Mass diffusion model including resistance

Using the computer programs discussed above, it is possible to extract from these breakthrough curves the effective local mass transfer coefficients as a function of CO2 concentration within the stable portion of the wave. These mass transfer coefficients are shown in Figure 15, along with the predicted values with and without the inclusion of the surface diffusion model. It is seen that without the surface diffusion model, very little change in the local mass transfer coefficient is predicted, whereas with surface diffusion effects included, a more than six-fold increase in diffusion rates is predicted over the concentrations measured and the predictions correspond very closely to those actually encountered in the breakthrough runs. Further, the experimentally derived results indicate that, for these runs, the assumption that micropore (intracrystalline) resistances are small relative to overall mass transfer resistance is justified, since the effective mass transfer coefficients for the two (1/8" and 1/4" pellets) runs scale approximately to the inverse of the square of the particle diameter, as would be expected when diffusive resistances in the particle macropores predominate. 

Both radial and axial diffusion can be taken into account and the final equations to be solved are relatively simpler than those of the continuum model. Although, the equations of the model at steady state are algebraic equations, the dimensionality of the system increases considerably. McGuire and Lapidus (1965) used this model for the study of the stability of a packed bed reactor which included both interphase and intraparticle mass and heat transfer resistances. 

A further generalization of the Glueckauf approximation is suggested by comparison of the moments for the simple linear rate plug flow model (model la) and the general diffusion model with axial dispersion (model 46). One may define an overall effective rate coefficient (k ) which includes both the effects of axial dispersion and mass transfer resistance ... 

On the other hand, customized models for catalytic reactors include all of the main processes taking place inside the catalytic reactors. The most important of these processes for catalytic reactors are those associated with the catalyst pellets, namely intrinsic kinetics (which includes chemisorption and surface reaction), intraparticle diffusion of mass and heat, external mass and heat transfer resistances between the catalyst surface and the bulk of the fluid, as well as all the heat production and heat consumption accompanying the catalytic reaction. 

By changing the outer boundary condition, external resistance can be included in the diffusion model, though this makes the mathematical resolution of the model more difficult. In this case, two parameters must be identified (i) the effective diffusivity (internal resistance) and (ii) the mass transfer coefficient (external resistance). In order to reduce the number of possible solutions, the effective diffusivity identified by neglecting the external resistance can be used as the start value of diffusivity. 

In this book a combination of the principles of separation processes, process modelling, process control and numerical methods is used to describe the dynamic behaviour of separation processes. The text is largely mathematical and analytical in nature. Adsorption processes are commonly operated in a cyclic manner involving complex sequences of individual steps which are dynamic in nature and three chapters in this book specifically address this separation process. Chapter 11 covers the fundamentals of adsorption processes and includes physical adsorption of pure gases and mixtures, mass transfer by convective transport and the roles of pore and surface diffusion in the adsorption process. Chapter 12 addresses the separation of multicomponent mixtures by the use of adsorption columns and includes the Gleuckauf, film resistance and diffusion models and adiabatic operation of a fixed bed adsorption column together with periodic operation. Chapter 14 addresses the thermodynamics of the physical adsorption of pure gases and multicomponent gas mixtures. 

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

In many industrial reactions, the overall rate of reaction is limited by the rate of mass transfer of reactants and products between the bulk fluid and the catalytic surface. In the rate laws and cztalytic reaction steps (i.e., dilfusion, adsorption, surface reaction, desorption, and diffusion) presented in Chapter 10, we neglected the effects of mass transfer on the overall rate of reaction. In this chapter and the next we discuss the effects of diffusion (mass transfer) resistance on the overall reaction rate in processes that include both chemical reaction and mass transfer. The two types of diffusion resistance on which we focus attention are (1) external resistance diffusion of the reactants or products between the bulk fluid and the external smface of the catalyst, and (2) internal resistance diffusion of the reactants or products from the external pellet sm-face (pore mouth) to the interior of the pellet. In this chapter we focus on external resistance and in Chapter 12 we describe models for internal diffusional resistance with chemical reaction. After a brief presentation of the fundamentals of diffusion, including Pick s first law, we discuss representative correlations of mass transfer rates in terms of mass transfer coefficients for catalyst beds in which the external resistance is limiting. Qualitative observations will bd made about the effects of fluid flow rate, pellet size, and pressure drop on reactor performance. 

Pellet [35] and Rasmuson and Neretnieks [36] extended the solution of Rosen by including axial dispersion, but still assuming that the kinetics of adsorption-desorption is infinitely fast. Later, Rasmuson [37] extended the earUer solution and calculated the profile of a breakthrough curve (step bormdary condition, or frontal analysis) in the framework of the general rate model (Eqs. 6.58 to 6.64a), which includes axial dispersion, the film mass transfer resistance, the pore diffusion, and a first-order slow kinetics of adsorption-desorption. 

Discrepancies between experimentally obtained and theoretically calculated data for cadmium concentration in the strip phase are 10-150 times at feed or strip flow rate variations. These differences between the experimental and simulated data have the following explanation. According to the model, mass transfer of cadmium from the feed through the carrier to the strip solutions is dependent on the diffusion resistances boundary layer resistances on the feed and strip sides, resistances of the free carrier and cadmium-carrier complex through the carrier solution boundary layers, including those in the pores of the membrane, and resistances due to interfacial reactions at the feed- and strip-side interfaces. In the model equations we took into consideration only mass-transfer relations, motivated by internal driving force (forward... 

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