### The equilibrium-dispersive model of chromatography

Lots of theories and models have been proposed for describing liquid chromatography mathematically with various equilibrium and mass transfer parameters. In this work, the equilibrium-dispersive model was used to present the chromatographic process. Therefore, the mass balance equation for each component in the mobile phase can be written as [[9]]

\frac{\partial {\mathit{c}}_{\mathit{i}}}{\partial \mathit{t}}+\mathit{F}\frac{\partial {\mathit{q}}_{\mathit{i}}}{\partial \mathit{t}}+\mathit{v}\frac{\partial {\mathit{c}}_{\mathit{i}}}{\partial \mathit{z}}={\mathit{D}}_{\mathit{L}}\frac{{\partial}^{2}{\mathit{c}}_{\mathit{i}}}{\partial {\mathit{z}}^{2}}

(1)

where *c*_{
i
} and *q*_{
i
} are the concentrations (component *i*) in the mobile phase and the stationary phase, respectively, *F* is the phase ratio and it is related to the total column porosity *ε*_{
t
}_{,}\left(\mathit{F}=\frac{1-{\mathit{\epsilon}}_{\mathit{t}}}{{\mathit{\epsilon}}_{\mathit{t}}}\right), *v* is the velocity of interstitial mobile phase, *D*_{
L
} represents the axial dispersion coefficient, *t* is the time, and *z* is the space coordinates.

For a linear driving force approximation,

\frac{\partial \mathit{q}}{\partial \mathit{t}}=\mathit{k}\left({\mathit{q}}^{*}-\mathit{q}\right)

(2)

where *k* is the mass transfer coefficient, *q** is the equilibrium concentration of stationary phase, and the solution of concentration *c*_{
i
} is given by the linear isotherm of Equation 3.

{\mathit{q}}^{*}={\mathit{K}}_{\mathit{i}}{\mathit{c}}_{\mathit{i}}

(3)

*K*_{
i
} represents the equilibrium constant of component *i*, Equation 2 supposes the driving force of the mass transfer of *q*_{
i
}*** is different with *q*_{
i
} and the mass transfer rate will increase with the driving force increasing. This linear driving force expression supposes that the main resistance in the particle diffusion step can be described by an overall mass transfer coefficient.

### Moment analysis

The method of moment analysis, used to describe the hydrodynamic characteristics, has been considered as the effective means to determine axial dispersion coefficient *D*_{
L
} and mass transfer parameters in pulse experiments [[10]–[14]]. As we all know, the experimental first (μ_{1}) and second (μ_{2}) moments are the average retention time and variance, respectively, which can be written as

{\mathrm{\mu}}_{1}=\frac{\mathit{L}}{\mathit{V}}\left[1+\left(\frac{1-{\mathit{\epsilon}}_{\mathit{T}}}{{\mathit{\epsilon}}_{\mathit{T}}}\right)\mathit{K}\right]

(4)

{\mathrm{\mu}}_{2}=\frac{2\mathit{L}}{\mathit{v}}\left\{\frac{{\mathit{D}}_{\mathit{l}}}{{\mathit{v}}^{2}}{\left[1+\left(\frac{1-{\mathit{\epsilon}}_{\mathit{T}}}{{\mathit{\epsilon}}_{\mathit{T}}}\right)\mathit{K}\right]}^{2}+\left(\frac{1-{\mathit{\epsilon}}_{\mathit{T}}}{{\mathit{\epsilon}}_{\mathit{T}}}\right)\frac{\mathit{K}}{{\mathit{k}}_{\mathit{m}}}\right\}

(5)

The first and second moments can deduce the height equivalent to a theoretical plate (HETP).

\mathrm{HETP}=\frac{\mathit{L}}{\mathit{N}}=\frac{{\mathrm{\mu}}_{2}\mathit{L}}{{{\mathrm{\mu}}_{1}}^{2}}=\frac{2{\mathit{D}}_{\mathit{L}}}{\mathit{v}}+2\mathit{v}\left(\frac{{\mathit{\epsilon}}_{\mathit{T}}}{1-{\mathit{\epsilon}}_{\mathit{T}}}\right)\frac{1}{\mathit{K}{\mathit{k}}_{\mathit{m}}}{\left[1+\left(\frac{{\mathit{\epsilon}}_{\mathit{T}}}{1-{\mathit{\epsilon}}_{\mathit{T}}}\right)\frac{1}{\mathit{K}}\right]}^{-2}

(6)

\mathit{N}=5.545{\left(\frac{{\mathit{t}}_{\mathit{R}}}{{\mathit{w}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)}^{2}

(7)

where *L* is the column length, *N* is the theoretical plate number, *w*_{
1/2
} is the peak width at half height, *k*_{
m
} is the overall mass transfer coefficient, and *K* is the equilibrium constant. Equation 6 implies that axial dispersion cannot be calculated by a simple linear regression because of its nonlinear relationship with HETP.

