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A nuclei size distribution model including nuclei breakage
Chemical Engineering Science 86 (2013) 19–24
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A nuclei size distribution model including nuclei breakage L.X. Liu a,n, L. Zhou b, D.J. Robinson c, J. Addai-Mensah b a b c
School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia Ian Wark Research Institute, ARC special Research Centre, University of South Australia, Mawson Lakes, Adelaide, SA 5095, Australia CSIRO Process Science and Engineering & Minerals Down Under National Research Flagship Australian Minerals Research Centre, PO Box 7229, Karawara, WA 6152, Australia
a r t i c l e i n f o
abstract
Article history: Received 6 October 2011 Received in revised form 7 March 2012 Accepted 5 April 2012 Available online 14 April 2012
In this work, a nucleation model that includes nuclei breakage/fragmentation is proposed. The model is based on the nucleation model of Hapgood and the Stokes deformation number calculated from the granule dynamic yield strength from the previously reported granule breakage work. It is proposed that breakage or fragmentation of primary nuclei from binder spray will occur if the Stokes deformation number exceeds a certain critical number. In the case where breakage occurs the model for secondary nuclei size distribution is proposed. To validate the model, the characteristics of the primary nuclei formed from nickel laterite ores with diluted sulphuric solutions as a binder were investigated. The nuclei were produced by dropping the binder solution onto a stationary powder bed. The mechanical integrity of the primary nuclei formed, the relationship between the nuclei diameter and binder drop diameter were studied. The Stokes deformation numbers for nickel laterite powders with different particle size in a lab scale drum granulator were calculated and the nuclei size distributions with different nickel laterite feed powders are predicted. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Agglomeration Nuclei size distribution model Nuclei breakage Primary nuclei Secondary nuclei Nickel laterite
1. Introduction Agglomeration of fine mineral particles as a precursor to heap leaching is an important means of enhancing leaching rates and metal recoveries, particularly in processing low grade ores. To fully understand the underlying mechanisms of agglomeration for better design and control of the agglomeration processes, it is necessary to establish a useful, predictive model based on feed and binder liquid characteristics right from the wetting and nucleation stage (Sanders et al., 2003). Useful rate parameters of the agglomeration mechanisms and kinetics may be extracted from appropriate agglomeration experiments and used for the optimization and scale-up, and also the benchmarking of our understanding on real ore agglomeration processes. In the initial stage of agglomeration process, binder liquids are sprayed onto the feed powder to form nuclei for further growth. The nucleation process in which the initial granules are formed, has been shown to be crucial in determining the final agglomerate size distribution (Schaafsma et al., 1997, 2000; Litster et al., 2001; Hapgood et al., 2009). Powder and liquid properties directly affect the nuclei formation and their properties and subsequently affect the final agglomerate properties. Inaccurate information or modeling of the nucleation mechanism directly affects the accuracy of
n
Corresponding author. Tel.: þ61 7 33658591; fax: þ61 7 33654199. E-mail address: [email protected] (L.X. Liu).
0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.04.009
subsequent agglomeration process modeling as it forms the initial conditions for the growth and coalescence model (Hapgood et al., 2009). In addition, the detailed properties of the primary particles/liquid can be captured in the nucleation model and the effect of these properties on the subsequent stage of coalescence is reflected through the nuclei size distribution. While both the mechanisms and modeling of agglomerate growth by coalescence have been studied for different solid-liquid systems by many workers (e.g., Sastry and Fuerstenau, 1970; Kapur and Fuerstenau, 1969; Gluba, 2003; Liu et al., 2000; Hounslow, 1998), nucleation are the least understood mechanism and least accurately modeled (Hapgood et al., 2009). Litster et al. (2001) and Hapgood et al. (2002) studied the liquid distribution using a dimensionless spray flux to describe the nucleation mechanism and Feise et al. (2008) and Van den Dries and Vromans (2004, 2009) investigated the liquid penetration in high shear mixer. Poon et al. (2008) used a mechanistic nucleation model in the 3-D population balance model for nucleation and aggregation and the nucleation model was included into the whole population balance model. However, the majority of the nucleation model used an average nuclei size. Hapgood et al. (2009) and Wildeboer et al. (2005) have developed a generalized nuclei size distribution model based on drop penetration and overlap by assuming no breakage on primary nuclei formed. Van den Dries and Vromans (2004, 2009) summarized the different ways that a liquid can be mixed with powder in high shear agglomeration process. That is, the dispersion mechanism
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where binder drops are smaller than the primary particles and the penetration mechanism where binder drops are larger than the particle size. One of the key penetration nucleation mechanism proposed by Van den Dries and Vromans is the destruction nucleation mechanism in which nuclei breakage was quantitatively proven. In drum agglomeration process, although the shear force and impact forces are lower than these in the high shear agglomeration process, nuclei breakage/fragmentation could still occur as the initial nuclei formed are very much dependent on the binder and primary particle properties. In this work, we propose a nucleation model which is based on the nucleation model developed by Hapgood et al. (2009) but includes the granule breakage model by Liu et al. (2009). The properties of the nuclei formed with nickel laterite minerals and sulphuric acid solution in a packed bed were studied and used to validate the nucleation model.
2. Theory The formation of nuclei from liquid drops in the wetting and nucleation stage of the agglomeration process is based on liquid penetration into a porous powder bed due to capillary forces. The process involves the drop penetration followed by the formation of nuclei. If the nuclei formed are strong enough to withstand the impact forces, the nuclei will then go through the consolidation and coalescence stage where growth occurs. On the other hand, if the nuclei formed are too weak, they will be fragmented into smaller particles. The model presented below will include all the possible scenarios based on the feed and liquid properties.
