Literature review
HPLC simulation
Figure 1.1: HPLC model classification 1.1.1 HPLC simulation: data – driven methods
There are numerous studies and software using empirical data to interpret the relationship between operation parameters and retention time, peak width, etc [1] [2] [3] [4] [5] [6] [7]
In 2013, Paul G Boswell and colleagues introduced an HPLC simulator designed as an effective educational tool for students in analytical chemistry This study utilized experimental data from 22 compounds analyzed on an Agilent Zorbax SB – C8 column, showcasing both gradient and isocratic modes The simulator effectively demonstrated the fundamental principles of High-Performance Liquid Chromatography (HPLC).
Theoretical Plate models General Rate models
Equilibrium Dispersive modelsData -driven methods
Two empirical equations were developed to calculate retention time and peak width based on operational parameters such as temperature, mobile phase composition, flow rate, injection volume, column length, and diameter Chromatograms displaying Gaussian peaks were generated for each compound, and these equations were integrated into a Java-based HPLC simulator The study evaluated the simulator's effectiveness as an educational tool for undergraduate analytical chemistry students, revealing that those who utilized the simulator scored significantly higher on a quiz assessing their understanding of HPLC principles (12.5/15 compared to 11.7/15) However, there were areas for enhancement; the simulator lacked the capability to provide information on the symmetry factor, a critical system suitability parameter, and was limited to predicting retention time and peak width for only 22 compounds in its library.
In a 2019 study by Angelo Antonio D’Archivio, an artificial neural network (ANN) model was developed to analyze the elution data of 16 amino acid derivatives under various organic modifier (φ) and pH gradients This research utilized empirical data from three original studies conducted by Pappa Louisi and colleagues, revealing significant insights into how solvent strength and pH influence retention behavior The experiments employed mobile phases consisting of acetonitrile and aqueous phosphate buffer at a total ionic strength of 0.02 M The first study generated 19 chromatograms across a range of fixed eluent pHs (2.80 to 7.80) while varying the organic modifier volume fraction between 0.2 and 0.5 The second study maintained specific organic modifier fractions (0.25, 0.27, 0.3, or 0.35) and explored 22 linear pH gradients within ranges of 2.8–10.7 or 3.2–9 Lastly, Zisi's research implemented a dual gradient approach, adjusting both pH and organic solvent fractions simultaneously, resulting in 27 experiments at fixed initial solvent fraction (0.25) and pH (3.21).
A three-level preliminary design was employed to vary the pH levels (4.68, 5.86, and 7.86), organic modifier fractions (0.35, 0.40, and 0.50), and gradient times (10, 20, and 30 minutes) for the calibration of a model predicting the retention behavior of analytes in reversed phase HPLC Utilizing Artificial Neural Networks (ANN) as a regression tool resulted in a highly accurate model, with training coefficients of determination (R²) between 0.9980 and 0.9999, and prediction coefficients (Q²) ranging from 0.9799 to 0.9984 The model effectively predicted retention times, achieving mean errors of 1.4% for pH gradients, 1.1% for organic modifier fractions, and 2.5% for combined pH/solvent gradients However, further investigation is needed to assess other critical system suitability parameters, such as symmetry factor, plate count, and peak width, which significantly influence chromatography separation.
In 2020, S.I Abba and his team utilized artificial intelligence to simulate High-Performance Liquid Chromatography (HPLC) through a data-driven approach, exploring three methodologies: artificial neural networks (ANN), adaptive neuro-fuzzy inference systems (ANFIS), and multi-linear regression (MLR) The study focused on pH and the volume fraction of organic modifiers, specifically methanol, as key input parameters to model the retention behavior and peak widths of amiloride and methyclothiazide in ion chromatography Various mobile phase programs, including isocratic, gradient, and multistep gradients, were employed The predictive accuracy of these models was assessed using statistical parameters such as correlation coefficients (R), coefficients of determination (R²), root mean squared error (RMSE), mean squared error (MSE), and mean absolute percentage error (MAPE) The results demonstrated that ANN and ANFIS outperformed MLR, with ANFIS showing a slight advantage over ANN in terms of predictive performance.
ANN The research is one of a few research implementing different data – intelligent methods and comparing the results
Quantitative structure-retention relationships (QSRR) play a crucial role in chemical and biological research A study by Soo Hyun Park and colleagues utilized QSRR to forecast the retention behavior of low molecular weight anions in ion chromatography, employing the equation log k’=a – b*log[E], where k’ represents the retention factor and [E] denotes the eluent concentration The model's parameters, a and b, were derived using an evolutionary algorithm-multi linear regression (EA-MLR) approach, inspired by Darwin's theory of evolution, which evaluates a population of proposed solutions to find optimal values This QSRR model effectively predicted the parameters for novel molecules, leading to accurate retention behavior forecasts, as validated by strong agreement with experimental data Notably, QSRR models offer a significant advantage over traditional data-driven methods by enabling the prediction of analyte retention behaviors beyond existing databases.
Data-driven approaches are the leading models for simulating High-Performance Liquid Chromatography (HPLC), due to the system's inherent complexity, which is straightforward to describe but challenging to calculate Techniques for data interpretation, including Artificial Neural Networks (ANN), Adaptive Neuro-Fuzzy Inference Systems (ANFIS), Multiple Linear Regression (MLR), and Quantitative Structure-Activity Relationship (QSAR), are diverse and continuously evolving As information technology progresses, data processing methods hold significant potential for advancement However, these methods face challenges when addressing novel analytes and often require extensive datasets, which may not be readily available to all users.
1.1.2 Theory – based approaches for HPLC simulation
HPLC can be effectively simulated using physicochemical theories to develop equations that illustrate fluid dynamics, adsorption, and diffusion principles These equations, known as partial differential equations (PDE), can be solved through numerical or algebraic methods Alternatively, algorithms can be created based on these theories to compute the required parameters A key benefit of these models is their minimal reliance on extensive experimental data, requiring only a small number of experiments for model calibration and result validation.
