## 1 Introduction

Protein purification and separation are major concerns in the downstream processing of pharmaceutical industries (carta2010protein; scopes2013protein). They require a series of processes aimed at isolating single or multiple types of proteins from complex intermediates of process productions. Choosing a purification method, depending on the purpose for which the protein is needed, is a critical issue. Chromatography is a prevailing separation technology. Simulated moving bed (smb) chromatography (broughton1961continuous) is an excellent alternative to the single column batch chromatography, because of its continuous counter-current operation, its potential to enhance productivities and to reduce solvent consumption (seidel2008new; rajendran2009simulated). According to the position of the columns relative to the ports (i.e., feed, raffinate, desorbent, extract), the process is divided into four zones (i.e., I, II, III, IV), each has a specific functionality in the separations (see Fig. 1).

Multicolumn continuous chromatography is also commonly used in the protein separation and purification. For instance, there are sequential multicolumn chromatography (smcc) (ng2014design), periodic counter-current packed bed chromatography (pcc) (pollock2013optimising), gradient with steady state recycle (gssr) process (silva2010new), multicolumn solvent gradient purification (mcsgp) (aumann2007continuous) and capture smb (angarita2015twin).

However, the prominent features of smb processes are based on the prerequisite that optimal operating conditions can be determined, which can be very challenging in practice. Determination of operating conditions is predominantly based on classical and extended triangle theories (storti1993robust), which are originally derived from the ideal chromatographic model with linear isotherm. Powerful criteria to design processes with nonlinear isotherms (e.g., Langmuir and Bi-Langmuir) have been elaborated in the past few years (charton1995complete; mazzotti1997optimal; antos2001application). Versatile results on triangle theory for the design of smb processes have been published (nowak2012theoretical; kim2016combined; lim2004optimization; kazi2012optimization; bentley2013prediction; bentley2014experimental; sreedhar2014simulated; toumi2007efficient; silva2015modeling; kiwala2016center).

Substances separated by smb processes have recently been evolved from monosaccharides to proteins and other macro-molecules. The adsorption behaviors of macro-molecules, as observed both in experiments (clark2007new) and molecular dynamic simulations (dismer2010structure; liang2012adsorption; lang2015comprehensive), are much more complex than that described by the linear isotherm, the Langmuir kinetics or the steric mass-action (sma) model (brooks1992steric). Macro-molecules can undergo conformation changes in the mobile phase, orientation changes on the functional surfaces of stationary phase and aggregation in the whole process. clark2007new used an acoustically actuated resonant membrane sensor to monitor the changes of surface energy and found that the changes persist long after mass loading of the protein has reached the steady state. Therefore, more sophisticated adsorption models, such as the multi-state sma (diedrich2017multi) based on the spreading model (ghosh2013zonal; ghosh2014zonal), have been proposed to describe dynamic protein-ligand interactions. Although these kinetics describe more accurately the adsorption behaviors by taking orientation changes into consideration, the strong nonlinearity of kinetics exerts tremendous difficulties in deriving analytical formulae to calculate flowrates, as it is in the triangle theory.

Ion-exchange (iex) mechanism is commonly used to separate charged components in chromatography. In iex chromatography (both single column and smb processes), an auxiliary component acting as a modifier is used to change the electrostatic interaction force between functional groups and macro-molecules; sodium chloride is most frequently used. The higher the salt concentration is, the lower electrostatic force is, such that the binding affinity is decreased and bounded components are successively eluted (lang2015comprehensive). In the single column iex processes, this principle can be either implemented in an isocratic mode or a linear gradient mode. The linear gradient mode has frequently been applied to achieve better separation capability (osberghaus2012optimizing). While in the iex-smb processes, the isocratic mode (i.e., the salt strength is identical in all zones) was originally applied. The gradient mode can be applied, by using a salt strength in the desorbent stream that is stronger than that in the feed (cf. Fig. 1). Thus, high salt concentration in zone I and II (high salt region), low salt concentration in zone III and IV (low salt region). As shown in the studies, applying the gradient mode can further improve the separation performance (antos2001application; houwing2002effect; houwing2003positioning; li2007proteins). The salt profiles of the gradient mode are referred to as two-step salt gradient in this work; and derivations of multi-step gradient to ternary separations are intuitive. However, the construction of the multi-step salt gradient imposes difficulty on the design of iex-smb processes.

