A finite difference method using high order schemes for modeling non linear chromatography

Trang 1

HO CHI MINH CITY, February 2023

HO CHI MINH UNIVERSITY OF TECHNOLOGY

CAO HÀ THÀNH

A FINITE DIFFERENCE METHOD USING HIGH – ORDER SCHEMES FOR MODELING

NON – LINEAR CHROMATOGRAPHY

PHƯƠNG PHÁP SAI PHÂN HỮU HẠN SỬ DỤNG CÔNG THỨC BẬC CAO ĐỂ MƠ PHỎNG

SẮC KÝ PHI TUYẾN TÍNH

Major: Chemical Engineering Major ID: 8520301

Trang 2

ii

Assoc Prof PhD Nguyen Tuan Anh Reviewer 1: (Signature)

Assoc Prof PhD Nguyen Quang Long Reviewer 2: (Signature)

PhD Ly Cam Hung

Master’s Thesis was defended in HCMC University of Technology, VNU-HCM on February 14th, 2023

The participants of the Mater’s Thesis Defend Council includes:

1 Chairman: Assoc Prof PhD Nguyen Đinh Thanh 2 Reviewer 1: Assoc Prof PhD Nguyen Quang Long 3 Reviewer 2: PhD Ly Cam Hung

4 Participant: PhD Đang Bao Trung 5 Secretary: PhD Đang Van Han

Verification of the chairman of the Master’s Thesis Defense Council the Head of Chemical Engineering Faculty

CHAIRMAN OF THE COUNCIL DEAN OF CHEMICAL ENGINEERING FACULTY

Trang 3

iii

MASTER’S THESIS ASSIGNMENT Full name: Cao Ha Thanh Student ID: 1970161 Date of birth: 08/01/1996 Place of birth: TP.HCM Major : Chemical Engineering Major ID : 8520301 I TITLE

A FINITE DIFFERENCE METHOD USING HIGH – ORDER SCHEMES FOR MODELING NON – LINEAR CHROMATOGRAPHY

PHƯƠNG PHÁP SAI PHÂN HỮU HẠN SỬ DỤNG CÔNG THỨC BẬC CAO ĐỂ MÔ PHỎNG SẮC KÝ PHI TUYẾN TÍNH

II ASSIGNMENT AND CONTENT

 Establishing high- order approximation schemes to simulate HPLC  Verifying the schemes using relavent test functions

 Applying the schemes in simulating the two major model of HPLC, namely GRM and EDM

 Analysing the effect of operation parameters on parameters of the chromatographic peak

 Validation by comparing the simulation and the experiments III ASSIGNMENT DELIVERY DATE : 14/02/2022

IV ASSIGNMENT COMPLETION DATE : 10/12/2022

V THESIS SUPERVISOR : Assoc Prof PhD Nguyen Tuan Anh

HCM city, _

THESIS SUPERVISOR (Full name and Signature)

HEAD OF DEPARTMENT (Full name and Signature)

Trang 4

iv

First and foremost, I would like to express my sincere gratitude to my thesis supervisor, Associate Professor Dr Nguyen Anh Tuan In 2019, he taught me Modeling and Simulation Thanks to his inspiring teaching, I learned the miracle of the numerical method, which can help engineers solve any differential equation Since then, my passion for discipline began to grow, and I decided to research this field His comprehensive expertise, extensive experience, and enthusiastic guidance are essential factors in helping me do this research Besides, I also appreciate the intellectual support of my friend, Nguyen Van Vinh Ha His experiments and knowledge in analytical chemistry are valuable to me And last but not least, I would like to thank all my friends, family, and colleagues who have given me indispensable emotional support

Lời cảm ơn

Trang 5

v Abstract

Trang 6

vi

Trang 7

vii

I proclaim that this thesis is original and that any other source is cited appropriately Most of the figures in this work are originally illustrated, and the others are adapted from the identified source

Lời cam đoan

Tôi xin cam kết luận văn này là nguyên bản, tất cả các tài liệu tham khảo khác đều được trích dẫn đúng cách và ghi nguồn cụ thể Các hình ảnh số liệu phần lớn được lấy từ kết quả của luận văn này, các hình ảnh từ nguồn khác đều được trích dẫn phù hợp

(Ký tên)

