Water hammer in pressurized pipelines In a closed water-conveyance pipe system, the transient pressure rise (surge) produced when a flowing stream is decelerated by valve closure depends primarily on which factors?

Introduction / Context:
Water hammer is a classic unsteady-flow phenomenon in hydraulics. When a valve is closed or a pump trips, the flowing water column is suddenly decelerated, generating pressure waves that can far exceed normal steady-state pressures. Understanding what controls the surge magnitude is essential for safe pipeline design, surge-tank sizing, and valve-closure specifications.

Given Data / Assumptions:

• Rigid or elastic pipes carrying water under pressure.
• Valve operation causes a transient from an initial steady velocity.
• Waves propagate at c (celerity) determined by water and pipe elasticity and geometry.

Concept / Approach:

The Joukowsky relation gives the ideal instantaneous surge: Δp = ρ c ΔV. The wave speed c depends on bulk modulus of water, pipe wall modulus, diameter, and wall thickness. The resulting maximum pressure rise also depends on valve-closure time relative to the pipe period (2L/c) and on reflections governed by pipeline length and end conditions.

Step-by-Step Solution:

Verification / Alternative check:

Compare quick vs slow closure using method of characteristics: shorter closure yields higher initial surge; more elastic pipes reduce c, reducing Δp for the same ΔV.

Why Other Options Are Wrong:

Each single factor (a, b, c, d) affects the surge, but not exclusively. The correct comprehensive choice is that all listed factors contribute.

Common Pitfalls:

Ignoring pipe-wall elasticity; assuming only valve speed matters; neglecting reflections that can compound pressures at tees or dead ends.

Final Answer:

All of the above