Filling water cup/bucket: which way is faster?

[rugerts]

Hello all,
I've been thinking about this while I fill up my water cup or when I fill up a bucket from a hose.

In the attached images are the two scenarios I present.

My question is, which scenario would be faster (meaning in which scenario does the bucket or cup become full in the shortest amount of time), and what physics formulas are related to this phenomenon?
My intuition says that if I bring the cup closer to the (constant) water source, it will fill up faster. The same goes for bringing the hose closer to the bottom (although I'd be interested in what suction effects may have that alter the rate of filling up for the hose scenario).
Thanks for your time!

 

[dRic2]

If the cup is very far from the hose of curse you have to take into account the time water requires to get from the hose to the cup. But I doubt it is something you can see when you are washing plates (cups) in the sink.

Also I don't think there will be any differences if hose is near the bottom or at the top pf the cup.

 

[Nugatory]

My intuition says that if I bring the cup closer to the (constant) water source, it will fill up faster.

Easy to test this... you need a cup and a stopwatch.

 

[rugerts]

Easy to test this... you need a cup and a stopwatch.

True, but I'm looking for a more mathematical approach if possible. I'd like to see if there are any related laws/principles that can help here.

 

[Nugatory]

True, but I'm looking for a more mathematical approach if possible. I'd like to see if there are any related laws/principles that can help here.

Here's one:
Water is incompressible. Therefore if you consider any region of space, either the amount of water flowing into it per unit time is equal to the amount of water flowing out or the amount of water within that region is increasing or decreasing. Conversely, if the amount flowing into the region per unit time is equal to the amount flowing out, the amount of water in that region must be constant over time.

Try applying this principle to the region whose bottom surface is the top of the cup and whose top surface is at the faucet, and which completely enclosed the stream of water...

 

[CWatters]

If the flow rate from the tap is say 10 cups per min then 1 cup takes 6 seconds regardless of how close or far it is from the tap (provided all the water goes in).

However in the case of a hose in a bucket... The pressure at the bottom of the bucket increases as it fills up. This could in theory reduce the flow rate if the end of the hose is submerged and the supply is low pressure. For example you can't completely fill a barrel with water using a hose connected to another barrel at the same height. At best you end up with two barrels half full then the water stops flowing.

 

[sophiecentaur]

The result has to depend on the source pressure and the friction in the supply pipe and stuff. The syphon effect of an extra few cm of pressure difference may or may not compensate for the extra friction within the short hose.
Domestic water supplies can often be regarded as 'constant current' sources but, when someone else in the house is running a bath or when there's an irrigation system going on in parallel, the effective supply head of water can be very limited. Water supply companies in the UK are obsessed with the worry of actual negative supply pressure and the risk of unpleasant stuff getting back into the system. Under those circs, the hose to the bottom of the bucket would win easily.
You'd need to specify all the variables involved if you want a serious answer.

 

[rugerts]

If the flow rate from the tap is say 10 cups per min then 1 cup takes 6 seconds regardless of how close or far it is from the tap (provided all the water goes in).

However in the case of a hose in a bucket... The pressure at the bottom of the bucket increases as it fills up. This could in theory reduce the flow rate if the end of the hose is submerged and the supply is low pressure. For example you can't completely fill a barrel with water using a hose connected to another barrel at the same height. At best you end up with two barrels half full then the water stops flowing.

Thanks for the reply.
How do you know that the cup takes 6 seconds to fill regardless of distance from tap?

Also, for the case of the hose, are you saying that the hose being submerged reduces its constant pressure to a lower one, thus making it fill slower?

 

[rugerts]

The result has to depend on the source pressure and the friction in the supply pipe and stuff. The syphon effect of an extra few cm of pressure difference may or may not compensate for the extra friction within the short hose.
Domestic water supplies can often be regarded as 'constant current' sources but, when someone else in the house is running a bath or when there's an irrigation system going on in parallel, the effective supply head of water can be very limited. Water supply companies in the UK are obsessed with the worry of actual negative supply pressure and the risk of unpleasant stuff getting back into the system. Under those circs, the hose to the bottom of the bucket would win easily.
You'd need to specify all the variables involved if you want a serious answer.

