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 by Monica H. Lamm and Laura R. Jarboe is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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