The difficulty can be detoured by using the open-loop multicolumn chromatography, where it is possible to apply linear salt gradients. The open-loop feature and possibility of linear gradient make the application potential of multicolumn continuous chromatography in the downstream processing unarguable (faria2015instrumental)

. As dealing with the model complexity induced by linear gradients is much easier than that by multi-step ones; moreover, the open-loop feature renders more degrees of freedom in the process designs.

Conventional four-zone smb processes are tailored for binary separations. Ternary separations are requested in many applications; the selection of network configurations is application-specific. Ternary separations, also referred to as center-cut separations, can be achieved by cascading two four-zone smb units (wooley1998nine; nicolaos2001application; nowak2012theoretical). Cascade schemes require designing and operating two four-zone smb processes, which can be laborious and costly. Hence, integrated schemes have been developed using a single multi-port valve and fewer pumps than cascade schemes (seidel2008new; da2016evaluation). Moreover, process design is simplified due to simultaneous switching of all columns. Due to the complexity of network configurations, in addition to the kinetic nonlinearity and multi-step gradient, a comprehensive column model, fast numerical solver and efficient optimizer are required.

In this study, we shall present a model-based process design of iex-smb units for separating a protein mixture of ribonuclease, cytochrome and lysozyme, using cation-exchange columns packed with SP Sepharose FF beads. Two network configurations for the ternary separation will be used, that is, a cascade scheme and an eight-zone scheme. For comparison, a conventional single column batch system, with the same column geometry and resin utilization, will be studied. In this study, smb processes are modeled by weakly coupling individual models of the involved columns, i.e., one large equation system is set up and sequentially solved (he2018efficient)

. The code has been published as open-source software,

cadet-smb.The operating conditions of the network configurations will be optimized by a stochastic algorithm, Markov Chain Monte Carlo (mcmc), with respect to conflicting objectives of purity and yield. Pareto fronts are computed for illustrating the best compromises between the two conflicting performance indicators. Unlike multi-objective optimization algorithms (e.g.

, non-dominated sorted genetic algorithm, strength Pareto evolutionary algorithms) that try to eliminate all the non-dominated points during optimization,

mcmc serves on the sampling purpose, which is interested in sampling the Pareto optimal points as well as those near Pareto optimal. For samping purpose, mcmcnot only accepts proposals with better objective value, but also accepts moves heading to non-dominated points with certain probability. Therefore,

mcmc renders greater information for process design. Besides, uncertainties of parameters that is related to the robustness of process design can be examined, though the information is not involved in this study. As shown in the Pareto fonts, the performance indicators of the single column are dominated by that of both cascade and eight-zone schemes. The performances of the two iex-smb schemes are quite similar, though the cascade scheme has a slight advantage over the eight-zone scheme.## 2 Theory

This section introduces the transport model, binding kinetics, load-wash-elution mode of the single column, smb network connectivity, performance indicators and optimization algorithm.

### 2.1 Transport model

The transport behavior of proteins in the column is described by means of the general rate model (grm), which accounts for various levels of mass transfer resistance (guiochon2006fundamentals).

grm considers convection and axial dispersion in the bulk liquid, as well as film mass transfer and pore diffusion in the porous beads:

(1a) | ||||

(1b) |

where , and denote the interstitial, stagnant and stationary phase concentrations of component in column . Furthermore, denotes the axial position where is the column length, is the radial position, is the particle radius, time, column porosity, particle porosity, interstitial velocity, is the axial dispersion coefficient, the effective pore diffusion coefficient, and the film mass transfer coefficient. At the column inlet and outlet, Danckwerts boundary conditions (Barber1998Boundary) are applied:

(2) |

where is the inlet concentration of component in column ; the calculation of inlet concentration in smb chromatography is deferred to Eq. (7). The boundary conditions at the particle surface and center are given by:

(3) |

### 2.2 Binding kinetics

The steric mass-action (sma) model has been widely reported for predicting and describing nonlinear ion-exchange adsorption of proteins; the relationship between the stagnant liquid and the stationary phase is described as follows:

(4) |

where and denote adsorption and desorption coefficients, is the characteristic charge of the adsorbing molecules, and the shielding factor. , , denote the salt concentration in the three different phases. Moreover, denotes counter ions that are not shielded and available for protein binding,

(5) |

It can be expressed with electro-neutrality condition,

(6) |

where is the ionic capacity.

