Analysis of a Chromatographic Model with Irreversible and Reversible Reactions

### The Phase Ratio

This is a fundamental property of a chromatographic model, identified as the ratio between mobile phase’s volume section to the volume section of column’s stationary phase.

### Column Efficiency

It informs us about how well the separation is progressing. The plate hypothesis is used to assess section proficiency

(1)

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

### Adsorption Isotherm

This technique produces an adsorbate film on surface of the adsorbent. Adsorption is the process of particles from a liquid or gas adhering to a strong solid surface. Isotherms are sometimes used to explain the adsorption process. The adsorption isotherm describes the equilibrium connection between molecules adsorbed onto the surface of stationary phase at particular temperature and molecules of solute in mobile phase. The adsorption isotherm has a significant impact on the chromatogram.

### Linear Chromatography

Linear chromatography is a separation technique that works at low concentrations, has a small sample size, and the volume absorbed at equilibrium during the stationary phase is directly proportional to the solute concentration in mobile phase. In other way, equilibrium isotherms are straight lines that start at the origin. Linear chromatography is quite useful when the sample contains a small amount of injected components. The band thickness is dependent on energy of mass exchange and axial dispersion of the column, but not on equilibrium state. The peak width is reduced, the peak height is increased, and the amount of separation between each peak is changed using linear chromatography.

(2)

Here for absolute number of columns in mixture, denotes concentration, represents the adsorption of ith term in moving phase, and defines Henry’s constant of the ith term.

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

### Nonlinear chromatography

A component’s concentration in stationary phase differs from it’s concentration in mobile phase. Concentration of all other compounds in the solution affects the equilibrium

isotherms of each chemical. The equilibrium isotherms are not linear. The connection of individual band profiles begun by the dependency of the amount of column absorbed to the groupings of all species in arrangements makes nonlinear chromatography problems extremely complicated. To simplify the nonlinear chromatography analysis, we must dis criminate between nonideal and ideal models. There are various models to represent non equilibrium isotherms such as Flowler, Langmuir, Freudlich and Bilangmuir models, the Langmuir isotherm model is the simplest and most practical. Langmuir isotherm model is:

(3)

Where denotes the nonlinearity coefficient.

### Models of Chromatography

There are many chromatographic models in writing, While describing the chromatographic technique, take in mind that there are several levels of difficulty. The models are mostly use for the simulation of the chromatography technique are, GRM, EDM, and LKM.

### Analysis of a Chromatographic Model with Irreversible (Part 4)

To derive these mathematical models the following assumptions are used:

I. The chromatographic process must be isothermal.

II. Band broadening is caused by axial dispersion.

III. In the column, radial concentration gradients can be ignored.

IV. In liquid chromatography, the mobile phase is assumed to be incompressible.

V. There is no connection between the mobile and solid phases.

VI. The bed is uniform, and the stationary phase packaging material consists of porous spherical particles of equal size.

### The Equilibrium Dispersive Model

The EDM deals with mass transfer kinetics and axial dispersion. In liquid chromatography, the dynamics of band profiles are studied using EDM. All contributions of mass the transfer processes to spreading of the profile are lumped into axial dispersion coefficient Dα. In a simple words, model assumes that adsorption desorption kinetic and kinetic mass transfer is extremely fast.

Here represents the equilibrium concentration of j-th component in liquid phase and *Dα* is dispersion coefficient.

### The General Rate Model

The general rate model is regarded as most authentic and reliable chromatographic model. Mass balance bed filled with columns specifies particle transport via the column, the outer film and axial dispersion in the GRM. There are two mass balances in the GRM for solute, one for the internal particles and one for the outside particles.

Here cp,j , cj for the stagnant mobile phase and bulk of fluid concentrations of j-th component respectively, F is for phase ratio, u is inertial velocity, kext is the coefficient of mass transfer and Dj is for coefficient of axial dispersive.

### Lumped Kinetic Model

The LKM takes into account the rate of change in local solute concentrations in solid phase, as well as local deviations from equilibrium concentrations. The LKM represents variation rate of solute concentration in adsorbents at the midpoint by accepting a straight main thrust starting from the deviation from equilibrium focus and presenting a mass exchange coefficient km. This coefficient combines the two commitments associated with internal and external mass transport resistances.

