What is Triangle Theory?
Triangle Theory is a graphical short-cut design methodology for Simulated Moving Bed (SMB) chromatography, developed by Mazzotti, Storti, and Morbidelli in the 1990s. It provides explicit criteria for choosing optimal operating conditions to achieve complete separation of binary mixtures.
The theory is based on equilibrium theory applied to a True Moving Bed (TMB) model, which is then mapped to SMB operations. It defines a triangular region in the m₂-m₃ operating parameter plane where complete separation is achievable.
The Separation Region
Flow Rate Ratios (m values)
The key design parameters are the flow rate ratios in each zone:
mj = (Qj · ts - V · εe) / (V · (1 - εe))
where Qj is the liquid flow rate in zone j, ts is the switch time, V is the column volume, and εe is the external porosity.
Complete Separation Criteria
- ▸Zone 1 (m₁): Must be above HA (Henry constant of A) to regenerate the solid phase
- ▸Zone 4 (m₄): Must be below HB (Henry constant of B) to regenerate the liquid phase
- ▸Zones 2 & 3: Operating point (m₂, m₃) must be within the triangular separation region
Supported Isotherm Models
Linear Isotherm
The simplest model where adsorption is directly proportional to concentration. The separation region forms a perfect triangle.
Langmuir Isotherm
Accounts for saturation effects and competitive adsorption. The separation region has curved boundaries determined by feed concentration.
Modified Langmuir Isotherm
Combines linear and Langmuir terms to model systems with both non-selective and selective adsorption sites.
Key Parameters
| Parameter | Symbol | Description |
|---|---|---|
| Flow Rate Ratio | mj | Dimensionless ratio of net liquid to solid flow in zone j |
| Henry Constant | Hi | Slope of isotherm at infinite dilution for component i |
| Switch Time | ts | Time interval between port switching in SMB |
| External Porosity | εe | Void fraction between particles (interparticle) |
| Internal Porosity | εi | Void fraction within particles (intraparticle) |
| Saturation Capacity | Ni | Maximum adsorption capacity (Langmuir) |
| Equilibrium Constant | Ki | Adsorption equilibrium constant (Langmuir) |
Key References
Foundational Papers
Mazzotti, M., Storti, G., Morbidelli, M. (1997)
"Optimal operation of simulated moving bed units for nonlinear chromatographic separations"
Journal of Chromatography A, 769, 3-24
Storti, G., Mazzotti, M., Morbidelli, M., Carrà, S. (1993)
"Robust design of binary countercurrent adsorption separation processes"
AIChE Journal, 39, 471-492
Mazzotti, M., Storti, G., Morbidelli, M. (1994)
"Robust design of countercurrent adsorption separation: 2. Multicomponent systems"
AIChE Journal, 40, 1825-1842
Review Articles
Rajendran, A., Paredes, G., Mazzotti, M. (2009)
"Simulated moving bed chromatography for the separation of enantiomers"
Journal of Chromatography A, 1216, 709-738
PubMed →Seidel-Morgenstern, A., Keßler, L.C., Kaspereit, M. (2008)
"New developments in simulated moving bed chromatography"
Chemical Engineering & Technology, 31, 826-837
Educational Resources
ETH Zürich - Separation Processes Laboratory
SMB Design Lecture Notes by Prof. Marco Mazzotti
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Limitations & Considerations
- ⚠Equilibrium Theory: Triangle Theory assumes equilibrium conditions and does not account for mass transfer limitations or axial dispersion.
- ⚠Binary Mixtures: The classic theory is developed for binary separations. Extensions exist for multi-component systems but are more complex.
- ⚠Initial Design: Results should be used as starting points for more detailed simulations or experimental optimization.
- ⚠Safety Margins: In practice, operating points should be chosen with safety margins inside the separation region to account for disturbances.