### 2.3 Single column batch system

There are various possibilities to operate a single column in batch manner. The specific one described here is used for comparison with smb processes; we refer you the protocol to osberghaus2012optimizing. Apart from regeneration steps, an operating cycle of the single column could consist of three phases: load, wash and elution, see Fig. 2. In phase I, the column is equilibrated with the running buffer, then the protein solution is injected to the column for a time. It is then followed by a wash step (II), that is, pumping through the column with running buffer for a time. Afterwards, gradient modes (involving zero gradient) are set to elute the bounded protein for a time. Elution (III) can be implemented by multi-step solvent gradients. As depicted in Fig. 2 as an example, there are two parts (i.e., IIIa, IIIb) with linear gradients but different slopes (, ), and eventually an isocratic part (IIIc). Multi-step gradients are used, because in center-cut separations peak overlaps at the front and at the back of the target component need to be minimized. With the regeneration of the column (IV), it comprises an operating cycle of the single.

### 2.4 SMB network connectivity

In smb chromatography two adjacent columns, and , are connected via a node , see Fig. 3. A circular smb loop is closed when two column indices point to the same physical column (i.e., by identifying column with column ), see Fig. 1. In this work, node is located at the downstream side of column and upstream of column . At each node only one or none of the feed (F), desorbent (D), raffinate (R), or extract (E) streams exists at a time. The inlet concentration of component in column is calculated from mass balance of the node :

(7) |

where denotes the outlet concentration of component in column , the volumetric flowrates and the column diameter. Meanwhile, determines the current role of node (i.e., F, D, R, E or none):

(8) |

where and are the component concentrations at feed and desorbent ports, and , , , the volumetric flowrates at the feed, desorbent, raffinate, and extract ports, respectively. indicates nodes that are currently not connected to a port (i.e., in the interior of a zone); this occurs when more columns than zones are present, such as eight columns in a four-zone scheme. As there might be multiple feed, desorbent, raffinate, and extract ports in smb processes (e.g., the cascade scheme and the integrated eight-zone scheme), an extension of the indexing scheme is required. Column shifting is implemented by periodically permuting each switching time .

### 2.5 Performance indicators

Purity, yield and productivity are commonly used for evaluating the performance of chromatographic processes. All indicators can be defined within one collection time (bochenek2013evaluating). In the case of smb systems, is the switching time , while in single column systems it is the length of pooling time interval .

In this study, performance indicators are all defined in terms of component, , withdrawn at a smb node, , or collected at the outlet of single column systems, , within the pooling interval. The definitions of performance indicators are all based on concentration integrals of component at node ,

(9) |

In smb, is the starting time of one switching interval and all performance indicators are calculated when the system is upon cyclic steady state (css); it is the starting point of the pooling time interval in single column.

The purity, , is the concentration integral of component relative to the integral sum of all components, Eq. (10). The yield, , is the ratio of the withdrawn mass and the feed mass, Eq. (11).

(10) |

(11) |

In smb, when the solution stream is continuously fed via the feed node and the extracts and raffinates are continuously withdrawn, and differs with . While in the single column, the flowrates are the same at the inlet node and outlet node, ; and . The productivity, , is the withdrawn mass of the component per collection time relative to the total volume of the utilized packed bed in all columns,

(12) |

In single column systems, . According to the definitions of and , both indicators increase with increasing amounts of the product collected, i.e., . However, can also be improved by reducing the collection time .

### 2.6 Optimization

Generally, operating conditions of both smb and single column processes are systematically optimized by numerical algorithms such that the above performance indicators are all maximized. In studies of center-cut separations, the middle component

is of interest. To specifically account for a vector of the conflicting performance indicators of

(referred to as hereafter) in this study, multi-objective optimization is applied.A set of objectives can be combined into a single objective by adding each objective a pre-multiplied weight (the weighted method (marler2010weighted)), or keeping just one of the objectives and with the rest of the objectives constrained (the -constraint method (mavrotas2009effective)). The latter method is used in this work, that is, maximizing the yield of the target component of the processes with the purity constrained to be larger than a threshold :

(13) |

where is a vector of optimized parameters with boundary limitations of . The inequality is lumped into the objective function using penalty terms in this study, such that it can be solved as a series of unconstrained minimization problems with increasing penalty factors, ,

(14) |

In Eq. (14), the penalty function is chosen as .

