Mr. Farooq Wahab,

I came upon this question rather late and I see that there are very many good answers. I cannot add much to them but since you seem to need a more physical answer, I will try to give you one. You need to draw a figure thus: Take a plane wall for simplicity and line up the spheres of the packing. The second layer of spheres will sit in the saddle points of the first layer. This will continue into the bed. If you determine the local voidage you will see the pattern you wish to understand. If you cannot see it, please write back and I will draw it for you. In the meantime, please look up this (rather dated) paper: Kubie J, Broughton J (1975) A model of heat transfer in gas fluidized beds, Int. J Heat and Mass Transfer, Vol 18, pp. 289-299.

This lower packing density near the tube wall is just an edge effect: the spheres can no more pack densely because the presence of the tube wall prevents them to do so. This perturbation has an influence on a limited but non negligible distance, depending on the both diameter of the tube of that of the spheres. For getting the same packing density near the tube wall, it would be necessary to intercalate smaller spheres betwen the bigger ones, a somewhat difficult thing to do just in this special area of the packing.

Alain

Dear Prof. Celzard, Thanks for your reply. One can understand that the particles cannot pack very well along the wall. I wanted to know about the main reason behind the "oscillatory" behaviour of porosity as a function of radius. The oscillations are greatest along the walls. Why does porosity oscillate?

OK ! Oscillatory behaviour is extremely frequent in packings. The same behaviour is observed when looking at the pair correlation function. Salvatore Torquato at Princeton published a lot of papers about packings and predicted and quantified this kind of behaviour. As this corresponds to a rather high level of mathematics for me, better looking directly at his papers.

Alain

Dear Prof. Alain,  Dr. Torquato's work seems to be very interesting but is beyond my expertise. Are you aware of qualitative explanation for this oscillation. If you look at the figure I attached in the question, you will see an interesting pattern. The maximum void fraction is indeed at the wall but minimum at the particle radius. The second maximum is found at the first particle radius. Do you agree with the following qualitative explanation given in a 1958 paper (pasted below). If we accept this view, this will imply that the wall region is more tightly packed (i.e. particle layer just next to the walls is the tightest). I was under the impression that wall region is usually less dense than the bulk. Is it always true?

 "In each case the void fraction has the limiting value of 1 at the wall, reaches a minimum at 1 particle radius in from the wall and a maximum at 1 particle diam. from the wall, and continues cycling for some distance into the bed. These results were rather unexpected, in that it had not been anticipated that the wall effect would continue to have an effect beyond 1 particle diam. from the wall. The explanation however is quite simple and consists of recognizing that because of the wall a layer of spheres is oriented on the wall. In fact each sphere in the bed which touches the wall is thus uniquely oriented. The net effect will then be that more spheres than usual are located adjacent to the wall, resulting in a concentration of sphere centers at 1 radius from the wall and a consequent minimum in porosity at that point. Similarly at 1 diam. from the wall there will tend to be a greater porosity than normally because this is the point of

maximum porosity for a sphere which is oriented at the wall. "

Looking at your figure, it's pretty much what I would expect to see.  

Think about it like this:  At the wall, the particles line up like a ball bearing system i.e. they touch each other and form a nearly linear chain (assume the particle diameter is much less than that of the container)  around the edge of the container.  In the limit, the contact points with the wall are all vanishingly small points which leads to a low solid packing fraction at the wall.  

Conversely, one particle radius inside the wall, the particles contact along diameters which leads to a very high solid packing fraction.  In 2D (i.e. packing of circles) the limits are respectively 0 and 1, but the packing in 3D is not so extreme due to the curvature of the particles in the z direction of the bed.

As you move further into the bed, the ordering effect of the wall becomes less strong and I would guess that you approach something like random close packing in the bulk of the bed.

(This is essentially rephrasing what you are quoting from the 1958 article) 

Having said all of this, it is debatable what use it is to calculate void fraction on this kind of length scale.  It's a continuum property, hence it is only really meaningful when averaged over a number of particle diameters.

Since this is a geometric effect, yes, it is always true (at least when you have a mono-dispersed particle size distribution and any kind of planar surface.)

In principle you could imagine a metastable state in which it didn't occur (these are thermal systems, so they will unlikely achieve an equilibrium state), but it's spectacularly unlikely that you will observe it for any reasonable number of particles. 

Hope this helps?

Adam

At the wall there is hexagonal grid of spheres.If the diameter of the tube is big enough then the next layer of balls is arranged in accordance with the layer near the wall.

But if the diameter of tube is not big then the second layer does not fit exactly to the first layer because of the curvature .  Part of the second layer fits and  on some parts the second layer does not fit.

The further away from the wall and closer to the axis of the tube then more successive layers do not fit and alignment becomes random.

