The continuity equation is simply a mathematical expression of the principle of conservation of
mass. For a control volume that has a single inlet and a single outlet, the principle of
conservation of mass states that, for steady-state flow, the mass flow rate into the volume must
equal the mass flow rate out. The continuity equation for this situation is expressed by Equation
(rAv)inlet = (rAv)outlet
For a control volume with multiple inlets and outlets, the principle of conservation of mass
requires that the sum of the mass flow rates into the control volume equal the sum of the mass
flow rates out of the control volume. The continuity equation for this more general situation is
expressed by Equation 3-6.
One of the simplest applications of the continuity equation is determining the change in fluid
velocity due to an expansion or contraction in the diameter of a pipe.
Continuity Equation - Piping Expansion
Steady-state flow exists in a pipe that undergoes a gradual expansion from a diameter of
6 in. to a diameter of 8 in. The density of the fluid in the pipe is constant at 60.8 lbm/ft3.
If the flow velocity is 22.4 ft/sec in the 6 in. section, what is the flow velocity in the 8
From the continuity equation we know that the mass flow rate in the 6 in. section must
equal the mass flow rate in the 8 in. section. Letting the subscript 1 represent the 6 in.
section and 2 represent the 8 in. section we have the following.