Basic Mass Transfer Models

Many mass transfer models exist, but most of them depend on three assumptions and are simplified versions of actual mass transfer mechanisms, many of which occur simultaneously. The first assumption is that the different phases and the phase interface offer resistance to mass transfer in series, in a similar manner to heat transfer resistances. The second assumption maintains that mass transfer is controlled by the phase equilibrium near the interface, which changes more quickly than the bulk phase equilibrium (Azbel, 1981). In other words, mass transfer occurs at the microscale level (van Elk et ak, 2007). Finally, gases are assumed to be single component. Multiple component problems are more complicated because each individual gas component making up the mixture has to be considered for the limiting gas-liquid mass transfer step. The complexity grows further once the relationships between each gas component and, for example, the bacteria in a bioreactor are considered. 

Molecular diffusivity and film thickness are combined into the liquid-phase mass transfer coefficient ki such that 

This model has several limitations. The film model assumes that mass transfer is controlled by the liquid-phase film, which is often not the case because the interface characteristics can be the limiting factor (Linek et al., 2005a). The liquid film thickness and diffusivity may not be constant over the bubble surface or swarm of bubbles. Experiments also indicate that mass transfer does not have a linear dependence on diffusivity. Azbel (1981) indicates that others have shown that turbulence can have such a significant effect on mass transfer such that eddy turbulence becomes the controlling mechanism in which diffusivity does not play a role. In most instances, however, eddy turbulence and diffusivity combine to play a significant role in mass transfer (Azbel, 1981). 

The border diffusion layer model was introduced as an amendment to the film model to present a more realistic description. It accounts for an undefined film thickness, turbulence effects, and the role of molecular diffusion. When the flow is turbulent, the flow around the bubble is split into four sections the main turbulent stream, the turbulent boundary layer, the viscous sublayer, and the diffusion sublayer. Eddy turbulence accounts for mass transfer in the main turbulent stream and the turbulent boundary layer. The viscous sublayer limits eddy turbulence effects so that the flow is laminar and mass transfer is controlled by both molecular diffusion and eddy turbulence. Microscale eddy turbulence is assumed to be dominant in the viscous sublayer. Mass transfer in the diffusion sublayer is controlled almost completely by molecular diffusion (Azbel, 1981). 

This model can be used as a rough estimate. It is still plagued by the steady-state assumption which is oftentimes an inadequate description of mass transfer. Tlie diffusion sublayer is a function of viscosity (v), diffusivity (Z)), and viscous sublayer thickness (Sq), bnt realistic measurements are difficult, if not impossible, to obtain. An empirical correlation has been suggested for S (Azbel, 1981)  

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Isoparaffin alkylation reactions are very fast and they suffer from severe pore diffusion limitations. As a result, when catalyst particle size is increased from 100 pm (for slurry reactors) to 1.6 mm for fixed-bed reactors, the catalyst activity reduces by 10-fold according to basic mass transfer models using experimental values of the intrinsic rate constant, as shown in figure 4. To match the catalyst productivity of a slurry reactor, one would need to build a fixed-bed reactor with ten times the volume - not practical for a commercial scale system. In addition to using a fixed-bed reactor, we wanted to ensure that the solid-acid catalyst was both robust as well as benign (i.e. environmentally fiiendly). 

The remaining sections of this chapter provide examples of mass transfer models, presented using the systems approach described above. In many cases, the models are of such importance that they are regarded as theories in their own right. These basic models are also the foundation for the more specific applications in the subsequent chapters of this book. 

Two Basic Descriptions of Transport by Random Motion Mass Transfer Model and Gradient-Flux Law... 

This method has been introduced in electrochemistry by Feldberg [124]. Basically, its purpose is to calculate the concentration vs. distance profile in the diffusion layer by simply digitizing the mass transfer model into events that occur within small increments, At, of time in small... 

The fundamental principles of the gas-to-liquid mass transfer were concisely presented. The basic mass transfer mechanisms described in the three major mass transfer models the film theory, the penetration theory, and the surface renewal theory are of help in explaining the mass transport process between the gas phase and the liquid phase. Using these theories, the controlling factors of the mass transfer process can be identified and manipulated to improve the performance of the unit operations utilizing the gas-to-liquid mass transfer process. The relevant unit operations, namely gas absorption column, three-phase fluidized bed reactor, airlift reactor, liquid-gas bubble reactor, and trickled bed reactor were reviewed in this entry. 

The 15 chapters fall into three parts. Part I (Chapters 1-6) deals with the basic equations of diffusion in multicomponent systems. Chapters 7-11 (Part II) describe various models of mass and energy transfer. Part III (Chapters 12-15) covers applications of multicomponent mass transfer models to process design. 

The concentration difference can also be re-defined in different ways, thus there exists a variety of modifications of the basic mass transfer coefficient definition as well. Therefore, care should to be taken to ensure that the mass transfer coefficient parameterizations adopted for modeling purposes, correspond to the model formulation used. 

Separation processes rely on various mechanisms, implemented via a unit operation, to perform the separation. The mechanism is chosen to exploit some property difference between the components. They fall into two basic categories the partitioning of the feed stream between phases and the relative motion of various chemical species within a single phase. These two categories are often referred to as equilibrium and mass transfer rate processes, respectively. Separation processes can often be analyzed with either equilibrium or mass transfer models. However, one of these two mechanisms will be the limiting, or controlling, factor in the separation and is, therefore, the design mechanism. 