Trang 8

viii

MASTER’S THESIS ASSIGNMENT iii

Acknowledgement iv

Lời cảm ơn iv

Abstract v

Tóm tắt vi

Declaration vii

Lời cam đoan vii

List of figures xi

List of tables xiii

Notation and glossary xiv

List of Abbreviations xvii

Chapter 1: Literature review 1

1.1 HPLC simulation 1

1.1.1 HPLC simulation: data – driven methods 1

1.1.2 Theory – based approaches for HPLC simulation 5

1.2 Finite difference methods 10

1.3 Other studies 11

1.4 Relevance and motivation 13

Chapter 2: Simulation and Modeling 14

2.1 Model assumptions 14

2.2 Modeling 14

2.2.1 Mass conservation equation 14

2.2.2 Adsorption models 17

Trang 9

ix

2.3 Simulation 22

2.3.1 Deriving approximation schemes for first order derivatives 22

2.3.2 Deriving approximation schemes for second order derivatives 27

2.3.3 Injection simulation 29

2.3.4 Courant number and alpha number 29

2.3.5 Central difference scheme for diffusion 30

2.3.6 Approximation of derivative of concentration with respect to distance in the Equilibrium Dispersive Model 30

2.3.7 Approximation of derivative of concentration with respect to distance in the General Rate Model 35

2.3.8 Approximation of derivative of concentration with respect to time 36

Chapter 3: Calculation and Experiment 37

3.1 Software and codes 37

3.2 Chemicals and equipment 37

3.3 Scheme’s verification 37

3.4 Experiment 39

Chapter 4: Results and discussions 40

4.1 Verification of the schemes’ accuracy 40

4.1.1 Derivative of the surface concentration 40

4.1.2 Derivative of concentration in the mobile phase 43

4.2 Simulation result 44

4.2.1 Distribution of the analyte in the column 44

4.2.2 Effect of Approximation schemes 46

4.2.3 Effect of diffusion coefficient 49

Trang 10

x

4.2.6 Effect of adsorption constant 54

4.2.7 Effect of flow rate 56

4.2.8 Effect of injection concentration 57

4.2.9 Effect of injection volume 58

4.2.10 Multi – component separation 59

4.3 Model validation 61

4.3.1 Assessment of mass conservation 61

4.3.2 Simulation of non – retained substance 62

4.3.3 Single injection 62

4.3.4 Simulation of a sample set 64

Chapter 5: Conclusions 70

List of Publication 71

References 72

Appendix A: Algorithm to define approximation schemes 78

Appendix B: Algorithm for validating schemes to approximate derivative of adsorption constant 79

Appendix C: Algorithm for validating schemes to approximate derivative of concentration 81Appendix D: Verification results the approximation schemes 88

BIOGRAPHY 90

STUDY EXPERIENCE 90

Trang 11

xi

Figure 1.1: HPLC model classification 1

Figure 2.1: HPLC diagram 14

Figure 2.2: Effect of the isotherm curve on the peak shape (recreated from [44]) 17

Figure 2.3: Procedure of derivative calculation 20

Figure 2.4: Injection pulse 29

Figure 4.1: Test function and the respective verification results 40

Figure 4.2: Relative error and logarithm of relative error as functions of ∆C 41

Figure 4.3: Verification result of different test function with different n0 42

Figure 4.4: Distribution of concentration in the column using EDM 44

Figure 4.5: Distribution of concentration in the column using GRM 45

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 46

Figure 4.7: Chromatogram at different approximation scheme for simulating the derivative of concentration with respect to distance in non – linear range using GRM 47

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 48

Figure 4.9: Chromatogram and suitability parameters at different diffusion coefficients using EDM 49

Figure 4.10: Chromatogram and suitability parameters at different diffusion coefficients using GRM 50

Figure 4.11: Chromatogram and suitability parameters at different mass transfer coefficient using GRM 51

Figure 4.12: Chromatogram and suitability parameters at different capacity of the stationary phase using EDM 52

Trang 12

xii

using EDM 54

Figure 4.15: Chromatogram and suitability parameters at different adsorption constant using GRM 54

Figure 4.16: Area recovery of peaks produced by the EDM and GRM 55

Figure 4.17: Chromatogram and suitability parameters at different flow rate using GRM 56

Figure 4.18: Chromatogram and suitability parameters at different injection concentrations in linear range using GRM 57

Figure 4.19: Chromatogram and suitability parameters at different injection concentrations outside linear range using GRM 57

Figure 4.20: Chromatograms and suitability parameters at different injection volumes in linear range using GRM 58

Figure 4.21: Chromatograms and suitability parameters at different injection volumes outside linear range using GRM 58