I hadn't thought about the friction of the hose. Is it okay to assume that it's negligible? Why do you say extra friction in the short hose? How does the siphon effect come into play here whilst the hose is underwater? I am assuming the pressure from the hose is constant (at least at first), and am interested as to how the hose being underwater affects that pressure (and therefore the flow rate?). Could you elaborate a little more? Thanks for your reply and time.

 

[sophiecentaur]

I hadn't thought about the friction of the hose.

It would depend on the hose diameter. Not too much of an effect with 15mm internal diameter.

How does the siphon effect come into play here whilst the hose is underwater?

Being under the water or not will make no difference, There is still the same difference in atmospheric pressure between the top and the bottom of the length of hose (ignoring the mm or so when a small gap is left at the bottom. If you bear in mind that most house supplies are more than +1Atmosphere of pressure, (that's at least 10m) the 50cm of hose will make very little difference. That's about 5%. which would probably be a greater effect than internal friction.
For a uniform pipe all the way from a tank at 10m - the easiest model I` can think of - I would think a 5% flow increase could be a reasonable first stab. I'd be interested in other opinions on that.

 

[Nugatory]

How do you know that the cup takes 6 seconds to fill regardless of distance from tap?

Whatever the rate at which water flows out of the tap has to be the rate at which water flows into the cup (because there's there's nowhere else for the water to go) and the flow from the tap is not going to be affected by the position of the cup.

The hose/bucket problem is different because the length of the hose may affect the flow rate at the tap, as both @sophiecentaur and @CWatters have pointed out above. Although the effect may be so small as not to be interesting, the theoretical prediction is that a longer hose will reduce the flow and increase the fill time.

So you have your theoretical predictions. If this were a serious problem involving some incompletely understood physics, the next step would be experimentally test these predictions to see how good our understanding really is.

 

[rugerts]

Whatever the rate at which water flows out of the tap has to be the rate at which water flows into the cup (because there's there's nowhere else for the water to go) and the flow from the tap is not going to be affected by the position of the cup.

The hose/bucket problem is different because the length of the hose may affect the flow rate at the tap, as both @sophiecentaur and @CWatters have pointed out above. Although the effect may be so small as not to be interesting, the theoretical prediction is that a longer hose will reduce the flow and increase the fill time.

So you have your theoretical predictions. If this were a serious problem involving some incompletely understood physics, the next step would be experimentally test these predictions to see how good our understanding really is.

Very interesting. That's a little counter-intuitive (for me at least). I would have thought that since the water is closer to the source, it travels less distance and therefore fills up faster. Isn't there some time delay before the water hits the cup if it's further away? But, I guess if the stream is constant, then the rate at which it fills up is constant makes sense.

The hose scenario is also very interesting. Essentially there we've got that the longer the hose, the slower the fill rate because it travels a further distance? If friction and diameter are related, does this mean a larger diameter will make the bucket fill faster? Right now I'm thinking about electricity and wires, and how thicker wires are generally better for reducing resistance (which is sort of like friction?). What about decreasing diameter increasing pressure though? Wouldn't that mean less is coming out, but it's coming out at a faster rate?

Thanks for your replies!

 

[DaveC426913]

Very interesting. That's a little counter-intuitive (for me at least). I would have thought that since the water is closer to the source, it travels less distance and therefore fills up faster. Isn't there some time delay before the water hits the cup if it's further away?

Depends on where/when are you measuring empty and full from.

If you start the stopwatch from the moment the water leaves the hose/tap then sure, there will be a measurably longer time.

