Pipes in series (equivalent single pipe): A compound pipeline made of segments with different lengths and diameters carries the same discharge through each segment. If replaced by one equivalent pipe of diameter D and length L_eq with the same head loss and discharge, what is L_eq?

Introduction / Context:

Pipelines in series often change diameter due to design constraints. To analyze head losses compactly, engineers replace the compound line with an equivalent single pipe carrying the same discharge and experiencing the same total friction head loss.

Given Data / Assumptions:

• Steady, incompressible flow with uniform roughness and friction factor f taken equal in all segments (or appropriately averaged).
• Darcy–Weisbach equation for friction losses.
• Pipes are in series → discharge Q is the same in every segment.

Concept / Approach:

For a segment i, head loss h_fi = f * (L_i / D_i) * (V_i^2 / (2g)). With common discharge Q, velocity V_i = 4Q/(π D_i^2). Therefore, h_fi ∝ (L_i / D_i^5) * Q^2. Equating the sum of segment losses to the single equivalent pipe loss yields the L_eq formula.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional consistency: L_eq has units of length; D^5 multiplies a sum of L_i / D_i^5 (dimensionless inside except L_i), giving length overall.

Why Other Options Are Wrong:

• Σ(L_i * D_i^5)/D^5 inverts the dependency; larger D_i should reduce loss, not increase it.
• Linear diameter combinations (Σ L_i / D_i) ignore the D^5 sensitivity from velocity-squared and area effects.
• Heuristic averages (options d, e) lack the correct fifth-power relation for series flow.

Common Pitfalls:

• Mixing Darcy–Weisbach with Hazen–Williams forms (different exponents).

Final Answer:

L_eq = D^5 * Σ(L_i / D_i^5)