# [ASK] Integral - Draining a Pipe

• MHB
• bleedpurple
In summary, the equation for calculating the time to drain a pipeline, tank, or vessel through an orifice is straightforward, and can be integrated using the orifice equation.
bleedpurple
My first post, and first use of Latex. Here goes.

The engineering problem of calculating the time to drain a pipeline, tank, or vessel through an orifice is fairly straightforward using the orifice equation.$$Q=CA_{o}\sqrt{2gh}$$

With C being the coefficient of discharge for the orifice, Ao being the area of the orifice, g is the acceleration of gravity, and h the energy 'head' or elevation of the fluid above the orifice all in feet.
This equation can then be integrated by expressing Q as the change in volume over time.

$$Q=\frac{dv}{dt}$$

And expressing V as a function of head, h.
For a pipeline with constant gradient or slope the volume equals the cross sectional area of the pipe, Ap x length, and the length is the elevation change or head, h divided by the slope.

Therefore:

$$\frac{dv}{dt}=\frac{A_{p}}{slope} \frac{d_h}{dt} =C\,A_{o}\sqrt{2gh}$$and

$$\int{h^{-1/2}}dh=C\,A_{o}\sqrt{2gh} \frac{slope}{A_{p}} \int{dt}$$

or

$$\int{dt} = \frac{A_{p}}{slope} \frac{1}{CA_{o}\sqrt{2g}}\int{h^{-1/2}}dh$$

from this the result is

$$t=\frac{2 A_{p}}{slope} \frac{1}{CA_{o}\sqrt{2g}}(h_{1}^{1/2}-h_{0}^{1/2})$$

Now the case where there is also a constant flow in addition to the orifice flow.

$$Q=CA_{o}\sqrt{2gh}+K$$

resolves to the integral

$$\int{dt} = \frac{A_{p}}{slope CA_{o}\sqrt{2g}} \int\frac{1}{h^{1/2}+\frac{K}{CA_{o}\sqrt{2g}}}dh$$

from this the result is ?

Hi bleedpurple, welcome to MHB! (Wave)

Good job with the $\LaTeX$!

I take it you're wondering how to evaluate $\int \frac{dh}{\sqrt h + B}$?
If so, let's substitute $u=\sqrt h + B\Rightarrow h=(u-B)^2 \Rightarrow dh=2(u-B)du$:
$$\int \frac{dh}{\sqrt h + B} = \int \frac{2(u-B)\,du}{u} = 2\int (1-\frac Bu)\,du = 2(u-B\ln u) = 2\Big(\sqrt h + B - B\ln(\sqrt h + B)\Big)$$

## 1. What is the purpose of draining a pipe?

Draining a pipe is necessary for various reasons, such as repairs, maintenance, or to prevent freezing. It involves removing any liquid or debris from the pipe to ensure it functions properly.

## 2. How is a pipe drained?

To drain a pipe, you must first locate the drain valve, which is usually located at the lowest point of the pipe. Once found, open the valve and let the liquid or debris flow out. You may also need to use a pump or vacuum to remove the contents of the pipe.

## 3. How long does it take to drain a pipe?

The time it takes to drain a pipe depends on various factors, such as the size of the pipe, the amount of liquid or debris inside, and the method used to drain it. It can range from a few minutes to several hours.

## 4. Are there any safety precautions to take when draining a pipe?

Yes, there are several safety precautions to take when draining a pipe. Make sure to wear protective gear, such as gloves and goggles, to avoid any potential hazards. Also, be cautious when handling any tools or equipment, and make sure the surrounding area is clear of any obstructions.

## 5. Can I drain a pipe on my own or should I hire a professional?

The answer to this question depends on the complexity of the draining process and your level of expertise. Simple draining tasks can be done on your own, but for more complicated situations, it is best to hire a professional for safety and efficiency reasons.

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