Equilibrium-dispersive Model Research Articles

WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modeled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modeling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (2018) [3] to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.

Simulation of Fixed-Bed Chromatographic Processes Considering the Nonlinear Adsorption Isotherms.

This paper presents the numerical approximation of a nonlinear equilibrium-dispersive (ED) model of multicomponent mixtures for simulating single-column chromatographic processes. Using Danckwerts boundary conditions (DBCs), the ED is studied for both generalized and standard bi-Langmuir adsorption isotherms. Advection-diffusion partial differential equations are used to represent fixed-bed chromatographic processes. As the diffusion term is significantly weaker than the advection term, sophisticated numerical techniques must be applied for solving such model equations. In this study, the model equations are numerically solved by using the Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The technique is designed to handle sudden changes (sharp discontinuities) in solutions and to produce highly accurate results. The method is tested with several case studies considering different parameters, and its results are compared with the high-resolution finite volume scheme. One-, two-, and three-component liquid chromatography elutions on fixed beds are among the case studies being considered. The dynamic model and its accompanying numerical case studies provide the initial step toward continuous monitoring, troubleshooting, and effectively controlling the chromatographic processes.

The Competitive Adsorption of Water and Methanol on a Hybrid Silica Stationary Phase in Supercritical Fluid Chromatography

This study investigates the adsorption of methanol, water, and their mixture in a hybrid silica stationary phase with supercritical carbon dioxide as a mobile phase in supercritical fluid chromatography (SFC). The adsorption isotherms of methanol and water were determined by two dynamic methods: the elution by characteristic point (ECP) method and the inverse method (IM). Both the single-component and competitive bi-Langmuir models were pre-selected for the inverse method. The initial parameters of the single-component isotherm for both methanol and water were estimated with the ECP method by fitting the experimental data to the bi-Langmuir isotherm model. Then, using the inverse method, we refined the single-component isotherm parameter values, which were then further used for determining the competitive isotherm of the methanol–water mixture. The elution profile of the (methanol–water) mixture sample was calculated by the equilibrium-dispersive (ED) model. The results indicated that there is a good agreement between the experimental band profile and the calculated band profile, which was obtained from the parameters of the competitive bi-Langmuir isotherm model, revealing a competition between methanol and water to reach the adsorption sites. Furthermore, the saturation capacity of the adsorption sites in the stationary phase decreased in the case of the mixture sample compared to those for the single-component sample.

Theoretical Study of Non-Isothermal Gradient Elution Liquid Chromatography.

A two-component model of gradient elution chromatography is investigated to theoretically study the effects of simultaneous variations in temperature and solvent strength on the retention behaviors of elution profiles in thermally insulated liquid chromatographic columns. The gradient elution technique is based on the gradual increase or decrease in eluent strength during the chromatographic operation by varying the composition of the mobile phase. The enthalpy of adsorption is primarily responsible for internal temperature variations inside the column, as heat adsorbs during the adsorption process and releases in the desorption phase. Both types of variations change the propagation speeds of moving pulses inside the column which can lead to better separation of the components and a reduction in the recycling time for the next injection. The equilibrium dispersive model (EDM) coupled with the energy balance equation for temperature and transport equation for the volume fraction of the solvent is utilized to simulate this complex process. The resulting nonlinear model equations are approximated by applying a semi-discrete second-order finite volume scheme. The numerical solutions are used to study the impact of a gradient starting and ending times, volume-fraction of the solvent, solvent strength parameter, the slope of gradient, enthalpy of adsorption, injection temperature, and the ratio of specific heats on the concentration profiles.

On the apparent dispersion coefficient of the equilibrium dispersion model: An asymptotic analysis

To model chromatography, researchers have developed several approaches. These cover a broad range of applications and, depending on the assumptions adopted, have different levels of accuracy. In general, the most suitable modelling approach is the simplest that can describe a process with the desired accuracy. A model that often meets this criterion is the equilibrium dispersion model (EDM). This features one mass balance equation per analyte, including an axial dispersion term, and assumes the analyte concentrations in the mobile and stationary phases to be in local equilibrium. To account for the finite mass transfer rate between the phases, the model employs an apparent dispersion coefficient. Two expressions are available for this coefficient, one being used much more frequently than the other. In this paper, we aimed to clarify which one should be favoured. A desirable feature of simple models is that they can be derived from more general ones with appropriate physical assumptions and rigorous mathematical methods. Thus, to answer our research question, we derived the EDM from the more general pore diffusion model (POR), using an asymptotic method. The expression obtained for the apparent dispersion coefficient does agree with one of the two reported in the literature – the less frequently used. To test the validity of this expression, we simulated elution profiles using the two versions of the EDM and compared the results against those from the POR model. The simulations were conducted in the range where the POR and EDM models should be essentially equivalent, their results confirming the outcome of the asymptotic analysis. This work offers a solid theoretical grounding for the EDM, clarifies which formulation of the model is correct, and provides usable applicability conditions for the model.

