Dimensional Analysis and Scale-up in Chemical Engineering

The chemical engineer is generally concerned with the industrial implementation of processes in which chemical or microbiological conversion of material takes place in conjunction with the transfer of mass, heat, and momentum. These processes are scale-dependent, i.e. they behave differently on a small scale (in laboratories or pilot plants) and a large scale (in production). They include heterogeneous chemical reactions and most unit operations (e.g., mixing, screening, sifting, filtration, centrifugation, grinding, drying, and combustion processes). Understandably, chemical engineers have always wanted to find ways of simulating these processes in models to gain insights that will assist them in designing new industrial plants. Occasionally, they are faced with the same problem for another reason: an industrial facility already exists but will not function properly, if at all, and suitable measurements have to be carried out to discover the cause of the difficulties and provide a solution.

All the essential (“relevant”) physical quantities (variables, parameters) which describe a physical or technological interrelation must be known before this process can be described with a full set of dimensionless numbers. This demands a thorough and critical appraisal of the process being examined.

In the Introduction it was pointed out that using dimensional analysis to handle a physical problem, and thus to present it in the framework of a complete set of dimensionless numbers, is a sure way of producing a simple and reliable scale-up from the small-scale model to the full-scale technical plant. The theory of models states that:

Two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe them have the same numerical value (Π_i = idem).