Using Computational Fluid Dynamics To Study the Dynamic Behavior

Sep 9, 2008 - CFD flow fields were used to obtain the system dynamic response to a tracer ... shear thinning fluids with yield stress such as pulp sus...
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Ind. Eng. Chem. Res. 2008, 47, 7465–7475

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Using Computational Fluid Dynamics To Study the Dynamic Behavior of the Continuous Mixing of Herschel-Bulkley Fluids Salwan Saeed, Farhad Ein-Mozaffari,* and Simant R. Upreti Department of Chemical Engineering, Ryerson UniVersity, 350 Victoria Street, Toronto, Ontario M5B 2K3, Canada

A continuous-flow mixer was designed and built to study the mixing of xanthan gum solution, a pseudoplastic fluid possessing yield stress. The extent of flow nonideality was quantified using a dynamic model that incorporated two parameters: channeling and fully mixed volume in the vessel. Dynamic experiments were made using the frequency-modulated random binary input of a brine solution to determine the magnitude of nonideal flow parameters. The same experiments were simulated using a computational fluid dynamics (CFD) package (Fluent 6.2). CFD flow fields were used to obtain the system dynamic response to a tracer injection applied at conditions identical to the experimental ones. The extents of channeling and effective mixed volume were determined using the CFD model and then compared with the parameters obtained experimentally. Validated CFD flow fields enabled us to effectively monitor the effect of various operating conditions on flow nonideality, to relate flow pattern and cavern dimension to flow nonideality, to compare the efficiency of impellers, and to provide a pictorial synopsis of continuous-flow mixing operation. Introduction Continuous-flow mixing is a vital component to many processes including polymerization, fermentation, wastewater treatment, and pulp and paper manufacturing because of its demonstrated ability to improve product uniformity and minimize shutdown, loading, and unloading times. Pseudoplastic or shear thinning fluids with yield stress such as pulp suspensions, food substances such as ketchup and mayonnaise, paints, cement, pigment slurries, certain polymer and biopolymer solutions, and wastewater sludge are commonly encountered in the aforementioned processes.1,2 Continuous-flow mixers have traditionally been designed on the basis of ideal flow assumption;3 however, dynamic tests conducted on these mixing vessels4 have shown different nonideality parameters quantified by channeling, recirculation, and dead zones. Channeling and the dead volume can contribute up to 60% of the vessel volume.5-7 Most of the studies7-13 conducted on continuous-flow mixing indicated poor mixing conditions identified by the presence of short-circuiting and dead zones. They recommended special remedies such as the use of bottom input and overflow type output,11 employing multiple feed locations,13 and changing the output location and impeller type7 to minimize flow nonideality. A significant improvement in the design of mixing vessels can be achieved by developing models that take into account the actual flow fields in a mixing vessel. Computational fluid dynamics (CFD) made the development of such models possible by simulating impeller motion using such methods as multiple reference frames (MRFs) and sliding mesh (SM).14 In the mixing of non-Newtonian fluids, CFD has been intensively used to investigateflowfields,15-20 validatetheMetzner-Ottoconcept,21-25 monitor the pressure distribution on the impellers,26 observe cavern formation,27-30 and measure the mixing time.31,32 However, the use of CFD to study flow nonideality in continuous-flow mixers has been reported in a few publications. Using residence time distribution, CFD, and visual tracing, Samaras et al.6 studied the hydrodynamics of a continuous * To whom correspondence should be addressed. E-mail: [email protected]. Fax: (416) 979-5083

