In certain applications, for example crystallization, length is considered as the relevant particle property. It is therefore interesting to rewrite the PBE (2.20) in term of the number density function which uses particle length l as the internal property coordinate.
To this end it is necessary to invoke some assumptions about the relationship between particle volume x ∈ R+ and particle length l ∈ R+. Fore example one can consider the case x=l3. In this case we have the following relationship between the two expressions of the number density function
f(t, x)dx=f(t, l3) 3l2dl =n(t, l)dl , (2.25) where n(t, l) ≥ 0 is the length based number density and f(t, x)≥ 0 is the volume based number density. Moreover, let G′(t, l), β′(t, l, l′), b′(l, l′), S′(l) respectively denote the length based growth rate, aggregation kernel, breakage kernel and selection kernel.
After multiplying equation (2.20) on both sides by 3l2, we obtain
∂n(t, l)
∂t =− ∂[G′(t, l)n(t, l)]
∂l +Knuc+ (t, l) +K±agg(t, l) +K±break(t, l), (t, l)∈R2+, (2.26) where G′(t, l) =G(t, x)/3l2 and according to derivations in Appendix A.2 we obtain
K±agg(t, l) =l2 2
Z l
0
β′(t,(l3−l′3)13, l′)
(l3−l′3)23 n(t,(l3−l′3)13)n(t, l′)dl′
− Z ∞
0
β′(t, l, l′)n(t, l)n(t, l′)dl′, (2.27) K±break(t, l) =
Z ∞
l
b′(t, l, l′)S′(l′)n(t, l′)dl′ −S′(l)n(t, l), (2.28) K+nuc(t, l) =nnuc(t, l)Bnuc(t). (2.29) It is interesting to note thatβ′(t, l, l′) andS′(l), being intensive properties of the particulate system, can be converted easily from the volume-based to the length based number density functions, whereas b′(t, l, l′) results in
b′(t, l, l′) = 3l2b(t, l3, l′3) and Z l
0
l′3b′(t, l′, l)dl′ =l3. (2.30) In order to apply a finite volume schemes one can rewrite equation (2.26) in the following form
l3∂n(t, l)
∂t =−l3∂[G′(t, l)n(t, l)]
∂l +l3Knuc+ (t, l) +l3K±agg(t, l) +l3Kbreak± (t, l). (2.31)
Let us define ˜n(t, l) :=l3n(t, l) and using the relation l3 ∂[G′(t, l)n(t, l)]
∂l = ∂[l3G′(t, l)n(t, l)]
∂l −3l2G′(t, l)n(t, l)
= ∂[G′(t, l)˜n(t, l)]
∂l −3G′(t, l)˜n(t, l)
l (2.32)
in (2.31), we get
∂n(t, l)˜
∂t =−∂[G′(t, l)˜n(t, l)]
∂l + 3G′(t, l)˜n(t, l) l + ˜K+nuc(t, l)−∂F˜agg′ (t, l)
∂l +∂F˜break′ (t, l)
∂l , (t, l)∈R2+, (2.33) where ˜n:R≥0×R+→R≥0. Here ˜K+nuc(t, l) =l3K+nuc(t, l) and
F˜agg′ (t, l) = Z l
0
Z ∞
(l3−u3)13
u3β′(t, u, v)n(t, u)n(t, v)dvdu , (2.34) F˜break′ (t, l) =
Z l
0
Z ∞
l
u3b′(t, u, v)S′(v)n(t, v)dvdu . (2.35) If we compare (2.31) and (2.33) it comes out that
K±agg =−1 l3
∂F˜agg′
∂l , and K±break= 1 l3
∂F˜break′
∂l . (2.36)
The left hand side relation in (2.36) is verified in Appendix A.3. The proof for the right hand side relation is analogous to (2.16), therefore we skip its proof.
The jth moment µj(t) of this number density is defined as µj(t) =
Z ∞
0
ljn(t, l)dl . (2.37)
As mentioned in Chapter 1, in crystallization processes the population balance models are coupled with the mass (mole) balance of the liquid phase which is an ordinary differential equation (ODE) for the solute massm(t). In that case the growth and nucleation are also functions of m(t). Furthermore, an ODE for the temperature T(t) which can be obtained from the energy balance of the crystallizer may also exist and coupled with the correspond-ing PBE and mass balance equation of the liquid phase. In Chapter 3 and 4 we will give the detailed models for the batch and preferential crystallization processes.
