Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

### Analysis of the EDM and LKM for Rectangular Pulse injection

The EDM and the LKM are two liquid chromatographic models that are examined in light of linear isotherms. And two different sets of BC’s at inlet and outlet. () for a rectangular single solute with limited length concentration pulses and Within linear limitations, a revolution of curves () is infused. The Laplace transformation is used to determine the analytical solution. The temporal moments up to the fourth demands for studying the solute transport conduct. We analyzed the models using Laplace-transformed configurations. Our analysis included both exact and numerical results, which we compared for validation. For linear adsorption isotherms, we described the results of various experiments.

We also determined the temporal moments to understand numerically and analytically.### LKM

The LKM ties combine internal and outside mass transport precautions to generate a mass exchange coefficient .

in these equations, represents the equilibrium liquid phase concentration. We use for liquid phase concentrations and for solid phase concentrations. The interstitial velocity is denoted by the letter , represents the porosity, is for time, denotes axial dispersion, denotes mass transfer coefficient, and is for axial coordinate.

The isotherm describes the equilibrium relationship between mobile and stationary phase concentrations. This thermodynamic detail is crucial for accurately predicting the column’s concentration profile. We often model this relationship using the convex nonlinear Langmuir isotherm, expressed as follows:

Here, represents the isotherm’s nonlinearity, and denotes the Henry coefficient.The isotherm becomes linear. For low concentrations,

The dispersive coefficient, , is often expressed as a dimensionless Peclet number, *Pe*. This dimensionless parameter represents the ratio of advective transport to dispersive transport. The *Pe* can be calculated by,

where *L *represents the length of the column. The IC’s for a column that is uniformly preequilibrated are:

### EDM

The EDM is based on the assumption that mass transfer kinetics are infinitely quick, . The visible dispersion coefficient is the aggregated of all band-expanding commitments. The mass equilibrium equation for a chromatographic column in the equilibrium dispersive model is,

Where the phase ratio depending on the porosity . We can express the apparent dispersion coefficient, *D*, in terms of the Peclet number, *Pe*, as follows:

When the column efficiency is high, the EDM accurately predicts chromatographic profiles, i.e. for huge peclet number *Pe*. The initial and BCs of the model are the same as LKM given by Eqs. 3.8-3.11.

### Analytical solution of linear EDM and LKM

In this section we will discuss about exact solution for EDM and LKM for the two different pairs of BCs.

### Analytical Solution of LKM

In this section, we’ll analyze the linear isotherm using the single-component LKM model. For the given boundary conditions, LT utilized to find analytical solution. We can express the LT as follows:

(1)

(2)

For solid phase, the governing equations are

(3)

(4)

In above equations, denotes the liquid phase concentration, represents the solid phase

concentration and is the mass transfer coefficients of component \textit{i} and denotes the solid phase reaction coefficient for first and second components of mixture , where \textit{i=1,2} and . Now the initial conditions are given below,

(5)

### Part 1: Analysis of a Chromatographic Model with Irreversible

where and represents the initial concentrations of components of the mixture in solid and liquid phase, respectively in column.

Let’s define some dimensionless variables to make the analysis easier.

(6)

Where L represents the length of column. Now use these variables in equations 1 to 4 , we get

and

Use these above equations in eq 1, we get,

From initial condition\

we obtain

(7)

Now putting

and

in equation 2, we obtain,

(8)

Now we have

put into equation 3 we get

multiplying above equation with and simplify then the equation becomes

(9)

### Read Part 2: Analysis of a Chromatographic Model with Irreversible and

similarly we can obtain the equation

(10)

ICs in non-dimensionalize form are given as

(11)

Apply the Laplace transformation in domain in equation 7 and 8 and eliminate the solid concentrations

applying Laplace transformation on equation 7

multiplying above equation with and use ICs then equation becomes

(12)

Now apply Laplace transformation to equation 9 then\

where

use ICs and simplify then above expression becomes

so use this above value in equation 12

### Part 3: Analysis of a Chromatographic Model with Irreversible and

simplify and then rearrange the above equation

### Part 1: Analysis of a Chromatographic Model with Irreversible

so

(13)

Now equation 8 takes the form by appliyang Laplace transformation

from the ICs

(14)

Now apply Laplace transformation on equation 10

now use the ICs and putting the value of

using the value of in equation 14

(15)

### Part 3: Analysis of a Chromatographic Model with Irreversible and

where

(16)

The concentrations of mixture components in liquid phase are represented by and in the Laplace domain. Equations 13 and 15 takes the form in matrix form

(17)

where the square matrix is represented by the and the column matrix is represented by the . Thus, on the left hand side of equation 17, a combined reaction coefficient matrix [D] is given as

(18)

The linear transformation matrix [A] is computed. The eigenvectors of the combined reaction coefficient matrix [D] should be the columns of [A].

Now we calculate the eigenvectors and eigenvalues of [D] for eigenvalues the characteristic equation

where [I] is identity matrix

### Part 1: Analysis of a Chromatographic Model with Irreversible

takes determinant

applying quadratic formula to calculate the values of

and

where and are the eigenvalues of the matrix [D] now we find eigenvectors we have

for

we have

for

we get