stirred tank for water. Aubin et al.13 studied the effect of the pumping direction, feeding rate, and number of feed inlets on the mixing quality in a continuously stirred tank using CFD. Ford et al.17 simulated the flow dynamics in a rectangular pulp mixing chest previously measured by Ein-Mozaffari.4 Little information is available in the open literature on the effects of flow nonideality on continuous-flow mixers for shear thinning fluids with yield stress. It is therefore highly desirable to investigate the dynamic behavior of such vessels under realistic conditions, aiming to incorporate flow nonideality parameters (e.g., channeling and dead zones). Building upon Saeed and Ein-Mozaffari’s7 investigation of the dynamic responses of continuous-flow mixers under different operation conditions (including the impeller speed and type, feed flow rate, fluid rheology, and vessel configuration), in this study we aim to simulate the same flow fields and dynamic responses obtained experimentally using a CFD package (Fluent 6.2). Velocity fields obtained from steady-state CFD simulations are used to simulate the dynamic response of the vessel to a tracer injection. CFD dynamic responses are employed to estimate flow nonideality parameters using the Kammer et al.33 method. Experimental Setup The experimental setup previously described by Saeed and Ein-Mozaffari7 was used for this study as shown in the schematic diagram in Figure 1. A flat-bottom cylindrical tank with a diameter of 0.40 m was used as the mixing vessel. The tank was equipped with four baffles spaced at 90° intervals against the vessel wall. The mixing tank was equipped with a top-entering impeller driven by a 2 hp variable speed motor. The impeller torque and speed were measured using a rotary torque transducer with an encoder disk. The fluid height in the tank and the impeller clearance were 0.41 and 0.15 m, respectively. During the experiment, the fluid was pumped from the feed tank through the mixing tank and on to the discharge tank. The dynamic testing was performed by exciting the system and observing its input and output over a specified time interval. A saline solution (as a tracer) was injected through a computercontrolled solenoid valve into the feed stream using a metering pump. The tracer was mixed with the fresh feed solution in the

10.1021/ie800496x CCC: $40.75  2008 American Chemical Society Published on Web 09/09/2008

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Figure 1. Schematic diagram of the continuous-flow mixing process.

feed pump prior to being pumped to the mixing vessel. Conductivity (C) variations in the input and output streams were then measured using flow-through conductivity sensors. A data acquisition system controlled by LABVIEW software was used to record the impeller torque, impeller speed, and both input and output conductivity signals. Xanthan gum solution at 0.5%, 1.0%, and 1.5% mass concentration was used in this study. As a pseudoplastic fluid possessing yield stress, its rheology can be described by the Herschel-Bulkley model:7,29,34 µ)

τy

.

+ K|γ|n-1

(1) γ where µ is the apparent viscosity, γ˙ is the shear rate, τy is the yield stress, K is the consistency index, and n is the flow behavior index. It must be mentioned that for multidimensional flow of non-Newtonian fluids the apparent viscosity is a function of all three invariants of the velocity gradient tensor (the rate of deformation tensor). However, the first invariant is zero for incompressible fluids, and the third invariant is negligible, in particular for shearing flows. It is well accepted to consider the viscosity of non-Newtonian fluids as a function of the second invariant, which is the base of many non-Newtonian models including the Herschel-Bulkley model.34 The non-Newtonian viscosity becomes unbound at a small shear rate for the Herschel-Bulkley fluids. This behavior causes instability during computation.17 Thus, the modified Herschel-Bulkley model was used to avoid the numerical instability.17,20,29,30,32 At low shear stress, xanthan gum solution acts as a very viscous fluid with viscosity µo. When shear stress exceeds yield stress, the xanthan gum behavior is described by a power law fluid. Therefore, the apparent viscosity of xanthan gum is described as follows:29,30,32 .

µ ) µo for τ e τy

[ ( )]

τy + K γ ˙n -

τy µo

(2)

n

for τ > τy (3) γ ˙ Table 1 summarizes the Herschel-Bulkley rheological model parameters of xanthan gum solutions reported by Saeed and µ)

Table 1. Rheological Properties of Xanthan Gum Solutions concn (wt%) 0.5 1.0 1.5

τy (Pa)

K (Pa · sn)

n

µo (Pa · s)

F (kg/m3)

1.789 5.254 7.455

3 8 14

0.11 0.12 0.14

13.300 22.613 32.360

997.36 991.80 989.76

Ein-Mozaffari.7 The density of the xanthan gum solutions was measured using a 25 mL pycnometer. Dynamic Model. The dynamic model developed by EinMozaffari4 for continuous-flow mixers was used in this study. This model is based on the assumption that, once entering the vessel, fluid can follow two paths: (i) the channeling path comprising the first-order transfer function with a delay and (ii) the mixing path comprising the first-order transfer function with a delay and feedback for recirculation. The combined transfer function for flow paths in a continuous time domain can be expressed mathematically as follows:

G)

