Thread: Fluid dynamics: continuity equation

Hi, I'm trying to learn about the 'continuity equation' at the moment in fluid dynamics, and according to this page, it is based on the idea of a control volume, of arbitrary sides, Δx, Δy, and Δz, somewhere in the middle of a fluid stream:



...they then explain how the sum of mass flow rates into, and out of, each face of the cube will be equal to the rate at which mass accumulates within the it. According to them, the rate of mass entering a face of the cube, is a product of density, velocity, and the face area.

My first question...

How can they use density interchangeably with mass in their equations? Since density equals mass/volume, then for density and mass to be numerically equal, then the volume must be equal to 1 (it must be a unit volume). So using SI units, (expressing the mass in kg) then the unit volume must be 1m^3, but if Δx, Δy, and Δz, are supposed to be very tiny dimensions, then how can this be the case?.... I mean the entire amount of mass occupying a cubic metre, is not passing through a face of this tiny cube is it? only a tiny amount.

Hopefully you understand my question, if not just ask, and I'll try to rephrase,
thanks, bit4bit.

Noone knows?

The density is a property of the fluid and is not determined by the size of the control volume. Don’t think of the control volume as being 1m^3. It could be 1km^3 or 1mm^3. If the only inlet to the control volume is through the xz plane, then it doesn’t matter what the shape or size is of the control volume. All the geometry you need to know to find the mass flow rate into it is the area of the inlet plane.

The mass rate of fluid entering the control volume equals the area of the face it enters through, times the velocity times the density, which you already know.

I hope this helps.

Hi Bunbury, thanks for the reply,

If the only inlet to the control volume is through the xz plane, then it doesn’t matter what the shape or size is of the control volume. All the geometry you need to know to find the mass flow rate into it is the area of the inlet plane.

Ok, I'm fine with that, if mass is to enter or exit the control volume, then it can only do so through the surface of the volume. So long as the surface of the volume is closed, then geometry is irrelevant, but a simple cube is most likely the easiest example to use, and to define the theory on I suppose. (Also I think a further step to this theory is then to take the limits as Δx, Δy, Δz, and Δt tend to 0, i.e. to take a partial derivative, so this is probably easiest with the simple cube)

So basically they give 6 equations, each describing the mass flow rate through one of the 6 faces of the cube. They then assign 3 faces of the cube a positive sign (to represent the rate of mass flow IN), and the respective opposite 3 faces a negative sign (to represent rate of mass flow OUT). I think the idea of using very small increments (Δx, Δy, and Δz) for the cubes dimensions, is so that any particular path of fluid through it will be approximately linear (as opposed to being parabolic for instance, and exiting out of the same face that it entered), so the above method of signing 3 faces positive, and the other 3 negative, will hold true.

What I don't understand is how they actually mathematically define the mass flow rate through a SINGLE area of the cube. So considering now, a single face of the cube (e.g. ΔxΔy) how do they arrive at the expression for rate of flow of mass?...

rate of flow of mass = ρvΔxΔy

I mean, density is referring to the amount of mass per unit volume, so exactly what volume are they talking about here? for the mass and density to be numerically equal, the volume must be equal to 1 on whatever set of units we are talking about. If we are talking about the volume of the cube, then we are referring to the density inside of it, which doesn't make any sense if the mass is flowing from the outside to the inside, because we need to be talking about the density outside of the cube.

Another worry is that what if the mass passing through the lower half of the area is more concentrated than the upper half for instance. are we talking about an averaged density within the whole volume (single scalar value), or a physical 'scalar field' of density (a value for density at each point), not that the latter would make much sense either, as each density has to correspond to a particular volume anyway.

Anyway I hope you see what I'm gettin at... just trying to visualise this process really,

thanks, bit4bit

I mean, density is referring to the amount of mass per unit volume, so exactly what volume are they talking about here?

They are talking about an abstract standard volume, not any volume of the actual physical system.

For incompressible flow (e.g. liquid water) the density is constant, which simplifies the continuity equation. For compressible flow (gases) the density changes due to the pressure drop resulting from flow.

Does this help?

Thanks, so for an incompressible fluid, there would be a constant density, and we could pick any arbitrary volume, A, in the middle of the fluid stream, and obtain a value for the amount of mass contained by it. If we move A to another part of the stream, then the amount of mass contained by it will be the same as before.

But what about compressible flows then? By placing A at different points throughout the fluid stream, the amount of mass contained by it would be different in each case. so how exactly would we obtain a value for density in this case? In fact what type of value would this be? Is it represented in any special way?

I thought about having a scalar field for density - a value at each point in space and time. but then for this to make sense physically we would have to use some infinitesimally small volume at each point too? But then with the equation, density = mass / volume, then for an infinitesimally small volume we would have a infinitely large mass, which also can't be the case.

