Mass transfer model equations

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

The mass transfer model. In our previous work [6] the mass transfer model equations and their mathematical treatment have been described extensively. The relevant differential equations, describing the process of liquid-phase diffusion and simultaneous reactions of the species according to the penetration theory, are summarized in table 1. Recently we derived from this penetration theory description a film model version, which is incorporated in the evaluation of the experimental results. Details on the film model version are given elsewhere [5]. 

Figure 17.11 Validation of the mass transfer model (Equation E17.3.3) for the Kolbe-Schmitt carbonation of /3-naphthol.

Generally speaking, most of the existing mass transfer processes involve fluid flow, heat, and mass transfer. Thus, the process simulation using CMT should comprise momentum, heat, and mass transfer model equation sets for coupling computation as given below. 

For the numerical solution of the mass transfer model equations using the finite element method, the system must first be clearly defined. The finite element method is based on the numerical approximation of the dependent variables at a specific nodal location, where a set of simultaneous linear algebraic equations is produced that can be solved either directly or iteratively. 

The equations below represent the boundary conditions involved in the mass transfer model equations. 

To simulate the empirical concentration profiles, an appropriate mass-transfer model has to be used. One of the simplest models is the model based on the equilibrium-dispersive model, frequently used in column chromatography [1]. It can be given by the following equation ... 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

Roult s law is known to fail for vapour-liquid equilibrium calculations in polymeric systems. The Flory-Huggins relationship is generally used for this purpose (for details, see mass-transfer models in Section 3.2.1). The polymer-solvent interaction parameter, xo of the Flory-Huggins equation is not known accurately for PET. Cheong and Choi used a value of 1.3 for the system PET/EG for modelling a rotating-disc reactor [113], For other polymer solvent systems, yj was found to be in the range between 0.3 and 0.5 [96],... 

Before an in-depth discussion of mass transfer models and coefficients we need to be explicitly clear that all mass transfer models are approximations that allow us to solve the partial differential equations (pde) describing an adsorption problem. There are a great many sources that derive and present the partial differential equations that describe adsorption of gases appropriate for column separations. The Design Manual For Octane Improvement, Book I [7] was among the earlier works to show them. The forms as presented by Ruthven [2] are shown here owing to the consistent and compact nomenclature that he has employed. There are a wider array of forms to choose from in the literature including [6, 7]... 

The mass balance equations for the epilimnion and hypolimnion look like Eq. 21-38, except for the air-water exchange fluxes which are replaced by the vertical fluxes across the thermocline, 7) EH and 7) HE. According to the general form of mass transfer models (Eq.18-4), we can express these fluxes as ... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Mathematical models derived from mass-conservation equations under unsteady-state conditions allow the calculation of the extracted mass at different bed locations, as a function of time. Semi-batch operation for the high-pressure gas is usually employed, so a fixed bed of solids is bathed with a flow of fluid. Mass-transfer models allow one to predict the effects of the following variables fluid velocity, pressure, temperature, gravity, particle size, degree of crushing, and bed-length. Therefore, they are extremely useful in simulation and design. 

The gas-phase mass transfer model (i) is based on molecular diffusion across a laminar boundary layer as described in Equation 5.9. 

Mass transfer models have been used to describe the leaching of soluble substances from porous particles into solution. Such models include a concentration difference that drives the concentration of soluble components in the solids and solution to equilibrate (14). Application of one such model (14) to track release from a solid into solution results in the following equations when applied to xylan conversion in a batch system ... 

Penetration theory equations for the mass transfer model (boundary conditions as usual in penetration theory [6 ]). ... 

For CO2 an expression analoguous to equation (6) can be derived. The values of the enhancement factors in (6) and its CO2 analogon, fjj2g and fc02 resPectively, are obtained from our mass transfer model and account for the interaction between H2S- and C02-amine reactions. 

The algorithm of the kinetics and mass transfer model is a system of algebraic equations that are developed in the following way. The key variable is AN, the change in the amount of a particular reactant or product component involved in a reaction step during an interval At. The material balance of a step is,... 

The solute is disseminated in a solid matrix in the most of the supercritical extractions of natural products. If the interactions between solute and solid matrix are not important, the mass transfer models can be developed from the equations of microscope balances to a volume element of the extractor. If the mass transfer resistance is in its solid phases, the mathematical models must consider the solute transport within the solid particles or the surface phenomena. 

We will now proceed to show how this type of question can be answered using the packed-bed, mass transfer-limited reactors as a model or example system. Mere we want to learn the effect of changes of the various parameters (e.g., temperature, particle size, superficial velocity) on the conversion. We begin with a rearrangement of the mass transfer correlation, Equation (11-49), to yield... 

Mass transfer model that accounted for the mass transfer both inside and outside the emulsion globules, the reaction between the diffusing component and the internal reagent in the globules jointly. A perturbation solution to the resulting nonlinear equations contained the parameters Bi and Da. 

Since Eq. 2.2 contains two functions, C, and another equation or relationship between them is necessary for its solution. Depending on the model of chromatography used, Eq. 2.2 will be accompanied by a mass balance in the stationary phase and a kinetic equation, by a lumped mass transfer kinetic equation, or by an adsorption isotherm (Section 2.1.3). 

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 first-order mass transfer model can be readily interpreted in terms of the various diffusion-based models and several researchers have done so Isee Brusseau and Rao (1989a) and references cited therein]. A straightforward means of equating the two models is to define the mass transfer constant in terms of the aqueous diffusion coefficient, shape factor, and diffusion path length characterizing the porous medium. Ball (1989) reported the following equation, equating k2 from the first-order bicontinuum model to the RIPD model... 

The mass transfer model presented in Equation (30) was applied by Tsoligkas et al. [117] to interpret experimental results characterizing the hydrogenation of 4-nitrobenzoic acid to 4-aminobenzoic acid. The reaction was conducted in a capillary with a circular cross-section and a washcoat incorporating an alumina-supported palladium catalyst. 

---