FLUID DYNAMICS ANALYSIS AND SCALE MODELLING OF MIXING AND AGITATING VESSELS
J. Wu, P. C. Bandopadhayay, L. Pullum and I. Shepherd
Advanced Fluid Dynamics Laboratory, CSIRO, Highett, Graham Rd, Vic 3190, Tel. (03) 96252 6173, Fax (03) 9252 6240, email: Jie.Wu@dbce.csiro.au
Abstract
Mixing and agitation with impellers in a vessel is a common technique found in the alumina industry. The fluid flow in the vessel is primarily a function of the slurry rheology, and the design and operation of the impellers. In general, the fluid flow in a vessel determines the homogeneity of mixing, mixing time, solids residence time, suspension of solids, rate of mass transfer.
Techniques have been developed at CSIRO Advanced Fluid Dynamics Laboratory to investigate slurry fluid flow in mixing vessels using smallscale physical models. Software has been developed using theoretical analysis and experimental data to predict the pumping performance of impellers, suspension of solids and tank flow parameters.
FLUID DYNAMICS ANALYSIS AND SCALE MODELLING OF
MIXING AND AGITATING VESSELS
J. Wu, P. C. Bandopadhayay, L. Pullum and I. Shepherd
1.0 INTRODUCTION
Mixing and agitation with impellers in a vessel is a common technique found in precipitation, desilication and many other processes of an alumina refinery. The slurry fluid flow of a vessel determines the homogeneity of mixing, the rate of mixing, solids residence time, suspension of solids and the rate of mass transfer. A poorly designed or operated vessel can have problems such as existence of dead fluid zones and "roping" of the feeding slurry through the vessel without a full utilisation of the tank volume (hence a shortened residence time), buildup of solids at the tank bottom, excessive scale growth in the wall region and poor mass transfer due to inadequate dispersion of solid particles or gas bubbles.
The objective of this paper is to show how fluid dynamics analytical and physical modelling tools are used to investigate the flows of mixing vessels at CSIRO Advanced Fluid Dynamics Laboratory (AFDL). Stateofart physical modelling methods and measurement facilities have been developed to quantify mixing vessel fluid flow field. Theory and experimental data have been incorporated into Windows software analysis tools. Some new research findings are also presented.
2.0 Mixing Vessel Fluid Dynamics Analysis
2.1 Impeller Pumping Performance Evaluation
An impeller can be characterised by a set of performance curves, i.e. the head vs. flow, power vs. flow and efficiency vs. flow curves. A blade element method is used to calculate the performance curves of an axial flow impeller.
Blade element analysis for an axial flow impeller
To predict the performance of an axial flow impeller, the flow around the blade can be analysed in a cylindrical plane containing the twodimensional flow field around an airfoil section, as shown in Figure 1. The velocities at the inlet (denoted by a subscript _{1}) and the outlet (denoted by a subscript _{2}) of the airfoil include the absolute velocity V observed in the fixed frame of reference, the relative velocity W observed in the rotating frame of reference and the impeller velocity U.
By equating the total tangential force acting on the blade section to the change of the fluid angular momentum and nondimensionalising all the parameters:
(1)
(2)
where
V_{a} is the bulk velocity, Q is the flow rate through the impeller, D is the impeller diameter, d is the hub diameter, S is the velocity shape function, s is the blade solidity. C_{L} and C_{D} are lift and drag coefficients, and they are a function of the angle of attack. For more detailed information, readers are referred to (Wu et al. 1998).
At a given radial position r, the nondimensional swirl velocity distribution e q _{2} at the impeller outlet can be found through iterations using (1) and (2). Usually 6 or 7 iterations are needed to converge with an error less than 0.02 degrees for b_{m}.
Once the swirl velocity distribution at the impeller outlet is found, the performance curves can be calculated (Wu et al. 1998):
where N_{H} and N_{Q} are head and flow numbers respectively, as shown in Figure 2(a) calculated for three different axial flow impellers. The power number can also be determined (Figure 2(b)). The intersection of the resistance curve and the impeller head curve gives the operating point of the impeller operating in the mixing vessel. The resistance curve can be determined experimentally by measuring N_{Q} of an impeller with its performance known (e.g. calculated using the present technique, refer to Wu et al. 1998).
