Using computational fluid dynamics to analyze the performance of the Maxblend impeller in solid-liquid mixing operations

Accepted Manuscript

Using computational fluid dynamics to analyze the performance of the Maxblend impeller in solid-liquid mixing operations Prakash Mishra , Farhad Ein-Mozaffari PII: DOI: Reference:

S0301-9322(16)30413-X 10.1016/j.ijmultiphaseflow.2017.01.009 IJMF 2529

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

16 July 2016 7 January 2017 23 January 2017

Please cite this article as: Prakash Mishra , Farhad Ein-Mozaffari , Using computational fluid dynamics to analyze the performance of the Maxblend impeller in solid-liquid mixing operations, International Journal of Multiphase Flow (2017), doi: 10.1016/j.ijmultiphaseflow.2017.01.009

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Highlights CFD model provided detailed insight into the particle distribution within the stirred tank.

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Optimal stirrer speed was determined as a function of operating conditions.

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Use of baffles significantly improved the solid suspensions with the Maxblend impeller.

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Physical characteristics of solid particles greatly influenced the local mixing quality.

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Effect of the radial concentration gradients was not significant for the Maxblend impeller.

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Using computational fluid dynamics to analyze the performance of the Maxblend impeller in solid-liquid mixing operations Prakash Mishra, Farhad Ein-Mozaffari* Department of Chemical Engineering, Ryerson University, 350 Victoria Street, Toronto, Canada M5B 2K3

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Abstract Solid- liquid mixing is prevalent in several industries due to its wide range of utilization in various unit operations. The goal of this work was to examine the performance of the Maxblend impeller for the particle suspension in a slurry tank employing the computational fluid dynamics

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(CFD) technique. The CFD model was created using the Eulerian and Eulerian (E-E) method, standard k-ε turbulence model, and sliding mesh (SM) technique for simulating the two-phase fluid flow, turbulence effects, and stirrer rotation, respectively. The CFD model was validated by

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comparing and analyzing the results obtained from the generated model to the experimental data in terms of stirrer torque, just suspension speed, and degree of homogeneity obtained from the

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tomography technique. The validated model was utilized to obtain the particle concentration profiles and to determine the local particle distributions attained by the Maxblend impeller. The

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data were utilized to analyze the impacts of various important parameters such as the agitation

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speed, particle concentration, particle diameter, specific gravity of the particle, and the use of baffles on the mixing efficiency of the Maxblend impeller in terms of the extent of homogeneity.

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A new mixing index was adopted to understand the effect of both axial and radial concentration gradients on the mixing performance of the Maxblend impeller. The particle distribution in the slurry reactor furnished with a Maxblend impeller was also assessed through clouding height approach in this study.

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Keywords: Solid-liquid mixing; Computational fluid dynamics (CFD); Maxblend impeller; Homogeneity; Solid suspension; Mixing index. 1. Introduction

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Mixing operations are prevalent in several industries due to their extensive applications. Paul et al. (2004) identified those industries as fine chemicals, pharmaceuticals, agrichemicals, petrochemicals, paints, consumer goods, food processing, water treatment, paper and pulp and minerals processing. Mechanically stirred tanks are employed for various purposes such as

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maximizing homogeneousness of the system taking concentration gradients into considerations (Tatterson, 1991), and improving mass transfer among the different phases. The liquid-solid mixing is a pivotal mixing operation because of its wide range of utilizations in several unit processes and operations (Mak, 1992) such as an activated sludge process, ion-exchange, solid

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catalyzed reactions, suspensions polymerization, dispersions of solid particles, leaching and

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dissolution, precipitation and crystallization, and adsorption and desorption ( Zlokarnik, 2001). Several factors such as quality and quantity of products, mixing time, power consumption,

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and cost effectiveness need to be taken into considerations while designing the mixing vessel and

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setting the operating parameters. Several research works have been reported in the literature to quantify the particle concentration and distribution inside the mixing vessels by experimental

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investigations in terms of just suspension speed (Zwietering, 1958; Yoshida et al., 2012), relative standard deviation (RSD) (Magelli et al., 1991; Barresi and Baldi, 1987), and clouding height (Bittorf and Kresta, 2003; Sardeshpande et al., 2009). Some researchers have employed intrusive (Sharma and Das, 1980; Bamberger and Greenwood, 2004; Shan et al., 2008) and non-intrusive (Guillard et al., 2000; Bolton and Primrose, 2005; Ren et al., 2008; Rodgers et al., 2009;

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Tahvildarian et al., 2011; Movafagh et al., 2016) particle concentration measurement techniques to assess the local mixing quality in solid suspensions. Due to the complex nature and various factors affecting solid-liquid mixing, sometimes it is impractical, time consuming, and expensive to examine and analyze the mixing quality in solid suspensions using these experimental

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methods.

