Numerical simulation of macro-mixing in liquid–liquid stirred tanks

Chemical Engineering Science 101 (2013) 272–282

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Numerical simulation of macro-mixing in liquid–liquid stirred tanks Dang Cheng, Xin Feng, Jingcai Cheng, Chao Yang n National Key Laboratory of Biochemical Engineering, Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T


The predictive capability for liquid– liquid flows is improved using EASM.
EASM predicts homogenization curves better than k−ε model.
The homogenization curves predicted by the EASM are very close to the LES ones.
The EASM gives better mixing time values than k−ε model.
Mixing time can be used to determine critical impeller speed.

art ic l e i nf o

a b s t r a c t

Article history: Received 21 December 2012 Received in revised form 26 April 2013 Accepted 11 June 2013 Available online 19 June 2013

Numerical simulations of turbulent immiscible liquid–liquid mixing processes in cylindrical stirred tanks driven by a Rushton turbine are carried out based on an Eulerian–Eulerian approach using in-house codes. An isotropic standard k−ε turbulence model and an anisotropic two-phase explicit algebraic stress model (EASM) are used for flow field simulations. Quantitative comparisons of the homogenization curve and mixing time predicted by the EASM are conducted with reported experimental data and other predictions by the standard k–ε model and large eddy simulation (LES). The comparisons show that the EASM predictions are in satisfactory agreement with experimental data and better than the k–ε model ones. The variation of the continuous phase mixing time with impeller speed can be an effective method to determine the critical impeller speed for complete dispersion of oil phase. The key features of the complex liquid–liquid mixing processes in stirred tanks have been successfully predicted by the EASM, which can be an alternative tool for practical engineering applications with economical computational cost and good accuracy. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Mixing Multiphase reactors Liquid–liquid Turbulence Computational fluid dynamics Transport processes

1. Introduction The stirred tanks involving two immiscible liquids are extensively used in chemical and metallurgical industries, such as suspension/emulsion polymerization, heterogeneous/phase-transfer catalytic chemical reaction and hydrometallurgical solvent extraction. Mixing plays a fundamental role in these systems, which controls the processes of blending different liquids, liquid– liquid mass transfer, and chemical reactions etc. The quality of product, yield and economy of the processes is hence significantly affected by mixing. Insufficient or excessive mixing may lead to

n

Corresponding author. Tel.: +86 10 62554558; fax: +86 10 82544928. E-mail address: [email protected] (C. Yang).

0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.06.026

wastage of processing time and raw materials and/or the formation of by-products (Yeoh et al., 2004). Mixing is a very complex process in a turbulent stirred tank, which occurs as a result of fluid motion at two (macro- and micro-) typical scales. The presence of a second phase (gas, oil drop or solid) makes the flow and mixing process of the continuous phase even more complicated, especially for high dispersed phase loadings. For liquid–liquid systems, the macro-mixing determines the environmental concentrations for micro-mixing in the continuous phase, which affects the course of chemical reactions directly. It is thus believed that the information related to macro-mixing is very important to control the performance of chemical reactions occurring in the continuous phase in the presence of immiscible oil drops. The macro-mixing is usually characterized by mixing time, i.e., the time required to achieve certain degree of homogeneity of an inert tracer

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injected into a stirred tank. Moreover, mixing time is a simple and powerful measure to assess the effectiveness of a mixer and one of the most crucial parameters for design, optimization and scale-up of a stirred tank. Therefore, it is necessary to gain a detailed knowledge of the macro-mixing characteristics of turbulent liquid–liquid dispersions. A large number of experimental works have been devoted to studies on the mixing time in single phase (Nere et al., 2003; Grenville and Nienow, 2004), gas–liquid, solid–liquid and gas– liquid–solid systems (Cheng et al., 2011a), and relevant empirical correlations were well developed during last decades. Whereas, few research efforts have been focused on the mixing time in complex immiscible liquid–liquid systems. Recently, Zhao et al. (2011) measured the continuous phase mixing time in the presence of immiscible oil drops for a wide range of oil volume fractions and viscosities using four different impellers, and the results were combined to an empirical correlation as well. However, the correlation is based on laboratory scale measurements, and its extrapolative use to industrial-scale stirred tanks is risky. Furthermore, it conceals detailed localized information and cannot be used for prediction of homogeneity degree at various locations inside the tank (Jahoda et al., 2007). It is therefore essential to develop computational fluid dynamics (CFD)-based methods, which are powerful and capable of eliminating scaling-up/down problems by numerical solution of the fundamental equations governing fluid flow and tracer transport. Various alternative methods can be employed to model turbulent macro-mixing processes. The most popular are the Reynoldsaveraged Navier–Stokes (RANS) approach with turbulence models and the large eddy simulation (LES) approach. Although the LES method was generally revealed to be able to mimic the transient experimental responses of probes monitoring local tracer concentrations quite accurately and give more realistic mixing time values (Van den Akker, 2006; Jahoda et al., 2007), its tremendous computational cost, e.g., more than 2 months for a typical job (flow field simulation plus mixing) in very small lab-scale stirred tanks (Yeoh et al., 2005; Hartmann et al., 2006) is still a major constraint for industrial/pilot scale applications and therefore is not yet fine-tuned for quick process design validation (Kasat et al., 2008). Overall, improvement is expected with the LES approach for flows in which the rate-controlling processes occur in the resolved large scales, while the appeal of LES is weak when the rate-controlling processes occur below the resolved scales (Pope, 2004). Moreover, the flow field simulations by LES have not yet been quantitatively validated for the complex immiscible liquid– liquid flows in stirred tanks. Coroneo et al. (2011) performed systematic and stringent evaluation of the contribution of numerical issues to the accuracy of the most widespread k−ε model, and confirmed that Reynolds averaging of the convection–diffusion equation was an acceptable approximation. For these reasons, the computationally efficient RANS approach with appropriate turbulence models might be the main tool in practical industrial applications. The RANS approach with a turbulence model based on the fully predictive strategy, i.e., the sliding mesh or the multiple frames of reference (MFR) framework, or sometimes a combination of the two, or the inner-outer iterative procedure, has been widely used to model the turbulent macro-mixing processes in stirred tanks. Most of them were devoted to single phase systems (Jaworski and Dudczak, 1998; Osman and Varley, 1999; Jaworski et al., 2000; Do et al., 2001; Bujalski et al., 2002a, 2002b; Murthy Shekhar and Jayanti, 2002; Montante and Magelli, 2004; Montante et al., 2005; Kukukova et al., 2005; Mostek et al., 2005; Javed et al., 2006; Kumaresan and Joshi, 2006; Ochieng et al., 2008; Coroneo et al., 2011), and very few were focused on multiphase (gas–liquid and solid–liquid) stirred reactors (Khopkar et al., 2006b; Jahoda et al.,