-T1s

fe + 1 + λ1s

(1 - f)(1 - R) 1-

e-T2s 1 + λ2s

ReT2s 1 + λ2s

(4)

where G is the transfer function of the vessel, f is the portion of stock channeled in the mixing vessel, and R is the portion of the stock recirculated within the mixing zone. λ1 and λ2 are the time constants for the channeling and mixing zones. T1 and T2 are the time delays for the channeling and agitated zones, respectively. Ein-Mozaffari et al.12 identified the recirculation in the industrial mixing vessels. However, they did not observe R in a laboratory-scale pulp mixing tank.3,5 Had the recirculation existed in the mixing vessel, a series of first-order exponential responses would have appeared in the dynamic responses.4,33,35 The effect of recirculation was not observed in the dynamic responses of our laboratory-scale mixing vessel. Thus, R was set to zero in this study: G)

e-T2s fe-T1s + (1 - f) 1 + λ1s 1 + λ2s

(5)

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To describe the performance of the continuous-flow mixer, we employed the parameters f, the degree of channeling in the vessel, and Vfully mixed/V, the ratio of the fully mixed volume in the vessel to the total volume of the vessel. The fully mixed volume is given by5,12,36 Vfully mixed Qλ2(1 - f) ) Vtotal Vtotal

(6)

where Q is the solution flow rate through the mixing vessel. Channeling and the fully mixed volume in the mixing vessel were obtained from experimental data by minimizing the cost function associated with the system transfer function using a numerical method developed by Kammer et al.33 Two distinct stages were used during the minimum search process: an efficient but less accurate least-squares minimization for the optimal delays, followed by an accurate gradient search for all parameters using the sequential quadratic programming (SQP) method.37,38 The identification experiment was performed by exciting the system and observing the input and output conductivities over a time interval. The experimental procedure consists of the following steps:3,35 (1) exciting the system by a rectangular pulse, (2) designing a frequency-modulated random binary signal on the basis of the response of the system to the rectangular pulse by concentrating the excitation energy at frequencies where the Bode plot is sensitive to parameter variations,39 (3) exciting the system by the frequency-modulated random binary signal, and (4) validating the dynamic model with a new input imposed by another random exciting signal to generate a new output and then comparing the measured and predicted outputs.7 CFD Simulation. Fluent 6.2 was used to simulate xanthan gum flow within the mixing vessel. Continuity and momentum equations were solved in a steady-state domain. Boundary conditions imposed on the system included (1) nonslip at the vessel wall and baffles (V ) 0), (2) zero normal velocity (Vn ) 0) at the free surface, (3) an inflow boundary condition for the vessel inlet, defining the inlet velocity from the volumetric flow rate of the feed, i.e, V ) 4Q/πd2 (where Q is the volumetric flow rate and d is the pipe inlet diameter), and (4) an outflow boundary condition for the vessel outlet, implying zero normal diffusive flux for all flow variables (∂φ /∂n ) 0).40 The computational domain was discretized using Gambit 2.2 to generate a three-dimensional mesh. The final three-dimensional mesh of the model generated by Gambit 2.2 and the boundary conditions are presented in Figure 2. About 98% of the cells generated had a skewness smaller than 0.6, indicating an excellent mesh formation, and 70% of the cells were tetrahedral. The mesh for the CFD calculations was fine-tuned on the basis of the velocity profile calculated in front of the impeller, which is a zone of large velocity gradients. The minimum grid size in the mesh was achieved by reducing the size to a final value below which the changes in the velocity profile were insignificant. Mesh refinement was accomplished using the impeller mesh growth factor function. This factor controls the mesh density by allowing the mesh elements to grow slowly as a function of the distance from the impeller blade to the vessel walls. The optimal mesh used in the simulations, which consists of 161 464 cells, captured large gradients of velocity existing below the impeller region. The MRF technique was used to model the geometry of the impeller in the mixing vessel.41 A rotating frame was used for the region containing the impeller, while a stationary frame was used for regions that are stationary containing the tank walls. The momentum equations inside the rotating frame were solved in the frame of the enclosed impeller,

