Linear Driving Force Approximation Research Articles

Linear driving force approximation in cyclic adsorption processes: Simple results from system dynamics based on frequency response analysis

The linear driving force (LDF) approximation for cyclic adsorptive processes is discussed on the basis of model equivalence with the homogeneous diffusion model (HDM), the pore diffusion model (PDM) and the intraparticle diffusion and convection model (IDCM). Model equivalence is based on the frequency response of the adsorbent particle, namely on the equality of the amplitude ratio and the phase-lag functions. The analysis of the continuous stirred tank adsorber (CSTA) and of the plug flow adsorber (PFA) is addressed.

Adsorption of phosphorus in saturated slag media columns

Abstract Adsorption of phosphorus (P as phosphates) in saturated slag media (dust and cake) columns was studied to assess the slag media's capability in removing P from wastewater. Prior to the experiments, slag media were completely washed to rinse off the soluble metal ions which are generally responsible for forming insoluble precipitates. Experimental data revealed that the adsorption capacities of washed slag media are still much higher than that of a sandy roam soil. Two dynamic models were tested for simulating the adsorption behavior of phosphorus in columns packed with slag media. The model employing the intraparticle transport through surface diffusion is successful, while a simplified model based on the linear driving force approximation (LDFA) could not predict the column behavior properly at low concentrations, particularly on the cake slag.

Study of a six‐bed pressure swing adsorption process

AbstractA mathematical model of a six‐bed, ten‐step PSA operation is developed. The process cycle considered resembles an industrial hydrogen recovery process from the refinery fuel gas. Three hydrocarbon impurities (methane, ethane, and propane) are considered in the feed gas. The adsorbent used is activated carbon. The nonisothenmal, bulk separation PSA model adopts the linear driving force approximation for particle uptake and the extended Langmuir isotherm to represent adsorption equilibrium. A transient pressure equation is incorporated to account for the dynamics in uariable pressure steps. The model was verified using experimental results from a computer‐controlled, laboratory‐scale PSA unit. Considering the complexity of operation, comparison of various stream flow rates, concentrations, bed pressure and temperature profiles indicates that the model provides a sufficiently accurate prediction of the PSA performance. Parametric studies further show that its product purity declines relatively quickly with increasing cycle time and decreasing high operating pressure due to the breakthrough of a relatively sharp methane front. There is no real advantage in operating the PSA unit beyond a high operating pressure of about 18.0 bar.

Multicomponent Transport Effects in Sorbent Particles under Pressure Swing Conditions

A comprehensive theoretical study of the problem of multicomponent adsorption−desorption in sorbent particles under pressure swing conditions is carried out in this study. A rigorous mathematical model is formulated for the process in which the dusty-gas model and the commonly used Fickian model are employed to describe the coupling of the mass fluxes and the partial pressures in the pore space, with or without viscous flow. The dusty-gas model is used as a standard against which the performance of the Fickian model is tested. Computations are carried out for binary and quaternary mixtures. It is found that the rigorous model (the one based on the dusty-gas model) predicts faster responses for the partial pressures of the components in the mixture. Faster responses are also observed when the viscous flow is accounted for in the transport model, with the effect being considerably stronger in the case of the Fickian model. The obtained results are also used to investigate the performance of the Glueckauf linear driving force approximation for multicomponent systems. It is concluded that like the Fickian diffusion model on which it is based, this approximation cannot predict, even qualitatively, the dynamic responses of the partial pressures in a multicomponent system.

