For fully developed laminar (viscous) flow in a circular pipe, the momentum correction factor β equals:

Introduction / Context:
The momentum correction factor β adjusts between the idealized “plug” velocity assumption and the actual velocity distribution when computing momentum flux. In laminar flow through a circular pipe, the parabolic velocity profile makes the momentum flux larger than if the mean velocity acted uniformly; β corrects for this discrepancy.

Given Data / Assumptions:

• Steady, incompressible, laminar flow in a straight circular pipe.
• Parabolic velocity profile: u(r) = umax * (1 − (r/R)^2).
• Mean velocity Um = Q / A.

Concept / Approach:

By definition, β = (∫_A rho * u^2 dA) / (rho * Um^2 * A). For the parabolic profile, integration over the circular cross-section yields β = 4/3 = 1.333… . This exceeds 1 because the squared velocity weights the faster core more heavily than the mean.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Known results: α (energy correction) = 2 and β (momentum correction) = 4/3 for laminar circular pipe; for turbulent, both approach ≈1 due to flatter profiles.

Why Other Options Are Wrong:

1.25, 1.50, 1.66, and 2.00 do not match the exact integral result for laminar flow (though α = 2.00 is the energy correction factor, not β).

Common Pitfalls (misconceptions, mistakes):

Mixing up α and β; assuming β = 1 (valid only for uniform or very flat profiles typical in high-Re turbulent flow).

Final Answer:

1.33