Question

A very simplified schematic of the rain drainage system for a home is shown in 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 $\mathit{M}$below the basement, which carries the water to an even larger pipe below the street. In Figure, a floor drain in the basement is also connected to drainage pipe $\mathit{M}$. Suppose the following apply: $\mathbf{1}\mathbf{.}$The downspouts have height ${\mathit{h}}_{\mathbf{1}}\mathbf{=}\mathbf{11}\mathit{m}$, $\mathbf{2}\mathbf{.}$the floor drain has height${\mathit{h}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}\mathit{m}\mathbf{,}\mathbf{}\mathbf{3}\mathbf{.}$pipe $\mathrm{M}$has radius $\mathbf{3}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}\mathbf{,}\mathbf{}\mathbf{4}\mathbf{.}$the house side is width $\mathit{w}\mathbf{=}\mathbf{30}\mathit{m}$and front

length $\mathit{L}\mathbf{=}\mathbf{60}\mathit{m}\mathbf{,}\mathbf{}\mathbf{5}\mathbf{.}$all the water striking the roof goes through pipe $\mathbf{M}$, $\mathbf{6}\mathbf{.}$the initial speed of the water in a downspout is negligible, $\mathbf{7}\mathbf{.}$the wind speed is negligible (the rain falls vertically). At what rainfall rate, in centimetres per hour, will water from pipe $\mathit{M}$ reach the height of the floor drain and threaten to flood the basement?

Short answer

The rainfall rate in $\mathrm{cm}/\mathrm{hr}$as the water from pipe $M$reaches the height $h_2$of the floor drain and threatens to flood the basement is$7.8\mathrm{cm}/\mathrm{hr}$.

Step-by-step solution

Listing the given quantities.

The downspouts height $h_1=11m.$

The floor drain height ${\mathrm{h}}_2=1.2\mathrm{m}.$

Radius of pipe $\mathrm{M}$is

$$\begin{gathered} {\mathrm{R}}_{\mathrm{m}}=3\mathrm{cm} \\ =0.03\mathrm{m} \end{gathered}$$

Side width of the house is$\mathrm{w}=30\mathrm{m}.$

Front length of the house is$\mathrm{L}=60\mathrm{m}.$

All water striking the roof goes through pipe$\mathrm{M}.$

Initial speed of the water in a downspout is negligible.

Wind speed is negligible.

Understanding the Bernoulli’s Principle.

By using Bernoulli’s equation, we find the speed ${\mathrm{V}}_{\mathbf{2}}$of water in pipe M and by putting this value of ${\mathbf{V}}_{\mathbf{2}}$in equation of continuity, we can find the rainfall rate $\mathit{c}\mathit{m}\mathbf{/}\mathit{h}\mathit{r}$in as the water from pipe $\mathbf{M}$reaches the height ${\mathbf{h}}_{\mathbf{2}}$ of the floor drain and threatens to flood the basement.

Formula:

Bernoulli's equation,

$\mathrm{P}+\frac{1}{2}{\mathrm{\rho V}}^2+\mathrm{\rho gy}=\mathrm{constant}$

Equation of continuity

$\mathrm{av}=\mathrm{constant}$

Explanation.

By applying Bernoulli’s equation, we get,

${\mathrm{P}}_1+\frac{1}{2}{{\mathrm{\rho V}}_1}^2+{\mathrm{\rho gh}}_1={\mathrm{P}}_2+\frac{1}{2}{{\mathrm{\rho V}}_2}^2+{\mathrm{\rho gh}}_2=\mathrm{constant}$

When the water level rises to high ${\mathrm{h}}_2$just on the verge of flooding, the speed $V_2$of the water in pipe $M$is given by

${\mathrm{\rho gh}}_1=\frac{1}{2}{{\mathrm{\rho V}}_2}^2+{\mathrm{\rho gh}}_2$

$\{\xcancel{}{V}_1$is negligible and${\mathrm{P}}_1={\mathrm{P}}_2\}$

$\rho g(h_1-h_2)=\frac{1}{2}\rho {V_2}^2$

$V_2=2g(h_1-h_2)$

$V_2=\sqrt{2g(h_1-h_2)}$

$=\sqrt{2x9.8m/s^2(11.0m-1.2m)}$

$=13.8593\mathrm{m}/\mathrm{s}$

Now, using the equation of continuity, we get

$V_1{A}_1=V_2{A}_2$

$V_1=\frac{V_2{A}_2}{A_1}$

Here, areas $A_1$and $A_2$are

${\mathrm{A}}_1=\mathrm{w}\mathrm{x}\mathrm{L}$

$=30mx60m$

$=1800{\mathrm{m}}^2$

${\mathrm{A}}_2={{\mathrm{\pi R}}_{\mathrm{m}}}^2$

$=3.14\mathrm{x}{(0.03\mathrm{m})}^2$

$=0.002826{\mathrm{m}}^2$

Putting these values in the equation of continuity, we get

${\mathrm{V}}_1=\frac{13.8593\mathrm{m}/\mathrm{s}\mathrm{x}0.002826{\mathrm{m}}^2}{1800{\mathrm{m}}^2}$

$=2.176\mathrm{x}{10}^{-5}\mathrm{m}/\mathrm{s}$

But we need the rainfall rate in $\mathrm{cm}/\mathrm{hr},$

Conversion factor is

$1\mathrm{m}/\mathrm{s}=359999.999972\mathrm{cm}/\mathrm{hr}$

We get

${\mathrm{V}}_1\mathrm{in}\mathrm{m}/\mathrm{s}\mathrm{x}359999.999972={\mathrm{V}}_1\mathrm{in}\mathrm{cm}/\mathrm{hr}$

${\mathrm{V}}_1\mathrm{in}\mathrm{cm}/\mathrm{hr}=2.176\mathrm{x}{10}^{-5}\mathrm{m}/\mathrm{s}\mathrm{x}359999.999972$

$=7.8336\mathrm{cm}/\mathrm{hr}$

${\mathrm{V}}_1\mathrm{in}c\mathrm{m}/\mathrm{s}=7.8336cm/hr$

$=7.8\mathrm{cm}/\mathrm{hr}$

The rainfall rate in $\mathrm{cm}/\mathrm{hr}$as the water from pipe $M$reaches the height $h_2$of the floor drain and threatens to flood the basement is$7.8cm/hr.$