I Solving the Mystery: Which Tank Fills First?

[fahraynk]

I originally thought tank K because the area in tank J is much larger than the area in the pipe connecting J to I
But then I realize Bernoulli's equation does not include volume but only includes height.
H_1, H_2 = height of water tank J, height of water in pipe between tanks J and I
P, p, V = pressure, density, velocity
$$P_1+pgH_1+\frac{1}{2}pV_1^2=P_2+pgH_2+\frac{1}{2}pV_2^2$$
Pressure terms are the same, so is density. Velocity = 0, gravity cancels out $H_1=H_2$
So even in the small pipe connecting J to I, the water level is going to be the same as the height of water in the tank J.

Is this correct?

Also note the pipe from C to D is blocked off.
Also I assume A would be trivial solution so water must exit pipe faster than the tap fills tank A

 

[jbriggs444]

The water is dripping in. Flow velocity will be negligible.

 

[Regla]

It really depends on what point do you consider the tank is 'full', let's say that the point full is when the tank is overfilling. I think the tank H would get filled first because the connection JI goes higher than connection JL.

 

[jerromyjon]

I say F will fill up (overflowing) first!

 

[jbriggs444]

It really depends on what point do you consider the tank is 'full', let's say that the point full is when the tank is overfilling. I think the tank H would get filled first because the connection JI goes higher than connection JL.

There is no path into tank H.

 

[Regla]

There is no path into tank H.

whoops that's true. Then my guess is F. Thanks.

 

[CWatters]

Tank G fills up first. The water flows A B C D G. No water goes into J. No water goes into E.

Darn I didn't spot the blocked pipe. Ok so yes it's F.

 

[phinds]

It's G

 

[Nugatory]

It's G

You've missed the blocked pipe C to D.

 

[Nugatory]

A and B fill to the level of their outputs; C fills to the level of its outlet into J; J fills to the level of its outlet into L; L fills to the level of the top of F; and F fills and overflows. Once F is full, every drop of water into A causes another drop to overflow from F and the system is in steady state.

 

[phinds]

You've missed the blocked pipe C to D.

Damn ! Thanks.

CWaters even SAID it was there and I still couldn't see it.

 

[fahraynk]

A and B fill to the level of their outputs; C fills to the level of its outlet into J; J fills to the level of its outlet into L; L fills to the level of the top of F; and F fills and overflows. Once F is full, every drop of water into A causes another drop to overflow from F and the system is in steady state.

So, basically, the area doesn't matter of tank J, it won't push the water through the pipe higher than its own water level? Tank J is roughly 27 times the cross sectional area then the pipe... but it doesn't matter right?

 

[Nugatory]

So, basically, the area doesn't matter of tank J, it won't push the water through the pipe higher than its own water level? Tank J is roughly 27 times the cross sectional area then the pipe... but it doesn't matter right?

Not as long as the rate at which water enters the tank is small compared with the rate at which a single pipe can carry the water out - the water level will not rise above the height of the lowest flowing pipe. The illustration suggests that we're dealing with a dripping faucet which will easily be handled by a single pipe - think about an air conditioning unit's condensate drain tray.

 

[Gurjeet210]

It's K

 

[Nugatory]

It's K

Can't be - nothing flows from J to I because the level in J doesn't rise above the outlet from J to L.

 

[Stressil]

I choose L due to pressure and flow

 

[RonL]

I say F will fill up (overflowing) first!

I agree with you, F will overflow before any others

 

[davenn]

I choose L due to pressure and flow

no ... look again ... F is at a lower level than L which fills F

Like phinds and a couple of others, I originally was going for G as I didn't see the block between C and D

 

[DrGreg]

Here's the picture for the slow-drip steady-state solution:

Spoiler

 

[Dr_Zinj]

View attachment 208203

I originally thought tank K because the area in tank J is much larger than the area in the pipe connecting J to I
But then I realize Bernoulli's equation does not include volume but only includes height.
H_1, H_2 = height of water tank J, height of water in pipe between tanks J and I
P, p, V = pressure, density, velocity
$$P_1+pgH_1+\frac{1}{2}pV_1^2=P_2+pgH_2+\frac{1}{2}pV_2^2$$
Pressure terms are the same, so is density. Velocity = 0, gravity cancels out $H_1=H_2$
So even in the small pipe connecting J to I, the water level is going to be the same as the height of water in the tank J.

Is this correct?

Also note the pipe from C to D is blocked off.
Also I assume A would be trivial solution so water must exit pipe faster than the tap fills tank A

Tank F will fill up first

 

[The Wizard]

Flow Chart is A - B - C - J - L - F

No tank will fill before F and as F is open at the top it will constantly overflow, therefore no other Tank can fill fully.

