MSMPR

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. 

GSD Characteristics for MSMPR Crystallizers. The perfectiy mixed crystallizer described ia the preceding discussion is highly constrained and the form of crystal size distributions produced by such systems is fixed. Such distributions have the foUowiag characteristics. 

Moments of the distribution can be calculated for MSMPR crystallizers by the simple expression... 

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. 

A pair of kinetic parameters, one for nucleation rate and another for growth rate, describe the crystal size distribution for a given set of crystallizer operating conditions. Variation ia one of the kinetic parameters without changing the other is not possible. Accordingly, the relationship between these parameters determines the abiUty to alter the characteristic properties (such as dominant size) of the distribution obtained from an MSMPR crystallizer (7). 

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)] ... 

In the specific case of an MSMPR exponential distribution, the fourth moment of the distribution may be calculated as... 

Crystallizers with Fines Removal In Example 3, the product was from a forced-circulation crystallizer of the MSMPR type. In many cases, the product produced by such machines is too small for commercial use therefore, a separation baffle is added within the crystallizer to permit the removal of unwanted fine crystalline material from the magma, thereby controlling the population density in the machine so as to produce a coarser ciystal product. When this is done, the product sample plots on a graph of In n versus L as shown in hne P, Fig. 18-62. The line of steepest ope, line F, represents the particle-size distribution of the fine material, and samples which show this distribution can be taken from the liquid leaving the fines-separation baffle. The product crystals have a slope of lower value, and typically there should be little or no material present smaller than Lj, the size which the baffle is designed to separate. The effective nucleation rate for the product material is the intersection of the extension of line P to zero size. 

Although surface-cooled types of MSMPR crystalhzers are available, most users prefer crystallizers employing vaporization of solvents or of refrigerants. The primary reason for this preference is that heat transferred through the critical supersaturating step is through a boil-ing-hquid-gas surface, avoiding the troublesome solid deposits that can form on a metal heat-transfer surface. 

If an elutriation leg or other product-classifying device is added to a ciystaUizer of the MSMPR type, the plot of the population density... 

The mixed suspension, mixed product removai (MSMPR) crystaiiizer... 

The flow of slurry within all the agitated erystallizer vessels illustrated is elearly eomplex and mixed to a greater or lesser extent at the mieroseopie level. In order to ease theoretieal analysis a new type of vessel therefore had to be invented This idealized vessel has beeome known as the eontinuous MSMPR erystallizer, after Randolph and Lawson (1988). The MSMPR is the erystal-lization analogue of the CSTR (eontinuous stirred tank reaetor) employed in idealizations of ehemieal reaetion engineering. 

The CSD from the continuous MSMPR may thus be predicted by a combination of crystallization kinetics and crystallizer residence time (see Figure 3.5). This fact has been widely used in reverse as a means to determine crystallization kinetics - by analysis of the CSD from a well-mixed vessel of known mean residence time. Whether used for performance prediction or kinetics determination, these three quantities, (CSD, kinetics and residence time), are linked by the population balance. 

Figure 3.6 Schematic particle flows in the ideal MSMPR crystallizer at steady state...

The MSMPR erystallizer provides a partieularly elegant way to illustrate the derivation of the population balanee under eertain assumptions. Within the MSMPR erystallizer erystals leave any given size range either by growth or by outflow in the produet stream (Figure 3.6). 

A log-linear plot of the idealized eontinuous MSMPR population density versus erystal size is shown in Figure 3.7. 

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. 

Figure 3.7 Crystal population distribution from the MSMPR crystallizer...

A pilot-scale continuous MSMPR crystallizer of 10 litre capacity is used to crystallize potash alum from aqueous solution, supersaturation. This is being achieved using a 15-min residence time, a 100-ml slurry sample was taken and the crystals contained in this sample subjected to a size analysis. The results of this analysis are given below... 

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

The mass distribution from the idealized MSMPR crystallizer is thus a Gamma function, as shown in Figure 3.8b. 

Figure 3.8 MSMPR mass distributions a) cumulative, (h) differential...

The population balance analysis of the idealized MSMPR crystallizer is a particularly elegant method for analysing crystal size distributions at steady state in order to determine crystal growth and nucleation kinetics. Unfortunately, the latter cannot currently be predicted a priori and must be measured, as considered in Chapter 5. Anomalies can occur in the data and their subsequent analysis, however, if the assumptions of the MSMPR crystallizer are not strictly met. 

Figure 5.14 MSMPR size distriimtion on log-linear co-ordinates...

Evidence for secondary nucleation has came from the early continuous MSMPR studies. MSMPR crystallization kinetics are usually correlated with supersaturation using empirical expressions of the form... 

Table 5.2 Crys tailization kinetics obtained in some early MSMPR studies after Garside and Shah, 1980)...

Similarly, the dependenee j of nueleation rate B on magma density Mt and stirrer speed N in MSMPR erystallizers given by... 

Where most values of i a 1, within the range 0.14-1.07. Additionally, a dependenee on stirrer speed k is observed being in the range 0 < /c < 7.8 (Garside and Shah, 1980). Thus the observed nueleation rates in MSMPR erystallizers depend on both magma density and stirrer speed. They are thus likely to have been due to seeondary rather than primary nueleation. Further evidenee suggests these effeets result from erystal/solution interaetions (see Chapter 5). 

Garside elal. (1979) measured size distributions of seeondary nuelei and reported their variation with supersaturation. Signifieant inerease of nuelei with supersaturation is observed. Thus the proeess is not simply an attrition event alone, but is also related to the level supersaturation at whieh parent erystal is growing. Jones elal. (1986) also observed anomalous growth of seeondary nuelei in a study of the eontinuous MSMPR erystallization of potassium sulphate with eonsequenees inferred for seeondary nueleation rates. Girolami and Rousseau (1986) demonstrate the importanee of initial breeding meehanism in seeded potash alum bateh erystallization. The number of erystals... 

Mydlarz and Jones (1990a,b) determined agglomeration of potash alum in eontinuous MSMPR erystallization from aqueous solution and its effeet on slurry filterability (see Chapter 9). 

Figure 6.11 Idealized MSMPR CSD (a) Nucleation and growth only, (h) effect of agglomeration...

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. 

In the MSMPR crystallizer at steady state, the increase of particle number density brought about by particle growth and agglomeration is compensated by withdrawal of the product from the crystallizer. 

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