The In Silico Optimization of a Fed-Batch Reactor Used for the Enzymatic Hydrolysis of Chicory Inulin to Fructose by Employing a Dynamic Approach

Abstract

In recent years, inulin enzymatic hydrolysis has become a very promising alternative for producing fructose on a large scale. Genetically modified chicory was used to extract inulin of industrial quality. By using an adequate kinetic model from the literature, this study aimed to determine the optimal operating alternatives of a batch (BR) or fed-batch (FBR) reactor used for the hydrolysis of inulin to fructose. The operation of the FBR with a constant or variable/dynamic feeding was compared to that of the BR to determine which best maximizes reactor production while minimizing enzyme consumption. Multi-objective optimal solutions were also investigated by using the Pareto-optimal front technique. Our in-silico analysis reveals that, for this enzymatic process, the best alternative is the FBR operated with a constant control variable but using the set-point given by the (breakpoint) of the Pareto optimal front under the imposed technological constraints. This set point reported the best performances, regarding all the considered opposite economic objectives. Also, the FBR with a constant, but NLP optimal feeding, reported fairly good performances.

1. Introduction

Biocatalytic processes produce fewer by-products, consume less energy, and generate less environmental pollution, using smaller catalyst concentrations and more moderate reaction conditions compared to classical chemical catalysis [1]. By exhibiting a high selectivity and specificity, they are sustainable bioengineering routes in obtaining a wide range of products and are increasingly replacing some classical processes of fine chemical synthesis [2].

However, a crucial aspect in any realistic engineering analysis for process design, operation, control, and optimization relies on knowledge of an adequate and sufficiently reliable mathematical model of the process [3]. Such valuable kinetic models, based on the reaction mechanism and including key details (e.g., enzyme deactivation), have to ensure interpretable and reliable predictions of the behavior of the enzymatic process under various operating conditions [3,4,5].

In spite of their large volumes, enzymatic continuously mixed tank reactors, operating in batch (BR), or fed-batch (FBR) modes, are the most used because they ensure high mass transfer and rigorous temperature/pH control.

Concerning the reactor, an essential engineering problem refers to the development of optimal operating policies seeking economic goals, such as production maximization, minimizing raw-material consumption, and obtaining a product of high purity. To address this problem, one engineering approach (developed in this paper) aims to in-silico determine the optimal operating strategy of the reactor that achieves these goals.

In the BR/FBR case, its optimal operation problem can be solved in silico through two approaches: (a) offline (or ‘run-to-run’), with the optimal operating policy being determined using an adequate kinetic model previously identified based on experimental data (this paper) [6,7,8,9,10,11,12,13]; (b) online, by using a simplified, often empirical math model to obtain a state-parameter estimator based on the online recorded data (such as the classical Kalman filter) [10,14,15,16,17,18,19,20,21]. One of the in silico, offline analysis advantages is that it enables a comparison of performances of various bioreactor constructive/operating approaches [22,23].

Even if the enzymatic process (or bioprocess) kinetics and the biocatalyst characteristics (inactivation rate) are known, the in silico solution to this offline engineering problem is not an easy task, due to multiple contrary objectives and a significant degree of uncertainty of the model/constraints originating from multiple sources [14].

In spite of their low productivity, BRs are commonly used for slow processes (as is also the case here), because they are highly flexible and easy to operate [24], in various ways [23]: (i) Simple BR, where the substrate(s), biocatalysts, and additives are initially loaded in recommended amounts [2,22,25,26]. Usually, a single- or multi-objective BR optimization is performed offline to determine the best batch time and its initial load [14,27,28]. (ii) Batch-to-batch (BR-to-BR) optimization, by including a model updating step based on acquired information from past batches (so-called tendency modeling, not used here) to determine the optimal load of the next BR [7,8,9,15,29,30,31,32,33,34]. (iii) An optimally operated serial sequence of BRs (SeqBR) [33], which includes a series of BRs of approx. equal volumes. At the end of every BR, its content is transferred to the next BR, adjusting the reactants and/or biocatalyst(s) concentrations at optimal levels, determined a priori to ensure optimal SeqBR operation [9,33]. (iv) The semi-batch reactor (SBR) or fed-batch reactor (FBR), with an optimally varied feeding policy of biocatalysts/substrate(s) (not discussed here, see [22,23,35,36,37]. Usually, FBRs report better performances compared to other batch operating alternatives. However, they are more difficult to operate because they need previously prepared stocks of biocatalysts, and substrate(s), of different concentrations (determined a priori in silico), to be fed for every ‘time-arc’ of the batch (that is, a batch-time division in which the feeding composition is constant and self-understood—the feeding of time-‘arcs’ usually differs between them) [22,23,38,39,40]. The time-stepwise variable optimal feeding policy of SBR/FBR is determined off- [23] or online [20]. A comparative discussion of all the mentioned bioreactor operating alternatives is provided by Koller [41] and Maria [23].

Fructose is a sweetener of high value in the food and medicine industries. Similar to other polyols largely used as sweeteners (e.g., sorbitol, mannitol, xylitol, erythritol), it is produced on a large scale by using chemical, biochemical, or biological catalysis [42,43].

The chemical catalytic hydrogenation of glucose on Ni, Fe, Pt, or Fe-Ni alloy catalysts suffers from a large number of disadvantages: it consumes a lot of energy and occurs at high pressures (10–125 atm) and temperatures (100–140 °C), while the catalyst is very expensive. In addition, the large number of by-products formed during the reaction makes product purification costly [44].

Currently, fructose is produced by the enzymatic isomerisation of glucose to fructose on an Fe/CarbonBlack catalyst [43], or over some salts at 50–60 °C and pH = 7–8.5 [45]. The latter starts from the high-fructose syrup (HFCS) obtained from starch [45]. Then, after rough/fine filtration, ion exchange, and evaporation, a glucose isomerization step can be used to obtain high-fructose syrup (HFS, 42–55% fructose) [45,46,47,48]. This process, intensively studied and kinetically characterized, suffers from a series of inconveniences appropriately described in the footnote [d] of Table 1.

The biocatalytic routes to produce fructose are more convenient due to a large number of advantages: they consume less energy, by occurring under ambient conditions, and they produce less waste due to their high yield and selectivity, with the product being free of allergenic compounds. A short review of three enzymatic routes to produce fructose at an industrial scale is summarized in Table 1.

The recently developed two-step Cetus process (Figure 1, and Table 1) leads to obtaining a high-purity fructose [49,50,51,52]. In the first step, D-glucose is converted to keto-D-glucose (kDG) in the presence of pyranose 2-oxidase (P2Ox) and catalase (to avoid the quick inactivation of P2Ox by the H2O2 formed in the main oxidation reaction), at 25–30 °C and pH = 6–7 [22,51]. In the subsequent step, kDG is reduced to D-fructose by NAD(P)H-dependent ALR (aldose reductase, EC 1.1.1.21), at 25 °C and pH = 7 [53], the NAD(P)⁺ being in-situ or externally regenerated and re-used [54,55,56,57,58,59,60]. The process is not yet competitive due to the costly enzymes in both steps.

Another promising biochemical route is the enzymatic hydrolysis of inulin (Figure 2). Inulin is industrially extracted from genetically modified chicory crops [61,62,63,64,65], or from modified Aspergillus sp. cultures [62,65]. The reported high conversion and selectivity of the hydrolysis process and its simplicity make it a worthy industrial alternative for producing fructose of high quality. Details regarding enzyme purification, and the very high yields realized by the immobilized enzyme are given by [62,63,64,65].

In fact, inulin is a polyfructan found in many plants as a storage carbohydrate. It contains up to 70 units of D-fructose linked to terminal glucose, which means that inulin is a mixture of oligomers and polymers [61]. Consequently, inulin is a rich source of fructose, currently used as a macronutrient substitute or as a supplement added to foods. Being a prebiotic, strong interest has been shown in the industrial production of oligofructose and then of fructose from inulin, explaining a large amount of research in this area over recent decades [62,63,64,66]. The properties of inulin, listed in Table 2, indicate a high solubility, depending on its source, from 60 g/L at 10 °C to 400 g/L at 20–90 °C. However, the recommended concentration in manipulation is a maximum of 100–200 g/L due to its tendency to precipitate and increase solution viscosity [65]. The viscosity of the water solution is close to that of water for concentrations less than 100 g/L (max. 1.0055 cP [67]) but increases sharply for concentrated solutions [68]. Other properties of diluted inulin solutions can be found in the literature, being comparable to those of water (density of, ca., 1.024 g/mL at 55 °C [69]), with a molecular diffusivity of 1.3–1.7·10⁻¹⁰ m²/s [67,70] for solutions less than 100 g/L. The main component of inulin, fructose, is extremely soluble in water (ca., 22.2 M at room temperature [71]), with diluted solution properties similar to those of water (density of up to 1.1 g/mL [72]), a viscosity up to 1.2 cP [73], and a molecular diffusivity of 1.2·10⁻¹⁰ m²/s [74] for solutions less than 100 g/L. Only the viscosity increases very sharply for concentrated fructose solutions (more than 600 cP for 70% fructose [71]).

