5. The First Law of Thermodynamics for a Control Volume

5.5 Key equations

Constant-pressure and constant-volume specific heats

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

[latex]C_v=\displaystyle\frac{R}{k-1}  \qquad  C_p=\displaystyle\frac{kR}{k-1}[/latex]

 

Specific enthalpy

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

 

Mass conservation equations in a control volume

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

 

Energy conservation equations in a control volume

Transient flow [latex]\displaystyle\frac{dE_{CV}}{dt}\neq0[/latex]

[latex]\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*}[/latex]

Steady flow [latex]\displaystyle\frac{dE_{CV}}{dt}=0[/latex]

[latex]{\dot{Q}}_{cv}  +\sum{{\dot{m}}_i\left(h_i+\displaystyle\frac{1}{2}V_i^2+gz_i\right) \\={\dot{W}}_{cv} +\sum{{\dot{m}}_e\left(h_e+\displaystyle\frac{1}{2}V_e^2+gz_e\right)}}[/latex]

 

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

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