Continuity Equation

The continuity equation is merely a mathematical expression of the principle of conservation of mass. For a control volume which has a single inlet and a single outlet, the principle of conservation of mass defines that, for steady-state flow, the mass flow rate into the volume should equivalent the mass flow rate out. The continuity equation for this condition is stated by the equation shown below.

m_inlet - m_outlet

(ρAv)_inlet = (ρAv)_outlet

For a control volume with multiple inlets and outlets, the principle of conservation of mass needs that the sum of the mass flow rates into the control volume equivalent the sum of the mass flow rates out of the control volume. The continuity equation for this common condition is stated by the equation shown below.

Σ m_inlets = Σ m_outlets

One of the easiest applications of the continuity equation is establishing the change in fluid velocity due to an expansion or contraction in the diameter of a pipe.

Therefore by using the continuity equation, we find that the rise in pipe diameter from 6 to 8 inches caused a reduction in flow velocity ranging from 22.4 to 12.6 ft/sec. The continuity equation can also be employed to illustrate that a reduction in pipe diameter will cause a rise in flow velocity.

Example: Continuity Equation - Centrifugal Pump

The inlet diameter of the reactor coolant pump shown in figure below is 28 in. whereas the outlet flow via the pump is 9200 lbm/sec. The density of the water is equal to 49 lbm / ft³. Determine the velocity at the pump inlet?

Solution:

A_inlet = 4.28 ft²

m_inlet = m_outlet = 9200 lbm / sec