Chapter 4, Problem 4.1E

To determine

The method by which the control volume energy rate balance accounts for work done as mass flows across a boundary.

Answer to Problem 4.1E

  $\frac{d{E}_{cv}}{dt}=\dot{Q}-\dot{W}+{\dot{m}}_i\left(u_i+\frac{V_i^2}{2}+g{z}_i\right)-{\dot{m}}_e\left(u_e+\frac{V_e^2}{2}+g{z}_e\right)$

Explanation of Solution

The amount of a substance flowing through a channel per unit of time is known as the mass flow rate. Since energy is an extensive property, it can neither be created nor be destroyed. The energy rate balance of a control volume can be obtained using energy balance criteria and doing some modifications in the closed system energy rate balance equation as follows:

  $\frac{d{E}_{cv}}{dt}=\dot{Q}-\dot{W}+{\dot{m}}_i\left(u_i+\frac{V_i^2}{2}+g{z}_i\right)-{\dot{m}}_e\left(u_e+\frac{V_e^2}{2}+g{z}_e\right)$

Here, $E_{cv}$ represents the energy of the control volume, $\dot{Q}$ represents the net rate of energy transfer by heat, $\dot{W}$ is work across the boundary of the control volume, ${\dot{m}}_i$ and ${\dot{m}}_e$ represent the mass flow rates at the inlet and the exit respectively, $V_i$ represents is the velocity at the inlet, $z_i$ represents the elevation at the inlet, $V_e$ represents the velocity at the exit and $z_e$ is the elevation at the exit.