2.1. Nuclei size from single binder drop penetration from literature When a binder drop hits a porous powder bed, what is the size of the nucleus formed? In other words, what is the nucleation ratio which is defined as the diameter of the nucleus to the binder drop? This ratio is a function of both liquid properties (surface tension and viscosity) as well as powder properties. For a given fluid and powder, reducing the powder porosity creates larger nuclei, which may be due to the fact that liquid is able to spread further if pores are narrow as the capillary force is increased. Van den Dries and Vromans (2009) developed a model to relate the nuclei diameter with binder and nuclei properties at static conditions and assumed that it is equally applicable to dynamic situations in a high shear mixer. Hounslow et al. (2009) developed a model for predicting the nuclei volume based on the surface tension-driven flow at any time t: rffiffiffiffiffiffiffiffiffi! 1fcp t vnuclei ¼ vdrop 1 þ ð1Þ t max fcp with 1=3
t max ¼
18:75 mr 2drop ð1fcp Þ
gdp f3cp
ð2Þ
where vnuclei and vdrop are the volume of the nuclei and drops respectively, rdrop is the drop radius, fcp is the critical packing liquid volume fraction, tmax is the drop penetration time for an entire drop to penetrate, m and g are liquid viscosity and surface tension respectively, dp is particle diameter. tmax can also be calculated from the equation by Hapgood et al. (2002): 2=3
t max ¼
1:35m vdrop
g Ref f e2ef f cos y
ð3Þ
and Ref f ¼
j d32
eef f
3
ð1eef f Þ
ð4Þ
where Reff is the pore radius, d32 is the surface mean diameter of the particles, j is the shape factor of the particles, y is the contact angle, eeff is the effective porosity of the packed powder. With Eq. (1), the nuclei volume at any available penetration time in dynamic situations can be calculated. When a drop fully penetrates into the powder, the time t reaches the maximum so that t is equal to tmax. With that, the maximum nuclei volume is equal to: vnuclei ¼
vdrop
fcp
ð5Þ
Interestingly, if fcp is replaced by the nuclei porosity, the above equation is the same as that proposed by Van den Dries (2009) at static conditions. Therefore, the maximum nuclei volume is only related to the drop volume and the critical packing liquid volume fraction fcp, which is material (powder and binder) and particle size dependent. The lower the fcp value (larger particle size), the higher the vnuclei value. That is, smaller particles will undergo consolidation which results in smaller nuclei size in comparison to larger particles. 2.2. Destructive nucleation mechanism Depending on the powder and binder properties, the destructive nucleation mechanism proposed by Van den Dries and Vromans (2004, 2009) will most likely be applicable for weak nuclei in granulators with higher shearing force. That is, secondary nuclei will be formed from the fragmentation of primary nuclei. The Stokes deformation number Stdef, which is the ratio of kinetic energy over the granule dynamic strength, was used as the criterion for granule breakage. However, only viscous force was used to calculate the dynamic strength of the nuclei by Van den Dries and Vroman and the model did not include the nuclei size distribution after breakage of the primary nuclei. Liu et al. (2009) used both the capillary forces and viscous force to represent the granule dynamic strength dp: " # 1e gcos y 9 1e 2 9pmvp ð6Þ dp ¼ Sb 6 þ 8 dp 16 dp e e Where vp is granule velocity after impact, e is granule porosity, Sb is binder saturation in the pores, which is 100% in the case of wet nuclei, g is binder surface tension. Eq. (6) is equally applicable to calculate the nuclei strength. The Stokes deformation number Stdef can then be used as a measure of primary nuclei breakage: Stdef ¼
rg v2c 2dp
ð7Þ
where vc is the impact velocity in the granulator. Unlike higher shear mixer granulators where the impeller tip velocity is most commonly used as the impact velocity, there is not much report on the most commonly used impact velocity in drum granulator. Liu and Litster (2002) used the peripheral velocity in the drum, which is equal to oDdrum/2 (where o is the drum rotational speed and Ddrum is the drum diameter) as the granule collision velocity and it can be used here to represent the nuclei impact velocity vc. A critical Stdef value of 0.1 between breakage and non breakage was found by Tardos et al. (1997) and the value of 0.2 was found by Liu et al. (2009) with the dynamic strength from Eq. (6). If the Stdef number is higher than the critical value of breakage, we can assume that fragmentation takes place and the extent of nuclei fragmentation is proportional to the Stdef value. If the
L.X. Liu et al. / Chemical Engineering Science 86 (2013) 19–24
Stokes deformation number is lower than the critical Stokes number, one can assume that nuclei breakage/fragmentation is negligible and the model developed by Hapgood et al. (2009) can be used for predicting the nuclei size distribution by including the formation of nuclei from multi-drops. 2.3. Nuclei size distribution model incorporating destruction mechanism Eq. (1) gives a relationship between the nuclei size and the drop size, provided that one drop forms one nucleus. However, in real industrial process, the nuclei size distribution also depends on the spray rate of the binder which determines if the spray drops will overlap to form one nucleus. For the situation where nuclei fragmentation is negligible, Hapgood et al. (2009) have developed a nucleation model for predicting the nuclei size distribution based on the dimensionless spray flux and the nucleation ratio. The detailed derivation of the model can be found in Hapgood et al. (2009). The key features of the nucleation model is that one can calculate the nuclei size distribution from the dimensionless spray flux ca, which is the ratio of the wetted area covered by the nozzle to the spray area in the nucleation zone. That is:
ca ¼
3V 2Addrop
ð8Þ
where V is the volumetric flow rate of the binder, ddrop is the average drop size and A is the spray area. Poisson distribution was then used to predict the dimensionless size distribution in terms of the number of drops used to form the nucleus (Hapgood et al., 2009): n
P n ¼ expð4ca Þ
ð4ca Þ ðn1Þ!
