Evaporator model assumptions

In this section the model for a continuous evaporative crystallizer is discussed. The crystallizer is of the draft tube baffled (DTB) type and is equiped with a fines removal system consisting of a large annular zone on the outside of the crystallizer (see Figure 1). In order to vary the dissolved fines flow without changing the cut-size of the fines removal system, the flow in the annular zone is kept constant and the flow in the dissolving system is varied by changing the recycle flow rate. The model assumptions are ... 

Abstract Evaporation of multi-component liquid droplets is reviewed, and modeling approaches of various degrees of sophistication are discussed. First, the evaporation of a single droplet is considered from a general point of view by means of the conservation equations for mass, species and energy of the liquid and gas phases. Subsequently, additional assumptions and simplifications are discussed which lead to simpler evaporation models suitable for use in CFD spray calculations. In particular, the heat and mass transfer for forced and non-forced convection is expressed in terms of the Nusselt and Sherwood numbers. Finally, an evaporation model for sprays that is widely used in today s CFD codes is presented. 

The discussion of laminar diffusion flame theory addresses both the gaseous diffusion flames and the single-drop evaporation and combustion, as there are some similarities between gaseous and Hquid diffusion flame theories (2). A frequentiy used model of diffusion flames has been developed (34), and despite some of the restrictive assumptions of the model, it gives a good description of diffusion flame behavior. 

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

As already discussed, DFT can be used to predict the capillary condensation and capillary evaporation pressures for pores with homogeneous surface and well-defined geometry. To generate model adsorption isotherms for heterogeneous pores, it is convenient to employ hybrid models based on both DFT data for homogeneous pores and experimental data for flat heterogeneous surfaces [6-9]. Such model adsorption isotherms can be used to calculate PSDs in mesopore [6-9] and micropore [9] ranges. This approach is particularly useful for pores of diameter below 2-3 nm (micropores and narrow mesopores), where an assumption about the common t-curve for pores of different sizes is less accurate, which in turn makes the methods based on such an assumption (even properly calibrated ones) less reliable [18],... 

In the first monolayer of conjugated model material, a model molecular solid or a polymer adsorbate, assume that no chemistry (covalent bonding) occurs, since, in the absence of, for example, mechanical rupturing, the bonds at the surface of the molecular film are completely satisfied. This assumption is supported by the fact that, at least for condensed molecular solids, vapor-deposited films may be re-evaporated (removed) from the surface by gentle heating in UHV. 

Since this model was far too complex to serve any practical purpose, Brunauer, Emmet and Teller made some simplifying assumptions (the main one being that in all layers the evaporation-condensation mechanisms are identical) to derive their famous BET equation, to be used in the multilayer-adsorption region of the adsorption isotherm ... 

Beard(Z.) has developed a useful mathematical model of a convection dryer for studying the use of energy in a tenter frame. The model is based on a set of simultaneous differential equations which can be solved numerically to obtain fabric temperature and macroscopic moisture contents along the length of the dryer. The model considers the fabric as a moist layer of fabric sandwiched between two dry layers of fabric. Thermal energy is convected from the dryer to the external surface of the dry layer and then from the exterior of the fabric to the interface between the wet and dry layers. At the interface, the water is evaporated and diffuses as vapor through the dry layer to the surrounding hot make-up air. Assumptions in the model include ... 

Figure 6.21. The model of Laurent et al. (1988) for deoxidation of an A1 drop under high vacuum is based on the assumption that deoxidation is controlled by the rate of evaporation of a volatile oxide formed according to reaction (6.25). This is justified for disrupted oxide films (a) or for continuous but very thin films (b), through which diffusion is very fast, but not for thick oxide films.

An assumption will now be added to the model. It is that near or at the melting point, a hole is annihilated by the evaporation into it of one particle i.e., just one particle fills it. There is no violation of physical sense in this assertion, for use ofEq. (5.44) shows that the size of the holes predicted by the hole theory is near that of the ions which are assumed in the model to jump into them. Correspondingly, the work done to annihilate a hole is numerically equal to the work done in forming a surface of radius r, namely, An where y is the surface tension. Hence, if rij particles must jump into a hole to annihilate it at temperature T... 

Many numerical models make additional assumptions, valid if only some specific questions are being asked. For example, if one is not interested in the start-up phase or in changing the operation of a fuel cell, one may apply the steady state condition that time-independent solutions are requested. In certain problems, one may disregard temperature variations, and in the free gas ducts, laminar flow may be imposed. The diffusion in porous media is often approximated by an assumption of isotropy for the gas diffusion or membrane layer, and the coupling to chemical reactions is often simplified or omitted. Water evaporation and condensation, on the other hand, are often a key determinant for the behaviour of a fuel cell and thus have to be modelled at some level. 

Alternatively, the thin-skin model, which assumes an infinite diffusion resistance with only a thin layer on the droplet surface being heated and evaporated, can be used. This second assumption is more appropriate for cases in which the droplet reaches its boiling temperature immediately after the start of the evaporation process. All the other approaches (such as the diffusion-limited model), which are recommended in the case of mulhcompo-nent droplets, require some form of spahal discretization inside the droplet, resulting in much higher computational costs. To close the set of equahons, it is necessary to describe the evolution of the droplet temperature Tp as discussed in Section 5.2.3. For example, upon including the effect of evaporation on the heat flux for the rapid-mixing model, the particle-enthalpy balance yields... 

It will be appreciated that our description of the plant is, in reality, only an approximation covering as few features as we can get away with, while still capturing the essential behaviour of the plant. For instance, in the example above of the tank liquid level, no mention was made of liquid temperature, entailing an implicit assumption that temperature variations would be small over the period of interest. If it had been necessary to allow for temperature effects, perhaps because of fear of excessive evaporation or because of environmental temperature limits set for a waste water stream, then liquid temperature would have had to be included as an additional state variable, and the dimension or order of the plant as we modelled it would go up from 4 to 5. If we had needed to make an allowance for the temperature of the metal in the tank. 

The assumption that, for a nonuniform surface, E increases linearly with increase of coverage is unrealistic from a physical viewpoint it is however, a convenient postulate from a mathematical viewpoint, particularly when it is realized that a surface comprising a small number of homogeneous patches, each patch having different E values on which there may or may not be induced effects, gives rise to an adsorption rate which subscribes well to an Elovich equation this model is an acceptable physical description for adsorbents in the form of powders or evaporated films. Similarly, models comprising uniform surfaces, but with site creation or exclusion, may be analyzed and extended to give conclusions of the same natures as those derived from a variation of E over a nonuniform surface mathematically, however, the extension to, e.g., interaction effects between two different adsorbates is more cumbersome. 

It should he pointed out, however, that Eq. (11.3) assumes Newtonian behavior, which the complex polymeric resists and B ARC fluids do not necessarily exhibit. In particular, mass is not lost, neither from the radial flow of material nor from evaporation of solvent. Meyerhofer considered the effects of evaporation on the final film thickness. He reported that the final solid film thickness is inversely proportional to the square root of the rotational velocity. He also developed a model similar to that considered above, but allowed the solvent to evaporate during the spinning process. His central assumption was that the thinning process could be divided into two major stages, namely, one dominated by radial flow outward and another by evaporation of solvent. Effectively, he assumed a constant rate of evaporation and the viscosity concentration relationship expressed as... 

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