[socreate]

Dear all,

For particle flow in gas-fluidised bed or other kinds of fluid flow, why do we need to consider the pressure gradient force?

For stokes flow, the pressure difference around particle surface has been considered, so the pressure gradient has also taken into account. In this case, how to answer the above questions? One research asked me last week, and I found it was not easy to make him believe the pressure gradient force should be considered separately with the drag force.

Look forward to your help. Thank you very much.

Best wishes,
Sunny

[mprinkey]

There are different ways of looking that this. In gas-solids two-fluid continuous formulations, there are two forms...so-called Model A and Model B, per Gidaspow. One distributes the pressure gradient over all phases based on the volume fraction and the other does not. It really come down to how you want to define the drag coefficient. It is the same issue as comes up when discussing form drag versus skin friction:

https://en.wikipedia.org/wiki/Drag_%28physics%29

If the drag terms are formulated from first principles and the pressure and viscous effects can be separated, then this argument can be completely quantified and you can put each in their place. BUT, the drag terms are almost always empirical constructs. So, you have to understand what, if any, form drag correction was applied in its construction.

There is a lot more to say about these things--Gidaspow book is a good place to start. For large particles (or small computational domains containing small particles) it is possible to simulate the flow around the moving particles and capture the fluid and particle motion and these can shed more light on this discussion. But, the reality is that multi-particle coupling (cluster formation, particle "trains", etc.) tend to blow up the ideas of isolated particles with a prescribed drag. So, you seem to get back to a place were empiricism reigns.

Good luck.

[socreate]

Hi Prinkey,

Thanks to your reply. In both Model A and Model B, the pressure gradient force could be implemented in different form. In the Stokes flow, the drag coefficient on a single particles could be derived from the N-S equation and then revised to the DeFelice's form. In the case, the pressure difference around the particle boundary is considered. I think the key is the difference between the pressure gradient in pressure gradient force and stokes drag law.

I am still not clear and do't know how to answer above question in a simple way. Could I think that they are in different scale, such as macro-scale and particle scale.

[cryabroad]

I've been working with particle tracking for a long time and here is my opinion.

The pressure gradient we consider in the drag formulation is caused solely by the relative motion between the particle and the fluid. If there is no relative motion, there's no stagnation of flow at the front part of the particle, and thus there's no pressure gradient around the particle.

However, in a pressure-driven flow, imagine there is no relative motion between the two phases, the particle still feels the pressure difference on its surface. This is a fundamentally different mechanism from the pressure gradient caused by the relative motion mentioned above. This pressure gradient force is a external force field.