Computational fluid dynamics (CFD) has evolved as an efficient and effective tool to get detailed insight information of complex fluid flow in less time and resources (Kazemzadeh et al.,

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2016a and 2016b; Fathi Roudsari et al., 2016). This technique can be utilized to retrieve detailed information about velocity and concentration profiles in the mixing tanks. Although CFD cannot replace experimental measurements entirely, it can substantially reduce the number of experimental measurements and costs involved.

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It is imperative to understand different approaches in CFD modeling of fluid flow in solid

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suspensions. There are basically two approaches reported in the literature: (a) Eulerian and Lagrangian (E-L) method, and (b) Eulerian and Eulerian (E-E) method (Paul et al., 2004). In E-E

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description of fluid flow, the particle (dispersed) phase is conceived as a fluid (continuous) phase with the assumption that the particle phase interpenetrates and interacts with the liquid phase

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(Gidaspow, 1994). In this model, equations of continuity and motion are solved for both

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(continuous and dispersed) phases while sharing a single pressure between them. Several research works have been reported in the literature to examine and analyze liquid-solid mixing using the E-E approach (Sha et al., 2001; Montante et al., 2001; Ljungqvist and Rasmuson, 2001; Barrue et al., 2001; Khopkar et al., 2006; Montante and Magelli, 2005; Kasat et al., 2008; Hosseini et al., 2010a; Sardeshpande and Ranade, 2012). The E-E approach needs less memory space and computational times compared to the E-L technique. The E-E method can be 4

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employed for the higher particle concentrations. In the E-L approach, tracking of movement of an individual particle is done by solving the transport equation for each particle on an individual basis. However, for liquid phase (continuous phase) the Eulerian method is employed. The momentum transport is assessed by calculating the difference in momentum when each particle

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travels through a control element (Li et al., 2015). Some research works have been reported in the literature to analyze the solid-liquid mixing operations employing the E-L approach (Decker and Sommerfeld, 1996; Derksen, 2006; Tyagi et al., 2007; Srinivasa and Jayanti, 2007; Guha et al., 2007; Waghmare et al., 2011). The reliable results can be obtained by employing this

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approach only for the lower particle concentrations (Sommerfeld and Decker, 2004). The selection of a suitable approach for the particle (dispersed) phase significantly relies on the: (a) particle size (b) volume fraction of the particle (c) response time of the particle

and (d)

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characteristics time scales of the fluid phase (Pekker and Helvaci, 2008).

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In solid-liquid mixing operations, the fluctuations developed due to turbulence need to be taken into considerations while creating a flow model to describe the system. There are several

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models reported in the literature which utilize Reynolds averaging method to evaluate the effect of turbulence (Joshi et al., 2011; Van den Akker, 2010). Some of them are: (a) mixing length

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model (zero-equation modeling), (b) Spalart-Almaras model (one-equation modeling), (c) standard k-ε model (two-equation modeling), (d) renormalization group k-ε model (two-

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equation modeling), (e) realizable k-ε model (two-equation modeling), (f) k-ω model (twoequation modeling), (g) algebraic stress model (two-equation modeling), and (h) Reynolds stress model (seven-equation modeling). To understand and describe the liquid-solid mixing, it is inevitable to evaluate the effect of various forces such as the gravitational force, the buoyant force, the virtual mass force, the lift force, the drag force, and Basset force acting on particles 5

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flowing in liquid medium. Several researchers have taken only the effect of drag force into considerations while developing their models (Fan et al., 2005; Murthy et al., 2007; Panneerselvam et al., 2008; Scully and Frawley, 2011; Tamburini et al., 2013). Numerous drag coefficient correlations have been mentioned in the literature (Clift et al., 1978; Turton and

