Mass transfer analysis examples

The main complication with this technique is that the mass transfer analysis is nontrivial. For example, the change of velocity as a function of the distance down the jet was not taken into account in modeling the system. The LJRR has been used to study the diffusivities of benzene and toluene in water [41] and cupric ion extraction [42,49]. 

Many processes of heat and mass transfer, for example, fast-running processes, lead to the investigation of singularly perturbed boundary value problems with a perturbation parameter e. For example, those problems arise in the analysis of heat and mass transfer for mechanical working of materials, in particular, metals. The use of classical methods for the numerical solution of such problems (see, e.g., [1,11,12]) leads us... 

The required mercury capacities of sorbent as predicted by mass transfer analysis are presented in Table 1. Under a mass transfer limited process, a low capacity sorbent is required. For example for a 5 pm carbon particle size, the mass transfer capacity is only 217 pg/g carbon. When the mercury capacity of a sorbent is comparable to that of the mass transfer capacity, however the C/Hg ratio is determined by both mass transfer parameters and adsorbent capacity. Under some extreme conditions, the mercury capacity of the adsorbent could limit the removal efficiency, and the C/Hg ratio is determined by the sorbent capacity rather than the mass transfer capacities presented in Table 1. 

The coupling of reaction kinetics with transport processes is necessary to develop effective bioreactor systems. Further discussion of this topic is given later in this chapter (see Reacting Systems and Bioreactors and Illustrative Example for reactor design specifications and mass transfer analysis in encapsulation motifs, respectively). Heat and momentum transport are major topics discussed in other chapters in this section of the handbook. Brief comments on the necessity for these studies are presented as here. 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

In their analysis, however, they neglected the surface tension and the diffusivity. As has already been pointed out, the volumetric mass-transfer coefficient is a function of the interfacial area, which will be strongly affected by the surface tension. The mass-transfer coefficient per unit area will be a function of the diffusivity. The omission of these two important factors, surface tension and diffusivity, even though they were held constant in Pavlu-shenko s work, can result in changes in the values of the exponents in Eq. (48). For example, the omission of the surface tension would eliminate the Weber number, and the omission of the diffusivity eliminates the Schmidt number. Since these numbers include variables that already appear in Eq. (48), the groups in this equation that also contain these same variables could end up with different values for the exponents. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

From an analysis of the electrochemical mass-transfer process in well-supported solutions (N8a), it becomes evident that the use of the molecular diffusivity, for example, of CuS04, is not appropriate in investigations of mass transfer by the limiting-current method if use is made of the copper deposition reaction in acidified solution. To correlate the results in terms of the dimensionless numbers, Sc, Gr, and Sh, the diffusivity of the reacting ion must be used. 

The Grashof number given by Eq. (40) appears to have a weaker theoretical basis than that given by Eq. (37), since it is based on an analysis that approximates the profile of the vertical velocity component in free convection, for example, by a quadratic function of the distance to the electrode. The choice of an appropriate Grashof number, as well as the experimental conditions in the work of de Leeuw den Bouter et al. (DIO) and Marchiano and Arvia (M3), has been reviewed critically by Wragg and Nasiruddin (W10). They measured mass transfer by combined thermal and diffusional, turbulent, free convection at a horizontal plate [see Eq. (31) in Table VII], and correlated their results satisfactorily with the Grashof number of Eq. (37). 

Some of the methods of analysis of porosity are based on specific properties of porous and disperse materials, namely, thermoporometiy method is based on shifts of the temperature of phase transitions and permeametry methods are based on characteristics of mass transfer through porous media. Each method has its advantages, for example low cost of equipment and high performance. Each has its own range of optimal measurements. But, all the methods are really doomed for coexistence, and in many cases they supplement each other. 

The above analysis, which is exceedingly brief and simplified is designed to demonstrate how, even in a pre-mixed flame, the question arises as to what is the appropriate reaction volume (i.e. the flame thickness). In heterogeneous reactions, this is a question that will recur again and ain and the designer of reactors must not attempt to avoid it. It is interesting to note that, in the next but one example to be treated, the overall reaction rate (a flame speed cm s in the above) becomes a mass transfer coefficient (also cms" ) when considering the absorption of gas into a liquid with which it reacts quickly. Furthermore, exactly the same sort of analysis as the above leads to the dependence of the mass transfer coefficient fej on the reaction rate coefficient and the diffusivity, D, in the liquid phase, of ki o. (rD), cf. z a RKY above. 

In Chapter 3, the reactor models have been presented along with the hydraulic and mass/heat transfer analysis. In the following sections, the solutions of the reactor models are presented along with several examples. 

Why adsorption, ion exchange and heterogeneous catalysis in one book The basic similarity between these phenomena is that they all are heterogeneous fluid-solid operations. Second, they are all driven by diffusion in the solid phase. Thus, mass transfer and solid-phase diffusion, rate-limiting steps, and other related phenomena are common. Third, the many aspects of the operations design of some reactors are essentially the same or at least similar, for example, the hydraulic analysis and scale-up. Furthermore, they all have important environmental applications, and more specifically they are all applied in gas and/or water treatment. 

Figure 4.26 shows a flow reactor of diameter D in which the downstream portion of the walls is catalytic. Assume that there is no gas-phase chemistry and that there is a single chemically active gas-phase species that is dilute in an inert carrier gas. For example, consider carbon-monoxide carried in air. Assume further a highly efficient catalyst that completely destroys any CO at the surface in other words, the gas-phase mass fraction of CO at the surface is zero. Upstream of the catalytic section, the CO is completely mixed with the carrier (i.e., a flat profile). The CO2 that desorbs from the catalyst is so dilute in the air that its behavior can be neglected. Thus the gas-phase and mass-transfer problem can be treated as a binary mixture of CO and air. The overall objective of this analysis is to... 

This equation indicates that minimal separation time depends on plate numbers, capacity factor, and resistance to mass transfer. It should be pointed out that the analysis times calculated from Equation 2.121 also depend on the desired resolution. Our example calculations were made on the basis of resolution, =... 

Traditionally, CVD reaction data have been reported in terms of growth rates and their dependence on temperature. The data are often confounded by mass-transfer effects and are not suitable for reactor analysis and design. Moreover, CVD reaction data provide little insight, if any, into impurity incorporation pathways. Therefore, the replacement of traditional macroscopic deposition studies with detailed mechanistic investigations of CVD reactions is an area of considerable interest. A recent, excellent review of CVD mechanistic studies, particularly of Si CVD, is available (98), and the present discussion will be limited to highlighting mechanisms of Si CVD and of GaAs deposition by MOVCD as characteristic examples of the combined gas-phase and surface reaction mechanisms underlying CVD. 

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). 

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