A computer software program based on the present technique was developed and packaged into a CSIRO Windows software Agitator (to be described later). Typically, only a fraction of second is needed to calculate the performance of an impeller using the code. The performance curves predicted using Agitator in Figure 2(a) and (b) are for three axial flow impellers, ie. Lightnin A310, A 4bladed axial flow impeller (AFDL rot4) and 30degreepitched blade turbine (30PBT).
(a)
(b)
Figure 2
c
Performance evaluation of axial flow impellers operating in mixing vessel: (a) head number curve; (b) power number curve. Reynolds number is assumed to be sufficiently high so that flow is turbulent. The resistance curve was obtained using LDV data and found valid for D/T=0.350.5
Impellers 
N_{Q} (calculated) 
N_{Q} (measured) 
P_{0} (calculated) 
P_{0} (measured) 
Lightnin A310 
0.56 
0.56 
0.33 
0.31 
AFDL rot4 
0.45 
0.45 
0.19 
0.16 
4bladed 30PBT 
0.65 
0.64 
0.51 
0.52 
Comparison of prediction and experiments
The flow numbers at the operating points for the three impellers are listed in Table 1. It is readily seen that impellers with a large head number consume more power. The flow numbers integrated from the velocity profiles are listed in Table 1, together with the power numbers measured with the torque transducers. Predictions using Agitator are also listed in the table. Good agreement between predictions and measurements can be seen. Although not shown in this paper, the flow numbers of many other axial flow impellers were also measured in our laboratory and compared well with Agitator predictions.
2.2 Suspension of solids
A key step in suspending solid particles in a vessel is to ensure that the speed of an impeller is sufficiently high to keep solids from settling to the tank bottom. The minimum speed of an impeller required to achieve the justoffbottom suspension is a function of solids particle density, particle size, fluid density and viscosity, solids concentration, the geometry of the impeller and the vessel. A standard method of predicting of the just suspension speed (N_{js}) is to use Zwietering’s correlation (Zweitering 1958):
where n is kinematic viscosity, d is particle diameter, D is impeller diameter, X is solids loading (weight of solids/liquid ´ 100) and r_{s} is particle density, r_{L} is liquid density and Dr=r_{s}r_{L}, S is a nondimensional coefficient which is dependent on impeller type, D/T and C/T. S values of standard impellers can be found in literature (e.g. Nienow 1992).
An extensive review of the technique has been conducted to compare calculation with experimental data published by a number of authors (Buurman et al. 1986, Chapman et al. 1983, Myers et al. 1996, Nienow 1992 and Rao et al. 1988). It was found that to a good degree of accuracy the technique can be used to estimate the justsuspension speed N_{js}. S value in the correlation has to be chosen for calculation. Sometimes this can be difficult, as a particular type of impeller may not have been tested before. To overcome this problem, a new method was developed based correlating S with the impeller flow number, as detailed in the followings: For simplicity, we only consider axial flow impellers, but the result is also applicable to the radial flow impellers. At the justoffbottom suspension condition, the areaaveraged impeller’s discharging axial velocity is:
where N_{Q} is the flow number. It is reasonable to assume that suspension of solids is determined by V_{js}, the just suspension velocity. In other words, N_{js}*N_{Q} is expected to be insensitive to the impeller type.
(b)
Suspension of solids test, T=0.390m, C/T=1/3, D/T=0.42, water
was the fluid filled to a height of 0.390m
Experiments were carried out at AFDL to establish the influence of the flow number on N_{js}. Suspension of spherical glass particles of 1.780 mm in diameter at various concentrations were tested in a 390 mm diameter flat bottom mixing vessel (its detail will be described later). Two impellers (Lightnin A310 (N_{Q}=0.56) and 30PBT (N_{Q}=0.65)) were used in the test. Figure 4(a) shows that N_{js} is a function of solids loading. It can be concluded that N_{js}’s of the two impellers are substantially different. However, N_{js}*N_{Q} data as plotted in Figure 4(a) and (b) for the two impellers are almost identical.
This feature has been used to calculate the minimum speed required to suspend solid particles of nonstandard impellers and the code is incorporated in a CSIRO’s software Agitator.