The Equilibrium-dispersive model posits that the concentrations in the stationary and mobile phases reach immediate equilibrium, assuming negligible axial dispersion and mass transfer resistances This model is effective for predicting the behavior of high-performance systems with minimal mass transfer resistance, making it applicable for simulating HPLC and other high-resolution chromatography systems It can accurately predict retention times; however, it may lack precision in peak shape predictions when mass transfer resistances are significant Numerous studies and practical applications have explored the equilibrium theory.
The plate model is a fundamental principle in chromatography, consisting of two versions The first version resembles the tank-in-series model for non-ideal flow systems, conceptualizing the column as a series of tiny theoretical cells with complete mixing This approach leads to a set of first-order ordinary differential equations (ODEs) that illustrate the adsorption and mass transfer between the mobile phase and stationary phase particles Many researchers have utilized this method for HPLC simulation The second type of plate model defines the equilibrium of each compound in the theoretical cells using a distribution coefficient, moving away from the ODE framework.
Craig models are notable for their effectiveness in solving multi-analyte systems through recursive iterations, leveraging the blockage effect These models also provide insights into the column-overload problem Velayudhan and Ladisch successfully simulated the elution and frontal adsorption phenomena using a Craig model that incorporates a corrected plate count.
In 2011, J.J Baeza proposed an enhanced version of the Martin and Synge plate model, incorporating slow mass-transfer kinetics between the fluid and particle phases This model divides the HPLC column into N theoretical plates, enabling the simulation of slow mass-transfer processes The use of Laplace transform facilitated the solution of ordinary differential equations (ODEs) for each theoretical plate A key concept introduced was the kinetic ratio, which represents the proportion of kinetic constants for mass transfer in the flow direction versus that between the mobile and stationary phases Experimental methods estimated drift from equilibrium conditions through variances and retention times at varying flow rates, revealing a linear relationship between kinetic ratio and variance Higher kinetic ratios correlated with wider peak widths, while increased flow rates reduced peak variances Validation was achieved by comparing this model to diluted species transfer through theoretical plates, where the mobile phase migrates in modest steps before complete blending The application of Laplace transform to solve the ODEs from the theoretical plate model was a significant contribution of this study.
In 2015, Yaxiong Zhang proposed an innovative algorithm for modeling High-Performance Liquid Chromatography (HPLC), introducing a novel coefficient known as the "phase transfer probability factor" to illustrate the non-equilibrium distribution state This model utilized a genetic algorithm (GA) to enhance the fitting of simulated experimental data, while multiple layer perceptron artificial neural networks (MLP-ANNs) and GA were employed to effectively simulate the separation processes of mixture samples containing phenol, hydroquinone, and resorcinol.
4 – nitrophenol The study simulated HPLC column by dividing into N segments The
The "phase transfer probability factor" identifies the equilibrium of each segment, enabling the calculation of analyte concentration in both the mobile and stationary phases for the next time increment This recursive iteration method represents the initial theoretical plate model, capable of simulating non-linear and non-ideal analytical chromatography.
The rate model is an enhanced version of the equilibrium dispersive model that accounts for mass transfer resistances between the mobile and stationary phases Typically, it comprises two equations: one representing the concentration deviation in the mobile phase and the other for the stationary phase Numerous studies and applications have been conducted utilizing rate models of varying complexity.
Finite difference methods
Finite different methods are widely used in simulating advection and diffusion [34] [35]
Finite element methods are widely discussed in various academic papers and books, particularly in the context of advection and convection modeling A commonly referenced technique in this area is the first order upwind scheme, which is known for its straightforward formulation.
The simplicity of this method allows for easy implementation in practice, and it yields stable results without oscillation However, its truncation error is directly proportional to the spatial increment, leading to reduced accuracy This limitation can be addressed by decreasing the spatial increment, but doing so results in a larger number of distance segments and increased runtime.
The second – order upwind scheme used by some researchers [40] [38] has a better margin of error which is proportional to ∆𝑥
In 2019, Xu Yu and colleagues conducted a study simulating gas migration in coal using a finite difference method that integrated advection, diffusion, and adsorption dynamics They employed a first-order scheme for time approximation and a second-order scheme for the advection term Their proposed numerical method effectively modeled solid-fluid interactions in porous and fractured media, highlighting the significant impact of molecular adsorption and desorption on mass transfer between solid and fluid phases The method was validated, showcasing its robustness and suitability for modeling gas and mineral mass transfer behaviors in geological contexts.
The third-order upwind scheme is referenced in limited documents, and its origin and effectiveness as an approximation for first-order derivation are not immediately apparent.
The third-order upwind scheme offers greater accuracy compared to first and second-order schemes, with reduced oscillation levels than the second-order option However, it is not a definitive solution to the problem, and higher-order schemes may be employed to address its limitations.
Other studies
In 2015, P.G Aguilera conducted a study that utilized COMSOL Multiphysics software to predict fixed-bed breakthrough curves for hydrogen sulfide adsorption from biogas, solving the PDE of adsorption and diffusion The study incorporated both Langmuir and Freundlich isotherm models, with results evaluated against experimental data The model effectively matched breakthrough curves with experimental results, demonstrating significant advantages over ideal plug flow models By employing various adsorption constants, including those from the Langmuir and Freundlich models, the research provided flexibility in selecting the adsorption constant that best fit the experimental data.
In 2014, Elson Dinis Gomes conducted a study simulating various chromatographic reactors, including Reverse Flow, Fixed Bed, and Simulated Moving Bed types, to evaluate their efficiency in esterification reactions The research placed significant emphasis on the Reverse Flow reactor while analyzing the Simulated Moving Bed concept to a lesser extent Utilizing MATLAB's PDE Toolbox and its intrinsic solver, the study addressed the advection-diffusion-reaction-adsorption equation, highlighting the unique advantages and disadvantages of each reactor type Ultimately, the study successfully developed a sophisticated model for simulating complex interconnected systems.
