Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

### Read Part 1: Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

### Historical Background of Chromatography

It is modern separation technique that can resolve a multicomponent mixture into its individual fraction. In 1906, chromatography was first invented by Russian botanist and chemist Michael Tswett to separate green plant extract into its pigments by using a powered packed calcium carbonate column. Zechmeister’s first book published in 1937 on chromatography. Chromatography, as developed in the 1940s and 1950s, has found widespread application in various separation procedures and chemical analysis, especially in biochemistry.

Fundamental techniques of partition chromatography emerge with the work of Richard Synge and Archer Martin of various chromatography techniques such as gas-liquid chromatography, paper chromatography, and gas-solid chromatography, and with different column liquid chromatography techniques. The progress of chromatography to separates more complex mixtures continues to demonstrate further progress. Many chemists and scientists are working on this separation technique. Conventionally, we only had Silica and Aluminum but with the advancement we have more chemicals for separation process. It is one of the most economical separation techniques until now. With technological advances, the use of chromatography has increased dramatically.

### Read Part 1: Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

Column chromatography is another useful method of separation and determination. By the end of the 1960’s, the field of chromatography increased rapidly in high-performance liquid chromatography. A couple of decades ago, there was only silica and alumina. Now there are a lot of materials like titanium, hyphenia and germanium etc. This field never ends. In the future, the study of nanotechnology combined with chromatography is under consideration.

### Objectives and Motivations

Mathematical modeling is an important part of chromatography theory for understanding the behavior of dynamics inside the column and studying the process theoretically. Various chromatographic models exist in the literature.

We’ve conducted a comprehensive study of LC focusing on both gradient and isocratic elution techniques. Our research utilized core-shell particles and employed analytical techniques to solve the mass balance equations of the LKM.

**Key Findings:**

**Analytical Solutions:**We successfully derived analytical solutions in the Laplace domain to determine unknown moments.**Numerical Simulations:**We implemented a second-order accurate finite volume scheme to numerically approximate the nonlinear model equations.**Gradient Elution Benefits:**We compared gradient elution to isocratic elution and highlighted the advantages of gradient elution in various applications.**Preparative Chromatography:**We explored the potential of gradient elution in preparative chromatography.

### Research Methodologies

The following are the key points of this post:

1. Understanding of the physics involved in them, and their mathematical modeling, literature review of the considered research problem.

2. Finite volume scheme will be implemented.

Lumped kinetic model (LKM) will solved numerically and analytically to simulate gradient elution in a liquid chromatographic column packed with core beads.

### Laplace Transformation

There are many properties of Laplace transformations that are very useful in mathematics and engineering. One of them is conversion of convolution into multiplication.

### High Resolution Finite Volume Schema (HR-FVS)

In a FVS, the divergence theorem utilized to transform volumetric integrals having divergence terms to the surface integrals. There are several limiting functions in literature, each of them generates a unique numerical technique.

A researcher can utilized the system of ODE,s to get desired results. In this paper, we will use second order R-k technique to get required outcomes. Flux limiters are utilized to prevent numerical oscillations.

### Contents of this Post

This post contains the basic definitions, brief introduction to adsorption isotherms, and model parameters. It also explains the approaches we have to use to overcome the given model.

Considers both reversible A B and irreversible A → B reactions while implementing the two-component REDM solution technique for two-component RLKM. The model in this case consists of four PDEs, with reactions occurring in both the solid and liquid phases. A number of test issues are studied in order to validate the analytical and numerical solutions produced by HR-FVS.

This post provides a comprehensive overview of FBLC, exploring its fundamental concepts, derivations, and applications. By understanding the core principles and mathematical models governing FBLC, we can gain valuable insights into the behavior of mixed components within columns filled with solid stationary phases. In future posts, we’ll delve deeper into specific procedures, basic concepts, and the derivations of standard chromatographic models, providing a thorough understanding of this complex field.

### Chromatography

Components with strong interactions with stationary phase move more slowly inside of chromatographic column in comparison to components with poor interactions. The technique is appropriate for complex separations requiring high product purity.

### Key terminologies of chromatography

**Solute**: The substance dissolved in solvent to be separated.

**Solvent**: Substance in which solute get dissolved. Most often water is utilized as solvent.

**Stationary phase**: This is the column’s phase. The stationary phase in a chromatographic system is the solid or liquid material that selectively adsorbs the substances.

**Mobile phase**: The column is filled with a solvent, which is either a gas in GC or liquid in LC.

**Retention time**: The time it takes for a components in a mixture to leave the column.

**Mobile phase velocity**: Rate change of distance for mobile phase in chromatographic column.

**Adsorption**:The ability of a solid or liquid of drawing in and holding a gas, liquid, or solute to its surface. This method consists of two primary mechanisms known as adsorbate and adsorbent.

### Read Part 1: Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

**Adsorbent**: The adsorbent material on surface where adsorption occur.

**Convection**: An external force causes convection, which is defined as movement. Gravity or a pump are examples of external forces in chromatography.

**Dispersion**: Fluids travel via the pore gap within the column, which causes it to happen in porous media.

**Chromatogram**: Visual output representing detector response as a function of time.

### Parameters of Model

This section delves into key parameters influencing chromatographic models. We’ll explore column efficiency, porosity, and their relationship to the dispersion coefficient.

### Column Porosities:

Based on two phases of chromatographic column portrayed earlier as stationary phase and mobile phase. We can divide the column volume, , into two components: the mobile phase volume, , and the stationary phase volume, . The stationary phase volume can be further subdivided into the solid volume, , and the intraparticle pore volume, . Therefore, the total volume equation is:

(1)

The volume of the column is,

(2)

where R is the radius of the column and L is the length of the column.

(3)

internal porosities defined as

(4)

External porosities defined as

(5)