Derivation of Continuity Equation

The continuity equation says that the outcome of the pipe's cross-sectional area and fluid speed is always constant at any point along the pipe. This product equals the volume flow per second, often known as the flow rate. The continuity equation is as follows:

R = A v = constant

Where,

• R =volume flow rate
• A =area of the flow
• v =velocity of the flow

Continuity Equation Assumption

The following are the continuity equation's assumptions:

• The tube has a single entrance and exit.
• The fluid passing through the tube is viscous.
• The flow is inexhaustible.
• The flow of fluid is constant.

Consider how the fluid flows in the tube for a small period of time. Assume that a small span of time is denoted by t. At the end of the pipe, the fluid will travel a distance of x1 at a velocity of v1.

At this point, the fluid will have travelled the following distance:

Δx1 = v1Δt

The volume of fluid that will flow into the pipe at the lower end of the pipe will now be:

V = A1 Δx1 = A1 v1 Δt

It is well understood that mass (m) = density ()x volume (V). As a result, the fluid mass in the X1 area will be

Δm1= Density × Volume

=> Δm1 = ρ1A1v1Δt ——–(Equation 1)

The mass flux must now be estimated at the low end. Mass flux is the mass of a fluid flowing through any cross-sectional area per unit of time. The mass flux for the lower end with cross-sectional area A1 will be:

Δm1/Δt = ρ1A1v1 ——–(Equation 2)

Similarly, the upper-end mass flow:

Δm2/Δt = ρ2A2v2 ——–(Equation 3)

Here, v2 = velocity of the fluid through the top end of the pipe, i.e. via x2, in time (t), and A2 is the upper end's cross-sectional area. As far as the flow is steady, the density of the fluid between the lower and higher ends of the pipe stays constant. As a result, the mass flux at the lower end of the pipe equals the mass flux at the higher end of the pipe, resulting in Equation 2 = Equation 3.

Thus,

ρ1A1v1 = ρ2A2v2 ——–(Equation 4)

This can be written as:

ρ A v = constant

In fluid dynamics, the equation establishes the law of mass conservation. Furthermore, if the fluid is incompressible, the density will stay constant throughout the continuous flow. As a result, 1+1=2.

As a result, Equation 4 may now be expressed as follows:

A1 v1 = A2 v2

This equation can be written as:

A v = constant

Now, if R = volume flow rate, the above equation can be written as:

R = A v = constant

This is the derivation of the continuity equation.