Modeling and simulation of an enzymatic reactive absorption process in the internal zone of a rotating packed bed apparatus

Chemical Engineering & Processing: Process Intensification 189 (2023) 109409 Available online 3 May 2023 0255-2701/© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Modeling and simulation of an enzymatic reactive absorption process in the internal zone of a rotating packed bed apparatus Michał Blatkiewicz a , * , Justyna Wojtasik-Malinowska a , Dawid Zawadzki a , Marcin Piątkowski a , Ondˇ rej H´ ajek b , Milan Malý b , Ondˇ rej Cejpek b , Maciej Jaskulski a a Faculty of Process and Environmental Engineering, Lodz University of Technology, ul. Wolczanska 213, 93-005 Lodz, Poland b Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2, 616-69 Brno, Czech Republic ARTICLE INFO Keywords: rotating packed bed reactive absorption carbonic anhydrase kinetic model methyldiethanolamine ABSTRACT A kinetic model of carbon dioxide absorption within an internal of a rotating packed bed (RPB) apparatus was developed using experimental data. Kinetics of the chemical reaction between carbon dioxide and aqueous solution of N-methyldiethanolamine with and without carbonic anhydrase additive were determined in a flat phase contact surface batch stirred reactor and implemented into the model. Overall pressure drop and absorption efficiency within a metal foam internal were modeled and compared with experimental measurements. The model’s predictions of CO 2 absorption and pressure drop fit mostly within 20% error range. Validated model served as a tool for further absorption process simulations and sensitivity analysis. 1. INTRODUCTION Rotating packed bed (RPB) is an innovative device for mass transfer operations, first patented in 1981 [1]. It is designed to intensify interfacial mass transfer by substituting gravity with high centrifugal force, which leads to intense micro-mixing of the phases, resulting in mass transfer coefficients up to three orders of magnitude higher than of stationary packed beds [2]. Usually operating in counter-current mode, RPBs contact immiscible phases of different densities, e.g. gas and liquid, where the former is introduced at the outer rim of the device’s casing, and the latter at the center of the device [3]. The gas moves inward due to pressure difference, while the liquid flows outward due to centrifugal acceleration. The phases are vigorously mixed within the annular internal, which is the central part of any RPB apparatus, and can take a variety of sizes and types, including woven wire meshes [4], foams [5], blade rings [6], structured baffles [7], and other. Typically, the internal is placed between two rotor plates and carefully sealed to prevent by-passing of the media. Due to high mass transfer efficiency, compact design, portability and quick responsivity [8], RPBs have been gaining interest in the chemical industry during the last several decades [9]. A wide range of RPB processes has been reportedly investigated, most notably reactive absorption of CO 2 [10], but also distillation [11], wastewater treatment [12], nanoparticle production [13], removal of volatile compounds [14], and more. In reactive absorption processes, several absorbents can be considered go-to standards, which also prevail in the subject literature. Sodium hydroxide is attractive to the researchers due to high reaction rate coefficients, low viscosities and relatively low costs [6,15,16]. On the other hand, alkanolamines make attractive alternatives to inorganic sorbents thanks to high absorption efficiency and capacity, as well as easier and cheaper regeneration steps [4,10,17]. Popular amine absorbents include monoethanolamine (MEA), diethanolamine (DEA) and n-methyldiethanolamine (MDEA), as well as their blends [18]. Since primary and secondary amines, such as MEA and DEA, can react directly with CO 2 molecules, forming carbamates via covalent bonds, they offer very fast reaction kinetics up to 8.4 m 3 /(mol s) at 298 K [19]. Meanwhile, tertiary amines, like MDEA, are fully substituted, and react with CO 2 indirectly by accepting a proton from a water molecule and producing a bicarbonate ion. This reaction is significantly slower with the kinetic coefficient about 0.005 m 3 /(mol s) at 298 K [20], but with total absorption enthalpies up to two times lower when compared to analogous MEA solutions [21]. Since energy consumption is often the bottleneck of absorption processes, the trade-off between reaction kinetics and reaction enthalpy may, at proper conditions, favor MDEA. The problem of low reaction kinetics of MDEA solutions can be overcome by supplementing the mixture with reaction-accelerating compounds. One example of such additives is carbonic anhydrase (CA) (EC 4.2.1.1), which is an enzyme responsible for CO 2 transport and * Corresponding author. E-mail address: [email