REGISTRO DOI: 10.69849/revistaft/ra10202512202128
Anderson Rodrigo Geromel
Jornandes Dias da Silva
Abstract.
Experimental evaluation and dynamic modeling of liquid flow (H₂O with NaOH tracer) in trickle-bed reactors were performed. A dynamic gas-liquid-solid model was developed, treating the liquid phase and tracer as a continuum and solved using appropriate initial and boundary conditions. Tracer concentration measurements were carried out in a fixed-bed reactor under trickling flow across a range of operating conditions. Axial liquid-phase dispersion (Dax,L) and the overall liquid–solid mass transfer coefficient ((Ka)LS) were identified as key parameters. Model predictions were validated against experimental data from two cases, and optimal values of Dax,L and (Ka)LS were simultaneously obtained using a quadratic objective function. Empirical correlations were then proposed to describe both parameters as functions of the liquid volumetric flow rate.
Keywords: Trickle-bed reactor; NaOH tracer technique; Gas–liquid–solid flow; Dynamic modeling; Experimental analysis
1. INTRODUCTION
Trickle-bed reactors (TBRs) can be defined as fixed beds of catalyst particles in contacted with the cocurrent downward flows of gas and liquid phases at low superficial velocities. These reactors assume greater importance among the there-phase gas-liquid-solid reaction systems encountered in industrial processes. TBRs are extensively used in many process industries. These reactors are widely employed in petroleum refineries for hydrotreating, hydrodemetalization, and hydrocracking applications. On the other hand, they also are widely used for carrying out a variety of processes such as petrochemical, chemical, biochemical, and waste treatment. There are many works in the literature to model and describe the behaviour of processes of those TBRs. The behaviour to many of those works can be studied applying mathematical modelling.
Mathematical models of TBRs represent an ancillary tool for minimizing the experimental efforts required to developing this important equipment in industrial plants. Experiment and prototype development are the main requirements for accurate engineering design in any industrial process. However, mathematical modelling and numerical simulation are in continuous development, contributing in a growing form for the better understanding of processes and physical phenomena, and thereby for design. Besides, mathematical models require experiment in order to be validated and the required experiments involve complex measurements of difficult accomplishment. Therefore, mathematical modelling also represents an incentive for the development of new experimental methods and techniques.
There are various mathematical models of completely or partially wetted catalyst particles which may exist in TBRs. Each of those models is based on many assumptions and is forced to use simplifications in order to solve the complex equation systems. Normally, mathematical models of TBRs may involve the mechanisms of forced convection, axial dispersion, interphase heat and mass transfers, intraparticle diffusion, adsorption, and chemical reaction (Silva et al., 2003; Burghardt et al., 1995; Iliuta et al., 2002; Latifi et al., 1997).
Various flow regimes exist in a TBR depending on the liquid and gas mass flow rates, the properties of the fluids and the geometrical characteristics of the packed bed (Charpentier and Favier, 1975). A fundamental understanding of the hydrodynamics of TBRs is indispensable in their design, scale-up, and performance. The hydrodynamics are affected differently in each flow regime. The basic hydrodynamics parameters for the design, scale-up and operation are the pressure gradient and liquid saturation. The pressure gradient is related to the mechanical energy dissipation due to the two-phase flow through the fixed bed of solid particles. The liquid saturation, which partially occupies the void volume of the packed bed, is related to other important hydrodynamics parameters as the pressure gradient, the external wetting of the catalyst particles, the mean residence time of the liquid phase in the reactor and the heat and mass transfer phenomena (Specchia and Baldi, 1977).
The objective of the work is to estimate and describe the behaviour of the axial dispersion coefficient of the liquid phase, Dax,L, and overall liquid-solid mass transfer, (Ka)LS, coefficient using a set of experiments carried out in a laboratory scale TBR. Comparing the mathematical model with experimental data. Analyzing the correlations from the Dax,L and (Ka)LS as a function of the volumetric flow rate
2. MATHEMATICAL MODEL
To model the mathematical modelling according to the liquid-solid model, in which the liquid phase (H2O + NaOH tracer) was deal as a continuum in a fixed bed of solid particles. An one-dimensional mathematical model was adopted in which the axial dispersion, liquid-solid mass transfer, partial wetting and reaction phenomena are present. This model has been used for the liquid phase by using the NaOH as tracer and is restricted to the following assumptions: (i) isothermal system; (ii) all flow rates are constant throughout the reactor; (iii) the intraparticle diffusion resistance has been neglected; (iv) in any position of the reactor the chemical reaction rate within the solid is equal to the liquid-solid mass transfer rate.
