Fluid film diffusion

This model covers the case where we have combined resistances to diffusion (fluid-film and solid diffusion). In this case, the concentration in the main phase of the fluid (bulk concentration) is different from the one at the interface due to the effect of the fluid film resistance. The following equations can be used for Langmuir and Freundlich equilibrium equations (Miura and Hashimoto, 1977). The solutions of the fixed-bed model are the following ... 

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Resistance to transfer of mass between phases is assumed to be confined to that of fluid films between the phases. Let D = diffusivity... 

Thus, the analysis of the rate-determining step, as analyzed for heterogeneous processes in Section 3.1.2, is equally applied in adsorption and ion exchange. The only difference is that the diffusion processes in the fluid film and in the particle are followed by physical adsoiption or ion exchange and not by a reaction step as in catalysis. 

Eq. (4.140) is for liquid-film diffusion control and eq. (4.141) for solid diffusion control. The following equation is a solution of the fixed-bed model under the constant pattern and plug-flow assumption, for fluid-film diffusion control and the favorable Freundlich... 

The limiting cases of the analytical solutions for external fluid-film mass transfer controlling (c —> 0) and solid diffusion controlling -> oo ) are the following ... 

Figure 4.32 Characteristic C/C0 versus N (T - 1) curves for solid diffusion control (dotted line) and fluid-film diffusion control (La = 0.2).

Figure 4.33 Stoichiometric point curves (S solid diffusion control, F fluid-film diffusion control).

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

These groups have a definite, important, physical meaning. The Reynolds number is the ratio of inertial forces to viscous forces, the Sherwood number the ratio of mass transfer resistance in fluid film to mass transfer in bulk fluid, and Schmidt number the ratio of momentum diffusivity to mass diffusivity. 

The example furthermore shows that diffusion from the bulk fluid phase toward the volume near the IRE, which is probed by the evanescent field, has to be accounted for because it may be the limiting step when fast processes are investigated. The importance of diffusion is more pronounced when a catalyst layer is present on the IRE, because of the diffusion in the porous film is much slower than that in the stagnant liquid film. Indeed, the ATR method, because of the measurement geometry, is ideally suited to characterization of diffusion within films (50,66-68). Figure 16 shows the time dependence of absorption signals associated with cyclohexene (top) and i-butyl hydroperoxide (TBHP, bottom). Solutions (with concentrations of 3mmol/L) of the two molecules in cyclohexane and neat cyclohexane were alternately admitted once to... 

There are three distinct mass-transfer resistances (1) the external resistance of the fluid film surrounding the pellet, (2) the diffusional resistance of the macropores of the pellet, and (3) the diffusional resistance of the zeolite crystals. The external mass-transfer resistance may be estimated from well-established correlations (4, 5) and is generally negligible for molecular sieve adsorbers so that, under practical operating conditions, the rate of mass transfer is controlled by either macropore diffusion or zeolitic diffusion. In the present analysis we consider only systems in which one or other of these resistances is dominant. If both resistances are of comparable importance the analysis becomes more difficult. 

Generally all these considerations are also valid for the second fluid film phase, provided that reactions occur there (135). Both analytical and numerical solutions of the coupled diffusion-reaction film problem are analyzed at full length in Ref. 167 their particular applications are considered in Section 3. 

Figure 5.1. Diffusion mechanisms and corresponding concentration profiles, where C is the overall solid concentration, r0 is the mean radius of the particle, r is the linear distance along the particle radius from the radius surface, and 5 is the thickness of the stagnant fluid film. [From Liberti and Passino (1983), with permission.]...

FIGURE 4 Schematic diagram of a biporous adsorbent pellet showing the three resistances to mass transfer (external fluid film, macropore diffusion, and micropore diffusion). R9 pellet radius rc crystal radius. 

Consider an example from nucleation and growth of thin films. At least three length scales can be identified, namely, (a) the fluid phase where the continuum approximation is often valid (that may not be the case in micro- and nanodevices), (b) the intermediate scale of the fluid/film interface where a discrete, particle model may be needed, and (c) the atomistic/QM scale of relevance to surface processes. Surface processes may include adsorption, desorption, surface reaction, and surface diffusion. Aside from the disparity of length scales, the time scales of various processes differ dramatically, ranging from picosecond chemistry to seconds or hours for slow growth processes (Raimondeau and Vlachos, 2002a, b). 

The method can be applied to investigate the bidisperse pore structures, which consist of small microporous particles formed into macroporous pellets with a clay binder. In such a structure there are three distinct resistances to mass transfer, associated with diffusion through the external fluid film, the pellet macropores, and the micropores. Haynes and Sarma [24] developed a suitable mathematical model for such a system. 

Generally, there are four steps are included in the mass transfer mechanisms of the adsorption process. These steps are fluid-film transfer, pore diffusion, surface adhesion, and surface diffusion. The rate of surface adhesion for physical adsorption on the surface of porous adsorbents is very rapid, enough to be assumed instantaneous relative to the other transfer rates [5]. 

In this regard, Frank-Kamenetskii also describes that the main diffusion resistanee is localized within the so-called viscous sublayer, i.e., the fluid film [41]. 

In catalytic reactions mass transfer from the fluid phase to the active phase inside the porous catalyst particle takes place via transport through a fictitious stagnant fluid film surrounding the particle and via diffusion inside the particle. Heat transport to or from the catalyst takes the same route. These phenomena are summarized in Fig. 8.15. 

Proposed by Whitman,the film theory is based on the assumption that for a flowing fluid, there is a fictitious stagnant fluid film at the phase boundary, in which the entire resistance to mass transfer resides and the mass transport is, thus, completely by molecular diffusion. Therefore, the mass transfer coefficient is proportional to the ratio of the diffusivity to the thickness of the fictitious film. 

The number of transfer units for each mechanism can be estimated from known parameters and mass transfer correlations (4). For example, for a column with particles 0.01 cm in diameter, a superficial velocity of 0.01 cm/sec, and a solute bulk diffusivity of 7 x 10-7 cm2/sec, the estimated number of transfer units in a packed bed of length L for the four mechanisms, axial dispersion, external fluid film mass transfer, pore diffusion, and solid homogeneous particle diffusion,are... 

All facts established for a fluid film remain valid for a majority of problems on the diffusion boundary layer. Namely, near a gas-fluid or fluid-fluid interface, the dimensionless thickness of the layer is proportional to Pe-1 2 (for the diffusion flux we have j Pe1 2), and near the fluid-solid interface the thickness of the boundary layer is proportional to Pe-1 3 (the diffusion flux is j Pe1 3). 

Now let us consider mass transfer from a solid wall to a fluid film. We assume that the concentration on the surface of the plate is constant and is equal to Cs and that a pure fluid is supplied through the input cross-section. In the diffusion boundary layer approximation, the velocity profile near the surface of the plate can be approximated by the expression... 

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