The model's reliance on the MATLAB Toolbox presents a significant limitation, as both the MATLAB PDE Toolbox and its intrinsic solver exhibit certain weaknesses and may lack the necessary flexibility for adjustments.
Relevance and motivation
High-Performance Liquid Chromatography (HPLC) is a widely used analytical technique across various industries, including pharmaceuticals, food and beverage, and environmental science The complexity of HPLC involves numerous factors that significantly influence retention time and peak sharpness, ultimately affecting analytical outcomes Method development typically requires extensive experimentation and careful adjustment of operational parameters, making the traditional trial-and-error approach both time-consuming and costly Additionally, the use of toxic organic solvents like methanol and acetonitrile in HPLC poses environmental concerns due to their energy-intensive manufacturing processes To mitigate these issues, modeling and simulation can enhance method development by enabling more informed decision-making, thus reducing the number of required experiments, minimizing solvent use, lowering research costs, and decreasing environmental impact This thesis employs a comprehensive general rate model to simulate the HPLC system, utilizing advanced numerical methods such as finite element method (FEM) and finite volume method (FVM), while also exploring high-order finite difference methods (FDM) for improved accuracy Specifically, fifth-order upwind schemes were applied for advection, a fourth-order central difference scheme for diffusion, and a one-sided fourth-order scheme for time derivatives A novel method for deriving approximation schemes from Taylor expansion was developed, with the algorithm implemented in MATLAB to enhance flexibility without relying on external toolboxes.
Simulation and Modeling
Model assumptions
Mobile phase’s linear velocity is the same in the entire column during run time
Temperature in the column is homogeneous and does not change over time
Derivative of the concentration in the mobile phase with respect to the distant in the direction perpendicular to the flow direction is negligible
Solvent for preparing the sample is the mobile phase
Elution profiles have a direct proportion to the concentration
The stationary phase is a pack of homogeneous solid spherical particles without inner defectives Their shape and size do not change over time
The mobile phase is not compressible.
Modeling
Figure 2.1: HPLC diagram 2.2.1 Mass conservation equation
The elution process in an HPLC column can be modeled by dividing the column into infinitely small sites along the flow direction The change in analyte concentration within these minuscule sites over time is influenced by both the mobile and stationary phases.
15 the flow in and out of the site and the diffusion from the adjacent sites The mass conservation of the analyte can be expressed by the following equation:
Equilibrium theory posits that the concentration of an analyte on the stationary phase surface reaches equilibrium with the mobile phase concentration instantaneously Consequently, the concentration on the stationary phase is solely influenced by the mobile phase concentration Utilizing the chain rule for derivatives of composite functions, the derivative of the concentration on the stationary phase can be determined from the derivative of the mobile phase concentration through a specific equation.
Substituting the equation (7) to the equation (6) will gain a partial differential equation that can be solved by finite difference methods:
Equilibrium theory is effective in systems with minimal mass transfer resistance, making it suitable for simulating efficient chromatography techniques such as HPLC However, for slower mass transfer processes, a rate model is more appropriate This rate model comprises two key equations: the first illustrates advection and diffusion within the mobile phase, while the second details the mass transfer between the mobile phase and the stationary phase's surface.
The shape of the isotherm curve significantly influences the chromatogram's peak characteristics A linear isotherm curve results in a symmetrical peak, while a non-linear isotherm leads to an asymmetrical peak There are two primary categories of non-linear isotherms that affect peak shape.
If the isotherm curve is convex, the peak will be tailing And the concave isotherm curve will result in a fronting peak [44]
The formulation of Langmuir adsorption constant is based on the following assumption [44]:
There is only one layer of adsorption site that the analyte in mobile phase can bind to
Only one molecule of analyte can bind to each binding site of the adsorbent
There is no competition in binding with adsorption site between different analytes molecules
The adsorption constant of each analyte is homogenous in all binding sites
One way to increase the flexibility of the model is using the Bi – Langmuir isotherm [44]:
The Bi-Langmuir isotherm model suggests the presence of two distinct binding types with varying adsorption mechanisms In reversed phase HPLC, secondary interactions may arise between the analyte and residual silanol groups.
The Langmuir isotherm is grounded in solid theoretical principles, yet a modified version may be necessary to accurately align with experimental data The extended form of the Langmuir isotherm is represented by a specific equation.
For multi – compound separation, when considering the interaction between different analyte molecules, the competitive Langmuir adsorption constant [17] [44] is used in the form of the following equation:
The modified isotherm in case of competitive adsorption can be addressed by the following equation [44]:
2.2.2.3 Calculation of the derivative by numerical method
To effectively apply an adsorption model to the partial differential equation (PDE), it is essential to compute the derivative of the analyte concentration on the stationary phase surface (n) in relation to the analyte concentration in the mobile phase (C) Given the complexity of various adsorption models, calculating this derivative can be quite challenging This thesis employs a fourth-order central difference scheme to achieve high accuracy in the derivative calculation.
To validate the approximation scheme, it is essential to calculate the derivative of the analyte concentration on the stationary phase surface concerning the analyte concentration in the mobile phase, following the Langmuir model This validation will identify the optimal concentration increment, which allows us to select a specific concentration value for derivative calculations.
20 must calculate a set of concentrations around the concentration that we want to calculate the derivative
𝐶 − 2∆𝐶 𝐶 − ∆𝐶 𝐶 + ∆𝐶 𝐶 + 2∆𝐶 Then, we can calculate the value of n respective to C and finally, calculate the derivative
The diagram illustrates the process of determining the derivative of the analyte concentration on the stationary phase's surface in relation to the analyte concentration in the mobile phase at concentration Ci.
