Continuity Equation

Consider the elementary control volume shown in the figure below. In this figure, distance x is measured along the channel, as shown. At the midpoint of the control volume the flow and total flow area are denoted Q(x,t) and A_T, respectively. The total flow area is the sum of active area A and off-channel storage area S.

Elementary Control Volume for Derivation of Continuity and Momentum Equations.

Conservation of mass for a control volume states that the net rate of flow into the volume be equal to the rate of change of storage inside the volume. The rate of inflow to the control volume may be written as:

1)	$\begin{array}{l}\displaystyle \displaystyle Q- \frac{\partial Q}{\partial x} \frac{\Delta x}{2}\end{array}$

the rate of outflow as:

2)	$\begin{array}{l}\displaystyle \displaystyle Q+\frac{\partial Q}{\partial x} \frac{\Delta x}{2}\end{array}$

and the rate of change in storage as:

3)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial A_T}{\partial t} \Delta x\end{array}$

Assuming that $\begin{array}{l}\Delta x\end{array}$ is small, the change in mass in the control volume is equal to:

4)	$\begin{array}{l}\displaystyle \displaystyle \rho \frac{\partial A_T}{\partial t} \Delta x = \rho \left[ \left( Q-\frac{\partial Q}{\partial x} \frac{\Delta x}{2} \right) - \left( Q+ \frac{\partial Q}{\partial x} \frac{\Delta x}{2} \right) +Q_l \right]\end{array}$

Symbol	Description	Units
$\begin{array}{l}Q_i\end{array}$	is the lateral flow entering the control volume and ρ is the fluid density.

Simplifying and dividing through by $\begin{array}{l}\rho\end{array}$ yields the final form of the continuity equation:

5)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial A_T}{\partial t} + \frac{\partial Q}{\partial x} = q_t\end{array}$

Symbol	Description	Units
$\begin{array}{l}q_t\end{array}$	lateral inflow per unit length