Population Balance Equation: a Tool for Dynamic Modeling of Particulate Processes

Particulate processes are often encountered in industry. The polydispersity of some key properties characterizes particulate processes and makes them ill-suited to be modeled within the framework of conventional conservation equations only. In the mid 60s a tool called Population Balance Equation was developed to quantify the dynamics of particulate processes. This paper aims to give an introduction to the Population Balance Equation and review its state of the art. Introduction Particulate processes are encountered in many applications and are widely used in the production of valuable industrial products. Crystallization, biochemical processes, polymerization, leaching, comminution, and aerosols are just some examples of particulate processes. Such processes differ considerably from one another but they all are characterized by the presence of a continuous phase and a dispersed phase comprised of entities with a distribution of properties, such as size, chemical composition, etc. The entities typically interact with one another as well as with the continuous phase. Such interactions may vary from entity to entity. Therefore, the polydispersity of particulate processes affects significantly the behavior of such systems, thus affecting the quality of the final products. Moreover, the polydispersity makes particulate processes ill-suited to be modeled within the framework of conventional conservation equations only. This paper deals with the Population Balance Equation, a tool to quantify the dynamics of particulate processes, which was developed in the mid 60s and since then has experienced wide acceptance and use in the field of particulate processes. This paper describes the formulation of the Population Balance Equation, and reviews the strenghts, the weaknesses and the status of this approach. Formulation The Population Balance Equation was originally derived in 1964, when two groups of researchers studying crystal nucleation and growth recognized that many problems involving change in particulate systems could not be handled within the framework of the conventional conservation equations only, see Hulburt & Katz (1964) and Randolph (1964) . They proposed the use of an equation for the continuity of particulate numbers, termed population balance equation, as a basis for describing the behavior of such systems. This balance is developed from the general conservation equation Accumulation = Input Output + Net Generation (1) applied to the number of entities having a specified set of n properties ζi i = 1,2...n. The properties ζi to be considered for the number balance will depend on the application. Typical examples are the entity's size diameter, entity's chemical composition, entity's age... In equation 1, all the terms represent number of entities with the specified property in a given interval, each term being related to certain transport, generation or destruction processes. Thus the accumulation term is the change of number of entities in a given property interval by accumulation in the system, the input and output terms are related to convective flow, the generation term includes both generation and destruction by continuous or discrete processes. Examples of continuous processes are chemical reaction or precipitation. Discrete generation processes are those giving birth or death of entities in a given property interval such as nucleation (birth), agglomeration (birth) or breakage (death). The derivation of the Population Balance Equation is equivalent to the development of the conventional equations of change, i.e. the number balance shown in equation 1 is applied to a volume element of the system ∆x ∆y ∆z fixed in the space, the resulting equation is divided by the volume element and the limit as ∆x ∆y ∆z go to zero is taken. By doing this, the microscopic Population Balance Equation is obtained ( ) ( ) ( ) D B v