Figure 2.3: Procedure of derivative calculation
Setting value of concentration increment
Calculating a set of concentration arround the value Ci
Calculating the value of n for each value of C above
Porosity plays a crucial role in HPLC modeling and simulation Various methods exist to measure porosity using experimental data, one of which involves injecting a non-retained compound into the column In reversed-phase HPLC, thiourea is commonly used for this measurement The retention time of the compound reflects the time taken to traverse the guard column, the main column, and the entire HPLC system.
The linear velocities of mobile phase in the guard column and the column are depended on the inner diameter of the guard column and the column respectively
Substituting the linear velocities in the first equation, we will have the following equation:
When acetonitrile is the organic modifier in the mobile phase, equation for calculating eluent viscosity is: [45]
In a methanol – water mobile phase, viscosity is calculated by the following equation: [45]
To calculate diffusion coefficient, we need to calculate the solvent association factor and average molecular weight of the mobile phase by these following equations: [45]
Diffusion coefficient in mobile phase is calculated by the following equation: [45]
Simulation
2.3.1 Deriving approximation schemes for first order derivatives
Several methods exist to derive approximate schemes for first-order derivatives using Taylor expansion, with Randall J LeVeque employing a method of undetermined coefficients to define these approximations In this thesis, I introduce a novel approach to determine these coefficients effectively.
First, we start with the general formular of Taylor expansion:
𝜕𝑥 + ⋯ (35) When multiplying ∆𝑥 with each factor a, we will have:
𝑥 𝑥 − 3∆𝑥 𝑥 − 2∆𝑥 𝑥 − ∆𝑥 𝑥 + ∆𝑥 𝑥 + 2∆𝑥 For each factor 𝑎 , there is a respective Taylor’s expansion:
Suppose that we want to gain a formular from q position of 𝐶(𝑥 + 𝑎 ∆𝑥), we must multiply 𝐶(𝑥 + 𝑎 ∆𝑥) with respective coefficients 𝑏 , and add these equations together to obtain the following equation:
To approximate the first-order derivative from the q node \( C(x + a \Delta x) \) using the respective coefficient \( b \) and \( C(x) \), we observe that the coefficients of \( C(x) \) sum to \( \sum b \), which equals zero in the case of symmetry Additionally, \( \sum b a \) serves as the dividing factor in this approximation.
𝜕𝑥 ≈ 𝑏 𝐶(𝑥 + 𝑎 ∆𝑥) − 𝐶(𝑥) 𝑏 ∆𝑥 𝑏 𝑎 (43) The leftover term will be the truncation error:
To minimize truncation error, it is essential to eliminate as many terms as possible Increasing the value of m leads to a reduction in the respective term's value With q variables, we can effectively eliminate up to q – 1 terms, denoted as ∆.
To eliminate the terms with m ranging from 2 to q, we assume a set of bi that equates these terms to zero Consequently, we need to solve the following equations to determine the values of b.
We can choose a suitable value of 𝑏 so that all the coefficients are all integral numbers The truncation error will become:
The most significant term of the truncation error is:
The key term related to ∆𝑥 is crucial in understanding the concept of order of accuracy, denoted as q This allows us to calculate the error factor, enabling a comparison of the accuracies among different schemes that share the same order.
A MATLAB script was developed to calculate the coefficients of approximations and their associated truncation errors This script enables the generation of approximation schemes of any desired order of accuracy, accommodating an arbitrary set of positions.
The following table show the most accurate approximation schemes for calculating first order derivative from 1 st to 5 th order of accuracy:
Table 2.1: Approximation for first order derivatives
2.3.2 Deriving approximation schemes for second order derivatives
A similar approach could be used to establish approximation for second order derivative:
Then the system of equations that must be solved will be:
We can choose a suitable value of 𝑏 so that all the coefficients are all intergral The truncation error of second order derivative approximation is:
The most significant term is:
The coefficient of the most significant term could be calculated to compare accuracy of scheme at the same order of accuracy:
−2 (𝑞 + 2)! 𝑏 𝑎 𝑏 𝑎 Table 2.2: Approximation for second order derivatives
The injection process is complex, with the concentration at the left boundary varying over time, illustrating the injection pulse To simplify this model, it is assumed that the concentration signal remains relatively stable from the injector to the column and from the column to the detector Additionally, the delayed time of the injection pulse is attributed to the system's void volume The injection signal is represented as a Gaussian pulse function, described by a specific equation.
The standard deviation of the Gaussian pulse was calculated by the following equation:
2.3.4 Courant number and alpha number
2.3.5 Central difference scheme for diffusion
The diffusion approximation formula is derived by incorporating terms from two adjacent positions on either side, utilizing a central difference scheme This method is highly accurate as it reflects the inherent nature of diffusion, where the analyte equally exchanges information from both sides, facilitating its movement in and out of a given position.
To enhance accuracy, the central difference scheme should be derived using additional points This approach aims to eliminate the influence of the fourth-order derivative by incorporating two extra points from either side.