Figure 4.22: Chromatograms of multi – component separation with similar concentrations using GRM 59

Figure 4.23: Chromatograms of multi – component separation with different concentrations using GRM 60

Figure 4.24: Experiment and simulation result at Vinj = 10 µL using GRM 62

Figure 4.25: Retention time of experiment and simulation using EDM 64

Figure 4.26: Peak widths of experiment and simulation using EDM 64

Figure 4.27 : Symmetry factor of experiment and simulation using EDM 65

Figure 4.28: Plate count of experiment and simulation using EDM 65

Figure 4.29: Retention time of experiment and simulation using GRM 66

Figure 4.30: Peak widths of experiment and simulation using GRM 66

Figure 4.31: Symmetry factor of experiment and simulation using GRM 67

Figure 4.32: Plate count of experiment and simulation using GRM 67

Figure D.1: Verification result of the schemes used for simulating advection 88

Trang 13

xiii

Table 2.1: Approximation for first order derivatives 26

Table 2.2: Approximation for second order derivatives 28

Table 3.1: Chemical reagent’s information 37

Table 3.2: Experimental parameters 39

Table 4.1: Area recovery of the simulation peak using EDM 61

Table 4.2: Area recovery of the simulation peak using GRM 61

Table 4.3: SST parameters of the experiment and simulation peak using EDM 63

Table 4.4: SST parameters of the experiment and simulation peak using GRM 63

Table 4.5: Experiment and simulation results using EDM 68

Table 4.6: Simulation accuracy and bias of the system suitability parameters using EDM 68

Table 4.7: Experiment and simulation results using GRM 69

Table 4.8: Simulation accuracy and bias of the system suitability parameters using GRM 69

Table D.1: The schemes used for simulating advection 88

Trang 14

xiv

Character Unit Description

Ac 𝑚 Cross sectional area of the inner column An - Analytical result of the test function Asp 𝑚 Surface area of stationary phase particle

Ap - Approximation result of the scheme

AR % Area recovery of the peak

As - Symmetry factor

C 𝑚𝑜𝑙/𝑚 Concentration of analyte in mobile phase Cinj 𝑚𝑜𝑙/𝑚 Injection concentration

Cr - Courant number

D 𝑚 /𝑠 Diffusion coefficient

dp 𝑚 Diameter of stationary phase particle

EI - Error index

Er - Error of the approximation

F 𝑚 /𝑠 Flow rate of the mobile phase

i, j - Co – Ordinator

ID1 𝑚 Internal diameter of the guard column ID2 𝑚 Internal diameter of the column

k’ - Retention factor

KF - Freundlich adsorption constant

KL 𝑚 /𝑚𝑜𝑙 Langmuir adsorption constant KL1, KL2 𝑚 /𝑚𝑜𝑙 Bi – Langmuir adsorption constant

L1 𝑚 Length of the guard column

L2 𝑚 Length of the column

Ma 𝑔/𝑚𝑜𝑙 Molar mass of the analyte

MMP 𝑔/𝑚𝑜𝑙 Average molecular weight of the mobile phase Morg 𝑔/𝑚𝑜𝑙 Molecular weight of the organic modifier

Trang 15

xv

n 𝑚𝑜𝑙/𝑚

phase

n0 𝑚𝑜𝑙/𝑚 Monolayer capacity of stationary phase

NEP - EP plate count

nsat 𝑚𝑜𝑙/𝑚 Saturated concentration on surface of stationary phase’s surface

NUSP - USP plate count

nvoid 𝑚𝑜𝑙/𝑚 Concentration of empty space on stationary phase’s surface

r - Ratio of the amount of analyte in stationary phase and mobile phase

RE - Relative error of the scheme compared to the analytical result

RT min Retention time

RT0 𝑠 Retention time of an analyte that is not adsorbed by stationary phase

sp 𝑚 ⁄𝑚 Specific surface area of stationary phase particle

TC ℃ Column temperature

tinj 𝑠 Injection time

tvoid 𝑠 System void volume

u1 𝑚/𝑠 Linear velocity of mobile phase in guard column u2, u 𝑚/𝑠 Linear velocity of mobile phase in column

VC 𝑚 Volume of column limited by dx

Vinj 𝑚 Injection volume

Vm 𝑚𝐿/𝑚𝑜𝑙 Molar volume of the analyte

Vp 𝑚 Volume of stationary phase particle

wd s Standard deviation of the injection pulse Wd10 min Peak width at 10 % height

