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In this study, the flow field of a cylindrical pulp mixing chest equipped with a side-entering impeller was modeled using commercial computational flu...
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Ind. Eng. Chem. Res. 2007, 46, 2172-2179

GENERAL RESEARCH Using Computational Fluid Dynamics Modeling and Ultrasonic Doppler Velocimetry To Study Pulp Suspension Mixing Salwan Saeed, Farhad Ein-Mozaffari,* and Simant R. Upreti Department of Chemical Engineering, Ryerson UniVersity, 350 Victoria Street, Toronto, Ontario, M5B 2K3, Canada

In this study, the flow field of a cylindrical pulp mixing chest equipped with a side-entering impeller was modeled using commercial computational fluid dynamics (CFD) software (Fluent) with the rheology of the pulp suspension approximated using the Herschel-Bulkley model. To validate the model, CFD results for the power and velocity field were compared to experimental data. Ultrasonic Doppler velocimetry (UDV), a noninvasive fluid flow measurement technique for opaque systems, was used to measure pulp suspension velocity. In order to calculate the mixing time, an unsteady state solver in Fluent was applied to monitor the tracer species concentration as a function of time in the tank. The validated CFD model provided useful information regarding the mixing time and the formation of the cavern around the impeller in the mixing of pulp suspension. The size of cavern predicted by the CFD model was in good agreement with that calculated using Solomon’s model. Introduction Mixing is an integral part of the pulp and paper manufacturing process. For stock blending, consistency control, bleaching, chemical generation, and de-inking, effective mixing is vital to successful process results. Agitated pulp stock chests are of great importance as they impact almost all facets of pulp and paper manufacturing. They serve a number of functions. In pulping processes, the chests are used to ensure uniform flow ahead of many operations, including chemical addition in bleaching stages, washers, screens, and cleaners. In papermaking, the chests are used to mix two or more pulp streams, often with wet-end chemicals, dyes, fillers, or additives, as well as to provide a uniform feed of stock to the paper machine. In essence, the chests function as low-pass filters to reduce high-frequency variability in pulp properties, and thus compliment the action of control loops, which only attenuate slow disturbances.1 The attenuation of high-frequency disturbances in mixture composition, fiber mass concentration, and other quality factors ahead of many unit operations are critical to a successful process outcome. The current design of mixing chests is based on limited published information, proprietary criteria held by mixer vendors, and experience with installations. One common design method has been summarized by Yackel,2 and is based on matching the momentum flux generated by an impeller with that needed to achieve smooth surface motion in the vessel. Ein-Mozaffari et al.3 showed that even when complete surface motion is attained considerable stagnant regions exist in the chests below the suspension surface. Current chest design criteria do not allow attainment of ideal mixing.4 Tests made on both the industrial chest and the scale-model chest (a geometric scaledown of the industrial chest) have shown that their dynamic * To whom correspondence should be addressed. Tel.: (416) 9795000, ext 4251. Fax: (416) 979-5083. E-mail: [email protected].

performance is far from ideal, with a significant extent of nonideal flow (short-circuiting, recirculation, and stagnation) possible.5 In a number of cases, the suspension volume involved in active mixing was only 20-40% of that in the chest. These nonideal flows reduce the degree of upset attenuation produced by the chest.6 Since typical disturbances occur at frequencies higher than the cutoff frequencies of paper machine control loops, the disturbances are not fully attenuated and pass through to the process where they impact paper quality and machine runnability. The undesirable flows occur because of the complex rheology of the pulp suspensions. It may be noted that pulp suspensions are non-Newtonian and possess a significant yield stress.7-9 In order to create motion throughout the suspension, the shear stresses imposed on the suspension must be greater than the suspension yield stress. Computational fluid dynamics (CFD) is a useful tool for studying fluid flows, including those of mixing systems. It is particularly powerful where the ability exists to corroborate model results with experimental data. A CFD model can be used to augment design correlation and experimental data. It also provides comprehensive data that are not easily obtained from experimental tests. The CFD simulation provides detailed information on the velocity profile within the chest and allows the identification of the locations of poor mixing regions. Bakker and Fasano10 developed a CFD model for the mixing of pulp suspension in a rectangular chest equipped with a side-entering impeller. A combination of turbulent and laminar flow regimes was used in the tank. To calculate the velocity profiles, they used the rheological data given by Gullichsen and Harkonen7 for pulp suspension. These profiles were not compared against experimental measurements. Roberg11 modeled the flow of pulp suspension in a cylindrical tank with a side-entering impeller using CFD with the pulp suspension treated as a power-law fluid. He assumed laminar flow throughout the agitated chest. However, the CFD model was not validated with experimental

