Page 33 - Packed bed columns for absorption, desorption, rectification and direct heat transfer

28

The similarity of the initial and boundary conditions requires the initial
conditions and the conditions at the boundaries of the similar systems to be
similar too, i.e. all mentioned above similarities to be Mfilled for them.
Very important values in the similarity theory are the invariants of
similarity or criteria of similarity. If all similar values determining the state of
the similar systems are given in relative units, these dimensionless values for
both systems are equal. The values are made dimensionless by dividing all of
them by values of the same system similar for both of them.

-L = J- =.... = i l= const. (84)

rf = — = = i t= const. (85)
*2 T 2

U, u.
-!- = -*- =..... = i u= const (86)
f/ 2 u 2

Thus the ratio between the geometrical sizes, times, and physical values
in a given system are equal to the corresponding ratios of the similar system.
That is, the equality of the constant i tii T and i u for two or more different systems
guarantees that these systems are similar.
The dimensionless numbers i, defined in Eqs. (84) to (86) are called
invariants of similarity and are written as

i = idem.

The invariants of similarity which are ratios of simple homogeneous
values are called simplexes.
The invariants of similarity could be also expressed not only as ratios of
simple homogeneous values but also as ratios of more complicated
heterogeneous values. For example, according to Newton's law the resultant
force 0 acting on a given body is equal to the product of its mass (m) and

acceleration — :