Attainable region analysis for continuous production of second generation bioethanol
 Felipe Scott^{1, 2},
 Raúl Conejeros^{1, 2} and
 Germán Aroca^{1, 2}Email author
DOI: 10.1186/175468346171
© Scott et al.; licensee BioMed Central Ltd. 2013
Received: 31 May 2013
Accepted: 4 September 2013
Published: 29 November 2013
Abstract
Background
Despite its semicommercial status, ethanol production from lignocellulosics presents many complexities not yet fully solved. Since the pretreatment stage has been recognized as a complex and yielddetermining step, it has been extensively studied. However, economic success of the production process also requires optimization of the biochemical conversion stage. This work addresses the search of bioreactor configurations with improved residence times for continuous enzymatic saccharification and fermentation operations. Instead of analyzing each possible configuration through simulation, we apply graphical methods to optimize the residence time of reactor networks composed of steadystate reactors. Although this can be easily made for processes described by a single kinetic expression, reactions under analysis do not exhibit this feature. Hence, the attainable region method, able to handle multiple species and its reactions, was applied for continuous reactors. Additionally, the effects of the sugars contained in the pretreatment liquor over the enzymatic hydrolysis and simultaneous saccharification and fermentation (SSF) were assessed.
Results
We obtained candidate attainable regions for separate enzymatic hydrolysis and fermentation (SHF) and SSF operations, both fed with pretreated corn stover. Results show that, despite the complexity of the reaction networks and underlying kinetics, the reactor networks that minimize the residence time can be constructed by using plug flow reactors and continuous stirred tank reactors. Regarding the effect of soluble solids in the feed stream to the reactor network, for SHF higher glucose concentration and yield are achieved for enzymatic hydrolysis with washed solids. Similarly, for SSF, higher yields and bioethanol titers are obtained using this substrate.
Conclusions
In this work, we demonstrated the capabilities of the attainable region analysis as a tool to assess the optimal reactor network with minimum residence time applied to the SHF and SSF operations for lignocellulosic ethanol production. The methodology can be readily modified to evaluate other kinetic models of different substrates, enzymes and microorganisms when available. From the obtained results, the most suitable reactor configuration considering residence time and rheological aspects is a continuous stirred tank reactor followed by a plug flow reactor (both in SSF mode) using washed solids as substrate.
Background
Production of bioethanol from sugar and starch rich feedstocks, such as sugar cane (sucrose) or starchy materials (corn, wheat, sorghum) is done using microorganisms such as S. cerevisiae or Z. mobilis in a fermentation process [1]. Since, bioethanol has to be recovered from the mixture of water (as reaction media), residual sugars and nutrients, it is convenient to increase the concentration of initial sugars (for batch fermentations) or feed concentration (for continuous processes) in order to raise the bioethanol titers. Thus reducing the energy consumption and operating and capital expenditures in the distillation operation [2, 3]. However, microorganisms suffer from inhibition at both high sugar and bioethanol concentration [4]. For alleviating ethanol inhibition, batch bioreactors and plug flow bioreactors (PFR) are the best options because they do not present backmixing, which effectively reduces their timeaveraged product inhibition [5]. Traditionally, batch fermentation has been used in the bioethanol industry especially for small scalefacilities, and the MoillerBoinot process (a fed batch process with cell recovery) has been extensively used in Brazil [6]. For modern bioethanol production plants, the working volume of bioreactors is on the order of thousands of cubic meter. As an example, a total of 20 bioreactors, with a working volume of 3000 m^{3} each, were constructed in the Shandong province, China in 2003 [1]. For such large facilities, batch bioreactors are unattractive because of the longer operational downtimes associated with mash adding, broth harvesting and facility cleaning [1]. Continuous PFR conditions are difficult to achieve in a fermentation process due to its extended residence time and gas production, which induce mixing. In fact, residence time can be as long as 48 to 72 hours to achieve an ethanol concentration of 10 to 12% [7]. Since a cascade of continuous stirred tank reactors (CSTR) also contributes reducing endproduct inhibition, this strategy has been practiced in the bioethanol industry [8]. Generally, a train of four to six CSTR connected in series are preferred because such design presents an adequate tradeoff between the glucose fermentation kinetics and the capital investments for tank manufacture [1]. This widely known use of a cascade of CSTRs as a way to minimize the residence time of the system is theoretically valid only for processes with a fixed overall reaction stoichiometry, and that can be described by a single kinetic expression. Although this may hold for ethanol fermentation kinetics [8], for enzymatic saccharification and simultaneous saccharification and fermentation operations in lignocellulosic ethanol production, the reaction network cannot be reduced to a single kinetic expression. Hence, the classic graphical methods for residence time optimization of continuous bioreactors are no longer applicable.