10.1021/ie0607548 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/06/2007

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data. Wikstrom and Rasmuson12 assumed that pulp behaved as a Bingham fluid with laminar flow existing inside a cylindrical mixing chest equipped with a jet nozzle agitator. They showed that the flow field obtained from CFD calculations increasingly deviated from measured velocities (obtained using sonar Doppler velocimetry) as the distance from the impeller increased. They suggested that a Bingham fluid did not fully describe the rheology of the pulp suspension. Ford13 developed a CFD model for a rectangular pulp mixing chest equipped with a Maxflo impeller. The pulp suspension was treated as a Bingham fluid. She found that the power input predicted by the simulations was slightly higher (about 12%) than the measured power. The velocity field obtained from the CFD model was used to obtain the system’s dynamic response to a frequency-modulated random binary input signal. These data were then used as input to a dynamic model developed by Ein-Mozaffari et al.6 that treated flow within the chest as the following two streams: the one bypassing the mixing zone and the one entering it. The CFD model did not capture the mixing dynamics of the mixing chest fairly well in flow situations containing significant bypassing where the model overestimates the extent of mixing in the bypassing flow. This departure was attributed to the rheology of the suspension, which was not fully described by the Bingham model. The objective of this study was to investigate the flow field in a cylindrical pulp mixing chest equipped with a side-entering A-310 impeller (Lightnin Co.) using CFD and ultrasonic Doppler velocimetry. The ultrasonic technique has been in use in the medical field for many years, particularly for imaging. Recently, it has been adopted for use in fluid mechanics and fluid flow measurements. Ultrasonic Doppler velocimetry (UDV) uses pulsed ultrasound echography together with the detection of the instantaneous Doppler shift frequency to measure fluid velocity.14 UDV has the following advantages15,16 over conventional techniques such as laser Doppler anemometry (LDA): (1) an efficient flow-mapping process, (2) applicability to opaque liquids, and (3) a record of the spatiotemporal velocity field. UDV is a noninvasive and nonintrusive method of measuring velocity profiles,17 and can therefore be used to monitor the flow in stirred tanks or pipes without obstructing it. A noninvasive instrument does not breach the wall of the vessel or pipe containing the process medium being examined. “Noninvasive” is often synonymous with the term “noncontacting” or “nonwetted”. “Noninvasive” should not be confused with “nonintrusive”. The latter term means that the sensor does not protrude into the vessel or pipe.17 The ability of UDV to work in opaque fluids makes it applicable for studies of all liquids, emulsions, and slurries and therefore very attractive from an industrial perspective.18,19 Experimental Setup and Procedure A cylindrical tank equipped with a side-entering impeller was used to study the mixing of pulp suspension (Figure 1). Since the mixing tanks in the pulp and paper industry are too big, the most popular mixers are side-entering. They are always less expensive and generally require less power.2,20 Side-entering mixers are used for blending, storage, and off-bottom solid suspension applications. They are commonly employed in major industries such as petroleum, crude oil, gasoline, edible oil, asphalt, and pulp and paper mills. An axial-flow A-310 impeller (Lightnin Co.) 16.5 cm in diameter was used in this study with rotational speeds varying from 1000 to 1750 rpm. Axial-flow impellers have been designed to produce a high flow with low turbulence. They produce more flow per unit power than do

Figure 1. Schematic of the experimental setup. Ultrasound probe locations 1, 2, and 3 are delineated by their x, y, and z coordinates in centimeters.

radial impellers and are more cost-effective in flow controlled operations such as solid suspension and blending. The axialflow impellers (i.e., marine propeller, hydrofoil impeller, jet nozzle agitator) are widely used for the agitation of pulp suspension. Both the tank diameter and the suspension height inside the tank were fixed at 40 cm in all tests. Impeller torque and speed were measured using a rotary-torque transducer with an encoder disk (Staiger Mohilo, Germany). An ultrasound Doppler velocimeter (Signal Processing, Switzerland) was used to measure velocity profiles within the agitated chest. Figure 1 shows the locations of ultrasonic probes at three different points. Probes at locations 1, 2, and 3 were used to measure the velocity profiles across the impeller, below the suspension surface, and along the chest wall, respectively. In this study, the single probe technique was used. This technique uses one probe as both a transmitter and a receiver. The probe emits an ultrasonic pulse at a set frequency. This pulse travels through the liquid, and is reflected by solid particles in the suspension. The same probe receives the echo reflected from the particles. The time delay (td) between the emitted pulse and received echo allows the position of the particle (Γ) along the measurement axis to be calculated as14