protected] (M. Blatkiewicz). Contents lists available at ScienceDirect Chemical Engineering and Processing - Process Intensification journal homepage: www.elsevier.com/locate/cep https://doi.org/10.1016/j.cep.2023.109409 Received 24 January 2023; Received in revised form 27 April 2023; Accepted 2 May 2023 Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 2 management in animal blood and respiratory systems [22]. Supplementing the mixture with carbonic anhydrase may result in overall pseudo-first-order reaction coefficients by up to two orders of magnitude [23]. Although kinetic rate constants for CA can be found in literature, the enzyme kinetics should be used as rough estimations, as enzymatic activities are strongly impacted by a number of internal and external influences, such as enzyme host, long-term stability, additives, etc. It is therefore advised to always test absorption kinetics experimentally for reliable modeling purposes. Therefore, this work also includes measurements of the reaction rates between CO 2 and the absorbent used in the base study [24]. One of the major problems regarding advances in RPB technology is difficulty in predictive modeling of mass transfer phenomena. Since the RPB internals have annular shapes, and both phases move in the radial direction, crucial parameters, such as liquid holdup [25], effective interfacial area [26], and mass transfer coefficients [27] change across the internal due to changing radius and centrifugal forces acting on all elements of fluids. Additionally, high rotational speeds and equipment opacity hinder visual studies, which creates a need for sophisticated tools, such as gamma ray tomography [28], or computational fluid dynamics [29], which are costly and time-consuming. Various empirical correlations for the parameters have been suggested, but they lack universality, and their applicability for predictive modeling often boils down to the trial and error approach [30–33]. Despite major interest being focused on mass transfer within the internal, a number of RPB studies show that phase contact within the outer cavity zone, that is the space between the internal and the outer casing of the device, strongly contributes to the overall efficiency of the process [34–36]. Upon reaching the outer rim of the rotating packing, the centrifugally accelerated liquid enters the cavity zone in the shape of ligaments and fine spray droplets, which then splash on the annular casing, forming films and secondary droplets [36]. Therefore, for reliable modeling of rotating packed beds, neither the internal zone nor the outer cavity zone should be neglected. Numerous RPB mass transfer models exist, but they are prevalently based on total change in concentration within the whole apparatus, Nomenclature Abreviations CA carbonic anhydrase CSTR continuous stirred tank reactor MDEA N-methyldiethanolamine RPB rotating packed bed rpm rotations per minute Latin letters A surface area m 2 a specific effective mass transfer area m 2 / m 3 a p specific surface area of the packing m 2 / m 3 B empirical constant var. C molar concentration kmol/m 3 c mass concentration kg/m 3 D diffusion coefficient m 2 /s d p effective pore diameter m E enhancement factor - Er error - F molar flow rate mol/s g gravitational/centrifugal acceleration m/s 2 H Henry constant Pa/(m 3 mol) h height of the internal m K G overall mass transfer coefficient mol/(m 2 Pa s) k G gas-side mass transfer coefficient m/s k L liquid-side mass transfer coefficient m/s k CA specific reaction rate coefficient of CA m 6 /(kmol kg s) k MDEA specific reaction rate coefficient of MDEA m 3 /(kmol s) k OV overall pseudo-first reaction coefficient 1/s M molar mass kg/kmol m direction coefficient in a linear equation - N interfacial molar flux kmol/s N rot rotational speed rpm P total pressure Pa p partial pressure Pa R universal gas constant kJ/(kmol K) r radius m rr reaction rate kmol/(m 3 s) T temperature K t time s u superficial velocity m/s V volumetric flow rate m 3 /s X independent variable var. x molar fraction of CO 2 in liquid kmol/kmol Y dependent variable var. y molar fraction of CO 2 in gas kmol/kmol Greek letters γ liquid-solid contact angle deg Δ difference - ε porosity m 3 /m 3 ε G gas holdup m 3 /m 3 ε L liquid holdup m 3 /m 3 μ dynamic viscosity N/(m s) ν kinematic viscosity m 2 /s ρ density kg/m 3 σ surface tension N/m ψ sensitivity parameter - ω rotational speed rad/s Indexes * equilibrium G gas in Inlet L liquid N normalized out outlet OV overall p packing R reaction s stirrer δ bulk phase ∞ infinity Dimensionless groups Fr Froude number u2 g⋅dp Ga m modified Galileo number d3 p ρ μ ( ρ g−ΔP rout−rin) Ha Hatta number  kOV⋅DL √ kL Re Reynolds number u ρ ap μ Re m modified Reynolds number u ρ ap(1− ε ) μ Re ω rotational Reynolds number r2 avg ω ν Sc Schmidt number μ ρ D Sh Sherwood number kLds DG