Homogeneous suspension flows

HOMOGENEOUS SUSPENSION FLOW Governing Equations Numerical Simulation Techniques Solution Algorithm... 

The numerical results provided by the application of the described model to the simulation of a homogeneous suspension flow have been tested against both experimental data and analytical solutions obtained from well-known treatments of non-Newtonian flows. 

Consider a single, freely suspended axisymmetric particle in a homogeneous shear flow held of an incompressible Newtonian liquid. The free suspension condition implies that the net instantaneous force and torque on the particle vanish. There is, however, a finite net force along the axis that one half of the particle exerts on the other, as shown schematically in Fig. 7.25. 

Release of larger amounts of macrophage inhibitory factor (MIF) by homogeneous suspensions of DCs isolated from patients with colitis ulcerosa, due to the effect of Thl lymphocyte cytokines, in comparison with healthy human DC. It was observed that in an in vitro test MIF blocks the flow of mononuclear cells and granulocytes, while in in vivo it is released not only by lymphocytes but also by dendritic cells and leads to cell accumulation— occurrence of inflammatory infiltration. Disease development is also the result of the effect of TNFa, IL 1, IL 6, IL 8, and other cytokines released by macrophages and lymphocytes (Murakami et al., 2002). Thus, it has been proved that DCs can release MIF. 

Suspension in a Fluid Stream—While diffusional and eddy processes tend to keep a suspension more or less homogeneous, there are many occasions when in spite of these processes conveyed material tends to stratify or distribute itself in a non-uniform manner. This difficulty is not present when particles are conveyed vertically, and yet we must appreciate the fact that the movement of a fluid in any duct creates a velocity gradient which is a maximum along the axis and decreases to very low values next to the duct surface. Thus, all the particles do not move at a Uniform rate and in the case, of vertical motion the largest particles may move only along the duct axis. Those near the duct surface may, in fact, tend to fall unless projected by eddies and particle impacts toward the duct axis. In any case, it is generally easier to sample a suspension flowing in a vertical duct than in a horizontal duct. 

A homogeneous particle packing can be obtained when the suspension flow units do not change on a time scale of that of the coating process. In industrial practice it is often also desirable that the suspension properties are stable on a longer time scale (shelf life) for economic reasons. This means that sedimentation and ageing cannot be permitted amd must accordingly be prevented. 

Kramers (44) in 1944 published an elegant paper on the mechanical and optical properties of dilute suspension of bead-rod systems in steady-state, homogeneous, potential flow [see comments just after Eq. (3.1)]. This theory has recently been extended by Bird, Johnson, and Curtiss (5) to include bead-spring systems as well. The latter theory gives for a system of N beads with any kind of connectors ... 

Homogeneous suspension exists when the particle concentration and, for a range of sizes, the size distribution is constant throughout the tank. This condition is particularly desirable when a continuous and representative flow of solids from the system is required. 

Generally, emulsion or suspension flow was treated as in the case of a homogeneous fluid, and the only issue addressed was to estimate the rheological characteristics of the dispersion, either Newtonian or nonNewtonian. 

In the general case of arbitrary suspension flow, the local instantaneous values for mean flow variables at a given physical point do not coincide with their respective values specific to any stationary fluidized bed of the same particles. The truly homogeneous and stationary state for any point would be a state characterized by arbitrary uniform and constant values for the flow variables which coincide with their local instantaneous values at this point and at a flxed time moment. Such a homogeneous, stationary state is practically impossible to effect. Since this state does not adhere in almost all cases, the concept of a locally homogeneous state represents a certain idealization. The meaning of this idealization is explained next. 

Hydrodynamic stability of uniform vertical suspension flow has been theoretically treated for more than 30 years (see reference [15,20,29,32-34,43,47], and also reference [48-36]). Much of this work has been undertaken when analyzing the important problem of reasons causing the transition from homogeneous (particulate) fluidization to nonhomogeneous (aggregative) fluidization, and subsequently, providing for the spontaneous origination in a fluidized bed of cavities (bubbles) almost devoid of particles. 

The fact that the total number of particles must be conserved during the development of occasional disturbances in a uniform vertical flow or in a homogeneous fluidized bed in itself results in the formation of kinematic waves of constant amplitude, as was first demonstrated by Kynch [48]. Both particle inertia and the nonlinear dependence of the interphase interaction force on the suspension concentration cause an increase in this amplitude. This amounts to the appearance of a resultant flow instability with respect to infinitesimal concentration disturbances and with respect to other mean flow variable disturbances. Various dissipative effects can slow the rate at which instability develops, but cannot actually prevent its development. Therefore, investigating the linear stability of a flow without allowing for interparticle interaction leads inevitably to the conclusion that the flow always is unstable irrespective of its concentration and the physical parameters of its phases. This conclusion contradicts experimental evidence that proves suspension flows of sufficiently small particles in liquids to be hydrodynamically stable in wide concentration intervals [57-59]. Moreover, even flows of large particles in gases may be stable if the concentration is either very low or very high. 

Numerical analysis of heterogeneous flows is much more difficult. This is because a general method which can effectively simulate all types of heterogeneous flow behavior cannot be developed. Relatively simple types of heterogeneous suspension flows, such as stratified regimes (i.e., layered homogeneous flows) can, however, be successfully modelled [3]. 

In view of such difficulties this chapter is restricted to the description of the numerical modelling of steady, laminar non-Newtonian flow of homogeneous suspensions. 

Suspensions which consist of sufficiently uniformly dispersed solids of very small diameter (less than 25p) can be treated as the equivalent of homogeneous singlephase fluids [I]. Thus, the governing equations of continuity and momentum for steady laminar flow of a non-Newtonian homogeneous suspension in an axisym-metric cylindrical (r,z) co-ordinate system can be written as ... 

For settling slurries, at relatively high velocities the particles are carried in suspension and the flow behaviour approximates that of a homogeneous suspension. As long as the velocity is maintained above the standard velocity, the particle concentration gradient is minimized. As the superficial velocity is reduced below this transition or standard velocity, particle concentration gradients develop and the flow becomes heterogeneous. 

Slow sedimentation of particles will occur, for example, in an activated sludge or in fine particle catalyst suspensions. For those lands of systems, a homt eneous distribution of solids is characteristic. Here, the liftoff from the vessel bottom as well as the state of a homogeneous suspension can be achieved with a comparably low power input or only slight movement of the liquid. On the other hand, at higher solids concentrations a pseudoplastic flow characteristic of the suspension can occur. As an example, concentrations of only 6% of fibrous material - typically known from paper industry - can lead to this non-Newtonian behavior Frequently observed in suspensions with high solids concentrations is a Bingham plastic behavior. In this case, if a certain amount of shear is not introduced by agitation, the system behaves like an elastic solid body or a gel. 

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