Modeling of breakthrough curves in removal of Cr(VI) and methylene blue using immobilized activated carbon from natural rubber waste

The activated carbon fines, derived from rubber sludge, were effectively immobilized inside calcium alginate beads to remove Cr(VI) ions and methylene blue (MB) in a fixed bed reactor after characterization of beads (FTIR, FESEM-EDS, BET, and thermogravimetric analysis) and batch study. The Langmuir batch uptake capacity was 136.45 and 324.87 mg g−1 for Cr(VI) and MB, respectively, at 30 °C. Contrary to the usual interpretation, it was shown that fitting of PFO model to experimental batch kinetic data may result from diffusive mechanism. Conventional practice of comparison of Bohart-Adams, Thomas, and Yoon-Nelson model being erroneous, these models were replaced by a generic function to explain breakthrough (BTC) curves. The equilibrium dispersive model (axial dispersion coefficient: 0.35 × 10−4 - 2.90 × 10−4 m2 min−1) predicted BTC for intermediate flow rates (5–15 mL min−1) and inlet concentrations (5–15 mg L−1) better than the one based on adsorption kinetics.

Hybrid model development for parameter estimation and process optimization of hydrophobic interaction chromatography

Hydrophobic Interaction Chromatography (HIC) is often employed as a polishing step to remove aggregates for the purification of therapeutic proteins in the biopharmaceutical industry. To accelerate the process development and save the costs of performing time- and resource-intensive experiments, advanced model-based process design and optimization are necessary. Due to the unclear adsorption mechanism of the salt-dependent interaction between the protein and resin, the development of an accurate mechanistic model to describe the complex HIC behavior is challenging. In this work, an isotherm derived from Wang et al. is modified by adding three extra parameters together with an equilibrium dispersive model to represent the HIC process. To reduce the development effort of isotherm equations and extract missing information from the available data, a hybrid model is constructed by combining a simple and well-known multi-component Langmuir isotherm (MCL) with a neural network (NN). It is observed that the structure of the hybrid model is of critical importance to the accuracy of the developed model. During parameter estimation, a regularization strategy is incorporated to prevent overfitting. Furthermore, the impact of NN structures and regularization rates are comprehensively investigated. One of the interesting findings was that a simple NN with one hidden layer with two nodes and sigmoid as the activation function, significantly outperforms the mechanistic model, with a 62% improvement in accuracy in calibration and 31.4% in validation. To ensure the generalizability of the developed hybrid model, an in-silico dataset is generated using the mechanistic model to test the extrapolation capability of the hybrid model. Process optimization is also carried out to find the optimal operating conditions under product quality constraints using the developed hybrid model.

Design of frontal chromatography separation of 1-phenylethanol and acetophenone using a hydrophobic resin

Adsorption and chromatographic processes havegood potential in the purification of natural aromas produced via biocatalytic routes. This work deals with the recovery of 1-phenylethanol (PE) from mixtures containing different amounts of acetophenone (AP). Amberlite FPX68, an industrial adsorbent with high selectivity with respect to acetophenone, was chosen for the separation of PE and AP by frontal chromatography. Small-scale experiments were performed for two different feed compositions and various flow rates and bed lengths. Using the target purity of 99 %, promising yields close to 90 % were obtained. The breakthrough curves were fitted with the equilibrium dispersive model, which provided HETP values under the different conditions. The obtained dependences of HETP on flow velocity complied with the Van Deemter equation. The contributions of individual phenomena responsible for adsorption zone broadening were analyzed. The experimental and modelling results obtained for the small-scale column became the basis for the scale-up of frontal chromatography separation of PE with an annual capacity up to 500 kg.