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2009; Kasat et al., 2008). From the above survey, it is noted that nearly all the simulation works used the standard k−ε model to handle turbulence. In the past decades, the standard k−ε turbulence model has been the most commonly used model for stirred tank simulations. The primary weakness of the k−ε model is that it fails to predict accurately the flow in anisotropic turbulence regions for its assumption of isotropic turbulence and spectral equilibrium. Further, it is clear that modeling of the transport equations for k and ε leads to difficulties to account for streamline curvature, rotational strains, and other body-force effects (Joshi et al., 2011). The impeller zone is the main source of anisotropic turbulence in stirred tanks, where the agitation power is transferred into the tank. Zhao et al. (2011) reported that the shortest mixing time was observed when the tracer was injected into the liquid from the impeller zone as compared to the cases when the tracer was injected from the liquid surface or the tank bottom. This observation further illustrates that the impeller zone is very important for the mixing process, suggesting that modeling of anisotropic turbulence is especially crucial for prediction of the overall mixing performance of stirred tanks. As the anisotropic turbulence model is concerned, Reynolds stress model (RSM) and algebraic stress model (ASM) are widely shown to perform well in prediction of single phase flow fields (Murthy and Joshi, 2008). Since Reynolds stress components are solved directly from a differential equation or an algebraic equation rather than being modeled by an isotropic hypothesis like the k−ε model, anisotropic turbulence can be successfully predicted. However, both RSM and ASM are not computationally robust and have difficulty to reach converged solutions. To overcome these problems, Pope (1975) proposed an explicit algebraic stress model (EASM) for two-dimensional flows based on the RSM or the ASM by using a tensor polynomial expansion theory, in which the Reynolds stress components were expressed as an explicit algebraic correlation of mean strain rate tensor, rotation rate tensor and turbulence characteristic quantities. Following Pope's theory, Gatski and Speziale (1993) and Wallin and Johansson (2000) developed three-dimensional EASMs. Recently, Feng et al. (2012a, 2012b) simulated single phase and solid–liquid twophase flows successfully with improved computational stability and greatly reduced computational cost using Wallin and Johansson's explicit algebraic stress model (EASM). Considerably improved agreement with experimental data was found in terms of mean as well as turbulence quantities compared to those predicted by the k−ε and ASM models. Since the EASM performs well in predicting single and solid–liquid flows in stirred tank, it is expected to perform as well for description of immiscible liquid– liquid flows. Feng et al. (2012c) also made an attempt to simulate two-phase liquid–liquid flows in stirred tanks using the two-phase EASM, and comparisons with experimental data and k−ε predictions in terms of mean velocities and the dispersed phase holdup distributions were conducted. The results were encouraging, but the quantitative comparisons were limited owing to the experimental measurement and CFD simulation of immiscible liquid– liquid flows in stirred tanks were only rarely reported (Wang and Mao, 2005; Svensson and Rasmuson, 2004, 2006; Laurenzi et al., 2009; Cheng et al., 2011b). Therefore, the EASM deserves further evaluation in order to assess comprehensively its performance for describing turbulent liquid–liquid flows in stirred tanks. As the macro-mixing process is closely related to the mean flow field and the turbulence, the EASM can be further verified using macromixing experimental data, which can be easily measured. To the best of the authors’ knowledge, numerical simulation of the continuous phase mixing characteristics and assessment of the predicted macro-mixing data using both the isotropic standard k−ε turbulence model and the anisotropic EASM against experimental values of liquid–liquid systems have not yet appeared in the

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literature. The main purpose of this work is to make an attempt to fill this gap. On the other hand, the comparative studies between RANS approaches with different turbulence models and the LES are still insufficient in the literature. Consequently, the results obtained by the RANS-based models are compared with the LES predictions as well. The macro-mixing experiments of Zhao et al. (2011) are replicated as closely as possible in the simulations realized by in-house computer codes, in order to compare the simulated mixing data with experimental results. The details of the computational model/approach and the predicted results are presented in the following sections.

2. Stirred tank configuration and experimental conditions The macro-mixing experiments in the presence of immiscible drops by Zhao et al. (2011) are targeted in the present work. The tank with a standard Rushton turbine (D¼T/3, b¼0.25D and w¼0.2D) is a cylindrical, flat-bottom reactor with diameter T¼ 0.38 m and liquid height H¼T. Four vertical standard baffles of width BW¼0.1T are equally spaced mounted at the reactor wall. The model system is an oil-in-water dispersion at room temperature (25 1C), in which the tap water (ρc ¼997.7 kg/m3, μc ¼ 0.001 Pa s) is the continuous phase, and kerosene (ρd ¼ 789.5 kg/m3, μd ¼0.002 Pa s), paraffin oil (ρd ¼864.5 kg/m3, μd ¼ 0.052 Pa s) or methyl silicone oil (ρd ¼972.4 kg/m3, μd ¼ 0.565 Pa s) is used as the dispersed phase. Kerosene is used in most experiments, whereas paraffin oil and methyl silicone oil are just used to test the influence of the dispersed phase physical property. The conductivity method is used to measure the homogenization curve and mixing time. The time interval between the addition of the tracer (250 g/L NaCl) and the moment when the instantaneous conductivity at a specific monitoring position reaches within 95% of its final value is defined as the mixing time (tm). Three tracer injection locations (P1, P2 and P3) and two monitoring locations (probes A and B) are set the same as those used in the experiments of Zhao et al. (2011), and their exact coordinates are marked in Fig. 1. The injection and monitoring positions are all located mid-way between two consecutive baffles. Unless stated otherwise, the simulated mixing time (tm) used in this paper is calculated from probe A, which is the same as Zhao et al. (2011). Measurements are obtained in the Reynolds number (Re) range from 8:78  104 to 12:23  104 with Re ¼