Figure 2. 3D mesh generated by Gambit 2.2 and boundary conditions. Table 2. Operating Conditions and Design Parameters Used in Numerical and Experimental Measurements variable impeller velocity xanthan gum mass concentration xanthan gum flow rate impeller type input/output location (see Figure 3)

range 50-700 rpm 0.5-1.5% 227-896 L/h A200 (pitched blade turbine), A100 (marine propeller), and A310 input, configurations 1 and 2 (13.5 cm, 90°, 38 cm); output, configuration 1 (13.5 cm, 315°, 0 cm); output, configuration 2 (20 cm, 315°, 35 cm)

while those outside the rotating frame were solved in the stationary frame. A steady transfer of information was made at the MRF interface as the solution progressed. The discretized equations were solved using a segregated implicit scheme. CFD simulations were carried out using the followings schemes: (1) second order for pressure interpolation,40 (2) power law for momentum interpolation,42 (3) first order for temporal discretization,40 and (4) SIMPLEC for pressure-velocity coupling.43 The convergence history was monitored for the mass and x, y, and z velocities. The solution was considered converged when the scaled residuals were smaller than 1 × 10-5. Convergence was achieved in approximately 6000 trials. The CPU time required to achieve convergence was 7-8 h. The simulations were performed on a 3 GHz, 2 GB RAM, Pentium 4 computer. The simulations were carried out at the operating conditions summarized in Table 2. Figure 3 shows the input/output locations for two configurations investigated in this study. The locations of the inputs and outputs for these two configurations are listed in Table 2. To simulate the dynamic test, a user-defined function (UDF), written in C programming language, was linked to the Fluent 6.2 solver. This UDF defines the time at which the tracer was continuously injected into the mixing vessel. The UDF was linked to the inlet as a boundary condition that specifies the tracer concentration in the inlet stream. The outlet concentration was recorded for comparison with experimental data later. The

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Figure 3. Two inlet/outlet configurations.

Figure 5. Power number versus Reynolds number for the A200 impeller.

Figure 4. Tracer output response to a frequency-modulated random binary input signal simulated using the CFD model at two different time steps: 0.1 s, black line; 1.0 s, gray line.

mass fraction of the tracer in UDF was set to values analogous to the conductivity of the random binary excitation signal used in the experimental work. To verify the effect of the time step, dynamic data were obtained at 0.1 s. This sampling time was then increased to 1 s. The predicted dynamic responses overlapped as indicated in Figure 4, indicating temporal independency. The time step used in simulating the tracer injection was thus set to 1 s. Results and Discussion The values of the power number (Po) for the A200 impeller were plotted against those of the Reynolds number at different xanthan gum concentrations, as presented in Figure 5. The power number was estimated from the following relationship using the calculated power obtained from the measured torque: Po )

P FD5N3

(7)

where P is the power, F is the fluid density, and N and D are the impeller speed and diameter, respectively. The Metzner-Otto correlation44 was used to obtain the modified Reynolds number (Re) for Herschel-Bulkley fluids: Re )

FN2D2ks τy + K(ksN)n

(8)

where ks is the Metzner-Otto proportionality constant. ks was set to 10.44,45

Figure 5 shows good agreement between the calculated power number using CFD and the experimentally determined values up to Re ) 500. However, there is a deviation at Re > 500 which may be due to the use of the laminar flow model in the transitional flow regime. The use of the laminar flow model in the transitional regime is quite common in numerical modeling of the mixing processes as may be seen from the work of other researchers.25 At low impeller speed (in the laminar regime), the power number is inversely proportional to the Reynolds number. It can be seen that the flow was laminar at Re < 20. To evaluate the CFD ability to predict the dynamic behavior of the continuous-flow mixers, experimental and CFD input/ output conductivity curves for tracer injections were compared against each other. Noise associated with measured input signals was excluded from input signals used in CFD simulations. The output response predicted by CFD demonstrated good agreement with the experimental response at higher impeller speed (Figure 6a,b). However, the signal output computed using CFD deviated from the measured output in flow situations containing significant channeling (at lower impeller speed), quantified by a steep increase in conductivity (Figure 6c). The computed output signal shows an exponential conductivity response, while the experimentally measured response shows a more rapid step change. This is attributed to a high percentage of channeling in the system, with the CFD simulation predicting a greater extent of mixing in the channeling zone. Similar observations were reported in the literature for the mixing of a pulp suspension in a rectangular chest equipped with a side-entering impeller.17,46 Table 3 compares CFD and experimental results for the extent of channeling and fully mixed volume at two different impeller speeds (50 and 700 rpm). It can be seen that increasing the impeller speed decreases channeling and increases the fraction of fully mixed volume in the tank. The same behavior was observed for water by Khopkar et al.11 Figure 7 shows pathlines generated using the CFD model by releasing tracer particles from the inlet to the outlet at N ) 50 and 700 rpm. This figure indicates that the flow generated by the impeller at 50 rpm was not sufficient to disperse the injected tracer homogeneously within the vessel. The tracer particles were not distributed all over the vessel at 50 rpm, thereby suggesting the existence of a high percentage of dead zones and channeling within the mixing vessel. Data listed in Table 3 also show that decreasing the flow rate reduced the extent of channeling and increased