Bioethanol production from lignocellulosic substrates comprises a pretreatment of the feedstock to increase its reactivity to further enzymatic degradation [9]. These biocatalysts break the structure of cellulose and hemicellulose, producing sugar monomers and oligomers, which are subsequently fermented to bioethanol. Even at high solid concentration in the enzymatic hydrolysis step, glucose concentration at the beginning of the fermentation stage will not normally exceed 145 g/L, even considering full cellulose to glucose conversion of a pulp with 20% DW solid content with 65% of cellulose. This value is rather modest compared with first generation bioethanol production. Although, inhibition by ethanol or sugar concentrations is reduced in bioethanol production from lignocellulosics, the enzymatic hydrolysis process has its own inhibition effects. Glucose, cellobiose and xylose have been reported to inhibit the reaction rates of cellulolytic enzymes [10]. Considering that in conventional fermentation processes using sugar and starchy materials, the inhibition problems have been minimized using adequate reactor configuration, the following question naturally arises: which are the most advantageous reactor arrangements in the hydrolysis and fermentation areas for the production of bioethanol from lignocellulosic materials?
Since the conventional use of graphical methods for residence time minimization of a reactor network is no longer applicable to the system under study due to its high number of reactions, we focus on more general optimization methodologies. Optimization of reacting systems involves solving the following reactor network synthesis (RNS) problem as stated by Biegler et al. [11]: “Given the reaction stoichiometry and rate laws, initial feeds, a desired objective, and system constraints, what is the optimal reactor network structure? In particular: (i) What is the flow pattern of this network? (ii) Where should mixing occur in this network? (iii) Where should heating and cooling be applied in this network?” Question (i) addresses the mixing patterns of the reactors in the reactor network. In idealized reactors, two extremes exist: no axial dispersion inside the reactor (PFR) and full axial dispersion (CSTR) [5]. Question (ii) inquires about which reactors in the network should be fed with fresh feed (F) and which reactors should be fed with a mixture of intermediate product streams. Finally, (iii) refers to the heat supply or withdrawal in the network, e.g. to improve selectivity by increasing the rate of certain reactions over the rest of the reactions in the reaction network.
The problem of RNS can be addressed by an approach based in mathematical optimization of a reactor network superstructure or by graphical methods. Optimization based approaches start by proposing a reactor superstructure where all the possible reactors, mixing streams and heat streams are included. Then, optimal candidates are determined by searching in this superstructure. The first attempt using this strategy considered axial dispersion models and recycle PFRs [12] and the resulting candidate structures were found using nonlinear programming. Later, the concept of modeling the superstructure as a mixed integer nonlinear programming (MINLP) formulation was introduced [13]. Although this formulation allows a more natural modeling approach, the resulting optimization problems are generally nonconvex and, therefore, it is difficult to obtain a global solution. In recent years, research in this area has been devoted to overcoming difficulties associated with the nonconvexity of the optimization problems using global optimization techniques [14, 15].
Graphical methods for RNS include the Attainable Region (AR) analysis. This method has originated from the work of Horn [16], who defined the AR as the set of all possible values of the outlet stream variables which can be reached by any possible (physically realizable) steadystate reactor system from a given feed stream using only the processes of reaction and mixing[17, 18]. Horn [16] showed that once the AR is obtained, then an optimization problem with reactor output concentration as decision variables was essentially solved. The attainable region can be constructed for a given reaction network with n chemical compounds in an n dimensional space. Its construction is supported by the application of proposition and theorems [17, 19–22] that describe properties of the AR. Despite these powerful theoretical advances, there exist no sufficient conditions for the AR. Hence, the regions that are calculated applying the known necessary conditions are termed candidate attainable regions (AR^{c}). For two and three dimensions, graphical constructive methods can be derived from these propositions and theorems, thus greatly facilitating its application. A detailed treatment of the methods used in this work is given in the Methods section. For the readers acquainted with the existing theory and results of the AR, this section can be skipped. However, we recommend consulting the details concerning the kinetic models used for the enzymatic hydrolysis and fermentation reaction networks.
