Fickian formulation

Equation (6.245) can be inverted to find the generalized Fickian formulation... 

At this point, the variables C and u are now used to connote the average (macroscopic) tracer concentration and velocity, respectively, and the overbar is dropped for convenience. Implicit in this formulation is the belief that fundamentally Lagrangian (particle) dispersion can be modeled as a continuum Eulerian phenomenon in a fashion analogous to the Fickian formulation of molecular transport by Brownian motion. This is a useful fiction for simple modeling exercises, but must be used with caution (see the next section on isopycnal diffusion). 

As a convenience, we have assumed a Fickian formulation for the diffusive flux, with a common diffusivity, D, for all gas species. It is straightforward to incorporate a Maxwell-Stefan formulation see [34, 35] for the diffusive gas fluxes into the analysis presented here. 

Linear driving-force mass-transfer model. This model is a widely accepted (by chemical engineers) empirical formulation that can be extended to very difficult mass-transfer problems if enpirical correlations for the mass-transfer coefficient are available. Since it is usually based on the Fickian formulation, can fail where the Fickian model fails. This is the model most commonly used for separation problems. Use of the Maxwell-Stefan formulation for this model is advised for ternary systems. 

The computed mole fractions as a function of distance along the tube are shown in Fig. 12.14. The mole fractions exhibit a nearly linear drop between their equilibrium values at the surface and zero at the top of the tube. This behavior is not unexpected for this simple system, in which the very dilute species diffuse into one dominant species (air) that is present in great excess. In such a case we expect (and observe) Fickian behavior. That is, we could solve this problem using one of the mixture-averaged formulations discussed in Section 12.7.4 with very little error. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

The most rigorous formulation to describe adsorbate transport inside the adsorbent particle is the chemical potential driving force model. A special case of this model for an isothermal adsorption system is the Fickian diffusion (FD), model which is frequently used to estimate an effective diffusivity for adsorption of component i (D,) from experimental uptake data for pure gases.The FD model, however, is not generally used for process design because of mathematical complexity. A simpler analytical model called linear driving force (LDF) model is often used. ° According to this model, the rate of adsorption of component i of a gas mixture... 

Mechanical dispersion is assumed to mathematically follow a Fickian diffusion formulation, i.e., the mechanical dispersive flux is assumed to be linearly proportional to the concentration gradient. As such the hydrodynamic dispersion is the sum of diffusive and mechanical dispersive terms, and the total dispersive flux is written as ... 

The rigorous Fickian multicomponent mass diffusion flux formulation is derived from kinetic theory of dilute gases adopting the Enskog solution of the Boltzmann equation (e.g., [17] [18] [19] [89] [5]). This mass flux is defined by the relation given in the last line of (2.281) ... 

The multicomponent generalization of Pick s first law of binary diffusion is the second mass flux formulation on the Fickian form considered in this book. The generalized Pick s first law is defined b [72] [22] [62] [20] [96] ... 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

For porous membranes the mass transport mechanisms that prevail depend mainly on the membrane s mean pore size [1.1, 1.3], and the size and type of the diffusing molecules. For mesoporous and macroporous membranes molecular and Knudsen diffusion, and convective flow are the prevailing means of transport [1.15, 1.16]. The description of transport in such membranes has either utilized a Fickian description of diffusion [1.16] or more elaborate Dusty Gas Model (DGM) approaches [1.17]. For microporous membranes the interaction between the diffusing molecules and the membrane pore surface is of great importance to determine the transport characteristics. The description of transport through such membranes has either utilized the Stefan-Maxwell formulation [1.18, 1.19, 1.20] or more involved molecular dynamics simulation techniques [1.21]. 

In addition to molecular difiusion, Knudsen diffusion can also be a significant transport mechanism in the p)ore space of SOFC electrodes. In Knudsen diffusion, the interactions of gas molecules with the pore walls are of the same frequency as the interactions between gas molecules. Knudsen diffusion is typically formulated as Fickian diffusion [Eq. (26.2)], with the Knudsen diffusion coefficient being used in place of the binary diffusion coefiicient. The Knudsen diffusion coefficient of a species is independent of the other species in the system and is derived from the molecular motion of the gas molecules and the geometry of the pores [8, 11, 12). Owing to the small average pore radii of SOFCs ( 10 m [13-15]), diffusion in the pore space of the electrodes usually falls within a transition region where both molecular and Knudsen diffusion are important [16]. To model the transition region. Pick s law can be used with an effective diffusion coefficient to account for... 

Metoclopramide [53] is of intermediate distance (5.8) and showed intermediate properties. For high molecular weight (smaller HSP radius), the kinetics were fiiUy zeroth order with a long time lag before the drug diffused through the shell. For lower molecular weight formulations, there was a mixture of zeroth-order and Fickian diffusion. 

---