MSMPR crystallization model

A growing crystal moves with time along the size axis in the direction of increasing L. A dissolving crystal moves in the other direction. For computations in the MSMPR model a relation between population density n and size L is needed. 

Consider the dL crystals between sizes L and L + dL per unit volume of magma in the crystallizer. In the MSMPR model, each crystal of length L has the same age, and if is the age of a crystal. 

Assume, now, that of the n dL crystals per unit volume of liquid, An dL are withdrawn as product during time increment At. Since the operation is in steady state, withdrawal of product does not aifect the size distribution in either magma or product, and since in the MSMPR model the discharge is accurately representative of the magma, it follows that the fraction of particles withdrawn is identical to the ratio of the volume of product liquid taken out in time At to the total volume of liquid in the crystallizer. Then if g is the volumetric flow rate of liquid in the product and V. is the total volume of liquid in the crystallizer. 

Constant Crystal Growth Model. In this instance, crystals have an inherent constant growth rate, but the rate from crystal-to-crystal varies. The modeling of this phenomenon must be accomplished by use of probability transform techniques due to the presence of a growth rate distribution. The complete solution for the population density yields a semilogarithmic population density plot that is concave upwards similar to size-dependent growth (Berglund and Larson 1984). Since it is relatively difficult to handle, a moment approximation was developed for an MSMPR crystallizer (Larson et al. 1985). 

The majority of applications of crystal population balance modeling have assumed that the solution and suspension in the crystallizer are homogeneous, i.e., the Mixed-Suspension Mixed-Product Removal (MSMPR) approximation (Randolph and Larson 1988). (This is simply the analog of the Continuous Stirred Tank (CSTR) (Levenspiel 1972) approximation for systems containing particles. It means that the system is well mixed from the standpoint of the solute concentration and the particle concentration and PSD. In addition, the effluent is assumed to have the same solute concentration, particle concentration, and PSD as the tank.) This approximation is clearly not justified when there is significant inhomogeneity in the crystallizer solution and suspension properties. For example, it is well known that nucleation kinetics measured at laboratory scale do not scale well to full scale. It is very likely that the reason they do not is because MSMPR models used to define the kinetic parameters may apply fairly well to relatively uniform laboratory crystallizers, but do considerably worse for full scale, relatively nonhomogeneous crystallizers. 

The traditional study of suspension crystallization has been carried out using the MSMPR crystallization model. It has been found that uniform mixing in a commercial-size crystallizer, as required by the MSMPR model, is impossible to achieve. Therefore, the understanding of industrial crystallization is hampered by the use of the MSMPR model. Also, it is difficult to experimentally study the effects of mixing on crystallization, as described earlier in Section 64.2.5. Therefore, the CFD presents the means for local simulation in the tank. Furthermore, CFD simulation enables the tank to be designed so that the shape and the positioning of the impellers and the liquid velocity create the optimal level of supersaturation and mass transferrate in all locations. This is likely to result in a narrowing of the particle size distribution. 

The model predicts that due to negligibly small nucleation rates in the main body of liquid, the number of small crystals is reduced compared with the ideal MSMPR model. 

Falope etal. (2001) extended the MSMPR model of agglomerative crystal precipitation based on the Monte Carlo simulation technique to account for particle disruption by considering two alternative particle size reduction mechanisms - one representative of particle sphtting into two parts of equal volume, the other representative of micro attrition. 

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. 

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. 

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... 

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... 

Equation (18-31) contains no information about the ciystallizer s influence on the nucleation rate. If the crystallizer 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 valid, the contribution to the nucleation rate by the circulating pump can be calculated [Bennett, Fiedelman, and Randolph, Chem. Eng. Frog., 69(7), 86(1973)] ... 

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.). 

This misconception is particularly common in crystallization. The hypothesis of a perfectly mixed system is, for crystallization and precipitation processes, labeled as mixed-suspension, mixed-product removal (MSMPR). With diis model the crystalUzer is modeled with a spatially homogeneous NDF, generally called the crystal-size distribution (CSD). However, the fact that the CSD is constant through the vessel does not mean that the rates of crystal nucleation, molecular growth, aggregation, and breakage are constant. 

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, ... 

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