I am trying to model the multiphase flow of air with a suspension of liquid-solid particles. I thought that a logical first step would be to ignore the particles and model the flow of air in the system. I imagine that the validity or relevance of this model would depend on the volume density of particles in the air, the particle size distribution, the mass density of each particle, and other factors. My question is:

Is there a dimensionless number or parameter that would indicate the validity of ignoring the particles in a model of the multiphase flow?

Thanks in advance for any help.

Elliot

Hi THere,

THe study of particle motion in a fluid flow is surprisingly complicated, but if you consider the simplest case of (say) dust particles whose sizes are less than the mean free path in the ambient gas and whose typical density (say rho1) is much greater than the density of the gas around it (say rho0), then the equation of motion of the dust particle with position r (vector) is

d2r/dt2 = -gamma*(dr/dt -v(r,t)) (equation I)

where v(r,t) is the flow velocity of the ambient gas, 2 denotes the second derivative (acceleration here) and gamma is a parameter whose dimension is 1/time. gamma is known as the fluid drag parameter (it depends on the density of the fluid rho0, the density of the dust rho1 and the diameter of the dust particles). If one writes

gamma=1/tau (equation II)

then tau has unit of time. For large gamma (small tau) the particles are rather like tracers in the flow, i.e. you don't need to consider the particles in the model.

Now large gamma means small tau, this means that tau has to be much smaller than the characteristic time scale of the flow. For example tau has to be much less than the time it takes for a sound wave to propagate from one boundary to another. Or tau has to be much smaller than the time it takes for the flow itself to move from one boundary to another (for imcompressible flows).

For a two-dimensional flow (like a model of the terrestrial atmosphere on a very large scale - hundreds of miles/km) tau is given by the following:

tau=2*rho1*a/sigma (equation III)

where rho1 is the dust particle density (i.e. the density of the material out of which the particles of dust are made).

'a' is the typical radius of the dust particle

sigma is the surface density of the gas (density per unit of surface).

So this would work for dust particles most probalby, but not for small liquid drops in suspension in the air (because their size is typically larger than the mean free path in the gas). Also one has to check that external forces (like gravity) can be neglected (they are small compared to the right hand side term in equation I).

If this is not the case of your flow then you will have to go to references (that I can provide you if you wish) to see the full equations.

Patrick.

Patrick,

Thanks for your helpful and quick response to my question. You have understood my question exactly--I want to know whether the particles act like tracers in the flow and can be ignored in the model.

You gave a formula (Equation III) for tau in a 2-D flow. You explained that tau should have units of time, but the only variables in Equation III are rho1, a, and sigma, none of which would include time as far as I can tell. What would the units of rho1 and sigma be? Also...is there a similar formula for a 3-D flow?

You explained that to determine whether the particles act like tracers, tau should be compared to the time it takes for a sound wave to propagate from one boundary to another. Could the distance that the air travels from the inlet to the outlet divided by the characteristic speed be used as the characteristic time scale of the flow?

Finally, at the end of your posting you explain that the analysis would probably work for dust particles but not for liquid drops, which is what I have. Does that mean that I cannot use the equations you presented to determine if the particles behave as tracers?

Thanks again for your thoughtful response.

Elliot

HI, yes you're right. Sorry, the equations I have are completely non-dimensional, so let's me refer you to the original papers, this will answer also your question for 3D. Whether your drops will be represented by this formalism or not depends on the exact value of the parameters (density, size, etc..).

Here are the references I got:

Tanga and Provenzale, 1994, Physica, vol. 76, p. 202.

Maxey and Riley, 1983, Phys. Fluids, vol. 26, p.883

Crisanti et al. 1992, Phys.Fluids A vol. 4, p.1805. (pay attention to the A)

Also if you can find it: Provenzale 1999 (maybe 1998), Ann. Rev. Fluid. Mech., in press (or already appeared?).

Good luck, Patrick.

Patrick,

Thanks for all of the help. I'll look up the references.

Elliot

Hi concerning gamma=1/tau here is the correction

tau=rhod*a/(rhog*c)

where rhod is the dust particle

a is the radius of the dust

rhog is the gas density

and c is the sound speed

This gives tau the exact unit of time and it is good for 3D.

Cheers, Patrick.

In studying multiphase flow, the interaction forces between the air and the liquid-solid phases are of great importance. The interactions such as drag, lift spin, etc. must be evaluated. Depending on the problem be studied, inertial, viscous, and gravitational forces may also be of importance. Suggest that you read the Theory of Interacting Continua by K. R. Rajagopal.