Fluid flow, description

Answer This fluid flow description is consistent with elongational flow in the x direction, and simultaneous lateral contraction in the y and z directions, transverse to the primary flow. The process occurs at constant volume and constant density such that Poisson s ratio is... 

Where surface-active agents are present, the notion of surface tension and the description of the phenomena become more complex. As fluid flows past a circulating drop (bubble), fresh surface is created continuously at the nose of the drop. This fresh surface can have a different concentration of agent, hence a different surface tension, from the surface further downstream that was created earlier. Neither of these values need equal the surface tension developed in a static, equiUbrium situation. A proper description of the flow under these circumstances involves additional dimensionless groups related to the concentrations and diffusivities of the surface-active agents. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Gas-liquid-particle operations are of a comparatively complicated physical nature Three phases are present, the flow patterns are extremely complex, and the number of elementary process steps may be quite large. Exact mathematical models of the fluid flow and the mass and heat transport in these operations probably cannot be developed at the present time. Descriptions of these systems will be based upon simplified concepts. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

While the simple stirred tank and plug flow models are adequate to describe convective transport in many cases, a more complete description of fluid flow is sometimes needed. For example, an accurate description of tablet dissolution in a stirred vessel may require information about the changing fluid velocity near the tablet surface. Neither the stirred tank nor the plug flow models can address these velocity changes, since both assume that velocity is independent of position and time. In such cases, a more detailed description of fluid flow can be developed using the Navier-Stokes equations, which describe the effects of pressure, viscos-... 

There are two variables used in the description of fluid flow shear stress and shear strain. Stress is measured in units of Pascals and the strain is dimensionless. 

A basic starting point in the development of predictive absorption models is to review the mathematical descriptions of rate and extent of dmg absorption. A physical model for simultaneous fluid flow and intestinal absorption that applies broadly to idealised simulation experiments, animal studies, and in vivo studies in humans has been described by Ho et al. [30] and is depicted in Figure 2.4. 

Another key point of differentiation is the fact that nearly all PSA separations are bulk separations and any investigator interested in a high fidelity description of the problem of adsorption must solve a mass balance equation such as Eq. (9.9), the bulk separation equation, together with the uptake rate model and a set of thermal balance equations of similar form. In addition to the more complicated pde and its attendant boundary and initial conditions the investigator must also solve some approximate form of a momentum balance on the fluid flow as a whole. 

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

It is common within the industry to characterize chemical processes in terms of one or a few global reaction steps, assigning an Arrhenius rate expression to describe the rate of each reaction. If knowledge of the detailed chemistry is inadequate or the chemical scheme is to be combined with computational fluid dynamics for a complex flow description, a simplified chemistry may be necessary. It is important, however, to realize that such a chemical description can only be used for the narrow range of conditions (temperature, composition, etc.) for which it is developed. Any extrapolation outside these conditions may be erroneous or even disastrous. 

Analytically Reduced Mechanisms Some problems can be described by models that involve a full reaction mechanism in combination with simplied fluid dynamics. Other applications may involve laminar or turbulent multidimensional reactive flows. For problems that require a complex mathematical flow description (possibly CFD), the computational cost of using a full mechanism may be prohibitive. An alternative is to describe... 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

The tubule is a spatially extended structure, and it presents both elastic properties and resistance to the fluid flow. The dynamic pressure and flow variations in such a structure can be represented by a set of coupled partial differential equations [11]. An approximate description in terms of ordinary differential equations (a lumped model) consists of an alternating sequence of elastic and resistive elements, and the simplest possible description, which we will adopt here, applies only a single pair of such elements. Hence our model [12] considers the proximal tubule as an elastic structure with little or no flow resistance. The pressure P, in the proximal tubule changes in response to differences between the in- and outgoing fluid flows ... 

In collaboration with Alexander Gorbach, NIH, we have initiated a study of the spatial patterns in the nephron synchronization. This study involves the use of infrared cameras or other types of equipment that can measure variations in the blood supply by small (0.01°C) fluctuations in the temperature at the surface of the kidney. It is also of interest to study how the large amplitude oscillations in pressure, fluid flow, and salt concentration at the entrance of the distal tubule influence the delicate hormonal processes in that part of the kidney, to establish a more quantitative description of some of the mechanisms involved in the development of hypertension, and to examine the effects of various drugs. 

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