Chemical Engineering Science 74 (2012) 9–17
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Numerical simulation on micromixing of viscous ﬂuids in a stirred-tank reactor Ying Han, Jia-Jun Wang n, Xue-Ping Gu, Lian-Fang Feng Department of Chemical and Biological Engineering, State Key Laboratory of Chemical Engineering, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e i n f o
Article history: Received 21 January 2011 Received in revised form 7 February 2012 Accepted 14 February 2012 Available online 23 February 2012
Micromixing of viscous systems in a stirred-tank reactor with Rushton turbine is investigated numerically. It is characterized by the product selectivity of parallel competitive reactions. Flow ﬁelds inside the reactor are determined by the Reynolds Stress Model (RSM). A computational ﬂuid dynamics (CFD) method combining the standard E model and Finite-Rate/Eddy-Dissipation (FR/ED) model is implemented and is validated using experimental data in the literature. The simulations show that a higher agitation speed, a lower ﬂuid viscosity and/or a feeding location closer to the discharge area of the impeller favor micromixing and the reaction rate. The trajectory of the reaction plume also inﬂuences the micromixing performance. The value of the FR/ED model parameter determined by labscale experiments decreases with increasing ﬂuid viscosity. However, it is little affected by the agitation speed. These results provide useful guidelines for the scale-up of industrial reactors with complex chemical reactions. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Mixing Computational ﬂuid dynamics Chemical reactors Non-Newtonian ﬂuids Fluid mechanics Turbulence
1. Introduction Mixing processes in industrial reactors with high-viscosity media may have a signiﬁcant effect on the product yield and distribution, especially molecular weight distributions for polymers (Baldyga and Bourne, 1998; Bourne et al., 1989, 1995; Gholap et al., 1994; Verschuren et al., 2001). Macromixing describes the process for ﬂuid elements to break up and advect into a space of interest. Micromixing takes place below Kolmogorov and Bachelor scales, promoting molecular diffusion and therefore inﬂuencing the product distribution. The last decades have seen many studies on developing methods to characterize micromixing in aqueous and low-viscosity solutions. Consecutive reaction systems and competitive reaction ones are widely accepted (Akiti et al., 2005; Baldyga and Bourne, 1984; Bourne and Gholap, 1995; Bourne, 2003; Ottino et al., 1979; Vicum et al., 2004), because their product selectivity depends very much on micromixing (Brucato et al., 2000; Fournier et al., 1996). There are also studies on the effect of multiscale mixing by considering ﬁne-scale mixing together with mesomixing (Akiti and Armenante, 2004; Baldyga et al., 1997). Micromixing performance is a function of reactor geometry, chemical and physical characteristics of reacting system, operating conditions and feeding method of reagents. In the case of an unbafﬂed tank, feeding reactants at the upper trailing vortex of a
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0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2012.02.018
Rushton turbine (Assirelli et al., 2002) with a rotating feed pipe (Assirelli et al., 2005, 2008) show good micromixing. In semi-batch tanks, the ﬂuid viscosity affects the characteristics of the ﬁne-scale, a scale near Kolmogorov microscale, and therefore micromixing (Bourne et al., 1995; Gholap et al., 1994; Guichardon et al., 1997). Atibeni (2005) and Cong et al. (2006) found that an increase in the ﬂuid viscosity increased the selectivity of the side product. Kunowa et al. (2007) discussed the effect of micromixing in polymerization reactors and found that mixing performance decreased rapidly with increasing viscosity of reaction media. Yang et al. (2009) studied the micromixing performance of viscous media in a microreactor and found that the segregation index increased with increasing viscosity. A screw extruder is a special type of reactor for highly viscous systems for reactive extrusion processes (Cartier and Hu, 1998; Chen et al., 1995; Feng and Hu, 2004; Hu and Kadri, 1998; Hu et al., 1998; Hu and Cartier, 1999; Hu et al., 2003) in which micromixing may play a crucial role. Li et al. (2010) studied the micromixing performance