Modeling, simulation and operation of a simulated moving bed for continuous chromatographic separation of 1,1′-bi-2-naphthol enantiomers

The objective of this paper is to study the separation of enantiomers of 1,1′-bi-2-naphthol in 3,5-dinitrobenzoyl phenylglycine bonded to silica gel, using heptane–isopropanol (72:28) as eluent by simulated moving bed chromatography (SMB). A model for the prediction of the cyclic steady state performance of the SMB, based on the analogy with the true moving bed, is developed assuming axial dispersion flow, linear driving force approximation for intraparticle mass transfer and multicomponent adsorption equilibria. The SMB package allows the simulation of the pilot unit. The effect of several operating parameters on the SMB performance is analyzed. The performance is characterized by purity, recovery, solvent consumption and adsorbent productivity. The package is an important tool for learning and training operators, allowing the choice of best operating conditions. The operation of the SMB pilot unit was carried out for the separation of racemic mixtures using a 8-column configuration. Purities and recoveries higher than 95% in the extract and raffinate were obtained. Model and experimental results are compared and the package is also used to predict the steady state internal profiles for the SMB operation in good agreement with experimental results.

Modified linear driving force approximations for cyclic adsorption-desorption processes

This work deals with linear driving force (LDF) approximations to the intraparticle pore diffusion equation assuming linear equilibrium. Two approaches are used to calculate global rate coefficients for the LDF model: the usual way where the average intraparticle concentration is calculated from the LDF model and a new one that uses the solution of the diffusion equation. The cyclic steady-state solutions to various cyclic perturbations are presented and compared for both methods. Also the global rate coefficients for processes with half-cycles of different length subjected to square wave perturbations are presented. It is shown that the two rate coefficients needed to represent a cycle are practically equal. To avoid the difficulties associated with the non-universality, i.e. dependence on the perturbation type, of the global rate coefficients, an instantaneous rate coefficient is presented, valid for perturbations smoother than the square wave.

Purification of monoclonal antibodies by simulated moving-bed chromatography

A simulated moving bed (SMB) system has been developed for the biospecific purification of monoclonal antibodies. Adsorption and desorption of the desired product is performed under different conditions. To increase the purity and yield of the antibodies, two purge steps have to be introduced. The steady-state performance of the SMB system was modelled by solving the governing differential equations using a linear driving force approximation. The model parameters were determined independently in batch experiments. They were used to determine the operating conditions of the SMB system for the purification of monoclonal antibodies from cell culture supernatant. The antibodies could be isolated with a yield of ⩾90. SDS gel electrophoresis of the feed and product stream showed that more than 99% of the contaminating proteins were removed in a single step by SMB chromatography.

Mathematical modeling of the adsorptive separation of multicomponent gaseous mixtures

A general dynamic model describing the adsorptive separation of multicomponent gaseous mixtures is developed in this study. The dusty-gas model and D'Arcy's law are used to describe the diffusive and viscous mass transport in the adsorbing bed, respectively, and the local equilibrium assumption or the linear driving force approximation are used for the uptake rate. The versatility of the developed model is demonstrated by applying it to separation of binary, ternary and quaternary mixtures by pressure swing adsorption (PSA). The relative importance of the diffusive (bulk and Knudsen) and viscous mass transport and the effects of the different uptake rate representations are also investigated. For the PSA process, it is found that viscous transport dominates in the adsorbing bed and the inclusion of other modes of transport in the model equations has practically no effect on the solution. However, for dynamic processes occurring in porous media of smaller pore sizes (macroporous membranes, for instance) both the viscous and the diffusive modes of transport must be included in the overall model to predict the system behavior correctly. The model with only viscous transport in bed and the linear driving force approximation for the uptake rate is the recommended option for modeling PSA operations, provided that the LDF approximation is applicable, since it is easier to handle numerically and reduces to the equilibrium model in the limiting case of large adsorption rate constants.

Continuous biospecific affinity purification of enzymes by simulated moving-bed chromatography theoretical description and experimental results

A system for continuous biospecific affinity purification of enzymes by simulated moving-bed chromatography was designed. The performance of steady-state operation was predicted by a model based on a linear driving force approximation. The system was tested for the purification of bovine α-chymotrypsin on immobilized soybean trypsin inhibitor. The theoretical results showed rather good agreement with the experimental results. The recovery was found to be dependent on the extract purity. For higher purities and recoveries the columns must be purged with non-desorbing buffer before shifting from the adsorption to the desorption zone.

