Section 4: Soil and Water Conservation Engineering
Fluid Mechanics
Subtopic: Flow through orifices, weirs and notches
Question Statement:
Two cylindrical reservoirs 'A' and 'B' are connected by a 30 m long pipe of 250 mm internal diameter as shown in Figure below. The Darcy-Weisbach friction factor for the pipe is 0.025. Initially the reservoir 'A' was full at the indicated level and reservoir 'B' was empty. If the entrance and exit losses in this pipe are neglected, the time required to empty the reservoir 'A' in hour is ___________ (rounded off to 3 decimal places). Consider π = 3.14 and g = 9.81 m s⁻².
Reconstructed figure for Q.57 (schematic; not to scale — as stated in the original).
Official Answer Key:
3.30 to 3.40
Step-by-Step Solution & Explanation
Method: Applying the unsteady Bernoulli/Darcy-Weisbach equation between the two free surfaces (both varying with time as A drains and B fills), with head loss h = f(L/D)V²/(2g) (entrance/exit losses neglected as stated), the instantaneous discharge through the connecting pipe is Q(t) = A_pipe√[2g·h(t)/(fL/D)], where h(t) is the elevation difference between the free surfaces of A and B at time t. Mass conservation gives dz_A/dt = −Q/A_A and dz_B/dt = +Q/A_B, which can be integrated (analytically for the combined head, or numerically) from the initial levels shown in the figure until reservoir A drains to the elevation of the pipe outlet at A.
Discrepancy noted: using the elevation values read from the reconstructed figure (initial water-surface elevation in A, and the base/invert elevation at B), our independent transient solution of this coupled-tank drainage problem did not converge to a value inside the officially listed range of 3.30–3.40 hours; our computed values (under several plausible readings of the figure's elevation datums) ranged from about 0.4 to 0.6 hours. Because the exact datum references in the original scanned figure (which tank-bottom or pipe-centerline elevation each dimension line denotes) could not be pinned down with full certainty from the reproduced image, we were unable to fully reconcile our calculation with the official key through this data-extraction path alone. We therefore report the official key value, t ≈ 3.30–3.40 h, as the answer, while flagging that independent verification of the exact figure geometry is recommended before relying on the intermediate working shown here.
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