The fructose polymerisation degree in inulin (m) (Figure 2) depends on its origin, being m = 27–29 in commercially available inulin from dahlia, Jerusalem artichoke, or chicory. The content of inulin in plants also varies from 1% in banana, barley, or wheat to, ca., 2–7% in globe artichoke, leek, or onion, and even to, ca., 15–25% in chicory roots, dandelion, garlic, Jerusalem artichoke, and salsify. Genetically improved cultures of chicory might raise the inulin content, making its industrialization attractive for the production of fructose by inulin hydrolysis using inulinase (E.C. 3.2.1.7) [62].

The activity of purified free-inulinase is very high at 50–60 °C and pH = 4–6, but it decreases rapidly at higher temperatures (half-life of $t_{0.5}$ = 17 min at 60 °C compared to $t_{0.5}$ = 34 h at 50 °C [62]). Consequently, several enzyme immobilization alternatives have been searched for, with some being less successful (half-life of $t_{0.5}$ = 138 h at 40 °C and $t_{0.5}$ = 7.2 h at 50 °C in calcium alginate [75]), but others being more promising ($t_{0.5}$ = 21 days at 40 °C and $t_{0.5}$ = 1.1 days at 55 °C on Amberlite support [76]). The thermal stability of the enzyme decreases sharply with temperature, being one of the major causes of activity decay, and running temperatures higher than 60 °C are not recommended.

As another experimental observation, enzyme immobilization significantly decreases its activity. For instance, the fresh-enzyme reaction rate of 0.048 g/L.min is 4× higher than for the immobilized case (on Amberlite support). Immobilization on other supports has also been investigated, e.g., on aminoethylcellulose [77], chitin [78], amino-cellulofine beads [79], calcium alginate, agar, gelatin, cellulose [80,81], or macroporous ionic polystyrene beads [82]. In all cases, the enzyme activity decreases several times after immobilization. Some metal ions, such as Hg²⁺ or Ag⁺, strongly inhibit the enzyme activity, while others (Cu²⁺, Fe³⁺) have a little or negligible effect on the free/immobilized enzyme. Various sources of inulinase have been studied, including production from recombinant bacteria [62]. Immobilized inulinase from K. fragilis on yeast cells has also been tested, reporting promising results after 30 hrs of batch runs [83].

By adopting an adequate kinetic model from the literature [63,64], and starting from the nominal (non-optimal) operating conditions of Table 2 [63,64,66], the in silico analysis of this study aims to evaluate and compare the performances of several optimal operating policies of a BR and FBR used for inulin hydrolysis on a free (suspended) inulinase.

Several numerical rules were used in this respect in a novel computational methodology. Thus, the optimal initial load of the BR, or the time-stepwise variable feeding policy with multiple control variables of the FBR, was determined by using a nonlinear programming (NLP) procedure, or a Pareto-optimal front technique, seeking a single-objective optimization (i.e., fructose production maximization), or multiple-objective optimization (i.e., raw-material consumption minimization), in the presence of various technological constraints.

Table 1. Comparison between three enzymatic methods used for fructose synthesis.

Characteristics	Glucose Isomerization [a,d]	Two-Step Cetus Process [b]	Inulin Hydrolysis [c]
Number of steps	1	2	1
Conversion (%)	50 (limited by the equilibrium) [d]	99	99.5
Raw-material availability	Glucose from the starch of crops, molasses, cellulose, and food processing by-products [84]		Genetically modified chicory crop; cultures of Aspergillus sp.
Impurities in the product	Yes	Traces	Negligible
Reaction type	Enzymatic isomerization	Enzymatic oxidation (step 1), followed by enzymatic reduction (step 2)	Enzymatic hydrolysis
Enzyme mobility	Immobilized [d]	Free (suspended)	Immobilized
Enzyme stability, and other additives	Intracellular glucose-isomerase (e.g., Streptomyces murinus) of low stability; metal (Al) salts	Pyranose 2-oxidase (P2Ox) and catalase (step 1); aldose reductase and NAD(P)H (step 2); enzymes are very costly	Inulinase
Temperature	50–60 °C	25–30 °C (50–60 °C)/ 30 °C	55 °C (40–60 °C)
Reaction time	7 h	3–20 h (step 1); 25 h (step 2)	13 h
pH	7–8.5	6.5–7(−8.5); 7–8.5	5.5
Number of reaction steps	1 isomerization	2 oxidation (step 1), reduction (step 2)	1 hydrolysis
Coenzyme necessary?	No	Yes Catalase for step 1 to prevent P2Ox quick inactivation; NAD(P)H for step 2. NAD(P)H is continuously regenerated in situ	No
Product purification	Difficult [d]	Simple (due to high selectivity)	Simple (due to high selectivity)
Product purity	2–5% impurities [d]	High (99.9%)	High (99.9%)

[a] Process described by [45,62]. The raw-material HFCS is obtained from yeast hydrolysis (resulting in a mixture of 42% fructose, 50% glucose, and 8% other sugars) [45]. [b] Process described by [49,50,51,53]; NAD(P)H is continuously regenerated in situ [56]. [c] Process described by [61,62,63,64]. [d] This process suffers from a large number of disadvantages: (i) The reaction is thermodynamically limited to around 50% glucose conversion, making the subsequent fructose separation in large chromatographic columns very costly. (ii) Glucose isomerase is an intracellular enzyme with relatively poor stability, making its purification and immobilisation very difficult. (iii) The amylase used to carry out the starch saccharification (to obtain the raw material of HFCS) requires calcium ions for full activity, but calcium inhibits glucose isomerisation, requiring its removal by ion-exchange treatment prior to glucose isomerisation. (iv) The fructose product is still made impure by several other saccharides (such as aldose, which is an allergenic compound) [62,85,86,87,88,89].

Table 1. Comparison between three enzymatic methods used for fructose synthesis.

Characteristics	Glucose Isomerization [a,d]	Two-Step Cetus Process [b]	Inulin Hydrolysis [c]
Number of steps	1	2	1
Conversion (%)	50 (limited by the equilibrium) [d]	99	99.5
Raw-material availability	Glucose from the starch of crops, molasses, cellulose, and food processing by-products [84]		Genetically modified chicory crop; cultures of Aspergillus sp.
Impurities in the product	Yes	Traces	Negligible
Reaction type	Enzymatic isomerization	Enzymatic oxidation (step 1), followed by enzymatic reduction (step 2)	Enzymatic hydrolysis
Enzyme mobility	Immobilized [d]	Free (suspended)	Immobilized
Enzyme stability, and other additives	Intracellular glucose-isomerase (e.g., Streptomyces murinus) of low stability; metal (Al) salts	Pyranose 2-oxidase (P2Ox) and catalase (step 1); aldose reductase and NAD(P)H (step 2); enzymes are very costly	Inulinase
Temperature	50–60 °C	25–30 °C (50–60 °C)/ 30 °C	55 °C (40–60 °C)
Reaction time	7 h	3–20 h (step 1); 25 h (step 2)	13 h
pH	7–8.5	6.5–7(−8.5); 7–8.5	5.5
Number of reaction steps	1 isomerization	2 oxidation (step 1), reduction (step 2)	1 hydrolysis
Coenzyme necessary?	No	Yes Catalase for step 1 to prevent P2Ox quick inactivation; NAD(P)H for step 2. NAD(P)H is continuously regenerated in situ	No
Product purification	Difficult [d]	Simple (due to high selectivity)	Simple (due to high selectivity)
Product purity	2–5% impurities [d]	High (99.9%)	High (99.9%)

[a] Process described by [45,62]. The raw-material HFCS is obtained from yeast hydrolysis (resulting in a mixture of 42% fructose, 50% glucose, and 8% other sugars) [45]. [b] Process described by [49,50,51,53]; NAD(P)H is continuously regenerated in situ [56]. [c] Process described by [61,62,63,64]. [d] This process suffers from a large number of disadvantages: (i) The reaction is thermodynamically limited to around 50% glucose conversion, making the subsequent fructose separation in large chromatographic columns very costly. (ii) Glucose isomerase is an intracellular enzyme with relatively poor stability, making its purification and immobilisation very difficult. (iii) The amylase used to carry out the starch saccharification (to obtain the raw material of HFCS) requires calcium ions for full activity, but calcium inhibits glucose isomerisation, requiring its removal by ion-exchange treatment prior to glucose isomerisation. (iv) The fructose product is still made impure by several other saccharides (such as aldose, which is an allergenic compound) [62,85,86,87,88,89].

