How can the probability of a particle entering a fluid flow be modeled

[BernieM]

Ok so maybe this will take some work to wrap your heads around. Maybe not. I sure know that I have had some real doozies of headaches working with this and visualizing it. Rather than look at particle capture in a gold sluice as a CFD problem I have been trying to visualize it as a statistcal problem.

Long story short I developed a gold sluice that works better than just about anything else out there. Those who would like to look, check out my site (no it's not spam it's real site) www.hmresearch.net. Perhaps a look at the device will help to understand the problem.

Now most approaches to gold recovery using gravitational methods, model the flow of the fluid in the system and particle interaction in that flow. This can become super complex pushing CFD to the limits, especially when you further introduce particles of various sizes also creating turbulence, etc.

But I wanted to look at the whole issue from a different perspective. Statistical. Leaving as much of the CFD out of the picture as possible. So here is the basic problem stated as best I can. If you don't like how I worded it don't attack me like is common here, just don't respond and let someone else if they would like. So here is the basics in the model:

Perspective: A cross sectional view of a pocket that drops below the normal bottom surface level in a fluid channel.

What is the probability that a particle will be re-introduced into the flow once it has dropped below the surface, given that there is a varying amount of activity and exchange occurring in that pocket between the fuid above and the particles in the pocket below it.

Not sure that comes out right, but it's basically this. The pockets in my new fangled sluice are essentially spiral wells ranging in size from 1" diameter at the top of the well to 5/16" diameter at the bottom of the well and the wells are 7/16" deep.

So the diameter of the pocket shrinks with depth.

Fluids flow across the top of the pocket (this is the normal fluid channel bottom, the top of the pocket), moves particles ranging in size from perhaps 1/2" diameter to just a few microns in diameter. Their densities may range from 2.8gm/cc to 21 gm/cc, with the bulk of the particles either 2.8 gm/cc (90% probability) and 5.8gm/cc(9% probability). The remaining 1% of the particles will range in density from 2 to 21 gm/cc with varying distributions depending on geology of the particles source.

But let's just say that we have 1% gold in this sample. And it has a density of 19.3 gm/cc. This will simplify it some. So 90% particles that have a density of 2.8gm/cc, 9% particles that have a density of 5.8gm/cc and 1% particles that have a density of 19.3gm/cc. Ranging in size randomly from 1/2" to a few microns, with the number of particles that are smaller exponentially larger in number than the number of particles that are large.

Without going into complex CFD, I want to just view this model as a cross sectional view of the spiral well pocket. And show that at the top at the interface between the mostly fluid/some particles boundary that an exchange is occurring, (due to the fact that the pocket can never get totally full), and that heavy particles have a higher statistical chance to stay in the pocket vs lighter particles because they have a higher probability of dropping quickly into the material below and exit the turbulent exchange area, reducing their likeliehood of leaving the pocket. And that as particles move about in the upper zone of the spiral well where there is a lot of exchange going on between fluid flow, introduction of new particles, and particles leaving the well, that a heavier particle regardless of geometry (surface area to weight ratio), is more likely to drop deeper into the pocket as surrounding particles are disturbed and displaced by other moving particles than lighter particles.

Furthermore that the deeper the particle sinks into this well, that the more and more certain it is that it will remain in the spiral well and not dislodged or re-enter the particle stream. As well that a smaller particle has a higher probability to be captured in this well than a larger particle (reason being that less particles have to move out of the way for a small particle to find a space below to drop further into the pocket vs a larger particle which would require more particles simultaneously making space for it to drop.)

Also that the disturbance and movement of the material from stirring of the particles due to fluid flow above diminishes to near zero at the bottom of the spiral pocket.

Furthermore that there are a number of rows of these spiral wells between where the material and fluid are input and where they will exit.

So what is the simplest statistical model to give a fairly accurate description of the probabilities of particle capture based on it's size and density, having to pass n number of rows of these wells to exit the system. ?? Is that fair?

Anyone want to take a stab at it? If I am unclear just ask and I will do my best to clarify.

 

[mfb]

So what is the simplest statistical model to give a fairly accurate description of the probabilities of particle capture based on it's size and density, having to pass n number of rows of these wells to exit the system. ?? Is that fair?

I don't think you can avoid fluid dynamics or good experimental data to make a reasonable model for that.

 

[BernieM]

Just using a force like gravity for example, where you have a large particle above small ones being jostled around, say that the large particle is 7 times larger in area than the smaller ones, that in order for it to sink, 7 particles would have to move out of the way at the same time. A smaller particle say that was only 3 times larger than average particle size would then only have to have 3 particles move out of the way from underneath it to sink away to the 'next level'.

In this kind of a model the water is mereley a force to move things around and would be considered to be a fairly equal and homogenous force throughout the pocket, which however would diminish in forcefullness the farther down in the pocket one went.

So rather than get into where the water is moving in what way, just consider all fluid motion in the pocket as a particle mover which at the top of the pocket carries particles into the pocket from the source farther upstream, and at the bottom of the pocket carrying particles that stray too close to the bottom lip of the pocket on out of the pocket and on down the stream below.

So although some fluid dynamics might enter in, I am thinking it can be super-simplified.
Just like gravity being a uniform force acting on all particles pulling them down, that the fluid motion would be somewhat random in the pocket with a bias toward the bottom lip, and supply a fairly even uniform force of movement and jostling in the pocket.

 

[BernieM]

Ok so how about an even simpler model:

3 sizes and 3 possible densities of marbles.
Each density of marble has a color. White, Black and Gold, that corresponds to the density of the marble.
Gravity pulls on all equally and normally.
The marbles enter a container from one direction, leaving at the opposite side of the container provided they are at the rim of the container and able to get out a hole there (there is enough other marbles to fill the container so that the marble may exit through the hole).
There exists a mysterious force in the container that has a tendency to act on marbles near the top of the container more than those at the bottom of the container.
This mysterious force acts in a primarily random fashion with a determinant bias in moving marbles in the general direction of the exit hole.
At the bottom of the container the only force at play is gravity.

Is this so different in some ways to the ping pong ball machine used in gambling and lotteries?

I have seen statistical analysis of these kinds of machines before and fluid modeling was not required to calculate statistical bias or odds of a particular number coming up based on minor idiosyncrasies in the ping pong balls.

 

[mfb]

Is this so different in some ways to the ping pong ball machine used in gambling and lotteries?

Those still need some physical simulation or experimental data, as long as the balls are not all identical (identical balls are probably a good idea for a lottery).

 

[BernieM]

If I could just establish a reasonably accurate and simple model for the relationships and interactions, with variables in place of the actual data, that would be sufficient. That is what I am attempting to do here is to get a good simplification of the problem and solve the gold capture problem using statistics instead of fluid dynamics. Or relying very minimally on fluid dynamics. I mean, down on the molecular level, where solids and fluids mix and interact with boundary layer effects, molecular attraction, viscosity, etc., there is a whole lot of randomity going on and very little ideal fluid flow characteristics!

Kind of like a macroscopic view of the real world and how it appears (CFD) but in reality it's all quantum mechanics and probability down at the most basic levels of existence (Statistics).

 

[mfb]

there is a whole lot of randomity going on

Sure, but you still need values for probabilities of whatever process you want to consider. You cannot just make them up.