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Levenspiel, 1986; Thompson and Clark, 1991; Gidaspow et al., 1992; Brown and Lawler, 2003). CFD has been employed by some researchers to quantify the solid particles distribution in terms of just suspension speed (Kee and Tan, 2002; Murthy et al., 2007; Srinivasa and Jayanti,

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2007; Panneerselvam et al., 2008) and clouding height (Micale et al., 2004; Ochieng and Lewis, 2006; Sardeshpande and Ranade, 2012). Some researchers have utilized CFD to examine the effect of solid loading (Altway et al., 2001; Oshinowo and Bakker, 2002), stirrer type (Khopkar et al., 2006; Cokljat et al., 2006), particle shape (Fan et al., 2005; Scully and Frawley, 2011), and

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particle size (Sha et al., 2001; Montante et al., 2001) in solid-liquid mixing operations. Hosseini et al. (2010a) employed CFD to compare the mixing performances of the A100, A200, and A310

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axial-flow impellers. Tamburini et al. (2013) utilized CFD to analyze the solid particles

impeller).

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distribution in fully baffled mixing vessels furnished with the Rushton turbine (a radial-flow

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After a comprehensive literature review, it is essential to mention that no research work has

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been reported regarding the assessment of the local particle concentration in a slurry reactor equipped with a Maxblend impeller through CFD modeling. Hence, the major aim of this paper was to explore and analyze the mixing performance of the Maxblend impeller for the particle suspension in a slurry reactor employing the CFD technique. The solid suspension was characterized in terms of just suspension speed (Njs) and clouding height. The impacts of several important parameters such as solid loading, particle size, impeller speed, use of baffles, and 6

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particle specific gravity were studied to evaluate the effectiveness and efficiency of the Maxblend impeller in terms of the extent of homogeneity and mixing index.

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2. Experimental Setup A cylindrical stirred vessel (transparent and flat bottomed) of 400 mm diameter (T) and 600 mm height (H) was used for carrying out all the experimental tests as shown in Fig. 1. The Maxblend impeller with a diameter of 250 mm was employed in this work. Electrical resistance

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tomography (ERT, ITS P2+, UK) was utilized to measure the distribution of the dispersed phase (Babaei et al., 2015a and 2015b; Hashemi et al., 2016). Glass beads of different sizes were utilized as the particle phase and the tap water as the liquid medium. The liquid height (h) was

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maintained at 47.5 cm in all experimental tests. The experimental setup and procedures have

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been explained in details in our previous paper (Mishra and Ein-Mozaffari, 2016). 3. Mathematical modeling and CFD simulation

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Eulerian and Eulerian (E-E) multiphase fluid model was employed in this study for the formulation of the numerical model. Equations of continuity and momentum were applied for

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solid-liquid phase flow to obtain the resulting transport equations. The standard k-ε model was ) in this

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employed to incorporate the turbulent flow into the CFD model. Drag coefficient (

study was calculated using the equation proposed by Gidaspow et al. (1992). All equations used in this study for CFD modeling have been provided in an electronic annex. In order to model the mixing of solid particles inside the stirred tank, ANSYS FLUENT (V15.07) was employed in this study. The rotation of the Maxblend impeller in the stirred tank

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was simulated using the sliding mesh (SM) technique (Tamburini et al., 2013; Kazemzadeh et al., 2016). Since the SM technique was employed for the simulation, a separate rotating fluid zone was defined around the Maxblend impeller. Grid interfaces were defined to interchange the data. The unstructured tetrahedral grids were employed for discretization of the fluid flow

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domain in order to solve the equations of continuity and momentum. To obtain the precise and reliable flow information, very fine grid elements were created close to the walls. The growth rate of the mesh from the surface of the Maxblend impeller or the vessel wall was ensured by the mesh size function. The grid independence test was carried out to obtain optimal number of

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grids. To achieve this goal, three grid systems were created: 671,059 cells, 1,340857 cells, and 2,801,357 cells. The root mean square (RMS) deviations between the grid systems of 1,340,857 and 2,801,357 cells in terms of the velocity magnitude and the solid volume fraction were less

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than 1.28%. Hence, 1,340,857 grids were used for simulating the fluid flow domain generated by the Maxblend impeller in this study. No penetrations and no-slip boundary conditions were

