Lets say I have a 5000 Gallon cylindrical tank filled with fluid. If I use a pump to suck fluid from the bottom of the tank, force it through a short hose and dump it back in the top of the tank, how many gallons would the pump run before “all” the fluid will have passed through the pump? Because of the continous mixing I think theoretically the answer is never, but is there a limit reached? I would blindly guess 1.5 to 1.67 times the tank capacity.
It’s going to depend on the flow rate of the pump, and the flow characteristics of the tank. If you have plug flow through the tank, which is never as easy as you would think, it can turn over pretty quick.
If we assume continuous and complete mixing, which is also not as easy as you think, it can take a long time. For example, take your theoretical 5000 gallon tank, and a 100 gpm recirculation pump. Initially we have 5000 gallons that has never been through the pump. At the end of one minute we have 4900 gallons that has never been through the pump.
At the end of two minutes we have 4802 gallons that has never been through the pump, 198 gallons that has been through the pump once, and 2 gallons that has been through the pump twice. At the end of three minutes we’re down to only 4706 gallons that has never been through the pump.
At the end of an hour you’ll have pumped 6000 gallons, and there will still be 1488 gallons that has never been through the pump. Two hours, and although you’ll have pumped the tank capacity 2.4 times, there will still be 443 gallons that never saw the pump. At the end of 2.5 hours you’ll have reached 95% recirculation, 3.8 hours to get to 99%, and 4.4 hours to get to 99.5%.
How close do you want to get it? At the end of 5.7 hours you’ve reached 99.9% recirculation, after having pumped the tank volume 6.8 times over.
If you build a diaphram pump that will push 5000 gal in one pulse, the answer would be one pulse.
Of course the “short pipe” would probably contain more than 5000 gal, hmm.
Wait, no, there was no time restraints,
We take a 5000 gal capacity piston pump, place it inside the tank, then slowly force the fluid through 1/4" tubing to the top of the tank.
Im guessing we would lose more than the tubes capacity to evaporation by the time the operation was complete.
Bouncing off Bill Door’s answer (which is pretty damn good without resorting to calculus):
If you’ve really buggered the design and used a highly viscous fluid, you can form huge eddie currents in your tank. This will cause the fluid to rotate inside the eddie and (almost) never mix with the moving fluid - which can increase the cycle time by an arseload.
Anybody have a few weeks of time booked out at a research facilty for dynamic fluid flowmodelling?
I think that if liquid perfectly mixes with the rest of the tank as you pump it back in, that the fraction of the fluid that has never been through the pump will decay exponentially with a time constant that is equal to the tank volume divided by the pump flow rate.
This is the sort of problem students in a differential equations class are introduced to the first day. As posted up thread, the answer is never, and the function of unpumped fluid in the tank decays exponentially (e^(-t/T)) Where t is time, and T is a time constant determined by the volumn of the tank vs the pump flow rate…think of it as the time required to pump one tank volume.
The problem is common enough in engineering that there is a rule of thumb: After 5 time constants, call it good. At this point 99.3% of the liquid has been through the pump.
As stated by Kevbo
“As posted up thread, the answer is never”
Is why I call BS.
The device called a “pump” is never defined. If I design a “cylindrical” tank with an identicle “cylindrical” tank positioned below it, this one with a piston at one end, I could push all the water back up in one stroke, minus the volume of the return pipe.
Practicality is where its at. You hook a one horse power pool pump to filter your 50 gal aquarium you will cycle that water once every few seconds. Of course your fish would be living in a blender.
Also like Engineerly describes you could create a highway between the intake and outake unless you took the shape of the “tank” into consideration.
There was an interesting thread recently about fluid flow, which got me to thinking. Perhaps somebody can tell me if I am right, but it seems that this theoretical answer is only true for a uniform fluid. In reality, however, fuids are quantized into molecules. Eventually, the volume of unpumped fluid is less than the volume of a molecule. Then at some point, that last molecule goes through the pump.
I have long since forgotten all my math so I’d be interested in finding out if my intuition is correct.