Extension of the linear driving force-dusty gas model approximation to include surface or micropore diffusion

This paper presents linear driving force approximations for the dusty gas model in a bisolute system obeying linear or Langmuir isotherms, in two situations: when there is a nonnegligible surface flux in parallel with the diffusive and convective fluxes; and when there is a micropore resistance to mass transfer in series. The approximations were developed using a semi-empirical procedure that starts with the particle response to a square wave perturbation considering a simple diffusion/convection mechanism, and proceeds by consecutive corrections to this basic result. These corrections are obtained using superposition, similarity and pattern recognition. The approximations are good (average quadratic error ⩽10%) when representing the system cyclic steady state, whether the perturbations are sharp or smooth.

Linear driving force approximation for isothermal non-isobaric diffusion/convection with binary Langmuir adsorption

This paper presents several reduced models for the intraparticle diffusion/convection mechanisms in spherical homoporous particles under isothermal conditions. These semi-empirical linear driving force approximations were developed for the cyclic steady state that is attained when a cyclic concentration and/or total pressure perturbation is imposed at the particle surface. The models are applicable to linear systems or those obeying bicomponent Langmuir isotherms. The linear driving force approximations have two or three parameters for each component and are presented for the following cases: Knudsen diffusion, molecular diffusion, molecular plus Knudsen diffusion in series, viscous flow, and also for both diffusion mechanisms in series, in parallel with the convective flow, as in the dusty gas model. Comparison between the results obtained with these approximations and with the complete models shows that on average the relative quadratic error is ≈10% for non-smooth and 5% for smoother perturbations.

Linear driving force approximation for intraparticle diffusion and convection in permeable supports

The objective of the communication is to determine if a linear-driving force (LDF) approximation can be used instead of a more rigorous diffusion-convection on the breakthrough behavior under conditions of nonlinear adsorption with a favorable isotherm. Such an approximation is commonly used in the simulation of fixed-bed adsorption operations and chromatography (Ruthven, 1984). However, its use in simulating process in which intraparticle convection is important has not yet been established

95/02135 Linear driving force approximation for diffusion in spherical adsorbents with binary non-linear adsorption

95/02135 Linear driving force approximation for diffusion in spherical adsorbents with binary non-linear adsorption

Fixed-bed adsorption for bulk component system. Non-equilibrium, non-isothermal and non-adiabatic model

A mathematical model (non-equilibrium, non-isothermal and non-adiabatic) for bulk single component adsorption on a fixed-bed system is developed. The model takes account of both temperature effects and changes in flow rate due to adsorption of solute, and is therefore more general than previous models. This model is based on the use of linear driving force approximations for heat and mass transfer rates. The temperature-dependent Langmuir equilibrium isotherm is used to represent gas-solid equilibrium isotherm. The model consists of a set of coupled partial differential equations, and the differential equations representing the mass and energy balances are solved by the numerical method of lines. Based on the model simulation, the effects of heats and mass transfer parameters on the concentration profiles through the bed were investigated, and the model predictions were compared with the isothermal and adiabatic model calculations. Also, the adsorption breakthrough curves and temperature profiles of carbon dioxide on activated carbon were measured experimentally for bulk component system. The effects of total pressure, inlet composition and flow rate on breakthrough curves and temperature profiles were studied, and the experimental data were compared with the model calculations. The mathematical model provides a good representation of the experimentally observed behavior of the breakthrough curves and temperature profiles for carbon dioxide on activated carbon, suggesting that the model represents the essential features of the real systems fairly well.

Intraparticle transport of amino acids in a cation exchanger

Intraparticle transport of amino acids, phenylalanine and tyrosine, in a macroreticular cation-exchange resin, Amberlite 200, was investigated in a finite batch adsorber. A transport model was developed based on the consecutive diffusion of the amino acids through liquid-filled macropores and microspheres. The latter was further simplified by the linear driving force approximation (LDFA) since the microspheres are quite small. The model was used satisfactorily for simulating and predicting the performance of finite batch operations.