Wd5 min Peak width at 5 % height

WdUSP min USP tangent peak width

Trang 16

xvi

zorg -

modifier

α - Alpha number

β 1/𝑠 Mass transfer coefficient

γ 𝑚 /𝑚 Saturated distribution coefficient ε - Porosity of stationary phase particle η 𝑚𝑃𝑎 𝑠 Mobile phase viscosity

θ 𝑚 /𝑚 Ratio of stationary phase surface area to mobile phase volume

λ - Modifying factor of extended Langmuir isotherm

ρa 𝑔 𝑚𝐿⁄ Analyte density

Trang 17

xvii

ANFIS Adaptive – neuro fuzzy inference system ANN Artificial neural network

EDM Equilibrium dispersive model

EP European Pharmacopeia

FDM Finite difference method

FEM Finite element method

FVM Finite volume method

GRM General rate model

HPLC High performance liquid chromatography

IDQC HCM Institute of Drug Quality Control – Ho Chi Minh City

LC Liquid chromatography

MAPE Mean absolute percentage error

MLP – ANN Multiple layer perceptron artificial neural network MLR Multiple linear regression

MSE Mean – squared error

ODE Ordinary differential equations

OOA Order of accuracy

PDE Partial differential equations

QSRR Quantitative structure – retention relationships RMSE Root mean – squared error

SST System Suitability Test

Trang 18

1

Chapter 1: Literature review

1.1 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 his colleges proposed an HPLC simulator as an effective educational tool for a student studying analytical chemistry [1]The study was based on experimental data of 22 compounds on an Agilent Zorbax SB – C8 column in both gradient and isocratic mode Fundamental principles of HPLC were demonstrated in several

HPLC models

Theory -based methods

Theoretical

Plate models General Rate models

Trang 19

2

empirical equations which were used to calculate retention time, peak width from operation parameters like temperature, mobile phase composition, flow rate, injection volume, column length, and diameter, etc Chromatograms featured with Gaussian peaks could be plotted for each compound These equations were coded into an HPLC simulator in Java programming language The study also tested the effectiveness of the HPLC model as an educational tool for undergraduate analytical chemistry students The result shown that students who had been given access to the simulator outperformed those who hadn’t (score 12.5/15 compared to 11.7/15) in the quiz to assess their understanding of HPLC fundamental principles [1] Despite many achievements, there was room for improvement For instance, the simulator did not give students information about the symmetry factor which was quite important as a system suitability parameter The simulator used was limited for educational purposes because it could not predict retention time and peak width of a novel compound besides the 22 ones in the library

Trang 20

3

Trang 21

4

ANN The research is one of a few research implementing different data – intelligent methods and comparing the results

Quantitative structure – retention relationships are widely used in chemical and biology research [11] A study of Soo Hyun Park and co – workers used quantitative structure – retention relationships (QSRR) to predict retention behavior of low molecular weight anions in IC [12] The model was based on the well – known equation log k’=a – b*log[E] When k’ is the retention factor, [E] is the concentration of the eluent, values a and b are respectively the intercept and the slope of the linear solvent strength model The model used evolutionary algorithm – multi linear regression (EA – MLR) to obtain a and b values for small organic and inorganic ion Evolutionary algorithms were inspired by Charles Darwin’s theory of evolution A set of proposed solutions were created for the problem These solutions were called ‘population’ and they will be evaluated by calculating a set of parameters that indicate how good the solutions fit the reality The so – called population evolves over time in order to gain better solutions [13] The QSRR model was used to predict a and b of the novel molecules These values in turn were used to predict the retention behavior of the analytes External validation shown a good agreement between predicted retention time with experimental data QSRR models have a remarkable advantage over other data – driven methods which is the ability to predict the retention behavior of analytes outside the existing databases

Trang 22

5

1.1.2 Theory – based approaches for HPLC simulation

HPLC could be simulated by using physicochemical theories to formulate equations demonstrating principles of fluid dynamics, adsorption, and diffusion These equations are partial differential equations (PDE) that could be solved by numerical or algebraical methods [14] [15] [16] [17] Another approach is creating an algorithm from theories to calculate the desired parameters [18] The advantage of these models is not being dependent on a huge amount of experimental data, they just need minuscule number of experiments for calibrating the model and validating the results

