Mass transfer model equations boundary conditions

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

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. 

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

Peak profiles can be calculated with a proper column model, the differential mass balance equation of the compound(s), the adsorption isotherm, the mass transfer kinetics of the compound(s) and the boundary and initial conditions [13], When a suitable column model has been chosen, the proper parameters (isotherm and mass transfer parameters and experimental conditions) are entered into the calculations. The results from these calculations can have great predictive value [13, 114], The most important of the column models are the ideal model , the equilibrium-dispersive (ED) model , the... 

The boundary conditions used in conjunction with the above equations can vary and are to some degree simulation dependent. Normally, the current density, water flux, reference potential, and water chemical potential are specified but two water chemical potentials or the potential drop in the membrane can also be used. If modeling more regions than just a membrane, additional mass balances and internal boundary conditions must be specified. In addition, for modeling the membrane in the catalyst layers, rate equations are required for kinetics and water transfer among its various phases [40]. The above equations are also valid only for the steady-state case (the time-dependent terms have been ignored). 

Abstract Chapter 5 provides an examination of the numerical solutions of the dyeing models that can be applied to different conditions. Numerical simulation of the system involves the use of Matlab software to solve systems of highly non-linear simultaneous coupled partial differential equations. The finite difference and finite element methods are introduced The partition of the fibrous assembly geometry into small units of a simple shape, or mesh, is examined. Polygonal shapes used to define the element are briefly described. The defined geometries, boundary conditions, and mesh of the system enable solutions to the equations of flow or mass transfer models. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

The latter strongly depends on the specific reaction mechanism, the stoichiometry, and the presence or absence of parallel reaction schemes (69). The rate expressions for Rt usually represent nonlinear dependences on the mixture composition and temperature. Specifically for the coupled reaction-mass transfer problems, such as Eqs. (A10), it is always essential as to whether or not the reaction rate is comparable to that of diffusion (68,77). Equations (A10) should be completed by the boundary conditions relevant to the film model. These conditions specify the values of the mixture composition at both film boundaries. For example, for the liquid phase ... 

The kinetic equations for the volume phase of the solid body are equations of the diffusion type (63). Much attention has been given to them in the literature [154,155], therefore here will be reminded only those aspects of the theory of mass transfer for which the lattice-gas model has been used. These are problems involved in the construction of expressions for the diffusion the coefficients and boundary conditions of the diffusion equations. 

The spatial distribution of composition and temperature within a catalyst particle or in the fluid in contact with a catalyst surface result from the interaction of chemical reaction, mass-transfer and heat-transfer in the system which in this case is the catalyst particle. Only composition and temperature at the boundary of the system are then fixed by experimental conditions. Knowledge of local concentrations within the boundaries of the system is required for the evaluation of activity and of a rate equation. They can be computed on the basis of a suitable mathematical model if the kinetics of heat- and mass-transfer arc known or determined separately. It is preferable that experimental conditions for determination of rate parameters should be chosen so that gradients of composition and temperature in the system can be neglected. 

The problem of mass transfer from a moving Newtonian fluid to a swarm of prolate and/or oblate stationary spheroidal adsorbing particles under creeping flow conditions is solved using a spheroidal-in-cell model. The flow field through the swarm was obtained by using the spheroid-in-cell model proposed by Dassios et al. [5]. An adsorption - 1st order reaction - desorption scheme is used as boundary condition upon the surface of the spheroid in order to describe the interaction between the diluted mass in the bulk phase and the solid surface. The convective diffusion equation is solved analytically for the case of high Peclet numbers where the adsorption rate is also obtained analytically. For the case of low Pe a non-... 

The mass-, heat- and momentum-transfer equations and their corresponding boundary conditions discussed so far are obviously very complex and their solutions are not trivial to obtain. Moreover, the thickness, diffusivities and conductivity of each layer in the membrane element are difficult to measure. It is, therefore, convenient and reasonable to consider the permselective membrane layer and the support layer(s) as an integral region with effective thickness, diffusivities and conductivity for the composite region. And it is also desirable to search for simpler models which are capable of providing the... 

The rate model represented by equation 6.9 is particularly interesting since it will be shown shortly that a similar function arises from a consideration of formal diffusion theory. Therefore, providing it is established by experiment that the pseudo rate constant is truly constant over the range of experimental boundary conditions employed a, b, Xg) it remains perfectly valid to equate its value to appropriate mass transfer parameters required by diffusion theory. 

Takeuchi et al. 7 reported a membrane reactor as a reaction system that provides higher productivity and lower separation cost in chemical reaction processes. In this paper, packed bed catalytic membrane reactor with palladium membrane for SMR reaction has been discussed. The numerical model consists of a full set of partial differential equations derived from conservation of mass, momentum, heat, and chemical species, respectively, with chemical kinetics and appropriate boundary conditions for the problem. The solution of this system was obtained by computational fluid dynamics (CFD). To perform CFD calculations, a commercial solver FLUENT has been used, and the selective permeation through the membrane has been modeled by user-defined functions. The CFD simulation results exhibited the flow distribution in the reactor by inserting a membrane protection tube, in addition to the temperature and concentration distribution in the axial and radial directions in the reactor, as reported in the membrane reactor numerical simulation. On the basis of the simulation results, effects of the flow distribution, concentration polarization, and mass transfer in the packed bed have been evaluated to design a membrane reactor system. 

---