A Systematic Review of Enzymatic Kinetics in Microreactors

: Microreactors have become an efﬁcient tool for many enzymatic reactions because the laminar ﬂuid ﬂow within the microchannel enables precise process control, rapid mixing, and short residence time. This paper provides a systematic overview of the application of reaction kinetics and the mathematical modeling of enzymatic processes in microreactors. Rapid heat and mass transfer and a high surface-to-volume ratio are usually the reasons why reactions in microchannels proceed faster and with higher yields and productivity compared to conventional macroreactors. Since there are no radial diffusion limitations, microreactors are also an effective tool for determining the kinetic parameters of enzyme-catalyzed reactions. By eliminating the mass transfer effect on the reaction rate, the kinetics estimated in the microreactor are closer to the intrinsic kinetics of the reaction. In this review, the advantages and disadvantages of using microreactors are highlighted and the potential of their application is discussed. Advances in microreactors result in process intensiﬁcation and more efﬁcient biocatalytic processes in line with the advantages offered by the application of microreactors, such as (i) higher yields, (ii) a cleaner and improved product proﬁle, (iii) scale-independent synthesis, (iv) increased safety, and (v) the constant quality of the output product through (vi) accelerated process development. Furthermore, microreactors are an excellent tool for kinetic studies under speciﬁed mass transfer conditions, enhancing the capabilities of other methods.


Microreactors: Definitions and Some Basic Properties
According to the literature, a microreactor can be defined as a device made up of a series of interconnected microchannels (typically with an internal diameter less than 1 mm) in which small amounts of chemical can react [1][2][3]. The internal volume of microreactors can vary from less than 1 µL to several µL, while the internal volume of conventional macroreactors ranges from 100 mL to several liters [4][5][6]. Microchannels are basically straight tubes, but their structure can be extremely complex and includes, for example, micromixers. They are typically produced as circular or rectangular in their cross-sections. Microreactors are made from stable and inert materials such as glass, silicon, stainless steel, ceramics, or polymers. The material used in the construction of microreactors is determined by their use and/or the working parameters of the reaction that takes place. To prevent unwanted effects, materials must be useful and durable, have superior electrical and temperature resistance, and be chemically inert [7]. The surface properties of the reactor microchannels depend on the material construction and also on the method of fabrication. As is described and discussed in detail by Suryawanshi et al. [8], a variety of manufacturing processes can satisfy the general requirements for building an efficient and compact microreactor. The most commonly used manufacturing methods are: micromachining, wet etching, injection of the reactor microchannels depend on the material construction and also on the method of fabrication. As is described and discussed in detail by Suryawanshi et al. [8], a variety of manufacturing processes can satisfy the general requirements for building an efficient and compact microreactor. The most commonly used manufacturing methods are: micromachining, wet etching, injection molding, laser ablation microforming, soft lithography, photolithography, hot embossing, nanoimprinting, electroforming, and micro-electro-discharge machining [8].
To perform chemical or biochemical reactions, microreactors are equipped with a tunable pumping system, reagent reservoirs, and system for the selective collection of reaction products. When working with continuously operated systems, pumps are a very important part of the equipment as they must ensure stable and accurate flow rates [9], which is why syringe pumps are mostly used (Figure 1). The small dimensions of microchannels ensure microreactor systems have numerous advantages over macro-and mezoreactors. Some of the most significant of these are as follows: (i) fast mixing and efficient mass transfer due to no concentration gradients and mixing based on molecular diffusion [10][11][12]; (ii) high surface-to-volume ratio in range from 10,000 to 50,000 m 2 /m 3 that ensures efficient mass and energy transfer [13][14][15]; (iii) laminar flow characterized by a Reynolds number less than 100, which ensures the better control of reaction conditions and eliminates back mixing [16,17]; (iv) the usage of small substrate volumes what results in significant cost reduction [18,19]; (v) environmental friendly so-called "green technology", due to the small amount of chemicals used and due to limited energy consumption [20][21][22]; (vi) safe reaction conditions due to the adaptability of small volumes and production materials under extreme reaction conditions such as high pressure and temperature [23]; (vii) the enhanced selectivity of the specific product synthesis [24,25]; (viii) rapid reaction rates due to high surface-to-volume ratios [26,27]; and (ix) numbering-up instead of complex scale-up allows the simple replacement of broken units without disturbing other units [28][29][30]. However, microreactor systems are still not perfect and have some disadvantages, such as: (i) clogging, due to the accumulation of solid materials on the microchannel surface reduces the flow rate and can cause the complete blocking of the channel [31,32]; (ii) the production price of microsystems depends on the used material, production technique, and additional used equipment [19]; The small dimensions of microchannels ensure microreactor systems have numerous advantages over macro-and mezoreactors. Some of the most significant of these are as follows: (i) fast mixing and efficient mass transfer due to no concentration gradients and mixing based on molecular diffusion [10][11][12]; (ii) high surface-to-volume ratio in range from 10,000 to 50,000 m 2 /m 3 that ensures efficient mass and energy transfer [13][14][15]; (iii) laminar flow characterized by a Reynolds number less than 100, which ensures the better control of reaction conditions and eliminates back mixing [16,17]; (iv) the usage of small substrate volumes what results in significant cost reduction [18,19]; (v) environmental friendly so-called "green technology", due to the small amount of chemicals used and due to limited energy consumption [20][21][22]; (vi) safe reaction conditions due to the adaptability of small volumes and production materials under extreme reaction conditions such as high pressure and temperature [23]; (vii) the enhanced selectivity of the specific product synthesis [24,25]; (viii) rapid reaction rates due to high surface-to-volume ratios [26,27]; and (ix) numbering-up instead of complex scale-up allows the simple replacement of broken units without disturbing other units [28][29][30]. However, microreactor systems are still not perfect and have some disadvantages, such as: (i) clogging, due to the accumulation of solid materials on the microchannel surface reduces the flow rate and can cause the complete blocking of the channel [31,32]; (ii) the production price of microsystems depends on the used material, production technique, and additional used equipment [19]; and (iii) industrial applications are limited to pharmaceutical and fine chemical production due to low production volumes [33].
When enzymatic reactions in a microreactor are considered, an important factor is the microreactor itself and its shortcomings. According to Gojun et al. [34], the main disadvantages of their application can be: (i) clogging, (ii) production costs (the materials used, the production techniques, and the complexity of the reactor but also the use of additional equipment, such as pumps, sensors, etc.), (iii) handling due to the formation Catalysts 2023, 13, 708 3 of 25 of a solid substance, (iv) the influence of the surface (microchannel wall) and chemical adsorption on that wall, and (v) the potential of the industrial application in terms of production volumes. The kinetics of chemical and biological reactions in microreactors may differ due to hydrodynamic and transport limitations, as the nature of the variation in key concentrations depends on the choice of values for these parameters [35,36]. However, the general advantages overcome the limitations, namely scale-independent synthesis, product profile improvement, accelerated process development, increased safety, consistent product quality, cleaner product profile, and higher yields [37].
