1.1. Continuous counter current chromatography
Continuous industrial-scale adsorption processes are well known for their efficiency. Very often, the Height Equivalent of a Theoretical Plate (HETP) in a batch operation is roughly three times higher than one find for the continuous mode (Gembicki et al., 2002). The operation of continuous chromatographic counter current apparatus (here-by referred as True Moving Bed, TMB) in particular, maximizes the mass transfer driving force providing a better utilization of the adsorbent, and thus, allowing the use of lower selectivity materials (Ruthven and Ching, 1989) as to operate with an increased productivity (i.e., higher processed throughput using less packing material). A scheme of a TMB unit is shown in Figure 1.
Figure 1 – A four section True Moving Bed (TMB) unit for the separation of A and B with D as eluent or desorbent (Fructose/Glucose separation).
If we define section as the part of the TMB unit where the fluid flow rate is approximately constant (section limited by inlet/outlet streams), then, it is possible to find four different sections with different roles:
- Section I: Regeneration of the adsorbent (desorption of A from the solid);
- Section II: Desorption of B (so that, the extract is not contaminated by the less retained component);
- Section III: Adsorption of A (raffinate clean from the more adsorbed species);
- Section IV: Regeneration of the eluent/desorbent (adsorption of B from the fluid phase).
From Figure 1, one can observed that the counter-current movement of the solid, with respect to the fluid phase, allows continuous regeneration of both the adsorbent in section I as the eluent/desorbent in section IV. Also, the moving bed arrangement allows the achievement of high purity even if the resolution of the two peaks is not excellent, since only the purity at the two tails of the concentration profiles, where the withdrawal ports are located, is of interest. This is contrary to batch chromatography where high resolution is vital in order to achieve high purity.
Nevertheless, with this counter current mode of operation is necessary to circulate not only the fluid phase but also the solid. The solid motion inside of the column and the consequent recycle presents some technical problems, namely: equipment abrasion, mechanical erosion of adsorbent and difficulties in maintaining plug flow for the solid (especially in beds with large diameter). From a technical point of view, this clearly limits the implementation of such technology.
1.2. The Simulated Moving Bed (SMB) concept
In order to avoid this issue, a sequence of fixed bed columns was conceived (Broughton and Gerhold, 1961) in which the solid phase is at rest in relation to a fixed referential, but where a relative movement between both phases is experienced by switching the inlet and outlet fluid streams to and from the columns from time to time (in the direction of the fluid flow). In the simplest operating mode, the period that a certain operating configuration prevails is called the switching time, . Since the solid flow is avoided, although a kind of counter-current movement is created relatively to the fluid, this technology is called Simulated Moving Bed (SMB).
Consider that at certain moment in the operation of an SMB, the positions for the inlet of feed and desorbent and outlet of products is represented by Figure 2a. Assume also the simplest operating mode (synchronous advance of all streams) and one column per section. After a period of time equal to the switching time, the injection and withdrawn points all move one column in the direction of the fluid flow (Figure 2b). When the initial location of injection/collection of all the streams is reencountered, we have completed one cycle (in a four equally zoned SMB, it takes to complete one cycle, where is the number of columns in each one of the four sections). As it is possible to see in Figure 2, during one cycle the same column is in different sections, assuming therefore different roles in the separation process.
Figure 2 – Schematic representation of a 4 columns SMB unit operating over a complete cycle, from 0to (with representing the ports switching time); (a) period of the first switch; (b) period of the second switch and (c) a TMB unit.
As mentioned before, the continuous movement of inlet and outlet lines into and from the column is almost impractical, therefore discreet jumps (with the length of one bed, during ) have to be applied.
The analogy between SMB and the TMB is then possible by the introduction of the relative velocity concept, where , with the fluid interstitial velocity in each section in the TMB, the interstitial velocity in the SMB unit and the solid interstitial velocity in the TMB. The solid velocity is evaluated from the switching time interval value in the SMB as , being the column length. As consequence, The internal flow rates in both apparatus are not the same, but related by where and represent the internal liquid flow-rates in the SMB and TMB, respectively, is the bulk porosity and the column volume.
