Flow field models

Equations (33) - (35) are taken from Tardos et al (23), and are based on a low Reynold s number analysis. Eqn. (33) is the result of a "best-fit" of the theoretically computed values taken from Figure 7 of that same work. Similarly, Eqn. (36) for the electrical deposition, is obtained from a "best-fit" of the theoretically computed values taken from Figure 3 of Tardos and Pfeffer (21). Note that if the particle and collector charges are of the same sign, the electrical deposition efficiency becomes the negative of Eqn. (36). Consistent with the flow field models used in the development of Eqns. (33) - (36), the velocity employed is an assembly averaged velocity for each phase. For the multi-phase situation that exists in the fluidized bed, this is given by the superficial or empty-tower velocity divided by the phase volume fraction, ... 

Singh, J. R, Yueh, F. Y, Cook, R. L., Lee, J. J., and Linberry, J. T. "Comparison of CARS Temperature Measurements with Flow Field Model Calculations at the CFFF Diffuser." Applied Spectroscopy 46 (1992) 1649. 

Electrowetting and Droplets, Fig. 3 Flow field model for droplet motion in a square pattern. The actual three-dimensional electrowetting flow will be substantially different than this model but is expected to display similar chaotic advection with dependence on the period of the cycle. The flow model here is a time-dependent, high-Peclet-number (low diffusion), two-dimensional Marangoni flow where the surface tension around the droplet periphery is varied in four phases (a) for the first quarter of cycle a = (ai -i- — (cji — cj2)cos(0)/2 (b)... 

Figure 1 Experimental configuration (left) and flow field model (right) to study catalytic ignition of a stagnation point flow on a catalytic foil.

Two-dimensional models can be used to provide effective approximations in the modelling of polymer processes if the flow field variations in the remaining (third) direction are small. In particular, in axisymraetric domains it may be possible to ignore the circumferential variations of the field unlaiowns and analytically integrate the flow equations in that direction to reduce the numerical model to a two-dimensional form. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

One-equation models relax the assumption that production and dissipation of turbulence are equal at all points of the flow field. Some effects of the upstream turbulence are incorporated by introducing a transport equation for the turbulence kinetic energy k (20) given by... 

Method A Calculating Contaminant and Exhaust Velocities at All Points in the Flow Field Local exhaust hoods are used to remove contaminants at the point of generation before they escape into the workplace air. The efficiency of any local exhaust system is greatly affected by the flow field generated by the exhaust opening. Therefore, accurate modeling of this flow field is essential for reliable predictions. However, solving the airflow field is a formidable task and often must be done numerically. 

Another design method uses capture efficiency. There are fewer models for capture efficiency available and none that have been validated over a wide range of conditions. Conroy and Ellenbecker - developed a semi-empirical capture efficiency for flanged slot hoods and point and area sources of contaminant. The point source model uses potential flow theory to describe the flow field in front of a flanged elliptical opening and an empirical factor to describe the turbulent diffusion of contaminant around streamlines. 

Abe, K., Kondoh, T., Nagano, Y. A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows 1. Flow field calculations. Int. ]. Heat Mass Trans, fer, vol. 37, pp. 139-151, 1994. 

The terms zonal model and flow element are also used for the simplified characterization of the flow field in a single enclosure. There, a zone represents a partial volume of air in the enclosure, whereas in the multizone models described here, a zone represents a specific enclosure which is connected to other enclosures by air conductances (see The Airflow Network later). 

CFD is appropriate in cases where the detailed flow field is of interest in a configuration with mostly known or at least steady-state boundary conditions (surface temperatures). Combined thermal and ventilation modeling is more suited to cases where the dynamic behavior of the building masses and the changing driving forces for the natural ventilation are of importance. 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

This equation describes the change of population in a well-mixed system and is often used to model fully mixed crystallization and precipitation processes. If the system is imperfectly mixed, however, then the more complicated equation 2.88 can be used provided that the external flow field can be calculated e.g. by use of CFD (see later). 

Brayshaw, M.D., 1990. Numerical model for the inviscid flow of a fluid in a hydrocyclone to demonstrate the effects of changes in the vorticity function of the flow field on particle classification. International Journal of Mineral Processing, 29, 51. 

The phase-field model and generalizations are now widely used for simulations of dendritic growth and solidification [71-76] and even hydro-dynamic flow with moving interfaces [78,79]. One can even use the phase-field model to treat the growth of faceting crystals [77]. More details will be given later. 

The combustion-flow interactions should be central in the computation of combustion-generated flow fields. This interaction is fundamentally multidimensional, and can only be computed by the most sophisticated numerical methods. A simpler approach is only possible if the concept of a gas explosion is drastically simplified. The consequence is that the fundamental mechanism of blast generation, the combustion-flow interaction, cannot be modeled with the simplified approach. In this case flame propagation must be formalized as a heat-addition zone that propagates at some prescribed speed. 

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