Reactor, batch equations with diffusion

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

In this way, the diffusion/reaction equations are reduced to trial and error algebraic relationships which are solved at each integration step. The progress of conversion can therefore be predicted for a particular semi-batch experiment, and also the interfacial conditions of A,B and T are known along with the associated influence of the film/bulk reaction upon the overall stirred cell reactor behaviour. It is important to formulate the diffusion reaction equations incorporating depletion of B in the film, because although the reaction is close to pseudo first order initially, as B is consumed as conversion proceeds, consumption of B in the film becomes significant. 

In these equations D represents the corresponding diffusion coefficients, and Q the permeate flow rate. The first term of each equation gives the radial dispersion and the second one corresponds to the radial convection. The authors [5.103] used in their model, a biological kinetic rate expression (cp), which was obtained by independent experiments and analysis of a batch reactor, and also made an effort to account for and correlate the permeate flow decrease with the amount of produced biomass. The simulation curves obtained matched well the experimental results in terms of permeate flow rate evolution and product concentration. One of the important aspects of the model is its ability to theoretically determine the biomass concentration profiles, and the relation between the permeate flow rate and the calculated biomass concentration in the annular volume (Fig. 5.24). Such information is important since the biomass evolution cannot be determined by any experimental methodology. 

Equations 14.2-3 and 14.2-4 bear a striking resemblance to the mass and energy balances for a batch reactor, Eqs. 14.1-13 and 14. There is, in fact, good physical reason why these equations should look very much alike. Our model of a plug-flow reactor, which neglects diffusion and does not allow for velocity gradients, assumes that each element of fluid travels through the reactor with no interaction with the fluid elements before or after it Therefore, if we could follow a small fluid element in a tubular reactor, we would find that it had precisely the same behavior in time as is found in a batch reactor. This similarity in the physical situation is mirrored in the similarity of the descriptive equations. 

This is the mass balance equation for the batch reactor. The symbol t for clock time is replaced here by the more usual symbol 0 for batch residence time. For the continuous, completely mixed reactor, it is usdiil to start from the reduced continuity equation in terms of concentrations, analogous to Eq. 7.2.b-l (but with no diffusion term) ... 

The work is organized in two parts in the first part kinetics is presented focusing on the reaction rates, the influence of different variables and the determination of specific rate parameters for different reactions both homogeneous and heterogeneous. This section is complemented with the classical kinetic theory and in particular with many examples and exercises. The second part introduces students to the distinction between ideal and non-ideal reactors and presents the basic equations of batch and continuous ideal reactors, as well as specih c isothermal and non-isothermal systems. The main emphasis however is on both qualitative and quantitative interpretation by comparing and combining reactors with and without diffusion and mass transfer effects, complemented with several examples and exercises. Finally, non-ideal and multiphase systems are presented, as well as specific topics of biomass thermal processes and heterogeneous reactor analyses. The work closes with a unique section on the application of theory in laboratory practice with kinetic and reactor experiments. 

The rate of metal removal in the porous oxide sorbents can be described with a film transfer process and either surface and/or pore diffusion models. To simplify the mass transfer of adsorbate from bulk solution to the adsorbent surface, some studies assume a linear concentration gradient existing in a hypothetical film surrounding the adsorbent particle [25]. When film transfer limits the rate, which, for example, is likely with nonporous particles, the following equation can be used to simulate the film transfer in a batch reactor ... 

To evaluate the appropriate form of Equation (7.18) we must employ the design equation used to obtain the kinetic data. We studied the kinetics of adsorption using a batch reactor with recycle operation in the differential mode. The reactor consists of a packed column with the adsorbent between two layers of glass beads. Pore diffusion and mass transfer resistances were minimized by using small particle sizes (180 to 120 pm) and high flow rates. The design equation written for the aforanentioned metal cation extraction is... 

Xu and Chang (1996), have conducted kinetic measurements in a batch reactor for the esterification of dilute acetic acid with methanol in presence of Amberlyst 15 at 367 K. The internal mass transfer resistance was found to be insignificant. A kinetic equation was developed and used in the simulation and design of a catalytic distillation column for removing dilute acetic acid from wastewater. Xu and Chang (1997) have also presented a theoretical analysis to determine the effect of internal diffusion on second order reversible esterification. The results of the analysis showed that the catalyst effectiveness factors were above 0.94 for beads smaller than 0.6 mm diameter at a temperature lower than 353 K. However, the effectiveness factors were lower than 0.77 for the beads larger than 1.0 mm diameter at reaction temperature higher than 367 K. 

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