That would hardly be an accurate test though. You've started the stopwatch even though the bucket has not started to fill with water. Imagine, for a moment, the hose were so tall that it took 2 seconds to fall into the bucket. That's a 2 second error in your time.

You would more fairly start the stopwatch the moment the first drop of water hits the bucket.

 

[rugerts]

Depends on where/when are you measuring empty and full from.

If you start the stopwatch from the moment the water leaves the hose/tap then sure, there will be a measurably longer time.

That would hardly be an accurate test though. You've started the stopwatch even though the bucket has not started to fill with water. Imagine, for a moment, the hose were so tall that it took 2 seconds to fall into the bucket. That's a 2 second error in your time.

You would more fairly start the stopwatch the moment the first drop of water hits the bucket.

Thank you for the reply.
Yes, you're right I was thinking about timing it as soon as water left the tap. I guess since the whole goal is to measure how long it takes for the container to be filled, it's only being filled when water hits it, and so timing would start at when water hits the container. I see your point.
However, in everyday life, if I'm at a dining hall trying to fill up my water from the tap, and I'd like to do this as fast as possible, I would bring it closer to the tap, right?

 

[DaveC426913]

However, in everyday life, if I'm at a dining hall trying to fill up my water from the tap, and I'd like to do this as fast as possible, I would bring it closer to the tap, right?

The time it takes for water to fall a foot as opposed to a few inches is measured in milliseconds. i.e literally less than the time it takes for you to move the cup into a preferred position (which will happen slower than water can fall).

Remember, that "fall time" is only measured once. i.e if a cup filled from 4 inches takes 2 seconds, then a cup filled from a foot would take, say, 2.001 seconds.

If real-life efficiency were the goal, you'd do far better to concentrate on workflow (place the rack of cups as close as possible, hold a cup at the optimal distance to avoid splash but also avoid bashing the cup on the spigot, etc.)

 

[rugerts]

The time it takes for water to fall a foot as opposed to a few inches is measured in milliseconds.

i.e literally less than the time it takes for you to move the cup into a preferred position (which will happen slower than water can fall).

If real-life efficiency were the goal, you'd do far better to concentrate on workflow (place the rack of cups as close as possible, hold a cup a the optimal distance to avoid splash but also avoid bashing the cup on the spigot, etc.)

Are you saying then that being closer technically does fill it up faster, but the difference is so small it's not measurable?
For now, I've assumed that the water doesn't start until the cup is in the position I want it to be in.

 

[hmmm27]

Given we're talking about everyday household scenarios, I fail to see how the issue of fill time is a primary consideration.

The only scenario which it might make a noticeable difference would be an unpressurized jug on a shelf, with a hose with a spigot on the end. In that scenario, the liquid in the hose would help empty a near-empty jug faster.

What prompted the question in the first place, and what factors do you think are important, beyond the obvious ?

 

[rugerts]

Given we're talking about everyday household scenarios, I fail to see how the issue of fill time is a primary consideration.

The only scenario which it might make a difference would be an unpressurized jug on a shelf, with a hose with a spigot on the end. In that scenario, the liquid in the hose would help empty a near-empty jug faster.

What prompted the question in the first place, and what factors do you think are important, beyond the obvious ?

This came to mind when I was filling up my cup at the dining hall and there are people behind me waiting. I was wondering that if before I turned the water on, I brought the cup closer to the tap source (which is pretty constant) if this would result in my cup being full (full meaning "pretty much" to the brim, assume no spilling) in a less amount of time.

 

[hmmm27]

Ah... well, do the Guinness fill, but don't immerse the spigot in the water unless you like the taste of whatever cleaning solution they use.

More kinesiology with a bit of sociology thrown in, than physics, frankly. You'll want to use your trailing arm (assuming a generic sideways shuffle through the station) on the glass so you can move your body away from the station, fastest - thus giving the impression of best speed to the mob, behind. With a bit of practice - perhaps during non-busy hours - you could probably save upwards of a second, or so.

