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Ind. Eng. Chem. Res. 2009, 48, 1740–1748
KINETICS, CATALYSIS, AND REACTION ENGINEERING Numerical Simulation of Trickle-Bed Reactor Hydrodynamics with RANS-Based Models Using a Volume of Fluid Technique Rodrigo J. G. Lopes and Rosa M. Quinta-Ferreira* GERSEsGroup on EnVironmental, Reaction and Separation Engineering, Department of Chemical Engineering, UniVersity of Coimbra, Rua Sı´lVio Lima, Polo II-Pinhal de Marrocos, 3030-790 Coimbra, Portugal
A trickle-bed reactor (TBR) was modeled by means of the volume of fluid (VOF) model to provide a hydrodynamic behavior analysis in trickling flow conditions. Fluid dynamics of the TBR is characterized by poor liquid distribution and inefficient catalyst utilization and conventional modeling techniques are unable to address these key design issues. Therefore, the VOF code was used to investigate the major hydrodynamic parameters in a three-dimensional packed bed providing a more rigorous physical description of the underlying flow process. Several numerical solution parameters including different mesh densities, time steps, and convergence criteria were optimized in order to provide computational independent results. During the parametric optimization it was found that the VOF model is more sensible to mesh density and time step than with respect to convergence criteria. The computational fluid dynamic model was thoroughly validated by comparing the model predictions with the published experimental data for liquid holdup and two-phase pressure drop. After the VOF optimization, selected values for the numerical solutions parameters were used to perform the assessment of different turbulent flow models at two nominal gas flow rates. Afterward, several computational runs were performed in the evaluation of the influence of either gas or liquid flow rate on TBR hydrodynamics. 1. Introduction Trickle-bed reactors (TBRs) are extensively employed in several industrial operations, ranging from chemical and biochemical plants to wastewater treatment and agricultural manufacturing processes. These gas-liquid-solid systems have some distinct advantages over other methods of three-phase reactors (slurry reactors, fluidized bed reactors, and bubble fixed-bed reactors), such as low-pressure drop, low liquid hold-up, high catalyst loading, and high conversion as both gas and liquid flow regimes approach plug flow. Known disadvantages are often related with partial catalyst wetting, poor liquid-phase distribution, high intraparticle resistance, low mass transfer coefficient, and poor radial mixing, and temperature control can be difficult.1 Over the last decades, a noteworthy amount of literature has been published on the determination of the so-called hydrodynamic parameterssliquid holdup and pressure dropsin both laboratory and pilot scale TBRs. The design of commercialscale TBRs traditionally depends on expensive pilot-scale experiments and the mathematical formulation based on dimensional analysis (or even applying the neural networks concept) is mainly focused on improving the simulation of the steadystate operation by developing more suitable empirical correlations. Nevertheless, due to the fact that the majority of correlations for the estimation of liquid holdup, two-phase pressure drop, gas-liquid mass transfer and interfacial area, and catalyst wetting efficiency,1 have been developed under steadystate conditions and are intended to describe only steady-state reactor operation, most of the correlations are inappropriate for the description of the reactor performance under transient * To whom correspondence should be addressed. E-mail: rodrigo@ eq.uc.pt,
[email protected]. Tel.: +351-239798723. Fax: +351239798703.
conditions. Recent advances in the improvement of multiphase reactor models indicated that computational fluid dynamics (CFD) is a valuable tool to address the complete multidimensional flow equations coupled with chemical species transport and reaction kinetics instead of traditional TBR models reported in the literature considering isothermal operation in either pseudohomogeneous or heterogeneous models with plug-flow for gas and liquid phases (Al-Dahhan et al., 1997).1 Two approaches for CFD modeling of gas-liquid-solid flows have been implemented for the hydrodynamics predictions: the Lagrangian-Eulerian model2 in particulate flows simulation and the Eulerian-Eulerian model3-5 in trickle-bed reactors. Using the Lagrangian-Eulerian model, particle trajectories of the discrete phase are tracked by solving individual equations of motion, whereas the continuum phase is modeled using an Eulerian framework. As a consequence, the Lagrangian-Eulerian model requires large computational resources for large systems of particles. With the Eulerian-Eulerian model, the base assumption is that gas and liquid phases are interpenetrating continua. Therefore, the Eulerian-Eulerian model for gasliquid-solid flows is the more commonly used CFD model to predict the dynamic behavior of trickle-bed reactors.6 However, the Eulerian-Eulerian approach does not accomplish interface tracking by the solution of a continuity equation for the volume fraction of one (or more) phases, and for this reason is unable to capture the wetting characteristics at the gas-liquid interface in the operation of trickle beds. The contacting efficiency is directly related to the spreading of a liquid on either wet or dry catalyst solid surfaces. It is of paramount importance to address the wetting phenomenon to understand its impact on other hydrodynamic and reaction parameters. The interface between both phases is important in evaluating the performance of the reactor and therefore free-surface modeling is necessary.