M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 3 calculated as differences between measurements at the inlet and at the outlet [37–41]. However, the internal zone and the outer cavity zone are characterized by significantly different hydrodynamics, and therefore, mass transfer mechanics. Flows of both phases through porous internal can be described as counter-current radial plug flow. Meanwhile, in the outer cavity zone, highly turbulent gas phase, dragged by the quickly spinning rotor, is in contact with finely dispersed liquid spray in a relatively large volume. Therefore, the nature of phase contact in the cavity zone is more accurately described as a continuous stirred tank reactor (CSTR) type. With this in mind, it can be assumed that the gas concentration measured within the cavity zone is the inlet concentration for the rotating internal. In the previous work [24], mass transfer experiments were conducted with the use of three infrared detectors of CO 2 : one before the gas inlet, second within the outer cavity zone, and third at the gas outlet. With the CSTR simplification of the cavity zone, the data allows to determine mass transfer specifically within the rotating internal. 2. PROBLEM STATEMENT Table 1 shows published models of reactive absorption of CO 2 in amine solvents with the use of RPB devices. Most of them investigate fast-reacting MEA solutions. Two papers focus on MDEA as the solvent: one with pure MDEA solution [52], and one with a blend of piperazine and MDEA [48]. However, neither work includes additional reaction activators, such as enzymes. Each of the presented models is based on experimental data from RPBs equipped with wire mesh packings. To the best of our knowledge, this work is the first to take interest in amine-CO 2 reaction process with the use of a metal foam packing. Additionally, none of the aforementioned works focus on modeling of pressure drop, while high viscosity of amine solvents results in different hydrodynamics than thinner, water-like solvents. This work is focused on development of a predictive kinetic model of absorption of CO 2 and wet pressure drop within a porous RPB packing, based on experimental data collected by our group in previous work. The investigated absorbent is 30 wt.% MDEA aqueous solution with and without additional 0.2 wt.% carbonic anhydrase. The model focuses on hydrodynamics and mass transfer within an annular internal made of stainless steel foam. Functions of CO 2 removal from simulated flue gas, as well as overall pressure drop, are determined and validated with experimental data. Upon validation, model predictions are used for sensitivity analysis. 3. MATERIALS AND METHODS The chemicals used within this study, along with their purities and suppliers, are presented in Table 2. 3.1. Batch kinetic experiments To experimentally determine reaction kinetics, a thermostated batch stirred reactor with flat phase contact area was used (Figure 1). The reactor is equipped with two electronic pressure gauges: WIKA S-10, capable of measuring pressures between 1 and 3.5 bar, and APC-2000 by Aplisens, working in the range between 0 and 1.6 bar, as well as A-class Pt100 thermometer. Safety valve with 3 bar overpressure limit is used for security measures. Pressure loss measured over 24 hours at 2.7 bar starting pressure was under 3%. In every experiment the liquid phase volume was 485.7 cm 3 , and the gas phase volume was 376.3 cm 3 . The process variables of the experiments are presented in Table 3. 3.2. Rotating packed bed reactive absorption experiments The total inner diameter of the RPB unit used in this work is 632 mm. The apparatus is equipped with two stainless steel rotor plates with Table 1 Models of RPB CO 2 absorption in amine-based solvents available in literature. Mass transfer approach Solvent Original data Ref. Key findings and comments Two-film MEA no [30] Focused on testing varying combinations of a e , k L and ε L correlations available in literature against literature data provided by Jassim et al. [42]. Additionally, two reaction kinetic models were investigated [43,44] yes [45] RPB model based on Kang et al. [30] with analogous correlations. A system composed of an RPB and a packed bed column was investigated for absorption process optimization. no [46] Development of an effective mass transfer area model based on the Onda equation [47], adapted to high gravity environment, validated with data published by Thiels et al. [45]. Contributions of k L , k G and k OV at varying lean loads of solvent were investigated. PZ+MDEA no [48] The model was developed for piperazine-MDEA system investigated by Zhang et al. [49]. The packing in the model was discretized into concentrical rings. Benfield solution yes [50] The hydrodynamic model was developed with the assumption of droplet flow, and the liquid-side mass transfer coefficient was based on diffusion in a spherical droplet. Simulations proposed a mechanism of the end effect in the rotating packing. Surface renewal MEA yes [51] Based on the surface renewal approach to mass transfer, a dynamic model was proposed to describe RPB in non-steady state. The model allowed to simulate system responses to disturbances in the flow. The results suggested 70 wt.