Modelling mass transfer processes

A comparatively large amount of work has been carried out on the extraction of oil from ground oil seeds using near-critical solvents. The rate of extraction has sometimes been analysed in terms of a model in which a solvent film is limiting the rate of mass transfer. On the basis of this model, appropriate mass transfer coefficients have been calculated using equation (7.41) (or occasionally 7.43) below. Although useful, this model may conceal more complex mechanisms. The experimental findings are summarised below. 

The rate of extraction is highly dependent on the pressure and temperature of the solvent, and thus the equilibrium solubility C [80, 89, 90]. 

The rate of extraction depends on the particle size and method of preparation, for example, grinding, flaking or pressing, [91, 92] with the largest 

Empirical film mass transfer coefficients derived using equation (7.41) to model experimental results increase with increasing superficial velocity Us, and are independent of bed height L [4, 79, 80]. 

Ct remains virtually constant and the model equations, (7.41) and (7.43) can be applied until about 60 or 70% of the oil present has been extracted [4, 79, 80, 90]. Similar results have been obtained for other crushed oil seeds and it has been found that the extent of the extraction period for which Ct remains constant depends on the type of crushing used and hence on the particle diameter [91, 92]. 

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Haggerty R, Gorelick SM (1998) Modeling mass transfer processes in soil columns with pore-scale heterogeneity. Soil Sci Soc Am J 62 62-74... 

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

The modeling of fluidized beds remains a difficult problem since the usual assumptions made for the heat and mass transfer processes in coal combustion in stagnant air are no longer vaUd. Furthermore, the prediction of bubble behavior, generation, growth, coalescence, stabiUty, and interaction with heat exchange tubes, as well as attrition and elutriation of particles, are not well understood and much more research needs to be done. Good reviews on various aspects of fluidized-bed combustion appear in References 121 and 122 (Table 2). 

It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may he neglected. From the penetration theory, the concentration C, at a depth y below the surface at time r is given by ... 

The several industrial applications reported in the hterature prove that the energy of supersonic flow can be successfully used as a tool to enhance the interfacial contacting and intensify mass transfer processes in multiphase reactor systems. However, more interest from academia and more generic research activities are needed in this fleld, in order to gain a deeper understanding of the interface creation under the supersonic wave conditions, to create rehable mathematical models of this phenomenon and to develop scale-up methodology for industrial devices. 

In this chapter the simulation examples are described. As seen from the Table of Contents, the examples are organised according to twelve application areas Batch Reactors, Continuous Tank Reactors, Tubular Reactors, Semi-Continuous Reactors, Mixing Models, Tank Flow Examples, Process Control, Mass Transfer Processes, Distillation Processes, Heat Transfer, and Dynamic Numerical Examples. There are aspects of some examples which relate them to more than one application area, which is usually apparent from the titles of the examples. Within each section, the examples are listed in order of their degree of difficulty. 

Other researchers used flow between two parallel plates as the experimental and theoretical system to incorporate diffusion plus convection into their dissolution modeling and avoid film model approximations [10]. Though they did not consider adding reactions to their model, these workers did show that convection was an important phenomenon to consider in the mass transfer process associated with solid dissolution. In fact, the dissolution rate was found to correlate with flow as... 

In Figure 33, both the first-order uptake kinetics and the equilibrium plateau levels decrease with increased BSA concentration, indicating that protein binding plays an influential role in the mass transfer process. To put biophysical meaning and quantitation into these observations, the following model is presented. 

The model equations in Section II,A have been formulated to describe the energy and mass transfer processes occurring in two-phase tubular systems. The accuracy of these model equations in representing the physical processes depends on the parameters of the equations being correctly evaluated. Constitutive equations that relate each of the parameters to the physical properties, system properties, and dependent variables of the system are discussed in the following sections. 

The model equations in Section III,C, have been formulated to describe those energy and mass-transfer processes in two-phase tubular systems for which one cannot neglect phase change. Constitutive equations for the parameters in these model equations are discussed in this section. 

The design of two-phase contactors with heat transfer requires a firm understanding of two-phase hydrodynamics in order to model effectively the heat- and mass-transfer processes. In this chapter we have pointed out areas where further theoretical and experimental research is critically needed. It is hoped that design engineers will be motivated to test the procedures presented, in combination with their use of the details from the original references, in the solution of pragmatic problems. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Generally, 3-D models are essential for calculating the radial distributions of spray mass, spray enthalpy, and microstructural characteristics. In some applications, axisymmetry conditions may be assumed, so that 2-D models are adequate. Similarly to normal liquid sprays, the momentum, heat and mass transfer processes between atomization gas and metal droplets may be treated using either an Eulerian or a Lagrangian approach. 

Some opportunities of such approximations are well illustrated by considering two characteristic examples. The first example will be a dusty-gas model, where porous media is considered as one of components of a gas mix of huge molecules (or particles of a dust), mobile or rigidly fixed in space [249,252,253], Such a model allows a direct application of methods and results of kinetic theory of gases and is effectively applied to the description of mass transfer processes in PS. The history of such an approach, the origins of which can be found in the works by Thomas Graham (1830 to 1840) is considered in Ref. [249], Actually, the model was first proposed by James Maxwell (1860), further it was independently reported by Deryagin and Bakanov (1957), and then also independently reported by Evans, Watson, and Mason (1961 see Refs. [249,252]). 

The two-phase kinetic model developed by Karickhoff (65) is capable of fitting either the sorption or desorption of a sorbing solute. For linear isotherms, the mathematical description given by Karickhoff (1) and others (67, 70, 71) is virtually identical to that of a mass transfer process (72). 