1.3. SMB Applications
Industrially, SMB applications can be regarded as “Old” and “New”, associated with petrochemical and pharmaceutical/fine chemistry fields, respectively (Sá Gomes et al., 2006d). Among the first applications of SMB technology (back to 1960s) are the ones implemented by the UOP Inc. (Des Plaines, IL-USA) with the Sorbex® processes, such as: the Parex® unit for separation of p-xylene from mixtures with its C8-isomers (Broughton et al., 1970), separation also performed by the Aromax® process from Toray Industries (Tokyo, Japan) (Otani et al., 1973) and the Eluxyl® process by Axens/IFP (Institut Français du Pétrole, France) (Ash et al., 1994); the Ebex® for the separation of EthylBenzene (EB) from a mixed of C8-aromatic isomers (Broughton, 1981); the Molex® for the separation of n-paraffins from branched and cyclic hydrocarbons; and the Olex® process to separate olefins from parafins; the Cresex® and Cymex® for the separation of p-cresol and p-cymene from its isomers, respectively.
The application of SMBs in the sugar industry is also substantial, with the Sarex® process, for the separation of fructose from the corn syrup with dextrose and polysaccharides on polystyrene-divinylbenzene resins in calcium form (Broughton, 1983); or as patented by Japan Organo Co. (Japan), (Heikkilä et al., 1989); by Amalgamated Sugar Company LLC, also known as the Snake River Sugar Company (Boise, ID-USA), (Kearney and Mumm, 1990, , 1991).
In the last decade, particularly in the area of drug development, the advent of SMB has provided a high throughput, high yield, solvent efficient, safe and cost effective process option. Although it had long been established as a viable, practical, and cost-effective liquid-phase adsorptive separation technique, the pharmaceutical and biomolecule separations community did not show considerable interest in SMB technology until the mid-1990s. The application of the SMB concept to the fine chemical separations in the earlier 90s, led to the second “boom” on the number of applications of SMB technology (Negawa and Shoji, 1992; Nicoud et al., 1993; Kusters et al., 1995; Rodrigues et al., 1995; Cavoy et al., 1997; Francotte and Richert, 1997; Guest, 1997; Pais et al., 1997a; Pais et al., 1997b; Francotte et al., 1998; Grill and Miller, 1998; Lehoucq et al., 1998; Pais et al., 1998; Dapremont et al., 1999; Miller et al., 1999; Nagamatsu et al., 1999; Nicoud, 1999a, 1999b; Pedeferri et al., 1999; Strube et al., 1999; Juza et al., 2000; Kniep et al., 2000; Wang et al., 2001), among other “pioneers”.
Daicel Chemical Industries, Ltd (Japan) first published the resolution of optical isomers through SMB (Negawa and Shoji, 1992). Since then, several are the SMB based processes already approved by the Food and Drug Administration (FDA) and others regulatory agencies. Examples includes renowned products such as: Biltricide (Praziquantel) Cipralex/Lexapro (Escitalopram), Keppra (Levetiracetam), Modafinil/Provigil, Taxol (Paclitaxel), Xyzal (Levocetirizine), Zoloft (Sertraline), Zyrtec (Cetirizine), Celexa/Citrol/Cipram (Citalopram), Prozac (Fluoxetine hydrochloride), (Abel and Juza, 2007) o paper de real SMB e rajendran, among others biological separation, with a particular emphasis in protein separations meteer referencias a biologias e proteinas.
Given the importance of such technique, this work reviews different operating SMB modes; design, modeling and optimization techniques; and addresses an example of the design, construction and operation of an SMB unit.
2. SMB modes of operation
So far, only the so-called conventional SMB mode of operation has been considered, which indeed means that each section has a fixed number of columns and there is no variation on the pre-established inlet/outlet flow rates or the switching time value. However, over the last decades some non-conventional SMB operating modes were proposed, developing the range of the applications of SMB technology and extending further its potential. Some of these operating modes, worthy of note, are listened in the following Sections.
