For a given required volumetric flow and fluid, how does increasing pipe diameter generally affect pumping (energy) cost in steady single-phase flow?

Introduction / Context:
Pipeline design balances capital cost (larger diameter, more material) against operating cost (pumping energy). Understanding how frictional losses scale with diameter allows engineers to choose an economical size for a specified flow rate and fluid.

Given Data / Assumptions:

• Single-phase, incompressible flow at steady state.
• Fixed volumetric (or mass) flow requirement and fluid properties.
• Fully developed flow; roughness and friction factor depend on Reynolds number as usual.

Concept / Approach:
Frictional pressure drop in a pipe of length L is ΔP = f * (L/D) * (ρ v^2 / 2). For fixed volumetric flow, velocity v scales as 1/D^2, so ΔP scales approximately as 1/D^5 in laminar and roughly between 1/D^4 and 1/D^5 in turbulent (since f varies mildly with Reynolds number). Hence, larger diameter dramatically reduces pressure drop and thus pump power P ≈ ΔP * Q / η.

Step-by-Step Solution:
Increase D → decrease v = Q / A → reduce velocity head ρ v^2 /2.ΔP ∝ (L/D) * v^2 → strong inverse dependence on D.Pump power P ∝ ΔP for constant Q → P decreases as D increases, holding all else constant.

Verification / Alternative check:
Numerical examples with common friction correlations (e.g., Blasius for smooth turbulent) confirm the steep drop in ΔP with increasing D at constant Q.

Why Other Options Are Wrong:
(b) and (c) contradict fluid mechanics; (d) Newtonian vs. non-Newtonian can change f(Re) but not the fundamental inverse trend; (e) is not representative for a fixed Q.

Common Pitfalls:
Ignoring capital cost increase with D; forgetting that very large D may lead to low velocities and solids deposition in slurries.

Final Answer:
Pumping cost decreases.