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 $C_p=\left(\displaystyle\frac{\partial\ h}{\partial\ T}\right)_p$ 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*} Steady flow $\displaystyle\frac{dE_{CV}}{dt}=0$ ${\dot{Q}}_{cv} +\sum{{\dot{m}}_i\left(h_i+\displaystyle\frac{1}{2}V_i^2+gz_i\right) \\={\dot{W}}_{cv} +\sum{{\dot{m}}_e\left(h_e+\displaystyle\frac{1}{2}V_e^2+gz_e\right)}}$

 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 ${\dot{m}}_i={\dot{m}}_e=\dot{m}$ ${\dot{W}}_{shaft}=\dot{m}(h_e-h_i)$