In this work, we analyzed the process synthesis of the enzymatic hydrolysis and fermentation operations for bioethanol production, applying for the first time the concept of the Attainable Region to these systems. Two scenarios are analyzed: (i) conversion of washed pretreated material to bioethanol and (ii) production of bioethanol from the discharge stream of the pretreatment reactor (solids and reaction liquor), from this point on nonseparated pretreated material (nSPM). In each scenario, production of bioethanol from pretreated material is performed in one of two alternative configurations: continuous separated saccharification and fermentation (cSHF) or continuous simultaneous saccharification and fermentation (cSSF). In cSHF mode, pretreated corn stover is continuously fed to an enzymatic hydrolysis system and the stream leaving this operation is discharged to a continuous fermentation system. In cSSF mode, pretreated corn stover is hydrolyzed and the released sugars fermented in the same reactor. The main purpose of this work is to establish the most appropriate configurations for these systems. Our interest in investigating the effect of reactor configurations when washed and nSPM are used was motivated by the work of Hodge et al. [10], regarding the effect of sugars and acids released during pretreatment over the enzymatic hydrolysis. We believe that, since an important inhibitory effect over the enzyme activity is caused by the sugars in the pretreatment liquor [10], appropriate reactor configurations may mitigate this problem.
Results and discussion
Attainable region candidate for cSHF
Rate balance equations per compound for cSHF and cSSF operations
Processes  Operations  Rate equations ( r( c) vector) 

cSHF  (1) Enzymatic saccharification  r_{ S } =  r_{1}  r_{2} 
r_{ B } = r_{1}  r_{2}  
r_{ G } = r_{2} + r_{3}  
(2) Ethanol fermentation  ${r}_{x}={r}_{x}^{F}$  
${r}_{G}={r}_{G}^{F}$  
${r}_{P}={r}_{P}^{F}$  
cSSF  1 and 2  r_{ S } =  r_{1}  r_{2} 
r_{ B } = r_{1}  r_{2}  
${r}_{G}={r}_{2}+{r}_{3}{r}_{G}^{F}$  
${r}_{x}={r}_{x}^{F}$  
${r}_{P}={r}_{P}^{F}$ 
Attainable region candidate for glucose fermentation
Glucose fermentation must follow enzymatic hydrolysis in the cSHF operation. Figure 6 shows the candidate AR for bioethanol production using S. cerevisiae and the effect of cell feeding to the CSTR reactor. The feed stream to the PFR should always contain cells because cell growth is an autocatalytic reaction; in Figure 6B, cell concentration corresponds to 1 g/L. When no cells are supplied to a CSTR in the feed stream, no ethanol production occurs until residence time reaches 4 h. Before this residence time, the feed rate exceeds the growth rate of the cells and the culture is washed out from the fermentor.
From feed point A to the point marked B, the CSTR trajectory describes a nonconvex curve, so a mixing line connecting the feed composition to point B (line $\overline{\mathit{AB}}$) can be used to extend the AR. Point B coincides with the point on the curve of the CSTR where the rate vector starts pointing outside the AR. Thus, at point B the AR^{c} can be extended by a PFR with feed concentrations in B. The line $\overline{\mathit{AB}}$ and the CSTR followed by PFR trajectory define the boundary of the attainable region. Along this boundary lies the minimum residence time reactor configurations for a given bioethanol concentration (or yield).
Attainable region candidate for cSSF
Comparison of cSSH and cSHF operations with washed solids and nonseparated pretreated material
When washed solids and nonseparated pretreated material operations are compared in a common potentially obtainable glucose basis (15% solid fraction for washed solids and 20% for nonseparated pretreated material), cellulose conversion is higher for washed solids as it is shown in Figure 12.