As Fu et al. [38] stated, in the case of enzyme microreactors, it is extremely important to assess the viability and analyze the profitability and cost of the application [39]. The cost analysis must include the (i) the labor (cost and requirements), (ii) the process of chemicals, (iii) the equipment (cost, lifetime and utilization), (iv) the nanosprings (production and cost of non-functionalized one), (v) the sample dimensions, and (vi) the number of samples that can be treated in parallel [38,40]. In the process itself, the price of the biocatalyst is often critical, and its maximum cost depends on the process in which it is used, which directly affects the value of the final product. The current approach to chemical analysis focuses on minimizing chemicals, and precisely such an approach shows how it contributes to an additional cost increase in costs of about 5% [40]. In scientific research, enzyme immobilization is observed at the laboratory scale, where the catalyst/enzyme system is prepared on a small scale, and the cost itself is not specifically considered. However, in an industrial setting, the cost of catalyst preparation is as important as equipment and operating costs. All of the above makes it difficult to estimate operating costs. Any industrial process dealing with synthesis (such as catalyst preparation) involves equipment (installation) maintenance, equipment cleaning, product analysis, and particle characterization. All of this affects the price and duration of the process. The advantages of using enzyme kinetics in microreactors are as follows: (i) the amounts of enzyme and substrate used are extremely small; (ii) the enzyme stability is better; (iii) in addition to enzymes, other chemical species can be immobilized in the microchannel (and the enzyme involved in a specific reaction can be injected into the system); (iv) contact times are in the range of milliseconds to seconds; and (v) the use of resources, time, and labor is minimized [41,42]. Consequently, the application of microreactors offers a great advantage because very small amounts of immobilized enzymes are required and the methods are easily amenable to automated protocols, which shortens the analysis [41] and reduces the risk of the process contamination. In addition, they are considered to be more sustainable overall due to the possibility of reusing the catalyst several times [42].
As described by Verdnik et al. [43], production using microreactor technology has benefits not only in terms of selectivity, improved product efficiency, environmentally friendly processes, and simpler scale-up but also in terms of cost. The economic advantages are mainly due to the high yield and low amounts of by-products but also due to the plant safety, low energy consumption, and lower carbon footprint. However, switching from batch to continuous manufacturing generates costs, which is why an economic analysis must be part of the process development [44]. For example, Krtschil et al. [45] performed a cost analysis for an economical production of 4-cyanophenylboronic acid using microprocess technology. The analysis included investment costs (CAPEX) and operating costs (OPEX) of different production scenarios. Their results showed that the cost of microstructured reactors often represents less than 10% of the total plant-related expenditure and should therefore have a modest influence on choice of the innovative technology. Furthermore, the cost of increased production capacity was also analyzed. Total expenditures were decreased by 33% for the microchemical process with five-fold capacity and by another 25% for the process with ten-fold capacity. Results indicate that both process intensification and numbering-up are feasible methods to increase profitability. Benaskar et al. [46] performed a cost study for 14 process scenarios to analyze the cost impact of microprocessing and microwaves separately or in combination for two liquid-phase model reactions in fine chemicals synthesis, namely the Ullmann C-O cross-coupling reaction Catalysts 2023, 13, 708 4 of 25 and aspirin synthesis. Their results showed that the OPEX in the Ullmann synthesis were material-based (reactant excess, pretreatment, and catalyst synthesis), while the OPEX in the aspirin synthesis seemed to be more downstream-based (workup, waste treatment). The CAPEX in the Ullmann synthesis was minimal relative to OPEX, but CAPEX for aspirin synthesis was about 40%. Furthermore, various scenarios resulted in a cost-effective and profitable processes. Based on that, integrated microwave heating and microflow processing resulted in a cost-effective system using a micropacked-bed reactor compared to a wall-coated microreactor, with a 20% profit increase. Similarly, Budžaki et al. [47] performed a cost analysis of enzymatic biodiesel production in a small-scale packed bed reactor and found that a large part of the scale-up production cost is the enzyme cost and an even larger part is the cost of the support material. Thus, it is challenging to select a support material that both ensure reaction stability and is cost effective. In addition, the reactor system must be sized to ensure sufficient production rates [48]. Furthermore, Yusuf et al. [49] demonstrated that the continuous production of biodiesel in fabricated microreactor systems would allow more fuel to be produced per unit of labor and reduce costs on a larger scale, while Tongtummachat et al. [50] stated that in the development of the continuously operated microfluidic process, possible substrate pretreatments and product separations and purifications must be considered in the cost analysis.
In this paper, information about the basic principles of working with enzymatic microreactors is described. The advantages and disadvantages of microreaction systems with dispersed enzymes and systems with immobilized enzymes are also composed. Furthermore, the challenges of performing kinetic studies and the mathematical modeling of the biotransformation process in continuously operated microsystems, as well as future perspectives in the presented field, have been discussed.

Microreactors with Dispersed Enzymes as Biocatalysts
Flow chemistry is a straightforward idea, where fluid containing a reactant or reactants is continuously pumped through a reactor to produce a stream of product [51]. According to Roberge et al. [52], 50% of reactions in fine chemical and pharmaceutical production could benefit from continuous production, while for 40% of them, microreactors would be the best reactor system for production. Taking into account the requirements of green chemistry, chemical catalytic processes are replaced with biocatalytic processes in which heavy organic solvents are replaced by green organic solvents and enzymes are used as catalysts. Enzymes are non-toxic biocatalysts, biodegradable, recyclable, naturally made from renewable resources, and known as green catalysts [53]. It can be concluded that the combination of microreactors as environmentally friendly reactor systems and bioanalytical processes can be considered as "green technology". Biocatalysts can be either cell-free enzymes or whole cells. Free enzymes are considered more efficient than whole cells because there are no additional barriers between the substrates and the catalysts [51]. Nevertheless, enzymes, a strong catalyst utilized in biocatalysis, exhibit poor stability in solution over extended periods of time and a progressive reduction in activity during storage [54]. As described by De Santis et al. [55], by using free enzymes, the substrate and enzyme are fed through the reactor at the same time. The time-independent and location-dependent enzymatic reaction occurs inside the specific volume element, and the residence time is the most important element in achieving complete conversion. Depending on the selected physical properties of solvents for introducing both substrates and enzymes in a microreactor, two different flow patterns can be achieved, namely (i) parallel flow and (ii) segmented or slug flow [56]. The parallel flow is characterized by a layer of one phase flowing next to a layer of another phase, while in the slug flow, segments of one phase are dispersed in another phase in uniform and constant size and shape. To reduce the production cost and ensure the sustainability of the biotransformation process, a parallel flow pattern can be considered preferable due to the simple phase separation and possible reuse of the enzyme in the second cycle [57]. On the other hand, the segmented flow provides a larger interface and therefore can contribute to mass transfer [58]. When working with enzymes, a very important aspect is enzyme activity, which has to be preserved to achieve maximum production yield. Organic solvents used in biotransformations may reduce enzyme activity with long residence times. As mentioned earlier, in situ product separation has been presented as a means to enhance bioprocess performance and productivity and ionic liquids can be used as designer solvents to enhance simultaneous product separation. This approach has been described by Deng et al. [59] for the lipase-catalyzed hydrolysis of 4-nitrophenyl acetate using 1-butyl-3-methyl-imidazolium ionic liquids as the reactant solvent. The authors showed that continuous-flow biocatalytic reactions allow for highlevel product purification and the three-time recycling of the enzyme. The examples of certain biotransformations with free enzymes performed in the microreactor are listed in Table 1. Based on the presented results, it can be concluded that the application of microreactor systems with dispersed enzymes as biocatalysts ensures a significantly higher production yield compared to traditional batch processing at short residence times. The presented results indicate that microreactor systems can be considered optimal reactor geometries for biotransformation with dispersed enzymes. Continuously operated Y-shaped two inlets teflon microreactors with an internal volume of V = 392.5 µL.
