Galvanized iron drainpipes of diameter 50 mm are located at the four corners of a building, but

step 1 In this question, we have to find the maximum down poor rate it can handle for the galvanized dr

Galvanized iron drainpipes of diameter 50 mm are located at...

Galvanized iron drainpipes of diameter $50 \mathrm{mm}$ are located at the four corners of a building, but three of them become clogged with debris. Find the rate of downpour $(\mathrm{cm} /$ min) at which the single functioning drainpipe can no longer drain the roof. The building roof area is $500 \mathrm{m}^{2},$ and the height is $5 \mathrm{m}$. Assume the drainpipes are the same height as the building, and that both ends are open to atmosphere. Ignore minor losses.

Key Concepts

Pipe Flow Dynamics

Pipe flow dynamics involves analyzing how fluids move through conduits, considering factors such as the cross-sectional area, gravitational forces, and friction losses (which are sometimes neglected for simplicity). In this situation, the pipe's diameter and the effective head generate a specific volumetric flow rate that must be compared with the rainwater inflow to assess drainage capability.

Torricelli's Law

Torricelli's Law is used to estimate the speed of efflux of a fluid under the influence of gravity from an opening. It states that the exit velocity is proportional to the square root of twice the product of gravitational acceleration and the height of the fluid column, which is crucial for calculating the discharge capacity of the drainpipe.

Hydrostatic Pressure

Hydrostatic pressure arises from the weight of a fluid column and provides the driving force for flow through an open channel or pipe. In the drainpipe problem, the height of the water column (determined by the building’s height) generates the pressure difference that causes the water to flow through the pipe.

Continuity Equation

The continuity equation in fluid dynamics is based on the conservation of mass, stating that the rate of fluid entering a system must equal the rate exiting plus any accumulation. In this context, equating the rainwater inflow to the drainage outflow determines the critical rainfall intensity at which the drainpipe can no longer handle the water supply.

Rainfall Intensity

Rainfall intensity measures the depth of precipitation accumulated per unit time over a given area. In problems like this, it is used to determine the volumetric rate at which water is being added to a surface (i.e., the roof), serving as the inflow rate that must be balanced by the drainage capacity.

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