The standard approach in the development of a mathematical model for any separations scheme is to begin with the equations of mass continuity and then add additional terms (e. g., equations for equilibrium, mass transfer and reaction kinetics) as needed to accurately describe a specific process. Models developed from first principles are much more flexible than those obtained by regression or simply searching for equations that give the best ‘fit’ to empirical data. Since they are derived from fundamental principles and ensure energy and/or mass continuity, they can be used to estimate performance when system parameters are changed without the constraint of maintaining dynamic similitude. Nonetheless, the model must be vali­dated or compared to actual performance data in order to assess accuracy and/or modify terms to increase model robustness. Ideally, multiple itera­tions of this comparison are done for a range of operating parameters. This provides data for refinement and serves to increase confidence in the model and validate its utility for predictive purposes. The extent of this validation process is normally determined by several factors such as model complexity, availability of published material relating to similar models and systems, and the required accuracy of the model in terms of economic, safety and regulatory drivers. From an engineering perspective, modeling solid-phase extraction is quite similar to that of other separation processes wherein solutes are separated from a flowing fluid stream onto particles packed within a column (e. g., adsorption or ion exchange). Contacting the moving fluid with a stationary solid phase is often chosen for separating dilute solutes since the packed bed functions like a very large number of thin contactors in series. Obtaining solutions for models of these systems can, however, be difficult, since the active exchange zone is changing in both spatial and temporal dimensions during the process. Changes in concentra­tion with the radial component can usually be neglected and other simplify­ing assumptions may also be valid in order to minimize computational effort. The mass balance equation for a multi-component system to account for concentration changes in the axial direction with time is typically written as (Ruthven, 1984, Klein, 1982):

13.1

d=tF (c, q)

where:

є = bed void fraction

v = average interstitial fluid velocity

z = the distance from the bed inlet in the axial direction

D = axial dispersion coefficient for the bed

к = effective mass transfer coefficient

c, q = the fluid and solid phase concentrations, respectively

t = time

F(c, q) = a driving force to be specified With initial and boundary conditions as:

I. C. (c, q)|Z=0 = 0; starting with no sorbate on resin

B. C. c|Z=0 = C0 dC t

— = 0 13.3

dZz=L

Note: specific units are not included in this generic discussion; however, the reader should understand that choices for time, mass, length and volume units must remain consistent throughout.

A few analytical solutions have been obtained for very simple, single­component cases. Approximate solutions can also be attained in some cases by utilizing classical functions of series expansions, but solutions are best achieved by numerical integration. Numerical methods utilizing finite dif­ference, orthogonal collocation, method of characteristics or combinations thereof usually give very accurate solutions; and systems having <3 compo­nents are efficiently solved with simple finite difference algorithms (Ruthven, 1984, Tranter et al., 2009b, 2005, 2002). Due to the large increase in processing speed over the last two decades, many scientists and engineers now have adequate computational power for performing efficient numeri­cal integrations via desktop computers. Software to perform the necessary matrix operations (e. g., inversion, multiplication) are available commer­cially or can be written in user friendly languages (e. g., Visual Basic®, C++, or MATLAB®). It is evident from Eqns (13.1) and (13.2) that a driving mechanism or equation for mass transfer and equilibrium must be deter­mined along with the corresponding coefficients for mass transfer, equilib­rium and dispersion. It is best to determine the mass transfer and equilibrium parameters by separate batch experiments, since these are very significant to the accuracy of the model. Dispersion effects are often minor in com­parison and values obtained from published equations (Ruthven, 1984, Chung et al., 1968) often provide adequate correlation to experimental data. Though, the dispersion coefficient can be derived independently from tracer studies if dispersion effects are determined to be substantial for a given
system. Once the governing parameters and corresponding coefficients are obtained from independent experimentation, the values are used in the dynamic model (Eqn. 13.1 and 13.2) to predict breakthrough curves for a given resin and column configuration. The predicted breakthrough curves are then compared to data obtained from experimental column tests to assess the overall accuracy and rigor of the model.

Numerous expressions for the rate equation (Eqn. 13.2) have been pro­posed (LeVan et al., 1997, Ruthven, 1984). Although many were originally developed for ion exchange or sorption processes, the mathematics are based on the physical phenomena of diffusion to describe the resulting concentration gradients established within a porous media. They may there­fore be applied to describe solid-phase extraction processes with appropri­ate technical understanding and discretion. An evaluation of the literature shows that the assumption of a linear driving force typically provides suf­ficient model accuracy and has been used quite successfully by this author and numerous others. A linear driving force in terms of the solid phase concentration is thus used for purposes of this discussion and is written as:

where:

ce, qe = equilibrium concentration at the fluid-pore interface for the liquid and solid phases, respectively