The influence of geometry, surface roughness of the microchannel walls, and inner mixing in the microreactors on conversion was studied. In all used microreactors, the full conversion of benzaldehyde was achieved at residence times longer than 4 min. The microreactor with inner mixing (micromixers) showed higher conversion at residence times shorter than 2 min. [63] Hexanol oxidation with coenzyme regeneration at pH = 9 and T = 25 • C catalyzed by alcohol dehydrogenase from Saccharomyces cerevisiae Three continuously operated glass microreactor systems were used: (i) two Y-shaped inlets and outlets microreactor chips (V = 6 µL + 6 µL) connected in series, (ii) three Y-shaped inlets and outlets microreactor chips (V = 2 µL + 4 µL + 6 µL) connected in series, (iii) one Y-shaped inlets and outlets microreactor chip (V = 6 µL) with recirculation.
The 100% conversion of the coenzyme was achieved with a very short residence time (τ = 0.8 s). [64] Reversible oxidation of ethanol to acetaldehyde at pH = 9 and T = 25 • C catalyzed by alcohol dehydrogenase from Saccharomyces cerevisiae Continuously operated glass microreactor with two Y-shaped inlets and outlets with internal volume of V = 6 µL.
The 100% conversion of NADH was achieved for a residence time of τ = 0.8 s when the concentration of acetaldehyde was in excess. [57] Oxidation of hexanol to hexanal at pH = 9 and T = 25 • C catalyzed by alcohol dehydrogenase from Saccharomyces cerevisiae Continuously operated glass microreactor with two Y-shaped inlets and outlets with internal volume of V = 6 µL.
Significant improvement in the conversion of hexanol when compared to the classical macroscale process was obtained. In the 6 µL microreactor, 11.78% conversion of hexanol was attained after 72 s, while in the macroscale process 5.3% conversion of hexanol was reached after 180 s. [65] Catalysts 2023, 13, 708 6 of 25 Table 1. Cont.

Reaction and Reaction Conditions Microreactor Material and Structure Process Productivity Reference
Oxidation of catechol and L-DOPA at pH = 6.6 and T = 25 • C catalyzed by laccase from Trametes versicolor Continuously operated glass microreactor with two Y-shaped inlets and outlets with internal volume of V = 6 µL for catechol oxidation and V = 3.7 µL for L-DOPA oxidation.
For a residence time of 72 s and an inlet oxygen concentration of 0.271 mmol/dm 3 , catechol conversion of 41.3% was achieved, while approximately the same conversion of L-DOPA (45.0%) was achieved for an inlet oxygen concentration of 0.544 mmol/dm 3 . [66] Isoamyl acetate synthesis at T = 25 • C and 45 • C catalyzed by lipase B from Candida antarctica Continuously operated glass microreactor with two Y-shaped inlets and outlets with internal volume of V = 3.65 µL.
A 35% conversion at a residence time 36.5 s, temperature of 45 • C, and for 0.5 mol/L acetic acid and isoamyl alcohol inlet concentrations. [67] Transesterification of tert-butanol at T = 60 • C catalyzed by lipase from Thermomyces lanuginosus Six poly methyl methacrylate microreactors were used (four T-shaped with different lengths, one Y-shaped, and one Tesla-shaped).
The maximum conversion achieved was 68.90% at an inlet flow rate of 50 µL/min (per inlet) and 9:1 molar ratio of methanol to oil. [49] Aldol addition of dihydroxyacetone to Cbz-N-3-aminopropanal at pH = 7.5 and T = 25 • C catalyzed by D-fructose-6-phosphate aldolase overexpressed in Escherichia coli Continuously operated glass microreactor with two Y-shaped inlets and outlets with internal volumes of V = 4 µL and 13 µL.
The use of a microreactor did not have any beneficial effect on the reaction performance, the enzyme showed a decrease in its activity at longer residence times in one type of used microreactors (13 µL). [68]

Microreactors with Immobilized Biocatalysts
To ensure enzyme stability and, consequently, higher activity in the bioanalytical process, enzyme immobilization has been extensively used. As described by Hamidović et al. [69], nowadays, immobilized enzymes are preferred over free enzymes because immobilization allows enzyme recycling, making the process more cost-effective and ecologically beneficial. Furthermore, immobilization ensures long-term stability and temperature tolerance of the enzymes. The biocatalytic reaction rate is highly influenced by the quantity and activity of the immobilized enzyme in the microfluidic device. Enzyme immobilization is the binding of an enzyme to a supporting material, such as beads, gels, nanopores, magnetic nanoparticles, and/or polymer nanofibers [25]. Immobilization holds enzymes in situ all across the biochemical process, allowing them to be readily removed from the products and employed again for series of biochemical processes, allowing enzyme recycling. As described by Coloma et al. [54], here are three approaches for the preparation of immobilized enzyme microreactors, namely (i) the wall-coated type, where the enzyme is directly immobilized onto the inner wall surface of the microchannel; (ii) the monolithic type, where microchannels are filled with monolithic materials with interconnected meso-or microporous structures; and (iii) the packed-bed type, where enzymes are immobilized on polymeric or inorganic particles that are then packed into the microchannels of immobilized enzyme microreactors ( Figure 2). The enzyme is directly immobilized on the inner wall surface of the microchannel in its wall-coated form. The accessible surface areas of the microfluidic walls are extremely limited, and the enzyme-loading capacity is restricted. Moreover, the substrate diffusion pathway is quite long under these circumstances, resulting in a poor biocatalytic conversion rate [70,71]. Therefore, researchers are working to improve the surface-to-volume ratio of microchannels, which will enhance the enzyme-loading capacity [72][73][74]. To enhance the surface-to-volume ratio, enzymes can be immobilized on polymeric or inorganic materials, which are then packed into microchannels [42,75,76]. Microchannels can also be filled with linked meso-or microporous structures [77][78][79]. Monolithic reactors contain monoliths composed of permeable catalytic media or catalytic materials arranged in the tubes of an inactive monolithic base [80]. In addition to the advantages in using immobilized enzymes, such as cost reduction in downstream processes, immediate reuse, and enzyme stabilization in the immobilization matrix [81,82], there are also disadvantages and/or challenges in working with immobilized enzymes. Immobilization requires more time, equipment, and resources, which makes it more expensive to implement [83,84]. Moreover, the removal of system contaminants is expensive because the entire system must be shut down [81][82][83][84]. In recent years, several review papers have described and discussed current progress, advantages, problems, and challenges in working with microreactors with immobilized biocatalysts. For example, Zhu et al. [70] provided a thorough overview of the elements that influence the efficiency of continuous biocatalysis in microreactors with immobilized biocatalysts utilizing a top-down approach, while Bolivar and López-Gallego [71] presented the most important aspects when constructing immobilized enzyme reactors to ensure productive and stable flow conditions. They also investigate the efficiency and lifetime of immobilized enzymes incorporated into flow-reactors using protein loading, activity recovery, space-time yield, specific productivity, and turnover number. Furthermore, Thompson et al. [72] and Tamborini et al. [73] focused their reviews on the integration of enzyme immobilization and microfluidic devices for fine chemical production, with a particular emphasis on cutting-edge data. Romero-Fernández and Paradisi [74] presented most recent developments in continuous flow biocatalysis with an emphasis on novel enzyme immobilization and encapsulation technologies. De Santis et al. [55] focused on enzyme-catalyzed reactions, and Stojkovič and Žnidaršič-Plazl [75] described a method for immobilizing eukaryotic and prokaryotic cells onto the inner surfaces of microreactors made of various polymeric materials and glass. Some examples of microreactors for biotransformations with immobilized biocatalysts from recent years are listed in Table 2. In recent years, several review papers have described and discussed current progress, advantages, problems, and challenges in working with microreactors with immobilized biocatalysts. For example, Zhu et al. [70] provided a thorough overview of the elements that influence the efficiency of continuous biocatalysis in microreactors with immobilized biocatalysts utilizing a top-down approach, while Bolivar and López-Gallego [71] presented the most important aspects when constructing immobilized enzyme reactors to ensure productive and stable flow conditions. They also investigate the efficiency and lifetime of immobilized enzymes incorporated into flow-reactors using protein loading, activity recovery, space-time yield, specific productivity, and turnover number. Furthermore, Thompson et al. [72] and Tamborini et al. [73] focused their reviews on the integration of enzyme immobilization and microfluidic devices for fine chemical production, with a particular emphasis on cutting-edge data. Romero-Fernández and Paradisi [74] presented most recent developments in continuous flow biocatalysis with an emphasis on novel enzyme immobilization and encapsulation technologies. De Santis et al. [55] focused on enzyme-catalyzed reactions, and Stojkovič and Žnidaršič-Plazl [75] described a method for immobilizing eukaryotic and prokaryotic cells onto the inner surfaces of microreactors made of various polymeric materials and glass. Some examples of microreactors for biotransformations with immobilized biocatalysts from recent years are listed in Table 2. Table 2. Application of microreactors for biotransformations with immobilized enzymes.

