Orifice flow theory: What is the theoretical jet velocity at a section having head H (ideal, no losses)?
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Av = sqrt(2 * g * H)
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Bv = 2 * g * H
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Cv = sqrt(H / (2 * g))
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Dv = H / (2 * g)
Answer
Correct Answer: v = sqrt(2 * g * H)
Explanation
Introduction / Context:Torricelli's theorem gives the ideal (lossless) efflux velocity under a head H as if a particle fell freely through that head. It underpins orifice and nozzle calculations before applying discharge and velocity coefficients for real flows.
Given Data / Assumptions:
- Incompressible fluid.
- Negligible elevation differences aside from head H.
- No losses (ideal analysis).
Concept / Approach:Bernoulli between the free surface and the jet section with atmospheric pressure on both locations reduces to v^2/(2g) = H, so v = sqrt(2 * g * H). Real jets use v = Cv * sqrt(2 * g * H) with Cv < 1.
Step-by-Step Solution:
Apply Bernoulli: (p/ρg + z + v^2/(2g)) constant.Atmospheric pressure cancels; surface velocity ~ 0.Thus v^2/(2g) = H ⇒ v = sqrt(2gH).Verification / Alternative check:Dimensional check: g has m/s^2, H has m; product gives m^2/s^2; square root gives m/s as required.
Why Other Options Are Wrong:
- 2g*H: Missing square root; has wrong units.
- sqrt(H/(2g)): Inverts the relation.
- H/(2g): Again wrong units (seconds^2 per meter).
Common Pitfalls:Forgetting to take the square root; confusing head at vena contracta with upstream head; mixing up Cv and Cd.
Final Answer:v = sqrt(2 * g * H)