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considered on the shaft, the bottom, and the vessel wall. Normal velocity on the fluid surface was considered to be zero. Modeling of the near wall zones was accomplished by using the standard

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wall function. All the simulations were carried out at a time step of 0.001 s. The convergence for

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the simulation was attained for all transport equations with the scaled residual values below 10-3. The time required for computation of each simulation was about 60-72 h. HPCVL (High

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Performance Computing Virtual Laboratory) computing facilities were utilized to carry out all the simulations on 1.8GHz CPUs (12 dual core Sun Ultra-spark IV). The test conditions for CFD simulations are presented in Table 1. The impacts of several important parameters such as solid concentration, particle size, stirrer speed, use of baffles, and the particle specific gravity were analyzed to evaluate the effectiveness 8

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and efficiency of the Maxblend impeller in terms of the extent of homogeneity and mixing index. All the results are presented and discussed for baffled conditions unless unbaffled conditions are mentioned. Specific gravity (SG) of particle was considered to be 2.5 for all the results unless

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mentioned otherwise. The contours shown in this work were time-averaged. 4. Results and discussion

It is imperative to mention that the validation of the generated CFD model was done by comparing the results obtained from the simulation to experimental data in terms of stirrer

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torque, just suspension speed (Njs) and the degree of homogeneity. In order to validate the CFD model, four planes corresponding to four tomography planes were created to compare the particle distribution inside the stirred vessel at N = 420 rpm (Fig. 2a). As can be evident from Fig. 2a, the solid particles were uniformly suspended inside the stirred tank at N = 420 rpm. In

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order to understand the solid particles distribution for the non-homogeneous system precisely,

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the results obtained from the CFD simulation was compared to those retrieved from the ERT measurements at N = 240 rpm (Fig. 2b). As can be seen in Fig. 2b, the dispersion of the particles

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were not uniform inside the stirred vessel at N = 240 rpm. The results from the CFD model are

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consistent with the results from the conductivity tomograms. Fig. 3 illustrates the stirrer torque with respect to the stirrer speed for the Maxblend impeller.

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Significant change in the stirrer torque was evident as the particles were suspended inside the stirred vessel for both experimental and computed results. The computed torque was consistent with the experimentally calculated torque. The standard deviation value between computed and measured torque was noticed to be less than 1%.

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Several researchers have adopted the just suspension speed (Njs) method to examine the mixing quality in slurry reactors. The just suspension speed (Njs) is defined as the agitation speed needed to attain the complete (off-bottom) suspension inside the slurry reactors. The average solid volume fraction was measured at different agitation speeds at 1.5 mm above the base of the

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mixing vessel for a horizontal plane by employing the CFD model to determine Njs (Fig. 4). Two tangents (one at the point of minimum slope and another at the point of maximum slope) were drawn to the curve as shown in the Fig. 4. The just suspension speed (Njs) was the stirrer speed in rpm corresponding to the intersection point of these tangents. Mak (1992) and Hosseini et al.

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(2010a) employed the similar approach to estimate Njs. The just suspension speed (Njs) was found to be 215 rpm using this method for the Maxblend impeller. The power number (Po) method was employed to determine Njs from the experimental data. Njs was estimated to be 204

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rpm at the similar conditions for the Maxblend impeller (Mishra and Ein-Mozaffari, 2016). The relative error between these two Njs values was found to be 5%.

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The CFD simulation was utilized to obtain the distributions of solid particles for four

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horizontal planes in order to estimate the extent of homogeneity. It should be noted that these planes were at the same location as those for the tomography planes. The data obtained from the

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CFD modeling was utilized to calculate the level of homogeneity inside the stirred tank. The

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extent of homogeneity was calculated using the following equation (Hosseini et al., 2010b):

Homogeneity

√∑

(1)

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(

̅̅̅̅ )

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where

̅̅̅ and

are solid volume fraction on the given plane, the averaged solid volume

fraction inside the tank, and the number of planes, respectively. Fig. 5 illustrates the impact of the stirrer speed on the extent of homogeneity for the Maxblend impeller. The CFD results, in terms of the level of homogeneity, are consistent with those retrieved from the ERT

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measurements. The standard deviation value between computed and measured homogeneity was found to be less than 2.7%. As can be seen in Fig. 5, the stirrer speed had significant impact on the level of homogeneity for the Maxblend impeller. The level of homogeneity in the mixing tank was improved with an increment in the stirrer speed pointing out the uniform suspensions.