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Figure 7. Pathlines of tracer particles generated by the CFD model for configuration 1 at 1.5% xanthan gum concentration and at (a) N ) 50 rpm and (b) N ) 700 rpm. Figure 6. Input/output signals (experimental data in black and CFD results in gray) at 0.5% xanthan gum solution for the A200 impeller: (a) N ) 500 rpm, (b) N ) 250 rpm, and (c) N ) 50 rpm. Table 3. Effect of the Feed Flow Rate and Impeller Speed on the Extent of Fully Mixed Volume and Channeling Estimated from CFD and Experimental Data dynamic parameter Q ) 896 L/h f Vfully mixed/V Q ) 227 L/h f Vfully mixed/V

experiment

CFD

N ) 50 rpm

experiment

CFD

N ) 700 rpm

0.76 0.14

0.68 0.22

0.04 0.91

0.03 0.92

0.71 0.23

0.61 0.32

0.00 0.91

0.00 1.00

the fraction of fully mixed volume. In fact, increasing the flow rate reduces the mean residence time in the vessel, and this forces the material to leave the vessel faster. Ein-Mozaffari et al.3 found that the ratio of flow induced by the impeller (QP) to the flow through the mixing tank (Q) affects the extent of channeling. This behavior was attributed to competition between the pumping action of the impeller (which entrains the fluid in

Table 4. Effect of the Xanthan Gum Concentration and Impeller Speed on the Extent of Fully Mixed Volume and Channeling Estimated from CFD and Experimental Data dynamic parameter 0.5% Xanthan Gum f Vfully mixed/V 1.5% Xanthan Gum f Vfully mixed/V

experiment

CFD

experiment

CFD

N ) 50 rpm

N ) 500 rpm

0.68 0.21

0.58 0.33

0.09 0.91

0.07 0.94

0.76 0.14

0.68 0.22

0.18 0.83

0.15 0.87

the mixing flow) and the discharge flow (which removes fluid from the vessel). The effect of the xanthan gum concentration on the extent of nonideal flow was also examined. Table 4 compares CFD and experimental data for the percentage of channeling and fully mixed volume at 0.5% and 1.5% xanthan gum concentrations. It can be seen that the rheology of the fluid significantly affects the dynamic performance of the continuous mixing process. Increasing the xanthan gum concentration increases the solution

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Figure 9. Pathlines of tracer particles generated by the CFD model at N ) 150 rpm and 0.5% xanthan gum for configuration 2. Table 5. Effect of the Input/Output Location and Impeller Speed on the Extent of Fully Mixed Volume and Channeling Estimated from CFD and Experimental Data dynamic parameter Configuration 1 f Vfully mixed/V Configuration 2 f Vfully mixed/V

experiment

CFD

experiment

CFD

N ) 150 rpm

N ) 400 rpm

0.47 0.45

0.39 0.57

0.08 0.97

0.07 0.99

0.58 0.39

0.51 0.50

0.14 0.91

0.13 0.94

Table 6. Effect of the Impeller Type and Speed on the Extent of Fully-Mixed Volume and Channeling Estimated from CFD and Experimental Data dynamic parameter Figure 8. Pathlines of tracer particles generated by the CFD model for configuration 1 at N ) 150 rpm and at (a) 0.5% xanthan gum and (b) 1.5% xanthan gum.