 

[Ron G]

F fills first; L prevents any flow reaching I from J because inlet to I is higher than outlet to L. J must fill before any tank above (A, B, or C).
BUT ASSUMPTIONS ARE EVERYTHING.
F fills first only if 1) all the pipes are the same diameter and of the same material, 2) the heights of all tanks are identical, 3) the flow from the faucet does not exceed the pipe capacity (as indicated by the drip), and 4) therefore, flow through system is not sufficient to impact dynamic pressure caused by line friction and elbows. With these assumptions, the only law needed is water seeking lowest level.

Once you violate any of these assumptions, you will need to work through the system performance using the Bernoulli's equation; and the complexity of the system increases substantially, probably requiring a full dynamic modeling exercise.

EXAMPLE: Let's just change one assumption: that the faucet inflow exceeds the outflow capacity of Tank A, even when tank A is full.
When the faucet flow rate exceeds the outflow pipe capacity, A will begin to fill to over flowing. As it does, the pressure at the outlet will increase, thereby increasing the outflow rate. Thus, the inflow to i B will be greater than the outflow from B to C, and B will begin to fill. Since the outlet from B is lower relative to the tank height than is the outlet from tank A, at some point, the outflow from B may match that from A. The height difference of the two tanks as well as the depth of the outlet of the tank will determine whether B will over flow. It does appear that tank B is taller than tank A and that the A outlet is not as deep at the B outlet, thus the level in B will stabilize below over flow. This same exercise must now proceed through each box.Reference https://www.physicsforums.com/threads/which-tank-fills-first.921715/

 

[Matt Chipman]

No liquid will ever reach D, E or G...due to the closed off pipe from C to D...so none of them

 

[Mike Bergen]

View attachment 208203

I originally thought tank K because the area in tank J is much larger than the area in the pipe connecting J to I
But then I realize Bernoulli's equation does not include volume but only includes height.
H_1, H_2 = height of water tank J, height of water in pipe between tanks J and I
P, p, V = pressure, density, velocity
$$P_1+pgH_1+\frac{1}{2}pV_1^2=P_2+pgH_2+\frac{1}{2}pV_2^2$$
Pressure terms are the same, so is density. Velocity = 0, gravity cancels out $H_1=H_2$
So even in the small pipe connecting J to I, the water level is going to be the same as the height of water in the tank J.

Is this correct?

Also note the pipe from C to D is blocked off.
Also I assume A would be trivial solution so water must exit pipe faster than the tap fills tank A

Through observation, looks like G would fill first' others will just move the water to other places
IMHO

 

[jbriggs444]

Through observation, looks like G would fill first' others will just move the water to other places
IMHO

If you review the original post in this thread you will see why that is not correct. Several others have already missed seeing:

Also note the pipe from C to D is blocked off.

 

[RonL]

F fills first; L prevents any flow reaching I from J because inlet to I is higher than outlet to L. J must fill before any tank above (A, B, or C).
BUT ASSUMPTIONS ARE EVERYTHING.
F fills first only if 1) all the pipes are the same diameter and of the same material, 2) the heights of all tanks are identical, 3) the flow from the faucet does not exceed the pipe capacity (as indicated by the drip), and 4) therefore, flow through system is not sufficient to impact dynamic pressure caused by line friction and elbows. With these assumptions, the only law needed is water seeking lowest level.

Once you violate any of these assumptions, you will need to work through the system performance using the Bernoulli's equation; and the complexity of the system increases substantially, probably requiring a full dynamic modeling exercise.

EXAMPLE: Let's just change one assumption: that the faucet inflow exceeds the outflow capacity of Tank A, even when tank A is full.
When the faucet flow rate exceeds the outflow pipe capacity, A will begin to fill to over flowing. As it does, the pressure at the outlet will increase, thereby increasing the outflow rate. Thus, the inflow to i B will be greater than the outflow from B to C, and B will begin to fill. Since the outlet from B is lower relative to the tank height than is the outlet from tank A, at some point, the outflow from B may match that from A. The height difference of the two tanks as well as the depth of the outlet of the tank will determine whether B will over flow. It does appear that tank B is taller than tank A and that the A outlet is not as deep at the B outlet, thus the level in B will stabilize below over flow. This same exercise must now proceed through each box.Reference https://www.physicsforums.com/threads/which-tank-fills-first.921715/

It seems like you are trying to set the question of what flow from the faucet, with the assist from gravity, will all tanks reach the full mark before anyone of them overflows. I think (gut feeling) there might be an answer, though the math is far over my head.