2.3.6 Approximation of derivative of concentration with respect to distance in the Equilibrium Dispersive Model
The equilibrium model is simulated by solving the following equation:
2.3.6.1 The first – order upwind scheme
The first – order upwind scheme can be pulled out directly from the Taylor expansion:
Applying the first – order upwind scheme to simulate advection and second – order central difference scheme to simulate diffusion will give the following equation:
Normal positions will be calculated by the following equation:
1 + 𝑟(𝐶 − 2𝐶 + 𝐶 ) (67) Right boundary is calculated by following equation:
2.3.6.2 The second – order upwind scheme
Applying the second – order upwind scheme to simulate advection and second – order central difference scheme to simulate diffusion will give the following equation:
Position next to the left boundary is calculated by following equation:
The normal positions will be calculated by the following equation:
1 + 𝑟(𝐶 − 2𝐶 + 𝐶 ) (72) Right boundary is calculated by following equation:
1 + 𝑟(𝐶 − 𝐶 ) (73) 2.3.6.3 The third – order upwind scheme:
Applying the third – order upwind scheme to simulate advection and second – order central difference scheme to simulate diffusion will give the following equation:
Position next to the left boundary is calculated by following equation:
1 + 𝑟(𝐶 − 2𝐶 + 𝐶 ) (76) The normal positions will be calculated by the following equation:
1 + 𝑟(𝐶 − 2𝐶 + 𝐶 ) (77) Right boundary is calculated by following equation:
1 + 𝑟(𝐶 − 𝐶 ) (78) 2.3.6.4 The fourth – order upwind scheme
Applying the fourth – order upwind scheme to simulate advection and fourth – order central difference scheme to simulate diffusion will give the following equation:
Position 2 is calculated by following equation:
Position 3 is calculated by following equation:
Normal positions will be calculated by the following equation:
Position next to the right boundary is calculated by following equation:
Right boundary is calculated by following equation:
2.3.6.5 The fifth – order upwind scheme
Applying the fifth – order upwind scheme to simulate advection and fifth – order central difference scheme to simulate diffusion will give the following equation:
Position 2 is calculated by following equation:
Position 3 is calculated by following equation:
Normal positions will be calculated by the following equation:
Position next to the right boundary is calculated by following equation:
Right boundary is calculated by following equation:
2.3.7 Approximation of derivative of concentration with respect to distance in the General Rate Model
The rate model is simulated by solving the following set of equation:
The saturated concentration of the stationary phase surface is determined using the appropriate adsorption model equation The surface concentration for the subsequent time step is then calculated using this established equation.
2.3.7.1 The first – order upwind scheme
𝜕𝐶 = −𝜃𝛽∆𝑡(𝑛 − 𝑛 ) − 𝐶𝑟(−𝐶 + 𝐶 ) + 𝛼(𝐶 − 2𝐶 + 𝐶 ) (95) 2.3.7.2 The second – order upwind scheme
2 + 𝛼(𝐶 − 2𝐶 + 𝐶 ) (96) 2.3.7.3 The third – order upwind scheme
2.3.7.4 The fourth – order upwind scheme
2.3.7.5 The fifth – order upwind scheme
The nodes in the boundary and nearby, which did not exist was approximated by cubic polynomial extrapolation
2.3.8 Approximation of derivative of concentration with respect to time
The approximation of the derivative of concentration over time is determined using a one-sided scheme Initially, the concentration of the analyte across the entire column is zero The subsequent moment is calculated using a specific equation.
∆𝑡 → 𝐶𝑗= 𝜕𝐶 + 𝐶𝑗−1 (100) The third moment is calculated by the following equation:
The fourth moment is calculated by the following equation:
The fifth moment and after are calculated by the following equation:
Calculation and Experiment
Software and codes
The algorithm for modeling Partial Differential Equations (PDEs) was developed using MATLAB R2021a, with scripts designed to analyze the impact of various input parameters on SST parameters Key parameters examined include flow rate, injection volume, injection concentration, stationary phase capacity, adsorption constant, diffusion coefficient, and mass transfer coefficient Undefined parameters such as stationary phase capacity, adsorption constant, diffusion coefficient, and mass transfer coefficient were calibrated using experimental data to enhance the accuracy of the model.
Various approximation schemes based on Taylor expansion were developed to simulate the partial differential equation (PDE), with their outcomes presented for comparison These schemes were validated through the calculation of derivatives for specific test functions, and the numerical results were compared with the corresponding analytical solutions.
Chemicals and equipment
Experiments were conducted to calibrate the undefined parameters and validate the model using the HPLC Agilent 1260 Infinity II for sample analysis Details regarding the chemical reagents utilized in the sample preparation are provided in the accompanying table.
Distilled water HCMC IDQC NA
Scheme’s verification
The evaluation of approximation schemes involved calculating the derivatives of various testing functions and comparing the approximated results (Ap) with the analytical outcomes (An) The accuracy of these schemes was assessed through relative error (RE), determined using a specific equation.
The relative errors were extremely small, so they were a bit inconvenient for comparing the approximation schemes Error indexes (EI) were calculated by the following equation:
Error indexes were often negative numbers and the smaller they are the more accurate the approximation schemes are
A fourth-order central difference scheme was selected to accurately approximate the derivative of surface concentration on the stationary phase in relation to the mobile phase concentration This method proved to be more precise than an upwind scheme of the same order To validate the effectiveness of the approximation, differentiable functions resembling the geometric shapes of the Langmuir adsorption isotherm equation were utilized as test functions Various Langmuir equations with differing adsorption constants and capacities were employed for this validation The impact of these factors, along with concentration increments, was thoroughly assessed Additionally, all functions listed in Tables 1 and 2 were verified by calculating the derivative of Gaussian functions with comparable peak widths and heights to the simulated peaks, allowing for an evaluation of the effects of peak width, peak height, and temporal/spatial increments.
Experiment
To prepare a caffeine working standard, accurately weigh 25 mg and transfer it into a 100 ml volumetric flask Add 50 ml of water and sonicate for 15 minutes to ensure complete dissolution After dissolving, dilute the solution to the mark, mix thoroughly, and filter through a 0.45 µm filter.
Results and discussions
Verification of the schemes’ accuracy
4.1.1 Derivative of the surface concentration
The approximation scheme was verified by calculating the derivative of n with respect to
C in following equation and its analytical derivative: 𝑛 = ; ( )
Figure 4.1: Test function and the respective verification results
The first chart shows the graphs of the test function and the second chart show the effect of
The analysis of relative errors in the scheme reveals that as the value of K increases, the slope becomes steeper, as shown in the first chart Conversely, the second graph indicates that a higher adsorption constant correlates with increased relative errors This trend is attributed to the dominant term of the error, ∆, where the derivative of n increases with sharper changes in slope.
