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] |