 

[calinvass]

When the water level exceeds the height of the end of the pipe the water will flow through it. For this rule the tanks need to have no lid You only need to be careful a about details. (The answer is clearly F).
For example if you let the water go between C and D and block the pipe between D and G, after the water level nearly reaches the top of D it will start to run through the pipe and fill E as well. Then D and E will fill up to the top at the same time.
Placing a lid on C makes no difference on the result but placing a lid on F might prevent filling it up and the air pressure inside will go up.

 

[Skeptic Tom]

It is tank F, if it's not, I will eat my hat !

 

[davenn]

No liquid will ever reach D, E or G...due to the closed off pipe from C to D...so none of them

this is incorrect ... obviously you haven't read the earlier posts

It is tank F, if it's not, I will eat my hat !

you need to read the earlier posts as well

the answer "F" has been well established

 

[Number 42]

Obviously it is H
Just follow the lowest outlet of the tanks and end up in H
Assuming All pipes are about same dimensions.

Pipe to G is closed, assuming it is not a drawing error

 

[EspressoDan]

Depends on the flow-rate from the tap.

 

[jbriggs444]

Obviously it is H
[...]
Pipe to G is closed, assuming it is not a drawing error

As is the pipe to H.

 

[davenn]

Obviously it is H
Just follow the lowest outlet of the tanks and end up in H
Assuming All pipes are about same dimensions.

Pipe to G is closed, assuming it is not a drawing error

it's obviously not H
you really should read earlier posts and also look closely at the diagram

 

[Dadface]

There are so many variables to consider and it would help at the outset to be told what simplifying assumptions can be made. Just one variable is the radius of the outlet pipes from the tanks. Each pipe could have a radius such that more water enters the tank per second than leaves the tank per second and for all values of increasing pressure as the tank fills. In this case tank A could fill first.

 

[davenn]

Each pipe could have a radius such that more water enters the tank per second than leaves the tank per second and for all values of increasing pressure as the tank fills. In this case tank A could fill first.

this has also already been discussed and is irrelevant
The water is only dripping into the A tank

 

[Dadface]

this has also already been discussed and is irrelevant
The water is only dripping into the A tank

The fact that the water is "only dripping" into the tank does not make my point irrelevant and drip rate and drip size are yet other variables that determine what will happen.
Take another look at the diagram and compare the drop size to the radii of the tubes. I know the diagram is only a schematic but it is the main information we have to go on. And by comparing the sizes it seems that the tubes are capillaries with very small internal radii. If that's the case then the effect of surface tension on flow rate will be considerable.
The point I'm trying to make is that the question as presented is not specific enough.

 

[davenn]

The point I'm trying to make is that the question as presented is not specific enough.

it is presented well enough for a basic solving of the problem
all you are doing is turning it into something much more difficult
don't over think or over complicate it

 

[Dadface]

I know I'm probably over complicating it but I like to think of the problem in terms in terms of a real situation in addition to a situation where certain simplifying assumptions are made. Imagine water dripping into a bucket with a very narrow capillary tube sticking out the side, along with the question:
Will the bucket ever fill to the brim?
Without further information you couldn't give a definite answer but you could describe the conditions needed for it to fill to the brim.

 

[CWatters]

A reasonable assumption to make would be that the problem is intended to be solvable without additional info. It's unlikely the setter expected people to answer that is was "Unsolvable for the following 10⁶ reasons".

 

[davenn]

A reasonable assumption to make would be that the problem is intended to be solvable without additional info. It's unlikely the setter expected people to answer that is was "Unsolvable for the following 10⁶ reasons".

exactly :)

 

[sophiecentaur]

I know I'm probably over complicating it

Yep. The thread got out of hand after the point where we came up with the Obvious Answer. (But that's Physicists for you.)

 

[Dadface]

It seems that some people here think there's only one possible answer to the question. If that's the case what's the answer to the question below which illustrates the point I'm making and which illustrates what happens at tank A. Does the tank fill or does it not fill? The answer is not obvious and we can't tell what the answer is because what happens depends on the flow rates of water entering and leaving the tank. And we don't know what those flow rates are. So there are two possible answers and not just one. Similarly there are two possible answers to the original question.

 

[sophiecentaur]

It seems that some people here think there's only one possible answer to the question.

Of course there are dozens of answers, depending on the depth you want to discuss. But this puzzle is not a Physics Puzzle. You could find it in any old magazine or on Facebook. You could introduce string theory into it if you want but it would just exclude a vast number of people who might enjoy dealing with the initial problem - as presented.

