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Column Models

Column models are mathematical frameworks that describe the transport and separation of solutes in chromatography columns. These models vary in complexity, from detailed mechanistic descriptions to simplified representations that balance accuracy with computational efficiency.

Overview

In liquid chromatography, solutes are transported through packed columns where they interact with stationary phase particles. The separation process involves multiple mass transfer mechanisms operating at different length scales:

  • Convection through the interstitial (bulk) fluid between particles
  • Film diffusion across the boundary layer surrounding particles
  • Pore diffusion within the porous particle structure
  • Adsorption/desorption at the solid-liquid interface

Different column models capture these phenomena with varying levels of detail, leading to trade-offs between physical accuracy and computational cost.

Model Comparison

ModelMass Transfer ZonesKey AssumptionsComputational Cost
Lumped Rate Model (LRM)Bulk liquid + bound (lumped)No explicit mass transfer resistance; binding acts directly on liquid phaseLow (1D PDE, simpler)
Lumped Rate Model with Pores (LRMP)2 zones: bulk, particle (lumped)Film and pore resistances combined via effective diffusionMedium (1D PDE system)
General Rate Model (GRM)3 zones: bulk, film, poreRadial symmetry in particles, local equilibrium at solid surfaceHigh (2D PDE system)
When deciding which column model to use, start with the simplest option that fits your specific use case. If the selected column model is unable to predict the observed behavior, then switch to a more complex model.

Shared Mathematical Framework

All three models share a common axial transport (convection) structure based on mass conservation. The interstitial velocity is:

u=QA⋅εcu = \frac{Q}{A \cdot \varepsilon_c}

where QQ is the volumetric flow rate, A=π(d/2)2A = \pi (d/2)^2 is the cross-sectional area, and εc\varepsilon_c is the interstitial (column) porosity. The models differ in how they describe what happens inside and around the particles.

Axial Discretization

The column is divided into NN cells of uniform width Δz=H/N\Delta z = H / N, where HH is the column height. Using the interstitial velocity above, transport in each cell is discretized as follows.

Interior Cells

For cells j=1…N−2j = 1 \ldots N-2, axial transport is discretized using central differences for dispersion and first-order upwind for convection:

dcl,jdt∣transport=DiΔz2(cl,j+1−2cl,j+cl,j−1)⏟central difference dispersion−uΔz(cl,j−cl,j−1)⏟upwind convection\frac{dc_{l,j}}{dt}\bigg|_{\text{transport}} = \underbrace{\frac{D_i}{\Delta z^2}(c_{l,j+1} - 2c_{l,j} + c_{l,j-1})}_{\text{central difference dispersion}} \underbrace{- \frac{u}{\Delta z}(c_{l,j} - c_{l,j-1})}_{\text{upwind convection}}

Dispersion is component-specific (DiD_i per component). Convection uses first-order upwind, taking the concentration from the upstream cell.

Outlet Cell

At the outlet (j=N−1j = N-1), only convection with a half-cell width is applied. An implicit Neumann boundary condition ∂c/∂z=0\partial c / \partial z = 0 eliminates the dispersion term:

dcl,N−1dt∣transport=−u0.5 Δz(cl,N−1−cl,N−2)\frac{dc_{l,N-1}}{dt}\bigg|_{\text{transport}} = -\frac{u}{0.5 \, \Delta z}(c_{l,N-1} - c_{l,N-2})

Inlet Cell — Danckwerts Boundary Condition

The inlet cell (j=0j = 0) is treated as an algebraic constraint (not a differential equation). The Danckwerts boundary condition balances convective and dispersive fluxes at the column entrance:

cl,0 (u Δz+2Di)=cin,i u Δz+2Di cl,1c_{l,0} \, (u \, \Delta z + 2D_i) = c_{\text{in},i} \, u \, \Delta z + 2D_i \, c_{l,1}

This ensures consistency between the inlet profile and the first interior cell.

Lumped Rate Model (LRM)

The simplest column model. Binding acts directly on the liquid phase — there is no separate pore phase.