1.1.2.1 Equilibrium – dispersive model

Equilibrium – dispersive model assumes that the concentration on the surface of stationary phase and the concentration in the mobile phase get equilibrium immediately Axial dispersion and mass transfer resistances are considered negligible These assumptions are suitable for predicting the behavior of a high – performance system with insignificant mass transfer resistance HPLC and other high – resolution chromatography systems can be simulated by this model Retention time can be predicted by this model with meaningful accuracy However, it does not get sufficient precision in predicting the peak shape when mass transfer resistances are considerable [14] There are some studies and practical applications using the equilibrium theory [15] [19] [20] [21]

1.1.2.2 Plate model

Trang 23

6

systems, the solution is obtained by using recursive iterations [14] Craig models are among the most outstanding ones The distribution Craig models can be applied to multi – analyte systems by using the so – called blockage effect [25] Column – overload problem can be researched by using the Craig models Velayudhan and Ladisch had elution and frontal adsorption phenomenon simulated by a Craig model with a corrected plate count

Trang 24

7

equilibrium of each segment is identified by “phase transfer probability factor” and would be used to calculate the concentration of analyte in mobile phase and stationary phase in the next temporal increment This recursive iteration method is the first type of theoretical plate model [14] This model can simulate non – linear and non – ideal analytical chromatography

1.1.2.3 Rate model

Rate model is an advanced version of equilibrium dispersive model, it considers the mass transfer resistances between the mobile phase and the stationary phase They often have two or two set of equation, one presents the deviation of concentration in the mobile phase, the other is for the stationary phase [14] There are many studies and applications based on rate models with various complexity [14] [26] [27] [28] [29] [30] [31] [32]

Trang 25

8

some shortcomings The model used a first – order scheme for approximating advection, which is quite inaccurate This error was reduced by using extremely small time and space steps (∆t, ∆x) but this approach created another drawback – exceedingly long run time The study simply accepted the long run time as a minor inconvenience because they could have their computer run overnight However, this weakness can be overcome by using a higher – order scheme to save run time without sacrificing accuracy

A book written by Tingyue Gu in 2015 proposed a rate model to simulate and scale – up liquid chromatography [14] The book used rate models to simulate the elution of multi – component samples in the LC column The models not only investigated the main flow direction diffusion, interfacial film mass transfer between the mobile phase and the surface area of the stationary phase, but also inspected nonlinear multicomponent isotherms and intraparticle dispersal Several chromatographic processes like elution (including isocratic and gradient mode), breakthrough, and displacement could be studied and predicted by using the model The book provides several models for simulating LC elution of organic compounds (adsorption, reversed – phase, and hydrophobic interaction models), high molecular weight compound (size – exclusion model), inorganic compound (ion – exchange model), and protein (affinity model)

Trang 26

9

simulating a rate model of liquid chromatography reactor for two compounds and deriving semi – analytical solutions The validation showed a good agreement between analytical and numerical results However, the model is limited for linear general rate model

A study of Shamsul Qamar and co – workers in 2013 and 2016 derived an analytical solution of linear general rate model for liquid chromatography [32] [29] The models dealt with axial dispersion, mass transfer resistance between fluid phase and stationary phase, and intraparticle diffusion Laplace transformation and some mathematic conversions were applied to the partial differential equation and resulted in an ordinary differential equation with different variable Analytical solutions and moments of single component were derived from the ordinary differential equation The method worked out two different formulars for rectangular and continuous breakthrough inlets The models were validated by comparing the results of the analytical method and a second – order finite volume scheme The validation results show a good agreement between analytical and numerical solutions However, the studies did not examine competitive multi – compound elution and were confined in linear model

Trang 27

10 1.2 Finite difference methods

Finite different methods are widely used in simulating advection and diffusion [34] [35] [36] [37] There are many papers and books mentioning finite element methods in one way or another When it comes to advection/convection modeling, the first order upwind scheme is mentioned frequently [17] [38] [39] [40] [41] The formula is a bit straightforward:

𝜕𝐶𝜕𝑥 ≈

𝐶(𝑥) − 𝐶(𝑥 − ∆𝑥)∆𝑥

Its simplicity makes it very easy to come up with and apply in practices Another advantage is that the results it produces is very stable, or in other word, they do not have any oscillation However, this scheme’s truncation error is proportional the increment, making it poorly accurate This weakness can be overcome by decreasing spatial increment However, smaller spatial increment means bigger number of distance segment and longer run time

The second – order upwind scheme used by some researchers [40] [38] has a better margin of error which is proportional to ∆𝑥