When glucose yields at 100 h, for washed solids and nSPM, are plotted against the solid content, then negative slope straight lines are obtained with correlation coefficients of 0.9998 and 0.9996 for washed solids and nonseparated pretreated material respectively. This behavior was already observed for both SSF and enzymatic hydrolysis along several experimental data sets independently published by several authors and analyzed by Kristensen et al. [25]. It is interesting to point out that we are using a kinetic model published in 2004, and the observation of Kristensen et al. [25] was made on 2009, this means that with an appropriate simulation effort, this conclusion could have been drawn from in silico analysis several years earlier.
Results suggest that nonseparated pretreated material should only be used if a xylose cofermenting microorganism is available; otherwise, the strong inhibitory effect exerted by xylose over the cellulolytic enzymes causes an important reduction of cellulose conversion, and hence in the amount of bioethanol obtained from the cellulosic fraction of the pretreated material.
Validity of the results
Results presented so far suggest that a CSTR followed by a PFR has the minimal residence time for cSSF and bioethanol production, and a near minimal residence time for cSHF. Furthermore, this design entails significant benefits from a rheological point of view. However, our results were obtained with two among the many available reaction kinetics for the processes under analysis. Hence, we do not claim that the suggested reactor configuration will be the optimal case for any reaction network and kinetic expressions in the cSHF and cSSF systems. However, literature evidence supports that for autocatalytic reactions and productinhibited bioreaction networks, a combination of CSTR followed by PFR or a series of CSTRs often have the minimal residence time despite its particular kinetic parameter values [8, 26] for a reaction network that can be expressed as a single reaction kinetic.
From a practical point of view, the PFR operation it is not technically possible because of the gas production in the fermentation, thus a series of CSTR can be used to mimic this reactor.
Conclusions
An attainable region analysis was performed over the conversion of pretreated corn stover to bioethanol, considering two processes: SHF and SSF and washed and nonwashed material. Independent kinetic models were used for each operation, i.e.: enzymatic saccharification, fermentation, and simultaneous saccharification and fermentation, in continuous operation. Our aim was to identify the reactor network configurations that provide lower residence times for both processes. Due to the high number of chemical species involved in the reaction network, and hence the high dimensionality of the AR, it was expected that the bypass and/or DSR would shape the boundaries of the AR for minimum residence time, however these are not involved in the configurations that resulted in the lowest residence time.
For SHF, the saccharification reaction must be performed in a PFR to achieve the minimum residence time; however because it is unfeasible from a technical point of view due to the rheological restrictions of the system, the most adequate configuration with technical feasibility and with the closest residence time to the optimum is a CSTR followed by a PFR. For the fermentation operation, the minimum residence time is achieved in a reactor configuration of a CSTR followed by a PFR.
For SSF, the minimum residence time was obtained using a CSTR followed by a PFR, being the enzymatic saccharification and fermentation reactions carried out simultaneously in both reactors at isothermal conditions.
Regarding the effect of soluble solids in the reactor network feed stream; for cSHF, higher glucose concentration and yield are achieved for enzymatic hydrolysis with washed solids compared with nonseparated pretreated material. For cSSF, higher yields and bioethanol titers were obtained when using washed solids.
In this work, we demonstrated the capabilities of the attainable region analysis as a tool to assess the optimal reactor network with minimum residence time applied to the SHF and SSF operations for lignocellulosic ethanol production. According to the kinetic models used in this study, the most appropriate reactor configuration for ethanol production from pretreated corn stover is a CSTR followed by a PFR, both operating in cSSF mode, and with washed pretreated material as substrate. The methodology can be readily modified to evaluate other kinetic models of different substrates, enzymes and microorganisms when available.
Methods
All the methodology described in this section is oriented to construct the AR^{c} for the different scenarios described in the Background section. cSHF and cSSF AR^{c}s were constructed for washed solids and nSPM. Unless otherwise specified, the solid fraction is equal to 0.2 total dried solids. For enzymatic hydrolysis simulation the temperature was taken as 50°C, and for cSSF and fermentations temperature is 32°C. In both cSHF and cSSF operations, enzyme doses were established as 45 mg protein/g cellulose (CPN commercial cellulase, Iogen Corp., Ottawa, Ontario, Canada) [27].