2.3 CSIRO Windows Package: Agitator
When designing or evaluating mixing systems for the mineral and process industries the engineer is usually confronted with many imponderables e.g. "What is the effect of multiple impellers in this tank?", "How will the system operate in my slurry?", "How will this new impeller design effect my process?". These are not simple issues with simple answers and often the engineer must rely solely on the mixer manufacturers experience for an answer. Agitator attempts to address this problem by providing an easy to use tool capable of predicting the performance of mixing systems equipped with single or multiple axial flow impellers or predict the performance of a particular impeller design. In both cases the suspension to be mixed may be either Newtonian or nonNewtonian. General tank operating properties, e.g. power consumption, flow rates etc. as well as more detailed radial bottom and axial wall velocity profiles are readily calculated for the overall system using algorithms based on an extensive and detailed experimental program conducted at the AFDL. Impeller performance curves and operating conditions are also readily predicted given the impeller’s geometry and the surrounding fluid’s rheology. This last feature uses blade element theory and lift and drag coefficients established for the low Reynolds number flows typical of mixing systems, details of which are given elsewhere in this conference.
Agitator is an objectoriented code that performs three main tasks, (i) provide an online literature resource, (ii) calculate the performance of an axial flow rotor operating in the chosen rheology, (iii) calculate the performance of a mixing tank assembly given the rotor and tank geometries and fluid or suspension properties. This delineation of these tasks is summarised in Figure 4
Agitator's input and outputs
The basic philosophy behind Agitator is that it should be an easy to use tool capable of answering "what if questions" that arise when an engineer has to assess the impact that a new process might have on existing plant, or be required to design new plant, or assess how a new agitator design will fare. As well as encapsulating the algorithms and correlations required to do this extensive online help hyperlinked back to original reports and publications were also deemed essential. Consequently the ubiquitous help button accompanies all user interactions.
Mixing tanks in the mineral industries often have a nonNewtonian component. This is particularly true of the alumina and gold industries. Unfortunately most previous work done on mixing has been for Newtonian systems, and often just Newtonian fluids. Certainly the impeller characteristics quoted by mixer manufacturers will be for a Newtonian system. Furthermore most correlations are for standard configurations, i.e. where the tank’s working height equals its diameters, and where there is a single impeller with a diameter equal to a third of the tank’s diameter. Many plants, however, use multiple impeller systems where the aspect ratio is other than unity.
The effect of fluid rheology and tank configuration has been studied at the AFDL using many different industrial axial flow impellers in several different combinations. Powers and torques were measured and flow visualisation studies and LDV velocity traverses conducted to provide a substantial database of the interaction between rheology and mixer tank configuration.
3.0 Physical Modelling of Mixing and Agitation
3.1 Model Mixing Vessels and Measurement Facilities
Amongst the facilities used at AFDL for mixing vessel research are a large mixing vessel, four intermediate mixing vessels for flow visualisation and measurement, large and small water tunnels.
The intermediate mixing vessel dimension is 390 mm diameter x 1000 mm high and the large mixing vessel is 1070 mm in diameter and 1800 mm high. The tanks can be fitted with 4 standard baffles. Figure 6 is the photo of the large model mixing vessel.
A wide range of measurement facilities are available at AFDL, CSIRO. These include:
 Laser Doppler Velocimetry (LDV),
 Phase Doppler Velocimetry (PDV),
 Particle Image Velocimetry (PIV),
 Nuclei Magnet Imaging (NMI),
 Laser Particle Analizer (LPA),
 gastoliquid dissolved oxygen mass transfer measurement system,
 impeller relative flow imaging system,
 mixing time/residence time measurement system,
 laboratory or onsite rheometers,
 torque, shaft speed, gas holdup and many other vessel parameter measurement instruments.
Mixing vessel at AFDL, a fibre optical LDV probe is mounted on the robotic arm shown.
Axial velocity distribution underneath a downward pumping Lightnin A310 impeller, C/T=1/3, D/T=0.42, N=141rpm, T=1.070m. Measured using the TSI LDV system. Vz is the axial velocity (+ downward), Utip is the impeller tip velocity, r: radius and D the impeller diameter
Timemean velocity distributions were measured underneath a range of axial flow impellers pumping downward in a model mixing vessel rig using a TSI 2DLDV system as shown in Figure 6. Water and polymer water solutions (CMC) at different concentrations were used as fluid medium. A CMC water solution shows shearthinning nonNewtonian rheological property. Because of its transparent nature, its flow velocity can be measured using the LDV technique. Concentration of the solution can be adjusted so that the rheology of a slurry can be modelled. The results plotted in Figure 6 indicate that there was a reduction of the pumping velocity and decrease of the velocity in the wall region when CMC solutions were used, as compared with that when water was used.