Table-1: Summary of dimensionless variables
3. ANALYTICAL SOLUTION
The solution of transport problems in three-phase systems is very complex and usually numerical approximation methods are used. On the other hand, analytical solutions are used for the simple models. Although the analytical solutions are simple, the boundary conditions proposed for these models need a careful attention. The majority of the analytical solutions belong to infinite and semi-infinite field. The analytical solutions for the finite field have been developed by Feike and Torid (1998) and Dudukovic (1982). In these works, the authors adopt the analytical procedure in the finite field region (0 z L → 0 1) where the method of separation of variables is used. The S ( , ) was isolated from Eq. (10) and it was introduced in Eq. (6), reducing it to:
4. MATERIALS AND METHODS
The experiments were realized in a three-phase trickle bed reactor, which consists of a fixed bed with a height of 0.22 m and an inner diameter of 0.030 m with catalytic particles contacted by a cocurrent gas-liquid downward flow carrying the liquid tracer in the liquid phase. The experiments have been performed at conditions where the volumetric flow rates of the gas and liquid phases were maintained at such a level to guarantee a low interaction regime with QL in the range of 7.068 x 10-8 m3 s-1 to 2.122 x 10-6 m3 s-1 and QG in the range of 1.414 x 10-5 m3 s-1 to 3.181 x 10-4 m3 s-1 in pilot plant trickle be reactors (Ramachandran and Chaudhari, 1983).
Continuous analysis of the NaOH tracer, at a concentration of 10 mol m-3, were made using HPLC/UV-CG 480C at the outlet of the fixed bed. Results have been expressed in term of the tracer concentrations versus time.
The methodologies applied to evaluating the axial dispersion of the liquid phase and overall liquid-solid mass transfer effects for the (N2/H2O-NaOH/activated carbon) system were:
- Comparison of the experimental results with the Eq. (28) developed for the system;
- Evaluation of the Dax,L and (Ka)LS parameters of the mathematical model, in which the initial values are obtained from the correlations in Table (2);
- Optimization of the Dax,L and (Ka)LS parameters by the comparison between the experimental and calculated data by the Eq. (25).
The axial dispersion coefficient of the liquid phase and overall liquid – solid mass transfer coefficient have been determined simultaneously by the comparison between the experimental and theoretical data, obtained at the outlet of the fixed bed, subject to the minimization of the objective function (F), given by:
5. RESULTS AND DISCUSSIONS
The experimental results of the tracer in a laboratory scale TBR have been performed in twenty runs varying the volumetric flow rate (QL) of the liquid phase. Experimental procedures and results are presented and discussed in detail in the trickling flow regime. Results obtained by the mathematical model have been compared with both sets of experiments. An objective function (F) has been calculated and presented. Values of this objective function indicate a very good agreement between the mathematical model and both types of experiments. The computation methodology to optimize the axial dispersion coefficient (Dax,L) of the liquid phase and overall liquid-solid mass transfer (Ka)LS coefficients has to use an optimization subroutine with minimization of the objective function (Box, 1965).
Table -2: Correlations for the obtainment of the Dax, kLS and PWE, the initial values
The calculating of the NaOH concentrations from the mathematical model, for the system N2/H2O – NaOH / activated carbon, includes various fixed parameters. The values of the fixed parameters used in the parameter-fitting calculations are given in Table (3).
The Dax,L and (Ka)LS have been optimized from different volumetric flow rate of the liquid phase and it changes from 4.248 x 10-6 to 5.581 x 10-7 m3 s-1. The Dax,L of the liquid phase was varying from 3.186 x 10-5 to 0.735 x 10-5 m2 s1 using H2O and NaOH as liquid fluids. On the other hand, the (Ka)LS also was changing from 2.496 x 10-2 to 0.124×10-2 s-1. The range objective function range has been, respectively, from 1.572 x 10-4 to 1.212 x 10-4. The optimization of the Dax,L and (Ka)LS for all twenty runs has been performed with the minimization of the objective function, Eq. (25). Values of the Dax,L and (Ka)LS for all twenty runs differ in order of magnitude.
The tracer method has been developed in our laboratory to evaluate the external fluid-solid contacting efficiency because it produces rapidly contacting information (Silva et al., 2003). Therefore, the objective of the present work has been an experimental and theoretical investigation with the tracer introduced in liquid stream. First, experimentally obtained tracer concentrations have to be unified with the tracer answers obtained by the mathematical model. In experiments, the concentrations of the tracer have been determined in the time domain using HPLC/UV-CG480C at the outlet of the fixed bed TBR. The dimensionless dynamic concentration profiles are shown in Figs. (1a) and (1b). The agreement of these Figs. has been very well satisfactory especially between theoretical and experimental results. Figs. (1a) and (1b) have been obtained in different volumetric flow rate. These Figs. show clearly the validation of the mathematical model, Eq. (19), with physical reality.
