In this section results are presented for the wetness of agglomerates as a function of the number of particlesN, the minimum separation distancesbetween particles and the quantity of binderδ. The agglomerate model was solved using the parametersN = 50 (i.e. agglomerates between 4 and 50 primary particles) where 0≤s≤R (or 2R ≤σ ≤3R) and 0 ≤ δ ≤ 1. The number of steps used for the s and δ variables was 100 so that ∆s= 0.01R and ∆δ= 0.01. The resolution used for the polyhedra objects was k= 30.
The results below are presented in terms of a volume ratio between the binder fluid and the primary particles. For an N particle agglomerate, the fluid-to-solid volume ratio is defined as
V∗ = Vbinder
N Vsolid
(4.27)
‡The parallel computer used was the Massey Universitysisters Beowulf cluster which has 16 PIII
4.9 Results of the Model 119
whereVbinder is the binder volume and the volume of a primary particle isVsolid = 43πR3.
The surface areasAbinder,Aparticle and Atotal are also normalised as
A∗binder= Abinder N Asolid , (4.28) A∗particle= Aparticle N Asolid (4.29) and A∗total= Atotal N Asolid (4.30)
whereAsolid = 4πR2. From Equation (4.26),Atotal∗ =A∗binder+A∗particle.
The dependent variables areW, A∗binder and A∗total and the parameters are s, δ, N and
V∗. The graphs in Figures 4.14 - 4.18 show the response of the dependent variables to the parameters. Results for the complete parameter space may be viewed because the parameter along the contours is δ and subplots are used to show results for fixed values of one of the parameters.
The figures that are presented are as follows:
1. Figure 4.14 shows the relationship between W and s. The subplots are for fixed values ofV∗ and the curves represent agglomerates of a fixed size. The parameter along the curves isδ as indicated.
2. Figure 4.15 shows the relationship between W and sfor an agglomerate of a fixed size (N = 30). The curves represent fixed values of V∗ and δ.
3. Figure 4.16 shows plots of surface wetness with respect toN. Each subplot is for a fixed value ofsand the curves represent a fixed value of V∗. The parameter along the curves isδ.
4. Figure 4.17 plots the wet agglomerate surface areaA∗binder against N fors= 0 and
s= 0.2R. Each curve represents a fixedV∗ value.
5. Figure 4.18 plots the total agglomerate surface areaA∗
total against N fors= 0 and s= 0.2R. Each curve represents a fixedV∗ value.
Figures 4.14 and 4.15 were obtained by using sorting the dataset by particle sizeN and then using the Matlab contour command on each of these portions. Figures 4.16-4.17 were obtained by interpolating the data for fixed values ofV∗ and repeating this all the dataset segments (as sorted by size). The ‘jitter’ in these figures is because a non-constant
(a)V∗= 0.3 (b)V∗= 0.5
(c)V∗= 0.7 (d)V∗= 1.0
Figure 4.14: Plot of surface wetness W with respect to inter-particle separation distance s for
agglomerates composed of 10, 20, 30 and 50 primary particles. Values of V∗
used in this figure areV = 0.3,0.5,0.7 and 1.0.
4.9 Results of the Model 121
Figure 4.15: Volume contour plot of surface wetnessW with respect to separation distancesfor
anN = 30 particle agglomerate. There are two sets of contour labels in this figure. Contour labels
on the solid blue lines are the fluid to solid volume ratioV, while bold contour labels (drawn in
(a)s= 0 (b)s= 0.2R
(c)s= 0.5R (d)s= 0.7R
Figure 4.16: Graphs of wetnessW with respect to particle numberN for a range ofsand a range
of fluids to solid ratio valuesV∗
4.9 Results of the Model 123
(a)s= 0
(b)s= 0.2R
Figure 4.17: Plot showing the wet area of agglomeratesA∗
binderwith respect to number of particles
(a) Plot ofA∗
totalversusN fors= 0
(b) Plot ofA∗totalversusN fors= 0.2R
Figure 4.18: Plot showing the total area of agglomeratesA∗
4.9 Results of the Model 125
volume of fluid is added with each additional particle. (The volume of binder fluid added depends on the number and placement of the fluid segments.)
The following observations are made from the figures:
Observation(s) Shown in
Figure(s)
Physical Effect
1. W increases assis decreased for fixed N and fixed V∗. δ increases.
4.14, 4.15
Decreasingsdecreases the void space causing liquid to migrate to the sur- face of the particle.
2. W increases as V∗ is in- creased for fixedN and fixed
s. δ increases.
4.15
Fixing N and s fixes the void space. Additional liquid therefore increases the saturation state of the particle.
3. W decreases for increasing sizeN for fixedV∗ and fixed
s. δ decreases
4.16
The void space of the agglomerate in- creases withN. For fixedV∗,W and
δ therefore decreases.
4. The relative wet surface area
A∗binder decreases for increas- ing size N for fixed V∗ and
s. δ decreases.
4.17
The relative void space of the parti- cle increases with increasing size N. The fluid moves to this space which decreases the fluid surface area.
5. The relative total surface area A∗total decreases for in- creasing size N for fixed V∗
and s. δ decreases.
4.18
As above.
6. ForV∗ values below the ran- dom packing limit (approx- imately V∗ = 0.42 for s = 0) the total surface area in- creases withN until no solu- tion can be obtained.
4.18(a)
The fluid occupies the increased void space due to increasing N. This ex- poses portions of the spheres which proportionally have a higher surface area. No solution can be obtained when the binder volume is less than the void space.
The observation made in item 1 supports the theory of Wauters et al.[21], described in Section 1.2.2, where particles increase in surface wetness as consolidation occurs.
W(N, V∗, s). The extended population balance model in Section 5.2 requires a relation- ship of the formW =W(N, V∗) for the probability of particle coalescence. The original approach taken was to find the value ofs, for a givenV∗value, such that the particle had minimum wet binder surface areaAbinder. However, particles with highsand lowδvalues
were predicted which had a physically unrealistic appearance. The cause of this is due to the use of the approximate binder fluid surface. The case subsequently investigated was
s= 0 which corresponds to the case of maximum consolidation. The functions fitted to this case are given below and they used in the population balance simulation.