Table 2. Nominal operating conditions of the BR and its characteristics for the inulin hydrolysis case. Reaction conditions are those of Rocha et al. [66], and Ricca et al. [63,64] [a].

Operating Conditions	Value	Remarks
Reactor liquid volume	1 L (initial)	Up to 10 L capacity
Temperature/pressure/pH (buffer solution)	50–55 °C/normal/4.5–5	Batch time (tf) = 780 min.
Initial concentrations of Ricca et al. [64]	[S]o = 40 (g/L) [E]o = 97 (U/L) [W]o = 988.4 (g/L) [F]o = 0; [G]o = 0	To be optimized within imposed limits (this paper)
Optimization limits of control variables (initial BR, or in the FBR feeding) [63] [b,c,d]	[S]o; [S]in ∈ [40–200] g/L [E]o; [E]in ∈ [97–5500] (U/L) [W]o ∈ [98–4000] (g) FL ∈ [5 × 10−4–0.01] (L/min)	For FBR optimization, the W amount depends on the inlet feed flow rate (FL) of aqueous solution
Fructose polimerization degree in the inulin (m)	29 (adopted)	27–29 Inulin from chicory
Number of time-arcs for the optimized FBR (Ndiv)	5	FBR with variable feeding
Imposed inulin (S) conversion	Min. 90%
Inulin solubility [b]	60 g/L (10 °C) 160–400 g/L (50 °C), 330 g/L (90 °C)	[62,65,67]
Inulin solution viscosity, density [a]	Comparable to those of water	For [S] < 100 g/L [68,69]
Fructose solubility	4000 g/L (ca., 22.2 M) (25 °C)	https://en.wikipedia.org/wiki/Fructose (last accessing 5 May 2025)
Glucose solution solubility	5–7M (25–30 °C)	[90]
Glucose/fructose solution viscosity	Ca., 1–3 cps (for up to 0.3 M) and 1000 cps (4.5M, 30 °C)	[91]

[a] Physical liquid properties correspond to a solution of inulin (S) and fructose (F), with densities and viscosities given by [72,73,92]. Glucose is present in small concentrations, and its properties were assimilated with those of fructose. Molecular weights: M_F = 180.16 g/mol; M_S ≈ 504–5500 g/mol [https://link.springer.com/referenceworkentry/10.1007/978-3-319-03751-6_80-1 (last accessing 5 March 2025)]. [b] Higher concentrations of inlet inulin solution have also been reported (up to 200 g/L [66]). The inulin solubility depends on its source and purification method, varying from 60 g/L at 10 °C to 160–400 g/L at 50 °C, and to more than 330 g/L at 90 °C [62,65]. [c] Maximum enzyme activity is reported as, ca., 3000 U/mL (from K. marxianus var. marxianus CBS 6556) and, ca., 58,000 U/g (from Trametes multicolor [62]). [d] One unit (U) of inulinase activity is defined as the amount of enzyme necessary to produce 1 μmole of fructose by the hydrolysis of inulin over 1 min of reaction under the standard conditions of 60 °C and pH = 5 [63].

Table 2. Nominal operating conditions of the BR and its characteristics for the inulin hydrolysis case. Reaction conditions are those of Rocha et al. [66], and Ricca et al. [63,64] [a].

Operating Conditions	Value	Remarks
Reactor liquid volume	1 L (initial)	Up to 10 L capacity
Temperature/pressure/pH (buffer solution)	50–55 °C/normal/4.5–5	Batch time (tf) = 780 min.
Initial concentrations of Ricca et al. [64]	[S]o = 40 (g/L) [E]o = 97 (U/L) [W]o = 988.4 (g/L) [F]o = 0; [G]o = 0	To be optimized within imposed limits (this paper)
Optimization limits of control variables (initial BR, or in the FBR feeding) [63] [b,c,d]	[S]o; [S]in ∈ [40–200] g/L [E]o; [E]in ∈ [97–5500] (U/L) [W]o ∈ [98–4000] (g) FL ∈ [5 × 10−4–0.01] (L/min)	For FBR optimization, the W amount depends on the inlet feed flow rate (FL) of aqueous solution
Fructose polimerization degree in the inulin (m)	29 (adopted)	27–29 Inulin from chicory
Number of time-arcs for the optimized FBR (Ndiv)	5	FBR with variable feeding
Imposed inulin (S) conversion	Min. 90%
Inulin solubility [b]	60 g/L (10 °C) 160–400 g/L (50 °C), 330 g/L (90 °C)	[62,65,67]
Inulin solution viscosity, density [a]	Comparable to those of water	For [S] < 100 g/L [68,69]
Fructose solubility	4000 g/L (ca., 22.2 M) (25 °C)	https://en.wikipedia.org/wiki/Fructose (last accessing 5 May 2025)
Glucose solution solubility	5–7M (25–30 °C)	[90]
Glucose/fructose solution viscosity	Ca., 1–3 cps (for up to 0.3 M) and 1000 cps (4.5M, 30 °C)	[91]

[a] Physical liquid properties correspond to a solution of inulin (S) and fructose (F), with densities and viscosities given by [72,73,92]. Glucose is present in small concentrations, and its properties were assimilated with those of fructose. Molecular weights: M_F = 180.16 g/mol; M_S ≈ 504–5500 g/mol [https://link.springer.com/referenceworkentry/10.1007/978-3-319-03751-6_80-1 (last accessing 5 March 2025)]. [b] Higher concentrations of inlet inulin solution have also been reported (up to 200 g/L [66]). The inulin solubility depends on its source and purification method, varying from 60 g/L at 10 °C to 160–400 g/L at 50 °C, and to more than 330 g/L at 90 °C [62,65]. [c] Maximum enzyme activity is reported as, ca., 3000 U/mL (from K. marxianus var. marxianus CBS 6556) and, ca., 58,000 U/g (from Trametes multicolor [62]). [d] One unit (U) of inulinase activity is defined as the amount of enzyme necessary to produce 1 μmole of fructose by the hydrolysis of inulin over 1 min of reaction under the standard conditions of 60 °C and pH = 5 [63].

The paper presents a significant number of novelty aspects, as follows: (i) The engineering evaluation of this process with accounting for several reactor operation alternatives is a premiere in the literature. (ii) The way by which this difficult multi-objective optimization problem was successfully solved is a model that can be followed to solve similar enzymatic processes. (iii) The in silico (model-based) engineering analysis of a complex enzymatic process, leading to obtaining a Pareto-optimal operating policy of the approached BR is an approach seldom used in the literature. (iv) Confirmation that the Pareto-front ‘break-point’ choice proposed technique reported fairly good performances for a BR, from a multi-objectives perspective. (v) The scientific value of this paper is not virtual, as long as the numerical analysis is based on the kinetic model of Ricca et al. [63,64] constructed and validated by using extensive experimental data sets. (vi) The in-silico analysis suggests that, for this enzymatic process, the best alternative is the FBR operated with a constant, but using the set-point given by the (breakpoint) of the Pareto optimal front under technological constraints for the control variables. This set point reported the best performances, regarding all the considered opposite objectives. Also, the FBR with a constant, but NLP optimal feeding, reported very good performances.

2. The Experimental Enzymatic Reactor

The BR analyzed here was also used by [63,64,66] to study inulin hydrolysis and eventually derive a kinetic model of this process. The characteristics of this BR are presented in Table 2, while a reduced scheme is presented by Maria et al. [93]. The reactor includes a large number of components, so its operation is completely automated, with tight control of the pH, temperature, mixing speed, and feeding. Details about the reaction conditions and the control variable ranges are given in Table 2.

In the BR operating mode, substrate(s), biocatalysts, and additives are only initially loaded in optimum amounts, to be determined in silico by solving an optimization problem (product maximization here) in the presence of multiple technological constraints.

