O.K. Crosser

Modeling and simulation of the dynamic behavior of monoliths: Effects of pore structure from pore network model analysis and comparison with columns packed with porous spherical particles

A mathematical model is presented that could be used to describe the dynamic behavior, scale-up, and design of monoliths involving the adsorption of a solute of interest. The value of the pore diffusivity of the solute in the pores of the skeletons of the monolith is determined in an a priori manner by employing the pore network modeling theory of Meyers and Liapis [J. Chromatogr. A, 827 (1998) 197 and 852 (1999) 3]. The results clearly show that the pore diffusion coefficient, Dmp, of the solute depends on both the pore size distribution and the pore connectivity, nT, of the pores in the skeletons. It is shown that, for a given type of monolith, the film mass transfer coefficient, Kf, of the solute in the monolith could be determined from experiments based on Eq. (3) which was derived by Liapis [Math. Modelling Sci. Comput., 1 (1993) 397] from the fundamental physics. The mathematical model presented in this work is numerically solved in order to study the dynamic behavior of the adsorption of bovine serum albumin (BSA) in a monolith having skeletons of radius r(o) = 0.75x10(-6) m and through-pores having diameters of 1.5x10(-6)-1.8x10(-6) m [H. Minakuchi et al., J. Chromatogr. A, 762 (1997) 135]. The breakthrough curves of the BSA obtained from the monolith were steeper than those from columns packed with porous spherical particles whose radii ranged from 2.50x10(-6) m to 15.00x10(-6) m. Furthermore, and most importantly, the dynamic adsorptive capacity of the monolith was always greater than that of the packed beds for all values of the superficial fluid velocity, Vtp. The results of this work indicate that since in monoliths the size of through-pores could be controlled independently from the size of the skeletons, then if one could construct monolith structures having (a) relatively large through-pores with high through-pore connectivity that can provide high flow-rates at low pressure drops and (b) small-sized skeletons with mesopores having an appropriate pore size distribution (mesopores having diameters that are relatively large when compared with the diameter of the diffusing solute) and high pore connectivity, nT, the following positive results, which are necessary for obtaining efficient separations, could be realized: (i) the value of the pore diffusion coefficient, Dmp, of the solute would be large, (ii) the diffusion path length in the skeletons would be short, (iii) the diffusion velocity, vD, would be high, and (iv) the diffusional response time, t(drt), would be small. Monoliths with such pore structures could provide more efficient separations with respect to (a) dynamic adsorptive capacity and (b) required pressure drop for a given flow-rate, than columns packed with porous particles.

“Perfusion chromatography”. The effects of intra-particle convective velocity and microsphere size on column performance

A mathematical model describing the dynamic adsorption in columns with spherical bidisperse perfusive or spherical bidisperse purely diffusive adsorbent particles is presented and its solution is obtained numerically. The adsorption of bovine serum albumin on spherical anion-exchange porous particles was studied for different values of the intraparticle Peclet number, Peintra, and of the microsphere diameter, dm. The results show a departure from spherical symmetry of the isoconcentration profiles of the adsorbate in the porous adsorbent particle as Peintra increases. This spherical asymmetry increases the adsorbate availability in the pore fluid of the macroporous region and also increases the concentration of the adsorbate in the adsorbed phase in the upstream half of the spherical adsorbent particle. If, for example, the concentration profiles of the adsorbate in the adsorbed phase of the porous adsorbent particles could be determined experimentally and if these profiles show a departure from spherical symmetry, then this result could suggest that the porous adsorbent particles may have exhibited perfusion behavior under the conditions of operation of the column. The dynamic percentage utilization of the adsorptive capacity of the column increases as Peintra increases and dm decreases. The percentage breakthrough to be selected for column switching is influenced by the magnitude of Peintra because of its effect on the dynamic percentage utilization of the adsorptive capacity of the column.

OPTIMAL CONTROL OF THE PRIMARY AND SECONDARY DRYING STAGES OF BULK SOLUTION FREEZE DRYING IN TRAYS

Abstract The problem of operating a tray freeze dryer to obtain a desired final bound water content in minimum time is formulated as an optimal control problem with the use of the rigorous unsteady state mathematical model of Sadikoglu and Liapis [9] that has been found to describe satisfactorily the experimental dynamic behavior of the primary and secondary drying stages of bulk solution freeze drying of pharmaceuticals in trays. The heat input to the material being dried and the drying chamber pressure are considered to be control variables. Constraints are placed on the system state variables by the melting and scorch temperatures during primary drying, and by the scorch temperature during secondary drying. Necessary conditions of optimality for both the primary and secondary drying stages are derived and presented, and an approach for constructing the optimal control policies that would minimize the drying times for both the primary and secondary drying stages, is presented. The theoretical approach p...

