ANALYSIS AND MEASUREMENT OF CRYSTALLIZATION UTILIZING THE POPULATION BALANCE

ANALYSIS AND MEASUREMENT OF CRYSTALLIZATION UTILIZING THE POPULATION BALANCE 

The size and shape of particles in a particulate product are important for several reasons. The efficiency of any process for production of a particulate material relies on the size, shape, and size distribution. A very small mean size product is difficult to centrifuge, wash, and package. Many products must be dissolved for subsequent use and a broad size range leads to variation in time necessary for dissolution among product crystals. Shape can also be important in applications such as pharmaceuticals wherein different crystal faces have different dissolution rates, leading to variation in bioavailability for crystals of different shapes. Therefore, any comprehensive discussion of crystallization must include the genesis of size and size distribution, as well as its possible measurement and control. 

The population balance approach to analysis of crystallizers was formalized and presented by Randolph and Larson (1971, 1988). The technique parallels other balance approaches such as material and energy balances, which are familiar to process engineers. The population balance is used to account for both the size (an attribute to be described later in this section) and number of particles. Therefore, before a discussion of the development and application of the population balance can be enjoined, it is necessary to consider size and size distributions of particles. 

The concept of particle size and its measurement is both very simple and complex. We all know what we mean by size, but it is difficult to quantify in many instances. In the specific case of size definition for crystallization, we generally resort to a single linear dimension to establish size. This dimension, which we will denote the characteristic dimension, will be determined by the method of measurement to be discussed later in this section. 

Crystals do not occur in single sizes, but have distributions. Two types of crystal size distributions are normally considered, the density distribution and the cumulative distribution. The density distribution is defined as 

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