It is assumed that axial dispersion is influenced by two different mechanisms, among which one is molecular diffusion in the axial direction and the other is an eddy mixing term proportional to the fluid velocity. Therefore, the expression for the axial dispersion coefficient *D*_{
L
} is

{\mathit{D}}_{\mathit{L}}=\mathit{\eta}{\mathit{D}}_{\mathit{m}}+\mathit{\lambda v}

(8)

where *D*_{
m
} is the molecular diffusivity, *v* is the interstitial velocity, *η* is the tortuosity factor for a packed column, and *λ* is a constant dependent on flow geometry. In the liquid system, the molecular diffusivities are so small that Equation 8 can be simplified as [[15]]

{\mathit{D}}_{\mathit{L}}=\mathit{\lambda \nu}\phantom{\rule{0.25em}{0ex}}

(9)

Substituting Equation 9 into Equation 6,

\mathrm{HETP}=2\mathit{\lambda}+2\mathit{v}\left(\frac{{\mathit{\epsilon}}_{\mathit{T}}}{1-{\mathit{\epsilon}}_{\mathit{T}}}\right)\frac{1}{\mathit{K}{\mathit{k}}_{\mathit{m}}}{\left[1+\left(\frac{{\mathit{\epsilon}}_{\mathit{T}}}{1-{\mathit{\epsilon}}_{\mathit{T}}}\right)\frac{1}{\mathit{K}}\right]}^{-2}

(10)

In Equations 6 and 10, the overall mass transfer resistance (1*/k*_{
m
}) in theory contains three separate mass transfer mechanisms (the external film resistance, intraparticle diffusion resistance, and adsorption/desorption resistance). It is commonly assumed that the kinetics of adsorption/desorption are rapid. If we ignore the influence of adsorption/desorption resistance to the overall mass transfer resistance, the following relation between the film mass transfer and the lumped mass transfer coefficient and intraparticle diffusion coefficients can be represented by [[16]]

\frac{1}{{\mathit{k}}_{\mathit{m}}}=\frac{{\mathit{d}}_{\mathit{p}}}{6{\mathit{k}}_{\mathit{f}}}+\frac{{{\mathit{d}}_{\mathit{p}}}^{2}}{60{\mathit{D}}_{\mathit{p}}}

(11)

where *d*_{
p
} is particle diameter and *k*_{
f
} and *D*_{
p
} are the external film mass transfer coefficient and the pore diffusion coefficient, respectively. To us all, a function of liquid velocity can represent the external film mass transfer coefficient. When the liquid systems satisfy the condition of 0.0015 < Re < 55, *k*_{
f
} could be acquired by the Wilson and Geankoplis correlation [[17]].

\mathrm{Sh}=\frac{{\mathit{k}}_{\mathit{f}}{\mathit{d}}_{\mathit{p}}}{{\mathit{D}}_{\mathit{m}}}=\frac{1.09}{{\mathit{\epsilon}}_{\mathit{b}}}{\mathrm{Sc}}^{\frac{1}{3}}{\mathrm{Re}}^{\frac{1}{3}}\phantom{\rule{0.75em}{0ex}}

(12)

In general, *k*_{
f
} is larger than *D*_{
p
}, at least a few orders of magnitude, which had been confirmed by some researchers [[18]] at similar chromatographic systems. Thus, the influence of external film resistance can be negligible and *k*_{
m
} will not depend on liquid velocity in this study. It is clear that Equation 11 can be simplified as

\frac{1}{{\mathit{k}}_{\mathit{m}}}=\frac{{{\mathit{d}}_{\mathit{p}}}^{2}}{60{\mathit{D}}_{\mathit{p}}}

(13)

From Equation 13, *k*_{
m
} can be considered as a constant which is independent of liquid velocity. Therefore, the overall mass transfer coefficient *k*_{
m
} can be determined by the first and second moments.