Solution of Eqn. (13.4) requires additional equations and/or terms for calculating the mass transfer coefficient and fluid-solid pore equilibrium at each time step. The form of the equilibrium equation and corresponding equilibrium constant is specific to a given solid-phase extraction process and is selected to best represent the system and match experimental data. Equilibrium expressions from the extraction stoichiometry are often used. For example, Takeshita (1999) modeled a system for Ln extraction using a styrene-divinylbenzene copolymer impregnated with CMP (dihexyl-N-N — diethylcarbamoylmethylphosphonate) and used the following equilibrium model for calculating phase equilibrium concentrations:

where:

KLn = equilibrium coefficient for the given Ln element

Other researchers have used the equilibrium isotherm approach wherein solid and liquid phase equilibrium concentrations are obtained over a range

of interest by performing multiple, isothermal batch contacts at varying ratios of sorbent to extractable species. An appropriate isotherm equation is fit to the batch data and this equation is used in the model (Eqn. 13.1) to calculate the fluid-pore phase equilibrium concentrations for each time step in the numerical solution. The Langmuir or Freundlich isotherm models (LeVan et al., 1997) are frequently used to fit the equilibrium data for single component systems and extensions may sometimes be applicable to multi­component cases. Although the mechanism of extracting the species onto the solid phase differs from the sorption phenomena assumed for the deri­vation of these models, the isotherm models often provide a good fit to the equilibrium data and can simplify the numerical computation substantially; however, they are limited to the feed conditions under which they were generated and do not provide a straightforward way of accounting for changes of significant parameters in the liquid phase, e. g. nitrate concentra­tion or total ion activity. Ultimately the selection of a model to describe equilibrium, whether more fundamental or empirical, must be based on the complexity of the system, how well it is understood and the appropriate compromise between accuracy and computational difficulty.

The transfer of metal ion into the solid-phase extraction resin consists of several steps: 1) transport of the metal from the bulk fluid to the film inter­face between fluid and particle, 2) diffusion through the film layer, 3) dif­fusion through the particle — within the pores or gel phase — and 4) complex formation between the metal and extractant impregnated in the particle. The different regions of mass transfer are depicted in Figure 13.16.

Each of the above steps contributes to the total resistance for metal uptake by the resin; however, one or two of these processes are typically much slower than the rest and become the rate limiting steps for overall mass transfer. Transport through the bulk fluid and the complex formation reaction are usually fast in comparison to film or particle diffusion, thus coefficients for mass transfer of metal in the film and/or particle are most frequently needed for solving the rate expression of Eqn (13.5). Classical methods based on data from batch experiments (Helfferich, 1995) or column interruption tests (Tranter et al., 2009b, Kurath, 1994) may be used to determine whether film or particle resistance dominates, or if both are significant. Equations for each of these cases are written as (Ruthven, 1984):

к = kpa = 2е; particle controlled diffusion

Rp

3

к = kfa = kf ; film controlled diffusion

f f Rp

1 R R2

= — + p ; both resistances controlling

к 3kf 15De

where:

kf = mass transfer coefficient in the film layer kp = overall mass transfer coefficient in the particle Rp = average particle radius De = effective intra-particle diffusivity of the metal a = area per unit volume of a spherical particle

The above equations (Eqn. 13.6) are simplified in that they represent the overall or effective terms relating to mass transfer for a given component in each phase. It should be recognized that mass transfer in the particle takes place through multiple regions (i. e., macro, micro-pores and extract­ant), each with potentially different rates of diffusion. The solid portion of the exchanger matrix occupies a substantial portion of the medium and obstructs diffusion of the ions into and out of the matrix. Even if the aqueous phase diffusion coefficients for the ions of interest are known, predicting diffusion rates within the exchanger particle is difficult because of the varied rates within the different phases of the exchanger and the tortuous path the ions must take through the pores to reach the active exchange sites. A simplified approach is often taken by treating the exchanger as a quasi-homogeneous system and assuming an average or ‘effective’ diffusion coefficient within the particle phase. This approach is frequently used in engineering practice and usually works quite well for modeling purposes. The reader is referred to the comprehensive work by Ruthven (1984) for more detailed discussion and methods of mathematical analysis of mass transfer in sorbents for both liquid and gaseous systems.

