Question

State eqn of continuity and prove it

Answer 1

The continuity equation is an expression of a fundamental conservation principle, namely, that of mass conservation. It is a statement that fluid mass is conserved: all fluid particles that flow into any fluid region must flow out. To obtain this equation, we consider a cubical control volume inside a fluid.

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


An incompressible fluid flowing along a pipe.
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.
Pipes of different areas A1 and A2
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).