% of MEA as optimal concentration of the solvent. Penetration theory MDEA yes [52] According to Higby’s theory, the model allows to determine mass transfer coefficients with predictions of average lifetime of liquid film at the interface. Unlike the work of Yi et al. [50], the hydrodynamics were based on laminar films. MDEA concentrations of 10-30 wt. % were investigated. HTU/NTU MEA yes [53] Instead of kinetic approach, the model relied on determination of mass transfer units. In order to simulate real flue gas, high temperature gas phase with 40% water content was used. It was determined that under such operating conditions, K G a is mainly impacted by amine concentration. M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 4 outer diameter of 600 mm. The internal is a ring of isotropic metal foam made of stainless steel type NC1116 (RECEMAT B.V., Netherlands). The parameters of the packing are presented in Table 4. The RPB operates in counter-current. Gas mixture of 15 mol.% CO 2 and 100% humidified air is introduced at the rim of the casing in the perpendicular direction to the rotating internal. The liquid, introduced at the eye of the rotor, is distributed radially by three sets of nine nozzles, each of 0.9 mm in diameter. The total pressure drop is determined by two Aplisens APC-2000 electronic pressure gauges installed before the inlet and at the outlet of the gas phase. The setup is equipped with three Guardian NGDC devices by Edinburgh Sensors for online measurement of carbon dioxide in the gas phase: first in the inlet stream, second inside the outer cavity zone, and third at the gas outlet of the apparatus. Ambient air is compressed by Elektror HRD 2T FU fan, subsequently humidified by distilled water inside a 400 mm ×2000 mm (diameter ×height) packed absorption column, and eventually mixed with food-grade CO 2 stream controlled by a needle valve. Gas phase flow rates are calculated with the use of several calibrated orifice plates coupled with electronic thermometers. Liquid flow rate is controlled by a PID regulator in the setup’s electronic control system. The values of process conditions, under which the experiments were conducted, are presented in Table 5. A more detailed description of the experimental method, as well as the P&ID of the equipment can be found in [24]. All experiments were conducted at ambient temperature and under atmospheric pressure, noted at every experimental series for modeling purposes. Every experimental series was conducted in triplicates. The resulting data of each triplicate was averaged and had standard deviation calculated. 4. MODELING 4.1. Reaction kinetics In reactive absorption, the process of gas absorption is accelerated by chemical reaction occurring on the liquid side. In the two-film model, such acceleration leads to the shortening of the laminar film on the liquid side. Thus, the liquid-side mass transfer coefficient is multiplied by the so-called enhancement factor E. kLR =kL⋅E(1) Hatta number (Ha) is a dimensionless parameter describing the ratio of the components’ chemical reaction rate to their diffusion through laminar liquid films. As a general expression, it is defined as: Ha = 2 m+1⋅kmcm−1 Acn B⋅DA|L √kL (2) where m and n correspond to stoichiometric values determining the order of reaction toward reactants A (CO 2 ) end B (MDEA) respectively. Since absorbent concentrations change insignificantly during the reaction, the value of n can be assumed to be zero, while the reaction toward CO 2 is in first order (m =1). Thus the equation simplifies to: Ha = kOV ⋅DA|L √kL (3) where k OV is the overall pseudo-first reaction kinetic coefficient. Hatta number is a crucial factor in determination of the form of the enhancement factor. In two-film theory, the enhancement factor is calculated with Eq. (4). E=Ha tanh(Ha)(4) For values over 2, hyperbolic tangent asymptotically approaches 1, therefore for Ha>2 the enhancement factor is simply equal to the Hatta number. At the same time, the pseudo-first order reaction regime is limited by enhancement factor of instantaneous reaction E ∞ : E∞ Ha≫1 (5) where Table 2 List of chemicals Product CAS no. purity Supplier CO 2 124-38-9 >99.9% Air Products N 2 O 10024-972 >99.8% Air Products MDEA 105-59-9 >99% Brenntag 30 wt.% MDEA solution with 0.2 wt.