Question

In turbulent flow, the loss of pressure head is approximately (where v is the velocity of flow)

• A. Directly proportional to v
• B. Inversely proportional to v
• C. Directly proportional to \(\text{v}^2\)
• D. Inversely proportional to \(\text{v}^2\)

Hint:

Head loss scaling rules of thumb: - Laminar pipe flow: \(h_f \propto v^1\) - Turbulent pipe flow: \(h_f \propto v^2\) (or more precisely \(v^{1.75}\) to \(v^2\) depending on surface roughness). Therefore, doubling the velocity in a turbulent system increases pumping pressure head requirements by nearly four times.

Answer 1

The Correct Option is C

Concept: The loss of pressure head (\(h_f\)) caused by fluid friction over a pipe length \(L\) is evaluated using the Darcy-Weisbach formula: \[ h_f = \frac{f \cdot L \cdot v^2}{2 \cdot g \cdot d} \] Where:
• \(f\) = Darcy friction factor.
• \(L\) = Length of the pipe segment.
• \(d\) = Internal diameter of the pipe line.
• \(v\) = Mean velocity of fluid flow.
• \(g\) = Acceleration due to gravity. Let us examine how the head loss scales with mean velocity in different flow regimes:

Step 1: Comparing with Laminar Flow.
In laminar flow, the friction factor is inversely proportional to velocity (\(f = \frac{64}{Re} = \frac{64\mu}{\rho v d}\)). Substituting this into the Darcy-Weisbach equation shows that head loss scales linearly with velocity: \[ h_f = \left(\frac{64\mu}{\rho v d}\right) \frac{L v^2}{2gd} = \frac{32\mu L v}{\rho g d^2} \quad \Rightarrow \quad h_f \propto v \]

Step 2: Evaluating the Turbulent Flow Condition.
In turbulent flow, high-frequency velocity fluctuations and eddy mixing dominate momentum transfer, increasing shear resistance. For fully turbulent flow or flow in rough pipes, the friction factor \(f\) becomes nearly constant, becoming independent of the Reynolds number.

Step 3: Deducing the velocity proportionality.
Treating the friction factor \(f\), length \(L\), diameter \(d\), and gravity \(g\) as constant parameters for a given turbulent pipe flow system, the Darcy-Weisbach equation simplifies to: \[ h_f = \left( \frac{f \cdot L}{2 \cdot g \cdot d} \right) \cdot v^2 = \text{Constant} \cdot v^2 \] This demonstrates that in turbulent flow, the loss of pressure head is directly proportional to the square of the flow velocity (\(v^2\)). This corresponds to Option (3).