of highly viscous polymer melts in a twin-screw extruder upon developing a new parallel competitive system consisting of two macromolecular reactions. With the development of high performance computers, computational ﬂuid dynamics (CFD) methods have been widely used as a simulation tool to investigate macromixing and turbulence of non-reactive ﬂuid systems. Nevertheless, it is very challenging when it comes to a chemical reactor with interactions between micromixing and complex chemical reactions. The commonly used micromixing models are multi-environment (ME), engulfment (Eng), engulfment–deformation–diffusion (EDD), interaction by exchange with the mean (IEM), and direct
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quadrature method of moments interaction by exchange with the ¨ ncul ¨ et al., 2009) The formulation of the mean (DQMOM-IEM). (O implemented micromixing models can be based on the probability density function (PDF) approaches. The full PDF models are considered to be more powerful to solve micromixing and complex chemical reactions than the presumed PDF ones. (Baldyga and Makowski, 2004; Kolhapure and Fox, 1999; Marchisio and Barresi, 2003; Tsai and Fox, 1996; Wang and Fox, 2004) Nevertheless, the computational difﬁculty is further aggravated. The simpliﬁed EDD models, i.e., the standard or modiﬁed Engulfment model (E model) proposed by Baldyga and Bourne (1989a, 1989b) have been widely used for ideal reactors such as CSTR and plug ﬂow reactors owing to their simplicity and computational performance (Baldyga and Bourne, 1990; Gholap et al., 1994; Kougoulos et al., 2006; Vicum et al., 2004). Akiti and Armenante (2004) incorporated the E models into CFD to track the reaction zone by means of the Volume of Fluid (VOF) model upon taking detailed hydrodynamics into account. They assumed that there was little diffusion between different species. Although the VOF could show a distinct reaction zone interface, it has to treat the homogeneous volumetric reactions as heterogeneous interfacial ones, which is an inherent drawback. This work aims at studying the micromixing effect in viscous ﬂuids in terms of the ﬁnal product selectivity of a competitive parallel reaction system in a semi-batch stirred-tank reactor with a Rushton turbine. The standard E model and the Finite-Rate/ Eddy-Dissipation (FR/ED) model are used to describe volumetric reactions. The effects of the agitation speed, feeding location and ﬂuid viscosity on micromixing are simulated, revealing the different impacts of micromixing on different types of reactions and positions in the reactor. The numerical models are validated using experimental results in the literature (Atibeni, 2005).
micromixing (Brucato et al., 2000): k1
NaOHðAÞ þ 0:5CuSO4 ðBÞ! 0:5CuðOHÞ2 ðPÞk þ 0:5Na2 SO4 ðRÞ NaOHðAÞ þ CH2 ClOOC2 H5 ðCÞ
CH2 ClOONaðQ Þ þ C2 H5 OHðSÞ
where k1 and k2 are the reaction kinetic constants and are 107 and 0.023 m3/(mol.s), respectively. For convenience, Eq. (1) is called fast reaction and Eq. (2) slow reaction, although it is not slow actually. Q and S stand for the side products while P and R the primary ones. The selectivity of side product Q, XQ, as a quantitative index for describing the micromixing effect or performance, is sensitive to both ﬂuid mixing and reaction kinetics. If micromixing is perfect, which refers to complete mixing at the molecular scale, XQ is solely dictated by the kinetic constants of two reactions and is given by Eq. (3) and tends to zero as k1 is very large compared to k2. XQ ¼
k2 C C 0 k1 C B þ k2 C C
On the contrary, complete segregation refers to cases where no micromixing occurs. XQ is then not dictated by the reaction kinetics but by the initial concentrations of reactants. XQ is given by Eq. (4). The maximum value of XQ is 0.667 because the initial concentration of C, CCo, is twice that of B, CBo. XQ ¼
CQ C Co ¼ 0:667 CP þ CQ C Co þC Bo
In practice the value of XQ is between these two limits and is a measure of the interaction between ﬂuid mixing and reaction kinetics when the initial concentrations of reactants are ﬁxed.
2. Numerical models and simulation methods 2.2. Simulation conditions A micromixing model (standard E model) coupled with the turbulence reaction model (FR/ED model) and local hydrodynamics is developed to investigate the micromixing effect of viscous ﬂuids in a stirred-tank reactor.
