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Brazilian Journal of Chemical Engineering
Print version ISSN 01046632
Braz. J. Chem. Eng. vol.29 no.3 São Paulo July/Sept. 2012
http://dx.doi.org/10.1590/S010466322012000300014
PROCESS SYSTEMS ENGINEERING
Mathematical modeling of a threephase trickle bed reactor
J. D. Silva^{I, * }; C. A. M. Abreu^{II}
^{I}Polytechnic School, UPE, Laboratory of Environmental and Energetic Technology, Phone: + (81) 31837515, Rua Benfica 455, Madalena, CEP: 50750470, Recife  PE, Brazil. Email:jornandesdias@poli.br
^{II}Department of Chemical Engineering, Federal University of Pernambuco, (UFPE), Phone + (81) 21268901, R. Prof. Artur de Sá 50740521, Recife  PE Brazil. Email: cesar@ufpe.br
ABSTRACT
The transient behavior in a threephase trickle bed reactor system (N_{2}/H_{2}OKCl/activated carbon, 298 K, 1.01 bar) was evaluated using a dynamic tracer method. The system operated with liquid and gas phases flowing downward with constant gas flow Q_{G} = 2.50 x 10^{6} m^{3} s^{1} and the liquid phase flow (Q_{L}) varying in the range from 4.25x10^{6} m^{3} s^{1} to 0.50x10^{6} m^{3} s^{1}. The evolution of the KCl concentration in the aqueous liquid phase was measured at the outlet of the reactor in response to the concentration increase at reactor inlet. A mathematical model was formulated and the solutions of the equations fitted to the measured tracer concentrations. The order of magnitude of the axial dispersion, liquidsolid mass transfer and partial wetting efficiency coefficients were estimated based on a numerical optimization procedure where the initial values of these coefficients, obtained by empirical correlations, were modified by comparing experimental and calculated tracer concentrations. The final optimized values of the coefficients were calculated by the minimization of a quadratic objective function. Three correlations were proposed to estimate the parameters values under the conditions employed. By comparing experimental and predicted tracer concentration step evolutions under different operating conditions the model was validated.
Keywords: Trickle Bed; KCl Tracer; Modeling; Transient; Validation
INTRODUCTION
Mathematical modeling of threephase trickle bed reactors (TBR) considers the mechanisms of forced convection, axial dispersion, interphase mass transport, intraparticle diffusion, adsorption and chemical reaction. These models are formulated by relating each phase to the others (Silva et al., 2003; Iliuta et al., 2002; Latifi et al., 1997; Burghardt et al., 1995).
The tricklebed reactor is a threephase catalytic reactor in which liquid and gas phases flow concurrently downward through a fixed bed of solid catalyst particles where the reactions take place. These systems have been extensively used in hydrotreating and hydrodesulfurization in petroleum refining, petrochemical hydrogenation and oxidation processes, and methods of biochemical and detoxification of industrial waste water (AlDahhan et al., 1997; Dudukovic et al., 1999; Liu et al., 2008; Ayude et al., 2008; Rodrigo et al., 2009; Augier et al., 2010).
The flow regimes occurring in a tricklebed reactor depend on the liquid and gas mass flow rates, the properties of the fluids and the geometrical characteristics of the packed bed. A fundamental understanding of the hydrodynamics of tricklebed reactors is indispensable to their design and scaleup and to predict their performance (Charpentier and Favier, 1975; Specchia and Baldi, 1977).
The purpose of this work was to evaluate the transient behavior of a threephase trickle bed reactor using a dynamic tracer method to estimate the magnitude of the hydrodynamic parameters related to the operations, including the axial dispersion coefficient in the liquid phase, the liquidsolid mass transfer coefficient and the partial wetting efficiency. A dynamic phenomenological model was proposed and validated with experimental reaction data.
MATHEMATICAL MODEL
To represent the dynamic behavior of the tracer component, a onedimensional mathematical model was formulated considering the effects related to the axial dispersion, liquidsolid mass transfer, partial wetting and chemical reaction. The model was adopted for KCl, considered to be the tracer component in the liquid phase, and was restricted to the following hypotheses: (i) isothermal operation; (ii) constant gas and liquid flow rates throughout the reactor; (iii) moderate intraparticle diffusion resistance; (iv) the chemical reaction rate within the catalytic solid is equal to the liquidsolid mass transfer rate, in any position of the reactor. The mass balance for the tracer (A_{L}) in the liquid phase is written as:
Mass balance for the liquid;
The initial and boundary conditions for Eq. (1) are given as:
The equality of the mass transfer and reaction rates can be expressed by the following equations:
The kinetic model for the reaction was based on a firstorder reaction according to the following expression (Colombo et al., 1976):
where r_{KCl} is the consumption rate of the reactant, A_{s}(z, t) is the reactant concentration at the surface of the solid phase and k_{r} is the
Combining Equations (5) and (6), the rate of mass transfer is equal the rate of reaction at the surface of the solid phase as:
Equations (1) to (4) and (7) can be analyzed by employing the dimensionless variables in Table 1.
