Second Law for Steady-Flow Devices

Similar to the energy balance for open systems, where had:

$$\frac{d{E}_{\text{CV}}}{dt}=\dot{Q}-\dot{W}+{\dot{m}}_{\text{in}}{\left(u+\frac{1}{2}{V}^2+gz\right)}_{\text{in}}-{\dot{m}}_{\text{out}}{\left(u+\frac{1}{2}{V}^2+gz\right)}_{\text{out}}$$

we can also write an entropy balance for open systems.

Recall that $\Delta S_{\text{sys}}=\sum \frac{Q_k}{T_k}+S_{\text{gen}}$ and $S_{\text{gen}}\ge 0$. The entropy balance for an open system can then be written as:

$$\frac{d{S}_{\text{CV}}}{dt}=\sum \frac{{\dot{Q}}_k}{T_k}+{\dot{m}}_{\text{in}}{s}_{\text{in}}-{\dot{m}}_{\text{out}}{s}_{\text{out}}+{\dot{S}}_{\text{gen}}$$

where ${\dot{S}}_{\text{gen}}\ge 0$.