Of course you can artificially place balls in a regular hexagonal 3D grid with more random configuration near the tube wall.

Dear Dr. Adam,  Thanks for your reply. You interpreted porosity oscillation because of ordering effect of the particles along the walls and the random packing in the bulk. If we were to have an ordered packing of spheres into tube the oscillations would continue into the bulk as well without damping. Is this correct?

I always had an impression for a long time the overall the wall region (5-6 particle diameters) was loosely packed than the bulk. It seems that I was wrong. Can we say that wall region (5-6 particle diameters) is more tightly packed, in general, than the bulk? We can see the first layer of particles has the minimum porosity.

One case where I was wondering that one might obtain a loose ordering along the walls was when a shear thickening slurry is packed by pressure filtration into a tube. Since the shear rate is highest at the walls, would we get very loose disordered packing along the walls and the bulk would be tighter? Is this possible?

Hi Farooq,

Yes, if you had a completely ordered packing throughout the bulk then the oscillations continue throughout the system.  You can think of this as a kind of macroscopic crystal.

Generally the effect of the walls is to reduce the density of the packing.  This is because you cannot have a hexagonal packing because the wall would pass through the centres of the particles that would otherwise form the triangles.

The packing at the walls is highly ordered, it's just that the ordering that the walls impose is less dense than random close packing in the bulk.  The only way to increase the density of the wall packing that I am aware of is to dimple the wall in an ordered pattern that would seed something like an HCP packing instead.

Dear Dr. Adam, Thanks for this interesting discussion. In the chromatographic literature (packed bed columns), sometimes permeable walls are reported and sometimes very densely packed wall region. I guess this could be due to the fact all spheres are not of the same size and the small particles fit in the interstices. Are you aware of this possibility that the wall region can be denser than the bulk, perhaps due to wall friction, even if all spheres were of the same size?

Secondly, if the original suspension  (to be packed) were shear thickening, would it be very less dense around the walls, because the shear rate is highest at the walls?

One example, one question raised by late Georges Guiochon ten years ago seems to be still unanswered.

"Analytical HPLC columns are not radially homogeneous. On the contrary, a rather homogeneous core is surrounded by a thick layer of packing along the wall. The thickness of this layer is around 30 to 50 particle diameters. Why this layer is sometimes less permeable and probably denser than the core, as in our experiments and those of Baur et al. [4], while in other cases it is more permeable, as in the experiments of Knox and co-workers [1,2] and Eon [3], and why this is so, remains open for discussion."

Hi Farooq,

well, there is no true equilibrium state for the packing of the system, so in principle any geometrically accessible state is allowed and what you observe will depend on how you fill the bed.  Non-spherical, or poly disperse packings could certainly violate the argument that the packing density is lower at the (near) planar wall.

Re: The shear thickening argument - I suspect that (a) the particles will slip near to the wall, and, (b) the suspension will 'dilate' when sheared, which would suggest a lower packing density at regions of high shear,

Have you thought about the 'wetability' of the packing, and indeed the wall, as a possible additional cause of flow maldistribution?  

Dear Dr. Burbidge,

Are you aware of any studies which show the packing behavior (in terms of radial porosity distribution) of shear thickening and shear thinning suspensions?

You raised a very interesting point about the wettability of walls and the packing. Presumably, these effects should be negligible in a chromatography column. First, the percolating mobile phase is chosen to fully wet the porous silica. Secondly, a typical HPLC column is made of stainless steel which is almost mirror polished to perfection. I assume that most organic solvents and mixtures do wet the walls but as far as I know, nobody has reported quantitative contact angle studies for typical mixtures on such column materials.

Not off the top of my head, and not in this particular context.

 I would suggest that you start to look at some of the suspension rheology work, where both theoretical and experimental approaches have considered the effects of shear on local radial distribution function (i.e. the probability of finding another particle at a certain distance from the centre of the current particle).  Norm Wagner's group at U Delaware, Jan Mewis et al at KU Leuven for experimental stuff, and John Brady et al from Caltech for a theoretical perspective would be good names to start with. Anke Lindner at ESPCI and Philippe Coussot (can't remember where he is based) are also worth checking out for larger scale particle systems. 

One thing to bear in mind is whether the suspension is colloidal or not, since the behaviour will be controlled by different physics depending on the particle scale.

