Material balance equations, mass transfer model

In deriving the material balance equations, the dispersed plug flow model will first be used to obtain the general form but, in the numerical calculations, the dispersion term will be omitted for simplicity. As used previously throughout, the basis for the material balances will be unit volume of the whole reactor space, i.e. gas plus liquid plus solids. Thus in the equations below, for the transfer of reactant A kLa is the volumetric mass transfer coefficient for gas-liquid transfer, and k,as is the volumetric mass transfer coefficient for liquid-solid transfer. 

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,... 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

In this case, as shown in Figure 4, the subsystems are stoichiometry, material balance, energy balance, chemical kinetics, and interphase mass transfer. The mass transfer phenomena can be subdivided into (1) phase equilibrium which defines the driving force and (2) the transport model. In a general problem, chemical kinetics may be subdivided into (1) the rate process and (2) the chemical equilibrium. The next step is to develop models to describe the subsystems. Except for chemical kinetics, generally applicable mathematical equations based on fundamental principles of physics and chemistry are available for describing the subsystems. 

A model for an adiabatic HDS reactor (see Fig. 4-7) with a single quench is given by Shah et al.46 Under plug-flow conditions and assuming that there are no external mass-transfer resistances, the governing material and energy-balance equations are... 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

Note that these equations are simplified versions of the general model and based on an efifeetive diffusion eoefficient which is dependent on the slope of the adsorption isotherm. Principally speaking, the general model based on the material balance and all mass transfer resistances (concentration boundary layer around a pellet, macro and micropore diffusion, sometimes surface diffusion and laminar pore flow) has to be solved. 

In a number of adsorbents, the adsorbent particle is composed of a large number of microporous microparticles, with larger pores between them. If the dominant mass transfer resistance is within the microparticles, the adsorption process is controlled by the rate of micropore diffusion and the model is defined by the material balance on the microparticle level. For one-dimensional Fickian diffusion, it can be described by the following equation ... 

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