Reaction Enzyme Type of Immobilization and Process Conditions Process Productivity Reference
The activity of immobilized trypsin was evaluated with its commercially available substrate N-benzoyl-L-arginine ethyl ester (BAEE) Trypsin and chymotripsine from Pichia pastoris Two enzymes immobilized on a monolithic GMA-co-EDMA support in fused silica microreactor with volume of V = 1.8 µL. Reaction was performed at T = 30 • C and pH = 8.05 at different flow rates in range from 0.25 to 1 µL/min.
Volumetric productivity of up to 14.42 g/ (L·h) was achieved for residence time of 15 min. [77] Hydrolyzation of carboxymethyl cellulose Cellulase from Aspergillus niger Immobilization based on covalent binding of biocatalysts onto functionalized surfaces. Reaction was performed at T = 55-65 • C and pH = 6.
Glucose production per unit time in the cellulase-loaded microreactor is 97.2% higher than that in other reaction systems. [78] Hydrolyzation of wheat straw into monosaccharides Xylanase from Aspergillus oryzae and cellulase from Aspergillus niger Immobilization based on covalent binding of biocatalysts onto functionalized surfaces. Reaction was performed at T = 30 • C and pH = 6.
Increasing the glucose production by 89.7% comparing to conventional batch process. [79] Synthesis of α-glucose 1-phosphate Sucrose phosphorylase from Bifidobacterium longum Coated borosilicate microchannels with total volume of 83 µL and enzyme attached on the silica nanosprings.
Reaction was performed at T = 25 • C and pH = 7.
100% substrate conversion was achieved for residence time of 8 min, while maximum volumetric productivity of up to 19 mmol/(L·min) was achieved at flow rate of 80 µL/min.

[80]
Enzymatic synthesis of the biologically active sialic acid cytidine through monophosphate N-acetylneuraminic acid using enzyme cascade reaction N-acyl-D-glucosamine 2-epimerase, N-acetylneuraminate lyase, and CMP-sialic acid synthetase expressed in Escherichia coli Enzymes were immobilized into bulk hydrogels and microstructured hydrogel-enzyme dot arrays, which were then integrated into microfluidic devices. Reaction was performed at T = 25 • C and pH = 8 in a microreactor with active volume of 10.9 µL.
Immobilization efficiencies of up to 25% and pronounced long-term activity of the immobilized enzymes for several weeks were realized. Trypsin-immobilized monolithic enzyme microreactor had much higher tryptic digestion efficiency than the free trypsin in solution and digestion time was dramatically shortened from 12 h to 16 s. In the first two days, the reactor showed an average overall conversion of 98% to form 8% (R)-syn/anti-hydroketone with 90% of the wanted diol product, resulting in a diol space-time yield of 131 g/(L·day). [84] Catalysts 2023, 13, 708 High biodegradation efficiencies of 95 and 76% were achieved by soybean and potato peroxidase at 3 min residence time in a microreactor. After 10 cycles of reuse soybean peroxidase retained 65% of its initial activity, while potato peroxidase retained only 35% of its initial activity. [85] Glucose oxidation at low concentrations Glucose oxidase and horseradish peroxidase DNA-directed immobilization to fix glucose oxidase and horseradish peroxidase in the capillary, which had been functionalized with polyami-doamine dendrimer in a microreactor with volume of V = 2.25 µL. Reaction was performed with 100 mmol/L of glucose at T = 25 • C and pH = 6.
Prepared IMER ultimately retained approximately 70% of its initial activity for 25 days. After continuous enzymatic hydrolysis for ten cycles using the same capillary, enzymatic activity remained above 80% of its initial value. [86] Four proteins (myoglobin, lysozyme, hemoglobin, and albumin) in a wide size range ( TESB used for enzyme immobilization has the potential to facilitate a greater degree of enzymatic digestion with higher sequence coverage. [88] Digestion efficiency of serum albumin Porcine pancreas trypsin Covalent immobilization on poly(dimethylsiloxane) microchannel produced with pillars to increase surface-to-volume ratio. Reaction was performed at T = 30 • C and pH = 8.
Significant reduction of reaction time taking into account that digestion with the IMER requires less than 10 min, while in-solution digestion takes 16 h. The continuous biotransformation process using two layers of hydrogels integrated into a two-plate microreactor ensured high space time yield of 2.86 g/(L·h). [90] Transamination Amine transaminase expressed in Escherichia coli was immobilized together with the cofactor pyridoxal phosphate The enzyme and cofactor were retained in a porous copolymeric hydrogel matrix formed in a two-plate microreactor. Microreactor with active volume of 620 µL was operated at T = 25 • C and pH 8.
The space-time yield of the microbioreactor was 19.91 g/(L·h), while the highest achieved biocatalyst productivity was 5.4 mg/(mg enzyme ·h). Maximal conversion of approximately 95% was obtained in less than 0.8 min. [92]  70% conversion is achieved for residence time of 7 min. More than 80% of the initial productivity has been retained after a period of 21 days of the continuous miniaturized packed-bed reactor operation. [93] Kinetic resolution of racemic amines Transaminase expressing E. coli whole cells Transaminase expressing E. coli cells were immobilized on the surface of the continuously operated reactor with active volume of 816 µL. Microreactor was operated at T = 25 • C and pH 7.5.
50% conversion was achieved at flow rate of 40 µL/min and with substrate concentration of 20 mmol/L. [94] The transesterification of soybean oil for fatty acid methyl ester production Lipase from Candida antarctica Layer by layer coatings composed of polyethylenimine and lipase. Reaction was performed at molar ratio of oil to methanol of 1:7 and temperature of 50 • C 95% conversion rate achieved in 53 min. [95] Synthesis of glycerol ferulate Lipase from Candida antarctica 3D-printed polylactic acid microwell plates or tubular microfluidic reactors were modified with polyethylenimine and lipase was covalently immobilized in the interior of each structure.
