6.1 Model description
6.1.1 Estimating the probability of successful collisions
Chapter 6
Macroscopic modeling of the dominant size enlargement mechanism
This chapter is an extended version of Rieck et al. [172] and deals with estimation of the dominant size enlargement mechanism in a spray fluidized bed process. The underlying model is described in detail, before presenting results of a simulation study indicating the influence of the inlet gas temperature, viscosity, droplet size, and contact angle on the dominant size enlargement mechanism.
Furthermore, a classification of the dominant mechanism based on the probability of successful collisions is proposed, which follows from experimental data and simulation results and allows for regime maps to be created.
Chapter 6 Macroscopic modeling of the dominant size enlargement mechanism
collisions𝑃coll,suc, which is used in the present work to estimate the dominant size enlargement mechanism:
𝑃coll,suc =𝑃coll,wet ·𝑃coll,wet,suc. (6.1)
Depending on this parameter, the dominant size enlargement mechanism will then be:
dominant mechanism=
(layering 𝑃coll,suc →0,
agglomeration 𝑃coll,suc →1. (6.2)
For layering to be dominant, the probability of successful collisions must approach zero, which can be due to a low probability of wet collisions (𝑃coll,wet →0), a low probability of successful wet collisions (𝑃coll,wet,suc →0), or both. If the probability of successful collisions is sufficiently large (ideally if𝑃coll,wet →1), agglomeration will be dominant. This condition will certainly be fulfilled if both the probability of wet collisions and the probability of successful wet collisions approach unity (𝑃coll,wet →1 and𝑃coll,wet,suc →1). Note that this approach neglects the influence of liquid and solid bridge breakage. Consequently, a major model assumption is that breakage does not dominate the process and therefore the probability of successful collisions calculated using Equation (6.1) can be used as a suitable parameter to estimate the dominant size enlargement mechanism. Below, the calculation of the considered probabilities is presented.
Probability of wet collisions
The probability of wet collisions depends on the wet surface fraction𝛹wet. Following Rajniak et al.
[173], it is assumed that all particles are wet and have the same wet surface fraction. Since only binary collisions are considered, either one or two droplets can be involved in a collision. The probability of wet collisions is then comprised of the probabilities of both individual events (exactly one droplet and exactly two droplets take part in a collision):
𝑃coll,wet =2𝛹wet(1−𝛹wet) +𝛹wet2 =2𝛹wet −𝛹wet2 . (6.3)
Probability of successful wet collisions
The probability of successful wet collisions is calculated based on the Stokes criterion in this approach.
As described in Section 2.2.1, spherical, non-deformable particles, which are always larger in diameter than the sprayed droplets, are considered. Using these assumptions and the Stokes criterion for particles with rough surfaces given in Section 2.2.5, a critical particle size𝑥critcan be obtained:
𝑥crit = 9 4
𝜂 𝜚p𝑢coll
1+ 1 𝑒0
lnℎl ℎa
. (6.4)
6.1 Model description
0 0.1 0.2 0.3 0.4
0 2 4 6 8 10
𝑥 [mm]
𝑞0 mm−1
𝑞0 𝑥crit 𝜉
Figure 6.1:Plot of a particle size distribution (mean value and standard deviation equal to 0.2 mm and 0.05 mm, respectively) with𝑥critequal to 0.15 mm and the corresponding fraction of particles with a fulfilled Stokes criterion.
Here, the liquid film heightℎl is set to the height of the individual deposited droplets calculated according to Equation (3.10). Since the Stokes criterion is fulfilled, particles which are smaller than or equal to𝑥crithave the possibility to agglomerate upon a wet collision. The critical particle size is then used to calculate the fraction of particles with a fulfilled Stokes criterion from the number-based, normalized particle size distribution𝑞0:
𝜉 =
𝑥crit
∫
0
𝑞0d𝑥 (6.5)
Figure 6.1 shows a normalized particle size distribution (normal distribution with a mean diameter of 0.2 mm and a standard deviation of 0.05 mm) and the fraction of particles with a fulfilled Stokes criterion for𝑥crit =0.15 mm. In this example,𝜉 is equal to 0.16. The probability of successful wet collisions can then be calculated based on𝜉as presented below.
Since the Stokes criterion presented by Ennis et al. [69] is derived for spherical particles with equal sizes, the size of the colliding particles should be smaller than𝑥crit in order for agglomeration to occur. As described in Chapter 2, the criterion can be extended to spherical particles of unequal size when the harmonic mean of the particle sizes is used. Correspondingly, a wet collision results in agglomeration when the harmonic mean is smaller than or equal to𝑥crit:
2𝑥1𝑥2
𝑥1+𝑥2 ≤𝑥crit. (6.6)
Note that in this case many size combinations resulting in a successful collision are possible. For example, one of the two particles may be larger than𝑥crit, as long as the resulting harmonic mean is still smaller. Re-arranging Equation (6.6) yields𝑥2as a function of𝑥1. Since this function has a pole
Chapter 6 Macroscopic modeling of the dominant size enlargement mechanism
0 0.1 0.2 0.3 0.4 0
0.2 0.4 0.6 0.8 (a) 1
𝑥1 [mm]
𝑥2[mm]
𝑥crit
Agglomeration regime2
0 0.1 0.2 0.3 0.4 0
0.2 0.4 0.6 0.8 (b) 1
𝑥1 [mm]
𝜉(𝑥1)[−]
Figure 6.2:Plot of the particle size𝑥2(a) and the corresponding fraction of particles with a fulfilled Stokes criterion𝜉(b) as a function of𝑥1for the corresponding example.