Heating Policies during the Primary and Secondary Drying Stages of the Lyophilization Process in Vials: Effects of the Arrangement of Vials in Clusters of Square and Hexagonal Arrays on Trays

Abstract The dynamic behavior of the primary and secondary drying stages of the lyophilization process were studied when (a) single vials located at different positions on the tray were individually being dried, and (b) the vials on the tray are arranged in clusters of square and hexagonal arrays and all the vials on the tray are simultaneously being dried. For both cases (a) and (b), fast drying times and relatively more uniform distributions of temperature and concentration of bound water at the end of the secondary drying stage are obtained by heat input control that runs the lyophilization process close to the melting and scorch temperature constraints. The heating control policies for the systems of case (b) are found to be more conservative and significantly more complex than those for the systems of case (a), because in case (b) there are vials on the tray that are in their secondary drying stage while other vials on the same tray have not yet completed their primary drying stage. Furthermore, the analysis and synthesis of the results presented in this work (i) indicate the minimum number of vials and their relative locations on the tray that have to be monitored by sensors so that the dynamic drying state of all the vials being dried simultaneously on the tray, could be satisfactorily determined in real time and appropriate, if necessary, control action could be applied, and (ii) suggest changes in the design of the freeze drying equipment so that the production rate of the freeze dryer could be improved and the product could also have enhanced properties of stability and quality at the end of the lyophilization process.

Pore network modelling: determination of the dynamic profiles of the pore diffusivity and its effect on column performance as the loading of the solute in the adsorbed phase varies with time

A three-dimensional pore network model for diffusion in porous adsorbent particles was employed in a dynamic adsorption model that simulates the adsorption of a solute in porous particles packed in a chromatographic column. The solution of the combined model yielded the dynamic profiles of the pore diffusion coefficient of beta-galactosidase along the radius of porous ion-exchange particles and along the length of the column as the loading of the adsorbate molecules on the surface of the pores occurred, and, the dynamic adsorptive capacity of the chromatographic column as a function of the design and operational parameters of the chromatographic system. The pore size distribution of the porous adsorbent particles and the chemistry of the adsorption sites were unchanged in the simulations. It was found that for a given column length the dynamic profiles of the pore diffusion coefficient were influenced by: (i) the superficial fluid velocity in the column, (ii) the diameter of the adsorbent particles and (iii) the pore connectivity of the porous structure of the adsorbent particles. The effect of the magnitude of the pore connectivity on the dynamic profiles of the pore diffusion coefficient increased as the diameter of the adsorbent particles and the superficial fluid velocity in the column increased. The dynamic adsorptive capacity of the column increased as: (a) the particle diameter and the superficial fluid velocity in the column decreased, and (b) the column length and the pore connectivity increased. In preparative chromatography, it is desirable to obtain high throughputs within acceptable pressure gradients, and this may require the employment of larger diameter adsorbent particles. In such a case, longer column lengths satisfying acceptable pressure gradients with adsorbent particles having higher pore connectivity values could provide high dynamic adsorptive capacities. An alternative chromatographic system could be comprised of a long column packed with large particles which have fractal pores (fractal particles) that have high pore connectivities and which allow high intraparticle diffusional and convective flow mass transfer rates providing high throughputs and high dynamic adsorptive capacities. If large scale monoliths could be made to be reproducible and operationally stable, they could also offer an alternative mode of operation that could provide high throughputs and high dynamic adsorptive capacities.

EXERGY ANALYSIS FOR THE FREEZING STAGE OF THE FREEZE DRYING PROCESS

The mathematical expressions for exergy and the exergy analysis of the freezing stage of the freeze drying process are presented. The exergy analysis indicates that very substantial reductions in the magnitudes of the total exergy loss and of the exergy input due to the heat that must be removed during the freezing stage, can be obtained when the freezing stage is operated through the use of a rational distribution in the magnitude of the temperature of the cooling source. The rational distribution in the magnitude of the temperature of the cooling source should provide significant savings in the utilization of energy during the freezing stage of the freeze drying process as well as satisfactory freezing rates that form ice crystals that are continuous and highly connected and their shape and size are such that the pores of the porous matrix of the dried layer generated by sublimation during the primary drying stage, have a pore size distribution, pore shape, and pore connectivity that are appropriate to allow high rates for mass and heat transfer during the primary and secondary drying stages of the lyophilization process.