Thanks

You seem to be wandering into the realm of CFD - computational fluid dynamics - a tool that we use in engineering but leave the details to the CFD geeks. I'm not one of those, but you can read about CFD here:

http://en.wikipedia.org/wiki/Computa...fluid_dynamics

http://www.fluent.com/solutions/brochures/fluent.pdf

Thanks, I have heard of CFD, and briefly looked at it before. I think the reason for CFD though, is that for most fluid flow problems, we have to use statistical methods for finding certain values, which just wouldn't be practical to do by hand. We therefore use high FLOPS computers to crunch the numbers for us.

Nonetheless, surely this is only the case when dealing with a real practical use of the equations. At the core of any such problem is still the relatively simple continuity equation, for which purely theoretically, no statistical/numerical methods really enter into it....or do they? If so, how so?

I mean in the case of the question posed above about density, how would CFD be applicable?



Also, following the maths through, I thought of a possible answer to the question:

1.) We know that ρ = m/V.... i.e. Density is defined as the amount of mass per volume, and therefore to talk about density without a volume just doesn't make sense.

2.) From their expression of mass flow rate through a face:

mass flow rate = ρvΔxΔy
= (m/V)vΔxΔy

3.) The volume term in this equation will be equivalent to ΔxΔyΔz, (the volume of the cube), except it will represent a second cube attached to the side of the first at the face ΔxΔy. The density will therefore become (m/ΔxΔyΔz).

4.) The velocity of mass within this volume will be given by 'v = Δz/Δt', since Δz is the direction pointing towards the first cube, and the direction that mass will flow along into the cube.

5.) The equation becomes...

(m / ΔxΔyΔz) (Δz / Δt) ΔxΔy

= (m / ΔxΔyΔz) (ΔxΔyΔz / Δt)
= m / Δt (...volume cancels, leaving mass flow rate)

6.) You would have an extra hypothetical cube for each face of the cube, with the same principles as above... so six in total.

7.) When you take limits as Δx, Δy, Δz, and Δt tend to 0, then each extra cube will become infinitesimally small, just as the first, giving you a somewhat 'instant' value for density at a specific point...and it is still defined as being contained within a volume.


It's a bit long winded, and I'm still not sure if it would be right..what d'ya reckon?

Anyway, thanks

Good grief, that's a bit heavy for the post-turkey Thanksgiving afternoon. I wish I could help, but I'm not sure I understand where you're going with this.

We use semi-empirical equations for fluid flow and pressure drop and, as you correctly point out these are high level or statistical in nature. It is only if the problem is complex enough, or critical enough that we might get a bit more analytical, and then we generally just define the problem to a CFD spcialist who goes away for a month or two and comes back with some colored printouts, usually too late to be of any practical use because we had to buy the furnace or whatever without knowing the answer, to maintain the schedule. And that's no help to you.

Maybe someone else will jump in...

lol, what is thanksgiving anyway?

Well I've been doing some more reading, and I think I'm getting a better overview of it all.

We have three sets of equations describing fluid flow:

Continuity equation (conservation of mass)
Navier-Stokes equations (conservation of momentum)
Energy equations (...)

Each equation is based on a tiny control volume for which mass/momentum/energy conservation laws are defined on. This is then differentiated to give a differential equation representing the particular conservation law at any point within the fluid. (NSE is actually three equations, because momentum is a vector and had 3 spatial components)

We then need to find the density. For incompressible it's constant, for compressible we need a scalar field of density. So we can find a value for density at any point in the fluid. (corresponding to the point that we are using conservation laws on).

We can then find total changes in momentum etc, over particular paths/ surfaces, using path or surface integrals. within the fluid.

It seems the statistical side of things enters in because there is actually some kind of circular proofs within the laws, NSE anyway, so an exact definition isn't possible. I guess you're right - In the end, I'd have to plug something into a program to get a result, but I am genuinely interested in these equations nonetheless, and how they work.

Anyway, I'm gonna carry on reading, and look into some more programs (though most CFD programs I've looked at seem to be aimed at industry, and be quite expensive).

Thanks for all the help,
bit4bit

Originally Posted by bit4bit

lol, what is thanksgiving anyway?

I'm a foreigner here myself, but it seems to be some sort of harvest festival where we thank the Indians for giving us turkeys and they thank us for giving them smallpox.

Anyway, I'm gonna carry on reading, and look into some more programs (though most CFD programs I've looked at seem to be aimed at industry, and be quite expensive).

That is certainly true.

Thanks for all the help,
bit4bit

Sorry I couldn't help more.

I'm a foreigner here myself, but it seems to be some sort of harvest festival where we thank the Indians for giving us turkeys and they thank us for giving them smallpox.

Haha, I always thought it had something to do with Christmas..


Are there any programs out there, which could be considered for domestic use? For something like aircraft wing/compressor blade design (for RC aircraft/ gas turbines for example?).

Unfortunately I don't have a supercomputer either, but I could always leave a simulation running for a couple of days to complete