ð9Þ
Where Pn is the probability of n drops forming one nucleus. In Hapgood’s model, a constant relationship between the nuclei diameter and drop diameter is assumed. In this work, Eq. (1) is used to relate the nuclei diameter to the critical packing volume fraction and we also include processes in which drop penetration time tmax is longer than the powder circulation time t. For a nucleus formed from n drops, the relationship between the nuclei diameter dn and the drop diameter ddrop can be obtained: rffiffiffiffiffiffiffiffiffi!1=3 1fcp t dn ¼ nddrop 1 þ ð10Þ t max fcp The above equation results in the conversion of data from the number of drops to nuclei diameter, similar to what was ffi shown pffiffiffiffiffiffiffiffiffiffiffiffiffi in Hapgood’s model. The term ð1 þð1fcp =fcp Þ t max =t Þ1=3 in Eq. (10) is essentially equivalent to the nucleation ratio kd defined in Hapgood’s model (2009) and became ð1=fcp Þ1=3 if the powder circulation time is longer than the drop penetration time. On the other hand, if nuclei fragmentation (i.e., Stdef 4Stdef cri) takes place, one can use the population balance equation with breakage only to track the changes of nuclei size distribution after breakage: Z 1 @nðv,tÞ ¼ Sðu,tÞbðv,uÞnðu,tÞduSðv,tÞnðv,tÞ ð11Þ @t v where n(v,t) is the number density function of the nuclei with volume v( ¼ p/6 d3n), u is nuclei volume, S(u,t) is the selection rate constant, b(v,u) is the breakage function. Since nucleation happens in a much shorter time frame in comparison to the coalescence stage, one can assume that breakage of the primary nuclei, if any, occurs simultaneously. That is, the rate of change with time for n(v) during the nucleation stage can be assumed to be zero. With this assumption, the nuclei size distribution after
incorporating nuclei breakage/fragmentation is: R1 SðuÞbðv,uÞn0 ðuÞdu nðvÞ ¼ v SðvÞ
21
ð12Þ
n0 (u) is used to denote the nuclei size distribution without breakage. As the breakage rate increases with the increase of Stokes deformation number and possibly nuclei size, one can assume that the selection rate S(u) is proportional to the Stokes deformation number Stdef and nuclei volume u, similar to what was used in the work of Tan et al. (2004), i.e., SðuÞ ¼ ðSt def Þp uq
ð13Þ
where p, q are empirical parameters. The breakage function b can be taken as random binary breakage as proposed by Tan et al. (2004), i.e., the breakage function gives uniform probability of all fragment sizes on a mass scale: bðv,uÞ ¼
1 u
ð14Þ
Substituting Eqs. (13) and (14) into Eq. (12), the nuclei size distribution with breakage became: R1 ðSt def Þp uq1 n0 ðuÞdu nðvÞ ¼ v ð15Þ ðSt def Þp vq Since Stdef is only related to the feed and binder properties as well as the impact speed, we can further simplify Eq. (15) as: R 1 q1 0 u n ðuÞdu nðvÞ ¼ v ð16Þ vq Eq. (16) shows that, if breakage does occur, the resulted secondary nuclei size distribution is independent of the Stokes deformation number and the nuclei size distribution after nuclei breakage can be calculated from the primary nuclei size distribution n0 (u), which can be obtained from Eqs. (8)–(10). In summary, the nucleation model proposed can be summarized as follows: (1) Use Eqs. (6) and (7) to calculate the nuclei Stokes deformation number. If Stdef is smaller than Stdef cri, then nuclei breakage is negligible and Eqs. (8)–(10) can be used for the calculation of nuclei size distribution. (2) If Stdef is larger than Stdef cri, fragmentation is significant and one can use Eqs. (8)–(10) and (16) to calculate the nuclai size distribution after breakage. 3. Experimental 3.1. Materials and characterization Nickel laterite minerals (siliceous goethite) were used in the experiments. These nickel bearing minerals consist of quartz, goethite and iron oxides and also reactive, acid consuming clays (magnesium bearing silicates and iron rich clays). Four different size fractions, namely 2 mm ore, 106 mm, 2 mmþ106 mm and 53 mm were used. The 106 mm, 2 mmþ 106 mm and 53 mm size fractions were obtained by screening the 2 mm ore. The particle size distribution of the 2 mm ore was measured using Malvern Mastersizer. The size distribution of the 106 mm and 2 mmþ106 mm were calculated numerically by mixing input fractions. Table 1 summarizes the size distribution parameters for all the materials. The liquid used is 44% (w/w) H2SO4 solution in water. The minerals were found to be soluble in the sulphuric solution and the maximum amount dissoluble is 0.058 (w/w). The contact angle of the 44% H2SO4 solution on siliceous goethite was measured using Sessile drop method and is 101.
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L.X. Liu et al. / Chemical Engineering Science 86 (2013) 19–24
3.2. Ex-granulator experiments—drop penetration experiments Drop penetration experiments were carried out using 44% sulphuric acid (w/w) with the powders listed in Table 1. The powders were packed into a Petri dish with diameter 70 mm and depth of 17 mm and the packing density was measured. The liquid was filled into a syringe and positioned 10 mm above the surface of the powder bed. The drops were released onto powder bed surface by using the sessile drop technique (data physics OCA 20) and the penetration of each drop was videoed and the penetration time was recorded. A needle was attached to the syringe to generate the drops, the average drop size was measured by weighing the liquid in the syringe and the number of drops that produced out of certain amount of liquid. The average volume of the drops was 15.770.2 ml. The drop penetration time was obtained from the recorded video images between the time that the drop was in contact with the powder surface to that when it completely penetrated into the powder bed At each condition, 5 drops were measured and an average penetration time was obtained. The drop spread area (diameter) was measured using a ruler straight after the drop penetrates into the powder bed. The nuclei formed from the ex-granulator experiments were visually observed for their integrity.
the decrease of particle size. Smaller particles form smaller nuclei because capillary forces for smaller particles draw the particles together and therefore the nuclei formed are smaller and denser. Fig. 3 shows the measured drop penetration time versus the mean particle size (d32). The calculated drop penetration times using Eqs. (3) and (4) are also shown in Fig. 3. It is noted that as particle size increases, the rate of decrease of drop penetration time slows down. It is also noted that Eq. (4) predicts lower drop penetration time than experimental values, particularly with powders of small particle sizes. In general, the drop penetration time for the nickel laterite powders are in the order of seconds, which means that full drop penetration of droplets will occur in the drum agglomeration process. Table 2 lists the properties of the primary nuclei that were obtained from ex-granulator experiments. The physical properties of primary nuclei prepared from packed powder bed for siliceous goethite with drop volume of 0.0157 ml and its Stokes deformation number in a dynamic situation in a drum granulator are calculated using Eqs. (6) and (7) and are also listed in Table 2. The impact velocity vc used for calculating the Stokes deformation number is 0.942 m/s, which is half of the peripheral velocity of a
4. Results and discussion 4.1. Ex-granulator experimental results To find out how the nuclei size is related to the drop size at static conditions, the diameters of the ‘‘nuclei’’ after binder drop penetration are measured with different powders. Fig. 1 shows some examples of the nuclei. The ratios of the nuclei diameter to drop diameter are presented in Fig. 2. As can be seen from Fig. 2, the ratio of the nuclei diameter to drop diameter increases rapidly with the increase of primary particle size and then the rate of increase slows down as particle size increases further. If Eq. (5) is used to relate the ratio of nuclei diameter to droplet diameter, one can see that the ratio of dn/dd is proportional to (1/fcp)3, indicating that the critical liquid packing volume fraction increases with
Fig. 2. Ratio of nuclei diameter to binder drop diamter in ex-granulator experiments versus primary particle size.
