Page 40 - Packed bed columns for absorption, desorption, rectification and direct heat transfer

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c=fa+fy+rz. (Ill)

In this system of 3 equations x, y and z are unknown and it is easy to
determine them and to obtain the dimensionless number using Eq.(102),
It must be mentioned that if n=m the use of dimensional analysis is
impossible because at least one important value is forgotten. It must be also
always taken into account that the dimensional analysis requires knowing all
values important for the phenomenon, not only the variable ones. For example,
for many processes the influence of gravity is important and must be taken into
account thought it is a constant.
The equations obtained by dimensional analysis describe the
phenomenon for which they are obtained and, in case of similarity, the whole
class of similar phenomena.
The disadvantage of the dimensional analysis is that it is not helpful in
determining the uniqueness conditions.
The dimensionless numbers important for packed bed columns are
considered later in the respective parts of mis book.

1.3.3. Some additional remarks about the similarity theory and the dimensional
analysis

Here we discuss two important problems connected with the use of the
similarity theory and the dimensional analysis.
1. Is it obligatory to vary all parameters important for the process
during the experiments to be sure that the obtained equation describes the
phenomena?
2. Is it possible to use these methods for obtaining equations also in
case when full geometrical similarity does not exist? For example, is it possible
to use them for obtaining equations for calculating packings which are not
geometrically similar?
The first of these questions is very important because in literature there
is no generally shared view in this area. For example in [33, p.51-59] it is stated
that in order to obtain a valid equation, all physical parameters including those
in the similarity criteria have to be changed during the experiments. Practically
in case of similarity in the dimensionless equations dimensionless criteria are
used instead of simple physical parameters. That is why to obtain a criterion
equation valid in a given area, it is necessary to change just these numbers no
matter how, by changing only one or all of the physical parameters in them. As
already mentioned before, there are values, for example the gravity constant,
which practically cannot be changed. It is very important and we must always
keep in mind that no parameter that influences the process can be omitted.