The kinetic equation of the structure gives the relating mass equilibrium in the solid phase

as;

km,j (φ)(q∗(φ(t, z)) − qj ) = ∂qj . (2.17)

In above equation q∗ represents fixation in adsorbent at equilibrium,

ci stand for bulk concentration, t is for time, qj for the imbalance adsorbent concentration, z for distance along column. Dispersion coefficient along the axial coordinate represented by Dz,j and km,j stand for coefficient of mass-movement and Nc denotes the sample’s total number of mixture components.

kH,i(φ) = kHri e, (2.18)

kH,r is explicit Henry constant, α is parameter of dissolvable strength. We need more suitable boundary and initial conditions for model formulation.

### The Initial conditions (ICs) and Boundary conditions (BCs)

First we defined the initaial conditions (ICs) as

i (o, z) = 0, ci(0, z) = 0. (2.19) Furthermore, to solve model equations, appropriate inflow and outflow BCs are necessary.

### Robin or Danckwerts BCs

On the column inlet, Dankwerts boundary conditions are used.

(ci,inj , if 0 < t ≤ tinj , z=0, t > t (2.20)

### Neumann BCs

Neumann conditions on outlet flow of column length L, where τinj is injection time.

Dirichlet BCs

∂ci (t, L) = 0,

At the column inlet, the simplified Dirichlet BCs could be used instead.

(ci,inj , if 0 < t ≤ tinj , t > tinj . (2.21)

Here t total porosity and V • column flow rate are set to be constant.

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

### Ideal Model

Wicke, Wilson introduced this simple model first. In this model, we suppose that the column’s performance is limitless, implies that there is no axial dispersion and the mass transfer kinetics rate is unlimited. If we assume that the two phases are in constant and instantaneous equilibrium, and that there is zero axial dispersion i.e Dz,i = 0, then the mass balance equation 2.16 written as,

∂cj + u∂cj + (1 − ρ3) (2.22)

The effect of axial dispersion and mass transfer kinetics on band profiles is completely dismisses in this model. With the basic supposition of ideal model, The practical chromatograms acquired for large samples, with very efficient columns, accord well with the band profiles produced as the solution of this model.

### Temporal Moments

The moment of a set of points is a quantitative measure of its shape. The analysis of the moment is a useful way to get a general idea of how liquid chromatography elution works in a column. The kinetics of mass transfer and column retention equilibrium affect chromatographic behaviour such as elution peak profiles, column efficiency, sample retention, and other factors.

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

To characterise the density curves the zeroth moment, first moment, second moment, and third moment. Higher moments are generally less important for describing probability curves. The zeroth moment gives us the prediction of the total mass. These moments can be calculate from following formula, For zeroth moment

µ0 = lim(C(x = 1, s)). (2.24)

and

(C(x = 1, s)), n = 1, 2, …. (2.25)

The n-th moment at the end of the chromatographic bed of length x = 1 to derive numerical moments.

Mn = tnC(t, x = 1)dt, (2.26)

and the n-th central moment expressed as,

R = 0 (t − µ1) C(x = 1, t)dt. (2.27)

The first moment explain mean retention time and second moment for showing variation in concentration profiles. We can use the third moment for estimating skewness.

### Retention time

The retention time refers to how long it takes a component to depart a column. In other terms, it’s the amount of time the component spends inside the chromatographic column. Here in this thesis work first moment µ1 gives us the retention time.

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

### Mean

In the fields of statistical and probability theory, the term mean, also known as expected value, relates to a measure of probability distribution’s central tendency.

### Skewness

Extent of symmetry in distribution is skewness. In statistics and probability theory. The deviation in the dispersion around its mean is called skewness. It might have a positive or negative value. When the distribution is symmetrical around the mean, skewness is zero but in actuality, data points may not follow a perfectly symmetrical pattern.

### Varience

Measure of dispersion of a set of number is known as variance. If the value of variance is zero it’s means that there is no variation. Variance provides useful data on mass transfer in chromatographic columns.