8th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries SINTEF/NTNU, Trondheim Norway 21-23 June 2011
CFD11-126
IMPLEMENTATION OF THE QUADRATURE METHOD OF MOMENTS IN A 3D CFD PIPE GEOMETRY FOR GIBBSITE PRECIPITATION Kieran Hutton 1 2
1,3,*
1
, Darrin W. Stephens , Iztok Livk
2
CSIRO Mathematics, Informatics & Statistics, Clayton, VIC 3168, Australia
CSIRO Light Metals Flagship (CPSE)/Parker Centre, Waterford, Perth, WA 6152, Australia 3
CSIRO Light Metals Flagship/Parker Centre, Clayton, VIC 3168, Australia
* E-mail:
[email protected]
Bu C E G J k ka kb L M n
ABSTRACT A three node quadrature method of moments (QMOM) population balance model was implemented in an Algebraic Slip Multiphase model in the ANSYS CFX (release 12) Computational Fluid Dynamics (CFD) package for the simulation of gibbsite precipitation. The population balance model included the effects of agglomeration, molecular growth and secondary nucleation. Additional physics included buoyancy, viscosity and slip-diameter 2-way coupling. The computational domain investigated was a cylindrical pipe with 2.54cm internal diameter and a length of 10,000m with a residence time of 4h thus simulating 3D flow with time marching. The additional physics were added sequentially with the 0th and 3rd moments and average particle diameter being monitored. Response of the model to changes in temperature and initial suspended solids concentration was investigated. The model response to additional physics and variations in initial conditions yielded results consistent with what would be expected from a real precipitation system.
Nucleation Source Term [#/m3 s] Caustic Concentration [kg/m3] Activation Energy [J/mol] Linear Growth Rate [m/s] Rosenberg Healy Parameter [-] Moment Number [-] Aggregation Kernel Breakage Kernel Particle Diameter [m] Suspended Solids Concentration [kg/m3] Particle Number Density [#/m3]
Sub/Superscripts a Aggregation b Breakage G Growth N Nucleation * Saturation INTRODUCTION The precipitation stage of the Bayer Process is very important for the processing of bauxite ore to extract alumina. Approximately 80% of the processing time for a unit of bauxite is spent in the precipitation stage. This delay is in large part caused by the characteristic, slow molecular growth rate of alumina crystals in Bayer liquor. Given that the Bayer process is a very time and energy intensive process, any improvements that can be made to this stage may result in significant environmental and economic benefits. Precipitation of gibbsite is quite a complex process in terms of process response to variations in inputs such as suspended solids, temperature, species concentrations and particle size distribution (PSD). To combat this complexity and predictive difficulty it was deemed necessary to develop a computer model of the system with the aim of predicting its behaviour. The initial development work of Hutton et al was used as a start point for this work (Hutton et al, 2010). This previous
Keywords: Quadrature, ASM, Moments, CFD, Gibbsite, Precipitation
NOMENCLATURE Greek Symbols Fragment Distribution Function [-] β Energy Dissipation Rate [m2/s3] ε Particle Diameter [m] γ Shear Strain Rate [1/s] γ η Dynamic Viscosity [Pa s] Kinematic Viscosity [m2/s] ν Supersaturation Ratio [-] σ Quadrature Weight [#/m3] ω Agglomeration Efficiency [-] ψ Latin Symbols A Dissolved Alumina Concentration [kg/m3]
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K. Hutton, D. Stephens, I. Livk
work outlines the development of a 1D simplified timemarching implementation of the population balance model. The current model incorporates the effects of agglomeration, molecular growth and secondary nucleation as well as having breakage included but not activated.