2.1. Asynchronous shifting SMB (the Varicol® process)
The asynchronous shifting SMB or Varicol® process (Adam et al., 2000; Bailly et al., 2000; Ludemann-Hombourger et al., 2000; Ludemann-Hombourger et al., 2002) commercialized by Novasep (Pompey, France), became one of the more studied and used processes of the so-called non-conventional SMB modes of operation. Instead of a fixed unit configuration with constant section length, the Varicol® operating mode is performed by the implementation of an asynchronous inlet/outlet ports shift, providing a flexible use of each section length, Figure 3.
Figure 3 – [11.51.51] Asynchronous SMB for a complete cycle; section II has 1 column during the first half of the switching time and 2 columns in the remaining time (within a switching time period), thus 1.5 columns; the opposite happens to section III.
By means of Varicol® mode of operation it is possible to increase the productivity value up to 30% more than the classical SMB apparatus, principally when operating under a reduced number of columns (Toumi et al., 2002; Zhang et al., 2002b; Pais and Rodrigues, 2003; Subramani et al., 2003b, 2003a; Toumi et al., 2003; Yu et al., 2003b; Sá Gomes et al., 2006d; Mota et al., 2007b; Rodrigues et al., 2007a; Sá Gomes et al., 2007b; Zhang et al., 2007).
2.2. Partial-Feed, Partial-Discard
With the Partial-Feed mode of operation two additional degrees of freedom are introduced: the feed length and the feed time (Zang and Wankat, 2002a; Zang and Wankat, 2002b). Feed during a given feed length period will consequently influence the raffinate and extract flow rates are along the time. Also referred in the literature is the Partial-Discard (or partial withdraw) operating mode, where just a part of the outlet products is used in order to improve the overall purity (Zang and Wankat, 2002b; Bae and Lee, 2006), or with the partial recirculation of the outlet products back to the feed (Kessler and Seidel-Morgenstern, 2008a; Kessler and Seidel-Morgenstern, 2008b; Seidel-Morgenstern et al., 2008).
The ISMB (Improved SMB) mode of operating, commercialized by the Nippon Rensui Co. (Tokyo, Japan) and FAST – “Finnsugar Applexion Separation Technology”, now Novasep-France, is also well known (Tanimura et al., 1989). In this process, during a first step the unit is operated as a conventional SMB but without any flow in section IV; in the second step the inlet and outlet ports are closed and the internal flow through the four sections allowing the concentration profiles to move to adjust their relative position with respect to the outlet ports (Rajendran et al., 2009). Meter referencias do mazzotti e nova de sa gomes
Another novel non-conventional mode of operation, the Outlet Swing Stream-SMB (OSS) (Sá Gomes and Rodrigues, 2007), was developed under the framework of this thesis and is latter detailed in Chapter 3.
2.3. PowerFeed and ModiCon
The modulation of the section flow rates (PowerFeed) was originally proposed by Kearney and Hieb (1992) and later studied in detail by other authors (Kloppenburg and Gilles, 1999b; Zhang et al., 2003b; Zhang et al., 2004b; Kawajiri and Biegler, 2006b). Another SMB operating concept, based on the feed concentration variation within one switching interval, was suggested by Schramm et al., (2002; 2003b) known as the ModiCon. The use of auxiliary feed tanks, where section flow rate flows into a tank to dissolve solid raw materials and fed into section III, has also been studied (Wei and Zhao, 2008). The cross combination of PowerFeed, Modicon and Varicol modes of operation is also a recurrent research matter, principally of optimization studies (Zhang et al., 2004a; Arau?jo et al., 2006a; Rodrigues et al., 2007b), providing more degrees of freedom and allowing better performance values.
2.4. Two Feed or MultiFeed SMB and Side Stream SMB
Recently, the introduction of multi feed streams in the SMB area, by analogy with distillation columns, led to the formulation of the Two Feed SMB, or MultiFeed, operating mode presented by Kim (2005) and later studied by Sá Gomes and Rodrigues (Sá Gomes et al., 2007b; Sá Gomes and Rodrigues, 2007). Also multi extract/raffinate are referred in the literature (Mun, 2006), known as side stream SMB (Beste and Arlt, 2002). These techniques, combined with the distillation know-how for the optimum location of multiple feeds, can allow the development of more efficient SMB processes.