∂cl∂t+εcεtu∂cl∂z=εcεtD∂2cl∂z2−1−εtεtdqdt\frac{\partial c_l}{\partial t} + \frac{\varepsilon_c}{\varepsilon_t} u \frac{\partial c_l}{\partial z} = \frac{\varepsilon_c}{\varepsilon_t} D \frac{\partial^2 c_l}{\partial z^2} - \frac{1 - \varepsilon_t}{\varepsilon_t} \frac{dq}{dt}

where:

  • cl(z,t)c_l(z,t) — liquid phase concentration
  • uu — interstitial velocity
  • DD — axial dispersion coefficient
  • εc\varepsilon_c — interstitial (column) porosity
  • εt\varepsilon_t — total porosity
  • dq/dtdq/dt — adsorption rate

Lumped Rate Model with Pores (LRMP)

Adds a pore phase inside the particle. Film diffusion couples the liquid and pore phases, and binding acts on the pore phase rather than the liquid phase.

Liquid Phase

∂cl∂t+u∂cl∂z=D∂2cl∂z2−3(1−εc)kfεc rp(cl−cp)\frac{\partial c_l}{\partial t} + u \frac{\partial c_l}{\partial z} = D \frac{\partial^2 c_l}{\partial z^2} - \frac{3(1 - \varepsilon_c) k_f}{\varepsilon_c \, r_p} (c_l - c_p)

Pore Phase

∂cp∂t=3kfεp rp(cl−cp)−1−εpεpdqdt\frac{\partial c_p}{\partial t} = \frac{3 k_f}{\varepsilon_p \, r_p} (c_l - c_p) - \frac{1 - \varepsilon_p}{\varepsilon_p} \frac{dq}{dt}

where:

  • cp(z,t)c_p(z,t) — pore liquid concentration
  • kfk_f — film diffusion coefficient (per component)
  • rpr_p — particle radius
  • εp=(εt−εc)/(1−εc)\varepsilon_p = (\varepsilon_t - \varepsilon_c) / (1 - \varepsilon_c) — particle porosity
  • dq/dtdq/dt — adsorption rate

General Rate Model (GRM)

The most detailed column model. Adds radial discretization inside the particle with both pore diffusion and surface diffusion.

Liquid Phase

Same form as LRMP, but the film boundary condition couples to the outermost radial shell:

∂cl∂t+u∂cl∂z=D∂2cl∂z2−3(1−εc)kfεc rp(cl−cp∣r=rp)\frac{\partial c_l}{\partial t} + u \frac{\partial c_l}{\partial z} = D \frac{\partial^2 c_l}{\partial z^2} - \frac{3(1 - \varepsilon_c) k_f}{\varepsilon_c \, r_p} \left(c_l - c_p\big|_{r=r_p}\right)

Pore Phase (Radial)

εp∂cp∂t+(1−εp)∂q∂t=εpDp1r2∂∂r ⁣(r2∂cp∂r)+(1−εp)Ds1r2∂∂r ⁣(r2∂q∂r)\varepsilon_p \frac{\partial c_p}{\partial t} + (1 - \varepsilon_p) \frac{\partial q}{\partial t} = \varepsilon_p D_p \frac{1}{r^2} \frac{\partial}{\partial r}\!\left(r^2 \frac{\partial c_p}{\partial r}\right) + (1 - \varepsilon_p) D_s \frac{1}{r^2} \frac{\partial}{\partial r}\!\left(r^2 \frac{\partial q}{\partial r}\right)

where:

  • cp(z,r,t)c_p(z,r,t) — pore liquid concentration (function of axial and radial position)
  • q(z,r,t)q(z,r,t) — solid phase (bound) concentration
  • DpD_p — pore diffusion coefficient
  • DsD_s — surface diffusion coefficient
  • rr — radial coordinate within the particle

Boundary Conditions

Particle center (symmetry — zero flux):

∂cp∂r∣r=0=0\left.\frac{\partial c_p}{\partial r}\right|_{r=0} = 0

Particle surface (flux continuity — film diffusion, no surface diffusion flux across boundary):

εpDp∂cp∂r∣r=Rp=kf(cl−cp∣r=Rp)\left.\varepsilon_p D_p\frac{\partial c_p}{\partial r}\right|_{r=R_p} = k_f(c_l - c_p|_{r=R_p})
Ds∂q∂r∣r=Rp=0\left.D_s\frac{\partial q}{\partial r}\right|_{r=R_p} = 0

Further Reading

  • Guiochon, G., Felinger, A., Shirazi, D. G., & Katti, A. M. (2006). Fundamentals of Preparative and Nonlinear Chromatography (2nd ed.). Academic Press.
  • Ruthven, D. M. (1984). Principles of Adsorption and Adsorption Processes. Wiley-Interscience.
  • von Lieres, E., & Andersson, J. (2010). A fast and accurate solver for the general rate model of column liquid chromatography. Computers & Chemical Engineering, 34(8), 1180-1191.
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