When conditions inside a control volume change with time, energy can accumulate as well as cross the boundary. Let $E_{CV}$ denote the total energy currently stored inside the control volume.

A general fixed-control-volume balance can be written

$$\frac{dE_{CV}}{dt}=\dot Q-\dot W_s+\sum_{in}\dot m\left(h+\frac{V^2}{2}+gz\right)-\sum_{out}\dot m\left(h+\frac{V^2}{2}+gz\right).$$

For many tank-filling and tank-emptying problems, kinetic and potential energies are negligible. The balance then tracks the changing internal energy stored inside the tank together with enthalpy carried by entering or leaving streams.

Consider an adiabatic rigid tank initially empty that is filled from a large supply. With no outlet and no shaft work,

$$\Delta U_{tank}=m_{in}h_{in}$$

if changes in kinetic and potential energy are negligible.

This differs from a closed-system relation because matter crossing the control surface carries flow energy. Transient analysis is therefore essential for filling, blowdown, charging and startup processes even when the final state is steady.