2.5. Semi Continuous, Two and Three zones SMB
There are several semi continuous SMB apparatus that operate with two-zone, two or one-column chromatograph, with/or recycle, analogous to a four-zone SMB(Abunasser et al., 2003; Abunasser and Wankat, 2004; Arau?jo et al., 2005a; Arau?jo et al., 2005b; Jin and Wankat, 2005b; Mota and Arau?jo, 2005; Arau?jo et al., 2006b; Arau?jo et al., 2007; Rodrigues et al., 2008b), that allow a reasonable separation, some allowing centre cut for ternary or quaternary separations (Hur and Wankat, 2005b, 2005a, , 2006a, 2006b; Hur et al., 2007), under reduced equipment usage.
The discontinuous injection in a system with 2 or more columns, based on the concept of simulated adsorbent movement, as been applied by Novasep under the denomination of Cyclojet®, Hipersep®, Supersep™ (Supersep MAX™ with Super Critical CO2) and Hipersep®, (Grill, 1998; Valery and Ludemann-Hombourger, 2007).
2.6. Gradient SMB
As a further possibility for increasing the productivity, the introduction of gradients in the different separation sections of the SMB process is also described in literature. The gradient mode was suggested firstly for the SMB-SFC (SMB-supercritical fluid chromatography) process, where the elution strength can be influenced by a pressure gradient (Clavier and Nicoud, 1995; Clavier et al., 1996). Nowadays, there are more gradient-variants that allows the variation solvent elution strength by changing the temperature, the pH-value, the content of salt or the modifier concentration (Jensen et al., 2000; Antos and Seidel-Morgenstern, 2001; Migliorini et al., 2001; Abel et al., 2002; Antos and Seidel-Morgenstern, 2002; Abel et al., 2004; Ziomek and Antos, 2005; Mun and Wang, 2008a), or as in Rodrigues’s group with the purification of proteins by Ion Exchange-SMB (IE-SMB) (Li et al., 2007; Li et al., 2008). Also worth of note is the MCSGP (Multicolumn Counter-current Solvent Gradient Purification) process (Aumann and Morbidelli, 2006; Strohlein et al., 2006; Aumann and Morbidelli, 2007; Aumann et al., 2007; Aumann and Morbidelli, 2008; Müller-Späth et al., 2008), commercialized by ChromaCon AG (Zürich, Switzerland), which combines two chromatographic separation techniques, the solvent gradient batch and continuous counter-current SMB for the separation of multicomponent mixtures of biomolecules.
2.7. Hybrid-SMB: SMB combined with other processes
It is possible to improve the performance of SMB units by integrating it with other different separation techniques. The more simple application of this approach is to combine in series the two different processes and then recycle back the outlets between (or within) the different units (Lorenz et al., 2001; Amanullah et al., 2005; Kaspereit et al., 2005; Amanullah and Mazzotti, 2006; Gedicke et al., 2007). Among these, an interesting hybrid SMB was presented by M. Bailly et al., (2005; Abdelmoumen et al., 2006), the M3C process; or the similar process: Enriched Extract operation (EE-SMB) (Paredes et al., 2006), in which a portion of the extract product is concentrated and then re-injected into the SMB at the same, or near to, the collection point. The use of SMB-PSA apparatus is also referred in the literature for gas phase separations, (Rao et al., 2005; Sivakumar, 2007; Kostroski and Wankat, 2008). The use of two SMB units with concentration steps between, for the separation of binary mixtures, was also developed under the denomination of hybrid SMB-SMB process (Jin and Wankat, 2007a).
