Chapter 4.12, Problem 1E

To determine

The control volume energy rate balance for work.

Explanation of Solution

The below figure shows the control volume for applying the energy principle.

Figure-(1)

Write the expression for the conservation of the energy principle to the control volume.

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

Here, the mass flow rate is $\dot{m}$, the velocity is $V$, the internal energy is $u$, the acceleration due to gravity is $g$, the datum height is $z$, the time rate change of the energy contained within the control volume at time $t$ is $\frac{d{\dot{E}}_{CV}}{dt}$, the net rate at which the energy is being transferred out by work transfer at time $t$ is $\dot{Q}$.

Here, the term $\dot{W}$ in Equation (I) is the work and it consists of two separate contributions.

1. 1) Work associated with the fluid pressure as mass transfer occurs at the inlet and exit.
2. 2) Work associated with other contributions such as shaft work, displacement of boundary and electrical effects.

Write the expression for general energy rate balance for work as follows.

$\begin{array}{l}\dot{W}+\left(P_i{A}_i\right)V_i={\dot{W}}_{CV}+\left(P_e{A}_e\right)V_e\\ \dot{W}={\dot{W}}_{CV}+\left(P_e{A}_e\right)V_e-\left(P_i{A}_i\right)V_i\end{array}$

Here, pressure at inlet $i$ is $P_i$, area at inlet $i$ is $A_i$, velocity at inlet $i$ is $V_i$, pressure at exit $e$ is $P_e$, area at exit $e$ is $A_e$, velocity at exit $e$ is $V_e$, the time rate of the energy transfer by the control volume at exit $e$ is $\left(P_e{A}_e\right)V$, the time rate of energy transfer into the control volume at inlet $i$ is $\left(P_i{A}_i\right)V_i$, the other forms of work such as shaft work, displacement of boundary and electrical effects is ${\dot{W}}_{CV}$.

Thus, the control volume energy rate balance for the work can be accounted using equation.

$\dot{W}={\dot{W}}_{CV}+\left(P_e{A}_e\right)V_e-\left(P_i{A}_i\right)V_i$