First Law of Thermodynamics for Open Systems

For an open system where mass can transfer across boundaries, mass coming in and out of the system carries its energy with it ($e=u+\text{KE}+\text{PE}$). Note that for an open system, $m$ is not constant so we need to account for its rate of change, $\dot{m}$. This is called the mass flow rate.

Thus, the first law of thermodynamics for an open system can be written as:

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

When we have flow processes, where there’s mass flow across the boundaries of a control volume, it can be more convenient to use enthalpy, as it accounts for flow work done by the fluid entering or leaving the control volume.

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