Bernoulli principle and venturi effect.

[syphex]

I have a specific problem involving two reservoirs filled with water with a height difference Y and total head H, and was wondering if a venturi like device could be used to calculate the resulting pressure head x and if it will exceed the bottom of reservoir 1. Also the drain length is d.

The pipe is the same thickness 20mm diameter. I haven't included the horizontal length of the pipes because I assume v=$\sqrt{2gh}$

1.) Does the addition of reservoir 2 cause a difference in height at x?
2.) Would x rise to a height above the bottom of reservoir 1, because of H?
3.) What drain length d is needed to reduce x to below the bottom of reservoir 1?
4.) Additional reservoirs of the same height are connected together at the base adjacent to 1. Does this effect flow rate and thus increase x? Or can it be thought of as a larger reservoir which has the same head pressure anyway and thus doesn't effect x?


If someone would write me a formula relating these variables that would be a great help, thanks.

 

[sophiecentaur]

I'm answering this so you don't feel left out in the cold!
I can't do the sums for you as I can't remember enough of the topic but it looks like the familiar problem of 'drains'. 'Will the upstairs waste come up into the downstairs basin?' It hangs on the pressure difference caused by the drop d, of course. There are times when you cannot ignore a long horizontal run as it does constitute a resistive element. There is also the point that such a system is always at risk if there's a blockage, or even a constriction in the runs.
As a rule of thumb, I would say that the safe value for the height difference between the reservoirs should be no greater than the length d but you can get away with more by using a swept T union, I should think. (This assumes your problem is a real practical one. If it's a sneaky homework problem then you would have to come up with some actual numbers in your answer.)

 

[syphex]

I can see that without the drain, x would quickly overflow, but I cannot imagine even if d was 0 (the drain is still open to the atmosphere) that this would occur.

Actually Y is small compared to d (not drawn to scale), but I am curious as to how multiple resevoirs affect the pressure and flow rate, and how this influences x.

I'm also asking about the multiple reservoirs because although they may be at the same height and this exert the same amount of pressure, because there is more than one connected doesn't this alter the flow rate (flow in=flow out) and thus the height that x will rise (which is again related to pressure)?

What does the addition of reservoir 2 produce? Its at a different height to 1 so it will exert a different pressure, but surely it can't be ignored?

And no its not homework, it practical as x is actually a drain for a tray/sink for res 1, res 2 is the return res. I haven't studied in a while and am trying to refresh my memory also and I think this problem will help my understanding.

The thing I don't understand is, since the potential energy pgh is only the height above the pipe (not the width of the reservoir), then what about the pgh of another reservoir? Wont this add to the total potential energy of the system? But then the second reservoir and the first can be thought of as one combined res and thus the same energy as one. But if both are causing the same pressure and are joined together wouldent the flow rate, hence the speed, and conversely the pressure change? :S

 

[256bits]

As for your question about multiple resevoirs, replace each resevoir with a pipe similar to the one you have at x. Perhaps you can vsualize it better this way. As resevoir 1 drains then some of the water will flow to the left and into the left tube to a height x2, some water to the right and into the tube to height x, and some to the drain. Height x2 is a moving value and will fall as resevoir 1 empties.

If you now place a resevoir at 2, then you can see that if the height of water is below x2 then there will be flow into resevoir 2 from resevoir 1. If resevoir 2 level is above x2, then water will flow out of resevoir 2 and into resevoir 1. This is above and beyond whatever flow is going to the right to x and the drain. In both cases when level x2 is reached, both resevoirs will be draining out and x2 will fall accordingly

As Sophiecentaur says, level x will rise to whatever level due to resistance in the drain pipe. And that in turn will show its effect upon the level x2.

One resevoir drains at a certain rate. Simply put, all things being equal, two resevoirs at the same height connected to the same place at the pipe will then each drain at half the rate, giving the same exit flow out of the drain as one resevoir.

 

[syphex]

I agree that the height of resevoir 2 (I assume this is x2?) is a bit of a nomenclature, as it will naturally equalize with reservoir 1. Thank you for the clear explanation, I hope I understand it correctly.

Some of it was trying to understand if it made a difference if another reservoir was connected to the same pipe, but as I understand it now the pressure to the right created by 2 is canceled somewhat by the pressure to the left created by 1 and thus the resulting pressure is the same as if there was just reservoir 1?

Basically I am interested in draining 1 and 2 while stopping x from overflowing, but if res 1 and res 2 maintain the same level while draining, and if x is thought of as a res then obviously x will also reach the bottom of res 1 long before the drain is completed.

I was thinking of x as a venturi tube, which seems like it wouldn't overflow because of the dynamic flow beneath it, doesn't the above only apply to a static situation? How does this relate to the total pressure?

If its like a venturi tube can't I just reduce the thickness of the drain after res 1 because the pressure is lower in a constriction? No because that assumes constant flow rate and this would just be like a resistor effectively reducing the flow rate, so how can I solve this? How do you "drain the upstairs room without overflowing the downstairs basin"?

But the height a venturi tube rises to isn't equal to the height of head pressure providing the flow, because in the same pipe the venturi is lower in the constriction than the former.

I understand the res height is creating a head pressure, and the drain height is creating another pressure. The pressure from the res(s) is causing x to rise, and the pressure from the drain causes x to fall.. Does this mean the drain pressure only has to be equal to or greater than the res pressure? But the drain pressure will also have an effect on the res pressure.

I feel like I should be using the bernoulli equation at this point but its a little bit daunting until I understand some the above.

 

[syphex]

Still some unanswered questions above please.

 

[sophiecentaur]

As far as I can see, as long as d is significantly longer than h-x, the water from reservoir1 will not cause the pipe at x to overflow. Height x will be approximately H-d.
There is the small issue of possible resistance in the horizontal section, downstream of pipe x. I assume that the down pipe would be outside and R1, R21 and the x pipe are along the kitchen wall so the down pipe can extend right down into the hopper or gully pot drain. I (and many other people) have the same basic layout and it can't often be a problem or it would be discussed in all the DIY forums and even appear in the assembly instructions for sink wastes. I have certainly never seen it but there's no shame in backing up the theory of PF with the practical of one of those. All the common problems are discussed in them plus some really rare ones. In bathrooms with a single narrow outlet pipe, the sink outlet is way above the level of the bath and you don't find the water from the sink coming out of the bath drain when emptying a basin full.

Btw, I really don't think you can call it a "venturi tube" because a venturi tube is always placed at a constriction so you can measure a pressure drop due to flow. The bore is the same throughout so where could you get a dynamic pressure difference? Afaics, the only place that there will be a dynamic pressure difference is where the wide reservoir joins the narrow down pipe and the downward velocity of the water goes from slow in the reservoir to fast in the pipe. That's where a Bernouli effect can be seen (as when you get a gurgling sound when you take out the plug).

 

[syphex]

Actually turns out I need to reduce the drain pipe somewhere to about half the size. I know its not venturi because its not "measuring" anything, I was actually just wondering if I could lower the height x will reach by providing a constriction. But what effect will it have to constrict the rest of the pipe after x rather than just the section at x? (not the drain pressure or the friction factor). It seems like it would provide excess resistance.

The practical information is useful, but I am also interested in learning a bit.