Maximum productivity achieved with immobilized enzyme was 17.2 µg/(min·µg enzyme ). [96] Catechol oxidation Laccase from Trametes versicolor Enzyme covalently immobilized on the wall surface of glass tubular microchannel with active volume of 6 µL.
Microreactor was operated at T = 25 • C and pH 5.5.
25% substrate conversion was achieved for residence time of 25 s. [97] NADH + oxidation Alcohol dehydrogenase from Saccharomyces cerevisiae Alcohol dehydrogenase immobilized on the magnetic nanoparticles of maghemite in glass tubular microchannel with active volume of V = 6 µL.
Microreactor was operated at T = 25 • C and pH 9.
In the system with the oscillating magnetic field, 100% conversion was achieved for residence time of 3.5 min. [98] Propyl caffeate synthesis by transesterification of methyl caffeate and 1-propanol Lipase Novozym 435 Immobilized lipase Novozym 435 beads were incorporated into the channels. Microreactor was operated at T = 60 • C for 2.5 h.
The maximum yield (99.5%) in a microreactor was achieved in a short period of time (2.5 h) with a flow rate of 2 µL/min.
The maximum conversion rate of soluble substrate cellobiose digestion in the microreactor was 76% at a flow rate of 7 mL/min.

Kinetics of the Enzyme Reactions in Microreactors
The modeling of biotransformations as a technique for enzyme reaction engineering is fundamental for the design of enzyme-catalyzed reactions, both for large-scale production and microreactors, respectively [101]. Understanding enzyme kinetics enables the determination of optimal operating parameters and the identification of the most efficient mode of production system [41], as reaction kinetics is a key parameter for equipment design both in the laboratory and in industrial production. When it comes to determining the reaction kinetics of a chemical reaction, concentration, temperature, and pressure play an important role. Since most enzymatic reactions occur at mild reaction conditions, the concentration of substrate and products has the greatest influence. Therefore, to determine enzymatic kinetics, it is important to know the mechanism behind the interactions between enzyme and substrate and enzyme and product. To accurately determine the kinetics, it is important to accurately measure the concentration and reaction time. In addition, each reaction is also determined by mass transfer. According to Yan et al. [102] when the mass transfer is not favorable, the obtained rate can only reflect the characteristics of the mass transfer rather than the reaction itself.
When analyzing the kinetics of the enzyme-catalyzed reactions, it is necessary to consider the effects of the process variables, namely pH, temperature, substrate concentration, and enzyme concentration [1,18]. Each enzyme has an optimum pH and a change in pH outside this range will cause enzyme activity to slow down. Enzymes can be denatured by high pH [18,103,104]. Higher temperatures lead to more interactions between molecules, increasing the rate of a process. More collisions enhance the probability that the substrate will interact with the enzyme active site, increasing the rate of an enzyme-catalyzed process [105,106]. Higher substrate concentrations result in more molecular collisions and enzymes are more likely to come into contact with substrate molecules [41,92,[107][108][109]. Therefore, the reaction reaches its maximum rate when the active sites are practically constantly filled. Moreover, if the enzyme concentration is low, the reaction does not continue as fast as it normally would [41,107,108]. The importance of molecules that reduce the enzyme activity must also be considered when analyzing enzyme kinetics. Many compounds can inhibit the activity of enzymes and lead to complete enzyme inactivation or a reduction in enzyme activity [105,110]. Reactive chemicals could form covalent bonds with important amino acids and prevent the enzyme from performing its reaction process, resulting in inactivation. Inactivation is usually irreversible [111]. Chemicals that bind to an enzyme and limit its ability to catalyze a reaction cause reversible inhibition [110][111][112]. The binding is non-covalent and reversible, and removing the inhibitor restores normal activity. Several strategies of reversible inhibition defined by the interaction between the inhibitor and the enzyme's substrate are described in the literature [113][114][115][116]: (i) competitive inhibition occurs when an enzyme binds a substrate or an inhibitor but not both; (ii) non-competitive inhibition occurs if the inhibitor can bind to the enzyme whether the substrate is bound or not, implying that substrate binding has no effect on inhibition; (iii) non-competitive inhibition occurs if the inhibitor can only bind to the enzyme-substrate complex if the substrate is initially bound; and (iv) mixed inhibition is the combination of mechanisms that do not compete with any of the other processes.
Conventionally, the kinetic studies of enzyme-catalyzed reactions are performed in a cuvette or a batch reactor. The two most common methods involve using the initial reaction rate method and the determination of the change in the concentration of substrate or product in time. When it comes to moderate, conventional reactions that are not too fast, when the reaction system is homogenous, this approach is preferable, since it is fast and easy to perform. However, when the reactions are fast and heterogeneous, the previously mentioned approach has limitations. The analysis of the kinetics of rapid reactions is frequently challenging to examine in a batch reactor due to the system's poor mass transfer characteristics. Therefore, for such processes, the use of a microchannel reactor improves mass transfer rates and eliminates the mixing effect, which may ensure that the reaction works near to intrinsic kinetics [41,117]. Furthermore, with a constant substrate concentration, an inhibitor concentration gradient in a single microchannel, and a single starting inhibitor concentration, enzyme inhibition may be determined quickly [118]. Additionally, since the accurate measurement of concentration and concentration variation in reaction time are basic kinetic measurements, microreactors could be preferable tool. The residence time distribution in microchannel is very narrow, and sub-second and milli-second residence times can also be achieved [119], indicating that microreactors can be a strong tool for kinetic measurements [120]. According to Yen et al. [102], when it comes to mass transfer resistance in microreactor, two dimensionless parameters should be considered. The first one is the Hatta number, which represents the relative speed of the reaction rate to the mass transfer rate. The second is the Thiele modulus, often used to describe the effect of the internal mass transfer. In the microchannel, the Hatta number is less than 0.3, and the Thiele modulus is 6·10 −4 , indicating that mass transfer resistance and internal mass transfer can be ignored in a microreactor. As previously described by Tušek et al. [101], a thorough understanding of reaction kinetics is required for the reaction development and the collection of data on the catalyst under operational settings. Given that the settings in microreactors differ from those in traditional macroreactors (high mass transfer rate, short residence time, large volume-to-surface area ratio), the kinetic parameter values of enzyme processes must be established for microreactors as well. When it comes to the determination of enzyme kinetics, the use of microreactors has another advantage. Namely, most enzymes are expensive and hard to obtain. By applying microreactors, small volumes of reagents and enzymes are sufficient for the measurement.
When it comes to studying the enzyme kinetics in microfluidics, different strategies can be applied [121], such as multi-stream laminar flow parallel reaction chambers [122], droplet microfluidics [123], and microfluidic devices with continuous flow mixers [124][125][126]. As described by Sjostrom et al. [127], multi-stream laminar flow parallel reaction systems have two main drawbacks, slow mixing and Taylor dispersion, and therefore, dropletbased microfluidic systems are considered as efficient alternative technology. Dropletbased microfluidic systems technology can provide millisecond mixing and prevents dispersion [128,129]. Some examples available in the literature are given in Table 3.
