Laminar flow in a circular pipe: what are the correction factors for momentum and energy?

Introduction / Context:
Velocity in pipe flow is not uniform across the section. The kinetic energy and momentum fluxes are therefore larger than what would be computed using the mean velocity alone. The energy (α) and momentum (β) correction factors account for this non-uniformity and are standard results for laminar flow in a round pipe.

Given Data / Assumptions:

• Fully developed, steady, laminar flow in a circular pipe.
• Parabolic velocity profile u(r) with maximum at the centerline.
• Incompressible, Newtonian fluid.

Concept / Approach:

For a parabolic velocity profile, the derived factors are α = (∫A u^3 dA) / (A * U_m^3) = 2.0 and β = (∫A u^2 dA) / (A * U_m^2) = 4/3 ≈ 1.33, where U_m is the mean velocity. These increase the energy and momentum flux terms in Bernoulli and momentum equations, respectively.

Step-by-Step Solution:

Verification / Alternative check:

For turbulent flow with flatter profiles, α and β approach 1.0, matching intuition that uniform velocity fields need no correction.

Why Other Options Are Wrong:

(a) or (b) alone omit the other necessary factor. (d) and (e) contradict the well-known laminar results.

Common Pitfalls:

Using α = β = 1 for laminar conditions; forgetting these factors when computing head losses from measured velocity distributions.

Final Answer:

Both (a) and (b)