Mean crystal size

Although programmed cooling crystallization clearly results in a larger mean crystal size than that from natural cooling it is also evident that some fines i.e. small crystals are also present in the product. Since the solution was seeded these fine crystals must clearly have arisen from crystal attrition or secondary nucleation (see Chapter 5). 

Effect of residence time on mean crystal size... 

Figure 7.9 EJfect of mean residence time on mean crystal size parameter i = hjg, the relative nucleation to growth order index)...

Figure 8.32 Evolution of pH and mean crystal size, model versus experimental data of Jones etal. (1992) (Rigopoulos and Jones, 2001)...

The optimal network increases total residence time by 48 per cent when compared with an equivalent MSMPR of the same volume and throughput. This increase would translate into a similar increase in mean crystal size and a 78 per cent increase in yield. Exactly the same residence time as for the single crystallizer have been reported from simple cascade configurations previously designed for stage-wise crystallization processes for slight improvements in... 

Method of Moments. The moments of the population density can be used to characterize the CSD. The first four moments have physical Interpretations and the mean crystal size (Lp. ) and the Coefficient of Variation (CV) based on the mass distribution are functions of the moments ... 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

That is, Topt is proportional to As the average crystal size increases, the interface reaction rate becomes smaller because the degree of oversaturation becomes smaller. Hence, the coarsening rate dtopt/df for mean crystal size of 1 mm is 10 times slower than that for mean crystal size of 1 fim coarsening timescale for a rock to grow from a mean crystal size of 1 to 2 mm is lO times that for a rock to grow from a mean crystal size of 1 to 2 /im. 

Suppose coarsening is controlled by diffusion in the melt phase. If mean crystal size grew from 0.1 to 0.11 mm in 10 years, estimate the time needed for the mean crystal size to grow from 5 to 5.1 mm. 

From the value of /3 cos 0/X found, the mean crystal size, calculated as cube edge, was found to be... 

The mean crystal size is known to increase with increasing the volume of the insonated mixture. One explanation for this behaviour is that a fixed US wave in a larger container produces weaker penetrating and reflecting waves, so vibration and cavitation at some point in the liquid are lower. This results in fewer nuclei, and hence in larger crystals being formed. Also, increased liquid volumes provide larger free spaces for crystals to reduce collision and abrasion with each other. 

When controlling the PSD during crystallization, some characteristics of the PSD of the product are often the focus of processing rather than exactly matching the full size distribution. These characteristics include moments of the PSD, number-mean crystal size, weight-mean crystal size, variance of the distribution function, and the coefficient of variation. Weight-mean crystal size is the most common PSD characteristic used in practice. For the case where the particles have... 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

Several crystallization objectives have been recommended to favor downstream operations or product quality. One can maximize the number-mean or weight-mean crystal size, maximize the final size of crystals grown from seed crystals, or minimize the ratio of nucleated crystal mass to seed crystal mass.f ° Other particle size-related characteristics of product crystals that have been optimized during crystallization include the coefficient of variation and the crystal shape. Although the weight-mean crystal size is the most commonly used objective in optimal control studies, the weight-mean crystal size is too insensitive to the number of small crystals that can cause filtration problems when used as an objective to optimize the crystallization operations. 

Fig. 13 Optimal temperature profile calculated from a first-principle model for maximizing the mean crystal size for unseeded crystallization of paracetamol in water and the simulated change in mean crystal size during crystallization.

The mean size of the crystallites of linear-chain carbon films calculated from the width of their electron diffraction maximums depends on the film thickness (Table 11.1). As can be seen from Table 11.1 the mean crystal size of the films is comparable with the film thickness if that thickness is about 4.0 nm. In this case, the films become highly oriented and there are sharp reflections in their diffraction pattern (Figure 11.1(a)). If the thickness of the films increases to 16.0 nm, the orientation still takes place. In films with a thickness greater than 60 nm amorphization is observed, with the formation of inter-chain bounds. This is clearly seen by increasing the intensity of diffraction maximums corresponding to the three-dimensional amorphous carbon phase and finally by the complete disappearance of diffraction maximums corresponding to the sp phase. 

Figure 6.14. Measurement of the mean crystal size in the direction perpendicular to the (001) planes based on the Fourier analysis...

From these relatiorts and by applying eqiration [6.53], we get the mean crystal size ... 

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