\dot{Q}}_{cv}-{\dot{W}}_{cv} &+\sum{{\dot{m}}_i(h_i+\frac{1}{2}V_i^2+gz_i)} \\&-\sum{{\dot{m}}_e(h_e+\frac{1}{2}V_e^2+gz_e)} \end{align*}\]

Steady flow

$\displaystyle\frac{dE_{CV}}{dt}=0$

\[{\dot{Q}}_{cv} +\sum

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Mass and energy conservation equations for steady-state, steady-flow (SSSF) devices

SSSF device	Assumptions	Mass conservation	Energy conservation
Expansion device	Adiabatic flow; Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies	${\dot{m}}_i={\dot{m}}_e$	$h_i=h_e$
Nozzle and diffuser	Adiabatic flow; Negligible work transfer with the surroundings; Negligible change in potential energy	${\dot{m}}_i={\dot{m}}_e$	$h_i+\displaystyle\frac{1}{2}V_i^2=h_e+\displaystyle\frac{1}{2}V_e^2$
Mixing chamber	Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies	$\sum{{\dot{m}}_i=\sum{\dot{m}}_e}$	${\dot{Q}}_{cv}+\sum{{\dot{m}}_ih_i=\sum{{\dot{m}}_eh_e}}$
Heat exchanger	Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies	${\dot{m}}_i={\dot{m}}_e$ (for each of the hot and cold streams, separately)	${\dot{Q}}_{cv}+\sum{{\dot{m}}_ih_i=\sum{{\dot{m}}_eh_e}}$
Turbine	Adiabatic flow; Negligible changes in kinetic and potential energies	${\dot{m}}_i={\dot{m}}_e=\dot{m}$	${\dot{W}}_{shaft}=\dot{m}(h_i-h_e)$
Compressor	Adiabatic flow; Negligible changes in kinetic and potential energies	${\dot{m}}_i={\dot{m}}_e=\dot{m}$	${\dot{W}}_{shaft}=\dot{m}(h_e-h_i)$