Solid Diffusion Control

Alternate driving force approximations, item 2B in Table 16-12, for solid diffusion, and item 3B in Table 16-12, for pore diffusion, provide somewhat more accurate results in constant pattern packed-bed calculations with pore or solid diffusion controlling for constant separation factor systems. 

FIG. 16-16 Batch adsorption curves for solid diffusion control. The curve for A = 0 corresponds to an infinite fluid volume (adapted from Ruthven, gen. refs., with permission). 

This equation has the same form of that obtained for solid diffusion control with D,j replaced by the equivalent concentration-dependent diffusivity = pDpj/[ pn]Ki l - /i,//i)) ]. Numerical results for the case of adsorption on an initially clean particle are given in Fig. 16-18 for different values of A = = 1 - R. The upt e curves become... 

Parallel Pore and Solid Diffusion Control With a linear isotherm, assuming equilibrium between the pore fluid and the solid adsorbent, batch adsorption can be represented in terms of an equivalent solid diffusivity = ( pD i + ppD, )/( p + pp Q). Thus, Eqs. (16-96) and (16-99) can be used for this case with D, replaced by D. ... 

These expressions can also be used for the case of external mass transfer and solid diffusion control by substituting D, for 8pDpi/( p + ppK)) and/c rp/(ppK)D,i) for the Biot number. 

The diffusion of U and Th within a solid is, in general, very slow due to their large size and charge (Van Orman et al. 1998). Even at mantle temperatures, it is expected that a solid will not fully equilibrate with the surrounding phases (fluid, melt or other solid phases) if solid diffusion controls the equilibration. As yet, there have been no direct determinations of diffusion coefficients for any other decay chain element. 

This equation has the same form as that obtained for solid diffusion control with D replaced by the equivalent concentration-dependent diffusivity Da = - nf/nf)2]. Numerical results for the... 

In the following analysis, adsorption models for solid diffusion control are applicable for isotopic exchange, i.e. exchange of isotopes, while in the case of liquid diffusion control and the intermediate case, only adsorption models for linear equilibrium can be used for isotopic exchange. 

Infinite fluid volume and solid diffusion control Practically, infinite solution volume condition (w 1) amounts to constant liquid-phase concentration. For a constant diffu-sivity and an infinite fluid volume, the solution of the diffusion equations is (Helfferich, 1962 Ruthven, 1984)... 

Finite fluid volume and solid diffusion control The solution is the following (Perry and Green, 1999) ... 

Solid diffusion control and infinite solution volume In this case, the diffusion coefficient is not a constant but depends on the concentration of the ions in the solid phase. The basic diffusion equation to be solved is the following (Helfferich, 1962) ... 

These equations, for the case of solid diffusion-controlled kinetics, are solved by arithmetic methods resulting in some analytical approximate expressions. One common and useful solution is the so-called Nernst-Plank approximation. This equation holds for the case of complete conversion of the solid phase to A-form. The complete conversion of solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion, requires an excess of liquid volume, and thus w 1. Consequently, in practice, the restriction of complete conversion is equivalent to the infinite solution volume condition. The solution of the diffusion equation is... 

According to their analysis, if is zero (practically much lower than 1), then the liquid-film diffusion controls the process rate, while if tfis infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the so-called mechanical parameter represents the ratio of the diffusion resistances (solid and liquid film). The authors did not refer to any assumption concerning the type of isotherm for the derivation of the above-mentioned criterion it is sufficient to be favorable (not only rectangular). They noted that for >1.6, the particle diffusion is more significant, whereas if < 0.14, the external mass transfer controls the adsorption rate. 

In the case of solid diffusion control, even in the absence of agitation where the mass transfer coefficient is at its minimum value, sufficient agitation should be provided in order to avoid the negative effect of the liquid-film resistance. The effect of agitation should be taken into account in both the design and application stage. 

In this expression, U(t) is relative rate of uptake and Cx is relative to equilibrium, i.e. the sites available for ion exchange or adsorption for the specified ratio Vim. Thus, the absolute rate is a coupled result of kinetics and equilibrium. Note that in a solid diffusion-controlled process, U(t) is relative to the ease of movement of the incoming species in the solid phase (through Ds). 

Meshko et al. (2001) used a homogeneous solid model taking into account both internal and external diffusion. They found that the adsorption of the dye had not been significantly affected by the agitation speed, which indicated that the process was solid diffusion-controlled. Furthermore, for the specified conditions, they found that kf = 6.66 X 10 s m/s and /), =10 12 m2/s. 

The finite solution volume model for solid-diffusion control (Patterson s model) will be used (eq. (4.52)). Following the procedure presented in the section Design of a batch reactor system for adsorption and ion exchange (eqs. (4.119)-(4.125)), we obtain the results shown in Table 4.23. 

Assuming a solid diffusion-controlled process, use the Patterson equation and find the corresponding solid diffusion coefficient. Furthermore, find the impeller speed in order to have the reported value for k. ... 

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... 

In Figure 4.23, the model results for solid diffusion control (eq. (4.141)) and two different values of the Langmuir constant La) are presented. In Figure 4.24, the model results for solid diffusion and liquid-film diffusion control (eq. (4.140)) for La = 0.5 are presented. 

Finally, two models based on nonlinear driving forces will be presented. The first one covers the case of a pore diffusion control and the second one the case of solid diffusion control. Both models hold for the Langmuir-type isotherm. For the case of... 

For solid diffusion control, the model equations are the following (Inglezakis, 2002b) ... 

Note that in the case of a solid diffusion control, X- is replaced by X in the analytical solutions. The presented analytical solutions were developed primarily for solid diffusion control, also termed surface diffusion kinetics. However, the same equations can be used as approximations for pore diffusion as the controlling intraparticle diffusion step, if ksau is set equal to 15D (1 — e)/yi. ... 

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

In Figure 4.27, some examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5) are presented. As is clear, the curve corresponds to the case of equal and combined solid and liquid-film diffusion resistances ([ = 1) which is between the two extremes, i.e. solid diffusion control (l = 10,000) and liquid-film diffusion control ( = 0.0001). 

Using the Miura-Hashimoto model, calculate the time needed to reach a breakpoint concentration of 9.92 mg/L (10%). According to the experimental results given by Hashimoto et al., the time needed for the specified breakpoint concentration is 226 hr. What is the result if the solid diffusion control approximation is used ... 

This is only a difference of 0.45% from the experimental value. Constant pattern approximation does not negatively affect the final result. For solid diffusion control, and for this case, from eq. (4.184), 0T - XT = -1.277 (Xi = 0.1) and t. 227 h or a difference of 0.25% from the experimental value. 

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).

Model analysis The simple LDF model for solid diffusion control will be used, namely eq. (4.141). For the specified system with La = 0.1, the NS T- 1) versus (C/C0) is shown in Figure 4.34. 

It is obvious that for the whole flow-rate range, the rate-controlling mechanism is expected to be the solid diffusion control (Hi > 41). Furthermore, the flow can be characterized as ideal plug flow for flow rates above 2.15 BV/h, where PeL is higher than about 100. However, the liquid holdup is very low (56.83%) and this could be proved a serious problem, even with the use of a liquid distributor at the top of the bed. In order to have a satisfying liquid holdup, i.e about 80%, the relative flow rate should be about 5.62 BV/h. Then, by means of a liquid distributor at the top of the bed, it is possible to achieve a holdup near 100%. Thus, for downflow operation the limits of the relative flow rate are (BV/h)... 

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