Dimensionless numbers, from similarity analysis

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

This method can be easily used to show the logic behind the scale-up from original R D batches to production-scale batches. Although scale-of agitation analysis has its limitations, especially in mixing of suspension, non-Newtonian fluids, and gas dispersions, similar analysis could be applied to these systems, provided that pertinent system variables were used. These variables may include superficial gas velocity, dimensionless aeration numbers for gas systems, and terminal settling velocity for suspensions. 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

For the situation of forced flow, it follows from dimension analysis that a dimensionless form of the mass transfer coefficient, the so-called Sherwood number (Sh), is a given function of the Reynolds Re) and Schmidt (Sc) numbers, for geometrically similar arrangements ... 

This is similar to the analysis obtained by Ainsley and Smith (see Chhabra, 1992) using the slip line theory from soil mechanics, which results in a dimensionless group called the plasticity number ... 

Then the H-theorem states that the number of dimensionless groups resulting from analysis will be equal to the number of relevant variables minus the number of dimensions (corresponding to the number of fundamental units).If two systems are similar in the relevant sense (i.e. in terms of the mechanisms described by the variables occurring in the O s), then corresponding Il s must be the same in each case. In its most general form this can be written as (IIi,Il2,Il3,...) =... 

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