Note that in case of crystallization processes, Chapters 3 and 4, the crystal length will be considered as the internal property variable. For other particulate processes, Chapters 5 and 6, particle volume will be taken as the internal property variable.
Batch Crystallization
This chapter starts with a short introduction of crystallization process. Two different operational modes of industrial crystallization plants are briefly explained. We give the one-dimensional population balance models for the simulation of batch crystallization and prove the local existence and uniqueness of the solution of this model. For that purpose Laplace transformation is used as a tool. With the help of inverse Laplace transformation, we derive a new method which can be used to solve numerically the given Batch crystallization model. Afterward, the model is extended to the two-dimensional case. Moreover, the one and two-dimensional semi-discrete finite volume schemes are derived for the numerical solution of the these models. For the one-dimensional schemes the issues of positivity (monotonicity), consistency, stability and convergence are also discussed. To improve the numerical accuracy of the schemes further a moving mesh technique is introduced. Finally, several numerical test problems are considered for the validation of the schemes and are compared with the available analytical solutions.
3.1 Fundamentals of Crystallization
Crystallization is the process of formation of solid crystals from a homogeneous solution and is essentially a solid-liquid separation technique. It is an important separation and purification process used in pharmaceutical, chemical and food industries.
The crystallization concept is very simple and well known. A solution can become super-saturated either by cooling or by evaporation of solvent. The process consists of two major events, nucleation and crystals growth. In case of nucleation, the solute molecules dis-persed in the solvent come together to form stable clusters in the nanometer scale under current operating conditions. These stable clusters constitute the nuclei. However when the clusters are not stable, they re-dissolve. Therefore, for stable nuclei the clusters need to achieve a critical size. Such a critical size is dictated by the operating conditions (tem-perature, supersaturation, etc.). In stable nuclei the atoms are arrange in a defined and periodic manner defining the crystal structure. Note that crystal structure is a special
21
term that refers to the internal arrangement of the atoms, but not the physical external macroscopic properties of the crystal such as size and shape. The crystal growth is the subsequent growth of the nuclei that succeed in achieving the critical cluster size. During nucleation and growth the solute mass transfers from the liquid solution to solid crystals.
Consequently, nucleation and growth continue to occur simultaneously as far as the super-saturation exists.
Supersaturation is the driving force of the crystallization, hence the rate of nucleation and growth is driven by the existing supersaturation in the solution. Depending upon the conditions, either nucleation or growth may be predominant over the other. As a result, crystals with different sizes and shapes are obtained. The most significant property of crystals is their size. Crystal size distribution (CSD) is the crucial variable in industrial crystallizers. On the one hand, CSD helps in understanding the dynamics of crystallization plant. On the other hand, CSD is important due to its heavy influence on the product quality and down-stream processability. It influences properties such as filterability, the ability to flow or the dissolution rate of crystalline materials. Industrial crystallization plants can be operated either in continuous mode or in batch mode.
In case of continuously operated crystallization plant the solution is continuously fed to the crystallizer and product is continuously withdrawn. Continuous processes run for very long period of time and serve for the production of large amounts of bulk materials. They are desired to be operated at a steady state. Hence the product quality is determined by the steady state CSD. This quantity can be influenced by fines dissolution, i.e. the continuous removal and dissolution of small particles. Unfortunately, apart from the desired effect on the CSD this may lead to instability of the steady state. As a result damped oscillations in CSD and supersaturation occur, see [107, 108, 121].
On the other hand, batch cooling crystallization is used for the small scale production of high-value-added fine chemicals and pharmaceuticals. Figure 3.1 shows one of such crystal-lizers. In this case the product quality is determined by the CSD at the end of the batch, which can be influenced by the cooling profile, i.e. the temperature trajectory during the batch run. In contrast to the continuous crystallization, batch crystallization is a transient process and does not achieve steady state. In batch mode there are, obviously, no feed and product removal streams. Consequently the corresponding terms does not exist in the batch model. In contrast to the continuous crystallization where nucleation, crystal growth and attrition are described by detailed first principle models, in batch mode em-pirical relations are used to describe nucleation and growth rates. The kinetic parameters involved in these equations have to be determined by parameter identification techniques from experimental data. The parameters summarize dependencies on the chemical system, the crystallizer type, size and geometry and the operating conditions such as temperature range or stirrer speed.
Tc
Tc
Cooling Jacket Draft Tube
T, m, n(l)
0000000000000000 0000000000000000 1111111111111111 1111111111111111
Figure 3.1: Batch crystallizer.