Different types of methods can be applied to solve the minimization of

, such as deterministic methods and heuristic methods. Additionally, stochastic methods can be chosen. In order to use a stochastic method, the minimization of

is further formulated to maximum likelihood estimation:

(15) |

where the likelihood is defined as an exponential function of :

(16) |

### 2.7 Numerical solution

All numerical simulations were computed on an Intel(R) Xeon(R) system with 16 CPU cores (64 threads) running at .

The mathematical models described above for each column of smb processes are weakly coupled together and then iteratively solved. The open-source code has been published on Github, https://github.com/modsim/CADET-SMB.git. cadet-smb repeatedly invokes cadet kernel to solve each individual column model, with default parameter settings. cadet is also an open-source software published on Github, https://github.com/modsim/CADET.git. The axial column dimension is discretized into cells, while the radical bead is discretized into

cells. The resulting system of ordinary differential equations is solved using an absolute tolerance of

, relative tolerance of , an initial step size of and a maximal step size of .A stochastic multi-objective sampling algorithm, mcmc, is applied in this study to optimize the operating conditions. Specifically, the Metropolis-Hastings algorithm, incorporating with delayed rejection, adjusted Metropolis and Gibbs sampling, has been published as open-source software on Github, https://github.com/modsim/CADET-MCMC.git. For sampling purpose, mcmc not only accepts proposals with better objective values, but also accepts moves heading to non-dominated points with certain probability. Samples are collected until the Geweke convergence criterion (geweke1991evaluating) is smaller than or the desired sample size is reached.

The Pareto fronts in this study describe two-dimensional trade-offs between purity and yield. Thus, non-dominated stable sort method of Pareto front were is applied to generate the frontiers (duh2012learning).

## 3 Case

A protein mixture of ribonuclease, cytochrome and lysozyme is of interest in this study, . It is a prototype example used in the academic field for modelling purposes (osberghaus2012determination; osberghaus2012optimizing). The fractionation of this protein mixture is also referred to as center-cut separation, that is, the center component, cyt, is targeted and the other two components are regarded as impurities in separations. In the modelling study here, we have used the proteins on chromatographic columns packed with strong cation-exchanger SP Sepharose FF as an example system (see the column geometry in Tab. 1).

Catalogue | Symbol | Description | Value | Unit |

Geometry | column length | |||

column diameter | ||||

particle diameter | ||||

column porosity | 0.37 | |||

particle porosity | 0.75 | |||

Transport | axial dispersion | |||

pore diffusion | ||||

film mass transfer | ||||

Kinetics | ionic capacity | 1200 | ||

equilibrium constant | ||||

characteristic charges | ||||

steric factors |

Simplified models (e.g., tdm and edm) that rely on an assumption of equal concentrations in the particle pores and in the mobile phase, can not been applied here; because mass transfer dynamics of macro-molecules in ca. beads can be rate limiting (lodi2017ion). Hence, the grm model is used to describe the mass transfer in porous beads. In addition, the sma model inherently considers the impact of salt on binding affinity, the hindrance effect of macro-molecules. The mathematical modelling of the prototype example has been presented in teske2006competitive and puttmann2013fast; the transport and binding parameters from experiments are shown in Tab. 1. The geometry, transport and binding parameters are directly used in this study; and the operating conditions will be optimized.

In the pharmaceutical manufacturing usually at least purity of cyt is needed. Therefore, taking expensive computation cost in iex-smb simulations to generate Pareto optimal points with lower purities is also a trade-off. In this study, the in Eq. 13 is set to ; thus, only the Pareto optimal fragment with purity higher than is concerned in iex-smb. This implementation reduces the heavy computational burden of multi-objective sampling, as much fewer samples are requested to construct the frontiers. Pareto optimal fragment can be further extended by repeatedly solving Eq. (15) with varying values of . However, in the single column the whole Pareto optimal front is concerned as the computational cost is not expensive. This can be one of the advantages of the -constraint method.

