Analysis of a Chromatographic Model with (Part 6)

Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

Analysis of reversible and irreversible reactions in two-component RLKM

This chapter extendes the two-component REDM analysis to include. A two-component RLKM that describes reversible A B and irreversible A → B processes. The solution of RLKM converges to REDM for the large enough values of k.

Part 1: Analysis of a Chromatographic Model with Irreversible and

This chapter contains, reactions are carried out in both solid and liquid phases. Chromatographic separation in a chromatographic reactors is focused on on the simultaneous adsorption. And chemical reactions of solid particle catalysts in a reactor column that contains a mixture of adsorber and catalyst.

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

The procedure is similar to the reactive extraction, sedimentation, reactive distillation, and electrophoresis when combined with reaction. LT is utilized to get analytical solutions, and moments can be obtained from Laplace transformed solution. Laplace transformation considered as a basic tool for finding the temporal moments. In this work temporal moments and analytical solutions for the chromatographic models representing reversible A B. Irreversible A → B processes, as well as breakthrough curves.

Irreversible reaction (A → B) performed in chromatographic reactor

Let t stand for time coordinate and z for axial coordinate along length of the column. Furthermore, both components are considered to have a same apparent coefficient of dispersion D.

Part 3: Analysis of a Chromatographic Model with Irreversible and

The LKM takes into account in stationary phase rate of change in concentration of the solutes as well as back mixing in column because of dispersion. For fixed bed chromatography column, governing equations of 1-D linear RLKM are as follows, in above equations, ci represents for concentration of liquid phase, qi denotes concentration of solid phase, ai used for Henry constant of linear adsorption isotherm q∗ = ciai and ki is component’s mass transfer coefficients and vi denotes solid phase reaction coefficient for first and second components of mixture , where ∈ (0, 1) and i=1,2 is external porosity.

For the large value of k LKM changes into EDM.

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

Analytical solutions for linear isotherms

Analytical solutions for irreversible two-component reactions;

This section contains the following information, we considered a two component RLKM model described in Eq.4.1 and 4.4. We define some dimensionless parameters to simplify the analysis.

The eigenvectors of 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 D1, D2, E1, E2 are arbitrary constants we can find the values of these constants by using BCs at column outlet and inlet. We consider 2 types of BCs in this study as follows.

Case I : Dirichlet BCs at column inlet

In above equation γ is absolute constant that is greater than real component of all singularities of c¯i(x, s). Analytical Laplace inversions are not possible in this instance.

Case II : Dankwerts BCs at column’s inlet

In normalized form, the BCs are of the form. There is analytical Laplace inversion are not possible. So, In order to acquire the solution in real time domain, we can apply numerical Laplace inversion.

Temporal Moments

Moments analysis is an essential tool for determining crucial details relating retention equilibrium and mass transfer kinetics in a column. The LT used as fundamental tool to calculate the moments. We obtained complete elution profiles by applying inverse Laplace transforms to the equations. To determine the mass transfer coefficient and dispersion coefficient, we can compare the experimental moments to their theoretical values. We use these moments to compare the numerical and analytical moments.

We obtained numerical moments by integrating the profiles produced with HR-FVS. Sometimes analytical impression for µ0i and µ0i are very long so, we only plots these central moments. We can use Eqs. 4.48 and 4.49 to integrate concentration profiles of FVS or analytical solution in order to find zeroth, first, second, and third central moments numerically.

Analytical Moments

Analytical moments are provided for two sets of BCs. For moments calculation we suppose for i = 1, 2,c2,inj = 0, vi = 0, ci,init = 0, i.e. no reaction time in solid phase, only component injected to column and column is empty at start. Whenever vi = 0, then ri = r where i = 1, 2.

Conclusions and Future Work

For rectangular pulses of single components employing Danckwert BC’s, analytical/exact and numerical solution of LKM and EDM were reported with moment analysis. The LT was utilized as an fundamental tool for converting linear PDE to linear ODEs that in the Laplace domain might be solved analytically. Numerical Laplace inversion approaches gave reliable results for resolving the problem in the real-time domain without an analytical Laplace inversion. Under linear conditions, model’s moment analysis up to the fourth order was performed using analytical and numerical approach. The proposed numerical scheme was more accurate, and the analytical results were righter because of the high level of conformity between numerical and analytical results.

The LT was utilizes as fundamental way of obtaining solutions in a Laplace domain. Due to the analytical intractability of the solutions, we generated time-domain concentration profiles using numerical inversion techniques. To understand the behavior of the system, we derived analytical formulations for the first, second, and third temporal moments based on Laplace domain solutions.

Key Insights:

Elution Curve Analysis: We used the moments to investigate the effects of axial dispersion, film mass transfer resistance, intra-particle diffusion resistance, and core radius fraction on the elution curves.
Plate Height Equations: Employing Dirichlet and Danckwerts boundary conditions, we derived plate height equations for both totally core-shell and porous particles using the first two central moments.
Validation: We validated our analytical results using numerical findings from a second-order finite volume scheme (FVS).

Good agreement between analytical and numerically determined findings confirmed the analytical solutions’ and recommended numerical approach’s accuracy.

For the conditions investigated, increasing the core radius proportion of cored particles resulted in a small residence period and sharper elution curve peak. As a result, adding an inert core adsorbent to a column improves separation efficiency and sharpens and narrows the eluting peaks. This is ideal for separating mixtures chromatographically in a short amount of time. The analytical solutions of irreversible liquid chromatographic techniques were the focus of this thesis study.