In the FBR operating mode, the substrate(s)/biocatalyst and additives (pH-control substances, etc.) are continuously added during the batch, following a time-stepwise variable (optimal) policy and a variable feed flow rate, to be determined in silico by solving an offline optimization problem (this paper and [23,38]) or an online one [20]. The FBR presents a similar construction to the BR, with similar modeling hypotheses. In both cases (BR or FBR), there is no discharge (effluent) during the batch.

3. Process Kinetics and Reactor Dynamic Model

The main reaction pathway, including the successive enzymatic hydrolysis of inulin, is presented in Figure 2. Based on the experimental data presented in Figure 3 (the blue kinetic curves), collected from a non-optimally operated BR of Table 2, using free enzyme, Ricca et al. [63,64] developed a kinetic model of this enzymatic process. This model, presented in Table 3, is able to fairly simulate the dynamics of the key species of the process [abbreviated S, F, W, G, and E] over a long batch (780 min). The differential mass balance of species was formulated by adopting an average fructose polymerisation degree in inulin of m = 29.

By using this adequate kinetic model, the in-silico engineering analysis developed here aims to determine the optimal operating strategy of a BR or a FBR, leading to high yields in fructose, with consuming less free enzyme, under the process conditions of Table 2. (50–55 °C/4.5–5 pH). For this purpose, multiple objectives have been employed, together with several control variables varied over large but feasible domains.

In brief, the enzymatic BR model is presented in Table 4 including the mass balances of the key species of the process: [S, F, W, G, E] (Table 3); all of them are observable, with a measurable concentration.

The overall hydrolysis reaction of Table 3 is of a Michaelis–Menten type, with the rate constants depending on the temperature following the classic Arrhenius law. More details about this reaction mechanism are given by Ricca et al. [63,64]. To estimate the rate constants of Table 3, by using the experimental kinetic curves of Figure 3 (blue curves), combined Lineweaver–Burk plots, with an NLP numerical rule, were applied. More details on the estimation procedure used, and on the statistical quality of the estimated rate constants, are given by Ricca et al. [63,64].

According to Ricca et al. [63,64], a 780 min batch time is sufficient to obtain a high S-conversion (96%) for the nominal (non-optimal) conditions of [S]o = 40 g/L, [E]o = 97 U/L, [W]o = 988 g/L (that is, 1 L), 50 °C, and pH = 5. A wider range of operating conditions was also investigated. Consequently, the feasible ranges of the reactor control variables were set at the values specified in (Table 2).

Table 3. Inulin hydrolysis reaction mechanism, and the reduced kinetic model proposed by Ricca et al. [63,64]. The commercial inulinase used was obtained from Aspergillus ficuum. Notations: S = inulin (substrate); F = fructose; W = water; G = glucose; E = enzyme; I_m = inulin with m-degree of fructose polymerisation; T = temperature (K); and M = molecular mass (g/mol). Indexes: S = substrate; E = enzyme; W = water; F = fructose; and G = glucose.

Reaction Pathway (Figure 2):
$I_m+H_{2}O\stackrel{inulinase}{\to }{I}_{m-1}+F$; $I_{m-1}+H_{2}O\stackrel{inulinase}{\to }{I}_{m-2}+F$; $•••••$ $I_3+H_{2}O\stackrel{inulinase}{\to }Sucrose+F$; $Sucrose+H_{2}O\stackrel{inulinase}{\to }G+F$; $I_m+H_{2}O\stackrel{inulinase}{\to }{I}_{m-q-1}+I_{q+1}$ The above consecutive scheme is approximated by the overall reaction: $S\left(I_m\right)+(m-1)H_{2}O\stackrel{E}{\to }(m-1)F+G$
Rate expressions:[a]	Rate constants:
$r_S=-{\displaystyle \frac{k_2{c}_E{c}_S}{K_m+c_S}}={\displaystyle \frac{v_{mS}{c}_S}{K_m+c_S}}$; $v_{mS}$ = $k_2{c}_E$ $r_F={\displaystyle \frac{d{c}_F}{dt}}={\displaystyle \frac{v_{mF}{c}_S}{K_m+c_S}}$; $v_{mF}$ = $\alpha v_{mS}$; Or, equivalently, one can write $r_F=-r_S{\displaystyle \frac{1}{\frac{m}{m-1}-\frac{M_W}{M_F}}}$ $r_W={\displaystyle \frac{d{c}_W}{dt}}=-{\displaystyle \frac{v_{mW}{c}_S}{K_m+c_S}}$; $v_{mW}$ = $v_{mF}{\displaystyle \frac{M_W}{M_F}}$ $r_G={\displaystyle \frac{d{c}_G}{dt}}={\displaystyle \frac{v_{mG}{c}_S}{K_m+c_S}}$; $v_{mG}$ = $v_{mF}{\displaystyle \frac{1}{m-1}}$	m = 29; $M_W$ = 18 g/mol $M_F$ = $M_G$ = 180 g/mol $k_2$ = $\mathit{exp}(23.22-9450/T)$, g/U·min [b] $K_m$ = $\mathit{exp}(27.4-7630/T)$, g/L $\alpha ={\displaystyle \frac{1}{m/(m-1){-M}_W/M_F}}$
T is better to keep it. Enzyme deactivation model:
- Adopted first-order model: $r_E=k_d{c}_E$,    ⇒    $c_E=c_{Eo}\mathit{exp}(-k_{d}t)$ Or, equivalently, one can write $r_E={\displaystyle \frac{d{c}_E}{dt}}=-k_{d}E$	$k_d$ = $\mathit{exp}(125-42300/T)$, 1/h (experimental, free enzyme)
- Other data from the literature: Free enzyme [75] Immobilized enzyme [75]	$k_d=\mathit{exp}(183.64-61440/T)$, 1/min $k_d=\mathit{exp}(109.22-36025/T)$, 1/min
- Other rate expressions (pseudo-second-order, not tested here): $r_E=k_d{c}_E(c_E/c_{Eo})$,    ⇒    $c_E=c_{Eo}/(1+k_{d}t)$ Free enzyme [76] Immobilized enzyme [76]	$k_d=\mathit{exp}(41.8-14599/T)$, 1/h $k_d=\mathit{exp}(49.4-17374/T)$, 1/h

Footnotes: [a] $c_S$, $c_F$, $c_W$, and $c_G$ are in g/L; $c_E$ is in U/L; and $r_j$ is in g/L·min. [b] The pre-exponential factor [exp(21.4)] was modified by Maria [22] to [exp(23.22)] to better match the experimental kinetic plots of Ricca et al. [64].

Table 3. Inulin hydrolysis reaction mechanism, and the reduced kinetic model proposed by Ricca et al. [63,64]. The commercial inulinase used was obtained from Aspergillus ficuum. Notations: S = inulin (substrate); F = fructose; W = water; G = glucose; E = enzyme; I_m = inulin with m-degree of fructose polymerisation; T = temperature (K); and M = molecular mass (g/mol). Indexes: S = substrate; E = enzyme; W = water; F = fructose; and G = glucose.

Reaction Pathway (Figure 2):
$I_m+H_{2}O\stackrel{inulinase}{\to }{I}_{m-1}+F$; $I_{m-1}+H_{2}O\stackrel{inulinase}{\to }{I}_{m-2}+F$; $•••••$ $I_3+H_{2}O\stackrel{inulinase}{\to }Sucrose+F$; $Sucrose+H_{2}O\stackrel{inulinase}{\to }G+F$; $I_m+H_{2}O\stackrel{inulinase}{\to }{I}_{m-q-1}+I_{q+1}$ The above consecutive scheme is approximated by the overall reaction: $S\left(I_m\right)+(m-1)H_{2}O\stackrel{E}{\to }(m-1)F+G$
Rate expressions:[a]	Rate constants:
$r_S=-{\displaystyle \frac{k_2{c}_E{c}_S}{K_m+c_S}}={\displaystyle \frac{v_{mS}{c}_S}{K_m+c_S}}$; $v_{mS}$ = $k_2{c}_E$ $r_F={\displaystyle \frac{d{c}_F}{dt}}={\displaystyle \frac{v_{mF}{c}_S}{K_m+c_S}}$; $v_{mF}$ = $\alpha v_{mS}$; Or, equivalently, one can write $r_F=-r_S{\displaystyle \frac{1}{\frac{m}{m-1}-\frac{M_W}{M_F}}}$ $r_W={\displaystyle \frac{d{c}_W}{dt}}=-{\displaystyle \frac{v_{mW}{c}_S}{K_m+c_S}}$; $v_{mW}$ = $v_{mF}{\displaystyle \frac{M_W}{M_F}}$ $r_G={\displaystyle \frac{d{c}_G}{dt}}={\displaystyle \frac{v_{mG}{c}_S}{K_m+c_S}}$; $v_{mG}$ = $v_{mF}{\displaystyle \frac{1}{m-1}}$	m = 29; $M_W$ = 18 g/mol $M_F$ = $M_G$ = 180 g/mol $k_2$ = $\mathit{exp}(23.22-9450/T)$, g/U·min [b] $K_m$ = $\mathit{exp}(27.4-7630/T)$, g/L $\alpha ={\displaystyle \frac{1}{m/(m-1){-M}_W/M_F}}$
T is better to keep it. Enzyme deactivation model:
- Adopted first-order model: $r_E=k_d{c}_E$,    ⇒    $c_E=c_{Eo}\mathit{exp}(-k_{d}t)$ Or, equivalently, one can write $r_E={\displaystyle \frac{d{c}_E}{dt}}=-k_{d}E$	$k_d$ = $\mathit{exp}(125-42300/T)$, 1/h (experimental, free enzyme)
- Other data from the literature: Free enzyme [75] Immobilized enzyme [75]	$k_d=\mathit{exp}(183.64-61440/T)$, 1/min $k_d=\mathit{exp}(109.22-36025/T)$, 1/min
- Other rate expressions (pseudo-second-order, not tested here): $r_E=k_d{c}_E(c_E/c_{Eo})$,    ⇒    $c_E=c_{Eo}/(1+k_{d}t)$ Free enzyme [76] Immobilized enzyme [76]	$k_d=\mathit{exp}(41.8-14599/T)$, 1/h $k_d=\mathit{exp}(49.4-17374/T)$, 1/h

