A very simplified schematic of the rain drainage system for a home is shown in t

Question

A very simplified schematic of the rain drainage system for a home is shown in the figure. Rain falling on the slanted roof runs off into gutters around the roof edge; it then drains through downspouts (only one is shown) into a main drainage pipe M below the basement, which carries the water to an even larger pipe below the street. In the figure, a floor drain in the basement is also connected to drainage pipe M. Suppose the following apply: 1. the downspouts have height h1 = 15 m, 2. the floor drain has height h2 = 1.1 m, 3. pipe M has radius r = 3.0 cm, 4. the house has side width w = 22 m and front length L = 45 m, 5. all the water striking the roof goes through pipe M, 6. the initial speed of the water in a downspout is negligible, 7. the wind speed is negligible (the rain falls vertically). At what rainfall rate, in centimeters per hour, will water from pipe M reach the height of the floor drain and threaten to flood the basement?

Explanation / Answer

by continuity eqaution

v1 = (A2/A1)*v2

by bernouilli equation

P1 + rho*v1^2/2 + rho * g * h1 = P2 + rho*v2^2 /2 + rho * g* h2

When the water level rises to height h2 , just on the verge of flooding, v2 , the speed of water in pipe M , is given by

rho*g(h1-h2) = rho*v2^2/2

v2 = sqrt(2g(h1-h2)) = sqrt(2*9.8 * (15 - 1.1))

v2 = 16.5

A1 = 45 * 22

A2 = pi * (3 x 10^-2)^2

v1 = 4.71 x 10^-5 m/s

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