Section 4: Soil and Water Conservation Engineering
Dynamics of Flow & Bernoulli Equation
Subtopic: Energy and momentum correction factors
Question Statement:
Which of the following statements concerning kinetic energy correction factor ($\alpha$) and momentum correction factor ($\beta$) is/are TRUE?
Options:
(A) For fully developed laminar flow in a circular pipe, $\alpha = 2.0$ and $\beta = 4/3$
(B) For uniform velocity distribution across the entire cross-section, $\alpha = 1.0$ and $\beta = 1.0$
(C) In open channel flow, $\alpha$ is always less than or equal to $\beta$
(D) For turbulent flow in commercial pipes, $\alpha$ typically ranges between $1.02$ and $1.08$
Answer Key:
A, B, D
Step-by-Step Solution & Explanation
1. For parabolic laminar velocity profile in pipes: $\alpha = \frac{1}{A}\int (u/V)^3 dA = 2.0$, and $\beta = \frac{1}{A}\int (u/V)^2 dA = 4/3 \approx 1.33$ (A is true).
2. For ideal uniform flow: $\alpha = \beta = 1.0$ (B is true).
3. By Jensen's inequality / Schwarz inequality, $\alpha \ge \beta \ge 1.0$ always, never $\alpha \le \beta$ (C is false).
4. In turbulent pipe flow, velocity profiles are blunt, so $\alpha \approx 1.02 - 1.08$ (D is true).