Mass transfer relationships in a distillation column are based on the basic interphase transfer model for a differential slice of the cross section, as shown in Figure 12.57. The slice is taken from the packed bed or from the tray froth. For component i,... 

In this chapter, we have presented the more used theories to explain the mass transfer mechanism in a polymer solid (porous or tight, charged or not) membranes. It is clear that the membrane structure and the nature of the solution forms a system with complex interactions between the solutes, solvent, driving force, and the polymeric membrane. Often many basic mass transfer mechanisms intervene in certain membrane separations, and the mass transfer modeling is not very easy, because we must take into account any interactions between the components of the system. 

Basics and heat and mass transfer modeling. Food Eng. Rev. 4 89-106. 

This section contains a simple introduction to steady state and unsteady species mole (mass) diffusion in dilute binary mixtures. First, the physical interpretations of these diffusion problems are given. Secondly, the physical problem is expressed in mathematical terms relating the concentration profiles to the diffusion fluxes. Emphasis is placed on two diffusion problems that form the basis for the interfacial mass transfer modeling concepts used in reaction engineering. The basic theory is reviewed in many textbooks on chemical reaction engineering [6, 15, 27, 52, 87]. These texts may be recommended for complimentary studies. 

Based on the stoichiometry of the absorption process and the basic mass transfer equation, besi equations (21) and (22), the mathenratical model of die fnocess includes also the relations following from the mass balance ... 

The time constant R /D, and hence the diffusivity, may thus be found dkecdy from the uptake curve. However, it is important to confirm by experiment that the basic assumptions of the model are fulfilled, since intmsions of thermal effects or extraparticle resistance to mass transfer may easily occur, leading to erroneously low apparent diffusivity values. 

In this chapter, consideration will be given to the basic principles underlying mass transfer both with and without chemical reaction, and to the models which have been proposed to enable the rates of transfer to be calculated. The applications of mass transfer to the design and operation of separation processes are discussed in Volume 2, and ihe design of reactors is dealt with in Volume 3. 

Airlift loop reactor (ALR), basically a specially structured bubble column, has been widely used in chemical industry, biotechnology and environmental protection, due to its high efficiency in mixing, mass transfer, heat transfer etc [1]. In these processes, multiple reactions are commonly involved, in addition to their complicated aspects of mixing, mass transfer, and heat transfer. The interaction of all these obviously affects selectivity of the desired products [2]. It is, therefore, essential to develop efficient computational flow models to reveal more about such a complicated process and to facilitate design and scale up tasks of the reactor. However, in the past decades, most involved studies were usually carried out in air-water system and the assumed reactor constructions were oversimplified which kept itself far away from the real industrial conditions [3] [4]. 

Modelling can at least facilitate the determination of the most effective scale-up program. Information from three fields is needed for modelling (1) chemical kinetics, (2) mass transfer, and (3) heat transfer. The importance of information for different processes has been qualitatively evaluated (see Table 5.3-5). Obviously, sufficiently accurate information on heat transfer is needed for batch reactors, which are of great interest for fine chemicals manufacture. Kinetic studies and modelling requires much time and effort. Therefore, the kinetics often is not known. Presently, this approach is winning in the scale-up of processes for bulk chemicals. The tools developed for scale-up of processes for bulk chemicals have been proven to be very useful. Therefore, the basics of this approach will be discussed in more detail in subsequent sections. 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Fate and transport of organic leachates from SWMs/COMs in natural environments can be approximated by a series of laboratory tests or analyses. The basic approach is to measure the mass transfer of such chemicals under controlled conditions to determine rates that can be applied to specific mathematical models. 

An examination of the catalyst-layer models reveals the fact that there are many more cathode models than anode ones. In fact, basically every electrode-only model is for the cathode. This arises because the cathode has the slower reaction it is where water is produced, and hence, mass-transfer effects are much more significant and it represents the principal inefficiency of the fuel cell. In other words, while the cathode model can be separate from the anode model, the converse is not true due to the... 

Finally, there are some miscellaneous polymer-electrolyte fuel cell models that should be mentioned. The models of Okada and co-workers - have examined how impurities in the water affect fuel-cell performance. They have focused mainly on ionic species such as chlorine and sodium and show that even a small concentration, especially next to the membrane at the cathode, impacts the overall fuelcell performance significantly. There are also some models that examine having free convection for gas transfer into the fuel cell. These models are also for very miniaturized fuel cells, so that free convection can provide enough oxygen. The models are basically the same as the ones above, but because the cell area is much smaller, the results and effects can be different. For example, free convection is used for both heat transfer and mass transfer, and the small... 

Various models have been proposed to describe the facilitated mass transfer phenomena, although five basic categories of models have mostly been reported in the literature [29]. The same models can essentially be applicable for Type II facilitated transport. 

The transport equations for laminar motion can be formulated, in general, easily and difficulties may lie only in their solution. On the other hand, for turbulent motion the formulation of the basic equations for the time-averaged local quantities constitutes a major physical difficulty. In recent developments, one considers that turbulence (chaos) is predictable from the time-dependent transport equations. However, this point of view is beyond the scope of the present treatment. For the present, some simple procedures based on physical models and scaling will be employed to obtain useful results concerning turbulent heat or mass transfer. 

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