2.8. The SMBR multifunctional reactor
The integration of reaction and separation steps in one single unit has the obvious economical advantage of reducing the cost of unit operations for downstream purification steps. Besides reactive distillation, reactive extraction or membrane reactors, the combination of (bio)chemical reaction with SMB chromatographic separator has been subject of considerable attention in the last 15 years. This integrated reaction-separation technology adopts the name Simulated Moving Bed Reactor (SMBR). Several applications have been published considering the SMBR: the enzymatic reaction for higher-fructose syrup production (Hashimoto et al., 1983; Azevedo and Rodrigues, 2001; Borges da Silva et al., 2006; Sá Gomes et al., 2007a); meter a dos FOS the esterification from acetic acid and -phenethyl alcohol and subsequent separation of the product -phenetyl acetate (Kawase et al., 1996), or methyl acetate ester (Lode et al., 2001; Yu et al., 2003a); the synthesis and separation of the methanol from syngas (Kruglov, 1994); the esterification of acetic acid with ethanol (Mazzotti et al., 1996b); the lactosucrose production (Kawase et al., 2001); the MTBE synthesis (Zhang et al., 2001); the diethylacetal (or dimethylacetal) synthesis (Silva, 2003; Rodrigues and Silva, 2005; Silva and Rodrigues, 2005a; Pereira et al., 2008); the ethyl lactate synthesis from lactic acid and ethanol (Pereira et al., 2009a; Pereira et al., 2009b); the biodiesel synthesis (Geier and Soper, 2007) falta uma; or the isomerization and separation of p-xylene (Minceva et al., 2008) faltam os franceses, are examples that prove the promising potential of this technique. Depending on the reactive system some interesting arrangements of the general SMBR setup can be found in the literature, a more detailed review of several SMBR applications can be found elsewhere (Minceva et al., 2008).
2.9. Multicomponent separations
The application of SMB technology to multicomponent separations has also been an important research topic in the last years. The common wisdom for such multicomponent process is the simple application of SMB cascades (Nicolaos et al., 2001a, 2001b; Wankatt, 2001; Kim et al., 2003; Kim and Wankat, 2004); nevertheless, there are some non-conventional operation modes that proved to have interesting performance, as the one introduced by the Japan Organo Co. (www.organo.co.jp), called JO process (or Pseudo-SMB); this process was discussed in detail (Mata and Rodrigues, 2001; Borges da Silva and Rodrigues, 2006, , 2008) and (Kurup et al., 2006a). The process is characterized by a 2-steps operation: (a) in the first step the feed is introduced while the intermediary product is recovered with the whole unit working as a fixed bed; (b) during the second step the feed stopped, the unit works as a standard SMB and the less and more retained products are collected, see Annex I for details. The use of two different adsorbents (Hashimoto et al., 1993), two different solvents (Ballanec and Hotier, 1992), or a variation of the working flow rates during the switching period (Kearney and Hieb, 1992), were also proposed.
2.10. SMB – Gas and Super Critical phases
Most of the industrial applications of SMB technology operate in the liquid phase; nevertheless, SMBs can also be operated under supercritical conditions; where a supercritical fluid, typically CO2, is used as eluent offering a number of advantages namely reduction of eluent consumption, favourable physicochemical properties and lower pressure drop and higher column efficiency (Clavier and Nicoud, 1995; Clavier et al., 1996; Denet and Nicoud, 1999; Depta et al., 1999; Denet et al., 2001; Johannsen et al., 2002; Peper et al., 2002; Peper et al., 2007). Also in the gas phase the recent developments have been remarkable (Storti et al., 1992; Mazzotti et al., 1996a; Juza et al., 1998; Biressi et al., 2000; Cheng and Wilson, 2001; Biressi et al., 2002; Rao et al., 2005; Lamia et al., 2007; Mota et al., 2007b; Sivakumar, 2007; Kostroski and Wankat, 2008). Meter a do propano propylene
3. SMB design, modeling, simulation and optimization
Over the last 50 years, design, modeling, and optimization of chromatographic separation processes have been frequent research subjects. As consequence, several modeling methods, strategies and approaches have been developed, the more relevant are reviewed in this section.
3.1. Design strategies
The design of an SMB based separation involves taking decisions at many levels, from the configuration of the unit (number of columns per section, column and particle size) to operating conditions (feed concentration, switching time, internal flow rates). Although simulation can be exhaustively done until the right combination of parameters is found for the expected performance, it is useful to have a design method that will provide a preliminary estimation of the optimum operating point, followed by simulation and/or optimization, (Sá Gomes et al., 2009a).