If water is the drink of choice for many, and assuming that station is a bottleneck, leaving the tap on would be more efficient. Memos should be sent, meetings arranged.

 

[rugerts]

Ah... well, do the Guinness fill, but don't immerse the spigot in the water unless you like the taste of whatever cleaning solution they use.

More kinesiology with a bit of sociology thrown in, than physics, frankly. You'll want to use your trailing arm (assuming a generic sideways shuffle through the station) on the glass so you can move your body away from the station, fastest - thus giving the impression of best speed to the mob, behind. With a bit of practice - perhaps during non-busy hours - you could probably save upwards of a second, or so.

If water is the drink of choice for many, and assuming that station is a bottleneck, leaving the tap on would be more efficient. Memos should be sent, meetings arranged.

Hahaha. I guess. But I'm genuinely interested in the physics behind this. I'm a little shocked that the consensus is that it makes no difference. I'm still confused as to whether there actually is no difference or there is no measurable difference for the cup scenario.

 

[hmmm27]

What factors do you think would be relevant to making a noticeable difference ?

 

[rugerts]

What factors do you think would be relevant to making a noticeable difference ?

If you increase the distance between the cup and the source enough, I'd think that's the most obvious difference since there will be a delay before the water reaches the bottom of the cup.

 

[jbriggs444]

If you increase the distance between the cup and the source enough, I'd think that's the most obvious difference since there will be a delay before the water reaches the bottom of the cup.

Making the simplifying assumption that the water leaves the tap with zero velocity and accelerates downward under the acceleration of gravity, that's 1 second delay for a cup 16 feet below the tap and 125 milliseconds for a cup 3 inches below the tap.

 

[hmmm27]

1/4 second for a 1 foot drop, a little over 1/3 for a 2. You might get something out of the siphon effect, where the weight of the water stream helps pull the water out. So technically - spillage aside - your best time is going to be : open the tap, place the glass at the lowest point, then raise it abruptly at the end and close the tap.

 

[rugerts]

1/4 second for a 1 foot drop, a little over 1/3 for a 2. You might get something out of the siphon effect, where the weight of the water stream helps pull the water out. So technically - spillage aside - your best time is going to be : open the tap, place the glass at the lowest point, then raise it abruptly at the end and close the tap.

Interesting. Thanks for sticking with me. What if we placed the glass at a low point and raised it as it was being filled at a relatively constant upward velocity; would this be even better than the abrupt bringing up?

 

[jbriggs444]

Interesting. Thanks for sticking with me. What if we placed the glass at a low point and raised it as it was being filled at a relatively constant upward velocity; would this be even better than the abrupt bringing up?

Suppose that you have (for instance) a falling stream of water that carries enough water every second to fill the glass. You hold the glass 16 feet beneath the tap and turn on the water. When the water hits the glass you start your stopwatch, raise the glass 16 feet to the tap as fast as you can and stop the stopwatch. Depending only on how fast you can lift, you will have succeeded in filling the cup in no time at all.

And splashing water all over everything.

 

[Nugatory]

I'm a little shocked that the consensus is that it makes no difference. I'm still confused as to whether there actually is no difference or there is no measurable difference for the cup scenario.

In the cup scenario, there is no difference. Look at #5 above again...
Or equivalently, you can consider a stack of imaginary horizontal surfaces between the faucet and the cup, each a fixed distance apart. How much water flows through each surface in one second? It has to be the same for each, which means that amount of water flowing through the bottom surface into the cup in one second is equal to the amount of water flowing through the topmost surface from the faucet - and it doesn't matter how many surfaces are in between..

This is, of course, assuming that you start your stopwatch when the stream of water reaches the cup. @jbriggs444 and others have covered the correction if you start your stopwatch at the moment that the faucet opens - but note that you also have to close the faucet before the cup is full, or the water that is still in flight will overfill the cup. That's why the simplest case to consider, and the one that you should understand first, is the one where you start the stopwatch when the water reaches the cup and stop it when the cup is full; it's the most direct measurement of the phenomenon you're trying to understand.