at 𝑥crit2 , two cases are obtained:
𝑥2 ≥ 𝑥1𝑥crit
2𝑥1−𝑥crit for 𝑥1<
𝑥crit
2 , (6.7a)
𝑥2 ≤ 𝑥1𝑥crit
2𝑥1−𝑥crit for 𝑥1> 𝑥crit
2 . (6.7b)
Evaluating Equation (6.7) for the limit cases yields:
0<𝑥1< 𝑥crit
2 : 𝑥lim
1→0+
𝑥1𝑥crit
2𝑥1−𝑥crit =0 and lim
𝑥1→𝑥crit2 −
𝑥1𝑥crit
2𝑥1−𝑥crit =−∞, (6.8a) 𝑥crit
2 <𝑥1<∞: lim
𝑥1→𝑥crit2 +
𝑥1𝑥crit
2𝑥1−𝑥crit =∞ and 𝑥lim
1→∞
𝑥1𝑥crit
2𝑥1−𝑥crit = 𝑥crit
2 . (6.8b)
Both cases are shown in Figure 6.2a for the same example as in Figure 6.1. For𝑥1< 𝑥crit
2 ,𝑥2must be larger than the value calculated with Equation (6.7a). Since this value lies between 0 and−∞and only positive values for𝑥2are reasonable, the Stokes criterion is fulfilled for any value of𝑥2when 𝑥1 < 𝑥crit
2 , leading to agglomeration. For𝑥1 > 𝑥crit
2 ,𝑥2must be smaller than the value calculated with Equation (6.7b), which starts at∞and decreases asymptotically to𝑥crit2 . In this case, the particle size distribution plays a role in whether agglomeration occurs. The values of𝑥2for which Equation (6.7) is fulfilled are represented by the gray area in Figure 6.2a for this example.
The fraction of particles with a fulfilled Stokes criterion can be calculated similarly to Equation (6.5).
However, in this case𝜉 depends on𝑥1. The limit cases show that𝜉 equals unity for𝑥1 < 𝑥crit
2 . For 𝑥1 > 𝑥crit
2 , it can be calculated in a similar way as shown above, leading to:
𝜉(𝑥1)=
1 𝑥1< 𝑥crit 2 ,
𝑥2
∫
0
𝑞0d𝑥 𝑥1> 𝑥crit
2 . (6.9)
6.1 Model description For the current example,𝜉 is shown in Figure 6.2b as a function of𝑥1. For small values of𝑥1,𝜉is equal to unity and decreases to zero for increasing values of𝑥1.
The probability of successful wet collisions then follows from𝜉and the probability distribution of the particle size, which is given by the normalized particle size distribution𝑞0:
𝑃coll,wet,suc(𝑥, 𝑥1)=
∫∞ 0
𝜉(𝑥1)𝑞0(𝑥)d𝑥. (6.10)
Theoretical validation
The proposed model to calculate the probability of successful wet collisions can be validated theoret-ically by comparison to results obtained with a Monte Carlo model. In this model, a set of particles with different diameters is created according to a given particle size distribution using the algorithm described in Appendix C. In this example, the particle size distribution shown in Figure 6.1 is used to create a set of 108particles. Binary collisions between these particles are mimicked by randomly choosing two collision partners (according to their number). In each collision, both diameters are checked. A collision is then labeled “successful” when the harmonic mean diameter is smaller than a given critical value. In this example, 107collisions are performed and the fraction of successful wet collisions is calculated. Note that it is implicitly assumed in this example that each collision is
“wet” for simplification. Furthermore, besides checking whether a collision is successful, nothing else happens in this Monte Carlo model. Actual agglomeration, as in Chapter 5, is not considered since the purpose of this model is to validate Equation (6.10).
The Monte Carlo simulation was performed for different values of the critical particle size ranging between 0.15 mm and 0.4 mm. The results are then compared with the analytical model (Equa-tion (6.10)) in Figure 6.3. An increasing𝑥crit leads to larger values of the probability of successful wet collisions since the fraction of particles with a fulfilled Stokes criterion is increased. The results obtained from Equation (6.10) and the Monte Carlo model agree well for all investigated values of the critical particle size.
In order to calculate the probability of successful collisions, a process model providing the necessary parameters is required. The probability of wet collisions (see Equation (6.3)) depends on the wet surface fraction, which is calculated using a novel approach within a spray fluidized bed drying model. Looking at the processes from the point of view of layering, a population balance model for layering growth is used to obtain the transient behavior of the particle size distribution, which is necessary to calculate the probability of successful wet collisions according to Equation (6.10). The process model is presented in the following sections.
Chapter 6 Macroscopic modeling of the dominant size enlargement mechanism
0.15 0.2 0.25 0.4
0 0.2 0.4 0.6 0.8 1 1.2
0.12
0.56
0.94 1
0.12
0.56
0.94 1
𝑥crit [mm]
𝑃coll,wet,suc[−]
MCanalytical
Figure 6.3:Comparison of the probability of successful wet collisions obtained with the proposed analytical model and a Monte Carlo model.