required refinement there may need to be an additional 40 classes which translates into 40 additional transport equations. However, there is an alternative. Quadrature Method of Moments The quadrature method of moments was first introduced by McGraw (McGraw, 1996). This method results in a dramatic reduction in the computational cost required. This reduction is two-fold. Firstly the number of additional transport equations is reduced, and secondly the computational requirements to calculate the source terms to these equations are reduced. Integration of Equation 1 and 2 above multiplied by the length coordinate to an integer power will yield the moments form of the PBE. This form of the equation reduces the number of transport equations to 6 (for the case of a 3-node model), improving things from a transport equation perspective, but at the same time the actual PSD is lost through this integration. To counter this, a numerical integration technique known as quadratures is employed to close this equation set. Quadratures allow the calculation of the integrals of the PSD without knowing the PSD itself. This technique approximates the PSD with N particle diameters (abscissas, Li) and corresponding weighting factors (ωi), essentially representing the PSD with an N-bar bar chart. The resulting quadrature summation is as follows, k N dmk 1 N = ∑ ωi ∑ ω j L3i + L3j 3 ψ 10 k a (Li , L j ) dt 2 i =1 j =1
Population Balance (PB) modelling is a technique whereby the dynamics of change of a discrete phase, be it droplets or particles, may be captured and/or predicted. The rate of change of particle count due to pure aggregation and breakage is determined using a modified form of the original Population Balance Equation (PBE) (Smoluchowski, 1917). 1 ⎛ ⎞ k a ⎜⎜ L3 − λ3 3 , λ ⎟⎟ dn ( L) L ⎝ ⎠ n⎛⎜ L3 − λ3 = ⎜ 2 dt 2 ∫0 3 3 3 ⎝ L −λ 2 L
(
)
(
)
(
∞
− n(L )∫ k a (L, λ )n(λ )dλ
1 3
) ⎞⎟⎟n(λ )dλ ⎠
(1)
0
∞
+ ∫ β (L λ )k b (λ )n(λ )dv − k b (L )n(L ) L
Equation 1 may be extended to include the effects of molecular growth and nucleation by adding the following terms. dn( L ) dn( L ) (2) =G + δ ( L − Lcrit ) BU dt
dL
(
Typically, the PSD is divided into a number of size bins or classes and the rate of change of the number of particles in each class is calculated based on a range of system specific parameters such as chemical composition or temperature. The biggest issue with this technique is that the PSD must be divided into quite a large number of classes and the equation solved for each class. This may prove to be very computationally expensive depending on the level of accuracy required. Several alternative discretisation techniques have been published in the literature which attempt to address this issue by reducing the number of classes required to capture the dynamics of the process (Batterham et al., 1981; Hounslow et al., 1988; Marchal et al., 1988; Litster et al., 1995). These methods typically involve the PSD being broken up in a non-linear fashion. What is meant by this is that the resulting classes are not evenly distributed. For example, one of the most commonly used discretisation techniques is that developed by Hounslow et al in which the discretisation follows a power law approach where each class contains particles with a mass (averaged over the class) equal to twice the mass of the preceding class (Hounslow et al., 1988). It must also be noted that this technique requires the implementation of a modified PBE. This essentially results in a PSD which has fine detail at the smaller sizes and a rougher level of detail representing the larger classes. Overall, this discretisation method results in a significant reduction in the total number of classes required to represent a given distribution. Despite the reduction in the number of classes required, the application of this technique to CFD still poses a serious problem. For each size class included, there must be an additional transport equation included in the CFD. Depending on the width of the PSD and the
)
N
N
N
i =1
i= j
i =1
− ∑ Lki ωi ∑ψ 10 k a (Li , L j )ω j + ∑ k a (Li )β (Li , L j )ωi N
N
i =1
i −1
(3)
− ∑ Lki k a (Li )ωi + ∑ kLk −1Gωi + Lkcrit BU The additional term, ψ10, is the agglomeration efficiency term which accounts for the fact that not every particleparticle collision results in the successful formation of an aggregate The required number of nodes for acceptable accuracy is dependent on the kernels (physics) and the shape of the PSD (smoothness etc). For this initial implementation, three nodes (N=3) are used in the simulations. Marchisio et al. (2002) had shown for a precipitation problem that this number of nodes had resulted in acceptable accuracy for the computational effort required. Investigation of the effect of the number of nodes on the overall accuracy for this application is required. The result of this reduction to summation terms is a significant reduction in the computational effort required for population balance simulations from both a transport equation and source term perspective. MODEL DESCRIPTION Additional Variables and Mass Fractions The key process variables associated with this gibbsite precipitation model are the suspended solids concentration, dissolved alumina concentration, caustic concentration, temperature and the PSD. Although the concentration of impurities plays a significant role in the process, they are not included in the current model. 2
Implementation of the Quadrature Method of Moments in a 3D CFD Pipe Geometry for Gibbsite Precipitation
Essentially the liquor is taken to be synthetic for the purposes of this work. The suspended solids concentration was added as an additional mass fraction. The relative motion of the solid and continuous phases was captured using the Algebraic Slip Model (ASM), also known as the driftflux model. The slip velocity was calculated using the volume averaged particle diameter and this resulted in a 2-way coupling of the phases. The particle size distribution is not available with this technique and so the moments of the distribution are transported. The 6 moments used in this work were attached to and transported by the solid phase. Buoyancy effects were also included and were calculated using the Stokes settling model (Stokes, 1851). The effect of the total solids volume fraction on the apparent viscosity of the slurry, or mixture viscosity, was calculated using a modified Einstein viscosity model (Einstein, 1908).
