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Ind. Eng. Chem. Res. 2005, 44, 7892-7898
Point-by-Point Solution Procedure for the Computational Fluid Dynamics Modeling of Long-Time Batch Drying Roman G. Szafran and Andrzej Kmiec* Department of Chemical Engineering W3/Z7, Wroclaw University of Technology, ul. Norwida 4/6, 50-373 Wroclaw, Poland
Further results of simulations of the drying kinetics in a spouted bed dryer with a draft tube (ICFB) will be presented in this paper. As noted earlier [Ind. Eng. Chem. Res. 2004, 43, 11131124], the Eulerian-Eulerian multifluid modeling approach was applied to predict gas-solid flow behavior. The heat- and mass-transfer model was coupled with computational fluid dynamics (CFD) code FLUENT 6.1, through application of user-defined functions (UDF). The falling rate period of drying was described by the linear and nonlinear lumped-parameter models. The new, robust “point-by-point” solution procedure was proposed to predict the kinetics of long-time batch processes and to overcome a lack of sufficient computational performance. The results of the simulations were compared with experimental data and with values obtained from various correlations. The drying kinetics during the constant and falling rate periods of drying of inorganic particles was predicted with sufficient accuracy and efficiency for engineering calculations. The mean relative errors were 3.24% and 19.8% for drying periods I and II, respectively. The biological texture of rapeseed caused higher discrepancies; however, for both types of grain, the results from the CFD simulations were more similar to the experimental data than to the values obtained from correlations. A CFD modeling technique, coupled with classical drying kinetics models, provided useful results for engineering purposes and allows use of the model throughout all phases of research and development. Introduction Spouted beds were initially developed for the drying of wheat grain.1 Currently, they are used for various unit operations such as dispersion,2,3 coating/granulation,4 and drying.5 It allows one to obtain a product of high quality at a high energetic efficiency and maximum ecological safety simultaneously. Spouted-bed dryers also have been used for the coating of particles to obtain products with special properties. A special area of application for this type of equipment in the pharmaceutical industry is the coating of tablets6,7 and other functional particles,8-10 such as those used, for example, as injectable suspensions for cancer therapy.11 Drying is one of the stages that appears in all of the aforementioned technologies, and, in many cases, it limits the rate of process. A basic disadvantage of the spouted-bed apparatus is related to the scaleup and development of an industrial unit. He et al.12 proposed the modified Glickman’s scaling relationships for a spouted-bed scaleup. A force balance for particles in the annulus region of a spouted bed gave two nondimensional parameters: the internal friction angle and the loose-packed voidage to the original Glickman’s scaling relationships. As recently shown by Olazar et al.,13 stable operation, in particular for a light material, is required to establish welldelimited design operations for the contactor. Unfortunately, the correlational equations were proven to be valid only for the given types of material and in a narrow range of equipment configuration. The basic problem is to establish dependencies between the geo* To whom correspondence should be addressed. Tel.: +48713202838. Fax: +48713280475. E-mail: andrzej.kmiec@ pwr.wroc.pl.
