Derive an expression for the loss of head due to friction in a pipeline in terms of the velocity head, assuming that the frictional resistance per unit area of pipe wall is proportional to the square of the mean velocity of flow

To derive an expression for the loss of head due to friction in a pipeline, we start by considering the relationship between frictional resistance and flow velocity. In fluid mechanics, it is often assumed that the frictional resistance per unit area of the pipe wall is proportional to the square of the mean velocity of flow. This relationship can be expressed mathematically and leads us to the desired expression for head loss.

Understanding the Basics of Head Loss

Head loss due to friction in a pipeline is a critical concept in fluid dynamics, particularly when analyzing the efficiency of fluid transport systems. The head loss (\(h_f\)) can be defined as the energy loss per unit weight of fluid due to friction as it flows through the pipe.

Frictional Resistance and Velocity

Let’s denote the mean velocity of flow in the pipe as \(V\). According to the problem statement, the frictional resistance per unit area (\(f\)) can be expressed as:

• Frictional Resistance: \(f \propto V^2\)

This implies that we can write:

• Frictional Resistance: \(f = k V^2\)

Here, \(k\) is a proportionality constant that depends on factors such as the roughness of the pipe and the fluid properties.

Applying the Darcy-Weisbach Equation

To quantify the head loss due to friction, we can use the Darcy-Weisbach equation, which states:

• Head Loss: \(h_f = \frac{f L}{D} \cdot \frac{V^2}{2g}\)

In this equation:

• \(h_f\) = head loss due to friction
• \(L\) = length of the pipe
• \(D\) = diameter of the pipe
• \(g\) = acceleration due to gravity

Substituting for Frictional Resistance

Now, substituting our expression for \(f\) into the Darcy-Weisbach equation gives:

• Head Loss: \(h_f = \frac{k V^2 L}{D} \cdot \frac{V^2}{2g}\)

This simplifies to:

• Head Loss: \(h_f = \frac{k L}{2g D} V^4\)

Expressing Head Loss in Terms of Velocity Head

The velocity head (\(h_v\)) is defined as:

• Velocity Head: \(h_v = \frac{V^2}{2g}\)

We can relate the head loss to the velocity head by substituting \(h_v\) into our expression for \(h_f\). Rearranging gives:

• Velocity Head Relation: \(V^2 = 2g h_v\)

Substituting this back into our head loss equation results in:

• Head Loss: \(h_f = \frac{k L}{D} h_v^2\)

Final Expression

Thus, the final expression for the loss of head due to friction in a pipeline, in terms of the velocity head, is:

• Loss of Head: \(h_f = \frac{k L}{D} h_v^2\)

This equation shows that the head loss due to friction is proportional to the square of the velocity head, scaled by factors that depend on the length and diameter of the pipe, as well as the frictional characteristics of the flow. Understanding this relationship is crucial for designing efficient piping systems and predicting energy losses in fluid transport.