How fast can you drain a bath?

[chrishutt]

ChrisR, if atmospheric pressure did not bear down on the bath water (i.e. if the bath was a sealed container like a radiator) the water would not run out, at least not at anything like the same rate. Don't we all use that principle when changing rad valves without draining whole CH system?

Perhaps the way I expressed it wasn't technically correct, and I'm happy to stand corrected on such things, but I can't see that what I was saying was wrong in principle. But pray explain......

 

[rumplestiltskin]

I think you will find the force "pushing" the water from an open bath is
rho(density) X g(gravitational pull) X h(height). Because it is height dependent the flow will reduce as the bath empties.
It's really not anything to do with atmospheric pressure.
You can stop water running out of radiators and dead ends of pipework because you create a powerful vacuum which supports the weight of the fluid, again nothing to do with atmospheric pressure.

 

[ChrisR]

Quite so, R. If the bath of water, and its waste, were entirely in a room with no air, a vacuum, the water would still run out.

If the pressure on the surface of a bath were the driving force, you could point the waste upwards and you'd have a fountain. Back into the bath perhaps?

The plugged heating system doesn't let the water out, and it wouldn't if it were entirely in a vacuum. The reason - water is very strong, try compressing it. Same thing when you try to stretch it.

Don't tear yourself in two over it, that's Rumplestiltskin's fate.


If you really want to know how fast a box of given size would empty, taking into account the hole size, etc, you could check in a techie book in a library by someone like F.Hall. Lots of calculations of limited practical application!

 

[chrishutt]

Al right, I take the point. I overstated the significance of atmospheric pressure. Since atmospheric pressure applies at both ends of the pipework in question it's effect is neutralised, which is sort of what I said originally....

Atmospheric pressure also applies at the outlet end of the pipework so the net pressure differential depends on height difference between bath water level and the point at which discharge pipework becomes subject to atmospheric pressure.

However, if atmospheric pressure did not apply at one end (e.g. if a pump were fitted, as proposed by the poster) then surely the atmospheric pressure applied at the bath but not (fully) on the outlet becomes very significant. Is this not in fact the main driving force of such a pumped system?

 

[lorraine]

this is actually bernoulli's equation. as mentioned by a previous poster

rho x gravity x h = pressure at the bottom of the pipe (you need to define a reference point)

the other element is the flow rate which is

1/2 rho x V^2 which is actually dynamic pressure but you can use this to determine the speed of the water flowing out.

fully bernoulli is

pressure + dynamic pressure + pressure head = constant

in terms of the above

P + 1/2 x rho x V^2 + rho x G x H = constant

fiddling with the algebra a bit, and assuming that we are talking gauge pressures and not absolute pressures (absolute means including atmospheric)

1/2 rho V^2 = rho g H

so density terms cancel

and your left with V =sqrt( 2gh)

hope that made sense.

if you are pumping the water out then the pump is usually quoted with a pressure head, and you just subsititute that for the P term.

But please note Bernouli does not take into accoutn friction factors and surface roughness so the actual flow rate will be less, exactly how much less depends on the number of right angles, and length of pipe. Basically if anyone's interested the Reynolds number of the flow.

hope that made sense.

Lorraine's hubby!

 

[chrishutt]

Thanks Lorraine's hubby. Just what we needed . Actually all that algebra is a bit mind numbing for some of us , so, bearing in mind that we're talking exclusively about water (albeit dirty water), how would you express that in plain English. And was my simplified explanation more or less sound, or seriously misleading?

 

[lorraine]

okay, i'll have a go. note this is a bit untrue but should suffice

- the speed at which you drain at is proportional to the square root of height of the water column. (Unpumped). So as the water level drops, the flow rate will also fall.

(that doesn't necessarily mean the height of the bath water, but from the heighest point to your reference point which may be the botton of the drain pipe)

- so basically if you had a column B that was 4 times as high as column A, then column B would drain twice as fast.

- the effect of atmospheric pressure is zero as we are talking house heights, so the pressure on both ends of the column is the pretty much the same

- the effect of a pump is to artificially increase the pressure of the water, this causes a greater flow rate and can be likened to increasing the height of the column of water.

 

[chrishutt]

Thank you, that's very helpful. So I was wrong to bring atmospheric pressure into it.

However..... in practice we have waste water flows that do not fully occupy the full cross sectional area of the pipework (i..e. that are not full-bore), so atmospheric pressure applies to a greater or lesser extent at various points along the pipework. It is therefore difficult to say definitively where the "bottom" end of the pipe is, and therefore what the height differential is.

The design of waste pipework generally attempts to avoid full-bore flow beyond the trap, since it gives rise to syphonage of water seals and noise. I suggest that since in many cases the waste outlet (plug-hole) itself provides the major part of the resistance to flow, the length and discharge height of the subsequent pipework could be largely irrelevant.

In fact using smaller bore pipework (which would then run full-bore) that discharges at a much lower level may result in faster flow rate, since Bernoulli's equation would then apply to that height, rather than just to the level of the bath waste outlet.

Does that make sense to you?

 

[lorraine]

hi,

atmospheric pressure does have a part but when the height differential is so small it makes very little difference. Just a point worth noting as I see some people have stated that the atmospheric pressure pushes down on the bath water - this is true.

but it should also be noted that atmospheric pressure (around 1.01 bar) acts in all directions, so upwards downards, left right etc. hence it has an equal effect trying to push the water upwards.