Table 1 Size distribution parameters of the nickel laterite feed powders. Sample name
Sample description
d50 (mm)
d32(mm)
Siliceous goethite (SG) SG1 SG2 SG3
2 mm 106 mm SG sample, size fraction from 2 mm 2 mmþ 106 mm SG sample 53 mm SG sample
118.5 47.4 366 20.7
30.5 15.7 250 10.5
Fig. 1. Examples of the ex-granulator nuclei.
L.X. Liu et al. / Chemical Engineering Science 86 (2013) 19–24
rotational drum of 300 mm with a rotational speed of 60 rpm. The nuclei velocity vp in Eq. (6) is taken as 15% of the impact velocity and the d50 values were used in the calculations. It was observed that, even with gentle handling, the ‘‘nuclei’’ formed from the siliceous goethite samples SG and SG2 with 44% H2SO4 drops in packed bed were very weak and disintegrated into smaller particles without the formation of dry nuclei. For siliceous goethite with smaller particles (samples SG1 and SG3), weak nuclei are formed. When examining the calculated Stokes deformation numbers in a dynamic situation such as the initial binder spray process in a drum granulator, one can see that the Stokes deformation number for samples SG and SG2 are higher than the critical Stokes deformation number of 0.2, which indicates that breakage or fragmentation of the primary nuclei will most likely occur. On the other hand, minimum breakage of primary nuclei are expected for samples SG1 and SG3. For samples SG and SG2, it is expected that breakage of the primary nuclei will occur due to the fact that their Stokes deformation numbers are higher than the critical Stokes deformation number of 0.2. Among the physical factors that determine if nuclei breakage is significant, primary particle size plays a major part as the dynamic yield strength of the nuclei is inversely proportional to particle size. Therefore, it is expected that this model with breakage will most likely be applicable to systems with relatively large primary particles in the feed. It should be pointed out that care must be excised in extrapolating the nuclei properties observed from ex-granulator to ingranulator behavior and the in-granulator nuclei properties can only be verified from real granulator experiments.
4.2. Modeling nuclei size distribution with different feed powders
23
the calculated nuclei size distribution with a median drop size of 400 mm at the spray flux ca of 0.4 for the four different feed powders listed in Table 1. Another assumption in the model is that the drops are fully penetrated before circulated out of the spray zone, i.e., t ¼tmax in Eq. (10). Other parameters used in the models are the empirical parameter q¼0 and the experimentally measured ratios of nuclei size to drop size as shown in Fig. 1. Since the Stokes deformation numbers for feed powders SG and SG1 are higher than the critical Stokes deformation, breakage is expected to occur so Eq. (16) was used to calculate the final nuclei size distribution, together with Eqs. (9) and (10). The term du=u in the integration in Eq.(16) was approximated by the volume interval divided by the average volume in the size interval. To illustrate how the inclusion of the breakage of the primary nuclei affects the calculated nuclei size distribution, the comparisons of model calculations with and without breakage for feed powders SG and SG2 are shown in Fig. 5. Fig. 4 clearly shows how the feed powder size affects the nuclei size distribution. With a finer feed powder such as SG1 and SG3, the calculated nuclei size is smaller and the nuclei size distribution is also narrower. With coarser powders such as samples SG and SG2, although their medium size of the nuclei is similar, the spread of the nuclei size is broader than the finer powders. Fig. 5 shows that the model without considering breakage for the coarser powder predicted a much larger nuclei size than that with breakage included. It was found that the calculated secondary nuclei size distributions with different q values in Eq. (16) are not sensitive to the changes of q and therefore we can assume that q is equal to 0 and the selection rate S(u) in Eq. (13) is independent of the nuclei size, as found by Tan et al. (2004) in fluidized granulation. The model proposed here could be potentially used for predicting
To illustrate how the model developed can be used to obtain the nuclei size distribution at different conditions, Fig. 4 shows
Fig. 4. Modeled nuclei size distribution with different sized nickel laterite feed powder at a spray flux of 0.4.
Fig. 3. Drop penetration time versus average surface mean particle size.
Table 2 Physical properties of the ex-granulator nuclei and Stokes deformation number. Property 3
Packing density (g/cm ) Tapped porosity Drop penetration time (s) Ex-granulator nuclei property Calculated dynamic yield strength in drum (Pa) Stdef
SG
SG1
SG2
SG3
1.04 0.485 0.3217 0.055 No formation of nuclei with enough strength 2991
1.112 0.511 0.766 7 0.038 Strong enough for handling 8565
1.114 0.485 0.0837 0.016 No formation of nuclei with enough strength 1231
1.10 0.52 1.711 70.08 Strong enough for handling 16,131
0.207 (breakage)
0.091 (no breakage)
0.66 (breakage)
0.02 (no breakage)
24
L.X. Liu et al. / Chemical Engineering Science 86 (2013) 19–24
primary particle size plays a major role in determining whether breakage of primary nuclei occurs or not.
Acknowledgement Financial support provided under CSIRO Minerals Down Under Cluster project funding is gratefully acknowledged. References
Fig. 5. Comparisons of modeled nuclei size distribution with and without the breakage model for the coarser feed powders.
the effect of feed powder and binder properties as well as the operating conditions on nuclei size distribution. Parameters such as feed powder size, binder viscosity and contact angle, drum speed etc directly affect the Stokes deformation number and hence the breakage of the primary nuclei. In addition, feed powders with smaller particle sizes will lead to smaller nuclei size at the nucleation stage in comparison to a coarser feed powder. The calculated nuclei size distribution can then be used as the initial condition for the next stage growth model for predicting the final agglomerate size distribution. However, caution should be excised with the primary nuclei size distribution model (Eq. (9)) when the spray flux ca is larger than 0.5, as found out by Hapgood et al. (2009).