ND2 ρc μc

ð1Þ

It should be pointed out that the background tracer concentration at t ¼0 in the liquid will influence the exact shape of subsequent response curves as can be seen in Jahoda et al. (2007, 2009). However, numerical simulations are carried out in such a way that no tracer is present before the tracer injection. In order to match the numerical simulation as closely as possible, the experimental probe response curves used in this paper are recorded at the condition that no tracer is present beforehand. The experimental time traces are expressed in the form of

dimensionless concentration c¼

ct −c0 c∞ −c0

ð2Þ

where c is the tracer dimensionless concentration at time t, c0 is the tracer concentration at the beginning of the experiment, ct is the tracer concentration at time t, and c∞ is the homogeneous tracer concentration. In addition, as the LES simulation on liquid–liquid macromixing processes has not been found in the open literature, the single-phase experiments (a cylindrical vessel with T ¼100 mm, BW ¼0.1T, D ¼T/3, C ¼ T/3, H¼ T, N ¼36 rps) of Lee (1995) are also considered in order to compare the predictions by the k−ε turbulence model and the EASM with the predictions using the LES by Yeoh et al. (2005). The injection location (E) and two monitoring positions (F, G) adopted by Yeoh et al. (2005) are marked in Fig. 1 as well.

3. CFD simulations Two computational strategies are available to predict liquid homogenization and mixing time in a stirred tank. In the first approach, hydrodynamic equations and tracer transport equation are resolved in an uncoupled way. The hydrodynamic equations are solved first, and then the flow field is used to calculate the transient tracer transport equation. This method is mostly used in the published literature. In the second approach, the hydrodynamic equations and the tracer transport equation are solved simultaneously. Bujalski et al. (2002a) examined the effect that the two strategies had on the mixing time using the sliding mesh method to model the impeller and a low Reynolds number k−ε model to cope with turbulence based on the commercial CFD software CFX, and found that the two modeling strategies had little influence on the predicted mixing time. However, they pointed out with special emphasis that the computational demands of the second approach were about 10 times greater than those of the first approach. Jahoda et al. (2009) investigated numerically the homogenization process in a gas–liquid stirred tank using the RANS method with the k−ε mixture turbulence model. They revealed that the two computational strategies gave very similar homogenization curves and mixing time when the impeller was modeled using the MFR technique. Note that the experimental measurements were made after at least 1 h of stirring to obtain reliable and consistent data of mixing time in the oil–water system as Zhao et al. (2011) reported. On the other hand, Yeoh et al. (2005) also remarked that passive tracer did not influence flow characteristics. Therefore, the tracer field can be mathematically decoupled from the hydrodynamics equations. In order to replicate the experiments as closely as possible and to save computational loads, the hydrodynamic equations are solved first to get the turbulent liquid–liquid two-phase flow field, and then the tracer transport equation is solved. 3.1. Hydrodynamic model

Fig. 1. Geometric locations of injection points and monitoring positions.

The turbulent liquid–liquid flow is modeled using the threedimensional Reynolds averaged Navier–Stokes equations together with turbulence models based on an Eulerian–Eulerian approach. Although the standard k−ε model is widely used, such a model is incapable of describing the turbulence anisotropy which is commonly encountered in the impeller region and the impeller stream in stirred tanks. With respect to anisotropic turbulence models, Feng et al. (2012a, 2012b) employed Wallin and Johansson (2000)'s EASM to simulate single-phase and two-phase flows in stirred tanks successfully, showing that the EASM gave obviously better

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predictions than the k−ε model. To see the effect of the turbulence models on the macro-mixing processes, both the isotropic standard k−ε model and the anisotropic two-phase EASM are used to deal with turbulence. An improved inner-outer iterative procedure (Wang and Mao, 2002) is adopted to treat the impeller rotation in the fully baffled tank. The whole tank is considered as a solution domain since no symmetry in the tracer concentration field could be assumed. The grid independency tests are carried out by considering four grids: 24(r)  48(θ)  45(z), 36  72  72, 48  192  96 and 48  240  150 for the case of T¼0.38 m, H¼T, C¼T/3, D¼T/3, αd;av ¼ 5% and N¼400 rpm. The predicted quantity profiles with the coarse grids (24  48  45 and 36  72  72) deviate from the results of the finer grids. It is noted that the difference between the two grids of 48  192  96 and 48  240  150 is very small, suggesting that further grid refinement would have no obvious effect on the solution. Consequently, the grid 48  192  96 is chosen for all subsequent simulation works. The top free surface of the dispersion is treated as flat. At the top free surface of the vessel, the axial gradients of the dependent variables are set to zero, and along the axis the turbulent quantities are considered axisymmetric. The standard wall functions are applied to the near wall nodes. Still liquid and a homogeneous distribution of the dispersed phase within the computational domain are considered as initial conditions for each case of flow field simulation. The flow in the tank is assumed as an isothermal process with no mass transfer between phases and chemical reactions. For the details of the hydrodynamic model/numerical procedure used in the present work, refer to Wang and Mao (2005) for the standard k−ε model and Feng et al. (2012a, 2012b) for the EASM. The interphase interaction mainly involves four momentum exchange mechanisms such as Basset force, lift force, added mass force and drag force as discussed by Ranade (2002). However, the contribution of the drag force is dominant and the effect of the other forces is of little significance (Wang and Mao, 2005; Kasat et al., 2008; Jahoda et al., 2009). Considering this, only the drag force is incorporated in this work. The model of Barnea and Mizrahi (1975), which takes into consideration both the drop deformation and the wall effect, is adopted for the drag coefficient calculation. The drop size is an important parameter, and controls the increase of drop drag coefficient due to turbulence (Khopkar et al., 2006a). The prevailing drop size distribution in a liquid–liquid stirred reactor is controlled by many parameters such as the dispersed phase holdup, impeller speed and reactor configuration. The drop size distribution can be possibly accounted for by coupling the two-fluid computational model with population balance equations. However, the use of the two-fluid model coupled with the population balance equations increases the computational load by many times (Khopkar et al., 2006b). On the other hand, the available drop size distribution information in the present cases is unfortunately not adequate to calculate the parameters appearing in the coalescence and break-up kernels. Furthermore, the drop population balances cannot be applied in a satisfactory manner as the k−ε models and the EASMs cannot predict correctly the turbulent kinetic energy dissipation rate, thus causing a large deviation of drop break-up rates (Laborde-Boutet et al., 2009). Therefore, the implementation of drop population balance models appears premature at this time, and estimation of drop sizes using a correlation is a feasible choice (Laborde-Boutet et al., 2009). Wang and Mao (2005) estimated the local value of the drop diameter using Nagata’s (1975) correlation as follows:
0:196 −0:914 ηg d ¼ 10ð−2:316þ0:672αd Þ ν0:0722 ð3Þ c;lam ε ρc This correlation is improper as the RANS method underpredicts the turbulent kinetic energy dissipation rate. For simplification of the solution, some investigators used a uniform