The agglomeration efficiency term accounts for the fact that not all particle-particle collisions result in a stable doublet (Ilievski and Livk, 2006). Factors such as shear strain rate, fluid viscosity and the linear growth rate are taken into account which are key factors affecting sticking efficiencies. 0.00042 ψ 10 = (8) ⎛ γ 2 / U 2.25 ⎞ tip −19 1.5 ⎜ ⎟ 1 + 6.25 × 10 η 3 ⎜ (G / L ) ⎟ 10 ⎝ ⎠ The effects of breakage have not been accounted for in this model but breakage has been implemented for future use. Scaling of the moments used in the simulations was required due to large instabilities in the linear solver. It is thought that these instabilities were as a result of the relative magnitudes of the moments and their respective source terms in certain regions of the domain. The typical magnitude range of the moments was about 2223 orders of magnitude from ~1013 to ~10-10. The scaling is achieved by calculating the moments (for boundary and initial conditions) using a micron basis. The scaled moments are then transported by the CFD solver. Scaling using a micron basis reduced the moment magnitude range from the aforementioned 2223 to 7-8 orders of magnitude. The implemented QMOM algorithm uses the unscaled moments (so no change to the various kernels is required), conversion to scaled values is achieved with the scaling factor of 106k18 (where k is the moment number).
(
The dissolved alumina was added as an additional mass fraction. This quantity has zero drift with respect to the continuous phase and as such travels with the fluid. The caustic concentration was added as an additional variable and attached to the continuous phase. Inter-phase mass transfer is an integral part of this system, where the dissolved alumina is transported from the continuous phase to the solid phase due to molecular growth and nucleation. The rate of mass transfer is determined by the rate of change of the third moment, which is representative of the total solids volume fraction.
0.96197C 1+ J
(4) Figure 1: Pipe Precipitator Geometry (not to scale).
Growth of gibbsite crystals is known to be size independent (Misra and White, 1971). The linear growth rate of the solids (for a highly saturated environment) was calculated using the following equation (White and Bateman, 1988),
Figure 1 shows a diagram of the model geometry used in this work. A computational mesh consisting of 3,690,369 nodes, representing an axial spacing of 90 m was used for all simulations. The effect of mesh spacing was not investigated. The results reported were taken from a slice of the geometry at the YZ-plane at x=0, perpendicular to the direction of gravity. This corresponds to a plane tracing the centreline of the pipe. These tests are not backed up by experimental evidence but were conducted to ensure the correct implementation of the PB algorithm which is done by logical arguments and reasoning as well as the obvious metrics such as convergence and stability. Having said that, the model is based on a set of well founded empirical correlations and as such the physics of the process can be assumed to be correct.
2
⎛ − ΔEG ⎞ ( A − A *) G = kG exp⎜ ⎟ 2.5 ⎝ RT ⎠ C
(5)
Secondary nucleation was included using the model of Li et al and takes the following form (Li et al, 2003),
BU = K N σ 2 γ M
(6) Agglomeration was modelled using a combination of a collision kernel and an agglomeration efficiency term. The collision kernel chosen was the standard turbulent hydrodynamic kernel (Smoluchowski, 1917) and takes the form,
k a = 1.29
ε (Li + L j ) ν
)
Geometry and Initial Conditions
Model Kinetic Equations The driving force for precipitation is the supersaturation. Supersaturation is a measure of how much solute is available for transfer to the solid phase. The saturation concentration was calculated using the semi-empirical correlation of Rosenberg and Healy (Rosenberg and Healy, 1996). This correlation takes into account the chemical composition and temperature of the solution and takes the form,
A* =
CFD11-126
(7)
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K. Hutton, D. Stephens, I. Livk
RESULTS AND DISCUSSION Below are presented the results of the test simulations and discussions thereof. Each simulation was run in parallel using 20 cores and took an average computational time of 3 hours.