Γ)

ctd 2

(1)

where c is the speed of sound in the suspension. The maximum depth that can be measured is a function of the pulse repetition frequency (PRF). The PRF is the frequency of emitted bursts from the probe. The maximum depth is given by21

Γmax )

cTprf 2

(2)

where Tprf is the PRF period. The information that the probe receives from the reflected bursts, such as time delay and frequency shift, allows the ultrasonic velocimeter to determine the velocity profile. The displacement of the particle along the ultrasound beam (∆Γ) is

c ∆Γ ) Γ2 - Γ1 ) (t2 - t1) 2

(3)

If the particle is moving at an angle θ relative to the axis of the ultrasonic beam (Figure 2), its velocity can be measured

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Since the time difference is always very short (mostly less than a microsecond), it is advantageous to replace (t2 - t1) by the phase shift. Phase shift (δ in radians) is the product of the emitted angular frequency (ωe) and the time difference22where

identical to that quoted by the manufacturer. To check the accuracy of our measurements in this study, we calculated the flow number using UDV measurements. The measurement of the flow number for the A-310 impeller in water compared favorably with that reported by the manufacturer (0.55 ( 0.03). The ultrasonic probe can only project velocity profiles for particles that move on the same axis of transmission. In other words, the probe associates a negative velocity with the particles that travel toward it, and a positive velocity with those particles that travel away from it. A softwood pulp suspension at 3.3% fiber mass concentration was used in this work. Pulp suspensions are non-Newtonian fluids. They are continuous fiber networks that possess structure and strength resulting from interactions between neighboring fibers. To create motion throughout the suspension, the shear stresses imposed on the suspension must be greater than the network strength, which is measured as the suspension yield stress.7-9,25-27 Pulp suspensions are often treated as plastic materials; they show little deformation up to the yield stress, and only flow once the yield stress is exceeded. Fiber suspensions also exhibit viscoplastic properties, such as shear thinning.11,25 Therefore, pulp suspension rheology was approximated using the Herschel-Bulkley model,28 which contains a shearthinning parameter and a yield stress:

δ ) ωe(t2 - t1) ) 2πFe(t2 - t1)

τ ) τy + kγ˘ n

Figure 2. Ultrasonic Doppler velocimetry: single probe technique.

using the displacement between two emissions separated in time by Tprf:22

c Γ2 - Γ1 ) v(cos θ)Tprf ) (t2 - t1) 2

(4)

(5)

Fe is the frequency emitted from the transducer. Rearranging eq 5, we get

t2 - t1 )

δ 2πFe

(6)

Consequently, the velocity can be obtained by combining eqs 4 and 6

v)

cFd cδ ) 4πFe(cos θ)Tprf 2Fe(cos θ)

(7)

where Fd is called the Doppler shift frequency (the difference between the emitted frequency and the received frequency). The maximum velocity that can be measured depends on the PRF. The Nyquist theorem determines the maximum velocity (Vmax) that can be measured as23

Vmax )

c 4FeTprf

(8)

Combining eqs 8 and 2 gives

Vmax )

where τ, τy, k, n, and γ˘ are shear stress, yield stress, consistency index, power-law index, and shear rate, respectively. The yield stress of the 3.3% softwood pulp suspension was set to 475 ( 62 Pa as measured by Ein-Mozaffari et al.26 The consistency index was set to 0.001 Pa‚s,13 and the power-law index was set to 0.25.11 Numerical Model The commercial software Fluent was used to calculate the flow field within the mixing chest by solving the conservation of mass and momentum equations. Suspension flow observed in the experiments is primarily laminar.11-13 Although flow near the impeller might be turbulent, the fluctuation velocities diminish quickly in the areas outside the impeller zone due to the fibrous structure of the suspension. Considering the flow as laminar in the entire domain will not impact the flow field significantly. Further, in our study we used the non-Newtonian Reynolds number introduced by Gibbon and Atwood29 and Blasinski and Rzyski30 for pulp suspension given by

