Batch chromatography, binary separation, linear isotherms
We will use a simple batch separation with linear isotherms (and ideal conditions) to give a practical demonstration of the Unified Design method.
A rectangular fresh feed profile is introduced to the column. It will propagate at constant velocity without changing its shape. To construct the chromatogram, we need to know when the concentration steps at the front and the rear of each component profile elute from the column. The propagation velocities depend on the adsorption affinities, i.e. the slope \(H_j\) of the linear isotherm. The front and rear of component \(j\) elute at
Complete separation is achieved when the front of component 2 elutes later than the rear of component 1. We also need to perform the fractionation cut at time \(t_1^{rear} \leq t_{cut} \leq t_2^{front} \).
The corresponding dimensionless Unified Design operating parameters for cut events occurring at a these elution times are obtained from the definition
\[ s_{cut} = \left( \frac{t_{cut}-\Delta t^{FF}}{t_0}-1 \right) F^{-1} \]which gives
It is now clear that complete separation is achieved only if
\[ s_{cut,1}^{rear} \leq s_{cut,2}^{front} \Longrightarrow s^{FF} \leq H_2 - H_1 \]and 100% pure products are collected only if
\[ H_1 \leq s_{cut} \leq H_2 -s^{FF} \]Batch chromatography with a single cut event is thus described with two UD operating parameters, \(s_{cut}\) and \(s^{FF}\). They can be used as coordinate axes to visualise the feasible operating region as shown in the interactive charts below for isotherm parameter values \(H_1 = 1.5\) and \(H_2 = 2.5\). Any operating point \(\left( s_{cut}, s^{FF} \right) \) that conforms to the limits \(1.5 \leq s_{cut} \leq 2.5\) and \(s^{FF} \leq 1.0\) is found inside the feasible region, and will yield complete separation and 100% purity for components 1 and 2.
An alternative representation of the feasible region is obtained by using the auxiliary parameter, \(S\). Remembering that \(S_{cut} = s_{cut} + s^{FF}\), we can rewrite the limit for complete separation as
\[ S_{cut} \leq s_{cut} + \left(H_2 - H_1\right) \]
The cut event gives 100% purity in both product fractions when
\[ s_{cut} \geq H_1 \]
\[ S_{cut} \leq H_2 \]
The feasible region of the operating parameters is shown in the interactive charts. Any operating point \(s_{cut}, S_{cut}\) inside the green triangle yields complete separation and 100% purities with a single cut event.
Simulated moving-bed chromatography, binary separation, linear isotherms
Similar to batch chromatograpy, also the classical SMB chromatography is based on timed events. In the UD method, an SMB event includes two parts: flow of liquid for a given time (usually dubbed switching time, \( t_\text{switch} \) ), followed by moving the column against the direction of the fluid flow.
The distance compound \( j \) moves in a column in zone \( k \) during the SMB event (including the effect of moving the column) is calculated as \[ l_{j,k} = \frac{s_{\text{SMB},k} - H_j}{1+FH_j}F \]
The strongly adsorbed compound 2 will not reach the raffinate outlet in zone III when \( l_{2,\text{III}} \leq 0 \). The weakly adsorbed compound 1 stays away from the extract outlet in zone II when \( l_{1,\text{II}} \geq 0 \).
We will now drop the subscript "SMB" because the roman numerals II and III indicate a zone in the SMB process.
The maximum feasible value of \( s_\text{III} \) and the minimum feasible value of \( s_\text{II} \) are thus \[ s_\text{III} \leq H_2 \] \[ s_\text{II} \geq H_1 \]
The fresh feed into the SMB process is introduced between zones II and III. This means that \( s_\text{III} = s_\text{II} + s^\text{FF} \). Choosing \(s_\text{II} \) and \( s^{FF} \) as the operating parameters, we have get 100% purity in extract and raffinate in linear SMB under ideal conditions when \[ H_1 \leq s_\text{II} \leq H_2 - s^\text{FF} \]
It is readily observed that the feasible region of UD operating parameters in SMB is identical with the feasible reagion in batch chromatography. Moreover, when the batch and SMB processes are operated with same UD operating parameter values, both processes yield the same product purities under ideal conditions. This equivalence is not limited to linear isotherms but holds also in case of binary Langmuir isotherm.
An alternative representation of the operating point is obtained by using the auxiliary parameter \( S \). For the SMB, it is defined as \( S = s + \frac{1}{N_\text{SMB}}s^\text{FF} \), where \( N_\text{SMB} \) is the number of SMB events during the repeating sequence. For the classical SMB process, \( N_\text{SMB} \) = 1 because there are no "substeps" in the timing scheme, and the auxiliary operating parameter for zone II becomes \( S_\text{II} = s_\text{II} + s^\text{FF} \). The boundaries of the operating parameters are \[ s_\text{II} \geq H_1 \] \[ S_\text{II} \leq H_2 \]
Considering that \( S_\text{II} = s_\text{III} \) it is observed the feasible region on the Unified Design \( (s_\text{II}, S_\text{II}) \) plane is identical to that of the Triangle Theory on the \( (m_\text{II}, m_\text{III}) \) plane. This is because the UD operating parameters for the SMB were specifically chosen to be compatible with the Triangle Theory. As a consequence, all results obtained using the Triangle Theory are valid also in the frame of the Unified Design method.
Drag-and-drop the gray FB operating point on the UD operating parameter plane to see how the chromatograms and the cut event change.
Drag-and-drop the cut event operating point on the chromatogram to see how the FB operating point changes.
| Operating point | \( s_\text{cut} \) = | \( s^\text{FF} \) = | \( S_\text{cut} \) = |
| Purity | \( PU_1 \) = | \( PU_2 \) = | |
| Yield | \( Y_1 \) = | \( Y_2 \) = | |
| Productivity | \( PR_1 \) = | \( PR_2 \) = | |
| Eluent consumption | \( EC_1 \) = | \( EC_2 \) = |
Further reading:
- 1. Sainio, T., Unified Design of Chromatographic Separation Processes: Partial Separations and Waste Fractions, Chem. Eng. Sci. (2021), 116457. (open access)
- 2. Sainio, T., Unified design of chromatographic processes with timed events: ternary separations, Chem. Eng. Sci., 152(2016), 547−567.
- 3. Siitonen, J., Sainio, T., Unified design of chromatographic separation processes, Chem. Eng. Sci., 122(2015), 436-451.
- 4. Siitonen, J., Sainio, T., Explicit equations for the height and position of the first component shock for binary mixtures with competitive Langmuir isotherms under ideal conditions, J. Chromatogr. A, 1218(2011), 6379-6387
- 5. Siitonen, J., Sainio, T., Rajendran, A., Design of batch chromatography for separation of binary mixtures under reduced purity requirements, J. Chromatogr. A, 1286(2013), 55-68
- 6. Sainio, T., Kaspereit, M., Analysis of steady state recycling chromatography using equilibrium theory, Sep. Purif. Technol., 66(2009), pp. 9-18
- 7. Kaspereit, M., Sainio, T., Simplified design of steady-state recycling chromatography under ideal and nonideal conditions, Chem. Eng. Sci., 66 (2011), pp. 5428-5438
- 8. G. Storti, M. Mazzotti, M. Morbidelli, S. Carrà, Robust design of binary counter current adsorption separation processes, AIChE J., 39 (1993), pp. 471-492
