Sorption and Chromatography

Monica Lamm; Laura Jarboe

7 Sorption and Chromatography

Adsorption, Ion Exchange, and Chromatography

$c_i$ = concentration of species i in the mobile phase (mass volume^-1) or (mole volume^-1)

$k_i$ = empirical constant for species i for isotherms (units vary)

$K_i$ = adsorption equilibrium constant for species i

$n_i$ = internal parameter for isotherms (units vary)

$p_i$ = partial pressure of species i (pressure)

$q_i$ = amount of species i adsorbed per unit mass of adsorbent at equilibrium (mass mass^-1) or (mole mass^-1)

$q_{m_i}$ = amount of species i adsorbed per unit mass of adsorbent at maximum loading, where maximum loading corresponds to complete surface coverage (mass mass^-1) or (mole mass^-1)

linear isotherm:

(31.1) $\begin{equation*} q_i=k_ip_i \end{equation*}$

Freundlich isotherm:

(31.2) $\begin{equation*} q_i=k_ip_i^{1/n_i} \end{equation*}$

Langmuir isotherm:

(31.3) $\begin{equation*} q_i=\frac{K_iq_{m_i}p_i}{1+K_ip_i} \end{equation*}$

chromatography equilibrium:

(31.4) $\begin{equation*} K_i=\frac{q_i}{c_i} \end{equation*}$

Watch a video from LearnChemE for an explanation about the concept of adsorption: Adsorption Introduction (8:49)

Modeling Differential Chromatography

$\alpha_i$ = average partitioning of species i between the bulk fluid and sorbent (unitless)

$\epsilon_b$ = sorbent porosity, ranges from 0 to 1 (unitless)

$\epsilon^*_{p,i}$ = inclusion porosity, accounts for accessibility of sorbent pores to species i (unitless)

$\tau_f$ = sorbent tortuosity factor, usually approximately 1.4 (unitless)

$\omega_i$ = fraction of solute in the mobile phase, relative to sorbed solute, at equilibrium (unitless)

$A$ = cross-sectional area of the column (area)

$c_{f,i}$ = concentration of species i in the mobile phase (mass volume^-1) or (mol volume^-1)

$D_{e,i}$ = effective diffusivity of species i within the sorbent pores (length² time^-1)

$E_i$ = coefficient that accounts for axial diffusion of species i and non-uniformities of flow (length² time^-1)

$H_i$ = height of theoretical chromatographic plate for species i (length)

$k_{a,i}$ = kinetic rate constant of adsorption of species i to the sorbent (time^-1)

$k_{c,i}$ = mass transfer coefficient of species i in the mobile phase (length time^-1)

$k_{c,i,tot}$ = overall mass transfer coefficient of species i (length time^-1)

$K_{d,i}$ = equilibrium distribution coefficient of species i between the mobile phase and sorbent (unitless)

$L$ = length of column (length)

$m_{0_i}$ = amount of solute i fed to column (mass) or (mol)

$R_{1,2}$ = resolution of species 1 and 2 in the proposed operating condition (unitless)

$R_p$ = radius of sorbent particles (length)

$s_i$ = variance of the Gaussian peak of the distribution of species i along the column length (time)

$t$ = elapsed time since loading of the column (time)

${\overline t}_i$ = mean residence time of species i in the column (time)

$u$ = actual fluid velocity through the bed (length time^-1)

$u_s$ = superficial fluid velocity through the bed (length time^-1)

$z$ = position along the length of the column, in the direction of flow (length)

$z_{0,i}$ = mean position of species i along the length of the column as a function of time (length)

(32.1) $\begin{displaymath} z_{0,i}(t)=\omega_iut \end{displaymath}$

(32.2) $\begin{equation*} \omega_i=\frac{1}{1+\frac{1-\epsilon_b}{\epsilon_b\alpha_i}} \end{equation*}$

(32.3) $\begin{equation*} \alpha_i=\frac{1}{\epsilon_{p,i}^*(1+K_{d,i})} \end{equation*}$

(32.4) $\begin{equation*} u=u_s/\epsilon_b \end{equation*}$

(32.5) $\begin{equation*} \overline t_i=\frac{L}{\omega_iu} \end{equation*}$

(32.6) $\begin{equation*} c_{f,i}(z,t)=\frac{m_{0_i}\omega_i}{A\epsilon_b(2\pi H_iz_0)^{0.5}}\textrm {exp}\left(\frac{-(z-z_0)^2}{2H_iz_0}\right) \end{equation*}$

(32.7) $\begin{equation*} H_i=2\left[\frac{E_i}{u}+\frac{\omega_i(1-\omega_i)R_pu}{3\alpha_ik_{ci,tot}}\right] \end{equation*}$

(32.8) $\begin{displaymath} N_{{\rm Pe},i}=N_{\rm Re}N_{{\rm Sc},i}=\frac{2R_pu\epsilon_b}{D_i} \end{displaymath}$

if $N_{{\rm Pe},i}<<1$

(32.9) $\begin{displaymath} E_i=\frac{D_i}{\tau_f} \end{displaymath}$

else

(32.10) $\begin{displaymath} E_i=\frac{2R_pu\epsilon_b}{N_{{\rm Pe},E,i}} \end{displaymath}$

$N_{{\rm Pe},E,i}$ calculated by 15-61 or 15-62, Seader

(32.11) $\begin{equation*} \frac{1}{k_{ci,tot}}=\frac{1}{k_{c,i}}+\frac{R_p}{5\epsilon_{p,i}^*D_{e,i}}+\frac{3}{R_pk_{a,i}\epsilon_{p,i}^*}\left[\frac{K_{d,i}}{1+K_{d,i}}\right]^2 \end{equation*}$

(32.12) $\begin{equation*} s_i^2=\frac{\overline t_iH_i}{\omega_iu} \end{equation*}$

(32.13) $\begin{equation*} R_{1,2}=\frac{\textrm {abs}(\overline t_1-\overline t_2)}{2(s_1+s_2)} \end{equation*}$

Example

1.0 g of species A is added to a chromatography column of cross-sectional area 1.0 m² and length 1.0 m. Mobile phase is added at a flowrate of $4.0 \times 10^{-3}$ m³/s. Species A has a mass transfer coefficient of $2.0 \times 10^{-5}$ m/s in this solvent. The selected sorbent has a porosity of 0.40 m and average particle radius of $5.0\times 10^{-6}$ m. For species A in this sorbent, the inclusion porosity is 0.80, $K_d = 50$ , $E = 2.0\times 10^{-8}$ m²/s, $k_a = 100$ s^-1 and the effective diffusivity is $3.5\times 10^{-12}$ m²/s.

(a) When is mean expected elution time for species A?

(b) Plot the concentration profile for species A at 0.05 m increments along the column length in 10-minute increments, until all of the solute has eluted.

(c) Find the variance of the peak for species A in the proposed operating condition.

(d) The column feed also contains 1.0 g of species B. Species B has a mass transfer coefficient of $1.0\times 10^{-5}$ m/s in the mobile phase, inclusion porosity of 0.50, $K_d = 60$ , $E = 3.0\times 10^{-8}$ m²/s, effective diffusivity of $4\times 10^{-12}$ m²/s and $k_a = 200$ s^-1. What is the resolution of these two species in the proposed operating condition?

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Chemical Engineering Separations: A Handbook for Students Copyright © 2021 by Monica H. Lamm and Laura R. Jarboe is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.