 

[Dadface]

Of course there are dozens of answers, depending on the depth you want to discuss. But this puzzle is not a Physics Puzzle. You could find it in any old magazine or on Facebook. You could introduce string theory into it if you want but it would just exclude a vast number of people who might enjoy dealing with the initial problem - as presented.

Although the problem can be discussed in terms of physics a detailed knowledge of the subject is not necessary in order to understand it. It's simple and just needs a bit of common sense and general knowledge. It's something that people can experience in everyday life, for example leave a tap running and if the sink doesn't drain quickly enough you get an overflow. It happened my house recently the result being that there was a very wet bathroom floor to clean up.

Please look again at the question I posed in post 42. It describes a situation where there is no definite answer because the relevant flow rates are not known. So there are two possible answers to my question and by similar reasoning two possible answers to the original question. With the original question Tank F is one possible answer. But Tank A is the second possible answer.

I have been referring to physics in this thread because after all it is a physics forum. But that does not prevent any non physics person from tackling the question and seeing that Tank A is a possible answer. The main thing they need to know is that the water does not necessarily leave the tank at the same rate as it enters the tank.

 

[NTL2009]

Although the problem can be discussed in terms of physics a detailed knowledge of the subject is not necessary in order to understand it. It's simple and just needs a bit of common sense and general knowledge. It's something that people can experience in everyday life, for example leave a tap running and if the sink doesn't drain quickly enough you get an overflow. It happened my house recently the result being that there was a very wet bathroom floor to clean up. ... .

See post #2. The picture shows it dripping, not flowwing. You are overthinking it. It's already 'tricky' with levels and ports that are not obviously blocked. Plenty tricky for the intended audience w/o going into flow rates past the 'drip'.

 

[Dadface]

See post #2. The picture shows it dripping, not flowwing. You are overthinking it. It's already 'tricky' with levels and ports that are not obviously blocked. Plenty tricky for the intended audience w/o going into flow rates past the 'drip'.

I have seen post 2 and I ask you to please see my question in post fourty two where the water is dripping. The question mimics the situation at Tank A in the original question where in both cases water moves in and water moves out.
When a question is posted in a physics forum its reasonable to assume that the intended audience is a group of people many of whom have an interest in physics. Therefore it's reasonable to refer to some physics which has been done in this thread. But the question can also be tackled in terms of general knowledge.
Try to forget physics and please look again at my question in post fourty two. What would your answer be if you were told:
1. The hole is extremely big?
2. The hole is extremely small?
3. The size of the hole is unknown?

 

[fahraynk]

I have seen post 2 and I ask you to please see my question in post fourty two where the water is dripping. The question mimics the situation at Tank A in the original question where in both cases water moves in and water moves out.
When a question is posted in a physics forum its reasonable to assume that the intended audience is a group of people many of whom have an interest in physics. Therefore it's reasonable to refer to some physics which has been done in this thread. But the question can also be tackled in terms of general knowledge.
Try to forget physics and please look again at my question in post fourty two. What would your answer be if you were told:
1. The hole is extremely big?
2. The hole is extremely small?
3. The size of the hole is unknown?

2 situations are possible for the first tank. 1) the water is entering faster than it can exit through the pipe. Then the first tank wins and it is a trivial solution.
Case 2) the pipe can pass water faster than it is entered into the system. In this case, the obvious solution will win.

Am I wrong?

 

[Dadface]

2 situations are possible for the first tank. 1) the water is entering faster than it can exit through the pipe. Then the first tank wins and it is a trivial solution.
Case 2) the pipe can pass water faster than it is entered into the system. In this case, the obvious solution will win.

Am I wrong?

Yes, tank F wins but if water enters tank A faster than it leaves then tank A wins.

 

[shihab-kol]

I don't know much of this but shouldn't all of them (except blocked ones) have the same amount of water to maintain equilibrium ?
Most probably though I am wrong.

 

[jbriggs444]

I don't know much of this but shouldn't all of them (except blocked ones) have the same amount of water to maintain equilibrium ?
Most probably though I am wrong.

An "equlibrium" would not imply that all tanks have the same amount of water. Instead, it would mean that the situation remains the same from one moment to the next. In the problem at hand, we have an evolving situation that does not remain the same from one moment to the next. This is the opposite of equilibrium.

When one tank overflows, we will have an equilibrium situation. That equilibrium state is nicely portrayed in the spoiler by @DrGreg in response #19.

[Provided that we do not concern ourselves with the possibility of pipes narrow enough to restrict the flow unexpectedly]