Fickian diffusion model

Electrode transport is often represented by Fickian diffusion models. Such models, however, are applicable to bi-molecular transport, whereas for fuel cells it is often that more than two species exist, especially when considering the use of practical hydrocarbon fuels. For fuels such as natural gas, CH4, H2, H2O, CO and CO2 are all present, and the diffusion of one is coupled to the diffusion of all others. To properly analyze the transport across the electrode in such cases, the Stephan-... 

The model shows that the non-isothermal uptake curve for an adsorbent mass which has low effective thermal conductivity (k ) is identical in form to that of the isothermal Fickian diffusion model for mass transport. 

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 form of equation (21) is interesting. It shows that the uptake curve for a system controlled by heat transfer within the adsorbent mass has an equivalent mathematical form to that of the isothermal uptake by the Fickian diffusion model for mass transfer [26]. The isothermal model hag mass diffusivity (D/R ) instead of thermal diffusivity (a/R ) in the exponential terms of equation (21). According to equation (21), uptake will be proportional to at the early stages of the process which is usually accepted as evidence of intraparticle diffusion [27]. This study shows that such behavior may also be caused by heat transfer resistance inside the adsorbent mass. Equation (22) shows that the surface temperature of the adsorbent particle will remain at T at all t and the maximum temperature rise of the adsorbent is T at the center of the particle at t = 0. The magnitude of T depends on (n -n ), q, c and (3, and can be very small in a differential test. 

The kinetic reactions occurring in the sorption of Ni, Cd, and Zn on goethite during a period of 2 hours to 42 days at pH 6 were hypothesized to occur via a three-step mechanism using a Fickian diffusion model (1) sorption of trace elements on external surfaces (2) solid-state diffusion of trace elements from external to internal sites and (3) trace element binding and fixation at positions inside the goethite particle (Bruemmer et al., 1988). 

The Fickian diffusion models with constant effective diffusivities presented earlier and the rigorous dusty gas model presented in this section are not the only alternatives for modelling diffusion and reaction in porous catalyst pellets. Fickian models with effective diffusion coefficients which are varying with the change of concentration of the gas mixture can also be used. This is certainty more accurate compared with the Fickian model with constant diffusivities although of course less accurate than the dusty gas model. The main problem with these models is the development of relations for the change of diffusivities with the concentration of the gas mixture without solving the dusty gas model equations. Two such techniques are presented in this section and their results are compared with dusty gas model results. 

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. 

So far in this chapter the diffusion problem has been treated in a relatively simple manner using the Fickian diffusion models, which are the most widely used, although they are not rigorous for multi-component systems. In the present section the diffusion... 

When the Fickian diffusion model is used many reaction-diffusion problems in porous catalyst pellet can be reduced to a two-point boundary value differential equation of the form of equation B.l. This is not a necessary condition for the application of this simple orthogonal collocation technique. The technique in principle, can be applied to any number of simultaneous two-point boundary value differential equations. 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

The two major approaches to equation development for turbulent diffusion are the Fickian diffusion model and the description of the basic turbulent process by the theory of diffusion by continuous motion. Only the Fickian approach will be presented here. Fick proposed that the molecular diffusion of matter could be expressed in the same manner as the conduction of heat or electricity in a conducting body, as used by Fourier and Ohm, respectively, in their work. This can be stated as making the rate of diffusion in any direction directly proportional to the concentration gradient in that direction. Mathematically, this becomes... 

In Chapter 1 of this book we presented Eqs. fl-5a and b) that relate the rate of mass transfer/volume to a mass-transfer coefficient, the area/volume, and a driving force. This mathematical model is an attenpt to quantify a complicated situation. Although useful, this model and the other models presented later in this chapter can also be misleading. The term driving force inplies purpose or desire to transfer, and as noted, there is no purpose—the molecules are just moving randomly. With this caveat, let s look at the various models used to analyze mass transfer, starting with the Fickian diffusion model. 

One difficulty with the Fickian diffusion model should be obvious from the discussion in Section 1 S.2.3. One has to select a somewhat arbitrary basis velocity or plane of reference to calculate the convective and diffusive fluxes. The values and indeed the meaning of the convective and diffusive fluxes may change when the basis velocity is changed. Although irritating, this difficulty is not considered to be major because the total fluxes of A and B, which is the purpose of the calculation, do not change. 

For ionomeric systems in which the strong interactions between ionic sites and the penetrant (water) result in concentration dependent diffusion coefficients, the Fickian diffusion model, which assumes the solubility coefficient is independent of the concentration, is not valid. The commonly used Fickian diffusion constants actually contain mobility and solubility gradient contributions [19], In addition, the concentration dependent solubilities lead to nonlinear concentration profiles during steady state diffusion. Therefore, mobility measurements which generate average diffusion coefficients are generally not satisfactory. 

High-temperature interaction between gas and solid is normally treated as a diffusion problem in solid state science, using normal Fickian diffusion models. 

Loh et al. (2005) proposed a dual-uptake model based on the summation of two Fickian diffusion models. Physically this can be interpreted as two different uptakes processes operating in parallel, both of which are adequately described by Fickian diffusion. This model was... 

The rate of water ingress by diffusion through the adhesive is principally determined by its diffusion coefficient. An increase in temperature will rapidly increase the rate of water uptake in line with Fickian diffusion models as the diffusion coefficient will increase significantly with temperature. Such a hot-wet environment is generally regarded as a very aggressive form of... 

Rodriguez (2006) was able to locate numerous sets of data on both the soil turnover rates and depth of soil activity/disturbance for several types of macrofauna. These were converted to particle biodiffusion coefficients and summaries of the data appear graphically in Figure 13.2. Shown in this figure are cumulative probability distributions of this coefficient for earthworms, ants and termites and vertebrates. Included within the earthworm particle data set are four sorbed-phase chemical data points. These Dbs data are for PBCs, which were extracted from concentration profiles using a Fickian diffusion model. The PCB profiles were obtained in soils with abundant earthworm populations. The reader should note that the range of I>bs values for soils is within those for sediments see Figure 13.1 in comparison. 

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