5. Conclusions A nucleation model that includes nuclei breakage/fragmentation is proposed based on the nucleation model of Hapgood and the Stokes deformation number. The model proposed that breakage or fragmentation of primary nuclei from binder spray will occur when the Stokes deformation number exceeds a certain critical number. Model calculations with siliceous goethite minerals showed that breakage/fragmentation of primary nuclei from feed powders with large particle sizes will most likely occur in a real drum agglomeration process whereas powders with smaller particle sizes will not. The nuclei size distributions with different sized feed powders were calculated using the proposed model and the model showed that smaller nuclei size is expected with a finer feed powder. For the same type of binder and powder, the
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A nuclei size distribution model including nuclei breakage L.X. Liu a,n, L. Zhou b, D.J. Robinson c, J. Addai-Mensah b a b c
School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia Ian Wark Research Institute, ARC special Research Centre, University of South Australia, Mawson Lakes, Adelaide, SA 5095, Australia CSIRO Process Science and Engineering & Minerals Down Under National Research Flagship Australian Minerals Research Centre, PO Box 7229, Karawara, WA 6152, Australia
a r t i c l e i n f o
abstract
Article history: Received 6 October 2011 Received in revised form 7 March 2012 Accepted 5 April 2012 Available online 14 April 2012
In this work, a nucleation model that includes nuclei breakage/fragmentation is proposed. The model is based on the nucleation model of Hapgood and the Stokes deformation number calculated from the granule dynamic yield strength from the previously reported granule breakage work. It is proposed that breakage or fragmentation of primary nuclei from binder spray will occur if the Stokes deformation number exceeds a certain critical number. In the case where breakage occurs the model for secondary nuclei size distribution is proposed. To validate the model, the characteristics of the primary nuclei formed from nickel laterite ores with diluted sulphuric solutions as a binder were investigated. The nuclei were produced by dropping the binder solution onto a stationary powder bed. The mechanical integrity of the primary nuclei formed, the relationship between the nuclei diameter and binder drop diameter were studied. The Stokes deformation numbers for nickel laterite powders with different particle size in a lab scale drum granulator were calculated and the nuclei size distributions with different nickel laterite feed powders are predicted. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Agglomeration Nuclei size distribution model Nuclei breakage Primary nuclei Secondary nuclei Nickel laterite
1. Introduction Agglomeration of fine mineral particles as a precursor to heap leaching is an important means of enhancing leaching rates and metal recoveries, particularly in processing low grade ores. To fully understand the underlying mechanisms of agglomeration for better design and control of the agglomeration processes, it is necessary to establish a useful, predictive model based on feed and binder liquid characteristics right from the wetting and nucleation stage (Sanders et al., 2003). Useful rate parameters of the agglomeration mechanisms and kinetics may be extracted from appropriate agglomeration experiments and used for the optimization and scale-up, and also the benchmarking of our understanding on real ore agglomeration processes. In the initial stage of agglomeration process, binder liquids are sprayed onto the feed powder to form nuclei for further growth. The nucleation process in which the initial granules are formed, has been shown to be crucial in determining the final agglomerate size distribution (Schaafsma et al., 1997, 2000; Litster et al., 2001; Hapgood et al., 2009). Powder and liquid properties directly affect the nuclei formation and their properties and subsequently affect the final agglomerate properties. Inaccurate information or modeling of the nucleation mechanism directly affects the accuracy of
n
Corresponding author. Tel.: þ61 7 33658591; fax: þ61 7 33654199. E-mail address: [email protected] (L.X. Liu).
0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.04.009
subsequent agglomeration process modeling as it forms the initial conditions for the growth and coalescence model (Hapgood et al., 2009). In addition, the detailed properties of the primary particles/liquid can be captured in the nucleation model and the effect of these properties on the subsequent stage of coalescence is reflected through the nuclei size distribution. While both the mechanisms and modeling of agglomerate growth by coalescence have been studied for different solid-liquid systems by many workers (e.g., Sastry and Fuerstenau, 1970; Kapur and Fuerstenau, 1969; Gluba, 2003; Liu et al., 2000; Hounslow, 1998), nucleation are the least understood mechanism and least accurately modeled (Hapgood et al., 2009). Litster et al. (2001) and Hapgood et al. (2002) studied the liquid distribution using a dimensionless spray flux to describe the nucleation mechanism and Feise et al. (2008) and Van den Dries and Vromans (2004, 2009) investigated the liquid penetration in high shear mixer. Poon et al. (2008) used a mechanistic nucleation model in the 3-D population balance model for nucleation and aggregation and the nucleation model was included into the whole population balance model. However, the majority of the nucleation model used an average nuclei size. Hapgood et al. (2009) and Wildeboer et al. (2005) have developed a generalized nuclei size distribution model based on drop penetration and overlap by assuming no breakage on primary nuclei formed. Van den Dries and Vromans (2004, 2009) summarized the different ways that a liquid can be mixed with powder in high shear agglomeration process. That is, the dispersion mechanism
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where binder drops are smaller than the primary particles and the penetration mechanism where binder drops are larger than the particle size. One of the key penetration nucleation mechanism proposed by Van den Dries and Vromans is the destruction nucleation mechanism in which nuclei breakage was quantitatively proven. In drum agglomeration process, although the shear force and impact forces are lower than these in the high shear agglomeration process, nuclei breakage/fragmentation could still occur as the initial nuclei formed are very much dependent on the binder and primary particle properties. In this work, we propose a nucleation model which is based on the nucleation model developed by Hapgood et al. (2009) but includes the granule breakage model by Liu et al. (2009). The properties of the nuclei formed with nickel laterite minerals and sulphuric acid solution in a packed bed were studied and used to validate the nucleation model.
2. Theory The formation of nuclei from liquid drops in the wetting and nucleation stage of the agglomeration process is based on liquid penetration into a porous powder bed due to capillary forces. The process involves the drop penetration followed by the formation of nuclei. If the nuclei formed are strong enough to withstand the impact forces, the nuclei will then go through the consolidation and coalescence stage where growth occurs. On the other hand, if the nuclei formed are too weak, they will be fragmented into smaller particles. The model presented below will include all the possible scenarios based on the feed and liquid properties.