Footnotes: [a] $c_S$, $c_F$, $c_W$, and $c_G$ are in g/L; $c_E$ is in U/L; and $r_j$ is in g/L·min. [b] The pre-exponential factor [exp(21.4)] was modified by Maria [22] to [exp(23.22)] to better match the experimental kinetic plots of Ricca et al. [64].

Table 4. Mass balances of key species in the BR model by including the kinetic model of the enzymatic process together with the associated rate constants of Table 3. The ideal model hypotheses of Maria [23], Moser [94], and Dutta [95] assume a homogeneous liquid composition, with negligible mass transport resistance in the bulk phase.

Species	Remarks
Species mass balances: ${\displaystyle \frac{d{c}_j}{dt}}={\displaystyle \sum _{i=1}^{n_r}{\nu }_{ij}{r}_i}\left(\mathit{C}(t),{\mathit{C}}_{\mathit{o}},\mathit{k}\right)$; ‘j’ = species index (S, F, W, G, E), With the initial conditions of $c_{j,o}=c_j\left(t=0\right)$, where ‘i’ = (S, E, W) are to be optimized; $c_{j,o}$= 0, for j = (F, G).	The species reaction rate ($r_i$) expressions, the rate constants, and the stoichiometry (νij) are given in Table 3. Enzyme (E) deactivation is included in this dynamic balance. The optimal initial load of the BR (Table 6) is determined offline by solving in silico the associated NLP optimization problem (this paper). C = species concentration vector; k = rate constants vector.

Table 4. Mass balances of key species in the BR model by including the kinetic model of the enzymatic process together with the associated rate constants of Table 3. The ideal model hypotheses of Maria [23], Moser [94], and Dutta [95] assume a homogeneous liquid composition, with negligible mass transport resistance in the bulk phase.

Species	Remarks
Species mass balances: ${\displaystyle \frac{d{c}_j}{dt}}={\displaystyle \sum _{i=1}^{n_r}{\nu }_{ij}{r}_i}\left(\mathit{C}(t),{\mathit{C}}_{\mathit{o}},\mathit{k}\right)$; ‘j’ = species index (S, F, W, G, E), With the initial conditions of $c_{j,o}=c_j\left(t=0\right)$, where ‘i’ = (S, E, W) are to be optimized; $c_{j,o}$= 0, for j = (F, G).	The species reaction rate ($r_i$) expressions, the rate constants, and the stoichiometry (νij) are given in Table 3. Enzyme (E) deactivation is included in this dynamic balance. The optimal initial load of the BR (Table 6) is determined offline by solving in silico the associated NLP optimization problem (this paper). C = species concentration vector; k = rate constants vector.

Table 5. Mass balances of key species in the fed-batch bioreactor FBR model by including the kinetic model of the enzymatic process together with the associated rate constants of Table 3. The ideal model hypotheses of Maria et al. [96] assume a homogeneous liquid composition by neglecting the mass transport resistance in the bulk phase. The time-stepwise variable feeding is made over Ndiv time arcs (adopted Ndiv = 5 here), where Ndiv is the number of equal time arcs, in which the batch time (tf) is divided. The control variables are $C_{S,\mathit{inlet},j}$, $C_{E,\mathit{inlet},j}$, and $F_{L,j}$, with j = 1, … Ndiv.

Species	Remarks
Species mass balances: ${\displaystyle \frac{d{C}_i}{dt}}={\displaystyle \frac{F_{L,j}}{V_L}}\left(C_{i,inlet,j}-C_i\right)\pm {\mathit{r}}_{\mathit{i}}\left(\mathit{C}\left(t\right),{\mathit{C}}_{\mathit{o}},\mathit{k}\right)$; $C_{i,o}=C_i\left(t=0\right)$, for species ‘’' = S, F, G, E; $C_{i,o}=C_{i,\mathit{inlet},1}$, for ‘’' = S, E; $C_{i,\mathit{inlet},j}$, $F_{L,j}$= control variables. ‘’' = S,E; ‘j’ = 1,.., $N_{div}$; unknown time-stepwise values to be determined from the FBR optimization. For species W, the mass balance is [b] ${\displaystyle \frac{dW}{dt}}={\displaystyle \frac{F_{L,j}}{V_L}}\left(-W\right)+r_w\left(\mathit{C}\left(t\right),{\mathit{C}}_{\mathit{o}},\mathit{k}\right)+F_{L,j}{\rho }_W$ [W]o = 988 g/L; $c_{j,o}$= 0, for ‘j’ = (F, G).	For the optimal FBR with the adopted Ndiv = 5, the feeding policy is (Footnote [a]) $\left[C_{S,inlet}{;\mathit{C}}_{\mathit{E},\mathit{inlet}}\right]=\left\{\begin{array}{c}\left[C_{S,inlet,1}{;\mathit{C}}_{\mathit{E},\mathit{inlet},1}\right]\mathrm{if}0\le t<\mathrm{T}1\\ \left[C_{S,inlet,2}{;\mathit{C}}_{\mathit{E},\mathrm{inlet},2}\right]\mathrm{if}\mathrm{T}1\le t<\mathrm{T}2\\ \left[C_{S,inlet,3}{;\mathit{C}}_{\mathit{E},\mathit{inlet},3}\right]\mathrm{if}\mathrm{T}2\le t<\mathrm{T}3\\ \left[C_{S,inlet,4}{;\mathit{C}}_{\mathit{E},\mathit{inlet},4}\right]\mathrm{if}\mathrm{T}3\le t<\mathrm{T}4\\ \left[C_{S,inlet,5}{;\mathit{C}}_{\mathit{E},\mathit{inlet},5}\right]\mathrm{if}\mathrm{T}4\le t\le \mathrm{T}5=t_f\end{array}\right.$
Liquid volume in the reactor (footnote [c]): ${\displaystyle \frac{d{V}_L}{dt}}=F_{L,j}$; $F_{L,j}$= control variable; ‘j’ = 1, ..., $N_{div}$; unknown time-stepwise values to be determined from the FBR optimization. The unknown $F_{L,0}$ = $F_L$(t = 0) = $F_{L,1}$ is determined together with all the $F_{L,j}$ values.	For the optimal FBR with the adopted Ndiv =10, the feeding policy is (Footnote [a]) ${\mathit{F}}_{\mathit{L},j}=\left\{\begin{array}{c}{\mathit{F}}_{\mathit{L},0} \mathrm{if} 0\le t<\mathrm{T}1\\ {\mathit{F}}_{\mathit{L},1} \mathrm{if} \mathrm{T}1\le t<\mathrm{T}2\\ {\mathit{F}}_{\mathit{L},2} \mathrm{if} \mathrm{T}2\le t<\mathrm{T}3\\ {\mathit{F}}_{\mathit{L},3} \mathrm{if} \mathrm{T}3\le t<\mathrm{T}4\\ {\mathit{F}}_{\mathit{L},4} \mathrm{if} \mathrm{T}4\le t\le \mathrm{T}5=t_f\end{array}\right.$

Footnotes: [a] For the adopted N_div = 5, the j = 1, …, $N_{div}$ time-arc switching points are as follows: T1 = $t_f$/ $N_{div}$ = 156 min; T2 = 2 × T1; T3 = 3*T1; T4 = 4*T1; and T5 = $t_f$ = 780 min. [b] The water density is ${\rho }_W$ = 988 g/L (50 °C). [c] The $F_{L,j}$ time-stepwise feed flow rates are determined simultaneously with the other control variables (that is, $C_{E,\mathit{inlet},j}$ and $C_{S,\mathit{inlet},j}$) to ensure optimal FBR operation.