The equivalence between TMB and SMB can be quite useful in the SMB design procedure. Recalling the role of each SMB section (Figure 2c), one can state a set of constraints that will limit the feasible region and allow a complete separation (recover of the more retained species (A) in the extract stream, the less retained one (B) in the raffinate port, and regeneration of the solid in section I as fluid in section IV).
Where represents the solid flow rate, the average solid concentration of species in section and the bulk fluid concentration of species in section .
The flow rates constraints in Eq. 1b and 1.c will identify the separation region (section II and III), while Eq. (1 a) and Eq. (1 d) the regeneration one (section I and IV).
Usually, the fluid and solid velocities in each section are combined into one only operating parameter, such as the from Morbidelli’s group or the , as used by Ruthven (1989). The identification of constrains, Eq. (1 a) to Eq. (1 d), led to the appearance of several design methodologies, which are usually approximated and/or graphical, providing a better insight to the possible operating regions. From the plates theory and McCabe-Thiele diagrams (Ruthven and Ching, 1989); passing by the analytical solutions for a linear adsorption isotherms system in presence of mass transfer resistances (Silva et al., 2004); to the determination of waves velocities as in the Standing Wave Design (SWD) methodology (Ma and Wang, 1997; Mallmann et al., 1998; Xie et al., 2000; Xie et al., 2002; Lee et al., 2005). A particular emphasis should be given to the strategy developed for binary and multicomponent separations modeled by linear and non-linear isotherms as in (Storti et al., 1989b; Storti et al., 1993; Mazzotti et al., 1994; Storti et al., 1995; Mazzotti et al., 1996c; Mazzotti et al., 1997b; Chiang, 1998; Migliorini et al., 2000; Mazzotti, 2006b), the so-called “Triangle Theory”, where the term is treated by assuming that the adsorption equilibrium is established everywhere at every time (Equilibrium Theory, (Helfferich, 1967; Klein et al., 1967; Tondeur and Klein, 1967; Helfferich and Klein, 1970), resulting in a feasible separation region formed by the above constraints Eq. (1 b) and Eq. (1 c), which in the case of linear isotherms takes the shape of a right triangle in the plane, Figure 4, (or a triangle shaped form with rounded lines in non-linear isotherms case), and a rectangular shape in the plane.
Recently, this methodology was also extended for the design of SMB units under reduced purity requirements, in which the separation triangle boundaries are “stretched” to account for different extract and/or raffinate purities (Kaspereit et al., 2007; Rajendran, 2008).
Figure 4 – “Triangle Theory”, separation and regeneration regions for linear isotherms, where represents the Henry constant for linear adsorptions isotherms (A: the more retained and B: the less retained species), is the intraparticle porosity; case of (S,R)–Tetralol enantiomers, see Section 4.3.2.
Nevertheless, the inclusion of mass transfer resistances can deeply affect the result of the design. By taking into account all mass transfer resistances, and running successive simulations, it is possible obtain more detailed separation/regeneration regions, as well as the separation study carried out for three different sections (II, III and I) or (II, III and IV) allowing the analysis of solvent consumption or solid recycling, as proposed in the “Separation Volume” methodology, (Azevedo and Rodrigues, 1999; Rodrigues and Pais, 2004a), or the influence of the solid flow rate in the separation region (Zabka et al., 2008a).
3.2. Modeling and simulation
Generally, one can model a chromatographic separation process, and consequently an SMB unit, by means of two major approaches: by a cascade of mixing cells; or a continuous flow model (plug flow or axial dispersed plug flow, making use of partial differential equations derived from mass, energy and momentum balances to a differential volume element ), (Rodrigues and Beira, 1979; Ruthven and Ching, 1989; Tondeur, 1995; Pais et al., 1998). Each of these approaches can include mass transfer resistances, thermal, and/or pressure drop effects. Nevertheless, most of the recent literature concerning SMB processes just makes use of the continuous approach, detailing the particle diffusion and/or film mass transfer (the Detailed Particle Model), or using approximations to the intraparticle mass transfer rate in a similar way as the Linear Driving Force (LDF) approach presented by Glueckauf (1955a), (Guiochon, 2002).
