INTRODUCTION. EINSTEIN EQUATION AND ITS LIMITATIONS.
The non-expanded form of Einstein’s equation for the viscosity of suspensions in liquids is:
Where mu sub zero is the viscosity of the pure fluid, eta is the apparent viscosity and phi is the volume fraction of suspension in the fluid. The equation development can be found in many sources (e.g. Probstein, 1994, pp 127-130). It is based on the idea that the particles in a suspension cause a retardation of the flow due to their own velocity components. The equation is assumed to be valid for phi below 0.02, but it is used up to phi equal to 0.1.
I propose a restricted form of a viscosity equation for liquid suspensions, which unlike the main view that the particles cause a retardation effect in the velocity profile of the suspension, holds that the particles in a suspension cause retardation of the flow indirectly, by creating additional surface and an adjacent surface tension layer of higher viscosity and density that the bulk of the solution.
MODEL OVERVIEW.
The basic idea is that any surface ”breaks” the continuity of a liquid phase. As a consequence, the molecules at or near the surface experience ”pull” only from the direction of the liquid bulk (Probstein, 1994, p 306). What is not explicitly proposed is that as a consequence the liquid at the surface concentrates by becoming more densely packed, becoming both denser (tighter packing) and more viscid (intermolecular forces acting at shorter distances).
This is the basis of the proposed model: any suspended materials, any obstacles in the path of a liquid (like a membrane with pores), any break up in the continuity of the volume of the liquid increases the volume of liquid within a surface tension layer and therefore the volume fraction of the liquid with higher viscosity and density than the bulk. Within this view, the liquid in what is known in hydrodynamics as a ”boundary layer” is subject to viscid forces because of the adjacent discontinuity causes the bulk of the liquid compresses the liquid there, and not because of a dampening of the turbulence by the nearby wall. Conversely, turbulence is not generated near the wall and discomfits the flow structure of the liquid: turbulence would be the native state of the liquid flowing at a given velocity in an unrestrained environment. This “boundary layer” is what will be called here ”surface tension layer”.
It will be assumed that a suspension consists in fluid traveling in the bulk liquid, understood as the liquid outside the surface tension layers around the suspension particles, or the and the fluid travelling in the surface tension layers and in direct contact with the particles being carried along. Figure 1 shows an schematic example of different concentrations of particles flowing through a pipe, and what the concept of increasing particle concentration means in terms of taking up volume of liquid within the surface tension layers. Of course, to detect viscosity the fluid must be in motion, so matters of suspension stability in static conditions are not immediately relevant.
The thickness of the surface tension layer will be calculated from the critical value of the Reynolds number for the bulk phase viscosity and density of the liquid. As the liquid reaches the critical condition for all the film to be laminar, it will be assumed that the thickness of the two surface tension layers from the wall spans the whole diameter, and therefore that:
Therefore, the model for a suspension assumes that the solids do not count for the volume of liquid, but the volume around them which is their surface tension layer does count, as does the rest of the liquid. The surface tension layer around the particles will be a high viscosity/density region, and the remained will be a low viscosity/density region. This seems strange, because energy dissipation in liquid flows is accounted for by using turbulent viscosities. However, it makes sense if viscosity is solely linked to intermolecular forces and energy dissipation.
Upon understanding that viscosity means different things in each context it is possible to work out the difference between highly structured liquid as a consequence of fluid discontinuity (which causes a resistance to fluid motion) and highly disrupted liquid structure leading to dissipation of energy during fluid flow as a consequence of low level of structure in the liquid and subsequent turbulence (which causes resistance to fluid motion).
From a typical Moody chart it can be seen that pipe roughness has a consistent effect in increasing resistance to flow, which can be related to a larger surface tension layer due to a more developed surface. Figure 2 shows how the presence of the particles should be equated to the existence of a higher ratio pipe surface to volume of liquid, by restricting turbulent flow and hence restricting low structure and high fluidity in the bulk liquid.
MODEL EQUATIONS.
The model considers two region the surface tension layer region (tl) and the turbulent or bulk phase region (t), For these regions, where the viscosities are different, the velocity of will be:
.
where the total volume has been split between two virtual cylindrical regions of radius rt for the turbulent or bulk region and R for the total volume, where R-rt is the radius increment for the section corresponding to the surface tension layer.
Overall, the average velocity of the fluid will be:
from where the apparent mean viscosity of the fluid will be:
which simplifies as:
where as St/S and Stl/S are complementaries, if we make phi equal to Stl/S, and considering that (rt/R)2 = St/S and (R2 - rt2)/R2 = Stl/S then
and developing the parenthesis
which is similar to Einstein’s equation. Notice that phi has a different definition because:
- the total volume is NOT the volume of the suspension (Einstein), but is corrected substracting the actual volume of the suspended material, thus being smaller, and
- the suspension volume is NOT used, but instead the volume of the surface tension layer.
Some other factors to consider are:
- how specialty chemicals may affect the ratio of viscosities
- how they may affect the volume of the surface tension layer
- the surface/volume ratio of the material to be suspended
- the temperature of the system, as it influences the adsorption of specialty chemicals or their distribution between surface tension layer and