2.1. Reaction system The reaction system used is composed of parallel competitive reactions, of which one reaction is much faster than mixing while the characteristic time of the other is comparable with that of
Simulations are carried out in a fully bafﬂed (four bafﬂes) stirred tank, whose inner diameter and liquid height are equal and are 0.476 m. The reactor is equipped with a six-bladed Rushton turbine whose diameter is 0.190 m and whose blade width is 0.038 m. The clearance between the impeller and the bottom is 0.159 m. Fig. 1 shows the details of the apparatus. There are two feeding points. They are located at a vertical mid-plane between two adjacent bafﬂes. One (L1) is 0.100 m below the liquid surface and the other (L2) close to the discharge area of the Rushton turbine and 0.159 m above the bottom of the
Fig. 1. Schematic presentation and computational grid of the stirred tank reactor with Rushton turbine (Left: Tank; Right: Impeller details).
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Table 1 Model parameters for two non-Newtonian ﬂuids.
where the superscript ‘‘ ’’ denotes that the quantities are timemean values; Ri is the net rate of production of species i by chemical reactions, Ji is the diffusion ﬂux of species i which is generated due to concentration gradients, and uj is the jth component of ﬂuid velocity. (Vicum et al., 2004) In this work, turbulent diffusion ﬂux is written as: Ji ¼ rDi,m rY i ru0j Y 0i
tank. The distances between the feeding location and the axis of the tank are 0.070 and 0.096 m for L1 and L2, respectively. Two non-Newtonian viscous ﬂuids are formulated using different concentrations of hydroxyethyl cellulose (HEC) solutions as a thickener. The ﬂuid viscosity is described using a power law:
Z ¼ kZ gn1
where kZ is the ﬂow consistency index, n the ﬂow behavior index and g the shear rate. Table 1 gathers the values of the model parameters for non-Newtonian ﬂuids. The agitation speed, N, is varied between 180 and 420 rpm, corresponding to the generalized Reynolds number (Ren ¼ rND2/ Za, where Za is the apparent viscosity of the ﬂuid) from 975 to 2275 for Fluid-1. The rotation speed N for Fluid-2 is varied between 45 and 225 rpm with the generalized Reynolds number varying between 1934 and 9670. The operating conditions closely match those in the literature (Atibeni, 2005). The reactions are conducted at 293 K. The reactor is initially charged with well mixed aqueous solutions of two reagents: CuSO4 (B), CH2ClOOC2H5 (C) solutions with concentrations of 0.00675 and 0.0135 kmol/m3, respectively. Two liters of NaOH (A) solution with a concentration of 0.57 kmol/m3 is then added to the tank with a feeding time of 3500 s to preclude the effects of macromixing and mesomixing on the product distribution. A longer feeding time would not change the ﬁnal selectivity of the side product Q. In the simulations, it is difﬁcult to follow the experimental process with a very long feeding time due to the extremely high computational effort. This problem is solved by discretizing the feed (NaOH solution) into a ﬁnite number of portions (equal volume). (Akiti and Armenante, 2004; Baldyga and Makowski, 2004) For each portion, the numerical models are solved until it is exhausted completely. Then next portion of fresh feed is introduced into the reactor. Proper feed discretization number is carefully chosen to be equivalent to this feeding time of experiment. 2.3. Numerical models The ﬂow ﬁeld is calculated by solving the Navier–Stokes equation together with the Reynolds Stress Model (RSM) that accounts for turbulent effects. The RSM is found to be more accurate in turbulent ﬂows for turbulent kinetic energy, k, and turbulent energy dissipation rate, e, than the standard k–e model but at the expense of a higher computational cost. This is in agreement with the results of Akiti and Armenante (2004). Although the ﬂow is fully turbulent for most of the operating conditions in the discharge area of the impeller, other areas might be in the transitional regime in which the local Reynolds number (Rel) can be lower. To deal with the transitional ﬂow, low-Re modiﬁcations are incorporated into the linear pressure-strain model of the RSM. When chemical reactions are involved, several Reynolds-averaged species transport equations should be solved. The local mass fraction of each species, Yi, is predicted through the solution of a convection–diffusion equation for the ith species: @ ðrY i Þ þ rUðruj Y i Þ ¼ rUJ i þ Ri @t
