Head loss due to friction in a circular pipe of diameter D, under laminar flow, is inversely proportional to:

Explanation:

Laminar flow through a circular pipe:
In a constant diameter pipe, the pressure drops uniformly along the pipe length (except for the entrance region)
∵ we know that the average velocity through a circular pipe;\({V_{avg}} = \frac{1}{{8\mu }}\left( { - \frac{{\delta P}}{{\delta x}}} \right){R^2} = \frac{1}{{32\mu }}\left( { - \frac{{\delta P}}{{\delta x}}} \right){D^2}\)

\(\Longrightarrow \frac{1}{{32\mu }}\left( {\frac{{{p_1} - {p_2}}}{L}} \right){D^2} = {V_{avg}}\)

\(\Longrightarrow \frac{{{p_1} - {p_2}}}{{\rho g}} = \frac{{32\mu {V_{avg}}}}{{\rho g{D^2}}}\)

\(\Longrightarrow \frac{{{p_1} - {p_2}}}{{\rho g}} = \frac{{128\mu {Q}}}{{\pi ×\rho g{D^4}}}\)

Now, ΔP = γ × Hl

Putting ΔP, from the above equation, we get

Hl ∝ \(1 \over D^4\)

From the above expression, it is clear that hydraulic gradient is inversely proportional to D4

Confusion PointsThe above equation is called Hagen–Poiseuille equation, which is valid for only laminar flow in a circular pipe, (as asked in the question), and pressure or head loss is due to the viscous effect of the liquid.

While the Darcy formula, is valid for both laminar and turbulent flow in circular or noncircular sections, pressure loss is due to friction only.

So, when the Darcy formula is available, pressure difference Δ P  1/D5