3.2 Overall Mass Transfer in Sparged Mixing Tanks
Small scale, geometrically similar, models of agitated sparged mixing tanks can be used to predict the overall flow pattern and impeller shaft power and the performance of much larger tanks. A dynamic method using the measurement of dissolved oxygen concentration as a function of time, when the tank is sparged with gases with different oxygen concentrations, is used to determine the mass transfer rate k_{l} a for a given sparged tank (impeller / tank combinations). k_{l}. is the liquid side mass transfer coefficient a is the interfacial area.
Figure 7
Schematic of the test apparatus
Figure 8 shows the schematic of the experimental apparatus available at the CSIRO Advanced Fluid Dynamics Laboratory for studying the overall mass transfer rate in a sparged mixing tank. Two acrylic experimental sparge tanks, 390 mm diameter x 1000 mm high and 1070 mm diameter x 2000 mm high, are used for mass transfer experiments,. These tanks can be fitted with different wall baffles, impellers and sparging devices to model various configurations of sparged mixing tanks used in the industry. A bank of rotameter tubes are used to monitor the sparge gas flow rate. A variable speed three phase electric motor in conjunction with a speed reducer is used to drive the agitator via an inline torque transducer (ONO SOKKI). The agitator performance is monitored using a 486 PC based data acquisition system with a multichannel analoguetodigital converter. An optoelectronic transducer is used to measure the shaft speed while the inline torque transducer measured the shaft torque. A MILLTRONICS PL425 ultrasonic level monitor is used to measure the level of the free water surface in the tank.
A polarographic dissolved oxygen probe is used to measure the dissolved oxygen concentration (DOC) of the tank liquid. Compressed air and nitrogen are used as the two sparge gases with different oxygen concentrations. The temperature of the sample is measured using thermistor temperature sensor of the dissolved oxygen probe.
The DOC data as function of time is analysed to determine k_{l}.a based on the assumption that the mass transfer is proportional to the difference in the oxygen concentration in the liquid to that for saturated liquid at the same temperature.
dC/dt = k_{l}.a (C^{#}  C) (6)
where C is the DOC at time t and C# is the DOC of saturated liquid.
Or C =(C_{o} – C^{#}) e ^{–} ^{kl.a t} + C^{#} (7)
Figure 8 shows a typical set of mass transfer data obtained in the two different size mixing tanks using geometrically similar axial flow impellers with inshaft sparging. The mass transfer coefficients for the 390 mm tank (with a 130 mm high solidity axial flow turbine) and the 1070 mm diameter tank (with a 360 mm diameter high solidity axial flow turbine) were measured for different shaft speeds and two sparge rates of 0.42vvm and 0.66vvm. Figure YY presents the reduced mass transfer rate k_{l}.a/v_{s}^{0.4} where and v_{s} is the superficial gas velocity as a function of the specific shaft power of the impeller. It can be seen that geometrically similar physical models of different size can be tested to produce similar performance data of mass transfer in sparged tanks.
Figure 8
Reduced mass transfer rate for axial flow impellers in the large and small tank. k_{l}.a/v_{s}^{0.4} as a function of specific power for an axial turbine when tested in both the large (1070 mm diameter) and the small (390 mm diameter) tanks at sparge rates of 0.42vvm and 0.66vvm (D/T = 0.333, Zi/D = 1.0, Zn /D = 0.38 Zu/T = 0.82)
4.0 Concluding remarks
In this paper, examples of applying fluid dynamics to mixing and agitation at CSIRO Advanced Fluid Dynamics Laboratory have been demonstrated. Theoretical analysis has been used to develop software package to predict impeller performance and mixing vessel fluid dynamics. Physical fluid scale modelling and measurement facilities have been used to investigate mixing vessel fluid flows. The research results have been and will continue to be used to provide science and technology support to Australian alumina industry.
Acknowledgment
The authors would like to acknowledge the support from AMIRA P419 project sponsored by the following companies:
Lightnin Mixers  RGC Mineral Sands Ltd 
Normandy Mining 
Queensland Alumina Limited 
Westralian Sands Ltd 
Western Mining Corporation 
Alcoa of Australia 
BOC Gases 

Comalco Aluminium 
Placer Pacific 
ReferenceS
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