NRe ) 10 2

c 8FeΓmax

(9)

Equation 9 shows that the maximum velocity is inversely proportional to the maximum depth. Therefore, a compromise between measurable depth and velocity must be considered when measuring the velocity profile of a suspension. Ultrasonic Doppler velocimetry measures the velocity profile with good accuracies of about 5% for velocity and 1% for position.14 Bouillard et al.23 reported that the measured velocity profiles using UDV in a stirred tank agreed well with those obtained by the microimpeller method, and by the laser Doppler velocimetry technique. Ein-Mozaffari et al.24 measured the axial velocity profile using UDV in front of a Maxflo impeller in water and calculated the impeller flow number, which was

(10)

(NDm)

2

(11)

where m ) xτy/F. The Reynolds number calculated for our mixing system was below 1000 and, therefore, in the laminar regime according to the criterion used to characterize agitated vessels. To be able to solve the conservation equations numerically, all aspects of the process need to be discretized. For this purpose, a three-dimensional geometric mesh was generated. In order to capture the boundary layer flow detail, an increased mesh density was used near the tank wall and the rotating impeller. In order to have a very refined mesh in the vicinity of the blades, sufficient nodes that properly define the curvature of the blades were employed on the impeller edges, and a size function was used to control the mesh growth. This feature allows the mesh elements to grow slowly as a function of the distance from the impeller blades. The mesh for the CFD

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Figure 3. Calculated and measured power input vs impeller speed.

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. The final three-dimensional mesh of the model had 158 608 cells. Computations were carried out using a 3.0 GHz Pentium 4 CPU with 1.0 GB of RAM, and convergence was typically achieved in 4-5 h. To model the geometry of the impeller exactly, a threedimensional simulation must be performed. Several techniques have been used in the literature to treat that kind of problem, namely multiple reference frames (MRF) and sliding mesh (SM) methods. Reviews on the subject can be found in Brucato et al.31 and Deen et al.32 These techniques were designed to capture the motion of a rotating impeller in a stationary tank without requiring any empirical data. In the present work, the MRF technique was used to model the mixing of pulp suspension in a cylindrical tank equipped with a side-entering A-310 impeller. 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, 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. Results and Discussion To validate the model, CFD results for the power and velocity field were compared to experimental data. Figure 3 compares experimental and computational results for power input to the impeller. The power drawn by the impeller was computed using P ) 2πNT, where T is the moment vector about the center of the impeller. These results show good agreement between calculated power input and the experimentally determined values. The average deviation is about 7%. Velocity profiles across the impeller, below the suspension surface, and along the chest wall were measured using the UDV probes at locations 1, 2, and 3, respectively (see Figure 1). For three different locations, Figure 4 shows a comparison of the velocity data from the UDV measurements with the velocities computed using CFD. In this figure Vtip ) πND is the impeller tip velocity. It is observed that the CFD calculations pick up the features of the flow field, and the computed velocities agree with the measured data. How well the suggested model describes the rheology of the suspension is reflected in the deviation

Figure 4. Calculated and measured velocities: (a) in front of impeller at location 1, (b) below the suspension surface at location 2, and (c) along the tank wall at location 3.

between the calculated and measured data. Departure of the simulated and measured velocities may be attributed to the limitations of the rheological model chosen to describe pulp suspension rheology. It may be noted that pulp suspensions display unique rheology that is not well characterized,25,27 especially after the yield point. The accuracy of the measured rheological model parameters is another issue. For instance, some researchers have measured the yield stress of different

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Figure 5. Velocity vectors (m/s) on the plane x ) 0 (impeller plane).