metric configuration, operational parameters, and the rate of a given process and its efficiency.14 In the case of drying, it is the relationship between the moisture content of the material and its drying time. The classic application example of the computational fluid dynamics (CFD) modeling technique in drying is the modeling of spray dryers. Crowe applied this technique to spray drying in 1980.15 Recently, the Eulerian-Lagrangian approach was applied by Harvie et al.16 to predict the spray drying kinetics of skim milk. The main disadvantage of CFD models that restrains its broad application in research and development is a much greater cost in computing time than conventional methods. The present stage of computer technique development (even a parallel computing) allows simulations of time-dependent variables (e.g., grain moisture content) for a short period of process time (counted in seconds). The continuous, complete solution is available only for quick processes such as spray drying but is unattainable for long-time batch processes such as the drying of grain or coatings, which may require hours. The kinetics of processes in fluidized- or spouted-bed systems is dependent strongly on multiphase flow hydrodynamics. Among the three approaches (i.e., the pseudo-fluid (multifluid or Eulerian) approach, the discrete approach (discrete-particle model, DPM), and the pseudo-particle approach), only the first allows simulations of industrial-scale systems, because of its much lower computational requirement than the other two approaches.17 The CFD technique and the Eulerian-Eulerian approach were applied to predict hydrodynamics of flow in a fluidized-bed18 and spoutedbed19,20 apparatus with good results. Syamlal and O’Brien21 analyzed the catalytic decomposition of ozone in a bubbling fluidized bed using a two-fluid approach
10.1021/ie050340z CCC: $30.25 © 2005 American Chemical Society Published on Web 09/01/2005
Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7893 Table 1. Description of the Heat- and Mass-Transfer Model name
[
expression
mass flux of moisture evaporated from grain surface during period I of drying
1 - yˆ m,σ Mm(1 - yˆ m,0) + Mayˆ m,σ P M ln N ˆI)β + ln 1 - yˆ m,0 R ˜ g[(T0 + Tσ)/2] m Mm(1 - yˆ m,σ) + Ma5 y m,σ
mass flux of moisture evaporated from grain during period II of drying (linear model)
N ˆ II ) K(X h m - Xe)
mass flux of moisture evaporated from grain during period II of drying (nonlinear model)
Fd,s N ˆ II ) K(X h m - Xe)n As
surface-to-volume ratio of the particle
As )
Fd,s As
6 ds
in two-dimensional (2D) simulations. They confirmed the ability of multiphase hydrodynamic models to capture the effect of hydrodynamics on chemical reactions in a bubbling fluidized bed quantitatively. Hansen et al.22 performed three-dimensional (3D) simulations of ozone decomposition in the riser of a circulating fluidized bed (CFB). They also confirmed good agreement between the measured and simulated time-averaged ozone concentration at different elevations in the riser. Recently, De Wilde et al.23 applied the kinetic theory of granular flow and the Eulerian-Eulerian approach for simulating an industrial-scale dilute CFB with positive results. On the other hand, van der Hoef et al.24 proposed a new promising multiscale modeling strategy for dense gas-solid fluidized beds that combine advantages of the lattice Boltzmann model (LBA), the DPM model, the multifluid approach, and the discrete bubble model (DBM). In this paper, the new “point-by-point” procedure is proposed to shorten the computation time and to obtain a relation between the drying rate, grain moisture content, and drying time for a long-time batch process. Particularly, it is important to verify if the 2D simulations of an ICFB dryer based on the Eulerian-Eulerian approach and the heat- and mass-transfer model proposed in our previous publication25 are capable of predicting the drying kinetics of two types of grain (inorganic and organic) with sufficient accuracy for an engineering purpose, taking advantage of the newly proposed point-by-point solution procedure. Heat- and Mass-Transfer Model Drying could be described as the graveyard of academic theory.26 The subject is intellectually complex, involving simultaneous heat-, mass-, and momentumtransfer processes, giving a set of highly nonlinear governing equations. Numerous parameters that are dependent on flow hydrodynamics and solid properties affect the drying process. There is a large gap between drying theory and industrial drying practice. Manufacturers have had a tendency to rely on empirical scaleup rules that are based on a pilot-plant testing, rather than published theoretical models. This is because the theory has rarely been presented in the form of a stepby-step design procedure and also because of bad experience, when attempts were made to use the published theory in practical design.26 The CFD technique can change this situation. By integrating heat transfer and mass transfer with the CFD hydrodynamic model, one can achieve the necessary time-dependent distributions of scalar quantities in each phase, such as temperature and moisture
]
equation number 1
2 3 4