Table 5. Mass balances of key species in the fed-batch bioreactor FBR model by including the kinetic model of the enzymatic process together with the associated rate constants of Table 3. The ideal model hypotheses of Maria et al. [96] assume a homogeneous liquid composition by neglecting the mass transport resistance in the bulk phase. The time-stepwise variable feeding is made over Ndiv time arcs (adopted Ndiv = 5 here), where Ndiv is the number of equal time arcs, in which the batch time (tf) is divided. The control variables are $C_{S,\mathit{inlet},j}$, $C_{E,\mathit{inlet},j}$, and $F_{L,j}$, with j = 1, … Ndiv.

Species	Remarks
Species mass balances: ${\displaystyle \frac{d{C}_i}{dt}}={\displaystyle \frac{F_{L,j}}{V_L}}\left(C_{i,inlet,j}-C_i\right)\pm {\mathit{r}}_{\mathit{i}}\left(\mathit{C}\left(t\right),{\mathit{C}}_{\mathit{o}},\mathit{k}\right)$; $C_{i,o}=C_i\left(t=0\right)$, for species ‘’' = S, F, G, E; $C_{i,o}=C_{i,\mathit{inlet},1}$, for ‘’' = S, E; $C_{i,\mathit{inlet},j}$, $F_{L,j}$= control variables. ‘’' = S,E; ‘j’ = 1,.., $N_{div}$; unknown time-stepwise values to be determined from the FBR optimization. For species W, the mass balance is [b] ${\displaystyle \frac{dW}{dt}}={\displaystyle \frac{F_{L,j}}{V_L}}\left(-W\right)+r_w\left(\mathit{C}\left(t\right),{\mathit{C}}_{\mathit{o}},\mathit{k}\right)+F_{L,j}{\rho }_W$ [W]o = 988 g/L; $c_{j,o}$= 0, for ‘j’ = (F, G).	For the optimal FBR with the adopted Ndiv = 5, the feeding policy is (Footnote [a]) $\left[C_{S,inlet}{;\mathit{C}}_{\mathit{E},\mathit{inlet}}\right]=\left\{\begin{array}{c}\left[C_{S,inlet,1}{;\mathit{C}}_{\mathit{E},\mathit{inlet},1}\right]\mathrm{if}0\le t<\mathrm{T}1\\ \left[C_{S,inlet,2}{;\mathit{C}}_{\mathit{E},\mathrm{inlet},2}\right]\mathrm{if}\mathrm{T}1\le t<\mathrm{T}2\\ \left[C_{S,inlet,3}{;\mathit{C}}_{\mathit{E},\mathit{inlet},3}\right]\mathrm{if}\mathrm{T}2\le t<\mathrm{T}3\\ \left[C_{S,inlet,4}{;\mathit{C}}_{\mathit{E},\mathit{inlet},4}\right]\mathrm{if}\mathrm{T}3\le t<\mathrm{T}4\\ \left[C_{S,inlet,5}{;\mathit{C}}_{\mathit{E},\mathit{inlet},5}\right]\mathrm{if}\mathrm{T}4\le t\le \mathrm{T}5=t_f\end{array}\right.$
Liquid volume in the reactor (footnote [c]): ${\displaystyle \frac{d{V}_L}{dt}}=F_{L,j}$; $F_{L,j}$= control variable; ‘j’ = 1, ..., $N_{div}$; unknown time-stepwise values to be determined from the FBR optimization. The unknown $F_{L,0}$ = $F_L$(t = 0) = $F_{L,1}$ is determined together with all the $F_{L,j}$ values.	For the optimal FBR with the adopted Ndiv =10, the feeding policy is (Footnote [a]) ${\mathit{F}}_{\mathit{L},j}=\left\{\begin{array}{c}{\mathit{F}}_{\mathit{L},0} \mathrm{if} 0\le t<\mathrm{T}1\\ {\mathit{F}}_{\mathit{L},1} \mathrm{if} \mathrm{T}1\le t<\mathrm{T}2\\ {\mathit{F}}_{\mathit{L},2} \mathrm{if} \mathrm{T}2\le t<\mathrm{T}3\\ {\mathit{F}}_{\mathit{L},3} \mathrm{if} \mathrm{T}3\le t<\mathrm{T}4\\ {\mathit{F}}_{\mathit{L},4} \mathrm{if} \mathrm{T}4\le t\le \mathrm{T}5=t_f\end{array}\right.$

Footnotes: [a] For the adopted N_div = 5, the j = 1, …, $N_{div}$ time-arc switching points are as follows: T1 = $t_f$/ $N_{div}$ = 156 min; T2 = 2 × T1; T3 = 3*T1; T4 = 4*T1; and T5 = $t_f$ = 780 min. [b] The water density is ${\rho }_W$ = 988 g/L (50 °C). [c] The $F_{L,j}$ time-stepwise feed flow rates are determined simultaneously with the other control variables (that is, $C_{E,\mathit{inlet},j}$ and $C_{S,\mathit{inlet},j}$) to ensure optimal FBR operation.

4. Optimization Problem for BR and FBR

4.1. Selection of Control Variables

By analyzing the dynamic models of BR and FBR of Table 4 and Table 5, respectively, the chosen control variables are those related to the initial reactor load or its variable feeding:

BR case: Initial load of [S]o, [E]o, and [W]o (substrate, enzyme, and water, respectively).

FBR case: The feed characteristics for every time-division (arc), that is, $C_{S,\mathit{inlet},j}$, $C_{E,\mathit{inlet},j}$, and $F_{L,j}$, with j = 1, …, Ndiv (number of equal time arcs in which the batch time is divided).

4.2. NLP Optimization with a Single Objective Function (Ω)

Optimization of a BR operation translates to finding its initial load with the key species mentioned in Section 4.1 (that is, three unknowns in the present case, see the Table 4), by using a common nonlinear programming (NLP) optimization rule, seeking to determine the extreme of an objective function in the presence of multiple constraints. In the present case, this problem refers to the maximization of [F] (fructose) production:

Given [F]o = 0 and [G]o = 0,
find control variables [S]o, [E]o, and [W]o such that
Max Ω(C, C_o, k), where Ω = [F(t)]

(1A)

For the FBR case, the batch time is divided into Ndiv equal time- arcs (Ndiv = 5 adopted here). The control variables ${\mathit{C}}_{\mathit{S},\mathit{inlet},\mathit{j}}$, ${\mathit{C}}_{\mathit{E},\mathit{inlet},\mathit{j}}$, and ${\mathit{F}}_{\mathit{L},\mathit{j}}$, with j = 1, …, Ndiv, are kept constant over every time-arc at optimal values to be determined by solving the optimization problem shown in Equation (1B), with using the model of Table 5. The self-understood control variables may differ between different time-arcs. The time intervals of equal lengths Δt = t_f/N_div are obtained by dividing the batch time t_f into N_div parts t_j−1 ≤ t ≤ t_j, where t_j = j·Δt are switching points (where the reactor input is continuous and differentiable) [20,22,23,38,39,93,96]. In the present case, the switching points are presented explicitly in Table 5.

Given [F]o = 0 and [G]o = 0, find the following control variables:
C_S,inlet,j, C_E,inlet,j, and F_L,j, B
for j = 1, …, N_div, with the adopted N_div = 5 time-arcs, and the initial FBR condition of Table 5, to obtain
Max Ω(C, C_o, k), where Ω = [F(t)]

(1B)

The optimal FBR principle indicates obtaining an optimal feeding policy consisting of a time-stepwise variation in the control variables (i.e., feeding liquid flow rate and the concentrations of the added substrates and biocatalyst) over the adopted N_div = 5 equal ‘time-arcs’ of the batch [23,96]. This implies dividing the batch time into equal N_div time-‘arcs’, with the feeding being constant over each time-arc, but at different values between them. The suitable choice of a small N_div is discussed by Maria [23].