Figure 6. Contours of velocity (V(x,y,z), m/s) on the impeller surface.

types of pulp suspensions at different fiber mass concentrations.7-9,25-27 However, the values reported in the literature for the yield stress are significantly different even for the same pulp. For instance, the yield stresses measured by Ein-Mozaffari et al.26 are about 2 times greater than those reported by Bennington et al.8 for similar pulps. The discrepancy may result from differences in the fiber network made when different rotors were placed into the suspensions. Similar differences in the magnitude of reported yield stress as a function of measured technique have been noted by Bennington.27 Nonetheless, the validated CFD model provides detailed information on the velocity profiles and gives the locations of dead zones inside the chest. Figure 5 shows the velocity vectors on the yz plane, i.e., the

impeller plane. The impeller pushes the suspension toward the opposite wall, with the suspension moving along that wall from the bottom to the top of the chest. The pumping action of the impeller causes the suspension to return on the surface of the chest to the impeller suction. Dead zones (where pulp is stagnant or flows significantly slower than the bulk of the suspension) were identified at the top surface of the suspension, at the bottom of the chest, and near the wall. The velocity of pulp suspension in dead zones was considered to be less than 1% of the impeller tip velocity.12 Figure 6 depicts contours of velocity (V(x,y,z)) on the impeller surface. This figure shows how the velocity increases from the center of the impeller toward the tip of the impeller.

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Figure 8. Tracer concentration vs time for four monitoring points.

Figure 7. Location of the addition point and monitoring points for mixing time. These locations are delineated by their x, y, z coordinates, all in meters. (1) 0.10, 0.10, 0.35 (addition point and monitoring point); (2) 0.07, 0.00, 0.05 (monitoring point); (3) -0.10, -0.10, 0.36 (monitoring point); (4) -0.10, -0.10, 0.20 (monitoring point). Table 1. Cavern Diameters Predicted Using CFD Compared with Those Calculated Using Solomon’s Model cavern diameter (cm) impeller speed (rpm)

Solomon’s model

CFD calculations

1750 1600 1400 1100

21.4 20.5 18.9 16.7

18.5 17.9 16.6 16.0

The CFD model provides very useful information regarding the formation of the cavern around the impeller in the mixing of pulp suspension. Wichterle and Wein33 investigated the mixing of fluids with yield stress, and coined the term “cavern” to describe the well-mixed region that surrounded the impeller. Assuming that a spherical cavern was centered on the impeller with the shear stress along the cavern boundary equal to the yield stress of the fluid, Solomon et al.34 developed the following theoretical model for the cavern diameter:

( ) ( )( ) Dc 3 4P0 FN2D2 ) D τy π3

(12)

where P0 ) P/FN3D5 is the power number and P is power. The term FN2D2/τy on the right-hand side of eq 12 is often called the yield stress Reynolds number.35 A comparison between the cavern diameters calculated using Solomon’s model and those predicted using CFD simulations at different impeller speeds is given in Table 1. The outer boundary velocity for the cavern was considered to be less than 1% of the impeller tip velocity.12 The CFD predictions show good agreement with Solomon’s model with an average deviation of about 10%. Defined as the time taken to attain a certain degree of homogeneity, mixing time is usually combined with power consumption to obtain a measure of mixing system efficiency. Experimentally, it is measured by means of a suitable tracer, whose concentration is measured as a function of time using a suitable detector. Mixing time is widely used to quantify the effect of geometry (mixer and impeller) and operating conditions on mixing quality. Numerous publications are available regarding the mixing time.36 Since different experimental methods and

different methods of calculating mixing time have been used, it is not possible to directly compare data from different sources. Most reported work is on cylindrical tanks equipped with topentering impellers. Uhl and Gray37 reported the effect of fluid density, power, and ratio of impeller diameter to tank diameter on the mixing time of liquids of different densities in cylindrical tanks agitated by a side-entering impeller. Wesselingh38 studied the effect of the off-center angle of the impeller shaft, liquid height, and Reynolds number on the mixing time of water mixed in cylindrical storage tanks with side-entering impellers. In this study, we used the validated CFD model to calculate the mixing time of pulp suspension. An unsteady state solver in Fluent was used to monitor the tracer species concentration as a function of time in the tank. The first step in mixing time calculations was to define the tracer material. For the calculation of mixing time, a tracer material with properties similar to those of the bulk fluid was used. The injection point of the tracer was chosen at x ) y ) 0.1 m, and z ) 0.35 m shown as point 1 in Figure 7. At this point, a cluster of cells was marked and the tracer material was added to these cells in a process known as patching. Four points in the mixing system including the addition point were then defined as monitoring points (Figure 7). At these points, the concentration of tracer was monitored as a function of time. The concentration of tracer was normalized with the volume average equilibrium concentration (the concentration as time approaches infinity). Thus, the mixing system was considered homogeneous when the normalized tracer concentrations at all four monitoring points were equal to 1 (Figure 8). The mixing time was defined as the time required for the normalized tracer concentrations at all four monitoring points to reach 99% of the steady state value. The impeller momentum flux, which is proportional to N2D4, was introduced by Fox and Gex39 for mixing time correlation and used by Yackel2 for designing agitated pulp stock chests. This parameter was correlated with the mixing time (tm) of pulp suspension by Ein-Mozaffari et al.3

tm ) A(N2D4)B

(13)