10.1021/ie8014186 CCC: $40.75 2009 American Chemical Society Published on Web 01/27/2009
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1741
Volume of fluid (VOF) and level set approaches belong to the two best possible implicit free-surface reconstruction methods. This method was extensively used for many applications.7-9 The velocity field and bubble profile in a vertical gas-liquid slug flow inside the capillaries has been modeled with a VOF technique, and it was found to be in good agreement with published experimental measurements.10 To gain insight about the pertinent parameters that may affect the liquid-solid interface, several works have been published on the simulation of liquid drop impact with the solid surface.11-13 The experimental and simulation data for different contact angles14 and velocities13 indicated that few experimental and simulation studies were conducted at lower velocities that are characteristic of trickling flow regime. Therefore, additional simulation activities on the gas-liquid-solid interface at different flow regimes are needed for the meaningful knowledge of interaction between TBR hydrodynamics and reaction parameters. This work is devoted to the volume of fluid model for TBR modeling comprising the numerical validation in terms of well-known hydrodynamic parameters. Liquid holdup and two-phase pressure drop were selected for the parametric optimization of several models parameters including mesh aperture, time step, and different convergence criteria. The multiphase flow regime will be presented with several RANS turbulent flow models as well as the laminar one. The effect of gas and liquid flow rate on either frictional pressure drop or liquid holdup will be also examined under trickling flow conditions with three-dimensional packed bed geometry. 2. Modeling Approach and Mathematical Models Governing Equations for Multiphase Flow. Understanding the behavior of the fluid flow through the packed particle bed is important to enhance the performance of trickle-bed reactors from design and operating considerations. The solution of the hydrodynamic problem for multiphase reactors using a phenomenological approach should take into account several phenomena including the porous media flow and the interfacial effects, as well as the process parameters such as the bed package, the particles shape, and porosity.15 In general, trickle-bed reactors use regular shape catalyst particles, moderate to high gas and liquid flow rates and pressures, and can operate from trickling to pulsing flow regime. These characteristics have a strong influence on hydrodynamic parameters so that a reliable scale-up analysis is often performed with specific data for these processes. In the present study, the detailed information of momentum transport is investigated in a fixed bed by means of CFD codes with different turbulent flow models. A trickle-bed reactor of nonoverlapping spherical particles in cylindrical geometry was modeled with a specified void fraction and a set of fluid physical properties; the Navier-Stokes equations are solved for the velocity and pressure fields in the fluid phase of the void space. The computational geometry was designed so that the catalyst particles do not touch each other. The distance between two particles within 3% of the sphere diameter facilitates the grid generation, avoiding numerical difficulties that arise in the calculation of convective terms with the representative arrangement of catalyst particles shown in Figure 1. The purpose of this work is to develop a computational model to analyze the fluid flow through the cylindrical bed including the evaluation of liquid holdup and two-phase pressure drop predictions. In particular, the liquid-gas flow through a catalytic bed was considered composed of monosized, spherical, solid particles arranged in a cylindrical container of a pilot TBR unit
Figure 1. Configuration of catalyst particle arrangement for the trickle-bed used in VOF simulations.
(50 mmID × 1.0 mlength). The VOF method was used to compute velocity field as well as liquid volume fraction distributions. The multiphase flow is assumed to be vertical downward and incompressible, with the mathematical description for the flow of a viscous fluid through a three-dimensional catalytic bed based on the Navier-Stokes equations for momentum and mass conservation. The VOF model enables the computation of multiphase flows in which gas-liquid-solid interfaces are clearly identified. It is thus well adapted for calculating the breakup of liquid jets or films sheared by a gas flow16 or bubble dynamics.17,18 In the VOF model, the variable fields for all variables (pressure, velocity, etc.) are shared by both phases and correspond to volume-averaged values. It is thus necessary to know the volume fraction, Rq, of each phase, q, in the entire computational domain. This is possible through the resolution of the volume fraction equation for the phase q: ∂ (R F ) + ∇·(RqFqUq) ) 0 ∂t q q
with q ) g or l
(1)
where g and l denote, respectively, the gas and liquid phases and t is the time. The resolution of the momentum equation is shared by the two considered fluids ∂ (R F U ) + ∇·(RqFqUqUq) ) - Rq∇p + RqFqg + ∂t q q q ∇·Rq(τq + τt,q) + Iq
with q ) g or l (2)
where p, g, and the physical properties (density, F, and viscosity, µ) are determined by volume-weighted averages. Iq is the interphase momentum exchange term and τq and τt,q are, respectively, the viscous stress tensor and the turbulent stress tensor, defined as follows: 2 τq ) µq(∇Uq + ∇Utq) + λq - µq ∇UqI 3
(3)
2 τt,q ) µt,q(∇Uq + ∇Utq) - (kq + µt,q∇Uq)I 3
(4)
(
)
and
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The tracking of the interface is done in the cells where the volume fraction is different from 0 or 1 via the use of the geometric reconstruction scheme. This calculation scheme represents the actual interface as a piecewise-linear geometry. Free Surface Model: Surface Tension and Wall Adhesion. The surface tension is modeled by means of the continuum surface force model proposed by Brackbill et al.19 The pressure drop across the surface depends upon the surface tension coefficient, σ, and the surface curvature as measured by two radii in orthogonal directions, R1 and R2, as expressed by eq 5.
(
p2 - p1 ) σ
1 1 + R1 R2
)
(5)
where p1 and p2 are the pressures in the two fluids on either side of the interface. The surface curvature is computed from local gradients in the surface normal at the interface; n is the surface normal, defined as the gradient of Ri: n ) ∇Ri. The curvature, κ, is defined in terms of the divergence of the unit normal, nˆ: κ ) ∇ · nˆ where nˆ ) n/|n|. The forces at the surface are expressed as a volume force using the divergence theorem assuming the form of eq 6. Fj )
∑
pairs ij, i