% CA - - Evonik Figure 1. Experimental setup for kinetic measurements: thermometer (1), pressure gauge (2), CO 2 /N 2 O bottle (3), liquid supply (4), thermostated water outlet (5) and inlet (6), waste outlet (7), safety valve (8), outlet for manual decompression (9). Table 3 Conditions of batch reactor experiments Variable Unit Values Temperature [◦C] 20, 22, 25, 30 Stirrer rotational speed [rpm] 60, 90, 120, 150 MDEA concentration [wt.%] 30 CA concentration [wt.%] 0, 0.2 Table 4 Physical parameters of the RPB packing Parameter Symbol Value Unit inner radius r in 0.073 [m] outer radius r out 0.189 [m] height h 0.01 [m] porosity ε 0.922 [m 3 / m 3 ] specific surface area a p 1000 [m 2 / m 3 ] critical surface tension σ c 0.075 [N/m] Table 5 Investigated process conditions in RPB absorption experiments n rot 410, 535, 600, 750, 900, 1050, 1200, 1250, 1390 [rpm] V G 20, 40, 60 [m 3 /h] V L 0.12, 0.18, 0.27, 0.36 [m 3 /h] M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 5 E∞=1+(DB DA C∗ A Cδ B)(6) Due to the fact that pure gases were used for kinetic measurements, laminar film on the gas side has thickness of zero, and no mass transfer resistance is assumed: NCO2|G=d(pCO2) dt VG RT (7) Mass transfer on the liquid side is enhanced by chemical reaction, and concentration of CO 2 in bulk liquid is low enough to be approximated to zero, therefore: NCO2|L=AkL,R(C∗ CO2−Cδ CO2)=AkLE⋅HCO2pCO2(8) With the assumption of no accumulation of CO 2 on the interface, right-hand sides of Eqs. 7 and 8 are equal, with opposite direction of mass transfer, thus, with the assumption of pseudo-first order, noninstantaneous reaction, Eq. 9 is derived. ln(pCO2,1)= − A kOV DCO2|L √HCO2RT VG ⋅(t1−t0)+ln(pCO2,0)(9) where indexes 0 and 1 refer to beginning and end of the pressure decrease measurement respectively. Eq. (9) is approximated with a linear function for lean absorbent mixtures, and k OV can be calculated from its slope according to Eq. (10). kOV =(m⋅VG A⋅HCO2⋅RT)2 DCO2|L (10) A study by Kang et al. [30] suggests that under certain conditions the feasibility of pseudo-first order approach may be limited, and proposes the approach of Aboudheir et al. [43], but the problem is specific to high concentrations of fast-reacting MEA (see Table 1). 4.2. Mass transfer The RPB model was designed under the assumption of steady state of the process with plug flow of both phases at constant rates. The gas phase is treated as ideal gas. Since the air is saturated with water in the humidifier column (see Section 2.3), there is no water evaporation from the absorbent solution. Thus, only CO 2 is transferred between phases, and other gas components are treated as inerts. As stated before, due to different reaction mechanism, the reaction enthalpies of MDEA-CO 2 systems are significantly lower than in analogous processes using primary or secondary amines solutions [21]. Additionally, considering relatively low absorption rates (Δy<2 mol.%), the process is treated as approximately isothermal. It has been shown that in a rotating RPB the angular velocity of gas in the outer cavity zone approaches the angular velocity of the rotor [54], which, combined with fine spray of the liquid in the outer cavity zone, leads to a reasonable assumption that the outer cavity zone can be approximated as a continuous stirred tank reactor (see Figure 2). Thus, CO 2 concentration in the gas phase measured in the outer cavity zone can be assumed to be equal to the inlet concentration of the rotating internal. Eq. (11) describes the gas-side mass balance, and Eq. (12) describes the liquid-side mass balance. The general mass balance for both phases within an RPB are analogous to the system presented by Borhani et al. [55], adapted from mass balances for a packed column apparatus presented by Kvamsdal et al. [56]. However, due to the fact that the reaction cannot be treated as instantaneous, an additional expression for reaction rate in the liquid phase is considered in Eq. (12). Figure 2. Two mass transfer zones according to model simplification: internal zone (green) and outer cavity zone (orange). Red and blue arrows symbolize directions of gas and liquid flow, respectively. M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 6 ε G dyCO2 dt =1 2 π rh⋅ ∂ (FGyCO2) ∂ r−a⋅NCO2(11) ε L dxCO2 dt = − 1 2 π rh⋅ ∂ (FLxCO2) ∂ r+a⋅NCO2− ε L⋅rrCO2(12) With the assumption of steady state operation, equations (11-12) simplify to: d(FGyCO2) dr =a⋅NCO2⋅2 π rh (13) d(FLxCO2) dr = (a⋅NCO2− ε L⋅rrCO2)⋅2 π rh (14) The chemical reaction rate is defined as follows: rrCO2=kOV ⋅CCO2(15) Knowing that CO 2 flux N CO2 is described as: NCO2=KG(pG CO2|G−p∗ CO2)(16) and pCO2|G=P⋅yCO2(17) p∗ CO2=HCO2⋅CL CO2(18) The set of balance equations can be presented as follows: d(FGyCO2) dr =KGa(P⋅yCO2−HCO2⋅CL CO2)⋅2 π rh (19) d(FLxCO2) dr =(KGa(P⋅yCO2−HCO2⋅CL CO2)− ε L⋅kOV ⋅CL CO2)⋅2 π rh (20) Despite extensive research regarding hydrodynamics and mass transfer in RPBs, there still are no universal equations for mass transfer coefficient and effective phase contact area. Research literature positions suggest