2.1. Nuclei size from single binder drop penetration from literature When a binder drop hits a porous powder bed, what is the size of the nucleus formed? In other words, what is the nucleation ratio which is defined as the diameter of the nucleus to the binder drop? This ratio is a function of both liquid properties (surface tension and viscosity) as well as powder properties. For a given fluid and powder, reducing the powder porosity creates larger nuclei, which may be due to the fact that liquid is able to spread further if pores are narrow as the capillary force is increased. Van den Dries and Vromans (2009) developed a model to relate the nuclei diameter with binder and nuclei properties at static conditions and assumed that it is equally applicable to dynamic situations in a high shear mixer. Hounslow et al. (2009) developed a model for predicting the nuclei volume based on the surface tension-driven flow at any time t: rffiffiffiffiffiffiffiffiffi! 1fcp t vnuclei ¼ vdrop 1 þ ð1Þ t max fcp with 1=3
t max ¼
18:75 mr 2drop ð1fcp Þ
gdp f3cp
ð2Þ
where vnuclei and vdrop are the volume of the nuclei and drops respectively, rdrop is the drop radius, fcp is the critical packing liquid volume fraction, tmax is the drop penetration time for an entire drop to penetrate, m and g are liquid viscosity and surface tension respectively, dp is particle diameter. tmax can also be calculated from the equation by Hapgood et al. (2002): 2=3
t max ¼
1:35m vdrop
g Ref f e2ef f cos y
ð3Þ
and Ref f ¼
j d32
eef f
3
ð1eef f Þ
ð4Þ
where Reff is the pore radius, d32 is the surface mean diameter of the particles, j is the shape factor of the particles, y is the contact angle, eeff is the effective porosity of the packed powder. With Eq. (1), the nuclei volume at any available penetration time in dynamic situations can be calculated. When a drop fully penetrates into the powder, the time t reaches the maximum so that t is equal to tmax. With that, the maximum nuclei volume is equal to: vnuclei ¼
vdrop
fcp
ð5Þ
Interestingly, if fcp is replaced by the nuclei porosity, the above equation is the same as that proposed by Van den Dries (2009) at static conditions. Therefore, the maximum nuclei volume is only related to the drop volume and the critical packing liquid volume fraction fcp, which is material (powder and binder) and particle size dependent. The lower the fcp value (larger particle size), the higher the vnuclei value. That is, smaller particles will undergo consolidation which results in smaller nuclei size in comparison to larger particles. 2.2. Destructive nucleation mechanism Depending on the powder and binder properties, the destructive nucleation mechanism proposed by Van den Dries and Vromans (2004, 2009) will most likely be applicable for weak nuclei in granulators with higher shearing force. That is, secondary nuclei will be formed from the fragmentation of primary nuclei. The Stokes deformation number Stdef, which is the ratio of kinetic energy over the granule dynamic strength, was used as the criterion for granule breakage. However, only viscous force was used to calculate the dynamic strength of the nuclei by Van den Dries and Vroman and the model did not include the nuclei size distribution after breakage of the primary nuclei. Liu et al. (2009) used both the capillary forces and viscous force to represent the granule dynamic strength dp: " # 1e gcos y 9 1e 2 9pmvp ð6Þ dp ¼ Sb 6 þ 8 dp 16 dp e e Where vp is granule velocity after impact, e is granule porosity, Sb is binder saturation in the pores, which is 100% in the case of wet nuclei, g is binder surface tension. Eq. (6) is equally applicable to calculate the nuclei strength. The Stokes deformation number Stdef can then be used as a measure of primary nuclei breakage: Stdef ¼
rg v2c 2dp
ð7Þ
where vc is the impact velocity in the granulator. Unlike higher shear mixer granulators where the impeller tip velocity is most commonly used as the impact velocity, there is not much report on the most commonly used impact velocity in drum granulator. Liu and Litster (2002) used the peripheral velocity in the drum, which is equal to oDdrum/2 (where o is the drum rotational speed and Ddrum is the drum diameter) as the granule collision velocity and it can be used here to represent the nuclei impact velocity vc. A critical Stdef value of 0.1 between breakage and non breakage was found by Tardos et al. (1997) and the value of 0.2 was found by Liu et al. (2009) with the dynamic strength from Eq. (6). If the Stdef number is higher than the critical value of breakage, we can assume that fragmentation takes place and the extent of nuclei fragmentation is proportional to the Stdef value. If the
L.X. Liu et al. / Chemical Engineering Science 86 (2013) 19–24
Stokes deformation number is lower than the critical Stokes number, one can assume that nuclei breakage/fragmentation is negligible and the model developed by Hapgood et al. (2009) can be used for predicting the nuclei size distribution by including the formation of nuclei from multi-drops. 2.3. Nuclei size distribution model incorporating destruction mechanism Eq. (1) gives a relationship between the nuclei size and the drop size, provided that one drop forms one nucleus. However, in real industrial process, the nuclei size distribution also depends on the spray rate of the binder which determines if the spray drops will overlap to form one nucleus. For the situation where nuclei fragmentation is negligible, Hapgood et al. (2009) have developed a nucleation model for predicting the nuclei size distribution based on the dimensionless spray flux and the nucleation ratio. The detailed derivation of the model can be found in Hapgood et al. (2009). The key features of the nucleation model is that one can calculate the nuclei size distribution from the dimensionless spray flux ca, which is the ratio of the wetted area covered by the nozzle to the spray area in the nucleation zone. That is:
ca ¼
3V 2Addrop
ð8Þ
where V is the volumetric flow rate of the binder, ddrop is the average drop size and A is the spray area. Poisson distribution was then used to predict the dimensionless size distribution in terms of the number of drops used to form the nucleus (Hapgood et al., 2009): n
P n ¼ expð4ca Þ
ð4ca Þ ðn1Þ!