𝜕𝐶𝜕𝑥 ≈

3𝐶(𝑥) − 4𝐶(𝑥 − ∆𝑥) + 𝐶(𝑥 − 2∆𝑥)2∆𝑥

Trang 28

11

Only a few documents mention the third – order upwind scheme, [39] at first glance, it is not clear where the formula comes from and why it is a good approximation for the first – order derivation

𝜕𝐶𝜕𝑥 ≈

3𝐶(𝑥) − 6𝐶(𝑥 − ∆𝑥) + 2𝐶(𝑥 + ∆𝑥) + 𝐶(𝑥 − 2∆𝑥)6∆𝑥

The third – order upwind scheme is significantly more accurate than the first and second – order schemes The oscillation of the third – order scheme is less than the second-order scheme However, it is not the silver bullet for this problem Higher – order schemes could be used to offset its weaknesses

1.3 Other studies

In 2015, a study by P.G Aguilera predicted the fixed – bed breakthrough curves for Hydrogen sulfide adsorption from biogas [42] COMSOL Multiphysics software was used to solve the PDE of adsorption and diffusion of biogas through a fixed bed adsorbent The model considered two isotherm model, Langmuir and Freundlich adsorption model The isotherm models’ results were evaluated by comparing with experimental data The model has done a great job of matching breakthrough curves with experimental results and shown an obvious advantage over ideal plug flow models The model used several adsorption constants include Langmuir and Freundlich model with various exponential factors This gives the model flexibility in choosing the adsorption constant that fits the experimental data

Trang 29

12

Trang 30

13 1.4 Relevance and motivation

Trang 31

14

Chapter 2: Simulation and Modeling

2.1 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 2.2 Modeling

Figure 2.1: HPLC diagram 2.2.1 Mass conservation equation

Trang 32

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:

𝜕(𝜀𝑉 𝐶)𝜕𝑡 +𝜕(𝑛𝐴 )𝜕𝑡 = 𝐹𝐶 − 𝐹(𝐶 + 𝜕𝐶) + 𝜀𝐴 𝐷𝜕𝐶𝜕𝑥 (1) As 𝜀𝐶𝜕𝑉𝜕𝑡 = 0 𝑎𝑛𝑑 𝑛𝜕𝐴𝜕𝑡 = 0 Thus, 𝜀𝑉 𝜕𝐶𝜕𝑡 +𝐴 𝜕𝑛𝜕𝑡 = −𝐹𝜕𝐶 + 𝜀𝐴 𝐷𝜕𝐶𝜕𝑥 (2) 𝑠𝑖𝑛𝑐𝑒 𝑉 = 𝐴 𝜕𝑥; 𝑎𝑛𝑑 𝐴 = 𝑠 𝑉 = 𝑠 (1 − 𝜀)𝑉 = 𝑠 (1 − 𝜀)𝐴 𝜕𝑥 𝑇ℎ𝑒𝑛 𝜀𝐴 𝜕𝑥𝜕𝐶𝜕𝑡 +𝑠 (1 − 𝜀)𝐴 𝜕𝑥𝜕𝑛𝜕𝑡 = −𝐹𝜕𝐶 + 𝜀𝐴 𝐷𝜕𝐶𝜕𝑥 (3) ↔ 𝜕𝐶𝜕𝑡 +𝑠 (1 − 𝜀)𝜀 ×𝜕𝑛𝜕𝑡 =−𝐹𝜀𝐴 ×𝜕𝐶𝜕𝑥 + 𝐷𝜕 𝐶𝜕𝑥 (4) for 𝐴 = 𝜋𝐼𝐷4 ; 𝑠 =6𝑑↔ 𝜕𝐶𝜕𝑡 +6(1 − 𝜀)𝑑 𝜀 ×𝜕𝑛𝜕𝑡 =−4𝐹𝜀𝜋𝐼𝐷 ×𝜕𝐶𝜕𝑥+ 𝐷𝜕 𝐶𝜕𝑥 (5) 𝑓𝑜𝑟: 𝑢 = 4𝐹𝜀𝜋𝐼𝐷 ; 𝜃 =6(1 − 𝜀)𝑑 𝜀↔ 𝜕𝐶𝜕𝑡 + 𝜃𝜕𝑛𝜕𝑡 = −𝑢𝜕𝐶𝜕𝑥 + 𝐷𝜕 𝐶𝜕𝑥 (6)

Trang 33

16 𝜕𝑛𝜕𝑡 =𝜕𝑛𝜕𝐶×𝜕𝐶𝜕𝑡 (7)

