During sudden valve closure in a water-filled pipe with flow velocity v, the surge pressure (water hammer) intensity is directly proportional to which property?

Introduction / Context:Water hammer is a transient pressure rise that occurs when flow is rapidly changed. For a rigid column approximation with elastic fluid/pipe, the Joukowsky equation estimates the pressure rise: Δp = ρ * a * Δv, where a is the wave speed. Understanding how Δp scales with material properties is key for surge protection.

Given Data / Assumptions:

• Sudden closure so Δv ≈ v.
• Elastic fluid with bulk modulus K; pipe-wall elasticity may also matter, but the core dependence of a on K remains via a ∝ sqrt(K/ρ) (modified for wall effects).
• Density ρ approximately constant.

Concept / Approach:The wave speed a in a fluid–structure system is approximately a = sqrt( K / ρ / (1 + (KD)/(Ee)) ). With pipe rigidity large, a ≈ sqrt(K/ρ). Therefore, Δp ∝ a ∝ sqrt(K) for fixed ρ and fixed Δv. Hence, the direct proportionality to the square root of bulk modulus is the most accurate among the given choices.

Step-by-Step Solution:Start with Δp = ρ * a * v.Note a ≈ sqrt(K/ρ) (rigid pipe limit).Therefore, Δp ∝ ρ * sqrt(K/ρ) * v = v * sqrt(K * ρ) → proportional to sqrt(K) for fixed ρ and v.

Verification / Alternative check:Including wall elasticity changes only the prefactor in a; the dependence on sqrt(K) persists, confirming the chosen proportionality trend.

Why Other Options Are Wrong:Linear in K: contradicts wave-speed theory.Linear in specific weight: Δp depends on ρ via sqrt(ρ), not γ directly; γ also introduces g which is not part of the Joukowsky relation.None of these: incorrect because a √K dependence exists.

Common Pitfalls:

• Assuming incompressible (K → ∞) behaviour which would imply infinite a and unrealistic Δp.
• Ignoring the effect of valve closure time relative to 2L/a.

Final Answer:Square root of the bulk modulus of elasticity of water