Pretreated material
The pretreated material was assumed to be corn stover pretreated using dilute acid hydrolysis. The material composition was adapted from NREL’s 2011 report on biochemical conversion of corn stover to ethanol [28]. Only compounds taking part in the kinetic models used in this study were considered for calculations, with this consideration the soluble and insoluble compositions in the pretreated corn stover are given as follow (DW%): cellulose, 44.3; xylose, 27.9; lignin, 21.1; glucose, 6.0 and xylan, 0.7. Considering these compounds only, the total solid (soluble and insoluble) fraction is 0.148, the rest being water. When washed solids are used, the solid fraction is assumed to be composed only of cellulose, lignin and xylan. Subtracting the soluble solids from the composition given in NREL’s 2011 report [28], the washed solid is composed of (DW%): cellulose, 67.0; lignin, 32.0 and xylan, 1.1.
Reaction kinetics
In Eq. (4) G, X and P correspond to glucose, biomass and ethanol concentration respectively. In Eq. (4), μ_{ max }, P_{ max }, X_{ max }, Y_{ x } and Y_{ P } are functions of the fermentation temperature. Details regarding these expressions and the values of the constants in the model can be found elsewhere [29]. The above defined reaction rates describe the reaction processes that participate in the cSHF and cSSF operations. The particular reaction rates for each component in cSHF and cSSF processes are shown in Table 1.
We consider that the nonseparated pretreated material is free of fermentation inhibitors, because they were not produced due to optimized pretreatment conditions, or they were removed using suitable technologies. This allows us to concentrate our attention on the inhibitory effects of sugars over the enzymatic reaction rates as these compounds cannot be removed unless washed substrate is used. Additionally, the kinetic models used do not incorporate the effect of the inhibitors such as furfural or acetic acid. If, under these considerations the operation with nonseparated pretreated material results in worst results compared to washed material, then this simplification will not be important.
Attainable region: definitions and notation
Physically, a DSR corresponds to a PFR with a side feed stream distributed all along its length. It is interesting to note that, if α is equal to zero, then we have a PFR and if α is equal to 1/τ and the reactor operates in stationary state, then the reactor behaves as a CSTR.
Finally, extreme points are defined as points in R^{ n } that lie in a vertex of the convex hull. They can neither lie in the interior of the convex hull, nor in the interior of one of the hyper planes (lines) that bound the convex hull. In Figure 14 points A and B are not extreme points since they lie in the interior of the convex hull. Point C is not extreme either because it is along one of the lines between two vertices.
Now that the necessary terminology has been introduced, we are in position to present some necessary conditions that characterize the attainable region [17], this list is not exhaustive and more properties can be founded elsewhere [20]: (i) the AR must contain the feed point, (ii) the AR must be convex, (iii) all reaction rate vectors in the boundary of the AR (δAR) must be tangent, point inward or be equal to 0 and (iv) no negative of a rate vector in the complement (outside) of the AR, when extended, can intersect a point of δAR. Since, the feed point is attainable (even without mixing or reaction) condition (i) does not require further explanation. Condition (ii) is a consequence of the fact that a set of achievable points that is not convex can always be made convex by mixing. That is, mixing can fill in concave regions or spaces between two separates, yet achievable, regions. Recall the fact that a PFR follow a trajectory that is always tangent to the rate vector; then if condition (iii) is not satisfied, a vector in the AR frontier would point outwards the AR and hence using a suitable PFR it will be possible to extend the AR. Finally, if condition (iv) is not observed; then starting from a point on the AR, a CSTR could be used to reach the point in the complement of AR where the negative rate vector originates. That is, this vector and the vector defined by the difference between the outlet and feed concentrations would be collinear, and hence a CSTR can connect both points.
Conversion and yields definitions
Where f_{ SG }, f_{ SP } and f_{ GP } are stoichiometric coefficients equal to 1.111, 0.568 and 0.511 respectively. We also consider, for the sake of simplicity, that cellobiose and ethanol are not present in any feed stream and that the conversion of every reactor in the network is based on the values in the feed stream coming from the pretreatment reactor (either washed solids or nonseparated pretreatment material) as this stream represents the only feed stream of the reactors network.
