Page 31 - Packed bed columns for absorption, desorption, rectification and direct heat transfer

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phenomenon, it is necessary to limit the differential equation by additional
uniqueness conditions [33],
The uniqueness conditions can be:
-geometrical sizes of the system (installation) in which the processes take place;
-physical constants of the substances in the system;
-characteristics of the initial conditions as initial temperature, initial
concentration, initial velocity etc.;
-boundary conditions of the system.
The uniqueness conditions can be given in the form of equations which
connect the physical values important for the process. These conditions not only
separate the given phenomenon from the general class, but also supplement the
differential equation for more complete characterization of the process.
Moreover, the differential equations could be solved only together with the
uniqueness conditions at the boundary.
The solutions of the differential equations are analytical expressions
correlating the characteristics of the given phenomenon. They are usually used
as equations for calculating the apparatuses in chemical and similar branches of
technology. Unfortunately very often, tiiere are no mathematical methods to
solve the differential equations at the corresponding uniqueness conditions.
The simplest method, which can be used in this case, is to carry out
experiments and based on the obtained data to receive relations for calculation
of the apparatuses and processes. The disadvantage of this method is that the
obtained equations are valid only for the condition at which they are obtained.
Such types of equations are used in chemical engineering, but it is quite better
to be able to extend the results of the single experiment on a larger area of
phenomena. This possibility is given by the similarity theory and the
dimensional analysis.
The similarity theory proposes a method for obtaining of arithmetical
equations for calculation of a given class of processes. It is largely used in the
area of chemical engineering and similar branches of industry and reduces
strongly the necessary experimental information.
One of the basic principles of the similarity theory is separation of a
group of similar phenomena from the big class of phenomena described by a
general low, for example processes of fluid flow motion, diffusion processes,
heat conductivity, and so on.
According to the similarity theory, similar are these phenomena for
which the ratios between the similar characteristic values are constant.
The similarity can be geometrical similarity, time similarity, similarity
of the physical values, and similarity of the initial and boundary conditions.