Expressed in the dimensionless variables, the equations and the initial and boundary conditions can be rewritten as:
Equations (8) to (12) include the following dimensionless parameters:
The dimensionless concentration, , was isolated in Eq. (12) and introduced into Eq. (8), reducing it to:
where,
SOLUTION IN THE LAPLACE DOMAIN
Applications of the Laplace Transform (LT) to dynamic transport problems in threephase trickle bed reactors with tracer (liquid, gas) are employed to solve the linear differential equations. To complete the solution, the Laplace Transform inversion method is indicated, where numerical inversion is often employed. In the present work, the LT technique was applied to the partial differential equation, Eq. (16), as presented below:
where the overhead "s" indicate the LT and its domain variable, respectively.
The initial and boundary conditions in the Laplace domain are:
Eq. (18) is a secondorder nonhomogeneous ordinary differential equation. Its solution is expressed by Eq. (22) and is composed of the general solution of the homogeneous ordinary differential equation and a particular solution :
The secondorder homogeneous ordinary differential equation is expressed as:
Its general solution is given by the following function:
where and are defined as:
In term of hyperbolic functions, Eq. (24) was written as:
where f_{1}(s) _{2}(s) are expressed by f_{1}(s)=C_{1}(s)C_{2}(s), and f_{2}(s)=C_{1}(s)+C_{2}(s).
The particular solution was given by the expression:
The general solution has been presented as Eq. (22), in which and were attributed according to the result below:
where f_{1}(s) and f_{2}(s) are two arbitrary integration constants. Applying the boundary conditions from Eqs. (20) and (21) to the general solution, Eq. (27), led to the algebraic equations needed to find the arbitrary integration constants f_{1}(s) and f_{2}(s) in terms of known parameters. The expressions for these two constants have been found here as:
Eqs. (28) and (29) were introduced into Eq. (27) to obtain the general solution of the tracer concentration in the liquid phase:
For = 1 it was possible to obtain the concentration of the tracer at the exit of the fixed bed as follows:
Hence,
To obtain the concentration evolution of the tracer at the exit of the tricklebed reactor, the numerical fast Fourier transform (NFFT) technique was employed. In the NFFT operations, the Laplace variable "s" was changed to in the Fourier domain. This technique was applied considering a step concentration disturbance at the inlet of the fixed bed, whose expression is written as:
MATERIAL AND METHODS
The transient behavior in a threephase trickle bed reactor system (N_{2}/H_{2}OKCl/activated carbon, 298 K, 1.01 bar) was evaluated by using a dynamic tracer method. The experiments were realized in a stainless steel reactor which consists of a fixed bed (0.22 m in height, 0.030 m inner diameter) of spherical catalytic pellets of activated carbon (d_{p} = 0.00045 m, CAQ 12/UFPE). The bed was in contact with a concurrent gasliquid downward flow carrying the tracer in the liquid phase. Experiments were performed at constant gas flow Q_{G} = 2.50 x 10^{6} m^{3} s^{1} and with the liquid phase flow (Q_{L}) varying in the range from 4.25x10^{6} m^{3} s^{1} to 0.50x10^{6} m^{3} s^{1}. Under these conditions, the low interaction regime was guaranteed (Ramachandran and Smith, 1983; Silva et al., 2003). The evolution of KCl concentration was measured at the exit of the reactor as the response to a concentrations step at the reactor inlet.
Continuous analysis of the KCl tracer, fed at the reactor top at a concentration of 0.05M, was performed by using a refractive index detector (HPLC detector, Varian ProStar) at the exit of the fixed bed. The results were expressed in terms of the tracer concentration versus time.
The methodology applied to evaluate the order of magnitude of the axial dispersion, liquidsolid mass transfer coefficient and the partial wetting efficiency for the N_{2}/H_{2}OKCl/activated carbon system was:

Comparison of the experimental concentrations with the predicted concentrations based on the solutions of Eq. (33), developed for the system;

Evaluation of the initial values of the parameters

D_{ax}, k_{LS} and F_{M} from the correlations in Table 2;

Numerical optimization of the values of the model parameters employing, as the criterion, the minimization of a quadratic objective function expressed in terms of experimental and calculated concentrations, given by Eq. (34):
The operating conditions and the characteristics of the tricklebed system are presented in Table 3.