a variety of empirical equations describing such phenomena [31], but all of them are applicable only within a certain range of process conditions, RPB geometries and absorption systems. The initial approach in this work was akin to the approach of Kang et al. [30], with various models for k L , a and ε L for metal foam, mesh and other porous packings, but none of the combinations yielded satisfying results. Thus, the experimental data from the work of Wojtasik-Malinowska et al. [24] served as a base for development of the overall volumetric mass transfer coefficient. From the data presented, it can be seen that all the investigated process variables – gas flow rate, liquid flow rate, rotational speed – had non-negligible effect on the mass transfer efficiency. Gas and liquid flow rates in counter-current directly affect intimate contact inside the fast-rotating internal and the liquid holdup. Meanwhile, rotation of the packing not only accelerates the liquid phase, but also provides additional shear stress, affecting the micro-mixing. The proposed formula for the overall volumetric mass transfer coefficient is a function of liquid diffusion coefficient, porous packing geometry, liquid Schmidt, Reynolds and Froude numbers, and gas Reynolds number: KGa=f(DL,ap,dp,ScL,ReL,FrL,ReG)(21) 4.3. Pressure drop Modeling of pressure drop in an irrigated rotating packed bed equipment (wet pressure drop) has been the subject of several studies, which differ in approach and complexity level [57–61]. In this study, two models of different approaches were considered: a) An empirical model by Liu et al. [61], where the pressure drop is a function of the packing’s geometrical parameters, the gas capacity factor (FF), and a drag function (f dp ): ΔPI=FF2 2 ap ε 3fdp(rout −rin)(22) where: FF =UG⋅ ρ G √(23) fdp =72.93⋅Re0.042 L⋅Re−1.143 G⋅Re0.32 ω +97.84⋅Re0.536 G(24) b) A semi-empirical model by Lashkarbolooki et al. [59], where the overall pressure drop (Eq 25) is a sum of constituent pressure drops due to friction in the stationary packing (Eq 26), rotation (Eq 27) and gas-liquid interaction (Eq 28). ΔPII =ΔPIIa +ΔPIIb +ΔPIIc (25) ΔPIIa =a0.10 p(RemG ε 3 1− ε )0.74(VG 2 π h)ln(rout rin ) +a−1.25 p(RemG ε 3 1− ε )1.65(VG 2 π h)2(1 rin −1 rout)(26) ΔPIIb = ρ G ω 2 2(4.97+0.53⋅Re−3.72 mG (RemG Re ω )−1.65 −0.11(ap ε 3)0.46)2 (r2 out −r2 in) (27) ΔPIIc =(0.81⋅Re−0.35 mG Re0.69 ω +0.91(ap ε 3)1.25 Re−0.31 mG Re−0.25 ω Re0.06 mL )(uG−uL) (28) 4.4. Liquid holdup As described in Section 3.2, in the case of slow-reacting solvents, such as tertiary amines, the chemical reaction cannot be treated as instantaneous, which means that the mass balance equations cannot neglect the rate of the chemical reaction. Thus, liquid holdup needs to be taken into account in the model, as it determines the liquid retention time in the system. Unfortunately, liquid holdup is very difficult to measure, especially inside quickly rotating RPB rotors. Several empirical models of RPB liquid holdup have been proposed in the available literature [62–65], but, similarly to the mass transfer models, their applicability is usually limited. Groß et al. [28] conducted a set of experiments using gamma ray tomography to determine water holdup within an NCX1116 metal foam internal, which shares all the basic properties with the NC1116 type metal foam (see Table 4). Therefore, it may serve as a heuristic tool for selection of a best-fitting model from the available ones. Thus, simulations of water holdup inside a NCX1116 metal foam internal were carried out for the aforementioned holdup models. The best fit was achieved for the porous body model proposed by Lin et al. [63]: ε L=3.86 ε ⋅Re0.545 L⋅Ga−0.42 mL ⋅(apdp ε )0.65 (29) The model’s empirical parameters were re-fitted to better match the experimental data (Section 4.3). 4.5. Physicochemical properties To complete the model, the physicochemical properties of the system M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 7 need to be defined. All required empirical formulas are presented in Table 6. Gas density ( ρ G ) and gas viscosity (µ G ) are calculated for CO 2 -air mixtures from the ideal gas law. 4.6. Data evaluation and modeling software For basic data storage and input data calculation, Microsoft Excel was used (Version 2210). For model development, optimization and simulation, MATLAB (version R2021a) was used. For optimization of the algebraic formulas, the inbuilt GRG Nonlinear solver toolbox of Microsoft Excel, and the and least square nonlinear curve data-fitting toolbox (lsqcurvefit) of MATLAB were used. For solving sets of differential equations with opposite boundary conditions, MATLAB’s 4 th order boundary value problem solver (bvp4c) was used. All data diagrams were prepared in Microsoft Excel. 