## Modelling of Industrial Crystallizers

The development of an advanced control strategy requires a dynamic model that accurately describes the behaviour of the process. The derivation of such a model implies identification of the model structure and estimation of the model parameters using experimental data, and verification of the model. Parameter estimation falls beyond the scope of this article and will not be covered here.

The modelling of industrial crystallizers is dominated by the presence of the so-called population balance, which describes the evolution in time of the CSD in the crystallizer. The population balance provides a generally accepted approach to the modelling of dispersed phase systems and allows specification of the product quality in terms of the CSD. The general form of the population balance equation (PBE) is as shown in eqn :

ot aL

where the amount and the size of the crystals (or particles) are expressed in terms of number density n(L) and crystal length L. y is the suspension volume in the crystallizer, with m streams entering and n streams leaving the crystallizer at volumetric flow rates of <v. Gl(L) is the linear size-dependent growth rate, and B(L) and D(L) are birth and death rates respectively. Birth and death events can be caused by agglomeration and by the birth of small crystals, called nuclei. The classification function h(L) describes the relation between the CSD in the crystal-lizer and that in an outlet stream.

As the PBE is a partial differential equation with respect to time t and crystal length L, two boundary conditions are needed to solve it analytically:

n(L, 0) = initial distribution

are related as shown in eqn :

For the second boundary condition a seed population or a population formed by the outgrowth of primary nuclei can be substituted.

The mass balance is given by eqn :

dM, total dt

— ^v,feed(Sfeed Pieed ,liquid # (1 ^feed)pcrystal) ^v,product(£pliquid # (1 ^)pcrystal)

<v,vapour p vapour

The component balances are given by: dM,= (

dt < v,feed( ^feed pfeed,liquid Wfeed,liquid,i

# (1 ^feed)pcrystal^feedjcrystal,^ <v,product( liquid Wliquid,i

Mi = V(gpliquid^liquid,; + (1 - fi)Pcrystal^crystal*) 

are where component / = 1 is the main compound to be crystallized, and components i = 2, 3, •••,Ncomp the impurities present.

The distribution coefficients relate the impurity up^ take by the solid and the concentration of the impu rity in the liquid phase, as shown in eqn 

The enthalpy balance with the production rate or solids production P is given by eqn :

-H,product

-H,vapour

As primary and secondary nucleation typically involves the birth of small crystals, nucleation is often presented as the birth of nuclei at zero size. Instead of a birth term in the PBE for the nucleation event B(L) that happens over a size range 0 < L < y, the birth rate B0 given by boundary eqn  is used. These two in which H denotes the enthalpy of the crystallizer content and the enthalpy of the particular stream. Qheat is the effective heat input to the system including heat losses.

In order to complete the model, relations are needed for nucleation and growth. In many cases a size-independent growth rate is determined with a power law relation, as shown in eqn :

o where o is the relative supersaturation. For the secondary nucleation a comparable function is used (eqn )

B0 = kNohNhMT

LJp4

the population balance (eqn ) simplifies into eqn :

in which N is the stirrer frequency and MT is the total crystal mass in the crystallizer. Note that B0 and not B is calculated with this relation. Although these functions are still used, especially in controller design studies, it has been shown that in reality the growth rate function is more complicated and size dependent. In addition, the power law for the nucleation given in eqn  is not suitable for describing the dynamics in secondary nucleation dominated crystallization systems and improved relations have proven to give a much better description of these kinetic processes. Here the relations are used because they have been applied in controller design studies. The growth rate is given by eqn :

with similar boundary conditions. This equation can be reduced to a set of ordinary differential equations using the moment transformation (eqn ):

dmj mj j

in which L is the crystal size. In this relation the growth rate decreases beyond a certain crystal size La due to attrition, while Le is the maximum crystal size of the crystals. p6, p7 and p8 are model parameters, which have to be estimated. The relation for the secondary nucleation is shown in eqn :

Here B0 is calculated assuming tfc a negligible size. Note that only crystals beyond a certain size p4 contribute to the secondary nucleation. ## Solar Panel Basics

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