### 3.1 Single column

In the single column system, the equilibration, load-wash-elution and regeneration phases are performed. The interstitial velocity is ; the retention time for a non-retained component is . concentration of RNase, cyt and lyz is injected in the load phase. In both the load and wash steps, a salt concentration of is applied. The operating time intervals for the load and wash phases are and . The shapes of the multi-step elution phase, that can be characterized by the bilinear gradients (i.e., and ), the operating time intervals, and the initial salt concentration (i.e., ), are optimized. Since the symbols as illustrated in Fig. 2 are the elapsed times, the operating time intervals are calculated (i.e., and ). Thus, the optimized operating parameters are .

### 3.2 Smb

Depending on initial states, smb processes often undergoes a ramp-up phase and eventually enter into a css. In this study, the system is assumed to be upon css when a difference between two iterations falls below a predefined tolerance error, , see Eq. (17).

(17) |

In this study, the columns of iex-smb are initially empty except for the concentration of bound salt ions that is set to the ionic capacity , in order to satisfy the electro-neutrality condition. Additionally, the initial salt concentrations in column , , , , are set equal to the salt concentrations in the upstream inlet nodes. Taking the four-zone scheme as an example (cf. Fig. 1), the salt concentrations of the mobile and stationary phases ( and ) in zone III and IV are set to the value of ; while and in zone I and II are set to the value of :

(18) |

In contrast to the single column system, the chosen initial state of smb processes is irrelevant to the performance indicators calculated upon css.

A cascade of two four-zone units (named as and in the following content) and an integrated eight-zone scheme are applied, see Fig. 4. In both schemes, three single columns in each zone and 24 columns at all are designed. In this study, the volumetric flowrate of zone I is defined as recycle flowrate, . In each sub-unit of the cascade scheme, the optimized operating parameters are the switching time (same for both units as they are synchronously switched), recycle flowrate, the inlet and outlet flowrates, and the salt concentrations at the inlet ports, . While in the eight-zone scheme, the switch time, the recycle flowrate, the inlet and outlet flowrates, and salt concentrations at inlet ports are optimized, that is, . The flowrates and are calculated from flowrate balance.

## 4 Results and discussion

### 4.1 Single column

The gradient shapes of the single column process are first optimized in order to have a comparison with the iex-smb processes. Searching domain of the parameters, , is listed in Tab. 2. The intervals are based on literature values with additional safety margins. The maximal sampling length here is and the burn-in length is set to of the samples.

Symbol | Description | Value | Unit | |
---|---|---|---|---|

min | max | |||

elution interval one | ||||

elution interval two | ||||

elution gradient one | ||||

elution gradient two | ||||

initial salt conc. |

Fig. 5 shows the Pareto optimal front between purity and yield, as computed by Eq. (10) and Eq. (11). At a rather low purity requirement of , a yield of 0.9 can be achieved. However, at purity of , the yield drops dramatically to 0.1. The Pareto front provides full information of the single column process; the corresponding operating conditions, that render the Pareto optima on demand in application, can be chosen on purpose.

Three characteristic points (i.e., ) on the above Pareto front are exemplified and then compared. In Fig. 6, the corresponding chromatograms and multi-step salt gradients are shown; the grey areas indicate the pooling time intervals, , of the target components. The calculated performance indicators of the three characteristic points are as follows: is , and the productivity is of the point (see Fig. 5(a)). While they are and of the point (see Fig. 5(b)), and of the point (see Fig. 5(c)). The respective operating parameters are listed in Tab. 3.

The pooling time length in Fig. 5(b) is much bigger than that in Fig. 5(a) (i.e., ) and Fig. 5(c) (i.e., ). In this study, pooling time intervals were calculated from that the concentrations of other two components (i.e., RNase and lyz) are lower than a predefined threshold, . Based on the same operating parameters, if we set to a larger value, higher yield and lower purity will be obtained. Therefore, the selection is also a trade-off, which is illustrated in a Pareto front in Fig. 7, where was decreased from to with equivalent gap of using operating parameters listed in the column of Tab. 3.

Shorter pooling time interval means the cyt peak is more stiffly concentrated; it can result in high yield but larger overlap areas (thus low purity) as a conflict, see Fig. 5(a). When pooling time interval is longer, the cyt peak is more spread and it renders relatively lower yield but higher purity, see Fig. 5(b). As observed in Fig. 6, the peak of cyt always overlaps with the other two peaks, rather than thoroughly separated. In Fig. 5(b) and 5(c), even when lyz is hardly washed out, it begins to be eluted out with small concentration values quite early. Additionally, as lyz is almost not eluted out in Fig. 5(b) and 5(c), a strip step is needed before the regeneration step, in order to make the column reusable. The overlaps may denotes that cyt can not be collected with purity with the single column batch system.