where Di,m is the diffusion coefﬁcient for species i in the mixture; u0j and Y 0i are the local ﬂuctuating components of the jth component of ﬂuid velocity and mass fraction of species i. The Eddy–Dissipation (ED) model was proposed by Magnussen and Hjertager (1976). It assumes that the overall rate of reaction is controlled by turbulent mixing. The reaction process is said to be mixing-limited and the complex chemical reaction kinetics can be neglected. The net rate of production of species i due to reaction r, Ri,r, is given below, based on the mean concentration of the reacting species ! e YR ð8Þ Ri,r ¼ v0i,r Mw,i ar r min k R v0R,r M w,R where Mw,i and ni,r0 denote the molecular weight for species i and stoichiometric coefﬁcient for reactant i in reaction r, respectively. The symbols with subscript R refer to the properties of reactants. In the ED model, the chemical reaction rate is related to the largeeddy mixing time scale, k/e, and the empirical parameter, ar. The inertial convective mixing is largely inﬂuenced by the viscosity distribution due to the shear-thinning property of non-Newtonian ﬂuid and the transitional regime in most space of the tank. When the ﬂuid mixing rate is high and the kinetic rate is relatively low, the reaction kinetics controls the reaction rate. The ﬁnite-rate model (FR) calculates the chemical source terms using Arrhenius expressions and ignores the effects of turbulent ﬂuctuations. The net rate of creation/destruction of species i in reaction r is given by 0 1 N Y ðZ0j,r þ Z00j,r Þ A 00 0 @ Ri,r ¼ Mw,i ðvi,r vi,r Þ kr ð9Þ ½C j,r j¼1
where Cj,r, Zj,r0 , Zj,r00 , ni,r00 and kr are the molar concentration of species j, rate exponent for reactant species j and product species j, stoichiometric coefﬁcient for product i and reaction kinetic constant in reaction r, respectively. For the competitive parallel reaction system used in this work, both Zj,r0 and Zj,r00 are equal to 1 and kr is treated as a constant. Chemical reactions are molecular-level processes that take place at a scale much smaller than the CFD simulation cell. A micromixing model is required for Eq. (9) so as to predict the reaction process. Baldyga and Bourne (1984) developed an EDD model with a set of differential equations to express unsteady diffusion and reaction in deforming laminated structures formed by engulfment in a turbulent ﬂow. The EDD model could be simpliﬁed to the standard E model by neglecting deformation and diffusion for systems with a Schmidt number less than 4000 (Baldyga and Bourne, 1989a, 1989b). In this work, the standard E model is incorporated in the CFD and the micromixing is described by the engulfment parameter E from Eq. (10) e 1=2 E ¼ 0:058
where n stands for the kinematic viscosity. The engulfment parameter E is introduced to the mass balance equation to determine the source of generic species i undergoing engulfment, which is deﬁned as the micromixing-kinetics rate. The net source of chemical species i due to reaction r that involves micromixing
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The simulation process is composed of the following ﬁve steps:
is calculated by Eq. (11). 0 Ri,r ¼ M w,i ½Eð/C i SC i Þ þ ðv00i,r v0i,r Þ@kr
1 ½C j,r ðZj,r þ Zj,r Þ A 0
where Ci and /CiS are the molar concentrations of species i, in the reaction zone and the bulk ﬂuid surrounding the reaction zone, respectively. /CiS is kept constant throughout the integration of these differential equations for the entire time period, corresponding to the time the reactant added as a single feed element is nearly completely consumed. In other words, /CiS o10 7 kmol/m3 (Akiti and Armenante, 2004). Thereafter, another portion is fed the system with /CiS updated and the process repeated. It is found that when Eq. (11) alone is used as a reaction model, the agreement between the simulations and the literature work (Atibeni, 2005) is poor. Therefore it is thought that the effect of inertial convective mixing on the reaction and mixing process should be taken into account in the ED model for improving its predictability. For the competitive reaction system, both chemical kinetics and ﬂow turbulence are considered to solve the species transport equations. The