Mathematical modelling of the residence time distribution of CO2 tracer in a three- phase micro-packed bed reactor: An experimental analysis

This study reports the residence time distribution (RTD) using CO2 as tracer in Three-phase micro-packed bed (TPPB) reactor. Experimental measurements were obtained at the inlet and at the outlet from TP-PB reactor using the injection of small amount (3%) of CO2 tracer inside the sweep gas current. The dynamic model characterizes a diffusion-adsorption process of CO2 tracer in terms of mass transfer phenomena (external and internal). The mathematical model was validated against a set of experimental data where simulated results of CO2 tracer adequately matched the experimental measures at the outlet of the micro-packed bed.


Introduction
Three-phase micro-packed bed (TP-PB) reactors are an important and valuable device for process intensification where many applications are carried out with two or more immiscible fluids. When compared with conventional reactors, TP-PB reactors report several advantages, such as high surface-to-volume ratio and excellent mass and heat transfer performances Chen et al., 2016). These TP-PB reactor models of small catalyst (8-150m) particles became efficient laboratory-tools in the case of heterogeneously catalyzed gas-liquid reactions (Anjos et al., 2017;Chang et al., 2003). The literature has investigated some works on heterogeneous catalytic process in micro-packed bed reactors for two phase and/or three phase systems (Cruz & Silva, 2017;Dias and Silva, 2020;Figueroa et al., 2018). These studies reported results of processes for liquid-solid and/or gas-liquid-solid systems using simple models, but authors did not provide guidance when TP-PB reactors have been used to a full mathematical model (equation for the gas phase, liquid phase and solid phase) (Bonfim et al., 2020;Gray et al., 2008;Kongnoo et al., 2017;Joss et al., 2017). We fill this gap with the concepts indicated by these authors and extend an application for the Residence Time Distribution (RTD) using CO2 as tracer.
The analysis of the RTD from TP-PB reactor provides essential information about their overall performance (Silva, 2011, Moulijn et al., 2016. In addition, the RTD represents an useful tool in TP-PB reactor design, especially when deviations from ideal flow patterns (such as plug flow) play a key function in determining the reactor performance (Panariello et al. 2018;Silva et al., 2019). The RTD response is given by injecting a well-defined amount of the tracer into the inlet of TP-PB reactor at time t = 0 and in turn recording the response of the tracer concentration at a central position of bed outlet.
In this research, simulated measurements are carried out to study the RTD of CO2 tracers in TP-PB reactor. A mathematical model is developed to compare the experimental results of the RTD with that simulated results from the mathematical model at the outlet from TP-PB reactor. The mathematical model is validated by minimizing the differences between the experimental and simulated measurements of the RTD of CO2 tracer using as criterion an objective function.

Problem characterization
The hydrodynamic characterization of the flow pattern through the micro-packed bed configuration in TP-PB reactor is carried out by tracing the main gaseous phase. A schematic diagram (SD) is shown to simulate the RTD experiments using the stimulus-response methodology. This SD is shown in Figure 1 as follows. 5400 rpm), respectively. The heart from Figure 1 consists from a stainless steel TP-PB reactor with the inner diameter of 0.5mm, the outer diameter of 0.9 mm and bed length of 0.6 m.

Working model from TP-PB reactor
The governing equations of the developed mathematical model are described by mass balances limited to the region of the borders from TP-PB reactor. In this work, the mass balance equations are developed based of CO2 tracer in gas, liquid and solid phases. The developed mathematical model for CO2 tracer within phases is restricted to the following simplifications: (i) isothermal system, (ii) the gas and liquid phases are modeled as plug flow, (iii) the system operates with partial wettability of solid particles, (iv) intraparticle diffusion in the pores of catalytic particles, (v) adsorption equilibrium in the system and (vi) the system operates with a small concentration of tracer in order to minimize disturbances in operating conditions from TP-PB reactor. From above simplifications, the model equations can be described as follows.

Balance of CO2 tracer in the gas phase
The mass balance equation of CO2 tracer in the mobile gas phase inside TP-PB reactor is composed by three terms (see Eq. 1). The first describes the transient term of CO2 tracer, the second term expresses the convective mass flow towards the axial direction (z) in the gas phase, and the third term connects the gas-liquid mass transfer, respectively. All terms in Eq. 1 have unit of mol/m 3 sec. Thus, the mass balance of CO2 tracer in the percolating mobile (gas) phase around the solid particles inside TP-PB reactor is described as follows. The mass balance equation of CO2 tracer in the mobile gas phase inside TP-PB reactor is composed by three terms (see Eq. 1). The first describes the transient term of CO2 tracer, the second term expresses the convective mass flow towards the axial direction (z) in the gas phase, and the third term connects the gas-liquid mass Development, v. 10, n. 9, e23210917425, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i9.17425 transfer, respectively. All terms in Eq. 1 have unit of mol/m 3 sec. Thus, the mass balance of CO2 tracer in the percolating mobile (gas) phase around the solid particles inside TP-PB reactor is described as follows.
In Equation (1), hg (m 3 gas/m 3 reactor) is the gas holdup, t (sec.) is time, CCO2,g (kg/m 3 ) is the molar concentration of CO2 tracer in the gas phase, qg (m 3 /sec.) is the gas flow rate in the gas phase, d (m) is the diameter from TP-PB reactor, z (m) is the axial coordinate, respectively; kgℓ (m/sec.) is the gas-liquid mass-transfer coefficient, agℓ (m 2 / m 3 bed) is the gasliquid mass-transfer area per unit column volume, h (CCO2,g/CCO2,ℓ) is the Henry's law solubility constant, CCO2,ℓ (kg/m 3 ) is the molar concentration of CO2 tracer in the liquid phase, respectively.
The suitable initial and boundary conditions from Equation (1) are described as follows.