The initial conditions used in the first testing simulations whereby the extra physics were sequentially added are given below in Table. 1. Table 1: Initial conditions used in test simulations where additional physics are added.
Quantity A C M T γ Utip η
Value 170 240 250 65 350 2.12 (arbitrary) 3x10-3
Units kg /m3 kg / m3 kg / m3 o C s-1 m /s Pa s
The distribution chosen for this test work was obtained from experimental data obtained internally. The moments of this distribution, both raw and scaled, are presented below in Table. 2. These moments are on a moment/kg solids basis. Table 2: Initial raw and scaled moments.
Moment No. m0 m1 m2 m3 m4 m5
Raw (mk) 1.5370x10+11 1.8780x10+06 3.3548x10+01 7.8920x10-04 2.2766x10-08 7.6564x10-13
Scaled (µmk) 1.5370x10-07 1.8780x10-06 3.3548x10-05 7.8920x10-04 2.2766x10-02 7.6564x10-01
Figure 2: Effect of Additional Physics on the volume averaged particle diameter, d43.
Figure 2 above shows the effects of the additional physics on the volume averaged particle diameter. In Figure 2a the PB has been “switched off” and so the average diameter does not change. Addition of precipitation dynamics has the obvious effect of changing the average diameter. It can be seen that d43 increases over the length of the pipe indicating that both growth and agglomeration are dominant over nucleation. Addition of viscosity, Figure 2c , has the effect of lesser increase in d43. This is due to the fact that as the suspended solids concentration is taken into account, the viscosity is increased and the agglomeration efficiency is reduced. Coupling of the slip diameter has little impact on the average diameter. The impact of the slip diameter can only be seen as it goes above the 30µm. This would result in a higher solids concentration at the bottom of the pipe which should, in turn, increase the agglomeration rate and thus increase d43. However, by this time the supersaturation (not presented) is reduced to a point where the agglomeration efficiency is so low as to retard the agglomeration process and thus the slip has no impact on d43.
Addition of Physics In order to test the stability and convergence of the model, extra physics were added sequentially. This was done to isolate any potential issues with specific aspects of the model and also to build confidence in the role of each new set of physics. The 0th and 3rd moments, as well as the volume averaged particle diameter, d43, is presented. The cases below in Figure 2, Figure 3 and Figure 4 are labelled “a” to “d”. These represent the sequential addition of buoyancy effects, precipitation dynamics, viscosity coupling and slip diameter coupling respectively. For cases a, b and c the slip diameter was taken to be equal to constant 30µm. It must be noted that one of the limitations of QMOM and this work is the inability to capture differential settling. That is, the moments and solid phases have a slip velocity calculated from a representative local mean particle diameter. Whilst this may be viewed as a significant limitation, it is an improvement on previous CFD studies where the particle size was assumed to be mono-sized (Li et al., 2003). Additionally, the effects of differential settling are likely to be reduced by particle streaming (small particles following in the wake of large particles). Future enhancements to the model should consider the use of DQMOM or alternative population balance modelling techniques that allow for a distribution of particle velocities as well as length scales. Such models would also need to include the effects such as particle streaming.
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Implementation of the Quadrature Method of Moments in a 3D CFD Pipe Geometry for Gibbsite Precipitation
CFD11-126
reduced d43. This reduced d43 is accompanied by an increased surface area available for growth and nucleation, i.e. there are a greater number of particles. Inclusion of slip diameter coupling, Figure 4d, has the effect of increasing m3 locally, i.e. at the bottom of the pipe. This is caused by the increased settling rate. The effect on the total m3 across the pipe cross-section is negligible. Effect of Process Variables The above testing was conducted at only one set of process conditions. Modelling the effects of varying the system parameters on the final outputs was then necessary to ascertain whether the model correctly predicted the effects of such changes. The operating temperature and initial suspended solids concentration were varied from 65-70oC and 150-250kg/m3 respectively. All other system parameters were the same as that reported in Table 1 and Table 2.