content, as well as the heat- and mass-transfer coefficients and the heat and mass fluxes. Local timedependent heat- and mass-transfer conditions have a predominant role in the constant-rate period of drying when the performance of apparatus limits a drying rate. The drying kinetics in the falling-rate period is dependent mainly on the material properties; however, in some situations, equilibrium conditions may occur in the dryer25 that limit interphase heat transfer and mass transfer, so, in different zones of the apparatus, drying periods I and II may occur simultaneously. Only the CFD model can predict this mixed regime. However, there is only a limited range of materials, out of the millions of solid products, for which all parameters required in equations have been measured. Hence, for practical reasons, the parameters of the model must simply be determined experimentally, as it is in lumped-parameter models. In addition, the model must be simple enough to provide the best computational efficiency and fast solution convergence simultaneously. The heat- and mass-transfer models described in detail in our previous publications25,27,28 are based on an experimentally determined characteristic drying curve and combine the usefulness of a basic, classical model with the advantages of a sophisticated CFD technique. (See Table 1.) The Eulerian-Eulerian multifluid approach was used in these simulations. The continuity equation and conservation equations of momentum, mass, and energy for each phase were solved. Coupling was achieved through the pressure, momentum, mass, and energy interphase coupling terms. Additional closure laws are required to describe the rheology of the granular phase and the turbulence. To describe flow behavior of a fluid-solid mixture, the Gidaspow’s multifluid granular kinetic model was applied.29 The standard k- turbulence model30 was solved for each phase. Heat transfer and mass transfer was integrated with the hydrodynamic model by means of the user-defined function (UDF) programming technique. The local values of the heat- and mass-transfer coefficients were calculated from the Gunn correlation.31 The interphase momentum-transfer coefficient was calculated from the Gidaspow model32 that was proposed for a fluidized bed. The same model was used by De Wilde et al.23 to describe fluid-solid interactions in a dilute CFB with good results. Equation 1 in Table 1 describes a flux of moisture that is transferred between phases during drying period I. Equations 2 and 3 in Table 1 describe the flux of moisture that is exchanged between phases during drying period II. Equations 1 and 2 predict a linear dependence of the mass moisture fluxes on the moisture content. In the more complex
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Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005
Figure 1. “Point-by-point” procedure: (A) theoretical drying curve and (B) theoretical drying rate curve. Region I denotes the first (constant) drying rate period, whereas region II denotes the second (falling) drying rate period.
situation, the drying period II can be described by a nonlinear, monotonic function (eq 3). “Point-by-Point” Solution Procedure The major problem of the research and development cycle of a spouted-bed dryer is to establish a dependence between the moisture content of the grain and the drying time for a particular apparatus. Batch drying is a long-time process that sometimes lasts even for hours. Continuous, complete solution of the CFD drying kinetics model for such a system for the entire drying time currently is unattainable, because of insufficient computer performance. Instabilities of flow and circulations of phases involve using a short time step (1 × 10-3-1 × 10-4 s) to obtain solution convergence.33 This means that the simulation of a 1-h process could presently require even one year of computing time, but the acceptable time for research and development is only a few days. To overcome this problem, the new point-bypoint solution procedure is proposed. The main idea of the procedure is presented in Figure 1. To solve the problem, it is necessary to transform the time-dependent system of grain moisture content versus time (Figure 1A) to the time-independent system of drying rate versus grain moisture content (Figure 1B). The initial and finale values of moisture content of the grain, X0 ) X1 and Xe ) X6, respectively, are known: they are experimentally determined or assumed. Values X2-X5 are arbitrarily chosen between X0 and Xe. The inlet air humidity and its temperature, as well as the mass of the bed, are known for each point. The time-dependent parameters such as local volume fraction of phases, its velocities, temperatures, moisture content, etc., are known (or assumed) for the first point. For points 2-6, the initial distributions of the time-dependent parameters are unknown and are taken from simulation for the previous point. This simplification is appropriate for hydrodynamically stable or periodic processes. As was mentioned previously, the flow hydrodynamics in a spouted-bed dryer of fine particles is unstable. The fountain density and height change periodically over time. Particles in the entrainment region of the column move from the annulus into the gas jet to form clusters and are carried upward periodically, with a frequency of 5-10 Hz.33 The steady-state solution is unattainable; however, one can perform computations for a few seconds of process time (the mean residence time of air in the apparatus) to obtain the mean values of the drying rate for each starting point. The error of ap-