Balance of CO2 tracer in the liquid phase
The mass balance equation of CO2 tracer in the mobile liquid phase within TP-PB reactor is composed by four terms (see Eq. 5). The first describes the transient term of CO2 tracer, the second term expresses the convective mass flow towards the axial direction (z) in the liquid phase, the third term represents the gas-liquid mass transfer, and the fourth term connects the liquid-solid mass-transfer, respectively. All terms in Equation 3 have unit of mol/m 3 sec. Thus, the mass balance of CO2 tracer in the percolating mobile (liquid) phase around the solid particles inside TP-PB reactor is described as follows.
In Equation (5), hℓ (m 3 dissolved gas in the liquid phase/m 3 reactor) is the dissolved gas holdup in the liquid phase, qℓ(m 3 /sec.) is the gas flow rate in the liquid phase, respectively; fe (-) is the wettability factor, kℓs (m/sec.) is the liquid-solid mass-transfer coefficient, aℓs (m 2 /m 3 bed) is the liquid-solid mass-transfer area per unit column volume, Ci,p (mol/m 3 ) is the molar concentration of CO2 tracer in the solid phase, respectively.
The suitable initial and boundary conditions from Equation (5) are described as follows.

Balance of CO2 tracer in the solid phase
Conceptually, CO2 tracer is injected in the gas phase, dissolved in the liquid phase, and it is transferred towards the solid phase through the liquid film and then it diffuses in pores along the particle radius. The mass balance of CO2 tracer inside the solid particle is given by three terms (see Eq. (9)). The first term reports the transient term of CO2 tracer in pores of the solid particles, the second term describes the intraparticle diffusion of CO2 tracer inside pores of the solid particles, the third term expresses the equilibrium adsorption of CO2 tracer at the surface of the solid particles, respectively. All terms in Equation (9) have unit of mol/m 3 sec. Thus, the mass balance of CO2 tracer in the solid particles is given as follows.
In Equation (9) The suitable initial and boundary conditions from Equation (9)

Mass balance of CO2 tracer at the surface of the solid phase
A mass balance equation has been proposed to describe the equilibrium adsorption rate of CO2 tracer at the surface of the solid phase (Oliveira et al., 2020;Rios et al., 2014;Shen et al., 2011). The term (CCO2,ad/t) from Equation (7) describes the adsorbed amount of CO2 tracer at the surface of the solid particles. All terms in Equation (13) The suitable initial condition from Equation (13) is reported as follows.

Solution of the mathematical modelling
The basic concept of the coupled integral equations approach (CIEA) can be given by the Hermite approximation and can be found in Ref. . Thus, a general equation is reported as follows. Where, The function (f(x)) and its derivatives f () (x) are reported for all x  [xi-1, xi]. E,  is defined as the error in the approximation. In this context, it has been assumed that f () (xi-1) = fi () for  = 0, 1, 2, 3,…,  and f () (xi) = fi ( for  = 0, 1, 2, 3,…, . As result, this integration formula can give different levels of approximation that are traditionally called of Hα, β. Thus, approximations of order higher than H1,1 involve derivatives of order higher than one (Reis et al., 2018). These derivatives are avoided for the sake of simplicity of the methodology. Here, it was considered only two different approximations as follows.

Application of the CIEA methodology
For solving the equations system, i.e., Equations (1) It is possible to transform the equations system, Equations (1)- (14), applying Equations (20)-(21) using the boundary conditions of each EDP (Medeiros et al., 2021). Thus, transformed equations are reported as follows.
All coefficients from Equations (22)

Full solution's approximation
The choice of method is dependent on the desired accuracy as well as concerns about the stability and robustness of the system while maintaining computational efficiency. With respect to the transformed equations, Equations (22) Research, Society and Development, v. 10, n. 9, e23210917425, 2021 (CC BY 4.

Results and Discussions
The experimental results were measured at the inlet (ℓ1 = 0.05) and outlet (ℓ3 = 0.95) from micro-packed bed. As examples, Figure 2 shows the experimental RTD curves of CO2 experimentally obtained for different flow rates (gas (16x10 -6 m 3 /sec) and liquid (3x10 -6 m 3 /sec))) in terms of CCO2/CCO2,0 versus time. In Figure 2, we show corresponding results of the injection of small amount (3%) of CO2 tracer inside the sweep gas current. The operating temperature and operating pressure were kept constant at 300K in these experiments. From the injection of CO2 tracer, the experimental measurements are obtained at the inlet and outlet from TP-PB reactor as function of the operating time.

Conclusion
A dynamics modelling coupled the adsorption isotherm that describes the distribution of CO2 in TP-PB reactor was experimentally and numerically investigated. A computer code to simulate and validate the performance of the developed dynamic model for CO2 tracer in TP-PB reactor allowed the following conclusions: 1. The Residence Time Distribution (RTD) curves of CO2 tracer were experimentally obtained at the inlet and output of the micro-packed bed; 2. The experimental measures of CO2 tracer at the three different flow rates are in good agreement with simulated results of the developed mathematical model; 3. The residence time of CO2 tracer increases as the flow rate decreases.
As future works, novel tracers can be explored in the context to compute physical parameters. Moreover, new porous materials can be also used to improve the fluid-solid mass transfer. Solid open-cell foams consist in a kind of novel porous materials with low density and can be employed to enhance the mass absorption. On the other hand, solid open-cell foams are also used s potential candidates to generate efficiency products from chemical reactions.