Continuous-mixture model of extraction processes

Continuous-mixture theory is based on the assumption that when a large number of components make up a mixture, the frequency distribution can be approximated by a.continuous function. Here we apply this concept to multicomponent leaching or extraction from porous particles in a continuous-flow system. Desorption from the solid matrix and partitioning between the extraction fluid and an immiscible fluid in the pores of the matrix are included in the mathematical model. Intraparticle diffusion and external mass transfer are combined in a linear driving-force approximation. The mass balance equations for intra-particle and external fluid processes govern the time evolution of frequency distribution functions for different mixtures. The approach satisfactorily describes experimental data for supercritical extraction of triglycerides.

Linear driving force approximation for diffusion in spherical adsorbents with binary non-linear adsorption

Abstract This paper presents a new linear driving force approximation applicable to the cyclic steady-state in the case of pore diffusion in the presence of binary Langmuirian adsorption for a spherical particle under a periodic boundary condition in composition, total pressure or both. This approximation uses two parameters Ω, one for each solute, calculated for a square wave with the same period and a second pair of parameters that correct the approximation for the phase lags. An assessment of the quality of the approximation was made by measuring the differences between the cumulative fluxes calculated by the diffusion and the linear driving force models through an average relative quadratic error. It is shown that this error is less than 5% for equilibrium, diffusion and frequency conditions in the range of practical adsorption systems. The same approximation can be used in the case of homogeneous diffusion.

Numerical simulation of unsteady continuous countercurrent adsorption system with nonlinear adsorption isotherm

A theoretical dispersed plug flow model that includes a mass-transfer resistance represented by a linear driving force approximation is developed for unsteady-state continuous countercurrent adsorption systems having nonlinear equilibrium isotherms. The model is solved from unsteady state to steady state using Danckwert's boundary conditions. The system consists of two sections having 11 adsorption columns. The model is used to investigate the effects of various process parameters on the performance of the system. It is demonstrated that as the nonlinearity changes, the optimal choice of bed length, feed and eluent flow rates, and switch times must be suitably adapted. The results are compared with the “pore diffusion” and “equilibrium” models of Morbidelli et al. 14 and are found to lie between these two models. To establish the consistency of the model the results are compared with one of the limiting asymptotes, namely, the linear isotherm with the experimental data of Ching and Ruthven 12,13 and are found to agree well.

High-performance liquid chromatography of amino acids, peptides and proteins CXXVI. Modelling of protein adsorption with non-porous and porous particles in a finite bath

Analytical solutions for a mathematical model describing dynamic adsorption processes of proteins onto non-porous adsorbent particles in a finite bath are presented. The model, based on the Langmuir adsorption isotherm, has been applied to experimental data obtained with affinity and ion-exchange adsorbents. The external film mass transfer resistance, as well as the rate of surface interaction between proteins and adsorbents, have been taken into account. The model has been extended to the case of adsorption onto porous particles by employing a linear driving force approximation for describing mass transfer in the pore fluid. This approach enables the derivation of an effective overall liquid phase mass transfer coefficient, permitting subsequent adaptation of the analytical solutions developed for non-porous particles. The evaluation of the effective liquid phase mass transfer coefficients is also described. Examples of a comparison between predicted and experimental dynamic adsorption curves for both dye-affinity and ion-exchange systems are presented. The application of the model for predicting the optimum operating conditions is discussed.

The linear driving force approximation for cyclic mass transfer in spherical particles

An approach alternative to that of Alpay and Scott (1992) to obtaining an exact, purely periodic solution of the particle diffusion equation is presented. The solution is obtained via Laplace transform, and the intraparticle concentration profile and instantaneous flux at the surface are expressed as trigonometric series expansions with respect to time