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mono-dispersed drop size throughout the tank (Laurenzi et al., 2009; Abu-Farah et al., 2010), and our previous work (Cheng et al., 2011b) has quantitatively shown that the uniform drop diameter affected simulation results to a large extent by assuming several fixed diameters of the mono-dispersed drop size. Nevertheless, there is a wide distribution of drop sizes in a liquid–liquid stirred tank, which complicates the problem of selecting a suitable drop diameter if a uniform mono-dispersed drop size is used when simulating different operation cases. Also, the use of a uniform drop diameter seems to have the tendency to make the drop size a tuning parameter, which is unacceptable in CFD simulations. We thus make an attempt to use the well-developed empirical correlation, which is applicable to a spectrum of cases, to estimate the mono-dispersed drop size. The global average drop diameter in the stirred tank can be approximated by the Sauter mean diameter (d32) without notable error (Wang et al., 2006). The d32 is widely correlated by d32 ¼ Að1 þ γαd;av ÞðWeÞ−0:6 D

ð4Þ

The ð1 þ γαd;av Þ term reflects the effect of the dispersed phase holdup to account for turbulence dampening, and the average dispersed phase volume fraction (αd;av ) is applied in the present simulation. As indicated in the reviews of Davies (1992) and Peters (1997), the functional form of Eq. (4) has been found to work quite well by many workers. Many reported correlations show some differences in the values of constants A and γ. For a batch process, constants A and γ are 0.06 and 9 reported by Calderbank (1958), 0.051 and 3.14 by Brown and Pitt (1970), 0.047 and 2.5 by van Heuven and Beek (1971) and 0.058 and 5.4 by Mlynek and Resnick (1972). It is noted that, despite the diversity of systems, the reported A and γ values do not show considerable scattering. The average values are determined to be A ¼0.054 and γ ¼5.01 for these above-mentioned cases and are used in the present investigation. 3.2. Mixing model The tracer transport in the continuous phase is essentially governed by a convective–diffusion equation
 ∂ðαc cÞ ∂ðαc ui cÞ ∂ ∂αc c þ ¼ Deff ð5Þ þ Sc ∂t ∂xi ∂xi ∂xi where c is the local tracer concentration, ui is the resolved velocity component i, αc is the continuous phase volume fraction, Deff is the effective diffusion coefficient, and Sc is the source term which is zero in the present case of inert tracer. The effective diffusion coefficient Deff in Eq. (5) is the sum of molecular and turbulent diffusion coefficients, i.e., Deff ¼ Dmol þ Γ t . The turbulent diffusion coefficient is calculated from the turbulent kinetic viscosity (νe ) as Γ t ¼ νe =Sct , where Sct is the turbulent Schmidt number and its default value 0.7 is used. Dmol is assumed to be equal to 10−9 m2/s, a typical value for a solute in liquids (Montante and Magelli, 2004). The time step adopted for the simulation is equal to 0.001 s, which is much smaller than the experimental data acquisition rate (100 Hz). For transient mixing calculations, the initial tracer concentrations are designated as zero everywhere in the tank except at the feed cells, where the concentration is set equal to that of the tracer feed solution. A zero flux boundary condition at the walls is used for mixing calculations (Ranade et al., 1991). The time evolution of the tracer concentration is recorded at the same positions of the corresponding experimental probes to allow direct comparison with the experimental results. The in-house code is compiled in FORTRAN and all the simulations are run on a PC computer with a 4 Gb RAM and a 3.4 GHz clock frequency processor. Concerning the computational cost, it takes about 6–7

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days CPU time for a typical simulation work (flow field simulation plus mixing) when using the k−ε model, while the two-phase EASM about 8–9 days CPU time is necessary. The computational cost is quite acceptable in contrast to that of the LES for two-phase flows in stirred tanks as mentioned above.

4. Results and discussion 4.1. Flow field The flow field plays a very important role in prediction of the macro-mixing processes (Montante et al., 2005; Ranade et al., 1991). Ideally, the CFD simulations should be compared with the flow field data of kerosene–water systems used by Zhao et al. (2011) by detailed experimental measurement using a laser Doppler anemometry (LDA) or particle image velocimetry (PIV) technique. Unfortunately, these data are not available for this reactor and difficult to be obtained because the kerosene drops are not transparent and scatter light severely. Therefore, the flow field predictions are validated with the experimental data measured by Svensson and Rasmuson (2004, 2006) using LDA and PIV techniques. In their two experimental works, the stirred tank was a flat-bottom cylindrical tank with diameter T ¼0.14 m, and was equipped with four equally spaced vertical wall baffles with width BW¼T/12. The liquid depth in the tank was H¼ T. A six-bladed disk turbine with diameter D¼ T/3 was used, and the clearance of the impeller was C ¼T/3. The liquid–liquid system was composed of an aqueous NaI solution (ρc ¼ 1340 kg/m3, μc;lam ¼ 1.4  10−3 Pa s) as the continuous phase and silicone oil (ρd ¼ 940 kg/m3, μd;lam ¼11.0  10−3 Pa s) as the dispersed phase. The experimental conditions used by Svensson and Rasmuson (2004, 2006) are adopted for the k−ε model and EASM simulation validations. Our previous works (Wang and Mao, 2005; Feng et al., 2012c) show that the k−ε model predictions are in good agreement with experimental data on flow field, and the EASM predictions are closer to the experimental data than the k−ε model. However, previous comparisons concentrated only on the circulation zones (both above and below the impeller), which are mainly isotropic and the advantage of the anisotropic EASM is not sufficiently demonstrated. Comparison of velocity data in the impeller zone is thus desired as the impeller zone is the main source of strong anisotropy. Fig. 2 plots the comparison of the continuous phase mean velocity components at impeller disc height by the two turbulence models with the PIV experimental data (Svensson and Rasmuson, 2006). It can be seen that the mean velocity components predicted by the EASM are in excellent agreement with the experimental data, while there are apparent discrepancies between the k−ε model predictions with the experimental data near the impeller tip due to the poor assumption of isotropy of turbulence in the flow field. Unfortunately, the experimental data on the distributions of turbulent kinetic energy and dissipation rate of an immiscible liquid–liquid stirred tank have not been found in the open literature up to now, maybe due to measurement difficulties. The comparisons clearly show that the anisotropic two-phase EASM can provide notably improved predictions of flow field compared to those obtained by the k−ε model, suggesting that a better prediction of the macro-mixing process could be expected. 4.2. Homogenization For effective characterization of the macro-mixing process, both the time traces of tracer concentration and mixing time are required. For this reason, the dimensionless tracer concentration