Finite-Rate/Eddy-Dissipation Model (FR/ED Model) (Bakker et al., 2001; Fluent 6.2 User’s Guide, 2005) is a turbulence–chemistry interaction model. The calculated ﬁnite reaction rate and ﬂow turbulence rate are compared, and the smaller one is given to Ri,r as the actual species reaction rate. This work proposes a model that combines the standard E model and FR/ED one. Both the micromixing-kinetics reaction rate given by Eq. (11) and the eddy-dissipation one given by Eq. (8) are calculated. The net reaction rate is given by the smaller one of the two. Finally, the net source of chemical species i, Ri, in Eq. (6) is calculated as the sum over all the reactions the species participate in: X Ri,r ð12Þ Ri ¼ 2.4. Determination of parameters ar
ar is an empirical parameter for the Eddy-Dissipation (ED) model and accounts for the relationship between turbulent mixing and reaction kinetics in the system. The main assumption in the numerical methods is that small scale mixing only affects the fast reaction and hardly has an inﬂuence on the slow reaction. As a result, a1 is experimentally determined and a2 is set to be inﬁnity which means that the ED model does not work for the slow reaction any more. Decreasing a1 slows down the fast reaction and increases XQ. According to the simulations performed in this work, the value of a1 is sensitive to the ﬂuid viscosity but is almost independent of the agitation speed. Based on the experimental results in the literature (Atibeni, 2005), it is found to be 0.5 and 5 for Fluid-1 and Fluid-2, respectively. It is shown that a1 is in inverse correlation to the ﬂuid viscosity. Unfortunately, the effect of ﬂuid viscosity can not be decoupled from the parameter a1 due to the limitations of ED model.
(a) The velocity distribution in the stirred tank is numerically calculated by solving the Reynolds-averaged Navier–Stokes equation together with the RSM turbulence model. (b) The standard E model coupled with the FR/ED model is introduced to the transport equations by means of the user deﬁned function (UDF). They account for the interactions between reaction kinetics and ﬂow turbulence, when local micromixing is considered. (c) The total amount of NaOH (A) solution is numerically discretized into a ﬁnite number of equal volume portions. (d) A portion of NaOH solution is patched to the feeding location in the stirred tank. The mass balance for each reactant is performed with the proposed model till the NaOH is completely consumed. (e) Step (d) is repeated with renewed /CiS till all these NaOH portions are exhausted. Finally, the mass balance for the products over the entire feeding time is carried out to calculate the selectivity of product Q, XQ.
3. Results and discussion 3.1. Determination of feed discretization number In order to investigate the effect of micromixing, the effect of macromixing should be eliminated. This is done by discretizing the total feed solution into sufﬁcient elements. When the feed discretization number, s, is not large enough, it has an impact on the simulation results (Akiti and Armenante, 2004; Baldyga and Bourne, 1989a) and the computational cost grows proportionally with increasing s. Therefore, preliminary simulations consist in choosing an appropriate s. The volume of each identical element, Vo, is equal to 1/s of the total feed volume. Fig. 2 shows the ﬁnal XQ as a function of s for the poorest micromixing case, namely, the lowest agitation speed (180 rpm) and the feed location near the liquid surface (L1) in Fluid-1. It is seen that s has little impact on the ﬁnal XQ when it exceeds 20. In this work s is chosen as 20 for all subsequent simulations unless otherwise noted.
2.5. Simulation strategy 3D time-dependent CFD simulations are carried out using the CFD package, Fluent 6.2 (Ansys Inc.). The whole tank consists of 240,000 tetrahedral meshes as shown in Fig. 1. Different mesh sizes are tested in order to make the solution mesh independent and the areas near the impeller, bafﬂes and walls are reﬁned. The impeller zone is incorporated in the model by the Multiple Reference Frames (MFR) method. The Pressure-Implicit with Splitting of Operators (PISO) pressure–velocity coupling scheme is used.
Fig. 2. Effect of feed discretization number s on the predicted ﬁnal XQ for the worst mixing performance case (Feeding location: L1, N ¼180 rpm, Fluid-1, a1 ¼ 0.5).