A Finite Difference Method Using High-Order Schemes to Simulate an Equilibrium-Dispersive Model of Non-Linear Chromatography

High-performance liquid chromatography (HPLC) is a dynamic separation process with a lot of parameters having different roles. The equilibrium-dispersive model is relevant for simulating HPLC because it is relatively simple and suitable for high-efficiency processes. The partial differential equation was simulated in many different methods such as semi-analytical methods, finite element methods, and finite difference methods. Many studies using finite difference methods have used the first-order and second-order schemes, but higher-order schemes have not been reported yet. This work is about solving the equation of the equilibrium-dispersive model, using a finite difference method with high-order schemes. The fourth-order central difference scheme was used for estimating diffusion and the fifth-order upwind schemes were used for simulating advection. The model was evaluated by assessing the area recovery of the peak, testing the non-retained substance behavior, and comparing the calculation results with the experimental data. The solutions of the equation will indicate the effects of the operation parameters on the system suitability ones and can be used to predict the behavior of an HPLC system and calculate the system suitability parameters of a novel method set.

Separation of p-xylene and m-xylene by simulated moving bed chromatography with MIL-53(Fe) as stationary phase

The separation of p-xylene (PX) and m-xylene (MX) isomers with near boiling points is a worldwide problem. The metal-organic framework material is an ideal stationary phase for chromatographic separation because of its high porosity, homogeneous pore diameter and good chemical stability. In this paper, a simulated moving bed (SMB) chromatography system with MIL-53(Fe) as the stationary phase and petroleum ether-dichloromethane as the mobile phase was designed to separate PX and MX at ambient temperature. Firstly, according to the elution curves of a single column, nonlinear competitive Langmuir adsorption isotherm equation was confirmed by equilibrium dispersive chromatography model. Then, the SMB separation zone was determined based on triangle theory, and the SMB operating conditions were optimized. Finally, the purity, recovery and productivity of PX reached 100.0%, 99.1% and 93.1 g/L/h, respectively; the purity, recovery and productivity of MX reached 96.4%, 100.0% and 23.5 g/L/h, respectively; the solvent consumption was 0.42 L/g.

Establishment of Fuzzy Langmuir Adsorption Model and Prediction of Chromatographic Behavior

AbstractIn order to accurately predict the complex chromatographic behaviors of the components to be separated, the fuzzy Langmuir adsorption equations and the back propagation‐artificial neural network (BP‐ANN) are combined to establish fuzzy Langmuir adsorption models. Herein, the fuzzy Langmuir adsorption equations are deduced based on a series of different traditional adsorption equations such as with or without competition, one or two kinds of adsorbed sites, monomolecular or multimolecular adsorption, etc. The major adsorption parameter Ci (form concentration) is the function of the actual concentration ci, expressed in matrix form constructed by BP‐ANN and is obtained by solving the equilibrium dispersive chromatography model with the inverse method and genetic algorithm. Finally, the fuzzy Langmuir models are applied to study the chromatographic behaviors of m‐cresol and p‐cresol on MIL‐53 (Al) stationary phase. The results show that the models have excellent curve fitting ability, and can be used to determine the adsorption relationship and predict the chromatographic elution curves under complex or unknown adsorption mechanism.

Analysis and experimental demonstration of temperature step gradients in preparative liquid chromatography

The amount of substance adsorbed on solid surface depends on temperature. Therefore, the migration velocities of the solutes in a chromatographic column can be altered by introducing temperature gradients. Such gradients designed to change retention behaviours can be exploited to improve the separation performances in preparative chromatography. To describe key process features, we used analytical solutions of the equilibrium model with instant stepwise shift of temperature. To achieve a more realistic description, the equilibrium dispersion model was additionally applied to treat finite column efficiencies. The effect of temperature gradients was illustrated experimentally using two identical columns sequentially connected. Temperature of the second column was modulated by thermostats. Wide pulse injections of a single component led to instructive elution profiles in a preliminary investigation. The observations were found to be in qualitative agreement with predictions of the equilibrium dispersion model. Subsequently, the separation of a ternary model mixture was investigated considering a simple two-step temperature gradient. To support the quantitative analysis and to identify suitable switching and cycle times, the temperature dependencies of the Henry constants were determined by short pulse injections. A meaningful variation of the parameters of the temperature gradient is required for adjusting the cycle times, which is the time difference between two consecutive injections that needs to shorten. Decreasing this time is connected with a desirable increase in process productivity. The results achieved revealed that relatively simple to implement stepwise temperature gradients offer an option to improve and fine-tune the performance of repetitive batch chromatography.

