THE CONTINUITY EQUATION

The continuity equation is a statement of conservation of mass (covered in this blog entry as Equation 4.)

The flow quantity N becomes m, and n = m/m = 1. Thus, the general conservation equation becomes:

For a system of particles, since mass is neither created nor destroyed mass remains constant. Thus:

This means that the mass entering the control volume is equal to the summation of the mass stored and the mass leaving. For a steady flow with no storage, the mass entering is equal to the mass leaving. Thus, for a steady flow, equation 1 is expressed as:

For incompressible fluids (liquids), ρ is a constant, and the following equation is used for steady incompressible flow:

From Equation 3 we learn that:

• the volume of fluid entering the control volume per unit time is equal to the amount leaving.
• a velocity distribution normal to the control surface must be integrated over the cross-sectional area at inlets and exits to the control volume.

However, for many flow situations, a velocity distribution may not be known or derivable. Thus, in such cases it is more convenient to use average velocity at a cross section that is independent of area – assuming that it is 1-dimensional flow or at least flow that can be estimated as 1-dimensional. Therefore, by definition:

Where V = average velocity. The product AV = Q is known as the volume flow rate with dimensions of L³/T (m³/s or ft³/s). Also used in fluid mechanics is the term ṁ, defined as the mass flow rate, where ṁ = ρQ = ρAV. The mass flow rate has dimensions of M/T (kg/s or slug/s).

To understand average velocity, let us consider a flow in the pipe below with a velocity distribution, V_z:

where R is the pipe radius. The V_z expression or instantaneous velocity, the gives the velocity at any radial location within the pipe. At a control surface normal to the axial direction, V_z = V_n and the average velocity is:

dA = r dr dθ is used in the integral since we are dealing with cylindrical coordinates.

This result means that we can replace the velocity distribution with an average velocity so that the volume flow rate remains unchanged as shown in the velocity distribution figure above. In many practical situations, the volume flow rate and area are known but the velocity distribution isn’t but average velocity can be determined.

When we use the average velocity, the continuity equation becomes:

For steady flow systems where no mass is stored,