Figure 4.2: Relative error and logarithm of relative error as functions of ∆C
The most significant term of the error is ∆, leading to the assumption that as this error approaches zero, the logarithm of the error trends towards negative infinity Consequently, the concentration increment ∆C should ideally be minimized However, verification results indicate a different outcome; as the concentration increment decreases, errors initially align with expectations but eventually begin to oscillate randomly and increase This behavior is attributed to the limitations in the number of significant figures that the software can process A deeper understanding requires examining the approximation scheme in detail.
A decrease in ∆C leads to a reduction in the nominator, as the calculation involves subtraction When the difference becomes minimal in comparison to the minuends and subtrahends, the number of significant figures diminishes, resulting in decreased accuracy Therefore, it is essential to identify the optimal ∆C that minimizes relative error for more precise results.
Figure 4.3: Verification result of different test function with different n0
The verification results from the test function exhibit nearly identical outcomes despite variations in capacity, which may seem counterintuitive since increased capacity typically results in a stiffer slope, similar to the effect of the adsorption constant However, a critical distinction exists between the two factors A detailed examination of the test function and its derivative provides robust mathematical evidence indicating that changes in capacity do not influence relative errors Specifically, when the capacity shifts from n01 to n02, the value of n adjusts proportionally to n02/n01, affecting both the approximation of the derivative and the analytical derivative in a similar manner.
The approximation error for a test function with capacity n02 is influenced by a factor of n02/n01 When this error is divided by the analytical derivative, the n02/n01 term cancels out, resulting in a consistent relative error However, this relationship does not hold true in other scenarios.
The adsorption constant significantly influences the behavior of the adsorption curve, as changes in this constant lead to distinct variations in the value of n and its derivative, resulting in differing relative errors This indicates that the relative error is contingent upon the adsorption constant of the test function Additionally, the adsorption constant determines the geometric shape of the adsorption curve, affecting its curvature, while the capacity primarily influences the magnitude of the curve Consequently, curves with varying capacities can be viewed as scaled versions of one another, exhibiting geometric similarity.
To determine the optimal ∆C for accurate approximation of the adsorption constant, it is unnecessary to test every possible isotherm value Instead, focusing on the highest isotherm is crucial, as it tends to yield the largest relative error Therefore, identifying the ∆C that meets the requirements of the highest isotherm is sufficient for achieving accurate results.
4.1.2 Derivative of concentration in the mobile phase
The verification results for the approximation schemes used to derive concentration in the mobile phase indicated that while the magnitude of Gaussian peaks did not affect accuracy, peak width significantly influenced it Specifically, narrower peaks resulted in greater bias, making it more effective to test the thinnest peaks Detailed results can be found in Appendix E, which also demonstrated that higher-order schemes yielded lower relative errors Consequently, fifth-order upwind schemes were selected for simulating advection, while fourth-order central difference schemes were employed for diffusion, striking a balance between accuracy and complexity.
Simulation result
4.2.1 Distribution of the analyte in the column
Figure 4.4: Distribution of concentration in the column using EDM
Figure 4.5: Distribution of concentration in the column using GRM
The analyte's concentration distribution within the column exhibits a belt shape at a given time, often resembling symmetrical Gaussian peaks in the linear range, while showing slight tailing in the non-linear range As the runtime progresses, the analyte migrates toward the column outlet, resulting in a shorter and broader peak before it ultimately exits the column.
Figure 4.6: Chromatogram and suitability parameters at different approximation scheme for simulating the derivative of concentration with respect to distance in linear range using GRM
In the linear range, third-order accuracy and higher demonstrate superior performance in simulating chromatograms compared to first and second-order methods The first-order scheme produces peaks that are too short and yield low plate counts, while the second-order scheme generates peaks that, although similar in shape to other methods, lack symmetry and exhibit suboptimal area recovery As the order of accuracy increases, the parameters of the simulated peaks converge towards an equilibrium value, with no significant differences observed between third-order and higher results.
1 Order of accuracy is the order of the scheme used for approximation which is mentioned in section 2.3.1
Figure 4.7: Chromatogram at different approximation scheme for simulating the derivative of concentration with respect to distance in non – linear range using GRM
In the non-linear range, the performance of various numerical schemes is similar to that in the linear range; however, the second-order scheme exhibits significant oscillations at the peak base Further analysis reveals that while the fifth-order scheme maintains a normal shape, the third and fourth-order schemes experience some oscillations This behavior is attributed to the steep slope of the peak in the non-linear range, which negatively impacts the accuracy of lower-order schemes Consequently, the fifth-order scheme is identified as the most effective for simulating the derivative of concentration with respect to distance, making it the preferred choice for this analysis.
Figure 4.8: Chromatogram and suitability parameters at different approximation scheme for simulating the derivative of concentration with respect to time in linear range using GRM
Different schemes for simulating the time derivative of concentration yield similar chromatograms, with system suitability parameters converging to equilibrium values more quickly than the scheme for spatial increment (dx) The accuracy of these simulations relies on both the order of the scheme and the increment size; smaller increments reduce error To ensure result stability, the temporal increment (∆t) must be very small, while the spatial increment (∆x) should be relatively larger Consequently, the accuracy of the simulation scheme for the time derivative of concentration is optimized, leading to the selection of a fourth-order scheme to ensure high precision.
Figure 4.9 illustrates the chromatogram and suitability parameters associated with varying diffusion coefficients using EDM The overall trends observed in EDM and GRM are largely similar, with minor distinctions EDM can be viewed as a variant of GRM characterized by an infinite mass transfer coefficient, resulting in peaks with slightly higher plate counts, taller heights, narrower widths, and improved symmetry factors, indicating more tailing peaks Additionally, the peaks' base generated by EDM at the lowest diffusion coefficient exhibits a slight oscillation.
