Question

A pipe friction test shows that, over the range of speeds used for the test, the non-dimensional friction factor varies inversely with Reynolds number. From this, one can conclude that the

• A. Pipe must be smooth
• B. Flow must be laminar
• C. Flow must be turbulent
• D. Pipe must be rough

Hint:

Friction factor equations to remember: - Laminar Flow: \(f = \frac{64}{Re}\) (Purely inverse linear relationship, independent of pipe roughness). - Turbulent Flow (Smooth pipe): \(f = \frac{0.3164}{Re^{0.25}}\) (Blasius equation, much weaker dependence on $Re$).

Answer 1

The Correct Option is B

Concept: The mechanical energy loss due to friction in a fully developed pipe flow is modeled using the Darcy-Weisbach equation. This equation relates head loss (\(h_f\)) to the non-dimensional friction factor (\(f\)): \[ h_f = \frac{f \cdot L \cdot v^2}{2 \cdot g \cdot d} \] The characteristics of the friction factor \(f\) vary depending on the flow regime, which is determined by the Reynolds Number (\(Re = \frac{\rho v d}{\mu}\)). Let us analyze the relationship between the friction factor and the Reynolds number across different flow regimes:

Step 1: Analyzing the Laminar Flow Regime (\(Re < 2000\)).
In laminar pipe flow, the velocity profile is parabolic, and viscous shear forces dominate. The exact analytical solution derived from the Hagen-Poiseuille equation yields a direct equation for head loss: \[ h_f = \frac{32 \mu L v}{\rho g d^2} \] Equating this analytical head loss to the empirical Darcy-Weisbach equation allows us to isolate the friction factor \(f\): \[ \frac{f L v^2}{2 g d} = \frac{32 \mu L v}{\rho g d^2} \quad \Rightarrow \quad f = \frac{64}{\left(\frac{\rho v d}{\mu}\right)} = \frac{64}{Re} \] This mathematical result proves that in the laminar regime, the friction factor \(f\) is inversely proportional to the Reynolds number: \[ f \propto \frac{1}{Re} \] Crucially, this relationship depends solely on the Reynolds number and is entirely independent of the physical surface roughness of the pipe wall.

Step 2: Analyzing the Turbulent Flow Regime (\(Re > 4000\)).
In turbulent flow, momentum transfer is dominated by chaotic eddy currents. The friction factor becomes a complex function of both the Reynolds number and the relative pipe roughness (\(\epsilon/d\)), as described by the Colebrook-White equation or the Moody diagram: \[ \frac{1}{\sqrt{f}} = -2.0 \log_{10}\left(\frac{\epsilon/d}{3.7} + \frac{2.51}{Re\sqrt{f}}\right) \] For fully rough turbulent flow at high Reynolds numbers, the friction factor becomes constant and depends entirely on wall roughness, losing its dependency on \(Re\).

Step 3: Interpreting the test results.
Since the pipe friction test showed that the friction factor varies inversely with the Reynolds number across the full speed range, the flow field must be in the laminar regime. This matches Option (2).