Some additional examples of kinetic studies using free and immobilized enzymes in microreactor systems can be found in the literature. Kundu et al. [106] investigated enzyme-catalyzed ring-opening polymerization of caprolactone to polycaprolactone in microreactors. The procedure was carried out with the help of a commercially available immobilized lipase preparation containing Candida antarctica lipase B. Their findings demonstrated that continuously operated microreactors are a more efficient polymerization system than traditional macro-batch reactors. They also showed that conversion is temperatureand time-dependent. The efficiency of the used reactor systems was also compared based on the kinetic study. Experimental data were fitted to first-order reaction kinetics, and they showed that microreactors have an apparent rate of reaction that is at least an order of magnitude faster than the reaction rate achieved in batch reactors. Kundu et al. [106] proposed two possible explanations for that phenomenon: (1) because microreactors have a much smaller volume than batch reactors, reactants are forced to come into contact with enzyme active sites; and (2) because the surface-to-volume ratio in microreactors is much larger than in batch reactors, more enzyme active sites are available to reactants at any given time. Similarly, Jurinjak Tušek et al. [66] demonstrated that the reaction rate of the laccase-catalyzed oxidation of catechol in a microreactor was approximately two times that of batch experiments. They estimated kinetic parameters for the catechol oxidation reaction from the independent experimental results collected in a continuously operated microreactor. The parameters were estimated using non-linear regression of double substrate Michaelis-Menten kinetics (maximal reaction rate, saturation constant for catechol and saturation constant for oxygen). Tušek et al. [101] used the same approach for the kinetic analysis of the alcohol dehydrogenase-catalyzed oxidation of hexanol, where double substrate Michaelis-Menten kinetic constants were also estimated based on experimental data from independent microreactor experiments. For the estimation of the kinetic pa-rameters, the concentration of the first reactant was varied while the second was kept at a constant value and vice versa. The calculated parameter values show that the maximum initial rate (197.276 ± 79.334 U/mg) measured in the microreactor is much higher than the one seen in the cuvette (6.755 ± 0.089 U/mg). Because the microchannels lack radial diffusion constraints (small diameter compared to channel length), they are useful for determining the kinetic data of enzyme-catalyzed processes. Additionally, the enzyme alcohol dehydrogenase prefers hexanol and NAD + as substrates (low K m values). Since this parameter is a property of the enzyme utilized, the Michaelis-Menten K m constants for hexanol are almost identical when measured in a cuvette and in a microreactor.
Because of the growing interest in biodiesel production, a kinetic analysis of lipasecatalyzed transesterification was performed. Gojun et al. [107] used edible sunflower oil and methanol as substrates, while lipase from Thermomyces lanuginosus (Lipolase L100) was used as a catalyst for biodiesel synthesis by transesterification. Experiments were performed in a polytetrafluoroethylene (PTFE) microreactor with three inlets and in glass microreactors with two and three inlets. The kinetic parameters were estimated from independent experiments performed in a microreactor. The influence of fatty acids and the methanol concentration on the initial reaction rate was investigated to estimate the kinetic parameters of the Michaelis-Menten kinetic model. Similarly, Abdulla Yusuf et al. [108] used sunflower oil, methanol, and lipase from Thermomyces lanuginous for biodiesel production. Both studies showed similar results for kinetic parameters, indicating the better performance of the reactor in comparison to conventional reactors. Previously described examples discussed biocatalytic processed with dispersed enzymes. When working with immobilized enzymes, a different approach has to be used. A Michaelis-Menten model can represent the kinetics of an immobilized enzyme in a batch reactor. The experimental data are fitted to a Lineweaver-Burk relationship, where r is reaction rate, V max is the reaction's maximum velocity, K m is the kinetic constant, and S is the substrate concentration. The experimental results provided the values for K m and V max (Equation (1)).
The Lilly-Hornby model is an adaptation of the classic Michaelis-Menten kinetic model, which has been used to investigate the flow kinetics of enzymes immobilized in microreactor systems [96] (Equation (2)).
where f is the fraction of substrate converted to product during the reaction, Q is the flow rate of the substrate, [A 0 ] is the initial substrate concentration, C is the reaction capacity of the microreactor, and K m (app) is the kinetic parameter [109]. That methodology was used by (i) Bellou et al. [96] for the description of the hydrolysis of p-nitrophenyl butyrate catalyzed by lipase from Candida antarctica covalently immobilized on tubular microfluidic reactor, by (ii) Carvalho and Fernandes [110] and Carvalho et al. [111] for the description of packed-bed microreactor loaded with immobilized invertaze for the hydrolysis of sucrose into inverted sugar syrup, by (iii) Gong et al. [109] for description of isoquercitine production by grapheneimmobilized naringinase, by (iv) Abd Razak et al. [53] for description of glycerol dioelate by immobilized lipase form Candida antartica in a continuous flow packed-bed milli-reactor, and by (v) Abdul Halim et al. [112] for the description of 2-amino-1,3,4-butanetriol synthesis catalyzed by transketolaze and transaminase immobilized onto agarose beads and packed into tubes to enable multistep enzyme reactions. All researchers present similar results: K m (app) values decreased with increasing flow rates, whereas for higher flow rates, the K m (app) tends to stabilize at values close to those observed for the enzyme in its free form. Table 3. Strategies for enzyme kinetics studying in microfluidic devices.

Strategy Reaction Process Specificity Reference
Multi-stream laminar flow parallel reaction chambers Bioluminescent reaction between adenosine triphosphate (ATP) and firefly luciferase/luciferin Enzyme and substrate are introduced together in a co-flow microfluidic device. Analytical and numerical results showed that that the initial concentration of product changes with the distance x along the channel according to exponential expression.
[122] Droplet based microfluidic system β-galactosidase-catalyzed hydrolysis of resorufin-β-D-galactopyranoside Microfluidic system that combines a linear concentration gradient generated as a result of laminar diffusive mixing with parallel droplet formation in a T-junction. [123] Glucose oxidation by glucose oxidase Kinetics was obtained by probing droplets at multiple points over time. [103] β-galactosidase oxidation of resorufin β-D-galactopyranoside Proposed microfluidic system enables the scanning of multiple inhibitor concentrations and substrate conditions in a single, highly time-consuming experiment. [127] Cleavage of a fluorogenic substrate by ribonuclease A Rapid single-turnover kinetics of ribonuclease A was measured with better than millisecond resolution using sub-microliter volumes of solutions. [104] Decomposition of H 2 O 2 by catalase Droplet-based microfluidic chip with pneumatic valves for measuring millisecond enzyme kinetics using amperometric detection method. The best time resolution was about 0.05 s, and the reaction time measured was from 0.05 to 25 s. [119] (R)-1-Phenylethyl-(2-oxo-4-methyl-2H-chromen-7-yl) carbonate hydrolyzation by lipase and esterase Automatic microfluidic system for screening of enzymatic reactions based on droplets generation with the full control over the concentration of substrates, ratio of reagents, time of reaction, and the temperature control. [105]

Continuous flow mixers
Cleavage di-(b-D-galactopyranoside) (FDG) and β-galactosidase from Escherichia coli A multilaminar continuous flow mixer that enables the investigation of the dynamics of biochemical reactions with up to 2.5 s reaction time and a resolution of 500 µs. [124] β-amyloids formation Microfluidic continuous-flow mixer (MCFM) that is compatible with FT-IR imaging and can record biomolecular reactions over a wide range of timescales. The platform provides fast mixing (460 µs) to induce protein folding. [125] Although many kinetic measurements are typically performed on a single chip, microfluidic platforms have recently improved kinetic measurements significantly. Significant increases in throughput and minimized sample usage are the most important features of these systems. The superiority of the microfluidic platform, or "microfluidic mapper", was demonstrated in the paper presented by Rho et al. [122]. The authors created a platform that allowed them to map the enzymatic kinetics of 36 parallel biochemical reactions catalyzed by horseradish peroxidase with 36 combinations of two reagents (Amplex Red and hydrogen peroxidase). The authors obtained a 3D kinetic plot containing all information about the kinetics of the target enzyme system using a single experiment.