yield stress, resulting in an increasing degree of channeling and dead volume as energy dissipation becomes faster at higher yield stress.5 Pathlines generated using the CFD model (Figure 8) also show that, at higher xanthan gum concentration, the system is more susceptible to a high percentage of the channeling and dead volume. The input/output location has a significant effect on flow nonideality in continuous mixing processes.3,5 Compared to configuration 1, configuration 2 was more prone to a high degree of channeling and stagnant zones. Locating the output on the side of the vessel provokes more material to leave the vessel without entering into the mixing zone, as depicted in Figures 8a and 9. However, in configuration 1, the feed is forced to flow through the mixing zone before leaving the vessel. Table 5 summarizes the effect of the input/output location on the channeling and fully mixed volume calculated using both experimental and CFD data. These results show that the extent of nonideal flow can be reduced using the bottom output. Determination of the most effective impeller for a specific mixing application should be based on the understanding of

A100 Impeller f Vfully mixed/V A200 Impeller f Vfully mixed/V A310 Impeller f Vfully mixed/V

experiment

CFD

experiment

CFD

N ) 150 rpm

N ) 400 rpm

0.54 0.38

0.44 0.48

0.17 0.91

0.13 0.98

0.59 0.33

0.48 0.44

0.19 0.88

0.15 0.93

0.49 0.42

0.39 0.48

0.15 0.97

0.10 0.98

process requirements. Impellers share characteristics attributed to pumps, such as the ability to produce flow and velocity head (or shear). A comparison of CFD and experimental data in Table 6 reveals that the A310 impeller was able to reduce channeling and increase the fully mixed volume in the vessel. Many attempts have been made to define the impeller efficiency. The following two efficiencies are usually encountered when distinguishing between impellers.47-51 The pumping efficiency takes into consideration the impeller’s ability to pump fluid in the mixing vessel: ηp ) Fl/Po

(9)

where ηp, Fl, and Po are the impeller pumping efficiency, impeller flow number, and impeller power number, respectively.

Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7471 Table 7. Impeller Power Numbers A100 impeller

A200 impeller

A310 impeller

0.66

1.16

0.40

0.68

1.14

0.42

power number (experiment) power number (CFD)

Table 8. Impeller Flow and Circulation Numbers A100 impeller

A200 impeller

A310 impeller

0.42 0.89

0.47 0.92

0.37 0.77

flow number circulation number

Table 9. Impeller Pumping and Circulation Efficiencies

pumping efficiency, ηP ) Fl/Po circulation efficiency, ηc ) Flc/Po

A100 impeller

A200 impeller

A310 impeller

0.636 1.348

0.405 0.793

0.925 2.139

The circulation efficiency takes into consideration the impeller’s ability to circulate fluid in the mixing vessel: ηc ) Flc/Po

(10)

where ηc and Flc are the impeller circulation efficiency and impeller circulation number, respectively. To compare impeller efficiencies, the impeller power, flow, and circulation numbers must be estimated. Table 7 summarizes the impeller power numbers (eq 7) calculated using the CFD model and experimental data for the A100, A200, and A310 impellers. It can be seen that the power numbers for these three impellers obtained from experimental measurements are in good agreement with those calculated by the CFD model. The flow number can be estimated for axial flow impellers from the following equation using the impeller pumping flow rate, QP:48,52,53 Fl )

QP ND3

)

2π

∫

D⁄2

0

(rUaxial)|z1 dr ND3

(11)

where N is the impeller speed, D is the impeller diameter, r is the radial coordinate, Uaxial is the axial velocity, and z1 is the boundary of the impeller swept volume in the axial direction.8,48,54,55 The circulation number can be defined for axial flow impellers as follows:49,56,57 Flc )

Qc 3

ND

)

2π

∫

r|Uaxial turn

0

(rUaxial)|z1 dr

ND3

(12)

where Qc is the overall liquid circulation generated in the impeller swept volume and includes flow entrained by impeller discharge and r|Uaxial turn is the radius where axial flow velocity reversal (from downward to upward in downward pumping impellers) occurs.48,53 Axial velocity profiles were calculated at a plane exactly below the impeller blades (z1 ) 13.5 cm). Table 8 summarizes the flow and circulation numbers calculated using the CFD model for three impellers studied in this paper. Impeller power numbers (Table 7) were used along with flow and circulation numbers (Table 8) to calculate the pumping and circulation efficiencies for the A100, A200, and A310 impellers (Table 9). The data clearly show that A200 has the lowest pumping and circulation efficiencies, whereas the A310 impeller provides the best pumping and circulation efficiencies. The A100 impeller was reported to be superior to the A200 impeller by

