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tracker890 Source h
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- Homework Statement
- Confusing in the Conservation of Mass Flow Rate and Conservation of Flow Rate
- Relevant Equations
- Continuity Equation
The summation of the entering volumetric flows must be equal to the single leaving volumetric flow.tracker890 Source h said:Q： Why ρ1≠ρ2≠ρ3, but the answer says ρ1Q1+ρ2Q2=ρ3Q3 can be simplified into Q1+Q2=Q3?
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.Lnewqban said: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 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!tracker890 Source h said:I can understand it intuitively
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 said:Currently in search of a proof for the divergence theorem with varying densities.
Thank you for your detailed explanation, which reminds me of the concept of thermodynamics with its large systems and subsystems.pbuk said: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.
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).tracker890 Source h said:the answer says ρ1Q1+ρ2Q2=ρ3Q3 can be simplified into Q1+Q2=Q3?
Is this true for any two liquids? Can one liquid dissolve in another?haruspex said: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).
Quite possibly, but I would say it is unlikely for the two given here. Certainly the author doesn’t think so.gmax137 said:Can one liquid dissolve in another?
Yes, but this seems (to me) to be precisely the OP's doubt.haruspex said:Certainly the author doesn’t think so.
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.gmax137 said:Yes, but this seems (to me) to be precisely the OP's doubt.
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.gmax137 said:Is this true for any two liquids? Can one liquid dissolve in another?
That kind of puts a wrench in the problem statement...Robert Jansen said: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.
Robert Jansen said: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's what I was thinkingerobz said:That kind of puts a wrench in the problem statement...
The nearest liquor store has plenty of solutions of alcohol in water with varying concentrations and trace constituents.haruspex said:Quite possibly, but I would say it is unlikely for the two given here. Certainly the author doesn’t think so.
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.jbriggs444 said:The nearest liquor store has plenty of solutions of alcohol in water with varying concentrations and trace constituents.
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 said:that's what I was thinking
I think it is a bad question, and the OP should be commended for pointing out the problem with it.pbuk said: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.
Robert Jansen said: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.
Conservation of mass flow rate refers to the principle that the mass of fluid entering a system must equal the mass exiting the system, ensuring mass is conserved. Conservation of flow rate, on the other hand, generally refers to the volume flow rate, which is the volume of fluid passing through a section per unit time. While mass flow rate remains constant regardless of changes in fluid density, volume flow rate can change if the fluid density changes.
Mass flow rate is calculated using the formula: \[ \dot{m} = \rho \cdot Q \]where \( \dot{m} \) is the mass flow rate, \( \rho \) is the fluid density, and \( Q \) is the volumetric flow rate. This ensures that the mass of fluid remains conserved as it flows through different sections of a system.
Yes, the volumetric flow rate can change while the mass flow rate remains constant if there is a change in the fluid density. For instance, if a fluid is compressed or expands due to pressure or temperature changes, its density will change, affecting the volumetric flow rate even though the mass flow rate stays the same.
Confusing mass flow rate with volumetric flow rate can lead to significant errors in engineering calculations and design. For example, in systems where fluid density changes, assuming constant volumetric flow rate rather than mass flow rate could result in incorrect sizing of pipes, pumps, or other equipment, potentially leading to inefficiencies or system failures.
For incompressible fluids, the density remains constant, so the conservation of mass flow rate and volumetric flow rate are directly proportional. However, for compressible fluids, density can change with pressure and temperature, so while the mass flow rate remains conserved, the volumetric flow rate can vary significantly. This distinction is crucial in applications like aerodynamics and gas pipelines, where fluid compressibility cannot be ignored.