Equalising levels between two identical tanks via a small orifice Two identical cylindrical vessels (cross-sectional area A each) are interconnected by an orifice of area a and coefficient of discharge C_d. If the initial difference in liquid levels is H, what is the time T for the levels to equalise (neglecting losses other than the orifice)?
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AT = (A / (C_d a √(2 g))) * √H
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BT = (2 A / (C_d a √(2 g))) * √H
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CT = (A / (2 C_d a √(2 g))) * √H
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DT = (A^2 / (C_d a √(2 g))) * H
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ET = (A / (C_d a √(g))) * H
Answer
Correct Answer: T = (A / (C_d a √(2 g))) * √H
Explanation
Introduction / Context:Interconnected-tank transients are common in laboratory hydraulics and process engineering. When two identical tanks are connected by a small orifice, the level difference decays in time, and the equalisation time can be derived from unsteady continuity combined with the orifice equation.
Given Data / Assumptions:
- Two tanks, each of area A, connected by an orifice of area a with discharge coefficient C_d.
- Head loss confined to the orifice; inviscid elsewhere; free discharge between tanks.
- Initial level difference Δh(0) = H; final equalised difference Δh(T) = 0.
Concept / Approach:
Flow through the orifice is Q = C_d a √(2 g Δh). Because one tank empties while the other fills, the rate of change of difference is doubled: d(Δh)/dt = −(2Q)/A. This yields a first-order, separable differential equation in √Δh, leading to a closed-form time to reach Δh = 0 from H.
Step-by-Step Solution:
Q = C_d a √(2 g Δh).A d(h_1)/dt = −Q, A d(h_2)/dt = +Q ⇒ d(Δh)/dt = −(2 Q)/A.d(Δh)/dt = −(2 C_d a/A) √(2 g) √(Δh).Integrate: ∫ dΔh/√(Δh) = −(2 C_d a/A) √(2 g) ∫ dt.2 √(Δh) = −(2 C_d a/A) √(2 g) t + 2 √H.Set Δh = 0 at t = T ⇒ T = (A / (C_d a √(2 g))) √H.Verification / Alternative check:
Dimensional analysis: [T] = [A]/([a] √(g) √[H]) → seconds, consistent.
Why Other Options Are Wrong:
(b) and (c) have incorrect factors of 2; (d) is dimensionally inconsistent; (e) omits √2 in the orifice velocity.
Common Pitfalls:
Forgetting the factor of 2 because both tanks change level; misapplying Torricelli’s law without the discharge coefficient.
Final Answer:
T = (A / (C_d a √(2 g))) * √H