5. Results and discussion 5.1. Reaction kinetics In this section, the mass transfer characteristics of the flat-surface gas-liquid absorption reactor are determined, and the pseudo-first reaction rate coefficients are determined using the said reactor. To calculate the enhancement factors E, and thus the overall pseudofirst order reaction coefficients (Eq 10), the mass transfer characteristics of the reactor need to be determined with the use of an absorptive system without chemical reaction. Therefore, absorption of pure N 2 O was carried out in the same reactor under analogous conditions of further chemical absorption of CO 2 . The calculated liquid-side mass transfer coefficients served as input data to approximate the Sherwood correlation (Figure 3), which took the form of: Sh =21.053⋅Re0.1434 ω ⋅Sc0.1387 (30) Experiments regarding kinetics of CO 2 reactive absorption in pure MDEA solution (Figure 4a) showed the values of pseudo-first order reaction kinetic coefficient to be in line with literature data, with the formula presented in Eq. (31). kMDEA =3.306⋅109⋅exp(−6345 T)[ m3 kmol⋅s](31) Penders-van Elk et al. [23] showed that the specific rate of reaction facilitated by carbonic anhydrase is a function of the enzyme content and molar concentration of water. The formula for the specific CA kinetic coefficient, regressed from data shown in Figure 4b, is described by Eq. (32). kCA =5.707⋅1012⋅exp(−6088 T)[ m6 kmol⋅kg⋅s](32) Therefore, the overall pseudo-first order reaction rate coefficient of MDEA mixture supplemented with carbonic anhydrase can be expressed as: kOV =kCA⋅cCA⋅CH2O+kMDEA⋅CMDEA [1 s](33) kOV =5.707⋅1012⋅exp(−6088 T)⋅cCA⋅CH2O +3.306⋅109⋅exp(−6345 T)⋅CMDEA [1 s](34) Thus, the formula in Eq. (34) allows to calculate the overall pseudofirst reaction coefficient at the presence or absence of carbonic anhydrase additive and at any temperature in the investigated range. 5.2. Pressure drop For pressure drop prediction, two different models were considered – one semi-empirical [59] and one empirical [61], as described in Section 3.3. The pseudo-empirical equation of Lashkarbolooki et al. [59] takes into account three sub-models that contribute to the overall pressure drop. Upon entering experimental variables into the model, high deviations occurred, therefore MATLAB’s lsqcurvefit optimization function and Microsoft Excel’s Solver tool were used to re-fit the empirical parameters in accordance to the empirical data. Both fitting tools showed similar results, but the fitting of the MATLAB function was slightly better. Depending on the starting points, the optimization tool suggested several sets of parameters, for which the results converged within the evaluation limit. Therefore, the result with the lowest average error has been chosen for further consideration. In the best-fitting optimization result, several segments of the model could be neglected, as the empirical parameters tended to zero. Thus the simplified model can be presented as: ΔP1=aB1 p(RemG⋅ ε 3 1− ε )B2(VG 2 π h)⋅ln(rout rin ) + ρ G ω 2 2⋅(B3+B4⋅ReB5 mG⋅(RemG Re ω )B6 +B7⋅(ap ε 3)B8)⋅(r2 out −r2 in)(35) Although at this point the model managed to sufficiently predict the pressure drop for most of the data points, a portion of the results still showed high deviation from the experiments. Further analysis showed that high errors occurred for points characterized by low rotational speeds, more pronounced with higher flow rates of the gas and the liquid phase. This phenomenon of local minimum of the pressure drop as a function of rotational is known and has been reported in the literature [70]. Since liquid flow rate, gas flow rate, and rotational speed all contribute to this phenomenon, the additional empirical expression was designed as follows: ΔP2=B9⋅exp(B10⋅RemG⋅RemL Re ω )⋅(rout −rin)(36) The advantage of the presented model is that it is universally applicable to the whole scope of the operational window, and does not require calculation of conditional parameters, unlike the original version of the model [59]. The empirical parameters are presented in Table 7. For comparison, results of the empirical model by Liu et al. [61], re-fitted with the same methodology, are also presented in Figure 5, with Table 6 Empirical formulas for physicochemical properties used in the model Variable Formula Ref. Liquid density [kg/m 3 ] ρ L=1.03680 −4.2536⋅10−4⋅T −1.76785⋅10−6⋅T2 [66] Liquid viscosity [mPa s] μ L=5.44936 −0.01896⋅T+0.000608464⋅T2 [66] Diffusion of CO 2 in liquid phase [m 2 /s] DL=2.35 ∗10−6exp(−2119 T)( μ water μ L)0.8 [67] Diffusion of CO 2 in gas phase [m 2 / s] DG=1.883 ∗10−22 T 3 2 p∗ σ 2 i,j∗Ω(Ti,j)∗(1 Mi+1 Mj)0.5 [68] Henry constant for liquid phase [Pa/(m3 mol)] HCO2=101.325exp(S1+S2 T+S3 T2) S1=2.01874 −2.37638⋅101⋅cMDEA +2.90092⋅102⋅ c2 MDEA −4.80196⋅102⋅c3 MDEA S2=3.13549⋅103+1.54931⋅104⋅cMDEA −1.83987⋅ 105⋅c2 MDEA +3.00562⋅105⋅c3 MDEA S3= − 8.13702⋅105−2.4808⋅106⋅cMDEA +2.92013⋅ 107⋅c2 MDEA −4.70852⋅107⋅c3 MDEA [69] M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 8 empirical parameters shown in Table 8. It can be noticed that the fitting of both models is very similar, and in both cases the points that exceed the ±20% tolerance range behave similarly, which is probably caused by errors in experimental measurements. fdp =B11⋅ReB12 L⋅ReB13 G⋅ReB14 ω +B15⋅ReB16 G(37) The R 2 value is 97.37% and 96.58% for the first and second model, respectively, and the respective average absolute errors are 6.1% and 7.6%. 