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3.2. Micromixing at different operating conditions The ﬂuid viscosity, agitation speed and feeding location all have an effect on the micromixing performance characterized by the selectivity of product Q, XQ. In addition to the aforementioned factors, the feeding rate or the process time also inﬂuence the micromixing performance in the experimental and simulation work. (Baldyga and Makowski, 2004) This effect can be considered by changing the feeding discretization number in the simulations, which are equivalent to alter the feeding time. Figs. 3 and 4 compare the simulated and experimentally obtained selectivity (Atibeni, 2005) at different agitation speeds, feeding locations and ﬂuid viscosities. The agreement is reasonable when a1 is set to 0.5 and 5 for Fluid-1 and Fluid-2, respectively. From Figs. 3 to 4, an increase in viscosity results in a higher XQ. This is because when the viscosity is higher, the time for a portion of NaOH solution to reach Kolmogorov scale is longer. Therefore, the interface between the reactants is smaller and their diffusion slower, leading to a higher XQ. Fig. 5 shows the simulated effects of the ﬂuid viscosity and the feed discretization number on the consumption rate of NaOH solution for the feeding location of L1 and 180 rpm. The depletion
Fig. 3. Effect of agitation speed and feeding location on the ﬁnal XQ with simulated and experimental data from Atibeni (2005) (a1 ¼ 0.5, s ¼ 20, Fluid-1).
Fig. 4. Effect of agitation speed and feeding location on the ﬁnal XQ with simulated and experimental data from Atibeni (2005) (a1 ¼ 5, s ¼ 20, Fluid-2).
Fig. 5. Predicted volume averaged concentration of the ﬁrst portion of NaOH solution as a function of time for different ﬂuid viscosities and feed discretization numbers (Feeding Location: L1, 180 rpm, a1 ¼ 0.5 and 5 for Fluid-1 and Fluid-2, respectively).
process of the ﬁrst portion is illustrated. The gradient of CA (volume averaged concentration of NaOH solution), which characterizes the consumption rate, is much larger for Fluid-2. The poor micromixing in the highly viscous ﬂuid slows down the reaction rate. It takes more time to consume the reactant. The decrease in the feed discretization number leads to a little longer reaction time. Micromixing can be intensiﬁed by increasing the agitation speed, as shown in Figs. 3 and 4. An increase in the agitation speed brings about an increase in the turbulent energy dissipation rate and turbulent kinetic energy as well as a decrease in the ﬂuid viscosity in the discharge area of the impeller. They all improve the micromixing performance. The ﬂuid viscosity in the discharge area where the turbulent kinetic energy is the strongest decreases due to shearthinning (see Fig. 6). Although a1 is sensitive to the ﬂuid viscosity, it is set as a constant for the whole area of the tank, ignoring the ﬂuid viscosity distribution. It is one of the reasons for the discrepancy between the simulation and the literature work (Atibeni, 2005). The reaction process is expected to be mixing sensitive when the reaction time is much smaller than mixing time. In this work, the reaction time for one portion of reactant A (NaOH solution) is less than 4 s (s ¼20) while the mixing process takes more than 35 s. The NaOH solution is exhausted before the vessel content is homogenized. Fig. 7 shows the depletion process of ﬁrst portion of NaOH solution as a function of time at different agitation speeds. The reaction proceeds more quickly as the agitation speed increases and tends to reach an upper limit with a further increase in the agitation speed for both cases with the discretization numbers of 10 and 20. Also, the reaction time for both discretization numbers does not present a large difference. When the NaOH solution is fed at the discharge area (L2), XQ is generally lower than that at L1. The formation of the side product Q is favored by low local turbulent kinetic energy and high ﬂuid viscosity near the liquid surface (L1), shown in Fig. 6. Consequently, it is very important to choose the feeding location in order to control the product distribution, especially when the primary desired product and the secondary undesired by-product differ signiﬁcantly in formation rate. Fig. 8 shows the effects of feeding location and feed discretization number on the volume averaged concentration of NaOH solution following the addition of the ﬁrst portion. The reaction at feeding location L2 proceeds more quickly than that at L1. This is because in the former case micromixing is better where closing to the discharge area. In the case of feeling location L1, the reaction rate ﬁrst increases and then decreases. However in the case of L2,
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Fig. 6. Turbulent kinetic energy and ﬂuid viscosity distribution at 420 rpm in Fluid-1 (Left: Turbulent kinetic energy, k, (m2/s2); Right: Viscosity, Z, Pa.s).
