7

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:

\begin{equation}

\tag{31.1}

q_i=k_ip_i

\end{equation}

Freundlich isotherm:

\begin{equation}

\tag{31.2}

q_i=k_ip_i^{1/n_i}

\end{equation}

Langmuir isotherm:

\begin{equation}

\tag{31.3}

q_i=\frac{K_iq_{m_i}p_i}{1+K_ip_i}

\end{equation}

chromatography equilibrium:

\begin{equation}

\tag{31.4}

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)

\begin{displaymath}

\tag{32.1}

z_{0,i}(t)=\omega_iut

\end{displaymath}

\begin{equation}

\tag{32.2}

\omega_i=\frac{1}{1+\frac{1-\epsilon_b}{\epsilon_b\alpha_i}}

\end{equation}

\begin{equation}

\tag{32.3}

\alpha_i=\frac{1}{\epsilon_{p,i}^*(1+K_{d,i})}

\end{equation}

\begin{equation}

\tag{32.4}

u=u_s/\epsilon_b

\end{equation}

\begin{equation}

\tag{32.5}

\overline t_i=\frac{L}{\omega_iu}

\end{equation}

\begin{equation}

\tag{32.6}

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}

\begin{equation}

\tag{32.7}

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

\end{equation}

\begin{displaymath}

\tag{32.8}

N_{{\rm Pe},i}=N_{\rm Re}N_{{\rm Sc},i}=\frac{2R_pu\epsilon_b}{D_i}

\end{displaymath}

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

\begin{displaymath}

\tag{32.9}

E_i=\frac{D_i}{\tau_f}

\end{displaymath}

else

\begin{displaymath}

\tag{32.10}

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

\begin{equation}

\tag{32.11}

\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}

\begin{equation}

\tag{32.12}

s_i^2=\frac{\overline t_iH_i}{\omega_iu}

\end{equation}

\begin{equation}

\tag{32.13}

R_{1,2}=\frac{\textrm {abs}(\overline t_1-\overline t_2)}{2(s_1+s_2)}

\end{equation}