- #1

GwtBc

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## Homework Statement

A simplified schematic of the rain drainage system for a barn is shown in the figure. Rain falling on the slanted roof runs off into gutters around the roof edge; it then drains through several downspouts (only one is shown) into a main drainage pipe M below the basement, which carries the water to an even larger pipe. A floor drain in the basement is also connected to drainage pipe M.

Suppose the following apply:

1. the downspouts have height h1 = 14 m

2. the floor drain has height h2 = 1 m

3. pipe M has radius 2.7 cm,

4. the barn has a width w = 28 m and a length L = 45 m,

5. all the water striking the roof goes through pipe M ,

6. the initial water speed in a downspout is negligible,

7. the wind speed is negligible (the rain falls vertically).

At what rainfall rate, in centimeters per hour, will the water rise up to the level of the drainage pipe and threaten to flood the house?

## Homework Equations

AV = av

p1 + 1/2Rv1^1/2+ Rgh1 = p2 + 1/2Rv2^1/2+ Rgh2

## The Attempt at a Solution

I'm having trouble recognising what assumptions I can make and which ones I cannot. For example, am I supposed to assume that the water is up to height h1 in the downspout and that the flow rate ## v_{downspout} = 0 ##? If it's non-zero is it supposed to cancel out somehow, since there doesn't seem to be any way of calculating it (it would be very complicated, doubt it's what the question would ask for) That's the main thing really. How is the water supposed to be behaving in the downspout?

Making the aforementioned assumptions, I have ## v_{M} = \sqrt{2g(y_{1}-y_{2})} ##, where y1 is the height of the downspout, y2 is the height of the drain floor drain and vM is the speed of the water in pipe M. From here finding the rate of rainfall for y2 = 1 m should be straightforward, but I'm not sure if this is the correct expression.