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However, after attaining the maximum level of homogeneity, further raise in the stirrer speed had an adverse effect on the extent of homogeneity. Similar observations have also been reported in other research works (Hosseini et al., 2010a, 2010b; Tahvildarian et al., 2011). Hence, in order to

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achieve an enhanced mixing in the agitated tank, the determination of the optimum stirrer speed

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is crucial in solid-liquid mixing operations.

The CFD model was utilized to create particle concentration contours at different agitation

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speeds. Fig. 6 illustrates the particle concentration contours at various stirrer speeds for the Maxblend impeller. As can be evident from Fig. 6, the solid particles were accumulated at the

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base of the mixing vessel at the lower agitation speeds indicating that the particles were not completely suspended. The solid particles got uniformly suspended at the stirrer speed of 420

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rpm. As the agitation speed increased to 480 rpm, the zones with low particle concentrations were created, which ultimately reduced the homogeneity level in the stirred vessel. In order to understand the impact of the agitation speed on the mixing quality in the stirred vessel precisely, particle concentration contours were generated at four different horizontal planes using the CFD model (Fig. 7). At the stirrer speed of N = 180 rpm, the solid particles were accumulated at the 11

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lower region of the vessel. When the stirrer speed increased to 240 rpm, the solid particles reached to the top planes but they were not uniformly distributed. At the stirrer speed of N = 360 rpm, the solid particles got uniformly distributed pointing out the enhanced mixing in the mixing

presence of the particles at the top of the mixing tank (Fig. 8).

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vessel. Streamlines of the solid particles at the impeller speed of N = 420 rpm, confirmed the

The validated model was utilized to create axial particle concentration profiles at various agitation speeds to quantify the solid particles distribution in the stirred vessel for the Maxblend impeller. The solid volume fractions (

) computed for each horizontal plane was divided by the

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averaged solid volume fraction ( ̅ ) within the vessel to obtain the normalized solid volume fraction. Fig. 9 illustrates the axial concentration profiles at different agitation speeds for the Maxblend impeller. Homogeneity of the system increased as

approached ̅ . When

̅ ,

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the mixing in the stirred vessel resulted in the perfect mixing condition (homogeneity = 1).

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Several researchers have assessed the degree of particle suspensions in terms of clouding height. A distinct solid-liquid interface exists above which the solid particle concentration is

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almost nil for high particles loadings. The height (distance) of the clear interface from the base of the stirred tank is referred to as the clouding height (Bittorf and Kresta, 2003). This distinct

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interface distinguishes the clear liquid and solid suspension zones. In this work, clouding height was estimated by computing the particle concentration contours on a vertical plane using the

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validated CFD model. Fig. 10 shows the clouding heights at N = 150 and 420 rpm. The effect of the stirrer speed on the normalized clouding height was analyzed for the Maxblend impeller. As can be observed in Fig. 11, the normalized clouding height increased with an increment in the agitation speed indicating the increase of the suspension region. The clear liquid region vanished when the normalized clouding height reached the maximum value of 1. Fig 10 shows that the 12

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clear liquid region disappeared at the agitation speed of 420 rpm. Similar observations have been reported in numerous research works for other impellers (Hicks et al., 1996; Micale et al., 2004; Bittorf and Kresta, 2003).

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The Maxblend impellers can be utilized in mixing operations with and without baffles. In this work, the influence of baffles on the solid suspensions was examined using the validated CFD model. Fig. 12 depicts the impact of baffles on the extent of homogeneity for the Maxblend impeller at various impeller speeds for the solid-liquid mixing operation. As can be seen in Fig.

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12, the extent of homogeneity increased with the raise in the agitation speed for both unbaffled and baffled mixing tanks. The peak level of homogeneity achieved for the unbaffled tank was significantly lower than that for the baffled vessel at the same agitation speed. It should be noted that the extent of homogeneity reduced beyond the optimal agitation speed in an unbaffled tank.

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The particles were not dispersed uniformly throughout the unbaffled stirred vessel due to the formation of the vortex and dead zones at the higher stirrer speeds (Fig. 13). Baffles are

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essentially fixed to avoid the creation of vortex and dead zones, and to control the fluid flow

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inside the stirred vessel (Paul et al., 2004). It can be concluded that the use of baffles improved the solid suspensions inside the stirred tank equipped by a Maxblend impeller.