The diffusion coefficient significantly influences peak characteristics in chromatograms, as illustrated in Figure 4.10 An increase in the diffusion coefficient results in broader peaks and reduced peak height, leading to a decrease in plate count Additionally, a higher diffusion coefficient enhances peak stability, resulting in more symmetrical peaks Although the diffusion coefficient minimally affects retention time, a decrease in the symmetry factor can slightly increase it.
4.2.4 Effect of mass transfer coefficient
Figure 4.11: Chromatogram and suitability parameters at different mass transfer coefficient using GRM The mass transfer coefficient has the opposite effect as the effect of diffusion coefficient
An increase in the mass transfer coefficient leads to a rise in the symmetry factor, as a faster mass transfer process saturates the stationary phase's surface concentration more quickly, resulting in an earlier onset of the overload phenomenon Consequently, the retention time experiences a slight decrease due to the elevated symmetry factor Additionally, improvements in the column's efficiency contribute to higher plate counts and peak heights, while peak widths decrease with the rise in the mass transfer coefficient.
4.2.5 Effect of capacity of the stationary phase
Figure 4.12: Chromatogram and suitability parameters at different capacity of the stationary phase using EDM
Figure 4.13: Chromatogram and suitability parameters at different capacity of the stationary phase using GRM
The chromatograms produced by the EDM exhibit notable similarities to those of the GRM, though there are some distinct differences Specifically, the EDM chromatograms feature significantly higher peaks that are more tailing, thinner, and demonstrate a higher plate count due to the enhanced mass transfer efficiency of the EDM However, further analysis with more comprehensive data is necessary to determine if this behavior accurately reflects reality.
The retention factor is proportional to capacity, with retention time exhibiting a linear relationship to it As capacity increases, the symmetry factor tends to decrease due to the heightened effect of diffusion when components remain longer in the column, leading to more symmetrical peaks, except for unretained components Conversely, when stationary phase capacity decreases, diffusion effects lessen, resulting in increased peak symmetry Concentration overload, which occurs during interactions with the stationary phase, significantly influences symmetry; if the stationary phase capacity is zero, these interactions cease, eliminating concentration overload effects A smaller stationary phase capacity correlates with higher plate counts, but as capacity approaches zero, plate counts rise rapidly due to the nearly symmetrical peaks that emerge.
Figure 4.14: Chromatogram and suitability parameters at different adsorption constant using EDM
Figure 4.15: Chromatogram and suitability parameters at different adsorption constant using GRM
Figure 4.16: Area recovery of peaks produced by the EDM and GRM
The results from Electrodialysis Membrane (EDM) and Granular Resin Membrane (GRM) exhibit similar behaviors with slight variations, primarily characterized by the more pronounced tailing of peaks in EDM Typically, EDM chromatograms demonstrate higher plate counts due to minimal mass transfer resistance; however, in this instance, the symmetry factors lead to comparable plate counts with GRM Notably, GRM consistently achieves nearly 100% area recovery in peak mass conservation, while EDM shows significantly lower recovery in cases of high tailing peaks This indicates the theoretical superiority of GRM over EDM.
The retention factor is directly related to the adsorption constant, with retention time exhibiting a linear relationship that intersects the y-axis at the retention time of the non-retained component, used for calculating column porosity This relationship is valid as the tracer has an isotherm or capacity of zero To validate the model, one can simulate scenarios where either the isotherm or capacity is zero and compare the retention time with the dead time An increase in the adsorption constant results in a higher symmetry factor, as the adsorption constant curve becomes stiffer and curvier Additionally, a higher isotherm correlates with a lower plate count due to two main factors: longer retention times lead to greater peak broadening, and an increased symmetry factor is associated with reduced column efficiency, ultimately decreasing the plate count.
Figure 4.17: Chromatogram and suitability parameters at different flow rate using
The retention time in chromatography is inversely related to the flow rate, while the retention factor remains stable As the flow rate increases, both the retention time of the analyte and non-retained compounds decrease at similar rates Lower flow rates result in more symmetrical peaks due to increased retention time, whereas higher flow rates produce thinner peaks that initially raise the plate count but eventually lead to a decline This decline occurs because increased flow rates exacerbate peak tailing, negatively impacting column efficiency and overall plate count.
Figure 4.18: Chromatogram and suitability parameters at different injection concentrations in linear range using GRM
When injection concentrations are low and within a narrow range, key parameters such as symmetry factor, plate count, and retention time remain stable However, at extremely high concentrations, concentration overload occurs, leading to increased peak tailing, a reduction in plate count, and a forward shift in the peak.
Figure 4.20: Chromatograms and suitability parameters at different injection volumes in linear range using GRM
Injection volume significantly influences peak shape in HPLC, akin to the impact of injection concentration Unlike preparative chromatography, HPLC has a constrained injection volume range, preventing volume overload Within the linear range, increasing the injection volume marginally elevates retention time and slightly broadens peak width, resulting in a minor decrease in plate counts.
In multi-component separation using GRM, the injection of two components with similar concentrations leads to competitive adsorption, causing a slight forward shift in each peak on the chromatogram However, when the injection volume is minimal, the competition between the components diminishes, resulting in chromatograms from competitive and single injections that appear nearly identical.
Model validation
To validate the conservative properties of the model, the recovery of peak area compared to the injection pulse is calculated for each peak While simulations can generate visually appealing results, they may not always be accurate; thus, peak area recovery serves as a critical criterion for identifying false results The area of each peak is determined using the Composite Trapezoidal Rule, and peak area recovery is computed using a specific equation.
Table 4.1: Area recovery of the simulation peak using EDM
Area of injection pulse Peak area Recovery Bias
Table 4.2: Area recovery of the simulation peak using GRM
Area of injection pulse Peak area Recovery %
If the area recovery of each peak reached nearly 100%, the model would be deemed conservatively accurate However, the EDM's area recovery is marginally lower than that of the GRM, indicating that the EDM may be less effective than the GRM in preserving mass.
