A unifying feature of most chromatographic separation processes is that they are based on timed events. Examples of such timed events are starting and ending the collection of product fractions in batch chromatography and switching of column positions in SMB chromatography.

Unified Design method (UD) employs dimensionless operating parameters for a small set of such events. Simple and complex process options can be designed by constructing a sequence of UD events.

The approach is called the Unified Design method because it enables comparison of apparently quite different process schemes. The feasible regions of operating parameters are easily visualized. The UD method provides a means to transfer a feasible operating point from one process scheme to another (under ideal conditions).

Events

Fixed-bed events (FB) are operations where liquid flows into, inside, or out of a column for a certain duration of time. The column may be stand-alone or part of a multicolumn unit.

Cut events (cut) are performed outside the separation column to fractionate the effluent into product, waste, or recycle fractions.

Simulated moving-bed events (SMB) are similar to FB events but involve also a shift of the column(s) backwards relative to a reference position in a multicolumn unit.

The classical batch chromatography, for example, consists of two consecutive FB events and one or more cut events. These are feeding the fresh feed to the column, eluting the feed mass through the column, and performing one or more fractionation cuts. Classical SMB consists of consecutive SMB events only, whereas its many variants are combinations of FB events, SMB events and even cut events. In this sense the true moving bed (TMB) is not a good abstraction of practical SMB processes even though it has provided many important results.

Operating parameters \( (s) \)

The building blocks of the Unified Design method are the dimensionless operating parameters. Their key property is that they are scaled and aligned such that they are compatible with each other. The integrated forms we use in practical work are

\[ s_{FB} = \frac{Q\Delta t}{\left( 1-\epsilon \right) V_{col}} = \frac{\Delta t}{t_0}F^{-1} \]

\[ s_{cut} = \frac{Q\left( t_{cut}-\Delta t^{F} \right) -\epsilon V_{col} }{\left( 1-\epsilon \right) V_{col}} = \left( \frac{t_{cut}-\Delta t^{F}}{t_0}-1 \right) F^{-1} \]

\[ s_{SMB} = \frac{Q \Delta t_{switch} -\epsilon V_{col} }{\left( 1-\epsilon \right) V_{col}} = \left( \frac{\Delta t_{switch}}{t_0}-1 \right) F^{-1} \]

As seen in these definitions, the UD operating parameters are related to the volume of liquid pumped per unit volume of the stationary phase. Alternatively, they can be seen as timing of the events in dimensionless units and with the phase ratio included to account for the volume of stationary phase.

We frequently need to quantify the amount of fresh feed introduced to the chromatographic process. This is an FB event, and the corresponding UD parameter is

\[ s^{FF} = \frac{Q\Delta t^{FF}}{\left( 1-\epsilon \right) V_{col}} = \frac{\Delta t^{FF}}{t_0}F^{-1} \]

Note that FB and SMB events have finite durations but a cut event is instantaneous. This means that we cannot convert an FB event to a cut event. The difference between two cut events, on the other hand, defines an FB event. The size of a waste fraction, for example, can thus be presented by an \( s_{FB} \) value. An SMB event followed by an FB event can be interpreted as just one SMB event. Also carrying out an SMB event and an FB event simultaneously in the same column is just a single SMB event. New Unified Design operating parameters can be easily defined by combining several timed events and summing (or subtracting) the corresponding \(s_{FB}\), \(s_{SMB}\), and \(s_{cut}\) parameters.

If the chromatographic process is described by two UD parameters only, their feasible values can be presented on a two-dimensional plane. For example, batch elution chromatography with a single fractionation cut can be presented as an operating point \( \left(s_{cut}, s^{FF} \right) \). A simple practical example illustrates the use of the Unified Design method.

Auxiliary operating parameter \( (S) \)

Triangle Theory is the most well known method to design simulated moving bed chromatographic processes. A vast body of literature describes its background and applications. The Unified Design method was purposefully made compatible with the Triangle Theory. All the previous results are therefore directly applicable in the UD frame as well. This was achieved by 1) scaling and aligning the \(s\)-parameters and 2) by introducing an auxiliary parameter \(S\) for each event. The auxiliary parameters are defined as

\[ S_{FB} = s_{FB} + s^{FF} \]

\[ S_{SMB} = s_{SMB} + \frac{1}{N_{SMB}}s^{FF} \]

where \(N_{SMB}\) is the number of SMB events during a repeating sequence of steps in the process. This parameter is 1 for a classical SMB process but it could be 6, for example, in a so-called Japan Organo process.

When using the auxiliary parameter for compatibility with the Triangle Theory, the chromatographic process is presented by an operating point \( \left(s, S\right) \). The free tool for calculating the feasible regions in batch and SMB for binary separations displays the regions on both \( \left(s, s^{FF} \right) \) and \( \left(s, S \right) \) planes. A simple practical example illustrates the difference between the two alternative presentations.

Scaling and aligning

To mix and combine the three kinds of Unified Design operating parameters, they must be compatible with each other. This is achieved when they scale identically and are aligned such that process behavior is identical at least in certain limiting cases. Under isocratic conditions, the scaling condition is satisfied when

\[ \frac{1}{Q_{FB}}\frac{\mathrm d s_{FB}}{\mathrm d \Delta t} = \frac{1}{Q_{SMB}}\frac{\mathrm d s_{SMB}}{\mathrm d \Delta t_{switch}} = \frac{1}{Q}\frac{\mathrm d s_{cut}}{\mathrm d t_{cut}} \]

The s-parameters obtained by indefinite integration from this relation include unknown integration constants. They are used to properly align the s-parameters. As to the FB and SMB events, we let \( s = 0 \) correspond to zero propagation distance of a non-adsorbing species. This holds for the FB events when the duration of the event \( \Delta t \) is zero, and for the SMB events when the switch time equals the dead time

\[ s_{FB}\left( \Delta t^{F} = 0 \right) = 0 \]

\[ s_{SMB}\left( \Delta t_{switch} = t_0 \right) = 0 \]

We often use elution times relative to the end rather than the beginning of the feed when designing fixed-bed chromatographic processes (Sainio and Kaspereit, 2009; Kaspereit and Sainio, 2011; Siitonen et al., 2013). It is therefore useful to align \( s_{cut} \) such that \( s_{cut} = 0 \) means performing the cut event when the rear of a non-adsorbing compound elutes from the column. This corresponds to the sum of the dead time and the duration of the feed.

\[ s_{cut}\left( \Delta t^{F} + t_0 \right) = 0 \]

Transfer of design

Quite complex chromatographic process schemes can be described by constructing a set of the Unified Design events. In many cases we find that the feasible regions of the UD operating parameters of apparently different process schemes are identical. The most notable example is the equivalence between batch elution chromatography and the classical four-zone SMB process under ideal conditions (Siitonen & Sainio, 2015). In such cases it is - at least in principle - possible to transfer the design from one process scheme to another. If the fresh feed volume \( (s^\text{FF}) \) and cut time \( (s_\text{cut}) \) are found experimentally for the batch process, using the same numerical values for \( s^\text{FF} \) and \( s_\text{II} \) in the SMB gives the same purities if the amount of dispersion is the same. This of course limits the applicability of the transfer to relatively high column efficiencies and makes it approximate rather than exact.

It should be noted also that there may be other constraints, e.g. pressure drop, that make operating different process schemes with the same UD operating parameter values not possible.