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Some researchers utilized the axial concentration profiles to determine the extent of

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suspension for the liquid-solid mixing operations. However, the effect of the radial concentration gradients on the solid suspensions needs to be taken into considerations as well to understand the local mixing quality precisely inside the mixing tank (Michelletti et al., 2003). The validated CFD model was employed to generate velocity vectors to understand the flow pattern inside the mixing vessel furnished with the Maxblend impeller. As can be evident from Fig. 14, the Maxblend impeller generated both axial and radial flow patterns. A new mixing index was 13

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proposed in this work to quantify the solid suspensions in the stirred vessel taking the effects of both radial and axial concentration gradients into account:

where

̅̅̅ and

√∑

̅̅̅̅ )

(

(2)

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Mixing index =

are the solid volume fraction at each point location, averaged solid volume

fraction, and the number of contour points, respectively. The CFD model was utilized to obtain four vertical planes at spacing of 900. Then, 20 point locations were selected in each vertical

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plane. The total number of contour points selected in this study was 80. Similar approach was adopted by Carletti et al. (2014) to quantify the solid distributions in the mixing vessel equipped with PBT impellers using ERT. However, Carletti et al. (2014) selected mean conductivity values at 1264 pixels instead of solid volume fraction to develop the mixing index. Fig. 15

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illustrates the influence of the stirrer speed on the mixing index defined by Equation (2). It can

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be noted that the mixing index decreased with the increase of the agitation speed indicating the enhanced mixing in the mixing vessel. However, the mixing quality reduced with further

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increment in the stirrer speed beyond the optimum value. It is essential to mention that the

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mixing index increased at the impeller speed higher than the optimal impeller speed of 420 rpm. This result is in agreement with the results presented in Fig. 5. Hence, it can be concluded that

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the effects of radial concentration gradients are not significant for the Maxblend impeller. Physical characteristics of the particles particularly the average size of the solid particles

significantly impact the mixing quality in the solid suspensions. The impact of the average particle size (diameter) on the extent of homogeneity was analyzed in this paper. As can be observed in Fig. 16, the level of homogeneity increased with a decrease in the particle diameter.

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Fig. 17 demonstrates that the solid particles (111 µm) were distributed uniformly inside the mixing vessel. However, the solid suspension was not uniform for the particles with the average size of size 752 µm. Similar observations have been reported in previous research works (Godfrey and Zhu, 1994; Hosseini et al., 2010a, 2010b; Tahvildarian et al., 2011). The following

on solid-liquid suspensions:

where

and

(

)

]

(3)

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[

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equation developed by Perry and Green (1984) could shed light on the effect of particle diameter

are particle terminal settling velocity, the drag coefficient,

density of solid particle, liquid density, diameter of particle and gravitational constant, respectively. A decrement in average particle size reduces the particle terminal settling velocity

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(Eq. 3). Hence, settling of the finer solid particles would be slower in comparison to that for the

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larger ones indicating that the uniform distribution of the large solid particles would not be easier.

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The solid-liquid suspension is greatly influenced by the specific gravity (SG) of the solid

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particles. The validated CFD modeling was employed to assess the effect of the specific gravity (SG) on the extent of homogeneity for the Maxblend impeller. As depicted in Fig. 18, the level

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of homogeneity reduced with an increase in the specific gravity of the solid particles. The particles with the specific gravity of 5 were accumulated at the base of the tank whereas the particles with the specific gravity of 1.5 were uniformly suspended at the rotational speed of 300 rpm (Fig. 19). The solid particles with a higher specific gravity show more resistance to the fluid flow, which eventually reduced the level of homogeneity. Similar phenomenon was observed by

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Hosseini et al. (2010a) for the A310 impeller. It can be concluded that the particle specific gravity has pronounced influence on the local mixing quality in the stirred tank. The solid particle loading has a significant influence on the particle suspension in the mixing

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vessel. The impact of the solid loading on the level of homogeneity was examined using the developed CFD modeling. The solid particle loading was varied from 5 to 30 wt% in this study. Fig. 20 demonstrates the influence of the solid particle loading on the level of homogeneity at N = 480 rpm. The results clearly show that the highest homogeneity level was obtained at the

