Q: Confusing Conservation of Mass Flow Rate and Conservation of Flow Rate

[tracker890 Source h]

Homework Statement

Confusing in the Conservation of Mass Flow Rate and Conservation of Flow Rate

Relevant Equations

Continuity Equation

ref. answer
Q： Why ρ1≠ρ2≠ρ3, but the answer says ρ1Q1+ρ2Q2=ρ3Q3 can be simplified into Q1+Q2=Q3?

 

[Lnewqban]

Q： Why ρ1≠ρ2≠ρ3, but the answer says ρ1Q1+ρ2Q2=ρ3Q3 can be simplified into Q1+Q2=Q3?

The summation of the entering volumetric flows must be equal to the single leaving volumetric flow.
Otherwise, compression or expansion would be happening within the control volume indicated in dashed lines.
But those flowing substances are in liquid form, which are considered incompressible.

Imagine that we are doing the mixing of two volumes of liquids by hand in one second.
The mix will occupy a volume of 0.1 + 0.3 = 0.4 cubic meters, but its density will be a value between 1 SG and 0.8 SG.
As we are mixing 1 part of water and 3 parts of alcohol, the mix density should be closer to 0.8 SG.

 

[tracker890 Source h]

The summation of the entering volumetric flows must be equal to the single leaving volumetric flow.
Otherwise, compression or expansion would be happening within the control volume indicated in dashed lines.
But those flowing substances are in liquid form, which are considered incompressible.

Imagine that we are doing the mixing of two volumes of liquids by hand in one second.
The mix will occupy a volume of 0.1 + 0.3 = 0.4 cubic meters, but its density will be a value between 1 SG and 0.8 SG.
As we are mixing 1 part of water and 3 parts of alcohol, the mix density should be closer to 0.8 SG.

I can understand it intuitively, but I hope to find a mathematical derivation as well. Currently, I have only reached the divergence theorem for cases with equal densities.
Currently in search of a proof for the divergence theorem with varying densities.

 

[pbuk]

I can understand it intuitively

I don't think you do, because (assuming the liquids are incompressible and there are no leaks), the only thing to understand is 0.1 + 0.3 = 0.4!

Think of a litre of liquid as a ball. If during an interval you have 1 yellow ball and 3 red balls entering the system on the left, how many balls will exit the system on the right (assuming the balls are not being compressed or lost)?

Currently in search of a proof for the divergence theorem with varying densities.

No, the densities do not vary, they are constant. Different but constant. And the proof for different but constant densities is the same as the proof for equal densities.

 

[tracker890 Source h]

I don't think you do, because (assuming the liquids are incompressible and there are no leaks), the only thing to understand is 0.1 + 0.3 = 0.4!

Think of a litre of liquid as a ball. If during an interval you have 1 yellow ball and 3 red balls entering the system on the left, how many balls will exit the system on the right (assuming the balls are not being compressed or lost)?No, the densities do not vary, they are constant. Different but constant. And the proof for different but constant densities is the same as the proof for equal densities.

Thank you for your detailed explanation, which reminds me of the concept of thermodynamics with its large systems and subsystems.

 

[haruspex]

the answer says ρ1Q1+ρ2Q2=ρ3Q3 can be simplified into Q1+Q2=Q3?

No, it cannot be simplified to that. The two statements are independently true. One represents conservation of mass, while the other represents conservation of total volume. The second is true because the liquids do not interact in a way that would make the total volume anything other than the sum of volumes (assuming they enter at the same temperature).

 

[gmax137]

The second is true because the liquids do not interact in a way that would make the total volume anything other than the sum of volumes (assuming they enter at the same temperature).

Is this true for any two liquids? Can one liquid dissolve in another?

 

[haruspex]

Can one liquid dissolve in another?

Quite possibly, but I would say it is unlikely for the two given here. Certainly the author doesn’t think so.

 

[gmax137]

Certainly the author doesn’t think so.

Yes, but this seems (to me) to be precisely the OP's doubt.

 

[haruspex]

Yes, but this seems (to me) to be precisely the OP's doubt.

I was replying to the comment in post #1 wherein the OP seemed to think the author had derived the volume conservation from the mass conservation.

 

[Robert Jansen]

Is this true for any two liquids? Can one liquid dissolve in another?

From decades past: I recall a demonstration in my 8th grade Chemistry class when 500ml of water was combined with 500ml of methanol, resulting in 950ml of mixture.

 

[erobz]

From decades past: I recall a demonstration in my 8th grade Chemistry class when 500ml of water was combined with 500ml of methanol, resulting in 950ml of mixture.

That kind of puts a wrench in the problem statement...

 

[gmax137]

From decades past: I recall a demonstration in my 8th grade Chemistry class when 500ml of water was combined with 500ml of methanol, resulting in 950ml of mixture.

That kind of puts a wrench in the problem statement...

that's what I was thinking

 

[jbriggs444]

Quite possibly, but I would say it is unlikely for the two given here. Certainly the author doesn’t think so.

The nearest liquor store has plenty of solutions of alcohol in water with varying concentrations and trace constituents.

 

[haruspex]

The nearest liquor store has plenty of solutions of alcohol in water with varying concentrations and trace constituents.

Yes, I should have clarified that I was interpreting the question as being whether one liquid could take up another without changing volume. But of course, even water taking up salt probably changes volume.

Post #7 at https://www.physicsforums.com/threads/volume-contraction-water-methanol-mixture.522195/ makes an interesting point about the molecular interactions.

 

[pbuk]

that's what I was thinking

This is a question from an exam or exercise. When you answer questions in an exam or exercise you use the information in the question. There is no information in the question about any reduction in volume, so the answer does not require this to be taken into account.

 

[gmax137]

This is a question from an exam or exercise. When you answer questions in an exam or exercise you use the information in the question. There is no information in the question about any reduction in volume, so the answer does not require this to be taken into account.

I think it is a bad question, and the OP should be commended for pointing out the problem with it.

EDIT:
Because this is a common classroom demonstration:

From decades past: I recall a demonstration in my 8th grade Chemistry class when 500ml of water was combined with 500ml of methanol, resulting in 950ml of mixture.

 

[jbriggs444]

Laura Ingalls Wilder in "Farmer Boy" recalls that it is possible to add a cup of popcorn to a full cup of milk without having the milk overflow.

Apparently, this feat is reproducible.

 

[erobz]

In my opinion it's an interesting fact that the problem setter did not consider (nor did I). I also don't think the OP was aware of it i.e. it was not the crux of their issue. IMO The crux of their issue was not recognizing the independence of conservation of mass and conservation of volume. If it were another liquid besides alcohol, what is the likelihood we would be discussing that aspect...

With the extra knowledge of potential volume contraction, it wasn't the best choice for liquids to be mixed. However, without that extra knowledge the intent seemed clear (to me). When I said "that could throw a wrench into the problem statement" I was thinking in the sense of an increased level analysis.

In other words,... if it were on an intro fluids exam and some student knew of this effect, I would expect them to either completely ignore it, or alert the professor for clarification. Probably causing the prof to furrow their brow and the other students to shake their head in disbelief!

 

[gmax137]

I don't think continuity of volumetric flow should be taught, since eventually it will have to be unlearned (systems with varying temperature and / or pressure). Just my opinion.