Fig. 2. Comparison of predicted mean velocities with reported experimental data at impeller disc height with N ¼ 540 rpm and αd;av ¼ 7%. (a) Radial velocity. (b) Tangential velocity.

curves and mixing time at various conditions are presented in this paper.

4.2.1. Single-phase system The predicted dimensionless concentration curves by means of the traditional RANS approach combined with the standard k−ε turbulence model are compared with the more sophisticated approach of the LES by Min and Gao (2006) using a 2-narrow blade hydrofoil CBY impeller and Jahoda et al. (2007) using a sixbladed 451 pitched blade turbine and a standard Rushton turbine. They reported that the predicted tracer response curves by the LES were in much better agreement with experimental data than that obtained by the standard k−ε model. However, Min and Gao (2006) and Jahoda et al. (2007) both considered only a similar relative position between the injection point and the monitoring location (the tracer was added to the free surface of the liquid between two baffles and the detector was mounted at the position near the bottom of the tank opposite to the injection point). More knowledge might be expected if more relative positions between the injection point and the monitoring location are considered. The comparative studies between the RANS approaches with different turbulence models and the LES are still insufficient, especially for two-phase systems, e.g., the results obtained from the RANS approach coupled with the more accurate anisotropic turbulence EASM have not been compared with the LES predictions. Unfortunately, there is no macro-mixing data (i.e., homogenization curve and mixing time) by the LES available for liquid– liquid stirred tanks in the published literature. The LES results from Yeoh et al. (2005) for single-phase systems are therefore compared with the predictions by the EASM and the k−ε model. The numerical setups used by Yeoh et al. (2005) are adopted for the k−ε model and EASM simulations, which are described in Section 2.

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The simulated dimensionless concentration curves by the RANS approaches with the EASM and the k−ε model are shown along with the corresponding predictions by the LES in Fig. 3. Yeoh et al. (2005) provided the time traces of normalized concentrations against time in a large number of points. We consider two typical positions (points F and G in the same plane of the injection point) due to other positions with strong and very complex fluctuations, which make it difficult to extract the data from the reported figures without introducing obvious errors. It is clearly observed that the simulated mixing trends by the EASM are in satisfactory agreement with the LES predictions, and both the EASM and the k−ε model underestimate peak values. Though the k−ε model has captured the general mixing trend, it deviates from the LES predictions to a relatively large extent. It is noted that the fluctuations have not been captured by both the k−ε model and the EASM, which is due to the nature of RANS approaches. Overall, the findings are reasonable, considering that the EASM is superior to the standard k−ε model and has the predicted accuracy close to the LES in predicting the mean velocity components (Feng et al., 2012a) and the macro-mixing process depends largely on the mean flows. Patwardhan and Joshi (1999) made an attempt to quantify the contribution of circulation and eddy diffusion in the overall mixing process and revealed quantitatively that the bulk circulation played a dominant role in the blending process and the mixing time increased nominally even if the eddy diffusivity was reduced by a factor of 5. The insensitivity of the eddy diffusivity led to the conclusion that an error in its prediction (20–50%) would cause only a slight change (5%) in the mixing time (Nere et al., 2003). The comparisons illustrate that the EASM is an efficient and a promising tool with satisfactory accuracy in predicting macromixing processes in stirred tanks. However, its performance on description of macro-mixing processes in the continuous phase of liquid–liquid systems is desired.

4.2.2. Liquid–liquid system The mixing behavior differs substantially from location to location across the vessel where trace concentration is measured as revealed by Distelhoff et al. (1997) and Yeoh et al. (2005) in single-phase stirred tanks. Bujalski et al. (2002) also reported that the tracer feed position had a very profound effect on both mixing time and development of concentration field in CFD simulations. Zhao et al. (2011) selected several typical tracer injection and monitoring locations to show their impacts on the continuous phase macro-mixing processes in a liquid–liquid stirred tank, and further confirmed that the mixing time was sensitive to the relative position between tracer injection and monitoring locations. Local macro-mixing characteristics can be characterized by time traces of normalized concentrations. Fig. 4 shows the effect of injection and monitoring locations on the continuous phase homogenization process. In general, it can be observed that all experimental time traces first go up to peaks, then decay with time and eventually the tracer concentrations reach equilibrium values, c∞ , representing the fully mixed condition. However, the exact shape of the response curves differs noticeably as can be seen from Fig. 4a and b. For both probes A and B, the homogenization is obviously faster when the tracer is injected from P2 point than that from P3 injection point. It is because P2 is located in the impeller stream, where the liquid velocity and turbulence are higher and the tracer can be convected and dispersed to other regions more quickly. The time required for the tracer to travel from the injection point to the respective monitoring location depends on their relative positions in the tank. With P2 injection, the trace concentration remains 0 until t¼ 0.31 s for probe A and t¼0.08 s for probe B. With P3 injection, the trace concentration remains 0 until t ¼0.55 s for probe A and t¼0.43 s for probe B. Concentration peaks are related to the time periods that the tracer is present close to the probe either due to injection or because it

Fig. 3. Comparisons of predicted homogenization curves by different computational methods (the LES average line is a polynomial regression curve showing the average trend). (a) At F point. (b) At G point.