Behr and Obendorf [21] proposed a step-wise reaction model, according to which diethers are formed from monoether and isobutene and triether is formed from diethers and isobutene. In the simplified kinetic model no difference was considered between the two monoethers and the two diethers, and disproportion reactions and all side reactions were neglected (Fig. 10.6). The conversion rate was modeled without taking into account any mass transfer processes and phase... 

The authors applied this model to the situation of dissolving and deposited interfaces, involving chemically interacting species, and included rate kinetics to model mass transfer as a result of chemical reactions [60]. The use of a stochastic weighting function, based on solutions of differential equations for particle motion, may be a useful method to model stochastic processes at solid-liquid interfaces, especially where chemical interactions between the surface and the liquid are involved. 

Many industrial processes involve mass transfer processes between a gas/vapour and a liquid. Usually, these transfer processes are described on the basis of Pick s law, but the Maxwell-Stefan theory finds increasing application. Especially for reactive distillation it can be anticipated that the Maxwell-Stefan theory should be used for describing the mass transfer processes. Moreover, with reactive distillation there is a need to take heat transfer and chemical reaction into account. The model developed in this study will be formulated on a generalized basis and as a consequence it can be used for many other gas-liquid and vapour-liquid transfer processes. However, reactive distillation has recently received considerable attention in literature. With reactive distillation reaction and separation are carried out simultaneously in one apparatus, usually a distillation column. This kind of processing can be advantageous for equilibrium reactions. By removing one of the products from the reactive zone by evaporation, the equilibrium is shifted to the product side and consequently higher conversions can be obtained. Commercial applications of reactive distillation are the production of methyl-... 

With a good description of the mass transfer processes occurring in a CDPF now in place, it should be possible to predict the effects of PGM zoning and non-uniform aging on the performance of a CDPF. To illustrate the way in which this model can help in optimising the placement of the PGM washcoat in a CDPF systems, simulations were carried out over the European drive cycle for ... 

A new mathematical model based on moment techniques to describe micro- and macropore diffusion is used to study the mass-transfer resistances of Ci to C4 saturated hydrocarbons in H and Na mordenites between 127° C and 272° C. The intracrystalline diffusion coefficient decreases as the number of carbon atoms increases while the energy of activation increases with the number of carbons. The contribution from individual mass-transfer resistances to the overall mass-transfer processes is estimated. 

A knowledge of the velocity profiles within falling films under various flow conditions would be of very great value, making it possible to calculate the rates of convective heat and mass transfer processes in flowing films without the need for the simplified models which must be used at present. For instance, the analyses of Hatta (H3, H4) and Vyazovov (V8, V9) indicate clearly the differences in the theoretical mass-transfer rates due to the assumption of linear or semiparabolic velocity profiles in smooth... 

In the previous section the effects of poisons on reaction rates were related to the active component surface, while the influence of mass transfer was disregarded. It has long been recognized, of course, that the overall rate and selectivity of chemical reactions in porous systems involves the coupling of chemical reactions with convective and diffusive mass transfer processes. Beginning with the pioneering work of Thiele (67), an entire discipline has evolved in which model systems are used to... 

Diffusional mass transfer processes can be essential in complex catalytic reactions. The role of diffusion inside a porous catalyst pellet, its effect on the observed reaction rate, activation energy, etc. (see, for example, ref. 123 and the fundamental work of Aris [124]) have been studied in detail, but so far several studies report only on models accounting for the diffusion of material on the catalyst surface and the surface-to-bulk material exchange. We will describe only some macroscopic models accounting for diffusion (without claiming a thorough analysis of every such model described in the available literature). 

In summary, Ogwada and Sparks (1986c) developed a model and assumed that the adsorption of ions from solution by soil particles occurs in a series rather than a parallel reaction mode. Thus, mass-transfer processes and CR occur consecutively. Under the steady-state approximation, the rate of mass transfer is approximately equal to the rate of the reaction, so that instantaneous change in the concentration of CA with time approaches... 

The governing heat transfer modes in gas-solid flow systems include gas-particle heat transfer, particle-particle heat transfer, and suspension-surface heat transfer by conduction, convection, and/or radiation. The basic heat and mass transfer modes of a single particle in a gas medium are introduced in Chapter 4. This chapter deals with the modeling approaches in describing the heat and mass transfer processes in gas-solid flows. In multiparticle systems, as in the fluidization systems with spherical or nearly spherical particles, the conductive heat transfer due to particle collisions is usually negligible. Hence, this chapter is mainly concerned with the heat and mass transfer from suspension to the wall, from suspension to an immersed surface, and from gas to solids for multiparticle systems. The heat and mass transfer mechanisms due to particle convection and gas convection are illustrated. In addition, heat transfer due to radiation is discussed. 

Accurate modeling is only possible by the consideration of wavelength-dependent optical and temperature-dependent thermodynamic parameters and the correct application of the thermal accommodation coefficient which is dependent on the ambient particle conditions and is described in detail elsewhere (Schulz et al., 2006 Daun et al., 2007). Moreover, Michelsen (2003) suggested the inclusion of a nonthermal photodesorption mechanism for heat and mass loss, the sublimation of multiple cluster species from the surface, and the influence of annealing on absorption, emission, and sublimation. A more general form of the energy equation including in more detail mass transfer processes has been derived recently by Hiers (2008). For practical use, Equation (1) turns out to be of sufficient physical detail. 

The main physicochemical processes in thin-film deposition are chemical reactions in the gas phase and on the film surface and heat-mass transfer processes in the reactor chamber. Laboratory deposition reactors have usually a simple geometry to reduce heat-mass transfer limitations and, hence, to simplify the study of film deposition kinetics and optimize process parameters. In this case, one can use simplified gas-dynamics reactor such as well stirred reactor (WSR), calorimetric bomb reactor (CBR, batch reactor), and plug flow reactor (PFR) models to simulate deposition kinetics and compare theoretical data with experimental results. 

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