4.3.2 Simulation of non – retained substance
In the experiment, the solvent peak occurs at 2.73 minutes, which is essential for calculating column porosity, subsequently used in simulation calculations By setting all interaction parameters between the non-retained substance and the stationary phase to zero, a chromatogram can be simulated The retention times for the simulated tracer using the EDM and GRM are 2.7314 and 2.7296 minutes, respectively, closely matching the solvent peak retention time The negligible difference between the EDM and GRM results indicates that the tracer does not interact with the stationary phase, confirming the accuracy of both simulations.
The chromatogram results for the EDM and RGM at Vinj = 10 àL show nearly identical single peaks, with only minor deviations observed Notably, the EDM exhibits a larger bias of 6.8% compared to the experimental results, while the GRM demonstrates a smaller bias of 4.9%.
Table 4.3: SST parameters of the experiment and simulation peak using EDM
Table 4.4: SST parameters of the experiment and simulation peak using GRM
The model effectively simulates a single injection, yielding highly accurate results for system suitability parameters, including retention time, EP plate count, and tangent peak width, with deviations not exceeding 0.8% However, achieving this level of accuracy may require compromises in other parameters, such as symmetry factor and peak width at 5% and 10%.
After accurately simulating a single injection, we can project the influence of various operational parameters, such as injection volume, on system suitability metrics The simulation shows retention times with an error margin of less than 1%, while peak width accuracy exhibits a maximum bias of 6.5%.
Figure 4.25: Retention time of experiment and simulation using EDM
Figure 4.26: Peak widths of experiment and simulation using EDM
Re te nt io n tim e, m in
Injection Volume, uL Experimental Simualtion
Pe ak w id th , m in
Injection Volume, uLExperimental Wd 5 Experimental Wd 10 Experimental Wdv USPSimulation Wd 5 Simulation Wd 10 Experimental Wd USP
Figure 4.27 : Symmetry factor of experiment and simulation using EDM
Figure 4.28: Plate count of experiment and simulation using EDM
Sy m m et ry fa ct or
Injection Volume, uL Experiment Simulation
Injection Volume, uLExperimental EP Plate count Experimental USP plate countSimulation EP plate count Simulation USP plate count
Figure 4.29: Retention time of experiment and simulation using GRM
Figure 4.30: Peak widths of experiment and simulation using GRM
R et en tio n tim e, m in
Injection Volume, uL Experimental Simualtion
Injection Volume, uLExperimental Wd 5 Experimental Wd 10 Experimental Wdv USPSimulation Wd 5 Simulation Wd 10 Experimental Wd USP
Figure 4.31: Symmetry factor of experiment and simulation using GRM
Figure 4.32: Plate count of experiment and simulation using GRM
S ym m et ry f ac to r
Injection Volume, uL Experiment Simulation
Experimental EP Plate count Experimental USP plate countSimulation EP plate count Simulation USP plate count
Table 4.5: Experiment and simulation results using EDM
Table 4.6: Simulation accuracy and bias of the system suitability parameters using
The EDM and GRM exhibit comparable performance in fitting experimental data, with minor differences noted The maximum bias in predicting the symmetry factor for the EDM is 6.8%, slightly exceeding the GRM's 6.5% Additionally, the GRM demonstrates a lower maximum bias of 3.2% for simulated plate counts, compared to the EDM's 4.6% Furthermore, when predicting peak widths, the GRM outperforms the EDM, with the highest bias at 4.9% versus 6.5% for the EDM.
Table 4.7: Experiment and simulation results using GRM
Table 4.8: Simulation accuracy and bias of the system suitability parameters using
The model accurately predicts that plate counts increase as injection volume decreases, with a deviation of 3.8% or less However, the simulation shows a more pronounced decline in plate counts compared to experimental results Additionally, the model forecasts a decrease in symmetry factor with rising injection volume, aligning well with experimental data Nevertheless, the simulation achieves symmetry more rapidly than the experiments, resulting in a significant gap of approximately 6.8% between the two.
Conclusions
The analytical process of the HPLC system was solved using a finite difference method with high-order schemes, facilitated by a MATLAB algorithm designed to define the coefficients for these schemes This algorithm simplifies the establishment of schemes at any order with various node sets, allowing for the easy calculation of the most significant truncation error term Advection and axial diffusion were modeled using fifth-order upwind schemes and fourth-order central difference schemes, respectively, while the time derivative was approximated with a one-sided fourth-order scheme The accuracy of these methods was validated through tests with relevant functions, demonstrating that the fifth and fourth-order schemes significantly outperform their lower-order counterparts in simulation results.
Equilibrium theory and rate models are essential for simulating the HPLC process, with the choice of model depending on its fit to experimental data Factors such as injection volume and concentration significantly influence peak shape, with simulations indicating linear behavior at low concentrations and volumes, resulting in symmetrical peaks and stable retention times However, at higher volumes and concentrations, signs of concentration overload emerge, leading to increased symmetry factors and decreased plate counts In multi-component separations, the presence of additional components slightly reduces individual capacity, causing competitive adsorption that shifts peaks forward These observations align with established chromatography and adsorption theories Initial validation experiments show strong agreement between simulated and experimental results, although further improvements to the model and more comprehensive experiments are necessary for rigorous validation.
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Appendix A: Algorithm to define approximation schemes clc; clear all; format rat a=input('Input vector a (increasing sorting order): ');
Q=size(a);q=Q(2); disp('The order of accuracy: ');disp(q); i=0; b=zeros(1,q-1); for w=(1:1:q-1) i=i+1; b(i)=a(q)^(i+1); end b=b'; i=1;j=1;
Ci=-Ci; end i=1; for w=(1:1:q) if a(i)