Figure 10. Velocity magnitude contours at 1.5% xanthan gum: (a) N ) 50 rpm, (b) N ) 100 rpm, (c) N ) 150 rpm.

previous investigators,58,59 and the A310 impeller was found to be better than the A200 and A100 impellers.60-63 Motion in pseudoplastic fluids possessing yield stress ceases when impeller-imparted shear stress fails to exceed fluid yield stress. This leads to the formation of a cavern, a well-mixed region around the impeller surrounded by stagnant or slow moving fluid. Saeed and Ein-Mozaffari7 showed that the cavern formation significantly affects the dynamic performance of the continuous mixing processes. In this study, velocity contours (Figure 10) generated by the CFD model were used to analyze the formation of the cavern around the impeller. Velocity contours show the formation of a torus-shaped cavern, which is approximated by a cylindrical cavern around the impeller; therefore, the characteristics of the cavern can be described by the diameter and height of this region. The cavern diameter was measured on a horizontal plane cutting the impeller surface. The cavern boundary is usually defined by the locus of points where the local velocity magnitude is equal to 1% of the impeller tip speed.29,30,64-68 The cavern height was estimated on a vertical plane along the shaft and perpendicular to the impeller. The cavern size was found to increase with an increase in the impeller speed. With respect to the xanthan gum concentration, however, the cavern size was found to decrease. This can be attributed to the effect of yield stress. As increasing xanthan gum concentration increases yield stress, more stagnant regions are formed in the mixing vessel where shear stress fails to exceed yield stress. The cavern size continued to grow to the

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Figure 11. Dimensionless cavern diameter versus dimensionless Po(Rey) for the A200 impeller.

Figure 13. Dimensionless cavern diameter versus dimensionless Po(Rey) for the A100 and A310 impellers.

Figure 12. Dimensionless cavern height versus impeller speed for the A200 impeller.

point where the cavern diameter touched the vessel wall. The cavern height then increased until it filled the whole vessel. These observations were reported by many investigators.58,64,65,69,70 Different models65,71-73 have been proposed in the literature to estimate the cavern diameter, relating the ratio Dc/D to Po(Rey), where Dc is the cavern diameter, D is the impeller diameter, Po is the impeller power number, and Rey is the yield stress Reynolds number (Rey ) FN2D2/τy). A plot of the dimensionless cavern diameter, Dc/D, versus Po(Rey) should give a slope of 1/3, according to Elson’s model.73 This model describes the formation of a cylindrical cavern centered upon the impeller assuming that shear stress at the cavern boundary is equal to fluid yield stress. CFD data is plotted for all three xanthan gum concentrations in Figure 11. The slope of the Dc/D versus Po(Rey) line was 0.24 for the A200 impeller. The observed slopes were lower than expected from Elson’s model.73 However, Galindo and Nienow69 reported a value of 0.25 for the slope of the Dc/D versus Po(Rey) line for the axial-flow A315 impeller. Solomon et al.71 and Elson73 showed that after the cavern reached the tank wall the cavern height increased with the impeller speed as follows: Hc ∝ Np Dc

(13)

Figure 14. Dimensionless cavern height versus impeller speed for the A100 and A310 impellers: (a) before the cavern reached the wall, (b) after the cavern reached the wall.

where p is a function of the impeller type. Figure 12 demonstrates Hc/Dc versus impeller speed for the A200 impeller at three different xanthan gum concentrations. It can be seen that before the cavern touched the wall the ratio Hc/Dc was almost constant at 0.55 for all concentrations, which is equal to the value reported by Solomon et al.71 and Elson73 for the A200 impeller. Once the cavern reached the vessel wall, the cavern height became proportional to the 0.86th power of the impeller