Theoretical study of twin-column recycling chromatography with a solvent-gradient for preparative binary separations

Twin-column recycling chromatography with a solvent gradient (TCRC-SG) was investigated with the equilibrium-dispersive chromatography model. The solvent gradient caused by constant addition of a modifier between the two columns created a band compression effect to counterbalance band broadening, so that the target component band neither broadened nor shrunk. Meanwhile, band compression accelerated the separation but prevented excessive separation. Increasing the volume fraction of weak solvent in the modifier and reducing the modifier flowrate enhanced band compression and improved the separation. The effect of column efficiency (number of theoretical plates: 500–1500) on the separation was not significant. According to the separation behavior, a simple operation scheme is proposed to automatically control column switching without needing to determine the adsorption isotherm and designing operating conditions in advance. In comparison with simulated moving bed, TCRC-SG had a higher feed throughput, but consumed more solvent. The results showed that TCRC-SG is favorable for preparative separation.

Analysis of equilibrium dispersive model of liquid chromatography considering a quadratic-type adsorption isotherm

A single-component equilibrium dispersive model of liquid chromatography is solved analytically for a quadratic-type adsorption isotherm. The consideration of quadratic isotherm leads to a non-linear advection-diffusion PDE that hinders the derivation of analytical solution. To over come this difficulty, the Hopf-Cole and exponential transformation techniques are applied one after another to convert the given advection-diffusion PDE to a second order linear diffusion equation. These transformations are applied under the assumption of small non-linearity, or small volumes of injected concentrations, or both. Afterwards, the Fourier transform technique is applied to obtain the analytical solution of the resulting linear diffusion equation. For detailed analysis of the process, numerical temporal moments are obtained from the actual time domain solution. These moments are useful to observe the effects of transport parameters on the shape, height and spreading of the elution peak. A second-order accurate, high resolution semi-discrete finite volume scheme is also utilized to approximate the same model for non-linear Langmuir isotherms. Analytical and numerical results are compared for different case studies to gain knowledge about the ranges of kinetic parameters for which our analytical results are applicable. The effects of various parameters on the mechanism are analyzed under typical operating conditions available in the liquid chromatography literature.

Note of solving Equilibrium Dispersive model with the Craig scheme for gradient chromatography case

A modified method for the Craig scheme solution applied for a chromatographic column has been proposed and the corresponding implicit code is presented. The new approach improves the mass conservation problem frequently reported during the numerical solution of the Equilibrium Dispersive model with Craig scheme. The modified code has been successfully verified in gradient chromatography conditions by comparison with the reliable solutions of the Equilibrium Dispersive model by the Orthogonal Collocation on Finite Elements (OCFE). The errors obtained with the modified Craig method are less than about 1% in the case of retention times, and less than about 11% in the case of apparent number of theoretical plates, comparing to the OCFE solutions.

Separation of eicosapentaenoic acid and docosahexaenoic acid by three-zone simulated moving bed chromatography

Eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) are essential fatty acids for human body, which are widely used in the field of healthy food and medicine. Meanwhile, there are some differences in their physiological functions, such as “scavenger for blood vessel” of EPA and “brain protector” of DHA. In order to make full use of EPA and DHA, it is necessary to prepare their high-purity component. In this paper, EPA and DHA were separated and purified by three-zone simulated moving bed (SMB) chromatography with C18 used as stationary phase and ethanol-water as mobile phase. For the single column experiment, a separation unit of SMB, the effects of the ratio of ethanol to water, pH value and temperature on the separation were investigated. The equilibrium dispersion (ED) model was used to obtain the adsorption parameters of EPA and DHA by inverse method and genetic algorithm, and the accuracy of the adsorption parameters was verified by fitting the overloaded elution curves under different conditions. Based on the acquired nonlinear adsorption isotherms the complete separation region was found according to triangle theory. The effects of sample concentration, flow ratios of adsorption zone and rectification zone, and column distribution mode of SMB on the separation were investigated. Under the optimized SMB conditions, the experimental result was that without regard to the other components, the chromatographic purity and recovery values of EPA and DHA exceeded 99% with the productivity of 4.15 g/L/h, and the solvent consumption of 1.11 L/g.