### 4.2 Cascade scheme

Numerical optimization of the iex-smb schemes require suitable initial values.
To this end, empirical rules have been developed with big efforts to achieve the center-cut separation first.
With the operating conditions listed in the *empirical* column of Tab. 6, lyz is collected at the R port of with (see Fig. 7(a)).
In , cyt spreads towards the R port of , resulting low yield at the E port (i.e., ) and low purity at the R port (i.e., ) (the chromatogram is not shown).
Based on the initial values with safety margins, searching domain as shown in Tab. 4 is used for numerical optimization.

Symbol | Description | Value | Unit | |
---|---|---|---|---|

min | max | |||

recycle flowrate | ||||

feed one flowrate | ||||

desorbent one flowrate | ||||

extract one flowrate | ||||

feed two salt conc. | ||||

desorbent two salt conc. |

smb processes often undergo a long ramp-up phase first and eventually enter into a css. Elaborately choosing the initial state of columns can also accelerate the convergence (bentley2014experimental). In this case, a total of ca. switches was required for each iex-smb simulation to fall below the tolerance error of , see Eq. (17). Thus, gaining points for generating Pareto fronts in smb chromatographic processes is computational expensive. li2014using have proposed computationally cheap surrogate models for efficient optimization of smb chromatography. In this study, only the operating parameters of are optimized; the operating conditions of are taken directly from the empirical design. Moreover, only the Pareto optimal fragment with purity higher than is concerned, such that fewer points are requested. The maximal sampling length of mcmc is , and the burn-in length is . The Pareto front of the cascade scheme is illustrated in Fig. 9. As seen from the Pareto fronts, the cascade scheme outperforms the single column system in both purity and yield indicators.

Three characteristic points (i.e., ), ranging yields from high to low, on the Pareto front are compared. The corresponding operating conditions are listed in Tab. 6; and the resulting chromatograms are shown in Fig. 10. The combined axial concentration profiles in all columns are displayed at multiples of the switching time. The performance vector of the point is (cf. Fig. 9(a)), of the point (cf. Fig. 9(b)) and of the point (cf. Fig. 9(c)). After numerical optimization, the performance indicators tremendously increase from that of the empirical design, . The main difference between the chromatograms of and is that less RNase keeps retained to port E in , resulting in higher purity at port E. The productivities of cyt, , are , respectively. It indicates that productivity is high when the concentration values in the shadow areas at port E are high and flat. Though RNase and lyz are treated as impurities in the center-cut separation, they can be withdrawn with high performance indicators in this study. For instance, of the point , of the point .

The salt profiles, , along the two iex-smb units at time upon css, , are depicted in Fig. 11. In , the salt concentrations at inlet ports, , used for constructing the two solvent gradients are (cf. Fig. 7(b)). The salt concentrations at the inlet ports of , , are of the point ; of the point and of the point .

The salt gap of solvent between zones I,II and zones III,IV of (cf. Fig. 7(b)) facilitates to separate lyz from the mixture; RNase and cyt are still not rather separated in . Because of the salt profile in zones III and IV (i.e., ca. ), RNase and cyt have approximately the same electrostatic interaction forces with the stationary phase, causing rather similar desorption rates off the stationary phase. In order to separate RNase and cyt in , the salt concentrations at the inlet ports should be reduced, leading one component (i.e., cyt) to be strongly bounded. This can be achieved by diluting the bypass stream of with pure buffer (in the cascade scheme, the flow between the raffinate stream of and feed stream of is defined as a bypass stream):

(19) |

In numerical optimization, and are optimized, and are subsequently changed. As seen in Fig. 11, the salt concentration in is decreased to ca. in zones I,II and in zones III,IV. With these salt concentrations, RNase and cyt can be separated in . With the dilution, the protein concentrations are decreased as a side effect. Therefore, the peaks in are not as concentrated as have observed in ; they distribute over several zones, which makes the process design of the challenging.