it decreases all the time. In fact, the reaction rate depends not only on the feeding location of the NaOH solution but also on the trajectory a reaction plume follows. 3.3. Evolution of reaction plume
Fig. 7. Depletion process of the ﬁrst portion of NaOH solution for different rotating speeds of impeller and feed discretization numbers (a1 ¼ 0.5, Fluid-1, Feeding location: L1).
The trajectories of the reaction plume at different feeding locations can help shade insights into the characteristics of the reaction and micromixing processes (Figs. 9 and 10). The reaction plume zone is demarked by the critical concentration of NaOH solution (CA 410 7 kmol/m3). For the feeding at L1, the reaction plume moves to the discharge area and expands to its maximum at about 1 s. Thereafter it shrinks quickly. When the NaOH solution is fed at L2, the reaction zone is separated into two parts by the shear and advection of the discharge ﬂow, reaching its maximum volume at about 1 s. The upper part of the reaction plume moves toward the center of the surface with a low mixing performance, resulting in a long reaction time while the lower part is exhausted quickly. Therefore, in addition to the initial feeding location, the difference that exists among different feeding locations is related to different trajectories the reaction plume may have experienced in the tank, as a result of differences in ﬂuid velocity, viscosity and reagent concentration. Baldyga et al. (1997) proposed the so-called E model taking into account micromixing and mesomixing, in which the reaction plume volume grows exponentially till it occupies the entire tank volume or diminishes completely. Actually, the reaction plume expands its volume due to the processes like advection, engulfment, deformation and diffusion and it may shrink by the reaction. The mixing and reaction processes interact with each other, resulting in a maximum volume, as the NaOH solution is exhausted before the stirring tank is homogenized. The effect of different feed discretization numbers is also illustrated in Figs. 9 and 10. The smaller discretization number increases the reaction volume, which causes to the longer reaction time. 3.4. Interactions between mixing and reaction
Fig. 8. Depletion process of the ﬁrst portion of NaOH solution for different feeding locations and feed discretization numbers (a1 ¼5, Fluid-2).
For the purpose of investigating the relationship between mixing and reaction, the turbulent reaction rate from the Eddy-Dissipation (ED) model is compared with the micromixing-kinetics reaction rate estimated by combining the Finite-Rate Model with the standard E
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Fig. 9. Visualization of reaction plume of the ﬁrst portion of NaOH solution with the feeding location of L1 (Fluid-2, 135 rpm, a1 ¼ 5, Critical concentration is 1.0 10 7 kmol/m3).
Fig. 10. Visualization of reaction plume of the ﬁrst portion of NaOH solution with the feeding location of L2 (Fluid-2, 135 rpm, a1 ¼ 5, Critical concentration is 1.0 10 7 kmol/m3).
Fig. 11. Simulated volume averaged turbulent reaction rate and micromixing-kinetics rate of the ﬁrst portion of NaOH solution for different feeding conditions (Fluid-2, N ¼135 rpm, a1 ¼ 5, s ¼20). (a) Feeding Location L1, fast reaction. (b) Feeding Location L1, slow reaction. (c) Feeding Location L2, fast reaction. (d) Feeding Location L2, slow reaction. Solid lines represent the turbulent reaction rate and dash lines represent the micromixing-kinetics reaction rate.
micromixing model. Considering the fact that the fast reaction in Eq. (1) and the slow one in Eq. (2) differ greatly in reaction kinetics, they are expected to be affected by the mixing process differently.