On-line optimization of four-zone simulated moving bed chromatography using an Equilibrium-Dispersion Model: II. Experimental validation

In the first theoretical part of this work, an on-line optimization method using the equilibrium-dispersion model of chromatography was developed for the dynamic optimization of periodically operated four-zone simulated moving bed (SMB) processes. In this second part, we present the experimental validation of the proposed concept. To describe the behaviors of real SMB systems, the on-line control unit was modified. Unavoidable system void volumes, which can be described using delay functions corresponding to standard mass balance models for pipes and CSTs, were considered in addition to the chromatographic columns. Furthermore, a new sample collection method that does not interrupt product streams was introduced for process monitoring. The equipment including an external concentration analysis unit was automated via a tailor-made interface program. The on-line control unit estimated the parameters of the process model and optimized the future operating conditions ‘switch-by-switch’. The case study investigated in this work was the separation of racemic mixtures of two bicalutamide enantiomers using a chiral stationary phase applying the conventional and two advanced SMB operating modes.

On-line optimization of four-zone simulated moving bed chromatography using an equilibrium-dispersion model: I. Simulation study

Simulated moving bed (SMB) chromatographic processes have been successfully applied to petrochemical, pharmaceutical, and fine chemical industries to separate the target products with high purity and yield since it was introduced in 1960s. Conventional four-zone SMB chromatography and its advanced variants are nowadays the most promising continuous processes for the separation of chiral active pharmaceutical ingredients (APIs). However, the design and optimization of SMB processes are still challenging issues due to their structural complexity and operational sensitivity. In this work, the equilibrium-dispersion model with nonlinear adsorption isotherms was applied for model based on-line optimization. The proposed optimization technique estimates the process states including the isotherm and kinetic parameters. The operating conditions are dynamically optimized ‘switch-by-switch’ in order to separate binary mixtures up to 99% product purity.

Modeling and simulation of anion exchange chromatography for purification of proteins in complex mixtures

Ion exchange chromatography is extensively used in the purification of biological compounds. Reliable mathematical models describing this chromatographic technique are available and can be used to improve the performance of this separation step. However, the use of synthetic mixtures for model development hampers the application of this approach with real cell extracts processed in downstream operations. This work presents an original approach for handling non-synthetic genuine mixtures of proteins, which was applied in the purification of an untagged recombinant pneumococcal surface protein A (PspA4Pro). First, evaluation was made of the efficiency of steric mass action (SMA) and modified Langmuir isotherms, which were separately used together with the equilibrium dispersive model (EDM). The data used for parameter estimation and model validation were obtained from anion exchange chromatography runs (employing Q-Sepharose FF), applied to real cell extracts produced by different cultivation strategies. Simulations showed that the models were able to describe the complex mixtures of unknown proteins. Next, the EDM and SMA approaches were used to separately describe the profile of PspA4Pro and the pool of protein impurities eluted together. The simulations showed that PspA4Pro tended to elute at the beginning of the peak, enabling the establishment of an alternative elution schedule that provided a 34% increase in the purity achieved using the anion exchange chromatography.

Establishment of adsorption isotherms of m- and p-cresols in chromatographic process with aluminum terephthalate metal-organic framework as stationary phase

Close boiling points pose great challenges to the separation of the isomers m-cresol and p-cresol. Metal−organic frameworks (MOFs) material with uniform pore structure, high specific surface area and good chemical stability is a potential adsorbent. In this study, m-cresol and p-cresol were separated by a chromatographic process with aluminum terephthalate metal-organic framework (MIL-53(Al)) as stationary phase and acetonitrile as mobile phase. In view of the unique “breathing” effect of MOFs materials, according to the kinetic process and thermodynamic equilibrium of adsorption and desorption, the hybrid adsorption isotherm equations were established, which can reflect the “breathing” effect by combining the Langmuir adsorption process of narrow pore and the multi-stage adsorption process of large pore. The parameters of adsorption isotherm were determined by inverse method and genetic algorithm, and the adaptability of the equilibrium dispersive chromatography model based on this adsorption isotherm to the single-component adsorption process and two-component competitive adsorption process was proved under different injection concentration, injection time and flow velocity conditions. The separation process of m-cresol and p-cresol was optimized by the chromatographic model and genetic algorithm. Under the optimized conditions, the separated m-cresol and p-cresol were obtained with the purity values of 96.26% and 93.23%, and the recovery values of 96.93% and 96.59%, respectively.