Substituting the equation (7) to the equation (6) will gain a partial differential equation that can be solved by finite difference methods:

→ 1 + 𝜃𝜕𝑛𝜕𝐶𝜕𝐶𝜕𝑡 = −𝑢𝜕𝐶𝜕𝑥 + 𝐷𝜕 𝐶𝜕𝑥 (8) 𝑓𝑜𝑟 𝛾 = 𝜕𝑛𝜕𝐶; 𝑟 = 𝜃𝛾 → 𝜕𝐶𝜕𝑡 = −𝑢1 + 𝑟×𝜕𝐶𝜕𝑥 +𝐷1 + 𝑟×𝜕 𝐶𝜕𝑥 (9)

Equilibrium theory works in some systems that have insignificant mass transfer resistance This model can be useful for simulating highly efficient chromatography like HPLC However, in the case of a slow mass transfer process, a rate model will be better to describe the system [14] A rate model consists of two equations, the first equation demonstrates the advection and diffusion in the mobile phase The second equation describes the mass transfer between the mobile and the surface of the stationary phase: [14] [44]

𝜕𝑛

Trang 34

17 2.2.2 Adsorption models

Figure 2.2: Effect of the isotherm curve on the peak shape (recreated from [44]) The type of isotherm curve has a notable effect on the chromatogram If the isotherm curve is a straight line, the chromatogram will have a symmetrical peak When the isotherm is not linear, the peak will be asymmetrical There are two main types of non – linear isotherms If the isotherm curve is convex, the peak will be tailing And the concave isotherm curve will result in a fronting peak [44]

2.2.2.1 Langmuir adsorption constant

The formulation of Langmuir adsorption constant is based on the following assumption [44]:

Trang 35

18

 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 [𝐴𝑛𝑎𝑙𝑦𝑡𝑒] + [𝑉𝑜𝑖𝑑] ↔ [𝐴𝑛𝑎𝑙𝑦𝑡 − − − 𝑉𝑜𝑖𝑑] C nvoid n 𝑛𝑛 𝐶 = 𝐾 →𝑛𝑛 = 𝐾 𝐶 (11) 𝑛 = 𝑛 + 𝑛 (12) 𝑛𝑛 − 𝑛 = 𝐾 𝐶 →𝑛 − 𝑛𝑛 =1𝐾 𝐶 ↔𝑛𝑛 =1𝐾 𝐶 + 1 =1 + 𝐾 𝐶𝐾 𝐶 (13) → 𝑛𝑛 =𝐾 𝐶1 + 𝐾 𝐶→ 𝑛 =𝑛 𝐾 𝐶1 + 𝐾 𝐶 (14) → 𝜕𝑛𝜕𝐶 =𝑛 𝐾(1 + 𝐾 𝐶) (15)

One way to increase the flexibility of the model is using the Bi – Langmuir isotherm [44]:

𝑛 = 𝑛 𝐾 𝐶1 + 𝐾 𝐶+

𝑛 𝐾 𝐶

1 + 𝐾 𝐶 (16)

The ideal of Bi – Langmuir isotherm is that there are two different types of binding with different adsorption mechanisms [44] In reversed phase HPLC, the secondary interaction can occur between analyte and the remain silanol groups

The Langmuir isotherm has a solid theoretical foundation However, to better fit the experimental data, sometime a modified Langmuir isotherm must be used The extended form of Langmuir isotherm is presented in the following equation [44]:

𝑛 = 𝜆𝐶 + 𝑛 𝐾 𝐶

Trang 36

19

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:

𝑛 = 𝑛 𝐾 𝐶

1 + ∑ 𝐾 𝐶 (18)

The modified isotherm in case of competitive adsorption can be addressed by the following equation [44]:

𝑛 = 𝜆 𝐶 + 𝑛 𝐾 𝐶

1 + ∑ 𝐾 𝐶 (19)

2.2.2.2 Freundlich adsorption constant

𝑛 = 𝐾 𝐶 (20) → 𝜕𝑛𝜕𝐶 = 𝑚𝐾 𝐶 (21) → 1 +6(1 − 𝜀)𝜀𝑑 × 𝑚𝐾 𝐶𝜕𝐶𝜕𝑡 =−4𝐹𝜀𝜋𝐼𝐷 ×𝜕𝐶𝜕𝑥+ 𝐷𝜕 𝐶𝜕𝑥 (22)