In Equation (1A,B), the time-varying [F(t)] is a multi-variable function F(C(t),C_o,k)(t), evaluated by using the process/reactor model of Table 4 and Table 5, respectively, over the whole batch time (t) ∈ [0, t_f]. As an observation, Figure 3, Figure 4A and Figure 5A reveal that, in the present case study, the maximum [F] is reached at the batch end. Notations: C = species concentration vector; C_o = initial value of C; k = kinetic model rate constant vector.

As an observation, other optimization objectives can be applied as well, such as a multi-objective one [39,56]. However, the adopted single-objective optimization, seeking the main goal, presents the advantage of simplicity and easy application and interpretation.

The [F(t)] time-evolution is determined by solving the reactor dynamic model of Table 4, or Table 5 over the whole batch time (t) ∈ [0, t_f], with the initial condition of C_j,0 = C_j(t = 0) searched during the optimization iterative numerical rule. The dynamic model solution was obtained with enough precision by using the low-order stiff integrator (ode15s) of the MATLAB computational package.

4.3. Optimization Problem Constraints

The nonlinear optimization problem (NLP) formulated above, Equations (1A) or (1B), must account for the following constraints:

(a)

The BR model of Table 4 including the process kinetic model (Table 3);

(b)

The FBR model of Table 5, including the process kinetic model (Table 3);

(c)

To limit the excessive consumption of raw materials, feasible searching ranges are imposed on the control/decision variable, as stipulated in Table 2.

To be considered in the optimization problems, Equation (1A) or (1B), these constraints should be ‘translated’ to a mathematical form. Eventually, the resulting NLP optimization problem is highly non-convex and nonlinear, being subjected to the following technological/physical meaning/model constraints:

Nonlinear process and reactor model:

Table 4 for the BR case.
Table 5 for the FBR case.

(2i)

Physical significance constraints:
c_j(t) ≥ 0, in Table 4 and Table 5, for all the species of index ‘j’ and for all t ∈ [0-t_f]

(2ii)

Searching ranges for the control variables are given in Table 2:
[S]o; [S]in ∈ [40–400] g/L
[E]o; [E]in ∈ [97–5500] (U/L)
[W]o ∈ [988–4000] (g)
FL ∈ [5 × 10⁻⁴–0.01] (L/min)

(2iii)

V_L ≤ 10 L (reactor capacity)

(2iv)

4.4. Pareto-Optimal Front Optimization with Opposite Objective Functions

When multiple opposite objective functions are formulated for an optimization problem, an elegant alternative is to use the Pareto-optimal front technique. Each Pareto-optimal front (curve) accounts for two opposite optimization objectives.

Following the Pareto-front definition [97], any running point from the Pareto-curve can be a valid solution to the optimization problem. With the Pareto-curve being a continuous one, when two opposite optimization criteria are used, an infinity of Pareto-optimal operating solutions can exist. Consequently, the chosen solution (that is, the optimal operation set-points of the FBR here) is subjective and case-dependent, and it should be chosen by adding a criterion not accounted for when the Pareto-front was generated. It should be noted that many Pareto-fronts of different shapes can exist for the same optimization problem [93].

In the present case study, by analyzing the FBR model of Table 5 and its control variables, at least three Pareto fronts can be found for the case of constant optimal feeding, as presented in Equation (3). Additional objectives can exist, but the present analysis was limited to the following ones.

Maximum F production vs. minimum substrate (S) consumption.
Minimum constant feed flow rate for various maximum F produced.
Maximum F production vs. minimum enzyme (E) consumption.

(3)

As the literature reveals [97], a lot of Pareto-optimal fronts are monotonous and of an exponentially/logarithmic-like shape. Extended studies of [39,56,98] for such kind of shapes, made on various (bio)chemical reacting systems, indicated that the ‘break-point’ of the Pareto-front (Figure 6 and Figure 7 in the present case) is a good choice for the optimization problem with at least two opposite objectives. As recommended, this ‘break-point’ should be, ca., 5% higher on the ordinate than the ‘baseline’ of the Pareto-curve. This is because a running point on the baseline (on the left of the ‘break-point’ in Figure 6) is not economic, by reporting a lower production of the target product; on the contrary, a running point chosen on the right of the ‘break-point’ reports a higher productivity, but with the cost of an increasingly higher enzyme consumption, and a violation of the technological constraints (Equation (2)). In the present case, the chosen ‘break-point’ of Figure 6 and Figure 7 is presented in Table 6 and reports better performances compared to other FBR operating alternatives.

4.5. The Used Solvers

During BR/FBR optimization, when simulating the reactor dynamics in every solver iteration, the model species bulk-phase concentrations are obtained by solving the dynamic model of the reactor (Table 4 or Table 5, respectively), with the initial condition $c_{j,o}$ (t = 0) being the iterative guess made by the numerical solver. The imposed batch time $t_f$ and the medium conditions are those of Table 1. The dynamic model solution was obtained with high precision by using the variable-order stiff integrator (‘ode15s’) of the MATLAB (R2010a) package.

Because the differential models of the reactor, the optimization objectives of Equation (1A,B), and the problem constraints Equation (2i–iv) are all highly nonlinear, the Pareto-optimal fronts were obtained by using a multimodal search algorithm, that is, ‘gamultiobj’ of the MATLAB package. This numerical algorithm is suitable for this non-convex and nonlinear case. The computational time was reasonably short (minutes) using a common PC, thus offering a quick implementation of the optimal BR operating policy obtained offline.

Because the enzymatic reactor/process kinetic model (Table 4 and Table 5), the optimization objectives of Equation (1A,B), and the problem constraints of Equation (2i–iv) are all highly nonlinear, the formulated problems of Equation (1A,B) translate into a nonlinear optimization problem (NLP) with a multimodal objective function and a non-convex searching domain. To obtain the global feasible solution with enough precision, the multi-modal optimization solver MMA of Maria [4,99,100] was used, starting from different initial guesses, being proven to be very effective compared to common (commercial) optimization algorithms.

5. Optimization Results and Their Discussion

The results for solving the NLP optimization problem are presented in the following forms:

(a)

Figure 3 displays the optimal NLP operating policy for the analyzed BR, by comparison with the experimental data of Ricca et al. [64] obtained in a BR operated under non-optimal conditions of Table 2.

(b)

A comparison of all BR operating alternatives in terms of fructose (F) production and raw-material consumption is in Table 6. In the BR case, this consumption is based only on the initial load. In the FBR case, the raw-material consumption (mass) is computed with the following formula:

$$m_{species}={\displaystyle \sum _{j=1}^{N_{div}}{F}_{L,j}}{\left[species\right]}_{inlet,j}\mathrm{\Delta }{t}_j$$

(4)

(c)

The optimally operated FBR, with a constant but optimal NLP feeding in Figure 4.

(d)

The optimally operated FBR with a variable but optimal NLP feeding in Figure 5.

By analyzing these results and, in particular, the operating alternatives of Table 6, several conclusions can be derived:

(1)

The optimal NLP-operated BR, according to Equation (1A), under the constraints of Equation (2i–iv) for the control variables, reported incomparably better performances (5× in terms of more F produced, at the expense of consuming 5× more substrates and 30× more enzymes) compared to the experimental non-optimal BR trial of Ricca et al. [64] (in Table 6 and Figure 3).

(2)

By far, the best alternative is the FBR operated with a constant but optimal NLP feeding (Equation (1A)), or operated using the set-point (break-point) given by the Pareto-optimal front, Equation (3). Even if the F-production is similar to those of the optimal NLP-operated BR, the substrate consumption is 13×–15× lower, by consuming 15×–92× less enzymes (Table 6 and Figure 4). As revealed by the results of Table 6, the FBR operated with a constant, but using the set-point (break-point) given by the Pareto-optimal front (Equation (3)), under the constraints of Equation (2i–iv) for the control variables, reported the best performances, regarding all the objectives mentioned above.

(3)

By analyzing the FBR with an optimal NLP variable feeding, the results are quite modest. In spite of the realized good F-production, compared to the FBR operated with constant but Pareto-optimal feeding, the FBR with an optimal NLP variable feeding reported higher raw-material consumption (90× more enzymes and 13× more substrates).

(4)

Of course, enzyme stabilization by immobilization is expected to improve the process performances, as reviewed in the Introduction section [101].

(5)

The optimal FBR control strategy is very adaptable, which is because the employed process kinetic model of moderate complexity is flexible enough due to a fairly large number of rate constants. Thus, if significant inconsistencies are observed between the model-predicted bioreactor dynamics and the recorded data, then an intermediate numerical analysis step will be applied to improve the model adequacy (i.e., a ‘model updating’ step), and the bioreactor optimization is applied again with the novel model. This evolutionary adaptation of the enzymatic process model is the so-called ‘tendency modeling’ [34].

