Criterion for “sudden” valve closure in pipelines (water hammer) A valve closure in a pipeline is termed “sudden” when the actual closure time t_c satisfies which condition (L = pipe length, a = wave speed)?

Introduction / Context:
Water hammer analysis classifies valve closures as “sudden” or “gradual” depending on how the closure time compares to the characteristic time for a pressure wave to travel to the upstream reservoir and reflect back. This classification determines the peak transient pressure predicted by the Joukowsky equation.

Given Data / Assumptions:

• One-dimensional elastic water hammer model.
• Wave (celerity) speed a is constant.
• Uniform pipe of length L, liquid initially in steady motion.

Concept / Approach:

The critical time is 2 L / a, the round-trip travel time of a pressure wave in the pipe. If the valve closes in less time than this, the flow cannot adjust gradually; the transient is a “sudden” closure yielding the maximum pressure rise Δp = ρ a Δv (Joukowsky). For longer times, the event is “gradual,” and the peak pressure is smaller, depending on the closure law.

Step-by-Step Solution:

Verification / Alternative check:

Method of characteristics solutions show that if the boundary condition (valve opening) changes fully before the first reflected wave returns, the maximum Joukowsky rise occurs, consistent with the t_c < 2 L / a criterion.

Why Other Options Are Wrong:

Equal or greater than 2 L / a are not “sudden”; L / a and 4 L / a are not the accepted threshold in standard water hammer theory.

Common Pitfalls:

Confusing single-trip time L / a with the round-trip 2 L / a; overlooking that wave speed depends on both fluid compressibility and pipe wall elasticity.

Final Answer:

t_c < 2 L / a