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highest solid particle loading of 30 wt%. These CFD results are consistent with the results mentioned by some researchers (Godfrey and Zhu, 1994; Hosseini et al., 2010b). In fact, the density and viscosity of the slurry increase with an increase in the solid particle loading. At the higher solid loadings, the terminal settling velocity of the solid particles changes to the hindered

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settling velocity because of the increment in the density and viscosity of the slurry, interactions among solid particles and the contact of falling particles with the upward flowing fluid (Paul et

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al., 2004). The following correlation developed by Maude and Whitmore (1958) quantifies the

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relationship between the free solid particle settling velocity and the hindered settling velocity in

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(4)

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terms of the solid volume fraction:

where

and

(

)

are terminal settling velocity and hindered settling velocity of particles,

respectively. In Equation (4): at (5a) 16

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at (5b)

where

represents the particle Reynolds number.

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(5c) is the solid volume fraction. The extent

of homogeneity was improved at the higher particle loadings due to the effect of the hindered settling velocity. The uniform distribution of the particles increased with the increment in the

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solid loadings resulting in the improved mixing quality. 5. Conclusions

The performance of the Maxblend impeller was successfully analyzed for the liquid-solid

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mixing operations using the computational fluid dynamics (CFD) modeling. Grid independence

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tests were carried out in order to obtain the optimal number of grids used for the CFD simulation. The developed CFD model provided detailed insight into the particle concentration and

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distribution within the stirred tank. The generated CFD model was validated and verified by comparing the results obtained from the simulation to the experimentally determined values in

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terms of the stirrer torque, just suspension speed, and extent of homogeneity. The results

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obtained from the CFD modeling were consistent with those attained from the ERT measurements. The CFD modeling was then utilized to retrieve the particle concentration profiles and to assess the level of homogeneity and mixing index for the solid suspension inside the mixing vessel. The mixing quality was improved with an increment in the stirrer speed, in terms of extent of homogeneity and mixing index. However, further increment in the stirred speed after achieving the peak of the homogeneity had an adverse effect on the local mixing 17

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quality inside the mixing vessel. The particle distribution inside the mixing tank was also quantified using the clouding height approach. The results clearly showed that the use of baffles significantly improved the particle suspensions in the slurry reactor furnished with the Maxblend impeller. The effect of the radial concentration gradients was analyzed using a new mixing index

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for the slurry reactor. It should be noted that the dispersion of the solid particles was improved with the increase in the solid particle concentration due to the effect of the hindered settling velocity. The results obtained from the CFD simulation clearly demonstrated that the physical characteristics of the particle such as the specific gravity and the particle size had pronounced

Acknowledgements

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influence on the extent of homogeneity in a slurry reactor.

The financial support of the Natural Sciences and Engineering Research Council of Canada

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(NSERC) is gratefully acknowledged.

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Nomenclature

impeller off-bottom clearance, m

c

total number of contour points

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C

D dp

drag coefficient, dimensionless impeller (stirrer) diameter, m

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CD

solid particle diameter, µm

AC

gravitational acceleration, m/s2

H

height of the mixing vessel, m

h

height of the liquid inside the mixing vessel, m

N

impeller (stirrer) speed, s-

Njs

just suspended agitation speed, s-1

nt

a function of particle Reynolds number (Eq. (4)), dimensionless

Po

power number, dimensionless 18

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number of planes

Rep

particle Reynolds number, dimensionless

T

mixing tank diameter, m

Vt

free settling velocity of particle, m/s

Vts

hindered settling velocity of particle, m/s

X

solid particle weight fraction, dimensionless

Xv

solid volume fraction, dimensionless

Xvc

solid volume fraction at each contour point, dimensionless

̅̅̅

average solid volume fraction, dimensionless

Z

height of planes, m

fluid density, kg/m3

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Greek letters

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q

density of solid particle, kg/m3

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Abbreviations

computational fluid dynamics

ERT

electrical resistance tomography

PBT

pitched blade turbine

RSD

relative standard deviation

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CFD

SM

sliding mesh variable frequency drive

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VFD

specific gravity

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SG

References

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dispersions. Asia-Pac. J. Chem. Eng. 7 (4), 572-580. Zlokarnik, M., 2001. Stirring: Theory and Practice. Wiley-VCH, New York. Zwietering, T.N., 1958. Suspending of solid particles in liquid by agitators. Chem. Eng. Sci. 8 (3-4), 244-253.

