MSMPR model

Consider the erystal size distribution in a model MSMPR erystallizer arising beeause of simultaneous nueleation, growth and agglomeration of erystalline partieles. Let the number of partieles with a eharaeteristie size in the range L to L + dL be n L)dL. It is assumed that the frequeney of sueeessful binary eollisions between partieles (understood to inelude both single erystals and previously formed agglomerates) of size V to V + dV and L to Ll +dL" is equal to j3n L )n L")dL dL". The number density n L) and the eollision frequeney faetor (3 are related to some eonvenient volumetrie basis, e.g. unit volume of suspension. 

Once the theoretical yield from a crystallizer has been calculated from mass and energy balances, there remains the problem of estimating the CSD of the product from the kinetics of nucleation and growth. An idealized crystallizer model, called the mixed suspension-mixed product removal model (MSMPR), has served well as a basis for identifying the kinetic parameters and showing how knowledge of them can be applied to calculate the performance of such a crystallizer, ... 

The crystallizer model that led to the development of equations 44 and 45 is referred to as the mixed-suspension, mixed-product removal (MSMPR) crystallizer. 

Equation (18-31) contains no information about the ciystalhzer s influence on the nucleation rate. If the ciystaUizer is of a mixed-suspension, mixed-product-removal (MSMPR) type, satisfying the criteria for Eq. (18-31), and if the model of Clontz and McCabe is vahd, the contribution to the nucleation rate by the circulating pump can be calculated [Bennett, Fiedelman, and Randolph, Chem. E/ig, Prog., 69(7), 86(1973)] ... 

Now the speeial utility of the MSMPR population balanee model equation at steady state ean be elearly seen. Firstly, at known residenee time, t, the Growth rate, G, may be obtained from the slope (= —1/Gt) of the plot in Figure 3.7. 

Reliable kinetie data are of paramount importanee for sueeessful modelling and seale-up of preeipitation proeesses. Many data found in the literature have been determined assuming MSMPR eonditions, analogous to the CSTR model in reaetion engineering. Here, a method developed by Zauner and Jones (2000a) is outlined. 

The assumption of perfeet mixing applied in the bateh and eontinuous MSMPR erystallizer models does not always apply, however, espeeially as vessel size inereases. Methods for aeeounting for imperfeet mixing and seale-up are eonsidered in the next ehapter. 

A non-ideal MSMPR model was developed to account for the gas-liquid mass transfer resistance (Yagi, 1986). The reactor is divided into two regions the level of supersaturation in the gas-liquid interfacial region (region I) is higher than that in the main body of bulk liquid (region II), as shown in Figure 8.12. 

The model prediets that due to negligibly small nueleation rates in the main body of liquid, the number of small erystals is redueed eompared with the ideal MSMPR model. 

Falope etal. (2001) extended the MSMPR model of agglomerative erystal preeipitation based on the Monte Carlo simulation teehnique to aeeount for partiele disruption by eonsidering two alternative partiele size reduetion meehanisms - one representative of partiele splitting into two parts of equal volume, the other representative of miero attrition. 

Figure 8.27 Comparing Monte Carlo model predictions with MSMPR experimental data for calcium carbonate due to Hostomsky and Jones, 1991 (Faiope etal., 2001)...

Mydlarz, I. and Jones, A.G., 1989. On modelling the size-dependent growth rate of potassium sulphate in an MSMPR crystallizer. Chemical Engineering Communications, 90, 47-56. 

Sheikh, A.Y. and Jones, A.G., 1996. Dynamic flow sheet model for an MSMPR crystal-liser. In Industrial Crystallization 96. Ed. B. Biscans, Toulouse, Progep, 16-19 September 1996, pp. 583-588. 

Wojcik, J. and Jones, A.G., 1998a. Dynamics and stability of continuous MSMPR agglomerative precipitation numerical analysis of the dual particle coordinate model. Computers and Chemical Engineering, 22, 535-545. 

CSD modelling based on population balance considerations may be applied to crystalliser configurations other than MSMPR(37) and this has become a distinct, self-contained branch of reaction engineering)56,59,60 73). 

The development and refinement of population balance techniques for the description of the behavior of laboratory and industrial crystallizers led to the belief that with accurate values for the crystal growth and nucleation kinetics, a simple MSMPR type crystallizer could be accurately modelled in terms of its CSD. Unfortunately, accurate measurement of the CSD with laser light scattering particle size analyzers (especially of the small particles) has revealed that this is not true. In mar cases the CSD data obtained from steady state operation of a MSMPR crystallizer is not a straight line as expected but curves upward (1. 32. 33V This indicates more small particles than predicted... 

A number of investigators developed empirical growth rate expressions that included a size dependence. These models were siunmarized by Randolph Q2> 341 who showed that they all produced a concave upward semi-log population density plot thus are useful for empirical fits of non-linear MSMPR CSD data, lliese models however, supply no information on what is actually happening to cause the non-linear CSD. 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

In conclusion, the method of moments can be used to obtain a state space model for the dynamics of the moments of the CSD. The method is limited to MSMPR crystallizers with size-independent growth or size-dependent growth described by... 

A different approach in the use of orthogonal polynomials as a transformation method for the population balance is discussed in (8 2.) Here the error in Equation 11 is minimized by the Method of Weighted Residuals. This approach releases the restrictions on the growth rate and MSMPR operation, however, at the cost of the introduction of numerical integration of the integrals involved, which makes the method computationally unattractive. The applicability in determining state space models is presently investigated and results will be published elsewere. 

If an elutriation leg or other product-classifying device is added to a crystallizer of the MSMPR type, the plot of the population density versus L is changed in the region of largest sizes. Also the incorporation of an elutriation leg destabilizes the crystal-size distribution and under some conditions can lead to cycling. To reduce cycling, fines destruction is usually coupled with classified product removal. The theoretical treatment of both the crystallizer model and the cycling relations is discussed by Randolph, Beer, and Keener (loc. cit.). 

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