Dimensionality reduction techniques
Although it is natural to describe the dimensions of the AR in terms of the total number of species in the reaction network, this may be unnecessary because they are generally not independent. This dependence is a consequence of quantities that preserve their values during the course of a reaction. Among others, the atomic balance on the reacting species must always hold and the constraint imposed by this balance allows projecting the concentrations during the course of the reaction into a lower dimension space of independent species. That is, the constraints imposed by an invariable quantity introduce new equations that can be used to reduce the number of degrees of freedom to the extent that the remaining variables of the problem may be illustrated graphically in two or three dimensions. These projections build upon the concept of reaction invariants [30] and have been used previously to reduce the number of dimensions in which the AR must be constructed [31]. Here, we applied the same dimensionality reduction technique. Although, the method can be best explained by example, first we introduce some necessary notation. Additionally, a simpler but lengthy approach is presented in the Additional file 1.
Consider a reacting system with i components, being n_{ i } the moles of species i at any time of the course of the reaction. Each component i is formed by a_{ ij } atoms of element j. Let, ∆n be a vector of changes of the number of component moles and A the atom/component matrix with entries a_{ ij }. From the atomic balance, it follows that: A∆n = 0. Considering that ∆n and A can be partitioned as: Δn = [Δn_{ dep }Δn_{ ind }] and A = [A_{ dep }A_{ ind }]. Where the subindices dep and ind stands for dependent and independent components. Replacing the partitioned matrices in the atomic balance, and with minor rearrangements, the dependent components change of moles can be calculated as: $\mathrm{\Delta}{n}_{\mathit{dep}}={A}_{\mathit{dep}}^{1}{A}_{\mathit{ind}}\mathrm{\Delta}{n}_{\mathit{ind}}$. Clearly, A_{ dep } has to be square and nonsingular.
This demonstrates that the change of the number of moles of water and cellobiose during the reaction course can be calculated as a function of the changes of glucose and cellulose. This also means that the AR of the enzymatic hydrolysis reaction has to be constructed in a twodimensional space of glucose and cellulose concentration or cellulose conversion and glucose yield (and not in a fourdimensional one). Since we are interested in the residence time of the different reactor configurations, we add this variable as the third dimension of the AR. Hence, the AR of enzymatic hydrolysis must be build in the 3dimentional space {x_{ S }, x_{ G }, τ}.
In the original model of ethanol fermentation, the parameters m_{ s } and m_{ p } in Eq. (4), have values that are close to zero so in this study these values were taken as zero. Two reasons explain this simplification. Firstly, under SSF conditions glucose concentrations reach a very low value during the reaction course. This is caused by the greater glucose demand by the biomass compared with the rate of glucose production from cellulose. Clearly, in these conditions bioethanol rate is not controlled by the glucose to ethanol rate, but by the cellulose to glucose rate. However, if the parameters m_{ s } and m_{ p } are not zero, then the ethanol production rate (r_{ p }) will be larger than the glucose production rate, which is clearly impossible. Secondly, if m_{ p } and m_{ s } are equal to zero, no important differences in the model predictions are observed under the conditions used in this study. In fact, if 100 g/L of glucose are taken as the initial concentration in a PFR, the only effect is a 2% increase in the residence time required for total glucose consumption and a 0.88% decrease in ethanol yield at 32°C.
Finally, our ability to calculate X and S as a function of P allow us to also calculate the reaction rates as a function of P exclusively. In other words, for each value of P in the {P, τ} plane we can calculate a reaction vector {r_{ p }, 1} that uniquely determines the trajectories of the CSTR and PFR reactors from a given feed point.
Finally, to construct the AR^{c} for cSSF only three dimensions in the concentration space are required. Although a more rigorous analysis can be performed using the dimensionality reduction technique used by Omtveit et al. [31], the same results can be obtained by applying the following reasoning. If the AR^{c} for cSHF can be built in the bidimensional space of {x_{ S }, x_{ G }} and the AR^{c} for glucose fermentation can be reduced to only one dimension of ethanol yield, then as the two reaction networks are linked by a component present in both networks (glucose) then 3 dimensions are needed to build the AR^{c} for cSSF: {x_{ S }, x_{ G }, x_{ P }}. This result implies that every reaction rate in the cSSF network can be calculated from conversions and yields {x_{ S }, x_{ G }, x_{ P }}.
