For fully developed laminar flow in a circular pipe, what is the kinetic energy correction factor (α) used in energy equations to account for the non-uniform velocity profile?

Introduction / Context:
The kinetic energy correction factor α adjusts the V^2/2g term in the Bernoulli/energy equation to account for non-uniform velocity distributions across a cross-section. For laminar flow in circular pipes, the parabolic profile makes α significantly larger than 1.0.

Given Data / Assumptions:

• Flow is steady, incompressible, and fully developed.
• Laminar regime (Re < 2000 in a circular pipe).
• Velocity profile is parabolic: u(r) = umax(1 − r^2/R^2).

Concept / Approach:

α is defined as (∫A ρ u^3 dA) / (ρ A V^3). For a parabolic profile, integration yields α = 2.0. In contrast, for turbulent flows with flatter profiles, α is closer to 1.0 (often 1.03–1.10 in engineering practice).

Step-by-Step Solution:

Verification / Alternative check:

Textbook derivations universally report α = 2 for laminar circular flow; momentum correction factor β for the same case is 4/3 ≈ 1.33.

Why Other Options Are Wrong:

• 1.00 pertains to perfectly uniform velocity, not laminar parabolic.
• 1.33 is the momentum (not kinetic energy) correction factor in laminar flow.
• 1.50 and 1.67 do not match the exact integral result.

Common Pitfalls:

• Confusing α (energy) with β (momentum) correction factors.

Final Answer:

2.00