proximation is dependent on several chosen points, its distribution, and the duration of computation periods for each starting point. One can now start the CFD simulations from distinct initial grain moisture contents (X1-X6) to obtain appropriate values of mean drying rates (N1-N6) and local distributions of interesting parameters such as temperature and humidity of the air. In the simplest case, when the kinetics of drying period II is described by the linear function, as is shown in Figure 1B, only a few points are needed. The number of points must be minimally four, because the critical moisture content (Xcr) of the grain is not known and must be determined from the intersection of two lines (see Figure 1B). In practical cases, the number of points should be >4, to minimize the influence of computational error on obtained functions as well as to predict the nonlinearity of the process during drying period II. Assuming that the dependence of the drying rate on moisture content between each of the two points is linear, one can obtain n - 1 linear functions N ) f(X). By integrating each function, one can obtain a new one that describes the dependence of moisture content of grain on drying time between each two points. Starting from point p1(t1 ) 0, X1 ) X0), one can determine t2 from the reverse function X ) f1(t). Knowing the coordinates of point p2(t2, X2) from the previous step, one can determine t3 from the reverse function X ) f2(t). Going point by point, one finally obtains the total drying time tend that corresponds to the final grain moisture content Xend. Model Solution Procedure In the first step, it is necessary to determine the drying constant (KII), the exponent n, and the equilibrium moisture content Xe (eqs 2 and 3) experimentally. These parameters are dependent mainly on the material properties and the inlet air moisture content; therefore, the experiments could be performed in any apparatus with intensive convective conditions. In that case, one can assume that moisture transport in a grain limits the drying rate. These experiments could be performed even in a cabinet dryer in this particular case. Experimentally determined values of KII, n, and Xe are presented in Table 2 for each grain. In the next step, the distribution of starting points for CFD simulations (see Figure 1B) was chosen. Initial values of the grain moisture content were fixed between X0 and Xe for each type of grain: 0.03, 0.05, 0.08, 0.10, 0.11, 0.12, and 0.138 kg/kgi for microspherical particles
Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7895 Table 2. Initial Parameters Used in the Simulations parameter
value
Microspherical Porous Particles, Geldart’s A Group inlet velocity magnitude of air, normal to boundary 5.5 m/s inlet temperature of air 340 K inlet turbulence intensity 10% operating pressure 101 325 Pa mass of grain 0.2 kg initial height of bed 0.1 m diameter of particle, uniform 2.2 × 10-4 m packing limit 0.63 solids density 630 kg/m3 initial grain moisture content 0.14 kg/kgi grain equilibrum moisture content 0.02 kg/kgia initial air humidity 0.01 kg/kgi moisture diffusion coefficient in air 2.8 × 10-5 m2/s drying constant, KII 2.71 × 10-3 s-1b Rapeseed, Geldart’s D Group inlet velocity magnitude of air, normal to boundary inlet temperature of air inlet turbulence intensity operating pressure mass of dry grain initial height of bed diameter of particle, uniform packing limit solids density initial grain moisture content grain equilibrum moisture content initial air humidity moisture diffusion coefficient in air drying constant, KII for eq 2 for eq 3 exponent n (eq 3) a
14 m/s 349 K 10% 101 325 Pa 0.967 kg 0.155 m 2 × 10-3 m 0.63 1078 kg/m3 0.12 kg/kgi 0.001 kg/kgi,b 0.0284 kg/kgia 0.01 kg/kgi 2.8 × 10-5 m2/s 1.08 × 10-3 s-1b 3.50 × 10-3 s-1b 1.64b
Determined from the linear approximation of the experimental data for drying period II. b Experimentally determined.
and 0.025, 0.035, 0.04, 0.05, 0.07, 0.09, 0.10, 0.11, and 0.12 kg/kgi for rapeseed. The same 2D axisymmetric mesh that was tested in our previous investgations25,27,28 was used in the present work. The calculation domain was divided into 3985 (in a half of domain) control volumes with 2357 nodes. The numerical solutions of discrete governing equations were achieved by a control volume method. The CFD code FLUENT 6.1 was chosen to perform computer simulations. The pressure-velocity coupling was achieved by the SIMPLE algorithm. The second-order upwind discretization scheme of momentum, volume fraction of phases, energy, user-defined scalars, turbulence kinetic energy, turbulence dissipation rate, and the first-order implicit time discretization were chosen. The typical values of under-relaxation factors were 0.2-0.4. As criteria of convergence, the values of scaled residuals and volume integrals were monitored. The solution was considered to be converged when the scaled residuals were