ð9Þ
Where Pn is the probability of n drops forming one nucleus. In Hapgood’s model, a constant relationship between the nuclei diameter and drop diameter is assumed. In this work, Eq. (1) is used to relate the nuclei diameter to the critical packing volume fraction and we also include processes in which drop penetration time tmax is longer than the powder circulation time t. For a nucleus formed from n drops, the relationship between the nuclei diameter dn and the drop diameter ddrop can be obtained: rffiffiffiffiffiffiffiffiffi!1=3 1fcp t dn ¼ nddrop 1 þ ð10Þ t max fcp The above equation results in the conversion of data from the number of drops to nuclei diameter, similar to what was ffi shown pffiffiffiffiffiffiffiffiffiffiffiffiffi in Hapgood’s model. The term ð1 þð1fcp =fcp Þ t max =t Þ1=3 in Eq. (10) is essentially equivalent to the nucleation ratio kd defined in Hapgood’s model (2009) and became ð1=fcp Þ1=3 if the powder circulation time is longer than the drop penetration time. On the other hand, if nuclei fragmentation (i.e., Stdef 4Stdef cri) takes place, one can use the population balance equation with breakage only to track the changes of nuclei size distribution after breakage: Z 1 @nðv,tÞ ¼ Sðu,tÞbðv,uÞnðu,tÞduSðv,tÞnðv,tÞ ð11Þ @t v where n(v,t) is the number density function of the nuclei with volume v( ¼ p/6 d3n), u is nuclei volume, S(u,t) is the selection rate constant, b(v,u) is the breakage function. Since nucleation happens in a much shorter time frame in comparison to the coalescence stage, one can assume that breakage of the primary nuclei, if any, occurs simultaneously. That is, the rate of change with time for n(v) during the nucleation stage can be assumed to be zero. With this assumption, the nuclei size distribution after
incorporating nuclei breakage/fragmentation is: R1 SðuÞbðv,uÞn0 ðuÞdu nðvÞ ¼ v SðvÞ
21
ð12Þ
n0 (u) is used to denote the nuclei size distribution without breakage. As the breakage rate increases with the increase of Stokes deformation number and possibly nuclei size, one can assume that the selection rate S(u) is proportional to the Stokes deformation number Stdef and nuclei volume u, similar to what was used in the work of Tan et al. (2004), i.e., SðuÞ ¼ ðSt def Þp uq
ð13Þ
where p, q are empirical parameters. The breakage function b can be taken as random binary breakage as proposed by Tan et al. (2004), i.e., the breakage function gives uniform probability of all fragment sizes on a mass scale: bðv,uÞ ¼
1 u
ð14Þ
Substituting Eqs. (13) and (14) into Eq. (12), the nuclei size distribution with breakage became: R1 ðSt def Þp uq1 n0 ðuÞdu nðvÞ ¼ v ð15Þ ðSt def Þp vq Since Stdef is only related to the feed and binder properties as well as the impact speed, we can further simplify Eq. (15) as: R 1 q1 0 u n ðuÞdu nðvÞ ¼ v ð16Þ vq Eq. (16) shows that, if breakage does occur, the resulted secondary nuclei size distribution is independent of the Stokes deformation number and the nuclei size distribution after nuclei breakage can be calculated from the primary nuclei size distribution n0 (u), which can be obtained from Eqs. (8)–(10). In summary, the nucleation model proposed can be summarized as follows: (1) Use Eqs. (6) and (7) to calculate the nuclei Stokes deformation number. If Stdef is smaller than Stdef cri, then nuclei breakage is negligible and Eqs. (8)–(10) can be used for the calculation of nuclei size distribution. (2) If Stdef is larger than Stdef cri, fragmentation is significant and one can use Eqs. (8)–(10) and (16) to calculate the nuclai size distribution after breakage. 3. Experimental 3.1. Materials and characterization Nickel laterite minerals (siliceous goethite) were used in the experiments. These nickel bearing minerals consist of quartz, goethite and iron oxides and also reactive, acid consuming clays (magnesium bearing silicates and iron rich clays). Four different size fractions, namely 2 mm ore, 106 mm, 2 mmþ106 mm and 53 mm were used. The 106 mm, 2 mmþ 106 mm and 53 mm size fractions were obtained by screening the 2 mm ore. The particle size distribution of the 2 mm ore was measured using Malvern Mastersizer. The size distribution of the 106 mm and 2 mmþ106 mm were calculated numerically by mixing input fractions. Table 1 summarizes the size distribution parameters for all the materials. The liquid used is 44% (w/w) H2SO4 solution in water. The minerals were found to be soluble in the sulphuric solution and the maximum amount dissoluble is 0.058 (w/w). The contact angle of the 44% H2SO4 solution on siliceous goethite was measured using Sessile drop method and is 101.
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3.2. Ex-granulator experiments—drop penetration experiments Drop penetration experiments were carried out using 44% sulphuric acid (w/w) with the powders listed in Table 1. The powders were packed into a Petri dish with diameter 70 mm and depth of 17 mm and the packing density was measured. The liquid was filled into a syringe and positioned 10 mm above the surface of the powder bed. The drops were released onto powder bed surface by using the sessile drop technique (data physics OCA 20) and the penetration of each drop was videoed and the penetration time was recorded. A needle was attached to the syringe to generate the drops, the average drop size was measured by weighing the liquid in the syringe and the number of drops that produced out of certain amount of liquid. The average volume of the drops was 15.770.2 ml. The drop penetration time was obtained from the recorded video images between the time that the drop was in contact with the powder surface to that when it completely penetrated into the powder bed At each condition, 5 drops were measured and an average penetration time was obtained. The drop spread area (diameter) was measured using a ruler straight after the drop penetrates into the powder bed. The nuclei formed from the ex-granulator experiments were visually observed for their integrity.
the decrease of particle size. Smaller particles form smaller nuclei because capillary forces for smaller particles draw the particles together and therefore the nuclei formed are smaller and denser. Fig. 3 shows the measured drop penetration time versus the mean particle size (d32). The calculated drop penetration times using Eqs. (3) and (4) are also shown in Fig. 3. It is noted that as particle size increases, the rate of decrease of drop penetration time slows down. It is also noted that Eq. (4) predicts lower drop penetration time than experimental values, particularly with powders of small particle sizes. In general, the drop penetration time for the nickel laterite powders are in the order of seconds, which means that full drop penetration of droplets will occur in the drum agglomeration process. Table 2 lists the properties of the primary nuclei that were obtained from ex-granulator experiments. The physical properties of primary nuclei prepared from packed powder bed for siliceous goethite with drop volume of 0.0157 ml and its Stokes deformation number in a dynamic situation in a drum granulator are calculated using Eqs. (6) and (7) and are also listed in Table 2. The impact velocity vc used for calculating the Stokes deformation number is 0.942 m/s, which is half of the peripheral velocity of a
4. Results and discussion 4.1. Ex-granulator experimental results To find out how the nuclei size is related to the drop size at static conditions, the diameters of the ‘‘nuclei’’ after binder drop penetration are measured with different powders. Fig. 1 shows some examples of the nuclei. The ratios of the nuclei diameter to drop diameter are presented in Fig. 2. As can be seen from Fig. 2, the ratio of the nuclei diameter to drop diameter increases rapidly with the increase of primary particle size and then the rate of increase slows down as particle size increases further. If Eq. (5) is used to relate the ratio of nuclei diameter to droplet diameter, one can see that the ratio of dn/dd is proportional to (1/fcp)3, indicating that the critical liquid packing volume fraction increases with
Fig. 2. Ratio of nuclei diameter to binder drop diamter in ex-granulator experiments versus primary particle size.
