Help with math puzzle problem

[modamag]

=== Boundary Condition ===
1. Water pour into the bucket at rate of 100 lpm thru 10 inlet nozzle
2. Water is drained at rate of 100 lpm through 5 drain point

=== Exception Condition ===
1. Every 250 minutes one of the inlet nozzle need to be turn off for 1 minute
2. Every 700 minutes one of the outlet drain need to be plugged for 20 minutes

What is the MINIMUM volume of the storage tank to prevent overflow?

Thanx guys.

 

[paulr]

Water is entering faster than it is leaving on average, so bucket will eventually overflow, unless flow rate through drain points increases as pressure increases from height of water in the bucket.

 

[Dances with Flashlight]

I agree with Paulr. Think of it this way. Rates of flow are equal when none of the stated exceptions are in force. But the stated exceptions, restricting flow in and restricting flow out, do not apply equally, outflow being more restricted than inflow, resulting in a continuing net increase in water volume present in the tank over time. Since there are no stated exceptions regarding tank volume or pressure or any other factor which would alter the net flow rate, there is no specific tank volume that will suffice to prevent overflow. In other words, no minimum volume could prevent the eventual overflow. Increasing the volume, say to a zillion cubic liters, would only defer the inevitable as the length of time involved in the problem is not stated to be finite.

 

[Crenshaw]

thats one big bucket. After the first minute is will have 500l in there....?
Crenshaw

 

[jugg2]

+1 with everyone else. The bucket needs to be infinately large since the water level will continue to rise.

 

[DonShock]

The only way this would ever work is if you exactly balanced the gain and loss by adjusting the timing and quantity of the gains and losses for whenever the cycles happen to line up.

For example, given the 250/750 minute times above, the cycles will line up every 3500 minutes. During that time you will have the nozzles shut 14 times and the drains shut 5 times. If the nozzle shut volume for the total cycle is more than the drain shut volume for the cycle, the tank will eventually go empty. If the drain shut volume is more for the cycle, it will eventually overflow. Only when the total volumes are exactly equal for these two conditions will the tank not go either full or empty. If the totals were equal, then you would just have to allow for the mismatch due to the timing of the cycles.

To make this work you would need to lengthen the time the inlet nozzles are shut or shorten the time the drains are plugged.

This is basically what we have every day with a water supply system. Demand (draining) changes quantity all the time, and the available pumps (inlets) to fill the storage tanks have limitations. The normal means of control is to use the change in tank level to set when pumps start and stop or to change their speed. As the mismatch causes the tank to fill, the pumps slow or downshift as the level gets higher and higher. As the mismatch starts emptying the tanks, you force the pumps to speed up or additional pumps to start.

It's virtually impossible in the real world to have a static balance like this question would require. You need to have some means of dynamic control.

 

[LightInTheWallet]

Minimum volume for storage depends mainly on duration of intended use.

 

[Crenshaw]

Minimum volume for storage depends mainly on duration of intended use.

+1

 

[modamag]

OK. We all go the level 1 question correct. INFINITELY LARGE BUCKET!
Here's my reasoning.

inlet loss of 10 (nozzle) * 1 (minute) / 250 min period = 4% downtime loss
outlet loss of 5 (drain) * 20 (minutes) / 700 min period = 14.3% downtime loss.
inlet > outlet ~ overflow is inevitable.

The second part of the problem is.
add one more Exception condition to the inlet nozzle.
3. Every 100 minutes one of the inlet nozzle need to be turn off for 10.28 minutes.

The problem now becomes inlet loss = outlet loss
What is the minimum bucket size

 

[js]

The 250 min and 700 min cycles evenly fit into a 3500 min cycle, as mentioned.

That is 5 drain plugging cycles and 14 inlet restriction cycles.

Each drain plug cycle means 80 lpm for 20 mins instead of 100 lpm. So 5 of these 20 min cycles means a delta of 2000 liters

Each inlet restriction cycles means 90 lpm for 1 min. So 14 of these means 140 liter delta.

So for each 3500 min large cycle, we increase the water in the tank by 1860 liters. And thus it will eventually overflow.

Now if we add in the 100 min cycle, that means an extra delta of 35 cycles times 10 lpm delta times 10.28 minutes equals 102.8 * 35 = 3598 liter

Thus, the net loss is 1738 liters every 3500 minutes.

So in the long run, there is no danger of overflow. But what is the minimum size? We must examine the individual cycles.

At the start, inflow=outflow. But then 100 minutes in, one of the inlet nozzles is turned off. Big deal. The tank is still empty (assuming it started empty). Same for the 250 min cycle.

The 700 min cycles and the 100 minute cycles line up with each other. We never have a 700 min event without an accompanying 100 min event. Assume the 250 event is out of cycle with these two.

That means that for 20 minutes one outlet is shut off, and for 10.28 minutes one inlet is shut off. The imbalance is 10 lpm for 9.72 minutes = 97.2 liter tank minimum needed to prevent overflow.