Another successful example is the adaptation of an easy-to-use commercial microfluidic qPCR. In their paper, Rembeza and Engqvist, [113] adapted the mentioned system in order to determine the kinetic properties of three oxidases (lactate oxidase, glutamate oxidase, and glucose oxidase) in a resorufin-based fluorescence assay. When the results were compared to those obtained in the microfluidic chip and those reported in the literature, they were comparable, but the process was much faster when performed in plates.
A droplet based microfluidic platform was developed by Hess at al. [114]. The kinetics of four enzymes were studied: β-galactosidase, horseradish peroxidase, microperoxidase, and the engineered variants of haloalkane dehalogenases. When compared with conventional methods, their platform reduced assay volumes by six orders of magnitude and increased throughput to 9000 reactions/min.
With the premise "Go big or you'll get lost", Markin et al. [115] developed a highthroughput microfluidic platform, which they used to express and purify more than 1500 enzyme mutants in few hours and to perform their functional characterization in few days. For each mutant, they measured Michaelis-Menten kinetics for multiple substrates, inhibition constants, and effects on folding, in the end obtaining more than 5000 kinetic and thermodynamic constants from more than 670,000 total reactions.
On the other hand, despite the fact that microreactors improve the study of enzyme reactions, in the paper presented by Li et al. [116] the authors challenged the above mentioned strategies. They stated that, in the background of classical kinetic measurements in microreactors are complex experimental setup and procedures, equipment, and sophisticated microfabrication. All of this limits the application of these systems to a wide biological community. Additionally, they stress that all of the advantages of microreactors are outweighed by their complex operation and sample preparation outside the chip. In order to overcome the mentioned disadvantages, the authors report a one-step assay to measure enzyme kinetics using a 3D-printed microfluidic device, which eliminates the steps of preparing and handling multiple solutions, thereby simplifying the whole workflow significantly. In the system they proposed, the series of solutions with different substrate concentrations required by the Michaelis-Menten kinetics are all generated on-chip with minimal extra sample preparation steps outside the chip. In order to perform measurements, only flow rates need to be adjusted. The overall system was simplified and made more user friendly for non-microfabrication experts.
With the problems that are highlighted in the work of Li et al. [129] considering microreactor application in enzyme kinetic measurements, there are still some problems that should be solved in future work. Some of these are highlighted in the paper presented by Yan et al. [102], and although they refer to the determination of kinetics in chemical reactions, we can see some parallels with enzymatic kinetics. The first thing the authors mention is that most descriptions of the excellence of microreactors are still qualitative. So, more effort should be made to quantify the superior performance of microreactors. The authors' second point is that most kinetic research is performed on a single type of microreactor. If we assume that microreactors with different properties (e.g., the presence of micromixers, different surface roughness, etc.), such as different mass transfer and mixing properties, this may lead to different kinetic laws. On the other hand, this could be an advantage, as a more specific and tailored microreactor can be made for each reaction to achieve the best performance. The third problem is that, when it comes to kinetic studies in microreactors, they are all still focused on conventional reactions with clear mechanisms and pathways. When it comes to the kinetics of complex reactions, the equilibrium hypothesis and steady-state hypothesis, which are common for batch measurements, can be neglected. With the inline analysis, more information about reaction transition states and intermediates can be obtained, meaning that reaction mechanisms can be reflected more precisely.

Modelling of Enzyme Reactions in Microreactors
To obtain detailed insight into the performance of microreactor systems with dispersed or immobilized enzymes, mathematical modelling is necessary. A mathematical model describing enzymatic processes depends on a kinetic model for different biochemical transformations [130]. As described by Sudar et al. [68] each component in the reaction media, as well as the biocatalyst stability in the reactor, has an influence on the reaction rate, which must be specified for a good mathematical description. Once a mathematical model and its results are established, simulations may be highly beneficial for optimizing and improving a process, as well as evaluating the performance of a biocatalyst. Tišma et al. [131] stated that the mathematical modelling of enzymatic microreactors, including flow distribution, transport phenomena, and enzyme reaction kinetics, would allow process description and optimization. They listed the steps of the development of the mathematical model for the description of the dispersed enzyme biotransformation in a microreactor, namely (i) setting the velocity profile, (ii) the description of the mass transfer phenomena, and (iii) description of the enzyme kinetics. Several examples of such an approach are available in the literature. Tišma et al. [131] developed and validated a 2D mathematical model for the description of laccase-catalyzed L-DOPA oxidation in a microreactor. Two aqueous phases were fed into microreactor under laminar flow conditions and parabolic velocity profile was assumed. Furthermore, the concentration profiles of both reaction substrates (L-DOPA and oxygen) and enzyme were described by convection in the flow direction and diffusion in two directions considering the square cross-section of the microchannel. The kinetics of the enzyme reaction were described by double substrate Michaelis-Menten kinetics, and kinetic parameter values were adopted from the batch experiments. The developed model described the experiment data with high accuracy for all inlet L-DOPA and oxygen concentrations and for all applied flow rates. The same approach was efficiently used for modelling the catechol oxidation catalyzed by laccase in a microreactor [66] with the exception that kinetic parameter values were estimated based on microreactor experiments. The rate of catechol oxidation was modelled with a double substrate Michaelis-Menten equation (Equation (3)).
where V m is maximum reaction rate, K catechol m is saturation constant of catechol, K oxygen m is saturation constant of oxygen, γ laccase is enzyme concentration, and c is concentration of substrates. The mathematical model for steady-state conditions in a microreactor was composed of dimensionless partial differential equations for catechol, oxygen, and laccase and the corresponding boundary and initial conditions (Equations (4)-(6)): Oxygen: v ψ (ψ) ∂c oxygen ∂ξ where ν represents linear velocity; ξ and ψ represent independent dimensionless variables ξ = x/W and ψ = y/W; and x and y are coordinated in the length (L) and microchannel width (2W). D catechol/w , D oxygen/w , and D laccase/w are the diffusion coefficients for catechol, oxygen, and laccase in water. Aside from solving the set of partial differential equations for description of the laccase-catalyzed catechol oxidation, Jurinjak Tušek et al. [66,68] proposed a simple numerical approximation. A microreactor was described as an ideal plug flow reactor in a steady state. Mass transfer between the two plug flow reactors is modelled with diffusion, which is described as the quotient of the concentration gradient between the two positions in a microreactor and the distance between the selected positions. Their results showed that numerical simplification efficiently described the experimental data for the highest inlet concentration of oxygen at long residence times. Aside from the aqueous two-phase systems of enzyme-catalyzed biotransformations in microreactor systems, efficient mathematical models have also been developed for the description of the enzyme-catalyzed process in systems involving two miscible solvents. Žnidaršič-Plazl and Plazl [67] developed and validated a 3D model involving convection, diffusion, and enzyme reaction terms for n-hexane-water system with dissolved lipase B from Candida antarctica for conducting the reaction using isoamyl alcohol and acetic acid as reactants. Stojkovič et al. [132] proposed a 3D model for analysis of L-malic acid production from fumaric acid in a glass microreactor using surface-immobilized fumarase, while Gojun et al. [60,107] applied a 2D model to the biodiesel transesterification process in a microreactor. Gojun et al. [107] offered a comparison of multiple mathematical models used to describe biodiesel production in a lipase-catalyzed microreactor. The Bi-Bi pingpong mechanism is the most widely reported mechanism in the literature for biodiesel synthesis. Nonetheless, due to the low order of approximation, the authors stated that it is not suitable for microprocess characterization. Among the three mathematical process models investigated (the 2D mathematical process model, the process model of steady-state two parallel plug flow reactors, and the process model of steady-state plug flow reactors), the 2D mathematical process model and the process model of steady-state two parallel plug flow reactors demonstrated good agreement between experimental results and model simulation. Moreover, the transesterification and hydrolysis reaction rates were modelled using double substrate Michaelis-Menten kinetics, Bi-Bi ping-pong kinetics, and Hill kinetics. The Hill model was presented as the appropriate kinetic model for lipase-catalyzed biodiesel generation based on the model selection criterion. In contrast to the described results, Razak et al. [133] proposed a plug flow profile for modelling the hydrolysis of p-nitrophenylpalmitate by lipase using kinetic parameter values estimated based on experimental data from a microreactor. All the researchers mentioned agree with the necessity of a detailed kinetic study of the enzyme-catalyzed reaction in microreactors.