Lyophilization in Vials on Trays: Effects of Tray Side

Abstract: The effects of tray side on the performance of lyophilization of pharmaceuticals in vials arranged in clusters of hexagonal arrays on trays are studied and the results of this work indicate that it would be beneficial to operate the freeze drying process by employing freeze dryers whose trays have no tray sides and using heat input control that runs the lyophilization process close to the melting and scorch temperature constraints during the primary and secondary drying stages, respectively.

Pore network modelling of affinity chromatography: determination of the dynamic profiles of the pore diffusivity of β-galactosidase and its effect on column performance as the loading of β-galactosidase onto anti-β-galactosidase varies with time

A three-dimensional pore network model for diffusion in porous adsorbent particles was employed in a dynamic adsorption model that simulates the adsorption of a solute in porous particles packed in a chromatographic column. The solution of the combined model yielded the dynamic profiles of the pore diffusion coefficient of beta-galactosidase along the radius of porous adsorbent particles and along the length of the column as the loading of beta-galactosidase onto anti-beta-galactosidase immobilized on the surface of the pores of the particles occurred, and, the dynamic adsorptive capacity of the chromatographic column as a function of the design and operational parameters of the chromatographic system. It was found that for a given column length the dynamic profiles of the pore diffusion coefficient were influenced by (a) the superficial fluid velocity in the column, (b) the diameter of the adsorbent particles, and (c) the pore connectivity of the porous structure of the adsorbent particles. The effect of the magnitude of the pore connectivity on the dynamic profiles of the pore diffusion coefficient of beta-galactosidase increased as the diameter of the adsorbent particles and the superficial fluid velocity in the column increased. The dynamic adsorptive capacity of the column increased as (i) the particle diameter and the superficial fluid velocity in the column decreased, and (ii) the column length and the pore connectivity increased. In preparative affinity chromatography, it is desirable to obtain high throughputs within acceptable pressure gradients, and this may require the employment of larger diameter adsorbent particles. In such a case, longer column lengths satisfying acceptable pressure gradients with adsorbent particles having higher pore connectivity values could provide high dynamic adsorptive capacities. An alternative chromatographic system could be comprised of a long column packed with large particles which have fractal pores (fractal particles) that have high pore connectivities and which allow high intraparticle diffusional and convective flow mass transfer rates providing high throughputs and high dynamic adsorptive capacities. If large scale monoliths could be made to be reproducible and operationally stable, they could also offer an alternative mode of operation that could provide high throughputs and high dynamic adsorptive capacities.

Frontal chromatography of proteins : The effect on column performance of the restricted diffusion of molecules in porous chromatographic adsorbents

Abstract A theoretical formulation is presented that can be used to describe the dynamic behavior of frontal chromatography of proteins in columns packed with adsorbent particles in which restricted pore diffusion of the adsorbate molecules occurs. The results of this work clearly indicate that the time for breakthrough and the effective utilization of the adsorptive capacity of the chromatographic particles increase as (a) the size of the adsorbate and/or ligand (active site) molecule decreases, (b) the pore connectivity, nT, of the porous network of the adsorbent particles increases, and (c) the column length, L, increases.

Binary adsorption and desorption rates of propylene-propane mixtures on 13 X molecular sieves

Abstract The experimental adsorption and desorption rate data for pure propylene, pure propane, and propylene-propane mixtures on 13 X molecular sieves are presented for different temperatures and compositions; these rate data were efficiently determined using the differential sorption bed system described by Liapis et al. (Liapis et al. (1992) Rates and equilibria of adsorption and desorption of propane and propylene on 13 X molecular sieves determined experimentally using a differential sorption bed system. Sep. Technol. 2, 141–154). Propylene is the most preferentially adsorbed component, and the desorption rates of propylene and propane are slower than the adsorption rates. Mathematical models were constructed and solved, and it was found that the two-component model could provide a quantitative measure of the interactions between propylene and propane, and could describe the behavior of binary adsorption kinetics including the overshoot of the equilibrium loading by the less preferentially adsorbed propane. The theoretical results indicate that the desorption rates are not described by the same intracrystalline diffusion coefficients as the adsorption rates.