Water hammer basics: The minimum valve-closing time to qualify as “gradual closure” (so that the full Joukowsky rise does not occur) is expressed in terms of pipe length L and wave speed C as what?

Introduction / Context:

Water hammer is a transient pressure surge caused by rapid changes in flow. The characteristic time for a pressure wave to travel the pipe length and return (a round trip) governs whether a closure is considered “sudden” or “gradual” for surge calculations.

Given Data / Assumptions:

• Rigid-pipe approximation with wave speed C.
• One-dimensional transient behavior; reflection at reservoir/valve ends.
• Valve located at one end of a line of length L.

Concept / Approach:

If the valve-closing time t_c is less than 2L/C, the closure is “sudden”, and the Joukowsky equation Δp = ρ C ΔV applies for the maximum surge. If t_c ≥ 2L/C, the closure is “gradual”; the maximum head rise reduces and scales with t_c. Thus, 2L/C is the dividing line between sudden and gradual closures.

Step-by-Step Solution:

Verification / Alternative check:

Method of characteristics solutions and classic water-hammer charts corroborate the 2L/C threshold for end-valve systems.

Why Other Options Are Wrong:

• L/C (one-way travel) underestimates the interaction time.
• Other multiples (4L/C, L/2C) do not reflect the physics of wave reflection timing.

Common Pitfalls:

• Using local pipe segment length instead of the full length to the reflecting boundary.

Final Answer:

t_g = 2L / C