"why does the radial porosity profile oscillate near the tube walls and finally becomes a straight line as we move into the bulk?" If you consider the radial prosity profile in a very thin slice of porous medium, the porosity fluctuates all along the axis.  If  successive porosity profiles in packed beds of spheres  are averaged,  a typical profile is obtained which is shown in several works such as this one of R.F. Benenati, C.B. Brosilow,(AIChE J. 8  (3) 1962. 359.).  The averaged profile is different in the case of flat plates for example (see : Powder Technology 121  2001. 138–148). Considering spherical particles, the average radial porosity  profile can be flattened using mixing of spheres. For spheres with uniforme size, the porosity is higher at the vicinity of the wall, then the velocity of fluid is higher at the wall. Using an adequately corrugated wall can  flatten the porosity profile at the wall. Anyway, if the size of particles is very small compared to the diameter of the tybe (Dtube/dpart>50), the wall effect are usually neglectable.

In a random packing, the porosity is very high near the wall because many spheres touch it. Thus, the porosity at a distance of one sphere radius from the wall is very low. At two radii from the wall, the porosity is once again high, but not as high as at the wall. If the spheres are very small compared to the radius of the cylindrical container, these oscillations diminish fairly quickly. As noted above, Benenati and Brosilow show this effect clearly. Simulations of random packing eliminate wall effects by using periodic boundaries. See Tory et al.(Can. J. Chem. Eng. 51, 484-493, Errata ibid 622) and Jodrey and Tory (Phys. Rev. 32, 2347-2351, Errata ibid 675).   

Thank you for your input and the references. All of us agree that the porosity at (exactly at the walls) is highest because the spheres cannot fit into the wall. The next layer of particles we see a minima in porosity (at 0.5 particle diameter).  Then we see average porosity fluctuations for few particle diameters. However, there is still an unanswered problem. We are assuming that the flow in the wall region is higher than the bulk then this is what we do not see experimentally. In fact, people (Guiochon et. al) have shown that the flow at the walls in a packed chromatography bed is slower than the center. For example, a probe compound is injected to the column and if we place detector at various places at the end of the packed bed, we would see the probe comes out first in the center and then a small amount comes out from the wall region. Sometimes people see the opposite (in some cases). 

How would you account for the slowest velocity near the wall region (not at the walls), if the average porosity is higher at the wall regions in a packed bed. Certainly the velocity is not like the Poiseuille flow in a packed bed. It must be chaotic, because the particles are usually porous. The typical column diameter is 4.6 mm and the particle size is usually 5 micron (standard packed beds). This has been confirmed experimentally by many researchers. This seems like a paradox.

One factor that needs to be considered is the stick boundary condition at the wall.

Imagine a highly dispersed suspension of spherical is packed at high shear rates into a column- one may potentially observe that the probe compound injected into the column, the probe compound has a slightly higher velocity than the central bulk. I do not have any direct evidence but it is observed indirectly by the shape of the exiting band from the column.

Now the same suspension is packed slowly in the same dispersing medium. It is tested under same conditions, the probe compound band has a different shape which implies a slow velocity near the wall. If it were a stick boundary condition, then in the first case velocity at the walls would not be high. So why does the band "appear" to have a higher velocity in the first case. Have a look at the attached figure on the effect of packing bed at high shear rates from my previous work. Thanks

A solid wall, either flat or curved, has a smooth surface. The spaces between the wall and the nearby non-touching spheres cannot be filled. Thus, the density near the wall is always less. Also, many spheres are lined up at the same distance from the center of the cylindrical container. See Figures 2 and 3 in Benenati and Brosilow [A.I.Ch.E Journal, Vol.6, No. 3, pp 359-361]. This periodicity can be avoided by using periodic boundary conditions. See Figure 4 of Tory et al., Can. J.  Chem. Eng. pp 484-493. Also the floor effect on regularity can be reduced by making the floor rough. All spheres are supported and the packing density is about 0.58 To create a truly dense random packing, we can use a square box with periodic boundaries on all sides and use an iterative procedure (the Jodrey-Tory algorithm) to generate a packing with a density of about 0.64. See W.S. Jodrey and E.M. Tory, Physical Review A, vol. 32, pp 2347-2351. 

Mr. Farooq Wahab,

I came upon this question rather late and I see that there are very many good answers. I cannot add much to them but since you seem to need a more physical answer, I will try to give you one. You need to draw a figure thus: Take a plane wall for simplicity and line up the spheres of the packing. The second layer of spheres will sit in the saddle points of the first layer. This will continue into the bed. If you determine the local voidage you will see the pattern you wish to understand. If you cannot see it, please write back and I will draw it for you. In the meantime, please look up this (rather dated) paper: Kubie J, Broughton J (1975) A model of heat transfer in gas fluidized beds, Int. J Heat and Mass Transfer, Vol 18, pp. 289-299.

Hello all

in attached paper you can find detail answer about your question.

Analysis of Fluid Flow and Heat Transfer Model for the Pebble Bed High

Temperature Gas Cooled Reactor, S. Yamoah et,al