Table – 3: Summary of intervals of operating conditions for the particle-fluid (Colombo et al., 1976 and Silva et al., 2003)
In Fig. (1a), the fitted curve according to the experimental model has been compared to a transient mathematical model simulation for a volumetric flow rate from 3.751×10-5 m2 s-1. The axial dispersion coefficient of the liquid phase for this experiment was from 2.836×10-5 m2 s-1. The overall liquid-solid mass transfer coefficient for this run was from 2.196×10-2 s-1. The Peclet number for this experiment was from 149.692. The objective function for this experiment was from 1.423×10-4. It can be noticed that the computer time has been from 9s in the simulation. The saturation exhibits a propagation front experimental and theoretical behaviour around = 190, hence showing that the mathematical model of this work has been correctly presented.
In Fig. (1b), the fitted curve according to the experimental model also has been compared to a transient mathematical model simulation for a volumetric flow rate from 1.096×10-6 m3 s-1. The axial dispersion coefficient of the liquid phase for this run was from 1.105×10-5 m2 s-1. The overall liquid-solid mass transfer coefficient for this experiment was from 0.975×10-2 s-1. The objective function for this run was from 1.015×10-4. The Peclet number for this run has been from 112.256. It can be observed that the computer time was 6s in simulation. The saturation shows a propagation front experimental and theoretical behaviour around = 120, hence showing that the mathematical model of this work was correctly presented.
Analysis of the axial dispersion coefficient of the liquid phase and overall liquid-solid mass transfer coefficient in the concurrent gas-liquid trickle bed reactors has mostly been performed with empirical correlations in a specific range of operating conditions. The axial dispersion coefficient was calculated in terms of the volumetric flow rate of the liquid phase as well as the overall liquid-solid mass coefficient also has been computed in functions of the same volumetric flow rate of the liquid phase. It can be observed that the axial dispersion coefficient, Dax,L, of the liquid phase and overall liquid-solid mass transfer coefficient, (Ka)LS, increase with an increase in volumetric flow rate.
In the studied trickling flow regime, our experimental results for the axial dispersion coefficient of the liquid phase as well as the overall liquid-solid mass coefficient are well correlated by means of the following equations:
Fig (2) shows the variations of experimental results of the axial dispersion coefficient of the liquid phase and theoretical predictions computed from the Eq. (29) as function of the volumetric flow rate. It can be known that the agreement between theoretical and experimental values of the axial coefficient of the liquid phase is satisfactory, especially at all range of the volumetric flow rate. The mean relative error Dax,L between the predicted and experimental results of this hydrodynamic parameter has been calculated by the Eq. (31). On the other hand, the deviation of the relative error around the mean value Dax,L was quantified from the Eq. (33). The results of the mean relative error and quadratic deviation were obtained for twenty runs, respectively, from Dax,L = 8.57% and Dax,L = 6.21%.
Fig (3) presents the changes of experimental data of the overall liquid-solid mass transfer coefficient and theoretical predictions obtained from the Eq. (30). It can be seen that the comparison between theoretical and experimental results overall liquid-solid mass transfer coefficient is very satisfactory, especially at all variations of the volumetric flow rate. The mean relative error (Ka)LS between the predicted and experimental values of this parameter has been computed by the Eq. (32). Furthermore, the deviation of the relative error around the value (Ka)LS be seen calculated from the Eq. (34). The values of the mean relative error and quadratic deviation have been estimated for twenty experiments, respectively, from (Ka)LS = 9.71% and (Ka)LS = 6.83%.
6. CONCLUSIONS
The objective of this paper has been to realize an analysis of the axial dispersion coefficient of the liquid phase and overall liquid-solid mass transfer coefficient using the NaOH tracer in a TBR under trickling flow conditions. In a first step, it has been shown the analytical solution of the mathematical model developed for the NaOH tracer in the TBR. In a second step, it has been optimized the Dax,L and (Ka)LS simultaneously by the comparison between experimental results and the Eq. (19) through the Eq. (25). In a third step, it has been validated on the basis of a comparison between the experimental results and theoretical predictions obtained from the Eq (19) according to Figs. (1a) and (1b). In a fourth step, it has been proposed the correlations for the axial dispersion coefficient of the liquid phase and overall liquid-solid mass transfer coefficient as a function from the volumetric flow rate according to Eqs. (29) and (30). In a fifth step, it has been shown the behaviour for the axial dispersion coefficient of the liquid phase and overall liquid-solid mass transfer coefficient as a function from the volumetric flow rate according to the Figs. (2) and (3). Furthermore, the statistical analysis through confidence intervals showed that all the parameters involved in the correlations from the Dax,L and (Ka)LS were estimated very accurately.
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