2.2.2.3 Calculation of the derivative by numerical method

To apply an adsorption model to the PDE, we must calculate the derivative of the concentration of the analyte on the surface of the stationary phase (n) with respect to the concentration of the analyte in the mobile phase (C) Some adsorption models are very complicated, so calculating the derivative can be extremely tricky This thesis uses a fourth – order central difference scheme to calculate the derivative with high accuracy:

𝜕𝑛𝜕𝐶 ≈

𝑛 − 8𝑛 + 8𝑛 − 𝑛

12∆𝐶 (23)

Trang 37

20

must calculate a set of concentrations around the concentration that we want to calculate the derivative

𝐶 𝐶 𝐶 𝐶

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 following diagram presents the procedure of calculating the derivative of the concentration of the analyte on the surface of the stationary phase with respect to the concentration of the analyte in the mobile 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

Trang 38

21 2.2.3 Calculating porosity

The porosity of the column is extremely important in HPLC modeling and simulation There are several methods to measure porosity by experimental data [14] [44] One of them is injecting a non – retained compound into the column and using the retention time of this compound to calculate the porosity In reversed – phase HPLC, thiourea is a popular substance for this purpose The retention time consists of the time it takes to flow through the guard column, column, and the system of HPLC:

𝑅𝑇 = 𝑡 +𝐿𝑢 +

𝐿

𝑢 (24)

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

𝑢 = 4𝐹

𝜀𝜋𝐼𝐷 ; 𝑢 =4𝐹

𝜀𝜋𝐼𝐷 ; (25)

Substituting the linear velocities in the first equation, we will have the following equation:

𝑅𝑇 = 𝑡 +𝜀𝜋 𝐼𝐷 𝐿 + 𝐼𝐷 𝐿4𝐹 (26) → 𝑅𝑇 − 𝑡 = 𝜀𝜋 𝐼𝐷 𝐿 + 𝐼𝐷 𝐿4𝐹 (27) → 𝜀 = 4𝐹(𝑅𝑇 − 𝑡 )𝜋 𝐼𝐷 𝐿 + 𝐼𝐷 𝐿 (28)

2.2.4 Calculating diffusion coefficient

When acetonitrile is the organic modifier in the mobile phase, equation for calculating eluent viscosity is: [45]

Trang 39

22

In a methanol – water mobile phase, viscosity is calculated by the following equation: [45]

𝜂 = exp 𝜙 −4.597 + 1211𝑇 + 273.15 + (1 − 𝜙) −5.961 +1736𝑇 + 273.15+ 𝜙(1 − 𝜙) −6.215 + 2809𝑇 + 273.15(30)

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]

𝑧 = (1 − 𝜙) 2.6 − 𝑧 + 𝑧 (31)

𝑀 = 𝜙𝑀 + 18(1 − 𝜙) (32)

Diffusion coefficient in mobile phase is calculated by the following equation: [45]

𝐷 = 7.4 × 10 ×(𝑇 + 273.15) 𝑧 𝑀

𝜂𝑉 . (33)

2.3 Simulation

2.3.1 Deriving approximation schemes for first order derivatives

There are several ways to derive approximative schemes for first order derivatives from Taylor expansion Randall J LeVeque used a method of undetermined coefficients to define the approximations [39] In this thesis, I propose an approach to gain these coefficients:

First, we start with the general formular of Taylor expansion:

𝐶(𝑥 + ∆𝑥) = 𝐶(𝑥) + ∆𝑥𝑚!𝜕 𝐶𝜕𝑥 (34) ↔ 𝐶(𝑥 + ∆𝑥) = 𝐶(𝑥) + ∆𝑥𝜕𝐶𝜕𝑥 +∆𝑥2!𝜕 𝐶𝜕𝑥 +∆𝑥3!𝜕 𝐶𝜕𝑥 +∆𝑥4!𝜕 𝐶𝜕𝑥 + ⋯ (35) When multiplying ∆𝑥 with each factor a, we will have:

Trang 40

23

𝑎 – 3 – 2 – 1 1 2

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:

𝐶(𝑥 + 𝑎 ∆𝑥) = 𝐶(𝑥) + 𝑎 ∆𝑥𝜕𝐶𝜕𝑥+𝑎 ∆𝑥2!𝜕 𝐶𝜕𝑥 +𝑎 ∆𝑥3!𝜕 𝐶𝜕𝑥 +𝑎 ∆𝑥4!𝜕 𝐶𝜕𝑥+ ⋯ (37)