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The application of Appropriate Technology

# Part 3: Derivation of the Continuity Equation

Consider an incompressible fluid (water is almost incompressible) flowing along a pipe, as in Figure 1.

Figure 1

Its volume (V) is given by:

$V=A.L$

Therefore the volume passing per second (the volumetric flow rate Q) is given by:

$Q = V/t = A.L/t$

But we can write velocity as distance moved/time (see Equation (1)), so we can replace L/t by v:

$Q = A.v$

(9)

This is the FLOW EQUATION.

Now consider pipes of different areas A1 and A2 as shown in Figure 2.

Figure 2

The volumetric flow rate (Q) must be the same for both pipes, because we cannot gain or lose any fluid.

Therefore from Equation (8) above:

$Q = A_1.v_1 = A_2.v_2$

(10)

This is the CONTINUITY EQUATION and it is true for any number of changes in pipe diameter for a single pipe arrangement (a single flow path).