Figure 3: Effect of Additional Physics on the 0th Moment, m0.
Referring to Figure 3, the additional physics have quite a significant effect of the total particle count. It must also be noted that the 0th moments reported here are the scaled moments and should be multiplied by a factor of 1018 to be correct on a #/m3 basis. The effects of buoyancy can clearly be seen in Figure 3a. Switching on the PB has the effect of reducing the total particle count, m0, as particles are agglomerated. Addition of viscosity coupling results in a lower degree of agglomeration, which is due to the increased viscosity and reduced agglomeration efficiency. Addition of slip diameter coupling has the effect of causing more particles to settle out but has little effect on the overall count.
Case 2.1 – Effect of Changing Temperature In this case the simulations were performed at two temperatures, namely 65oC and 70oC with the suspended solids concentration set to 250kg/m3.
Figure 5: Effect of Temperature on the volume averaged particle diameter, d43. a) 65oC, b) 70oC.
Figure 5 above shows the effect of changing the system temperature from 65oC (Figure 5a) to 70oC (Figure 5b) on d43. It can be seen that increasing the temperature results in a larger increase in d43. This can be explained by the fact that the molecular growth rate increases with temperature and as a result the agglomeration efficiency is increased resulting in a higher agglomeration rate.
Figure 4: Effect of Additional Physics on the 3rd Moment, m3.
Figure 4 above shows the effect of the additional physics on the 3rd moment, m3. It can be seen that buoyancy has the effect of increasing m3 towards the bottom of the pipe. This is due to settling of solids on the pipe floor. There is no effect on the total m3 across the pipe cross section. Addition of precipitation dynamics, Figure 4b leads to an increase in m3 along the pipe length. This is due to inter-phase mass transfer as the particles grow. Addition of viscosity coupling, Figure 4c, has only a small positive effect on m3. This effect is due to the increased viscosity, which leads to a
Figure 6: Effect of Temperature on the 0th Moment, m0. a) 65oC, b) 70oC.
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K. Hutton, D. Stephens, I. Livk
It can be seen in Figure 6 that temperature has a significant effect on the total particle count. Increasing the temperature from 65oC to 70oC results in a higher growth rate which then translates to a higher agglomeration efficiency. This has the effect of reducing m0 along the entire pipe.
Figure 9: Effect of Suspended Solids Concentration on the 0th Moment, m0. SS = 150kg/m3.
Figure 7: Effect of Temperature on the 3rd Moment, m3. a) 65oC, b) 70oC.
It can be seen in Figure 7 that increasing the system temperature results in a decrease in the final value of m3. As the temperature increases the saturation concentration decreases and as a result the volume (or mass) of solute that can be transferred to the solid phase is reduced.
Figure 10: Effect of Suspended Solids Concentration on the 0th Moment, m0. SS = 250kg/m3.
The effect of the initial suspended solids concentration on the total particle count is difficult to see in these plots, Figure 10 and Figure 11. This is due to corresponding change in the viscosity and also the fact that the initial conditions are quite different. The relative change in m0 may shed some light on this matter. The relative change in m0 is ~49% and 43% respectively. From this it may be deduced that the increase in suspended solids has a negative impact on the degree of agglomeration. This is due to the changes in viscosity and also the fact that the agglomeration efficiency reduces quicker in the higher solids case.
Case 2.2 – Effect of Initial Suspended Solids Concentration In this case the simulations were performed at two suspended solids concentrations, namely 150kg/m3 and 250kg/m3 with the temperature set to 70oC.
Figure 8: Effect of Suspended Solids Concentration on the volume averaged particle diameter d43.a) 150kg/m3, b) 250kg/m3.
Figure 11: Effect of Suspended Solids Concentration on the 3rd Moment, m3. SS = 150kg/m3.
Figure 8 reveals the effects of changing the initial suspended solids concentration on the average aggregate diameter. It can be seen that increasing the initial SS causes a reduction in the final d43. This may be explained by the fact that since the initial solute concentration is the same in each case and the higher solids concentration yields a higher surface area, the amount of solute that can be deposited per particle is reduced. This causes a reduction in the average diametric increase and also a reduced agglomeration rate.