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List of Tables

Table 1. Tests conditions.

Range

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Variables

Baffled, Unbaffled

Impeller speed

180-600 rpm

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Geometrical configuration

Impeller type

Maxblend

Particle size

111-1,100 microns

Solid concentration

5-30 wt%

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T/8

Specific gravity

1.5-6.0

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Impeller clearance

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List of Figures

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Figure 1. Experimental setup for this study.

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Figure 2a. (a) Conductivity tomograms, and (b) particle concentration contours created by CFD for four different horizontal planes (Maxblend impeller, dp = 209 µm, X = 10 wt%, C = T/8, and

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N = 420 rpm).

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Figure 2b. (a) Conductivity tomograms, and (b) particle concentration contours created by CFD for four different horizontal planes (Maxblend impeller, dp = 752 µm, X = 20 wt%, C = T/8, and

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N = 240 rpm).

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Figure 3. Stirrer torque versus agitation speed for the Maxblend impeller (dp = 209 µm, X = 10

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wt%, and C = T/8).

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Figure 4. Estimation of just suspension speed using averaged solid volume fraction versus

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agitation speed for the Maxblend impeller (dp = 209 µm, X = 10 wt%, and C = T/8).

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Influence of the agitation speed on the extent of homogeneity for the Maxblend

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Figure 5.

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impeller (dp = 209 µm, X = 10 wt%, and C = T/8).

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Figure 6. Particle concentration contours created by CFD at various agitation speeds for the

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Maxblend impeller (dp = 209 µm, X = 10 wt%, C = T/8): (a) N = 120, (b) N = 150. (c) N = 420,

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and (d) N = 480 rpm.

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Figure 7. Employing CFD to create particle concentration contours for four different horizontal

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240, and (c) N = 360 rpm.

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planes at various agitation speeds (dp = 209 µm, X = 10 wt%, and C = T/8): (a) N = 180, (b) N =

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Figure 8. Streamlines of the solid particles for the Maxblend impeller ((dp = 209 µm, C = T/8, X

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= 10 wt%, and N = 420 rpm).

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Figure 9. Axial particle concentration profile calculated using CFD for the Maxblend impeller

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(dp = 209 µm, X = 10 wt%, and C = T/8).

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Figure 10. Clouding height for the Maxblend impeller measured (dp = 209 µm, X = 10 wt%, C =

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T/8): (a) N = 150, and (b) N = 420 rpm.

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Figure 11. Normalized clouding height at different agitation speeds for the Maxblend impeller

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(dp = 209 µm, X = 10 wt%, and C = T/8).

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Figure 12. Influence of baffles on level of homogeneity at different agitation speeds for the

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Maxblend impeller (dp = 209 µm, X = 10 wt%, and C = T/8).

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Figure 13. (a) Particle concentration contours at a vertical plane, and (b) particle concentration

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contours at different horizontal planes for the Maxblend impeller without baffles (dp = 209 µm, X

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Figure 14. Velocity vectors for solid particles for the Maxblend impeller ((dp = 209 µm, X = 10

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wt%, C = T/8, and N = 480 rpm).

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Figure 15. Impact of agitation speed on mixing index for the Maxblend impeller (dp = 209 µm,

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C = T/8, and X = 10 wt%).

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Figure 16.

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Influence of the solid particle diameter on the extent of homogeneity for the

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Maxblend impeller (C = T/8, X = 10 wt%, and N = 300 rpm).

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Figure 17. Particle concentration contours for the Maxblend impeller (C = T/8, X = 10 wt%, and

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N = 300 rpm): (a) dp = 111, and (b) dp = 752 µm.

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Figure 18. Extent of homogeneity versus particle specific gravity for the Maxblend impeller (dp

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Figure 19. Particle concentration contours for the Maxblend impeller (dp = 209 µm, C = T/8, X

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= 10 wt%, and N = 300 rpm): (a) SG = 1.5, and (b) SG = 5.

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Figure 20. Influence of the solid loadings on the extent of homogeneity for the Maxblend

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impeller (dp = 209 µm, C = T/8, and N = 480 rpm).

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Graphical Abstract

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