8.1 Continuity equation

Continuity equation

The continuity equation is an expression for the condition that mass is not created or destroyed during a flow process.

Figure 8b Example of one-dimensional flow.

Assuming that the flow is steady, the mass flow ṁ must be of equal magnitude everywhere along the pipe or the flow tube. In the case of one-dimensional flow in figure 8b,

ṁ = ρ_I * v_I * A_I = ρ_II * v_II * A_II      (Equ. 8a)
or for an incompressible liquid flow,
Q =v_I * A_I = v_II * A_II         (Equ. 8b)

where
Q = volume flow (m³/s)
v = Q/A = flow velocity (m/s)
A = cross-sectional area (m²)

When the cross-sectional area in a pipe reduces, then, to the continuity equation, the flow velocity increases. If the area is halved, the velocity doubles and so on.

Figure 8c Branching

Since there is no increase in mass at the point of branching, the mass flow entering will equal the total mass discharging per unit of time. Using the symbols of figure 8c.

Q_I = Q_II + Q_III      (Equ. 8c)

or

v_I * A_I = v_II * A_II + v_III * A_III      (Equ. 8d)