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speed. The value of p given in the literature was also 0.86 for the A200 impeller.71,73 Similar calculations were performed for the A100 and A310 impellers. Figure 13 shows Dc/D versus Po(Rey) on a log-log scale for the A100 and A310 impeller at 0.5% xanthan gum solution. The slopes of both lines for the A100 and A310 impellers were 0.24. These results were in good agreement with those reported in the literature for axial-flow impellers.69 As mentioned earlier, Hc/Dc is approximately constant before the cavern touches the wall. From Figure 14a, it can be seen that the cavern height to cavern diameter ratios were constant at 0.75 and 0.80 for the A100 and A310 impellers, respectively. Elson58 reported that Hc/Dc was constant at 0.75 ( 0.05 for the A100 impeller. No value for this ratio was found in the literature for A310. Once the cavern reached the vessel wall, the cavern height became proportional to the impeller speed on a log-log scale with a slope of p ) 0.76 and 0.70 for the A100 and A310 impellers, respectively (Figure 14b). Elson58 reported a value of p ) 0.76 for the A100 impeller. Figures 11-14 demonstrate that the ratios Dc/D and Hc/Dc for the A310 impeller are greater than those for the A200 and A100 impellers. The noticeable ability of the A310 impeller to enhance vessel dynamics (observed by decreasing channeling and increasing the fully mixed volume) is related to the formation of a cavern. Saeed and Ein-Mozaffari7 and EinMozaffari et al.5 found that the size of the cavern generated around the impeller has a significant effect on the extent of nonideal flow in continuous mixing of fluids with yield stress. As the impeller speed increases, the cavern surface moves toward the tank wall and the fluid surface within the mixing vessel. When the cavern does not reach the vessel wall and fluid surface, a high percentage of the feed stream is channeled easily to the exit without being entrained in the impeller flow. Thus, the continuous mixing system is prone to a high level of nonideal flow. Conclusion A CFD software package (Fluent 6.2) was used to simulate xanthan gum solution flow in a continuous-flow mixer with the rheology of the xanthan gum solution estimated using the Herschel-Bulkley model. The impeller power number computed using the CFD model matched the experimentally measured values very well. The flow field calculated by the CFD model was then employed to obtain the dynamic response of the continuous mixing system to a frequency-modulated random binary input signal. The calculated dynamic responses agreed fairly well with the measured responses. To analyze the performance of the continuous-flow mixer, the percentage of channeling and fully mixed volume were calculated using the dynamic responses. These results show that the performance of a continuous-flow mixer can be improved by increasing the impeller speed, decreasing the feed flow rate, and decreasing the yield stress (by reducing the solution mass concentration). The extent of nonideal flows (e.g., channeling and dead volume) within the mixing vessel can also be reduced by relocating the input/output locations and using the proper type of impeller. It was found that, to minimize the extent of channeling and dead volume in a continuous-flow mixer, an impeller with large pumping and circulation efficiencies such as the A310 impeller should be chosen. The ability of an impeller to improve the mixing dynamics was related to the size of the well-mixed region (cavern) around the impeller.

Acknowledgment The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. Nomenclature C ) conductivity, S d ) inlet pipe diameter, m D ) impeller diameter, m Dc ) cavern diameter, m f ) fraction of channeling, % Fl ) impeller flow number, dimensionless Flc ) impeller circulation number, dimensionless G ) transfer function of the continuous mixing process Hc ) cavern height, m K ) consistency index, Pa · sn ks ) proportionality constant in the Metzner and Otto relationship n ) flow behavior index, dimensionless N ) impeller rotational speed, s-1 P ) power, W Po ) impeller power number, dimensionless Q ) solution flow rate, m3/s Qc ) overall liquid circulation generated in the impeller swept volume, m3/s QP ) impeller pumping flow rate, m3/s R ) fraction of recirculation, % Re ) Reynolds number, dimensionless Rey ) yield stress Reynolds number, dimensionless r, θ, z ) cylindrical coordinates T1 ) time delay for the channeling zone, s T2 ) time delay for the mixing zone, s Uaxial ) impeller axial velocity, m/s V ) velocity, m/s Vn ) normal velocity, m/s V ) solution volume, m3 Vfully mixed ) fully mixed volume, m3 z1 ) boundary of the impeller swept volume in the axial direction, m Greek Letters γ˙ ) shear rate, s-1 ηc ) circulation efficiency, dimensionless ηP ) pumping efficiency, dimensionless µ ) fluid viscosity, Pa · s µo ) fluid yielding viscosity, Pa · s F ) fluid density, kg/m3 λ1 ) time constant for the channeling zone, s λ2 ) time constant for the mixing zone, s τy ) yield stress, Pa

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ReceiVed for reView March 29, 2008 ReVised manuscript receiVed July 4, 2008 Accepted July 12, 2008 IE800496X