Table 1 Size distribution parameters of the nickel laterite feed powders. Sample name
Sample description
d50 (mm)
d32(mm)
Siliceous goethite (SG) SG1 SG2 SG3
2 mm 106 mm SG sample, size fraction from 2 mm 2 mmþ 106 mm SG sample 53 mm SG sample
118.5 47.4 366 20.7
30.5 15.7 250 10.5
Fig. 1. Examples of the ex-granulator nuclei.
L.X. Liu et al. / Chemical Engineering Science 86 (2013) 19–24
rotational drum of 300 mm with a rotational speed of 60 rpm. The nuclei velocity vp in Eq. (6) is taken as 15% of the impact velocity and the d50 values were used in the calculations. It was observed that, even with gentle handling, the ‘‘nuclei’’ formed from the siliceous goethite samples SG and SG2 with 44% H2SO4 drops in packed bed were very weak and disintegrated into smaller particles without the formation of dry nuclei. For siliceous goethite with smaller particles (samples SG1 and SG3), weak nuclei are formed. When examining the calculated Stokes deformation numbers in a dynamic situation such as the initial binder spray process in a drum granulator, one can see that the Stokes deformation number for samples SG and SG2 are higher than the critical Stokes deformation number of 0.2, which indicates that breakage or fragmentation of the primary nuclei will most likely occur. On the other hand, minimum breakage of primary nuclei are expected for samples SG1 and SG3. For samples SG and SG2, it is expected that breakage of the primary nuclei will occur due to the fact that their Stokes deformation numbers are higher than the critical Stokes deformation number of 0.2. Among the physical factors that determine if nuclei breakage is significant, primary particle size plays a major part as the dynamic yield strength of the nuclei is inversely proportional to particle size. Therefore, it is expected that this model with breakage will most likely be applicable to systems with relatively large primary particles in the feed. It should be pointed out that care must be excised in extrapolating the nuclei properties observed from ex-granulator to ingranulator behavior and the in-granulator nuclei properties can only be verified from real granulator experiments.
4.2. Modeling nuclei size distribution with different feed powders
23
the calculated nuclei size distribution with a median drop size of 400 mm at the spray flux ca of 0.4 for the four different feed powders listed in Table 1. Another assumption in the model is that the drops are fully penetrated before circulated out of the spray zone, i.e., t ¼tmax in Eq. (10). Other parameters used in the models are the empirical parameter q¼0 and the experimentally measured ratios of nuclei size to drop size as shown in Fig. 1. Since the Stokes deformation numbers for feed powders SG and SG1 are higher than the critical Stokes deformation, breakage is expected to occur so Eq. (16) was used to calculate the final nuclei size distribution, together with Eqs. (9) and (10). The term du=u in the integration in Eq.(16) was approximated by the volume interval divided by the average volume in the size interval. To illustrate how the inclusion of the breakage of the primary nuclei affects the calculated nuclei size distribution, the comparisons of model calculations with and without breakage for feed powders SG and SG2 are shown in Fig. 5. Fig. 4 clearly shows how the feed powder size affects the nuclei size distribution. With a finer feed powder such as SG1 and SG3, the calculated nuclei size is smaller and the nuclei size distribution is also narrower. With coarser powders such as samples SG and SG2, although their medium size of the nuclei is similar, the spread of the nuclei size is broader than the finer powders. Fig. 5 shows that the model without considering breakage for the coarser powder predicted a much larger nuclei size than that with breakage included. It was found that the calculated secondary nuclei size distributions with different q values in Eq. (16) are not sensitive to the changes of q and therefore we can assume that q is equal to 0 and the selection rate S(u) in Eq. (13) is independent of the nuclei size, as found by Tan et al. (2004) in fluidized granulation. The model proposed here could be potentially used for predicting
To illustrate how the model developed can be used to obtain the nuclei size distribution at different conditions, Fig. 4 shows
Fig. 4. Modeled nuclei size distribution with different sized nickel laterite feed powder at a spray flux of 0.4.
Fig. 3. Drop penetration time versus average surface mean particle size.
Table 2 Physical properties of the ex-granulator nuclei and Stokes deformation number. Property 3
Packing density (g/cm ) Tapped porosity Drop penetration time (s) Ex-granulator nuclei property Calculated dynamic yield strength in drum (Pa) Stdef
SG
SG1
SG2
SG3
1.04 0.485 0.3217 0.055 No formation of nuclei with enough strength 2991
1.112 0.511 0.766 7 0.038 Strong enough for handling 8565
1.114 0.485 0.0837 0.016 No formation of nuclei with enough strength 1231
1.10 0.52 1.711 70.08 Strong enough for handling 16,131
0.207 (breakage)
0.091 (no breakage)
0.66 (breakage)
0.02 (no breakage)
24
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primary particle size plays a major role in determining whether breakage of primary nuclei occurs or not.
Acknowledgement Financial support provided under CSIRO Minerals Down Under Cluster project funding is gratefully acknowledged. References
Fig. 5. Comparisons of modeled nuclei size distribution with and without the breakage model for the coarser feed powders.
the effect of feed powder and binder properties as well as the operating conditions on nuclei size distribution. Parameters such as feed powder size, binder viscosity and contact angle, drum speed etc directly affect the Stokes deformation number and hence the breakage of the primary nuclei. In addition, feed powders with smaller particle sizes will lead to smaller nuclei size at the nucleation stage in comparison to a coarser feed powder. The calculated nuclei size distribution can then be used as the initial condition for the next stage growth model for predicting the final agglomerate size distribution. However, caution should be excised with the primary nuclei size distribution model (Eq. (9)) when the spray flux ca is larger than 0.5, as found out by Hapgood et al. (2009).
5. Conclusions A nucleation model that includes nuclei breakage/fragmentation is proposed based on the nucleation model of Hapgood and the Stokes deformation number. The model proposed that breakage or fragmentation of primary nuclei from binder spray will occur when the Stokes deformation number exceeds a certain critical number. Model calculations with siliceous goethite minerals showed that breakage/fragmentation of primary nuclei from feed powders with large particle sizes will most likely occur in a real drum agglomeration process whereas powders with smaller particle sizes will not. The nuclei size distributions with different sized feed powders were calculated using the proposed model and the model showed that smaller nuclei size is expected with a finer feed powder. For the same type of binder and powder, the
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