When will the house be flooded?

[GwtBc]

Homework Statement

[/B]

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.

 

[Simon Bridge]

I don't think you can assume the flow rate down the downspout is zero, nor that the water fills the downspout.
You can check if these are good assumptions though - since you are looking for the rate that just fills M to h2... will the downspout be full when this happens?
However, you are right to try to think in termsof what the course expects you to do ... nobody here can properly advise you: you are the one doing the course so only you have the correct context (and the others inn your class of course).

How is the water supposed to be behaving in the downspout?

It should be behaving like water in a downspout.
How does water normally behave in a downspout? ie. what is the physics here?

I'm not sure if this is the correct expression.

Define "correct" in this context?

You have been doing a course which has covered a topic using some sort of model, probably with worked examples.
Use those for context.

 

[kuruman]

Suppose the drain is full to the top, but water is not spilling into the barn. Here are some pieces of the puzzle that need to come together.
1. What should the pressure be at a point in the drain pipe directly below the drain? (Actually the pressure is the same everywhere in M, but I specified directly below to make it easier to see.)
2. Can you use Bernoulli's equation to link this point below the drain to the point of water entry at the top of the downspout? I think it is a fair assumption that all pipes are full and water is flowing in all of them.
3. Can you relate the speed of the water in M to the volumetric flow dV/dt? Volumetric flow is essentially how many cubic meters per second flow through.
4. Can you relate the volumetric flow in M to the rate of rainfall on the roof? Let β = rainfall rate in m/s (you easily can convert to cm/hour after you get a number)

You can work on these four pieces independently; you don't need one to find another.

 

[GwtBc]

Suppose the drain is full to the top, but water is not spilling into the barn. Here are some pieces of the puzzle that need to come together.
1. What should the pressure be at a point in the drain pipe directly below the drain? (Actually the pressure is the same everywhere in M, but I specified directly below to make it easier to see.)
2. Can you use Bernoulli's equation to link this point below the drain to the point of water entry at the top of the downspout? I think it is a fair assumption that all pipes are full and water is flowing in all of them.
3. Can you relate the speed of the water in M to the volumetric flow dV/dt? Volumetric flow is essentially how many cubic meters per second flow through.
4. Can you relate the volumetric flow in M to the rate of rainfall on the roof? Let β = rainfall rate in m/s (you easily can convert to cm/hour after you get a number)

You can work on these four pieces independently; you don't need one to find another.

1. The pressure should be such that it satisfies Bernoulli's equation. At h2 and h3 it will be $p_{0}$.
2. Yes. If the point below the floor drain is point_{1}, and the point at the top of the downspout is point_{3}, then
$p_{1}+\frac{1}{2}\rho v_{1}^2+\rho gh_{1} = p_{3} + \frac{1}{2}\rho v_{3}^2+\rho gh_{3}$
since $h_{1} = 0$ , $p_{1}+\frac{1}{2}\rho v_{1}^2= p_{3} + \frac{1}{2}\rho v_{3}^2+\rho gh_{3}$
if the question statement is to be believed, $v_{3}$ is also 0 but that makes no sense because then the continuity equation would not hold.
3. $\frac{\mathrm{d} V}{\mathrm{d} t} = A_{m}v_{1}$
4. $A_{m}v_{1} = A_{house}v_{rate}$

Still not sure where to go.

 

[kuruman]

1. The pressure should be such that it satisfies Bernoulli's equation. At h2 and h3 it will be p0 p_{0} .

It is best to call the height of the point on the floor (top of drain) h₂ which is what's given. You did not answer the question, what is the pressure directly below that point inside the drain pipe? Can you find an expression? You need to establish a relation between the pressure at the top of the drain and the pressure inside the pipe.
2. Why is h₁ zero? The statement of the problem says it it 14 m.
3. Right.
4. Right, if v_rate is what you are looking for.

So, you need to find the pressure inside the drain, correct the error in h₁ and put it together using the Bernoulli equation.

 

[GwtBc]

It is best to call the height of the point on the floor (top of drain) h₂ which is what's given. You did not answer the question, what is the pressure directly below that point inside the drain pipe? Can you find an expression? You need to establish a relation between the pressure at the top of the drain and the pressure inside the pipe.
2. Why is h₁ zero? The statement of the problem says it it 14 m.
3. Right.
4. Right, if v_rate is what you are looking for.

So, you need to find the pressure inside the drain, correct the error in h₁ and put it together using the Bernoulli equation.

I managed to solve the problem yesterday, Thanks for the help. It was actually fairly simple once you made the assumption that $v_{3} = 0$. I can't see how continuity would hold if this were true. Also, in my formulations $h_{3}$ is the height of the downspout, $h_{1}$ is the height of the point directly below the floor drain.

 

[kuruman]

I'm glad you managed to solve it. That v₃ is zero is not an assumption. It has to be the case otherwise water will be flooding the barn. I looked at it this way: You have a stationary column of water in the drain of height h₃ (h₂ in the figure). The weight of this column creates pressure p₃ = ρgh₃ inside the pipe. The rest is application of the continuity equation.