MSMPR equations

The ability to predict ciystal behavior in complex systems is ahead of our ability to manipulate crystal and liquid residence times. Generalizations of the MSMPR equations have been made to predict CSD with arbitrary process configurations and kinetics however, (here is no guarantee that an assumed process residence-tima cfisttfbetxm can be physically implemented to produce a customized product CSD. Size distributions In cascaded ciystallizers. for example, muhipte-effea evaporators, can be computed if kinetic... 

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. 

Many industrial crystallizers operate in a weU-mixed or nearly weU-mixed manner, and the equations derived above can be used to describe their performance. Furthermore, the simplicity of the equations describing an MSMPR crystallizer make experimental equipment configured to meet the assumptions lea ding to equation 44 useful in determining nucleation and growth kinetics in systems of interest. 

The dominant crystal size is given by = 3Gr. This quantity is also the ratio mJwhich is often given the symbol 2-(J) Prom the definition of the coefficient of variation given by equation 41, cv = 50% for an MSMPR crystallizer. Such a cp may be too large for certain commercial products, which means either the crystallizer must be altered or the product must be screened to separate the desired fraction. 

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. 

The general form of the population density funetion from the ideal MSMPR erystallizer (equation 3.15) has rather fortunate statistieal properties sueh that... 

The following equation of moments (Randolph and Larson, 1988) for an ideal MSMPR erystallizer... 

It was shown in Chapter 3 that the ideal eontinuous MSMPR erystallizer eould be analysed using the population balanee approaeh eoupled with mass balanees and erystallization kineties to yield equations deseribing erystallizer perform-anee in terms of the erystal size distribution, solids hold up ete. These eoneepts will now developed further to yield methods for eontinuous erystallizer design. Firstly, however, it is useful to eonsider how erystallization kineties and erystallizer performanee interaet. 

From Chapters 2, 3 and 5, the basie deseribing equations for the MSMPR erystallizer at steady state are as follows... 

Recall the design equations for the MSMPR Slurry density Mt =... 

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 ... 

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. 

From Equations a-7 and a-8, for the case of j=l, i.e., for MSMPR, the following well-known expressions,... 

Form MSMPR (j=l) and P=1 (B=0), from Equations a-14, a-13 and a-8, the following well-known equation is obtained,... 

Figure 7-8 shows a simplified information flow diagram for a continuous MSMPR crystallizer. Population balance equations (see Chapter 4) can be used to separate nucleation and growth effects. For particles keeping geometric similarity, the surface area of the particles for... 

MOMENT EQUATIONS. Equation (27.29) is the fundamental relation of the MSMPR crystallizer. From it diflerential and cumulative equations can be derived for crystal population, crystal length, crystal area, and crystal mass. Also, the kinetic coefficients G and are embedded in these equations. 

Equations (27.32) to (27.35) give directly the distribution of the crystals from the idealized MSMPR plant Thus, is the number distribution x , the size distribution Xi, H2 the area distribution x , and (i-i the mass distribution x . 

Use the moment equations for constant growth rate in a MSMPR crystallizer to calculate (a) the surface-volume mean size and (6) the mass average size, (c) Compare these values with the sizes where a maximum occurs in the corresponding distribution curves. 

Continuous MSMPR Precipitator. The population balance, which was put forward by Randolph and Larson (1962) and Hulbert and Katz (1964), provides the basis for modeling the crystal size distribution (CSD) in precipitation processes. For a continuous mixed-suspension, mixed-product-removal (CMSMPR) precipitator with no suspended solids in the feed streams, the population balance equation (PBE) can be written as (Randolph and Larson 1988)... 

Nevertheless, as discussed previously, the physical model for a crystallizer is an integro-partial differential equation. A common method for converting the population balance model to a state-space representation is the method of moments however, since the moment equations close only for a MSMPR crystallizer with growth rate no more than linearly dependent on size, the usefulness of this method is limited. The method of lines has also been used to cast the population balance in state-space form (Tsuruoka and Randolph 1987), and as mentioned in Section 9.4.1, the blackbox model used by de Wolf et al. (1989) has a state-space structure. 

Hashemi and Epstein (1982) linearized the set of ordinary differential equations (ODEs) resulting from the application of the method of moments on an MSMPR crystallizer model and used singular value decomposition to define controllability and observability indices. These indices aid in selecting measurements and manipulated and control variables. Myerson et al. (1987) suggested the manipulation of the feed flow rate and the crystallizer temperature according to a nonlinear optimal stochastic control scheme with a nonlinear Kalman filter for state estimation. 

Another objective in the study of the application of CFD in crystallization is to simulate the particle size distribution in crystallization. In order to solve this problem, the simulation should take into account the population balance. The internal coordinates of the population balance make it difficult to utilize it in the CFD environment. In addition, different-sized particles have different hydrodynamics, which causes further complications. Wei and Garside [42] used the assumption of MSMPR and the moments of population balance to avoid the above difficulties in the simulation of precipitation. In the CFX commercial application, the MUSIC model offers a method for solving the population balance equation in CFD and defines the flow velocity of different-sized particles... 

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