 

[rugerts]

In the cup scenario, there is no difference. Look at #5 above again...
Or equivalently, you can consider a stack of imaginary horizontal surfaces between the faucet and the cup, each a fixed distance apart. How much water flows through each surface in one second? It has to be the same for each, which means that amount of water flowing through the bottom surface into the cup in one second is equal to the amount of water flowing through the topmost surface from the faucet - and it doesn't matter how many surfaces are in between..

This is, of course, assuming that you start your stopwatch when the stream of water reaches the cup. @jbriggs444 and others have covered the correction if you start your stopwatch at the moment that the faucet opens - but note that you also have to close the faucet before the cup is full, or the water that is still in flight will overfill the cup. That's why the simplest case to consider, and the one that you should understand first, is the one where you start the stopwatch when the water reaches the cup and stop it when the cup is full; it's the most direct measurement of the phenomenon you're trying to understand.

Thank you again for replying. Ok, I believe I understand how filling up the cup works when you time it as it reaches the cup.
Would you comment on the other scenario, where we time once it leaves the faucet? Other than having to stop before the cup is full, what other flaws are there to this approach? To me, it seems that if I'm in line wanting to fill up my water cup as fast as possible, I'd be considering the case where I time once it leaves the faucet, and therefore bring the cup closer. In my own experiences, this seems to fill up faster.

 

[Nugatory]

Would you comment on the other scenario, where we time once it leaves the faucet?

If the cup has volume $V$ milliliters and the faucet flows $r$ milliliters per second... the faucet has to be open for $V/r$ seconds. Any less time and it won't release enough water to fill the cup, any more and it will release too much and the cup overflows. This is independent of the distance between cup and faucet.

 

[DaveC426913]

Are you saying then that being closer technically does fill it up faster, but the difference is so small it's not measurable?

Well yes, but you've changed the nature of your question.

For now, I've assumed that the water doesn't start until the cup is in the position I want it to be in.

In the real world scenario, you are not starting and stopping the clock at your convenience. Your goal is minimum duration of the whole task of getting water (because you have a hundred other things to do). That pretty much starts from when you finish your last task and reach for the glass.

Of the time it takes to complete the entire task, the difference between 'fill time from one foot' and 'fill time from 4 inches' is effectively zero. i.e. it makes no difference. Other facts will play a much larger in the take it takes to complete the entire task of getting someone a glass of water.

So, which scenario do you want to examine? The specific length of the time it takes to fill a container? Or the fastest way to complete the task of fetching water in the context of the real world? Two very different scenarios.

 

[DaveC426913]

This came to mind when I was filling up my cup at the dining hall and there are people behind me waiting. I was wondering that if before I turned the water on, I brought the cup closer to the tap source (which is pretty constant) if this would result in my cup being full (full meaning "pretty much" to the brim, assume no spilling) in a less amount of time.

Ah.

You'll shave about 1/5th of a second off your time - If you don't squander that 0.2s getting the cup into position. Do you think you can move it into position in 1/5th of a second?

Remember: it isn't about how fast you fill your cup - the clock starts the moment the guy in front of you steps out of your way - and stops the moment you step out of the way.

 

[rugerts]

Thank you all for the patience and great in depth replies. I think I'm satisfied with the answers now.

 

[hmmm27]

What if we placed the glass at a low point and raised it as it was being filled at a relatively constant upward velocity; would this be even better than the abrupt bringing up?

Technically of course, you should bring the glass to the top just as the water stream ends, and wait 'til the last moment before bringing it up to get the full advantage of the open-siphon effect, but a bit of showmanship might be achieved by smoothly bringing it up and catching the last drop in mid-air aways below the faucet.

In either case the important bit is keeping a straight face and seeing how long it takes before somebody asks what the heck you're doing.