Fig. 4. Effect of injection and monitoring positions on homogenization curves (N ¼ 329 rpm, C ¼T/3, αd;av ¼ 10%, Re¼8:78  104 ). (a) Probe A. (b) Probe B.

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has been circulated back to the probe (Yeoh et al., 2005). The time traces of the normalized concentrations by simulations are also presented against experimental measurements, which can provide a clearer picture to evaluate the capability of the k−ε model and the EASM to predict the homogenization process. The general mixing trends are well predicted by the k−ε and the EASM as can be seen in Fig. 4. The peak values predicted by the EASM are larger than those predicted by the k−ε model, though they are lower than the experimental data. It is noted that the homogenization predicted by the EASM is closer to the experimental data and faster than that by the k−ε model. It is worth noting that in the case of probe A with P3 injection, the predicted concentration profile by the EASM shows a distinct peak above the value of 1, but no peak above c ¼1 is observed in the k−ε model prediction. When the dispersed phase is changed from kerosene to paraffin oil, the experimental homogenization process is apparently slowed down at both probes A and B as illustrated in Fig. 5. This might be attributed to higher effective viscosity. The calculated curves by the k−ε model and the EASM contain a peak above c ¼ 1 for probe A, which are in reasonably good agreement with experimental response curves. For probe B, the calculated curves by the EASM are similar, and contain peak values above c ¼1, which corresponds to reality. However, the k−ε model gives no peak value, which is very similar to those reported by Kukukova et al. (2005), Min and Gao (2006) and Jahoda et al. (2009). We can see that the effect of the dispersed phase viscosity on the homogenization process has been successfully captured by both the k−ε model and the EASM. It can be concluded that the predicted mixing trend is closely related to the relative position between the injection point and the monitoring location. Overall, the EASM performs clearly much better than the k−ε model in the comparisons.

Fig. 5. Effect of the dispersed phase viscosities on homogenization curves (N ¼ 358 rpm, C¼ T/3, αd;av ¼10%, Re¼9:55  104 ). (a) P1 injection, probe A. (b) P1 injection, probe B.

4.3. Mixing time Mixing time is an important performance indicator for liquid– liquid stirred vessels. The continuous phase mixing time in the presence of immiscible oil drops measured by Zhao et al. (2011) was sensitive to many variables such as agitation speed, impeller clearance, oil volume fraction and oil viscosity etc. The continuous phase mixing time decreased as the impeller clearance decreased in the range T/3 to T/6. Ochieng et al. (2008) successfully elucidated the effect of the impeller clearance by CFD simulation using the RANS based standard k−ε turbulence model in singlephase stirred vessels. They has shown that at a low impeller clearance, a Rushton turbine generated a flow field that evolved from the typical two loops to a single loop flow pattern similar to that of an axial impeller, which resulted in a decrease in mixing time. The liquid–liquid situation is found to be consistent with that for a single phase system. Two findings reported by Zhao et al. (2011a), which were further confirmed by our recent work (Cheng et al., 2012), appear to be interesting, i.e., the continuous phase mixing time was found to first decrease then increase with kerosene holdup, and increase with the viscosity of the dispersed phase. The impact of the dispersed phase holdup/viscosity on the continuous phase mixing time values reveals a complex interaction between the continuous phase and the dispersed phase. It is believed that mixing time comparisons between experiments and predictions can facilitate a quantitative and rapid assessment of the difference of the two models (i.e., k−ε turbulence model and EASM) in description of the continuous phase macro-mixing with varying oil volume fractions and viscosities. 4.3.1. Effect of dispersed phase volume fraction Zhao et al. (2011) measured the continuous phase mixing time at 0, 3, 5, 7, 10, 15 and 20 vol% dispersed phase (kerosene). As shown in Fig. 6, the predictions by the k−ε model and the EASM are compared with experimental data. The experimental error bars representing the standard deviations are plotted with experimental data. It can be observed that the non-monotonic profile of mixing time is well predicted by the k−ε model and the EASM, though over-predicted. The simulated mixing time by the k−ε turbulence model is overestimated above the measured value with the mean relative deviation of 31%. A better agreement is reached by the EASM predictions with the average relative deviation of 18%. Zhao et al. (2011) analyzed the non-linear relationship to the reason that turbulence could be enhanced by vortex shedding, which was caused by the motion of the drops relative to the continuous phase at low volume holdups and dampened by the dispersed phase at higher holdups. This conclusion was supported by several previous reports (Laurenzi et al., 2009; Svensson and Rasmuson, 2004, 2006). However, note that the flow field

Fig. 6. Mixing time versus oil volume fraction (C¼T/3, N¼ 329 rpm, Re¼8:78  104 ).

D. Cheng et al. / Chemical Engineering Science 101 (2013) 272–282

measurements (Svensson and Rasmuson, 2004, 2006) also revealed that the velocity fields were apparently changed with the variation of oil holdups. Accordingly, it is not enough just to attribute the continuous phase mixing time variation to the turbulence change, and an additional analysis that includes the velocity field effect would be expected to shed new light on the mechanisms by which the oil volume fraction affects the continuous phase mixing time. From the viewpoint of the bulk flow model (Nienow, 1990, 1997; Nere et al., 2003), mixing time is closely related to the flow number and the circulation number for a given impeller at a constant rotational speed. Van de Vusse (1955) has shown that the mixing time in a stirred vessel was dependent on the pumping capacity. Cooper and Wolf (1967) further revealed an approximately linear relationship between the intensity of segregation and the reciprocal of pumping capacity. The numerical simulation method has the advantage of providing detailed information on flow field, which allows calculation of the flow number (Fl) and the circulation flow number (Flc) R w=2 R θ ¼ 2π −w=2 θ ¼ 0 ruc;r dθ dZ Fl ¼ ð6Þ ND3 R r0 Flc ¼