RESULTS AND DISCUSSION
Experiments were performed at constant gas flow Q_{G} = 2.50 x 10^{6} m^{3} s^{1} and with the liquid phase flow (Q_{L}) varying in the range from 0.50x10^{6} m^{3} s^{}1 to 4.25x10^{6} m^{3} s^{1}. The experiments carried out with liquid phase flows of (0.50, 0.75, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75, 4.25)x10^{6} m^{3} s^{1} were employing to fit the model equations, while operations with liquid phase flows of (1.00, 1.50, 2.00, 2.50, 3.00, 3.50, 4.00)x10^{6} m^{3} s^{1} were used for the model validation. Corresponding to the gas and liquid phase flows, the following superficial velocities were employed in the model equations: for the gas phase (nitrogen), V_{SG} was maintained at 10^{3} m s^{1}, and for the liquid aqueous solution of KCl, V_{SL} ranged from 2 x 10^{4} m s^{1} ^{4}m s^{1}.
The values of the axial dispersion, the liquidsolid mass transfer coefficient and the partial wetting efficiency were determined simultaneously by comparing experimental and predicted concentration data obtained at the exit of the fixed bed, subject to the minimization of the quadratic objective function (F), Eq. (34).
The numerical procedure used to optimize the values of the parameters involved the solution of Eq. (34) associated with an optimization subroutine (Silva et al., 2003, Box, 1965). The procedure started with initial values of the parameters until the final values were obtained, considered to be the optimized values of the three parameters when the quadratic objective function was minimized. The magnitudes of the parameters at different liquid phase flows are reported in Table 4.
The axial dispersion, the liquidsolid mass transfer coefficient and the wetting efficiency are influenced by changes in the liquid flow. To represent the behavior of D_{ax}, k_{LS} and F_{M}, their optimized values were employed and empirical correlations formulated as Eqs. (35), (36) and (37). These are restricted to the following operational conditions:
The parameter correlations were fitted by the leastsquares method. The mean relative errors (MRE) between the predicted and experimental parameter values of D_{ax}, k_{LS} and F_{M} in the k experiments were computed as follows:
p=D_{ax},k_{LS} . Figures 1, 2 and 3 present parity plots of the correlated results. The mean relative errors of D_{ax}, k_{LS} and F_{M} at different liquid flows are shown in Table 5.
A model validation procedure was established by comparing the predicted concentrations obtained with the values of the parameters from the proposed correlations (Eqs. (35), (36) and (37)) and experimental data not employed in the model adjustment. Table 6 presents the values of the parameters.
Figures 4 to 6 represent the model validations for three different operating conditions, where the parameter values were obtained from Eqs. (35), (36) and (37).
CONCLUSIONS
The transient behavior of the threephase trickle bed system N_{2}/H_{2}OKCl/activated carbon was evaluated via an experimental dynamic method and via predictions of a phenomenological mathematical model. Operating at 298 K under 1.01 bar with liquid and gas phases flowing downward under constant gas flow Q_{G} = 2.50 x 10^{6} m^{3} s^{1} and the liquid phase flow (Q_{L}) varying in the range from 4.25x10^{6} m^{3} s^{1} to 0.50x10^{6} m^{3} s^{1}, the concentration of KCl was measured at the exit of the reactor in response to a concentration step at the reactor inlet.
The solutions of the model equations predicted concentration profiles of the tracer employing optimized values of the parameters for the axial dispersion coefficient in the liquid phase, the liquidsolid mass transfer coefficient and the partial wetting efficiency. The magnitudes of the parameters were in the following ranges: D_{ax} = 6.986 x 10^{7} m^{2} s^{1} to 0.572 x 10^{7} m^{2} s^{1}, k_{LS} = 6.109 x 10^{6} m s^{1} to 0.286 x 10^{6} m s^{1} and F_{M} = 0.581 to 0.465. These results led to the proposal of three empirical correlations to quantify the influence of liquid phase flow rate changes on the axial dispersion, liquidsolid mass transfer and wetting efficiency in the low interaction regime.
Based on the values of the parameters indicated by the correlations, the model was validated by comparing their predictions with those obtained in different threephase operations with mean quadratic deviations between experimental and predicted concentrations on the order of 10^{4}.
ACKNOWLEDGMENTS
The authors would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for financial support (Process 483541/079).
NOMENCLATURE
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Submitted: August 5, 2010
Revised: July 27, 2011
Accepted: April 16, 2012
* To whom correspondence should be addressed