Attainable region construction
 (i)
Calculate the PFR trajectory starting from the feed point. This trajectory can be calculated by solving Eq. (7) up to a preestablished residence time.
 (ii)
If the PFR trajectory is not convex, find the convex hull of the PFR by drawing mixing lines to fill the nonconvex parts.
 (iii)
Next, check along the boundary of the convex hull to see if any reaction vector points outward. If the reaction vector point outward over certain regions, then find the CSTRs that extend the region the most. If no reaction vector point outward, check if there are vectors in the complement of the AR^{c} that can be extrapolated back into the AR^{c}. If this is the situation, extend the region using appropriate reactors.
 (iv)
Find the new, enlarged convex hull. If a CSTR lies in the boundary, the reaction vector at this point must point out of the AR^{c}, and a PFR with feed point on the CSTR will extend the region.
 (v)
Repeat steps (iii) and (iv), alternating between PFRs and CSTRs until no reaction vectors point out over the AR^{c}, and the necessary conditions are met.
As was stated by Glasser and Hildebrandt [17], this constructive procedure implies that for a two dimensional system, the boundary of the attainable region “must be achieved by a sequential process and must consist of alternate straight lines and plugflow trajectories”.
For cSSF and cSHF (considering the residence time), the AR^{c} must be built in a three dimensional space. For cSSF, we choose cellulose conversion, glucose and ethanol yields as these dimensions since they provide useful insights regarding: the liquefaction process, as this process depends on cellulose conversion; the yield and productivity of the product of interest, related to ethanol conversion and the glucose yield since glucose is the compound that links the enzymatic hydrolysis and fermentation processes.
The construction of a three dimensional AR^{c} is far more difficult than the previously described process for two dimensions. Regardless of these difficulties, powerful theoretical results were derived in a series of papers [20–22]. These theoretical results were recently used to formulate an automated algorithm for AR^{c} construction [32] and we follow this algorithm to analyze the cSSF and cSHF reaction networks and build the candidate attainable regions. The algorithm can be summarized in the following steps:

Calculate the PFR and CSTR trajectories from the feed point. Stop the calculations when the maximum user defined value of residence time is achieved. Calculate the convex hull formed by these trajectories.

Create a set of constant feed rate (α) values such that α = [0, α_{1}, α_{2}, …, α_{ large }]. Calculate the DSR trajectories (Eq. (9)) for each α value from each available extreme point (such as feed point and equilibrium points). Then calculate the convex hull of these trajectories, eliminate the interior points and store the extreme points. These extreme points lie on the extreme DSR as defined by Feinberg [21].

If necessary, refine the set of α values to produce more points in the extreme DSR trajectory. A stopping criterion suitable for automation of the algorithm is given elsewhere [32], however we refined the set of α values manually.

From each extreme point on the DSR extreme trajectory, generate PFRs with feed points along these points. Calculate the convex hull of the enlarged region created by these trajectories.
We verified our ability to apply the above described methodology by reproducing the results of Example 1: 3D Van de Vusse type kinetics in Seodigeng et al. [32].
Software and computational tools
MATLAB® was used to perform all calculations in this work. To solve systems of ordinary differential equations (ODE), such as the ODEs that define the PFR and DSR trajectories, we used the MATLAB builtin ODE45 algorithm based on explicit Runge–Kutta formula. Systems of algebraic equations, defining CSTR trajectories, were solved using fmincon solver and its builtin interior point method [33]. For convex hull calculation, the MATLAB convhull solver was used. This tool is based on the Qhull algorithm developed by Barber et al. [34].
Abbreviations
 ARc:

Candidate attainable region
 cSSF:

Continuous simultaneous saccharification and fermentation
 cSHF:

Continuous separated hydrolysis and fermentation
 DW:

Dry weight
 nSPM:

Nonseparated pretreated material
 RNS:

Reactor network synthesis.
Declarations
Acknowledgements
Financial support granted to F. Scott by CONICYT's scholarship program (Comisión Nacional de Investigación Científica y Tecnológica) is gratefully acknowledged. This work was funded by Innova Chile Project 208–7320 Technological Consortium Bioenercel S.A.
Authors’ Affiliations
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