Fluid flow & mixture

[Taylor]

This doesn't have any real application anywhere; it's just something I thought of the other day. After posing the question to my Aerospace Engineering class, I don't have much more clarity.

Suppose there are two five-gallon buckets set next to each other. The first bucket is filled with Fluid A, the second bucket is filled with Fluid B. The two fluids are different, but they have identical densities. Bucket 1 has a hose in it, siphoning the fluid out onto the ground. A second hose goes from Bucket 2 into Bucket 1, running at the same rate as the first hose. Essentially, the level in Bucket 1 neither rises nor falls for the duration of the experiment.

After some amount of time, Bucket 2 will be empty, and Bucket 1 will be full of some mixture of Fluids A and B.

My question is this: At that time, what is the makeup of the solution? What percentage is Fluid A, what percentage is Fluid B?

If the experiment were performed again, would the results be repeatable? To what degree?

[Lil' Lavery]

Don't have the time to fully think through the problem, but it seems like it would be setting up some differential equations to find the theoretical answer.

[EricVanWyk]

If the two fluids are perfectly and instantaneously mixed, the solution (ha!) is a lot easier. If they aren't, I'm not sure how I'd start.

Edit: If I have my math right, it would be 1/e = approx 37% remaining.

[Al Skierkiewicz]

Doesn't this problem rely on where the hoses are placed within the buckets? i.e. how much of bucket 1 passes through hose 1 and how much of bucket 2 flows through hose 1 in the transfer? Is there an assumption that the fluid from bucket 2 mixes thoroughly with bucket 1 before passing through the first hose onto the ground?

[billbo911]

This doesn't have any real application anywhere; it's just something I thought of the other day. After posing the question to my Aerospace Engineering class, I don't have much more clarity.

Suppose there are two five-gallon buckets set next to each other. The first bucket is filled with Fluid A, the second bucket is filled with Fluid B. The two fluids are different, but they have identical densities. Bucket 1 has a hose in it, siphoning the fluid out onto the ground. A second hose goes from Bucket 2 into Bucket 1, running at the same rate as the first hose. Essentially, the level in Bucket 1 neither rises nor falls for the duration of the experiment.

After some amount of time, Bucket 2 will be empty, and Bucket 1 will be full of some mixture of Fluids A and B.

My question is this: At that time, what is the makeup of the solution? What percentage is Fluid A, what percentage is Fluid B?

If the experiment were performed again, would the results be repeatable? To what degree?

Doesn't this problem rely on where the hoses are placed within the buckets? i.e. how much of bucket 1 passes through hose 1 and how much of bucket 2 flows through hose 1 in the transfer? Is there an assumption that the fluid from bucket 2 mixes thoroughly with bucket 1 before passing through the first hose onto the ground?

First off, I don't think this experiment can be reproduced.
Here is my thinking (flawed as it may be).
Suppose there are two five-gallon buckets set next to each other.
If these buckets are on the same level, which really isn't stated either way, then the level of the fluid in bucket #1 can not remain the same throughout the duration of the experiment.
With the density of the two fluids being the same, the flow through hose #2 could not be the same as through hose #1, unless acted upon by an external force other than gravity, such as a pump. Again, assuming the buckets are on the same level.

As Al mentioned, we would also need to know the height of the ends of the hoses and where they enter and exit each bucket.

Basically, there are too many variables at this point to answer the question.

[Taylor]

Is there an assumption that the fluid from bucket 2 mixes thoroughly with bucket 1 before passing through the first hose onto the ground?
Yes. Anything else would be impossibly complicated.
With the density of the two fluids being the same, the flow through hose #2 could not be the same as through hose #1, unless acted upon by an external force other than gravity, such as a pump. Again, assuming the buckets are on the same level.
Sure. Pump it.

Good points, all. I tried to leave the original design intentionally vague so that people would follow the spirit of it, not try to lawyer the details (where have we heard that before?), important as the details may be. My question lies not so much in the mechanical process of the fluid transfer as it does in the resulting mixture. While I do realize the process does affect the mixture, for the sake of this, assume these effects are minimized. I also did not put any of my thoughts in the original post as I did not wish them to cloud any opinions or discussion.

Please, proceed.

[EricVanWyk]

Here are the simplifications I made:

1) Instantaneous perfect mixing.
2) flow in = flow out.
3) Track the amount of A in bucket A only. Don't bother with the other bucket or the other fluid.
4) Make the volume of the bucket 1 unit.

This simplifies things down to A' = -A; The amount of A that leaves the bucket is proportional to the amount of A left (times the amount of B entering).

This becomes A(B) = e^-B

or, pulling back from 1 unit to 5 gallons

A(B) = 5 gallons * e^-(B/5 gallons)

[Ether]

Let Vdot be the volume flow rate in gallons per minute (gpm) in each hose.

Let B(t) be the amount (in gallons) of Fluid B in Bucket1 at time t minutes.

At any given time t, Fluid B is entering Bucket 1 at the rate Vdot gpm, and is draining out of Bucket 1 at the rate Vdot*B(t)/5 (assuming perfect and instantaneous mixing).

Thus we have the differential equation dB(t)/dt = Vdot(1-B(t)/5)

See attached screenshot "buckets.png" for the DE solution in Maxima*. Assuming instantaneous and perfect mixing, the answer is independent of flow rate Vdot. So pick Vdot=5 gpm and t=1 minute to calculate the solution

3.16 gallons (63.2%) of Fluid B in Bucket1.


*Or you could solve the DE manually by separating the variables and integrating. See attached screenshot "separate_vars.png".

[JamesBrown]

Let Vdot be the volume flow rate in gallons per minute (gpm) in each hose.

Let B(t) be the amount (in gallons) of Fluid B in Bucket1 at time t minutes.

At any given time t, Fluid B is entering Bucket 1 at the rate Vdot gpm, and is draining out of Bucket 1 at the rate Vdot*B(t)/5 (assuming perfect and instantaneous mixing).

Thus we have the differential equation dB(t)/dt = Vdot(1-B(t)/5)

See attached screenshot "buckets.png" for the DE solution in Maxima*. Assuming instantaneous and perfect mixing, the answer is independent of flow rate Vdot. So pick Vdot=5 gpm and t=1 minute to calculate the solution

3.16 gallons (63.2%) of Fluid B in Bucket1.


*Or you could solve the DE manually by separating the variables and integrating. See attached screenshot "separate_vars.png".





People should skip creating these individual engineering problem threads and just create one thread called "Stump Ether" seems to me regardless of discipline Ether is up to the challenge.

[DonRotolo]

Stump Ether would be a good challenge all right!

To the young'uns out there, after a lifetime as an engineer, the gentleman seems to have been paying attention to his surroundings, and applying the concept of continuous learning.

Some engineers learn more and more about a narrower and narrower field until they are experts. But give them a problem outside their field and they aren't engineers anymore, they're stumped.

Other engineers learn more and more, but on a variety of topics. They are very good at what their employer pays them for, but also reasonably competent in many other disciplines.

In a vary large company (or research facility), the first type does pretty well. In smaller companies, the latter do very well - or if they are in larger companies they become the managers of the first type.

Is one better than the other? No, but which do you think is more fun?