One can argue that an SMB unit is no more than the practical implementation of the continuous counter current TMB process, Figure 2. Consequently, the equivalence between the TMB and a hypothetical SMB with an infinite number of columns can be used in the modeling and design of SMB units. However
TMB model approach will just give reasonable results if a considerable number of columns per section is present.
The SMB model approach represents an SMB unit as a sequence of columns described by the usual system equations for an adsorptive fixed bed (each column ), thus represented by a PDE system. Nevertheless, the nodes equations can be stated to each section, making use of the equivalence between the interstitial velocity in the TMB and SMB, and thus:
The issue here is that, due to the switch of inlet and outlet lines, the boundary conditions to a certain column are not constant during a whole cycle but change after a period equal to the switching time.
Since the model equations are set to each column , one will obtain the concentration of species in the begin of each section , , from the following node mass balances:
Considering now . This set of equations continues to progress in a similar way (shifting one column per ), until , repeating then from the first switch.
As for the TMB model approach, both the Detailed Particle Model and LDF approach can be used with the SMB model approach; nevertheless, and for the sake of simplicity, just the last is detailed in this work.
The LDF approximation can now be obtained from , and thus obtaining for the bulk fluid mass balance:
and for the mass balance in the particle,
with the respective initial:
and boundary conditions:
where the adsorption equilibrium isotherm is:
As a consequence one obtains discontinuous solutions, reaching not a continuous Steady State but a Cyclic Steady State (CSS).
By applying the SMB model approach, both the Detailed Particle as LDF strategies, to the case study mentioned before, one obtains the CSS concentration profiles over a complete switching time, Figure 6.
3.3. Performance parameters
The performance of the SMB unit for a given separation is usually characterized by the following parameters: purity, recovery, productivity per the amount adsorbent volume and eluent/desorbent consumption per mass of treated product. The definitions of all these performance parameters, for the case of a binary mixture, are given bellow:
Purity (%) of the more retained (A) species in extract and the less retained one (B) in the raffinate stream, over a complete cycle (from to ):
Recovery (%) of more retained (A) species in extract and the less retained one (B) in raffinate stream, again over a complete cycle:
the productivity per total volume of adsorbent :
the eluent/desorbent consumption :
These parameters hold for both SMB and TMB model approaches; nevertheless, one can simplify: in the SMB model strategy the same equations can be stated for a switching time period (from to ) if the unit is symmetrical, i.e., there are no differences between each switching time period (either due to the implementation of non-conventional modes of operation, or to the use of more detailed models accounting for dead volumes or switching time asymmetries); in the TMB model approach there is no need of the integral calculation, since the solutions from this model strategy are continuous and thus, the performance parameters constant over the time (at the steady state).
Usually one can classify the optimization of SMB units according to the type of objective functions: (i) optimization of performance parameters (productivity, adsorbent requirements or desorbent/eluent consumption for given purities and/or recovery requirements); (ii) optimization based on the separation cost. In case (i) each objective function, based on a different set of performance parameters, can lead to a different optimum solution; therefore multi-objective functions procedure should be considered; in the second case (ii) all those different performance parameters can be homogenized/normalized by the separation cost, where separation dependent costs (adsorbent, plant, desorbent/eluent recovery cost, desorbent/eluent recycling, feed losses…) and separation independent costs (wages, labour, maintenance, among others) are taken into account and weighted by cost factors, which sometimes are difficult to characterize (Jupke et al., 2002; Chan et al., 2008).
To solve these problems, the use of powerful optimization algorithms, such as: IPOPT (Interior Point OPTimizer, (Wa?chter and Biegler, 2006), employed for liquid as gas phase SMB separations (Kawajiri and Biegler, 2006b, 2006a; Mota et al., 2007a; Mota and Esteves, 2007; Rodrigues et al., 2007b; Kawajiri and Biegler, 2008a, 2008b); the commercial package gOPT from gPROMS with a Single (or Multiple) Shooting-Control Vector Parameterization, used in the two level optimization of an existing Parex® unit (Minceva and Rodrigues, 2005), for ageing analysi
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