0

2πuc;zL rdr þ

R T=2

ND

r0 3

2πuc;zU rdr

ð7Þ

where r0 is the radial position of the center of the circulation loop. The subscripts U and L indicate the upper and the lower loops with reference to the turbine respectively. More details can be found in previous works (Costes and Couderc, 1988; Jaworski, 1996). The experimental values of Fl for single phase flow induced by a Rushton turbine ranged from 0.6 to 0.9 (Revil, 1982) or 0.73 to 1.02 (Strek, 1977). The experimental value of Flc reported by Jaworski (1996) was 2.10, and Revil (1982) gave a range for Flc from 1.37 to 2.05. The present Fl and Flc calculated are summarized in Table 1. Jahoda et al. (2007) reported that Fl for single phase systems was 0.74 by the k−ε model and 0.83 by the LES. The computed Fl is 0.73 by the k−ε model and 0.81 by the EASM at 0 vol% in the present work, which resembles the values of Jahoda et al. (2007). The Fl value calculated by the EASM is very close to that by the LES, indicating further that the EASM has the predicted accuracy close to the LES in predicting velocity components. The simulated flow data by the EASM are a little larger than that calculated from the k−ε turbulence model. Considering that the EASM has given better predictions on mixing time, this might suggest that the over-prediction of mixing time by the k−ε model and the EASM is mainly due to underestimation of overall velocity field. It is interesting to find out that Fl and Flc are clearly reduced at higher oil holdups (αd;av ≥10%) as shown in Table 1. This finding can help us to understand the effect of oil holdups on the continuous phase mixing time.

279

ranging from 7.86 s for 2 mPa s kerosene to about 10.84 s for 565 mPa s methyl silicone oil (Fig. 7). This observation was also confirmed by our recent work (Cheng et al., 2012), which suggests that the continuous phase macro-mixing process is affected by the viscosity of the dispersed phase. The simulations by the k−ε model generally capture this trend as shown in Fig. 7, and again the EASM is noticed being able to give more accurate description on such effects. The average deviation between the EASM predictions with experimental data is 14%. The simulated mixing time by the k−ε turbulence model is longer than the measured value with the mean relative deviation of 24%. The effect of the viscosity of the dispersed phase on Fl and Flc is also examined as shown in Table 2. It can be seen that both Fl and Flc fall as the dispersed phase changes from kerosene to methyl silicone oil. This finding illustrates that an obvious variation occurs in the velocity field when the dispersed phase species changes, and may shed new light on the effect of the dispersed phase viscosity on the macro-mixing quality of the continuous phase. As can be seen from above, the k−ε turbulence model has returned reasonable predictions of the continuous phase mixing time in the presence of immiscible drops, which is consistent with previous computational studies reported in the literature (Kukukova et al., 2005; Javed et al., 2006; Min and Gao, 2006; Jahoda et al., 2009; Ochieng et al. 2008, etc.). A notably improved agreement with experimental data is obtained using the EASM simulation, despite the drop coalescence and break-up have not been taken into consideration. Yeoh et al. (2005) have actually made comprehensive comparisons between the LES predictions with various measurements, empirical correlations and published experimental data available in the literature, and an average deviation of 18% was satisfactorily obtained. The average deviation is 20% between the EASM predictions with all the experimentally obtained values reported by Zhao et al. (2011), suggesting that the EASM has the prediction accuracy close to the LES in predicting mixing time. It may be noted that Yeoh et al. (2005) have reported a large discrepancy (+53%) between the LES prediction with the

4.3.2. Effect of physical property of dispersed phase The experimental data of Zhao et al. (2011) showed that higher viscosity oils gave longer mixing time in the continuous phase

Fig. 7. Mixing time versus viscosities of the dispersed phases (C ¼T/3, N ¼358 rpm, αd;av ¼ 10%, Re¼9:55  104 ).

Table 1 Simulated

Table 2 Simulated flow data versus the dispersed phases (C¼ T/3, N ¼ 358 rpm, αd;av ¼ 10%,

flow

data

versus

oil

volume

fraction

(C ¼T/3,

N ¼ 329 rpm,

Re¼ 8:78  104 ).

Re¼ 9:55  104 ).

Holdup

0%

3%

5%

7%

10%

15%

20%

Dispersed phase

Water

Kerosene

Paraffin oil

Methyl silicone oil

k−ε model

Fl Flc

0.73 1.96

0.73 1.96

0.74 1.97

0.73 1.98

0.68 1.95

0.61 1.86

0.52 1.81

k−ε model

Fl Flc

0.74 1.98

0.70 1.96

0.68 1.95

0.63 1.90

EASM

Fl Flc

0.81 2.13

0.82 2.13

0.82 2.14

0.81 2.15

0.78 2.11

0.73 2.04

0.65 1.98

EASM

Fl Flc

0.82 2.15

0.80 2.12

0.77 2.10

0.71 2.05

280

D. Cheng et al. / Chemical Engineering Science 101 (2013) 272–282

measured mixing time of Lee (1995). The relatively small average deviation between the EASM predictions with experimental data does not necessarily mean that the EASM is more accurate than the LES. The likely reason as analyzed by Yeoh et al. (2005) is that the tracer concentration variations in the entire vessel was taken into account in the LES numerical simulation, compared to the measurement in which only part of one plane of the vessel was considered. Distelhoff et al. (1997) has shown that even within the same plane, mixing time can vary considerably from point to point. In addition, several areas (in the vicinity of the impeller, shaft, vessel wall, lid and base) were excluded from the experimental image analysis due to the locally strong light scatter. However, the experimental information is strictly adhered to in the predictions of mixing time by the EASM and the k−ε turbulence model (e.g., the same mixing time definition, tracer feed location and monitoring position) in the present work, which may have partly improved the predictive accuracy. The appealing advantage of the EASM method is substantially lower computational cost with satisfactory results.