Fig.11 compares the calculated turbulent reaction rate and micromixing-kinetics rate for different feeding conditions. Rav is deﬁned as the volume-weighted average reaction rate for a single
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reaction. The overall reaction rate, which can be converted to Ri,r in Eq. (12), is the smaller one of the two and is shown in the insert. For the fast reaction shown in Fig. 11(a) and (c), the turbulent reaction rate controls the process. On the contrary, the micromixing-kinetics reaction rate dominates the process for the slow reaction shown in Fig. 11(b) and (d). From Fig. 11(a), when a portion is fed at L1 the fast reaction approaches its maximum rate in 1.0–1.5 s. From Fig. 9, the reaction plume moves to the discharge area and then very quickly spreads out into the whole tank. However, in the case of the slow reaction in Fig. 11(b) the reaction rate decreases and its maximum appears at the very beginning. The fast and slow reaction rates are of the same order of magnitude, favoring the formation of the side product. When the NaOH solution is fed at L2 (discharge area), the reaction rates of both the fast and slow reactions decrease with time (see Fig. 11(c) and (d)). The fast reaction rate is almost ten times the slow one, leading to a low XQ.
4. Conclusions A numerical method is developed to investigate the micromixing performance in a stirred tank with a Rushton turbine and viscous ﬂuids. It is characterized by the product selectivity of a series of fast, parallel competitive reactions. The Reynolds stress model (RSM) is used to simulate the ﬂow ﬁeld and the standard E model coupled with the Finite-Rate/Eddy-Dissipation (FR/ED) Model is proposed and validated by experimental results in the literature. When the feed is discretized into more than 20 portions, which is thought to be sufﬁcient to preclude the effect of macromixing, the ﬁnal selectivity was independent of discretization. For one portion, the decrease in the feed discretization number brings about a longer reaction time. Better micromixing is achieved with a ﬂuid of lower viscosity and a higher agitation speed. As a result, the formation of side products is disfavored and the reaction rates are accelerated. The side product selectivity is higher when the feeding location is under the liquid surface (L1) (worse micromixing) than in the discharge area (L2). The micromixing performance also depends on the trajectory of the reaction plume. The reaction plume zone exhibits a maximum volume in the mixing–reaction process as a result of the competition between the expansion by advection, engulfment, deformation and diffusion and the shrinkage by the reaction. The FR/ED model parameter a1, which is obtained experimentally, depends on the ﬂuid viscosity and is hardly affected by the agitation speed. This characteristic is very helpful for reactor design, scale-up and optimization. a1 can be assessed at the labscale for a given viscous reactant ﬂuid and is then directly applied to predict the mixing–reaction process on an industrial scale.
N Rav Ren Rel Ri Ri,r t u Y
agitation speed, (rpm) volume-weighted average of the reaction rate for a single reaction, (kmol/(m3.s)) generalized Reynolds number, dimensionless local Reynolds number, dimensionless net rate of production of species i by all chemical reactions, kmol/(m3.s) net rate of production of species i due to reaction r, kg/(m3.s) time, (s) velocity vector, (m/s) mass fraction, dimensionless
a e g Za Zj,r0 Zj,r00 r s n ni,r0 ni,r00
FR/ED Model parameter turbulent energy dissipation rate, (m2/s3) shear rate, (s 1) apparent ﬂuid viscosity, (Pa.s) rate exponent for reactant species j in reaction r rate exponent for product species j in reaction r density of ﬂuid, (kg/m3) number of discretized feed elements, dimensionless kinematic viscosity, (m2/s) stoichiometric coefﬁcient for reactant i in reaction r stoichiometric coefﬁcient for product i in reaction r
Subscripts i, j r R
species i, j reaction r reactant R
Acknowledgments This work was supported ﬁnancially by Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT), the Natural Science Foundation of China (NSFC) under Grant 50503020 and Zhejiang Provincial Natural Science Foundation of China (Y4090319). References
Nomenclature Ci /CiS D Di,m E Ji k kZ kr Mw n
molar concentration of species i, (kmol/m ) bulk molar concentration of species i, (kmol/m3) impeller diameter, (m) diffusion coefﬁcient for species i in the mixture, (m2/s) engulfment parameter, (L/s) diffusion ﬂux of species i, (kg/(m.s)) turbulent kinetic energy, (m2/s2) ﬂow consistency index. (Pa.sn) reaction kinetic constant, (m3/(mol.s)) molecular weight, (kg/kmol) ﬂow behavior index, dimensionless
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