Figure 9 shows mixing time calculated using CFD as a function of N2D4 for our mixing system. The values of A and B are 17.21 and -1.03, respectively. The values of A and B reported by Ein-Mozaffari et al.3 for 3.3% pulp suspension mixed in a rectangular chest equipped with a Maxflo impeller are 1.36 and -2.56.

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commercial CFD software package (Fluent). Pulp suspension rheology was approximated using the Herschel-Bulkley model, which contains a shear-thinning parameter and a yield stress. The multiple reference frame (MRF) technique was applied to capture the motion of the rotating impeller in the stationary tank without requiring any empirical data. The CFD calculations picked up the features of the flow field, and the computed velocities agreed well with the measured data. Departure of the simulated and measured velocities was attributed to the suspension rheology, which is not fully described by the HerschelBulkley model. CFD results for the power input to the impeller compared well with the experimental data. The validated CFD model was used to estimate the size of cavern surrounding the impeller. The outer boundary velocity for cavern was considered to be less than 1% of the impeller tip velocity. The cavern diameter predicted using the CFD model matched well with that calculated using Solomon’s model. The mixing time calculated from the validated CFD model was found to be a function of the impeller momentum flux. The performance rating was approximately constant in the laminar regime, as has been observed by previous researchers. The CFD simulation provides detailed information on the velocity profiles and gives the locations of dead zones inside the pulp mixing chest. Thus, once validated, CFD models can be effectively used to aid in the design and troubleshooting of pulp mixing chests.

Figure 9. Mixing time vs N2D4 for side-entering A-310 impeller.

Acknowledgment The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. Nomenclature

Figure 10. Performance rating versus mixing time Reynolds number.

The other criterion which is widely used to determine the homogenization efficiency is defined as follows:40-42

Ptm2 ηD3

)f

( ) D2F ηtm

(14)

where η is the apparent viscosity. The expression on the lefthand side is directly proportional to energy divided by the volumetric performance of the mixing equipment and is referred to as the performance rating. D2F/ηtm is the mixing time Reynolds number.40 Equation 14 enables a decision to be made on which agitator exhibits the lowest specific power consumption in mixing a given fluid in a vessel of given volume and required mixing intensity. The values of the dimensionless groups in eq 14 were evaluated using mixing time calculated by CFD, and Figure 10 shows the plot of Ptm2/ηD3 vs D2F/ηtm on a log-log scale. Since the flow regime within the agitated chest is laminar, it can be seen that the performance rating is approximately constant, as has been observed by previous researchers.40-42 However, it was not possible to compare Ptm2/ ηD3 values with those reported in the literature, because there was no geometry similar to ours for which researchers had conducted agitator efficiency studies. Conclusions A CFD model of a cylindrical pulp mixing chest equipped with a side-entering A-310 impeller was developed using a

c ) speed of sound, m/s D ) impeller diameter, m Dc ) cavern diameter, m Fd ) Doppler shift frequency, Hz Fe ) frequency emitted from the transducer, Hz h ) suspension height inside the tank, m k ) consistency index, Pa‚s n ) power-law index, dimensionless N ) impeller rotational speed, s-1 P ) power, W P0 ) power number, dimensionless R ) tank radius, m NRe ) Reynolds number, dimensionless t ) time, s td ) time delay, s tm ) mixing time, s T ) torque, N‚m Tprf ) PRF period, s v ) velocity, m/s Vmax ) maximum velocity, m/s Vtip ) impeller tip velocity, m/s Vx ) velocity component in x direction, m/s Vy ) velocity component in y direction, m/s Vz ) velocity component in z direction, m/s Γ ) depth, m Γmax ) maximum depth, m γ˘ ) shear rate, s-1 δ ) phase shift, rad η ) apparent viscosity, Pa‚s θ ) Doppler angle, deg F ) fluid density, kg‚m-3

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ReceiVed for reView June 13, 2006 ReVised manuscript receiVed November 16, 2006 Accepted February 1, 2007 IE0607548