Table 6. Efficiency of BR/FBR (Table 2) operated using various alternatives, for the enzymatic hydrolysis of inulin to fructose, by using a batch time of 780 min for all cases. A total conversion is realized in all the cases.

Reactor Operation				Raw-Material Consumption [b]		Max F (Fructose), (g) [b]	Final VL (L)
Type	Ndiv	Control Variables		S (Inulin), (g) (Equation (4))	E (Enzyme) (U) (Equation (4))		[a]
BR Non-optimal, Ricca et al. [64]	1	Nominal load [c,f] (Figure 4)		40	9.7 (poor)	41.05	1
		[S]o	40
		[E]o	9.7
		Wo	988.4
BR Optimal load NLP (this paper) [h]	1	Initial load [f,b,h] (Figure 3)		200	302 (fairly good)	213.7	2 [g]
		[S]o	200
		[E]o	301.87
		Wo	2000
FBR Constant but optimal NLP feeding (this paper) [d]	1	Optimal feeding [f,j] (Figure 4)		156	2145.9 (almost best)	426.9	1.4
		[S]in	400
		[E]in	5500
		FL,in	5 × 10−4
FBR Constant but Pareto-optimal feeding (this paper) [d]	1	Optimal feeding [f,j]		180.4	357.9 (best)	422.9	1.4
		[S]in	399.88
		[E]in	793.19
		FL,in	5.78 × 10−4
FBR Variable optimal NLP feeding (this paper) [e]	5	Optimal feeding [f,j] (Figure 5)		2393.7	3.29 × 104 (high consumptions and dilution)	428	6.98
		[S]in [40–400]	variable Figure 5E
		[E]in [97–5500]	variable Figure 5D
		FL,in [5 × 10−4–0.01]	variable Figure 5C

Footnotes: [a] Initial liquid volume VL,o = 1 L. [b] The displayed digits come from the numerical simulations. [c] The checked BR set-point of Ricca et al. [64]. [d] The FBR operation with a constant feeding over time for all the control variables (Table 5): $F_{L,1}=F_{L,2}=F_{L,3}=F_{L,4}$; $C_{S,inlet,1}=C_{S,inlet,2}=C_{S,inlet,3}=C_{S,inlet,4}$; $C_{E,inlet,1}=C_{E,inlet,2}=C_{E,inlet,3}=C_{E,inlet,4}$ with the only 3 variables to be optimized being the initial inlet values of $F_{L,1}$, $C_{S,inlet,1}$, and $C_{E,inlet,1}$, under the constraints of Equation (2i–iv). See the resulting optimal FBR operating policy in Figure 4. [e] The optimal FBR time-stepwise variable feeding policy is obtained by using the control variable limits of footnote [j]. In this FBR operating case, the control variables, $F_{L,j}$, $C_{S,inlet,j}$, and $C_{E,inlet,j}$, where j = 1,…(Ndiv -1), of Table 5 follow an uneven policy in being optimized (that is, 15 unknowns for Ndiv = 5). The optimal control variable policy is given in Figure 5. [f] The units are as follows: [S] g/L; [E] U/L; [W] g; FL L/min. [g] The volume corresponds to the water (W) mass required by the reaction. [h] Search intervals of the control variables are as follows: [S]in = [40–200] g/L; [E]in = [97–5500] U/L. [j] Search intervals of control variables are as follows: [S]in = [40–400] g/L; [E]in = [97–5500] U/L; FL,in = [5 × 10⁻⁴–0.01] L/min [62].

Table 6. Efficiency of BR/FBR (Table 2) operated using various alternatives, for the enzymatic hydrolysis of inulin to fructose, by using a batch time of 780 min for all cases. A total conversion is realized in all the cases.

Reactor Operation				Raw-Material Consumption [b]		Max F (Fructose), (g) [b]	Final VL (L)
Type	Ndiv	Control Variables		S (Inulin), (g) (Equation (4))	E (Enzyme) (U) (Equation (4))		[a]
BR Non-optimal, Ricca et al. [64]	1	Nominal load [c,f] (Figure 4)		40	9.7 (poor)	41.05	1
		[S]o	40
		[E]o	9.7
		Wo	988.4
BR Optimal load NLP (this paper) [h]	1	Initial load [f,b,h] (Figure 3)		200	302 (fairly good)	213.7	2 [g]
		[S]o	200
		[E]o	301.87
		Wo	2000
FBR Constant but optimal NLP feeding (this paper) [d]	1	Optimal feeding [f,j] (Figure 4)		156	2145.9 (almost best)	426.9	1.4
		[S]in	400
		[E]in	5500
		FL,in	5 × 10−4
FBR Constant but Pareto-optimal feeding (this paper) [d]	1	Optimal feeding [f,j]		180.4	357.9 (best)	422.9	1.4
		[S]in	399.88
		[E]in	793.19
		FL,in	5.78 × 10−4
FBR Variable optimal NLP feeding (this paper) [e]	5	Optimal feeding [f,j] (Figure 5)		2393.7	3.29 × 104 (high consumptions and dilution)	428	6.98
		[S]in [40–400]	variable Figure 5E
		[E]in [97–5500]	variable Figure 5D
		FL,in [5 × 10−4–0.01]	variable Figure 5C

Footnotes: [a] Initial liquid volume VL,o = 1 L. [b] The displayed digits come from the numerical simulations. [c] The checked BR set-point of Ricca et al. [64]. [d] The FBR operation with a constant feeding over time for all the control variables (Table 5): $F_{L,1}=F_{L,2}=F_{L,3}=F_{L,4}$; $C_{S,inlet,1}=C_{S,inlet,2}=C_{S,inlet,3}=C_{S,inlet,4}$; $C_{E,inlet,1}=C_{E,inlet,2}=C_{E,inlet,3}=C_{E,inlet,4}$ with the only 3 variables to be optimized being the initial inlet values of $F_{L,1}$, $C_{S,inlet,1}$, and $C_{E,inlet,1}$, under the constraints of Equation (2i–iv). See the resulting optimal FBR operating policy in Figure 4. [e] The optimal FBR time-stepwise variable feeding policy is obtained by using the control variable limits of footnote [j]. In this FBR operating case, the control variables, $F_{L,j}$, $C_{S,inlet,j}$, and $C_{E,inlet,j}$, where j = 1,…(Ndiv -1), of Table 5 follow an uneven policy in being optimized (that is, 15 unknowns for Ndiv = 5). The optimal control variable policy is given in Figure 5. [f] The units are as follows: [S] g/L; [E] U/L; [W] g; FL L/min. [g] The volume corresponds to the water (W) mass required by the reaction. [h] Search intervals of the control variables are as follows: [S]in = [40–200] g/L; [E]in = [97–5500] U/L. [j] Search intervals of control variables are as follows: [S]in = [40–400] g/L; [E]in = [97–5500] U/L; FL,in = [5 × 10⁻⁴–0.01] L/min [62].

6. Conclusions

To conclude, with the BR operation that was optimized in silico and offline, or with an FBR with constant but optimal feeding, even though there are simple alternatives to implement, they can offer a significantly improved reactor effectiveness, due to their high flexibility in using an easily adaptable process model, and due to the applied effective optimization rules (single-objective NLP).

The nominal, non-optimal BR operation used to derive the process kinetic model reported very poor performances. Thus, the same BR but optimally NLP-operated by also taking into account the technological constraints for the control variables reported incomparably better performances (5× in terms of more fructose produced, at the expense of consuming 5× more substrates and 30× more enzymes).

Our in-silico analysis reveals that, for this enzymatic process, the best alternative is the FBR operated with a constant control variable but using the set-point given by the (breakpoint) of the Pareto optimal front of Figure 7, under the imposed technological constraints Equation (2i–iv). This set point reported the best performances, regarding all the considered opposite economic objectives. Also, the FBR with a constant, but NLP optimal feeding, reported fairly good performances.

The present optimization analysis proves its worth by including multiple elements of novelty: (i) An optimally operated FBR, by using wider but feasible ranges for setting the control variables, can lead to high performances of the bioreactor. (ii) The major role played by the biocatalyst (enzyme concentration) as a control variable during FBR optimization (an option seldom discussed in the literature). (iii) The in silico (model-based) optimal operation of enzymatic reactors is a very important engineering issue because it can lead to consistent economic benefits, as proven by the results presented in this paper.