4.3.3. Effect of agitation speed Zhao et al. (2011) examined the effect of agitation speeds, which were sufficiently above the critical speed for complete dispersion of oil phase, on the continuous phase mixing time. Under such conditions, it has been clearly shown that the continuous phase mixing time decreased with increasing agitation speed. The dispersion quality of oil phase ranging from nondispersion to uniform dispersion varies with the impeller speed. Unavoidably, the continuous phase mixing time is closely related to the oil phase dispersion state. It is thus desirable to examine the mixing time behavior when the impeller speed decreases to sufficiently below the critical speed for complete dispersion of oil phase using the present CFD approach coupled with the more accurate anisotropic EASM. The dispersion quality can be specifically identified as incomplete/partial dispersion, complete dispersion and uniform dispersion with respect to the impeller speed in a stirred tank. In this work, the critical state for complete dispersion is that the dispersed phase lumps visually disappear from the tank, and no clear liquid is observed either at the top or the bottom of the mixing tank, which is the same as in the experiments by van Heuven and Beek (1970) and Skelland and Ramsay (1987). The state of uniform dispersion is defined as that the dispersed phase is homogeneously distributed throughout the tank. Several validated numerical criteria are available to characterize the dispersion quality in the tank. Abu-Farah et al. (2010) and Cheng et al. (2011b) proposed criteria to predict the critical impeller speed for complete dispersion and uniform dispersion. The continuous phase macro-mixing processes in the absence of oil (αd;av ¼0%) and a high oil volume fraction (αd;av ¼15%) are simulated for the impeller speed starting from N ¼150 to 800 rpm with P2 tracer injection and C ¼ T/3. The impeller speed is increased at small intervals until it is well above Nun. The air entrainment is not taken into consideration in the present simulations. The critical impeller speed for complete dispersion is Ncd ¼270 rpm for the case of αd;av ¼15% with C ¼T/3 by Criterion 1 (to inspect the local dispersed phase concentration at a specific monitoring point against impeller speeds and determine where a sudden change in the slope of the curve occurs) from our previous work (Cheng et al., 2011b). Fig. 8 shows the variation of mixing time (tm) with impeller speeds. It can be seen that the mixing time at αd;av ¼15% decreases with an increase in impeller speed as in the case without oil. It is noted that the values of mixing time at αd;av ¼ 15% are higher than those calculated in the clear liquid. This is because part of the

Fig. 8. Predicted influence of agitation speed on the continuous phase mixing (C ¼T/3, P2 tracer injection).

energy transferred from the impeller is dissipated at the liquid– liquid interface for keeping drops in suspension. Hence, less energy is available for the continuous phase mixing process (lower Fl and Flc as can be seen in Section 4.3.1), which thereby increases the mixing time of the continuous phase. According to Kasat et al. (2008), approximately 50% of the total energy delivered to the system in a stirred reactor was dissipated in the impeller discharge region. However, in a stirred slurry reactor, additional 25% of the energy was dissipated at the solid–liquid interface. At point P the oil phase layer floats on the surface. Along the line PQ, more and more oil are dispersed into the continuous phase with increasing impeller speed. At point Q, complete dispersion occurs. Further increase in impeller speeds leads to a change in the curve slope at point Q, and a continuous decrease in mixing time thereafter. The rotational speed where the curve of mixing time versus impeller speed changes its trend indicates the critical impeller speed for oil dispersion. The Ncd (290 rpm) obtained by this method is found to be within 8% of that (270 rpm) calculated by Criterion 1 mentioned above, suggesting that mixing time can be an effective method to determine Ncd, which is very useful for opaque stirred tanks. Similar liquid mixing time behavior was also widely observed in solid–liquid (Raghav Rao and Joshi, 1988) and gas–liquid–solid systems (Rewatkar et al., 1991).

5. Conclusion The macro-mixing process of the continuous phase in the presence of immiscible oil drops in a baffled stirred tank driven by a Rushton turbine has been simulated by means of fully predictive CFD approaches. The flow field is calculated using the RANS technique with both the isotropic standard k−ε model and the anisotropic EASM based on an Eulerian–Eulerian approach. The predicted homogenization curves and mixing time by the k−ε model and the EASM have been compared with the LES predictions and experimental data. The accuracy of the predicted homogenization curves by the EASM and the k−ε model is found to be closely related to the relative position between tracer injection point and monitoring location. Both the k−ε model and the EASM can capture the effect of the dispersed phase holdup and oil viscosity. The anisotropic EASM predicts homogenization curves and mixing time better than the k−ε model and very close to the LES ones, suggesting that the EASM has a great potential to be an alternative model to simulate the turbulent macro-mixing process in stirred tanks. The appealing advantage of the EASM method lies in using substantially lower computational time to obtain satisfactory results. The variation of the continuous phase mixing time with impeller speed can be an effective method to determine the

D. Cheng et al. / Chemical Engineering Science 101 (2013) 272–282

critical impeller for complete dispersion of oil phase, which is very useful for opaque stirred tanks. Overall, the complex liquid–liquid mixing processes in stirred tanks can be successfully predicted by the anisotropic EASM method. The model is promising in application to large/industrial scale liquid–liquid stirred reactors.

i lam m r t z tip θ

281

radial, tangential or axial directions laminar mixing radial direction turbulent axial direction impeller tip tangential direction

Nomenclature A b BW c c∞ C d d32 D Deff Dmol Fl Flc g H k N N cd N un r r0 R Re Sc Sct t tm T u utip w We z Z

constant impeller blade length, m baffle width, m tracer concentration, g/m3 homogeneous tracer concentration, g/m3 off-bottom distance, m diameter of drop, m Sauter mean diameter, m impeller diameter, m effective diffusion coefficient, m2/s molecular diffusion coefficient, m2/s flow number circulation number acceleration due to gravity, m/s2 liquid height, m turbulent kinetic energy, m2/s2 impeller agitation speed, rpm critical impeller speed for complete dispersion, rpm minimum impeller speed for uniform dispersion, rpm radial coordinate, m radial position of the center of the circulation loop, m radius of the stirred tank, m Reynolds number source term in tracer transport equation turbulent Schmidt number in tracer transport equation time, s mixing time, s tank diameter, m velocity component, m/s velocity of impeller tip, m/s impeller blade width, m impeller Weber number, We ¼ ρc N 2 D3 =η axial coordinate starting from the tank bottom, m vertical distance from tank bottom, m

Greek letters α ε μ μt ν νe γ ρ η Γt θ

dispersed phase holdup turbulent kinetic energy dissipation rate, m2/s3 dynamic viscosity, Pa s turbulent dynamic viscosity, Pa s kinetic viscosity, m2/s turbulent kinetic viscosity, m2/s constant density, kg/m3 interfacial tension, N/m2 turbulent diffusion coefficient, m2/s azimuthal coordinate,(deg)

Subscripts av c d eff

averaged continuous phase dispersed phase effective

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