5.3. Liquid holdup As described in Section 3.4, the liquid holdup model used in this work is based on the model proposed by Lin et al. [63]. Since the original empirical parameters did not fit the experimental data by Groß et al. [28] sufficiently, the parameters were recalculated with the lsqcurvefit optimization tool in MATLAB. The optimized parameters to Eq. (38) are presented in Table 9. ε L=B17⋅ ε ⋅ReB18 L⋅GaB19 mL ⋅(apdp ε )0.65 (38) Figure 3. Fitting of Sherwood correlation for the absorption reactor Figure 4. Arrhenius plots of specific reaction rate coefficients for MDEA (a) and CA (b) Table 7 Empirical parameters for Eqs. (35) and (36) B 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 B 9 B 10 0.5807 0.3657 5.099 0.4509 -4.289 -1.332 -0.1016 0.4986 80.01 55.27 M. Blatkiewicz et al. Chemical Engineering and Processing - Process Intensification 189 (2023) 109409 9 The comparison between the experimental data by Groß et al. [28] with model simulations are presented in Figure 6. Although the original experiments were conducted with water as the liquid phase, and thus the model lacks validation data for MDEA solutions, it is the best available tool to predict liquid holdup in the overall model, as it is determined for the used packing type, only with a different liquid, the physiochemistry of which is accounted for in the model variables. 5.4. Mass transfer According to the two-film theory, the compound subjected to interfacial mass transfer in the absorption process migrates from the bulk gas phase to the laminar boundary gas layer, where it diffuses toward the interface, and then, through the laminar liquid layer – toward the bulk liquid phase. In reactive absorption processes the thickness of the boundary layer behaves as thinner due to mass transfer enhancement from the chemical reaction (Eq. 1), as the compound is partially depleted in the diffusion stage. Depending on liquid-side mass transfer coefficients and reaction rate coefficients, in some cases it may be assumed that the concentration of the compound approaches zero in the bulk liquid phase. In this work, however, the vigorous micro-mixing of the phases in the RPB contactor results in high mass transfer coefficients, while the pseudo-first-order chemical reaction rates are relatively low with k OV =66 s −1 at 298 K, which is two orders of magnitude lower than in the case of absorption of CO 2 in 1M NaOH solution at the same temperature [71]. Therefore, the accumulation of unreacted CO 2 in the liquid phase cannot be neglected, as it decreases the driving force of the absorption process. Due to high centrifugal forces and process variables drastically changing over the packing’s radius, determination of universal volumetric mass transfer coefficients is very difficult, especially for different media and packing structures. Therefore, mass transfer models available in the literature are usually built on the physicochemical properties of the media, characteristics of the RPB packing, and a combination of applicable criterial numbers which describe the phase flows. From the data previously published by our team [24], it can be seen that liquid flow rate, gas flow rate and rotational speed all have significant influence on the process performance. Preliminary simulations of available models have shown that in this process setup the mass transfer resistance in the gas phase is negligible to the overall mass transfer resistance, however, the intensity of gas flow significantly affects the micro-mixing intensity, affecting the mass transfer resistance on the liquid side [24]. Thus, liquid and gas Reynolds numbers were included in the model. As the laminar effects of the liquid phase have to be considered, the square root of liquid Schmidt number is also included in the model. Finally, since both phases are subjected to centrifugal forces within the rotating internal (with the emphasis on the liquid phase, which is three orders of magnitude denser than the gas phase), the centrifugal effect also needs to be accounted for. This effect can be described either by the Grashof Figure 5. Comparison of experimental and modeled values of pressure drop Table 8 Empirical parameters for Eq. (37) B 11 B 12 B 13 B 14 B 15 B 16 3.041 E-07 0.1745 -2.0772 2.0464 3.9384 1.1788 Table 9 Empirical parameters for Eq. (38) B 17 B 18 B 19 7.50 0.64 -0.39 M. Blatkiewicz et al.