### 4.3 Eight-zone scheme

As in the design of the cascade scheme, the initial values were obtained from empirical designs with manual efforts.
The searching domain for the numerical optimization is based on the initial values with safety margins, see Tab. 5.
The dilution as described in the design of the cascade scheme was applied to the bypass stream (i.e., the connection between the raffinate-I of the first sub-unit and the feed-II of the second sub-unit).
With the operating conditions listed in the *empirical* column of Tab. 7, cyt spreads from zone VI to zone VII, RNase from zone V to VI, such that and (the chromatogram is not shown).

Symbol | Description | Value | Unit | |
---|---|---|---|---|

min | max | |||

switch time | ||||

recycle flowrate | ||||

desorbent one flowrate | ||||

extract one flowrate | ||||

feed one flowrate | ||||

raffinate one flowrate | ||||

desorbent two flowrate | ||||

extract two flowrate | ||||

feed two flowrate | ||||

feed one salt conc. | ||||

desorbent one salt conc. | ||||

feed two salt conc. | ||||

desorbent two salt conc. |

In this case, a total of ca. switches was required for each eight-zone simulation to fall below the tolerance error of . The convergence time from an initial state to the css, thus the total computational time, in the eight-zone scheme is longer than that in the cascade scheme. The maximal sampling length of mcmc is , and the burn-in length is . The Pareto optimal front with purity larger than is illustrated in Fig. 12; the Pareto fronts of single column and cascade systems are superimposed for comparison. The shadow areas of light blue show the sampling region.

By using the multi-objective sampling algorithm, mcmc, not only the Pareto front solutions but also the points near Pareto front can be studied. In this study, three points (i.e., ; the point is on the Pareto front), ranging purities from low to high, were chosen to illustrate the impact of the composition of R1 stream on the performance of the second sub-unit. The corresponding operating conditions are listed in Tab. 7; and the chromatograms upon css are shown in Fig. 13. The combined axial concentration profiles in all columns are displayed at multiples of the switching time. In Fig. 12(a), cyt is collected at the E2 node with , while in Fig. 12(b) and in Fig. 12(c). As seen from Fig. 13, the higher concentration of the cyt composition in the R1 stream, the higher of the yield and productivity indicators at the E2 port. To be specific, and of the point ; and of the point , and and of the point . Although in the center-cut separation, the other two components (i.e., RNase and lyz) can be viewed as impurities, they can rather be withdrawn with high performance indicators. Specifically, for RNase at the R2 node it is of the point , of the point and of the point ; for lyz at the E1 node it is of the point , of the point and of the point .

The salt profile, , along the eight-zone iex-smb unit at time upon css, , are attached in Fig. 14. The salt concentrations at inlet ports, , are of the point ; of the point and of the point . Separating lyz at port E1 is facilitated with the salt gap between zones I,II and zones III,IV; RNase and cyt are both weekly bounded in zones III,IV and to be separated in the second sub-unit. Meanwhile, the bypass stream was diluted as described by Eq. 19. As seen in Fig. 14, the salt concentration is decreased to ca. in zones V,VI and in zones III,IV. With these salt concentrations, RNase is weakly bounded and cyt is strongly bounded in the second sub-unit. As seen from Fig. 14, the deviation of the salt profile along the columns of iex-smb is small, but the performance indicators varied. This indicates process design of iex-smb processes is sensitive to operating conditions.

Eight-zone schemes have fewer degrees of freedom than cascade schemes. Hence, decreases from of the cascade scheme to , based on the same yield, 0.95. Moreover, in the cascade scheme, almost pure cyt can be collected, which can not be achieved with the eight-zone scheme. However, the productivity of the eight-zone scheme can be higher than that of the cascade scheme. The productivity of the cascade scheme, (]), is slightly lower than that of the eight-zone scheme, (). This suggests that the stationary phase is more efficiently utilized with integrated smb processes.

Three columns in each zone was chosen and used in both the cascade and eight-zone schemes. In the empirical designs, neither one column nor two columns in each zone could lead to good separation performance. Having more columns in each zone and less switching time can approximate the true moving bed chromatography. However, introducing more columns into iex-smb system makes challenges for computational resource. Three columns in each zone is a good compromise.

Symbol | Description | Empirical | Numerical | Unit | |||

feed flowrate | |||||||

raffinate flowrate | |||||||

desorbent flowrate | |||||||

extract flowrate | |||||||

zone I flowrate | |||||||

zone II flowrate |

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