Batch kinetic studies to determine mass transfer rates in several solid — phase extraction resins have indicated that film diffusion is the controlling resistance for only a very short time when the solute is first contacted with the sorbent phase — likely corresponding to a monolayer coverage or uptake of the of solute at the pore surface (Juang et al., 1995a, 1995b, Takeshita et al., 2000). The studies indicate that mass transfer resistance in the particle phase is the dominant controlling mechanism throughout most of the period of solute uptake. Since this is likely to be the case with many solid-phase extraction resins of interest, methods for determining the effective mass transfer coefficient in the particle phase will be the focus of this discussion. Kinetic experiments are typically performed in batch mode in order to obtain the necessary data to estimate the diffusion coefficient for a given metal in the solid-phase resin particle. One technique is to simply stir or mix a beaker of solution with known volume and concentration of metal containing a known mass and particle size of resin, and then remove small aliquots of the liquid phase to measure metal uptake as a function of time. This approach has been used successfully by many researchers, but one must to careful to ensure that attrition of the resin particles by the stirring blade does not take place since mass transfer in the particle is dependent upon particle diameter. A preferred method is to use a centrifugal contact­ing apparatus similar to that described by Helfferich (1995). A device con­sisting of a stainless steel wire cage housed within the center of a circular Teflon™ housing that is connected to a stirring motor can be easily made for this purpose (Tranter et al., 2009a). Diagrams of top and side views of this type of contactor are shown in Figs 13.17 and 13.18, respectively. The device is fashioned with a threaded top and bottom portion so that the contactor can be opened to allow access to the hollow inner portion holding the wire screen or cage. The solid-phase extraction resin is sieved to obtain a fraction of known particle diameter and a measured amount of the sized exchanger material is placed in the hollow space inside the contactor. The top portion of the apparatus is then threaded tightly back onto the body of the device and it is attached to a stirring motor via the vertical shaft, as shown in Fig. 13.18. The unit is immersed in a solution of known metal ion concentration and stirred at a fixed speed. Centrifugal action produces a rapid circulating flow of solution entering the device at the bottom and exiting the wall through the radial holes in the casing. The contactor allows for very high flow rates around the adsorbent beads, thus shrinking the film layer to a point that the diffusion resistance in the film will be very small compared to that within the particle. Uptake of the solute ion by the exchanger particle is then determined as a function of time by removing and analyzing small aliquots from the bulk solution. It is important that these aliquots are very small relative to the total volume of solution so that the bulk volume remains approximately unchanged, i. e. the total ion

13.17 Diagram (top, inside view) of centrifugal contacting device.

circulation 13.18 Diagram (side view) of centrifugal contacting device.

concentration in the fluid phase at any given point in time is not measurably reduced by sampling.

The experimental regime provides a well-stirred finite volume system and the solution method proposed by Crank (1975) for spherical particles may be applied to the kinetic data to solve for the effective intra-particle dif — fusivity. The following assumptions are made for this analysis:

• The particles are spherical and uniform.

• The particles are initially free of solute.

• The concentration of solute in the bulk phase is always uniform and is initially Co.

• The fluid velocity is sufficient to reduce the film layer to the point that diffusion in the particle is the rate limiting step for the uptake of solute.

Normalized fractional loading or uptake of the solute (F) by the exchanger during the kinetic test changes with time and is defined as:

F _ q 13.7

q

where:

qt = metal concentration in the solid phase at time t qo = metal concentration in the solid phase at equilibrium

The fractional attainment of equilibrium is expressed by the following equation (Crank, 1975):

F _ q_ _ 1 _y 6a>(a +1)exp(-Реф2пг/Rp) 13 8

q„ П=1 9 + 9Ю + ФІЮ2

where:

t = elapsed time, min

The parameter a> is calculated from the following relationship: mq„ _ 1

vc0~ ї+й

where:

m = mass of solid-phase extraction resin V = solution volume

C0 = initial metal concentration in the solution The terms for Фп are calculated from the non-zero roots of:

tan фп = 13.10

з+юф2

Equation 13.10 is solved for n = 1 to 12 and these values are then used to obtain the series solution for F. Beyond n = 12, the additional terms usually become insignificant to the solution. Equations 13.8 and 13.9, along with the 12 solutions from Eqn 13.10, are used in to fit the fractional equi­librium data from the kinetic batch experiment. The independent variable in this approach is the elapsed time t, the dependent variable is the frac­tional attainment of equilibrium F, and the adjustable parameter is the effective intra-particle diffusivity coefficient, De, for the metal of interest. The curve fits can be obtained quite easily via the numerous commercial statistical software packages now available for desktop computers. The value for De, derived by fitting the data from the kinetic batch tests, is then used in Eqn 13.6 to calculate the mass transfer coefficient for the metal in the particle phase, kp.

In summary, the general steps for model development are:

1) Perform independent batch experiments to obtain equilibrium data for the metal(s) and solid-phase extraction resin system of interest.

2) Use the equilibrium data to elucidate the best form for the equilibrium equation and derive the corresponding equilibrium coefficients.

3) Perform independent batch experiments to obtain kinetic data for the metal(s) and solid-phase extraction resin system of interest.

4) Use the kinetic data to elucidate the best form for the mass transfer equations and derive the corresponding mass transfer coefficients.

5) Numerically solve the coupled partial differential equations (Eqns 13.1 and 13.2), using the coefficients for equilibrium and mass trans­fer derived from steps 2 and 4, to generate a predicted breakthrough curve for a specified metal feed concentration and column size of solid-phase extraction resin.

6) Perform column breakthrough experiments with test conditions specified in step 5 and compare the experimental breakthrough data to the modeled results.

7) Perform several iterations of steps 5 and 6 with varying feed concen­trations, flow rates and column dimensions to evaluate the efficacy of the model.