Part 4: The Continuity Equation for Multiple Pipe System

The rule for multiple flow paths for incompressible fluids is:

$\text{Sum of the Flows in} = \text{Sum of the Flows out}$

This is written mathematically as:

$\sum Q_{in}=\sum Q_{out}$ (10)

Consider the pipe system shown below (in section) in Figure 3:

Figure 3

In this case the flow in is given by:

$\sum Q_{in}=A_1v_1$

And the flow out is given by:

$\sum Q_{out}=A_2v_2+A_3v_3$

So from Equation X we can write:

$A_1v_1 = A_2v_2 + A_3v_3$ (11)

This is true for any number of flows in and out.