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growth and aggregation”. AIChE J, Vol. 34, 11, 1821– 1832. HUTTON, K., STEPHENS, D. and LIVK, I. (2010) “QMOM-CFD Model Development for An Idealised Pipe Gibbsite Precipitator”, CHEMECA 2010, Adelaide, Australia, September 26 - 29. ILIEVSKI, D., LIVK, I. (2006) “An Agglomeration Efficiency Model for Gibbsite Precipitation in a Turbulently Stirred Vessel”, Chemical Engineering Science, 61, 6, 2010-2022. LI, TS., LIVK, I., LANE, G., ILIEVSKI, D. (2003) “Dynamic Compartment Models of Uniformly-Mixed and Inhomogeneously-Mixed Gibbsite Crystallisers”, Chem Eng Technol, 26, 3, 369-376. LISTER JD., SMIT, DJ., HOUNSLOW, MJ. (1995) “Adjustable discretised population balance for growth and aggregation”. AICheJ, Vol. 41, 3, 591-603. MARCHAL, P., DAVID, R., KLEIN, JP., VILLERMAUX, J. (1988) “Crystallization and precipitation engineering - I. An efficient method for solving population balance in crystallization with agglomeration”, Chem. Eng Sci, 43, 59–67. MARCHISIO, DL., VIGIL, RD., FOX, RO. (2002) “Quadrature Method of Moments for AggregationBreakage Processes”. Journal of Colloid and Interface Science, 258, 322-334. McGRAW, R. (1996) “Description of Aerosol Dynamics by the Quadrature Method of Moments”, Aerosol Science and Technology, 27, 2, 255-265. MISRA, C., WHITE, ET. (1971) “Crystallisation of bayer aluminium trihydroxide”, Journal of Crystal Growth, 8, 2, 172-178. ROSENBERG, SP., HEALY, SJ. (1996) “A thermodynamic model for gibbsite solubility in Bayer liquors”, Proceedings of 4th International Alumina Quality Workshop, Darwin, NT. Australia. SMOLUCHOSKI, MZ. (1917) “Versuch Einer Mathematischen Theorie Der Koagulationskinetik Kolloider Losunger”. Zeitschrift fur Physikalische Chemie, 92, 129-142. STOKES, GG. (1851) “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums”, Trans Cambridge Philos. Soc., Vol 9, 8–106. WHITE, ET., BATEMAN, SH. (1988) “Effect of caustic concentration on the growth rate of Al(OH)3 particles”, Light Metals, 157–162.
Figure 12: Effect of Suspended Solids Concentration on the 3rd Moment, m3. SS = 250kg/m3.
Figure 11 and Figure 12 show these effects the 3rd moment. Increasing the initial suspended solids concentration is equivalent to increasing the 3rd moment. The end result of this is negligible in that the increase in m3 is governed by the amount of solute available initially. The case of higher m3 however does consume the solute at a greater rate resulting in the higher case achieving steady state sooner. This can be seen from the gradient of the contours of m3. CONCLUSION The results presented here show that the implementation of a population balance model in a 3D CFD domain for gibbsite precipitation has been a success. The sequential addition of extra physics revealed results which are in keeping with what would be expected from a real particulate system. This confirms that the individual physical models are accurate in so far as they add to the overall model a better, more accurate picture of reality. Varying the input parameters, namely temperature and suspended solids concentration also revealed trends that would be obtained from an equivalent change in a real precipitation system. This adds further confidence in the model’s performance and the belief that it is an accurate descriptor of true gibbsite precipitation system behaviour. The next phase of the model development is the application to a stirred precipitator vessel and the acquisition or modification of kinetic parameters to describe a real life precipitation system.
REFERENCES BATTERHAM, RJ., HALL, JS., BARTON, G. (1981) “Pelletizing kinetics and simulation of full scale balling circuits”. Proceedings of the 3rd International Symposium on Agglomeration, Nurnberg, Germany, 136-150. EINSTEIN, A.. (1908) “Eine neue bestimmung der molekul-dimensionen”, Annalew der Physik, Vol 19, 289–306. HOUNSLOW, MJ., RYALL, RI., MARSHALL, VR. (1988) “A discretized population balance for nucleation,
ACKNOWLEDGEMENTS The support of the Parker CRC for Integrated Hydrometallurgy Solutions (established and supported under the Australian Government’s Cooperative Research Centres Program) is gratefully acknowledged.
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