As engineers we often talk in gauge pressure (which is what measuring instruments such as car tyre inflators measure).

absolute pressure = gauge pressure + atmospheric pressure

I think you are incorrect about the flow not being full bore after the trap, as soon as the plug is taken away, the entire length of pipe section should be full bore, the. Thats in theory but in practice I'm not too familiar with bathroom wastepipes

I would think that the effect of the plughole would be to reduce the flowrate down by adding turbulence to the water which bernouilli's equation cannot account for because it assumes the working fluid is inviscid (has no friction - therefore no turbulence)

I can definitely say that the use of smaller pipework will result in lower flow rate because flow rate is

Velocity x cross sectional area = Flow Rate

the Velocity is defined by your water column height, so a bigger cross sectional area results in larger flow rate.

I've made some best guesses here and some of the stuff may not be true in real life but the theory is good.

 

[chrishutt]

Thanks again. I really appreciate your input.

I have to disagree with you about one point. Flow along waste pipes (e.g. from a bath) is definitely not meant to be full-bore, as this would suck out the water seal from the trap at the end of the discharge (as sometimes happens - due to poor design). There are rules for pipework gradients, lengths and sizes (see Building Regs. H1) to avoid this situation arising.

It follows that discharge to correctly sized (or if you like oversized) pipework (where only waste outlet and trap run full-bore) will effectively be as if there were no pipework beyond the trap and the height differential were just 300mm or so (depending on depth of bath water). Whereas undersized pipework that ran full-bore and dropped down say 3 metres would surely result in a more rapid flow rate, at least if the pipework was the largest size that would run full-bore.

In fact if we apply Bernoulli's equation, as so ably explained by Lorraine's hubby, the 3 metre column, being 10 x the height, will have a flow rate of square root 10 (= 3.15) x the flow rate of 300mm bath to trap outlet column (have I got that right?). Of course allowing for resistance of the additional pipework would reduce that figure, but surely not to the extent of neutralising it.

 

[lorraine]

Thanks again. I really appreciate your input.

I have to disagree with you about one point. Flow along waste pipes (e.g. from a bath) is definitely not meant to be full-bore, as this would suck out the water seal from the trap at the end of the discharge (as sometimes happens - due to poor design). There are rules for pipework gradients, lengths and sizes (see Building Regs. H1) to avoid this situation arising.

It follows that discharge to correctly sized (or if you like oversized) pipework (where only waste outlet and trap run full-bore) will effectively be as if there were no pipework beyond the trap and the height differential were just 300mm or so (depending on depth of bath water). Whereas undersized pipework that ran full-bore and dropped down say 3 metres would surely result in a more rapid flow rate, at least if the pipework was the largest size that would run full-bore.

In fact if we apply Bernoulli's equation, as so ably explained by Lorraine's hubby, the 3 metre column, being 10 x the height, will have a flow rate of square root 10 (= 3.15) x the flow rate of 300mm bath to trap outlet column (have I got that right?). Of course allowing for resistance of the additional pipework would reduce that figure, but surely not to the extent of neutralising it.

Having put a bit more thought into it, I would say you are correct the flow beyond the trap would be whats known as a "slug", running down the sides of the pipe.

This follows on to what you said, the plug/trap has the smallest cross sectional area of the entire length of the pipework. So I would use the bottom of the trap for my h in bernoulli (so the height of the bathwater + 0.1m - i think), and also use the cross sectional area of the trap for working out flow rates. Using the bottom of the water column for Bernoulli would produce an accurate (theoretical) velocity at the bottom but the flow rate would be wrong..

Water is considered incompressible so the flow rate and velocity are governed by the area with the most resistance - which is the plug/trap. beyond this region the pipe gets wider but the water only occupies the same cross sectional area as the place with most resistance.

Your maths looks correct!

but to cut a long story short, IMO the speed at which the bath drains is a function of how wide the trap + plug is as opposed to how high the bath is from the drain. The maths would also support this.

Lorraine's Hubby

 

[chrishutt]

Thanks yet again, Lorraine's hubby (don't you have a name of your own? - it seems very sexist to refer to someone by their spouse's name). I guess we've taken this about as far as we usefully can. Real life situations have so many variables that theories tend to be of limited value, other than to understand the underlying principles. I rather doubt that many others were following these posts beyond the appearance of algebra! But I've certainly learnt a thing or two. Thanks.

 

[ChrisR]

CHanged - was a bit harsh earlier. I have studied this sort of stuff in the past but not had to retain it. I suggest that anyone interested gets hold of a basic book on fluid dynamics, written for plumbers rather than 14 year old physics students. It will soon be appreciated that it's rather a lot more complex than has been suggested. The maths isn't difficult but it's a bit beyond A level.

 

[chrishutt]

Oh dear, ChrisR seems determined to have a go at all of us. Perhaps we should have a closer look at some of his posts.

There is a useful web site http://www.engineeringtoolbox.com/bernouilli-equation-21_183.html which covers many aspects of fluid mechanics.

 

[ChrisR]

If you find something wrong in my posts I do hope you'd tell me - always looking for new information.