An interesting approach was presented by Elbinger et al. [134], in which the spatialtemporal dynamics of metabolite concentrations during the conversion of sucrose to glucose-6-phosphate was analyzed. The model included a system of diffusion equations together with boundary and initial conditions. The results of the proposed model simulations were used to design the microreactor prototypes. These microreactors consisted of compartments divided by membranes containing specific transporters for substrate input and product output. Multienzyme complexes formed on nanobeads were employed to carry out metabolic processes inside the reactor compartments. Furthermore, Tměj et al. [135] proposed a mathematical model of the enzymatic-separating microreactor, including the mass balances of all reaction components in the liquid phase and in the solid phase, with the enthalpy balance and the equation describing the thermal enzyme inactivation. The microreactor used was constructed in the form of a thin channel filled with a gel containing an immobilized enzyme and an adsorbent, in which the enzyme reaction, molecular diffusion, electro-osmotic flux and adsorption took place. The numerical analysis of the mathematical model demonstrated the potential of the presented reactor design and operation mode and allowed the process optimization to take place without experimental studies. The mathematical model of a biosensor based on an array of enzyme microreactors immobilized on a single electrode was efficiently used for the analysis of the influence of a microreactor geometry on the biosensor response [136]. The model included diffusion equations with a non-linear component related to Michaelis-Menten enzymatic kinetics. Iliuta and Larachi [137] developed a model to describe nicotinamide synthesis in laminar flow microreactors, in which the enzyme nitrile hydratase was uniformly immobilized on the microchannel wall and/or on the surface of microparticles dispersed in the liquid phase. The model was proposed assuming the laminar flow, where mass transport in the radial direction occurs by molecular diffusion and mass transport in the axial direction occurs by both advection and diffusion. The diffusion of particles, depending on the Brownian motion, was described by the Stokes-Einstein equation. The results of the simulations were used to select the optimal microchannel geometry to achieve maximum nicotinamide synthesis. Furthermore, Krull and Peterat [138] modeled the aerobic cultivation of Saccharomyces cerevisiae in a continuous chemostat using a multiphase microreactor. The model was based on a steady-state equation describing the concentration changes of biomass, glucose, and ethanol with high accuracy. Schilke et al. [139] applied a mathematical modeling approach to analyze the hydrolysis of o-nitrophenyl-β-D-galactosylpyranoside with β-galactosidase from Aspergillus oryzae immobilized on the thiolated nanosprings by reversible disulfide linkages in a microreactor system. The two-dimensional mass transfer and reaction of the species was modeled by Fickian convection/diffusion equation, and the simulation results were used to improve the microreactor performance. Interestingly, Elagli et al. [140] used a stochastic algorithm based on Michaelis-Menten kinetics to describe hemoglobin hydrolysis by pepsin. The motivation for model development was the idea to modulate the kinetics of the reaction in order to obtain a new kinetic behavior, leading to altered peptides appearance kinetics.
Mathematical modeling is critical in the creation of microreactor [141]. Computational fluid dynamics is a modeling approach with strong visualization capabilities that can interact with a wide range of geometry and boundary conditions [137]. Computational fluid dynamics modeling is required to offer knowledge of the operating characteristics of microreactors and to assess the benefits and drawbacks of various flow profiles. Computational fluid dynamics modeling is useful not only for efficient design but also for interpreting experimental results. Different CDF commercial softwares are being used for microreactors modelling. A computational fluid dynamics modeling engineering approach to novel reformer designs can save time and money by avoiding costly iterative experimental design [12,[142][143][144]. Venezia et al. [145] performed the computational fluid dynamics simulations of the hydrolysis reaction in a monolith channel coated by β-glucosidase adsorbed on a wrinkled silica nanoparticles. Their simulation results showed high agreement with the experimental results, demonstrating the validity of this model. Furthermore, simulations were carried out to estimate cellobiose conversion under different inlet conditions (i.e., mixture inlet velocity variation), without carrying out further experimental analyses, which is significantly more effective. Furthermore, Strniša et al. [106] used the mesoscopic lattice Boltzman method to analyze the resident time distribution within a micro-packed bed reactor composed in a two parallel-plate configuration with immobilized Candida antarctica lipase B used for the transesterification of vinyl butyrate and 1-butanol into butyl butyrate. Simulation results were in good agreement with experimental data and were used to compare the reaction efficiency of different micro-packed bed reactor geometries. Similarly, Peñaranda et al. [146] used CFD modelling to select the optimal geometry of magnetic microreactor used for phenol removal from wastewater. Laccasse used for the oxidation of phenol was covalently immobilized on amino-terminated, silanized magnetic nanoparticles. Multiphysics simulations revealed that particle mobility caused mixing patterns that were able to promote interactions between wastewater and the bionanocomposites.

Conclusions and Future Perspective
Microchannel devices have the benefits of decreased area and laminar flow compared to conventional reaction apparatuses. They can mimic a biological reaction apparatus, such as cellular surfaces and vascular systems. The further development of microreaction technologies will lead to improved process intensification and more efficient methods for biocatalytic processes. Particular research interest is focused on improving selective and reversible enzyme immobilization in microreactor systems [145], which would improve process performance. With the development of nanotechnology [25] and additive manufacturing or 3D printing [137], nanoscale support matrices are increasingly used in enzyme-based biotransformation processes [138][139][140]. Furthermore, with the development of analytical systems compatible with microfluidic devices, there is significant interest in the online measurements of the reaction occurring in microreactors. The real-time monitoring of chemical/biochemical reactions is an important tool in chemical/biochemical engineering. Online monitoring assists operators in process design and optimization and enables early intervention in case of unusual behavior [141,142]. Furthermore, microreactors are an excellent tool for kinetic studies under specific mass transport conditions, enhancing the capabilities of other methods and the future application of those systems will likely focus on the use of enzymes in multiphase systems to perform cascade reactions [143][144][145]. This is because, as noted by Zhang et al. [147] and Guo et al. [148], the continuous flow cascade of multistep heterogeneous catalysis is an important approach that has the potential to revolutionize stepwise catalytic synthesis, but it remains difficult to implement in practice.