Particle diffusion control

Subject to this approximation and assuming intracrystalline diffusion to be the rate-controlling mass transfer process the response of a crystal to a small differential step change in sorbate concentration at the external surface is described by the following set of equations  

The equilibrium relationship at the crystal surface is assumed to be linear  

If macropore diffusion within the particle is controlling, the equations are 

FIGURE 6.12. Theoretical uptake curves calculated according to Eqs. (6.67) and (6.68) showing the effect of heat transfer resistance. As oo the curves approach the limiting isothermal curve calculated from Eq. (6.4) (—) while for - 0 the curves approach the limiting case of heat transfer control given by Eq. (6.70) (- ) (From ref. 20, with permission.) 

FIGURE 6.13 Experimental uptake curves for CO2 in 4A zeolite crystals showing near isother ma behavior in the large (34- and 21.5-pm) crystals (/)as9x 10 cm s at 371 K and 

---

Rapid Adsorption-Desorption Cycles For rapid cycles with particle diffusion controlling, when the cycle time is much smaller than the time constant for intraparticle transport, the LDF approximation becomes inaccurate. The generalized expression... 

Finally, the agitation rate does not affect the uptake rate if the particle diffusion controls the process. However, the latter criterion may be not safe the agitation in solution may have attained its limiting hydrodynamic efficiency, so that a change in the agitation rate has no effect on the uptake rate even in film diffusion-controlled systems. 

Particle-Diffusion Control. Here the activation barriers are most pronounced in the condensate, and so is the concentration gradient. Equilibrium pertains at the surface, but the mass-transfer coefficient varies with time. Pressure in the vapor phase is constant. 

Collision Model. Figure 3 shows an instructive, efficient, and convenient method for treating the case of mixed surface and particle-diffusion control. It is instructive because it is easy to visualize. It is efficient because although it is a molecular approach, it does not require Monte Carlo calculations, which are expensive and should be avoided whenever possible. It is convenient because it leads to a set of differential equations... 

Concurrent Processes Involved in Ion Exchange 103 Rate Laws for Film and Particle Diffusion Phenomena 105 Differentiating between Film- and Particle-Diffusion-Controlled... 

Differentiating between Film- and Particle-Diffusion-Controlled Phenomena... 

Helfferich, F. (1962b). Ion exchange kinetics. III. Experimental test of the theory of particle-diffusion controlled ion exchange. J. Phys. Chem. 66, 39-44. 

Plesset, M. S., Helfferich, F., and Franklin, J. N. (1958). Ion exchange kinetics. A nonlinear diffusion problem. II. Particle diffusion controlled exchange of univalent and bivalent ions. J. Chem. Phys. 29, 1064-1069. 

The limiting step in the kinetics of ion exchange in the zeolite is the interdiffusion of the electrolyte ions A zi and ions of the species B [24], In the case where the solid ion-exchanger particle is spherical (see Figure 7.9) and the particle diffusion control is the rate-determining process, then Fick s second law equation in spherical coordinates is [47]... 

The author and co-workers have made an attempt to use this intraparticle kinetic model [66] for the explanation of results obtained by L. Liberti et al. for the IE kinetics of forward and reverse sulfate-chloride exchange with complex-forming ion exchangers [33]. It was shown later [67] that in this case the kinetic process was determined by combined film and particle diffusion control. In this connection it was noted [44] that despite their refinement the models often do not allow a mechanism to be identified unambiguously on the basis of experimental observations. 

If mass transfer in the resin determines the overall rate of ion exchange then the reaction is said to be particle or intraparticle diffusion controlled. The simplest models for particle diffusion control regard the resin as a homogeneous gel phase for which Mackie and Mears... 

For these and similar systems the original source, resin or solution, of the counter-ion being chemically consumed and the nature of the co-ion greatly influence the observed kinetics. The association-dissociation of weakly functional resins is of particular practical interest since in these instances a reactive and non-reactive core respectively forms within the resin which shrinks towards the bead centre as exchange proceeds. This Shrinking Core or Shell Progressive mechanism is usually particle diffusion controlled and explains why exchange on weakly functional resins is invariably flow-rate sensitive under column operation. 

The value of this quotient (Helfferich number) is 1 and 1 for film diffusion control and particle diffusion control respectively. For non-isotopic or non-trace exchange the values of D and D are best taken... 

An examination of the rate theory equations for film and particle diffusion kinetics reveals an overall dependence of the rate of reaction on 0 and respectively for gel ( homogeneous ) exchangers. For a macroporous exchanger the rate of exchange, under particle diffusion control, may be either independent of particle size or vary as depending upon whether diffusion within the gel microsphere structure or macropores respectively is rate controlling. 

Loading Free base form operation at a typical flowrate of 5-80 m m h . Unlike water treatment service cycles on strongly functional resins which are all film diffusion controlled the loading cycle on weakly basic anion exchange resins is often particle diffusion controlled and therefore rate sensitive. 

Interruption test suggests particle diffusion control. Even Na distribution between the particle and solution after 5 ntin. 

Petruzzelli, D., L, Liberti, R. Passino, F.G. Helfferich, and Y.L. Ywang. 1987. Chloride/sulfate exchange kinetics Solution for combined film and particle diffusion control. React. Polym. 5 219-226,... 

Spalding, G.E. 1971. Predictive theory of coion transport accompanying particle diffusion controlled ion exchange. J. Chem. Phys. 55 4991-4995. 

Figure 3.6 Efficiency minlmuni for single fiber removal efficiency for particles of finite diameter. For very small particles, diffusion controls according to f3.38) and ijr The different curves nesuli from the effects of velocity. In the interception range according to (3.39), rjtt is...

For the cases of interest, the rate of ion exchange is usually controlled by diffusion, either through a hydrostatic boundary layer, called film dififusion control or through the pores of the resin matrix, called particle diffusion control. 

In the case of particle diffusion control, the rate of ion exchange depends on the charge, spacing and size of the diffusing ion and on the micropore environment. When the resin particle size is large, the feedstream is concentrated, or when a batch system has vigorous stirring, the kinetics are controlled by particle diffusion. 

T he diffusion equation for a binary, particle diffusion controlled, ion-exchange system was solved by a finite difference method in 2 previous papers (4, 5) using an interdiffusion coefficient, DAb, which includes activity gradient terms and is given by... 

Steps 4 and S are the reverse processes of steps 2 and 1, respectively. The kinetics of ion exchange are governed by either a diffusion or mass action mechanism, depending on which is the slowest step. In general, the diffusion of ions in the external solution is teimed film diffusion control. This is a useful concept but hydrodynamically it is ill defined. The diffusion or transport of ions within the exchanger phase is commonly term particle diffusion control. Chemical reaction at the exchanger sites can be rate controlling in certain cases. 

---