Page 892 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix G: Dimensionless Numbers

Dimensionless groupings of variables have been derived as a different temperature, the viscosity can be obtained from
using the Buckingham pi theorem in both fluid mechanics a handbook for the temperature in question, the Reynolds
and chemical engineering. The former has included the Euler number may then be calculated, and for a given pipe material,
number, the Froude number, the Reynolds number, the Weber the friction factor may be obtained from a Moody chart.
number, etc. Chemical engineers have adopted these numbers Dimensionless numbers have a rational basis as well. Con-
and have added the Schmidt number, the Sherwood number, sider those that describe bulk fluid flow, such as E, F, R.
the Power number, the Peclet number, etc. The nomenclature These dimensionless numbers, the Euler number, the Froude
adopted, which is common, was to use bold fonts to indicate a (pronounced ‘‘fru-d’’) number, and the Reynolds number are,
given dimensionless number, i.e., the Euler number is E, the respectively, the ratios of inertia forces to pressure forces,
Froude number is F, the Reynolds number is R, and so on. inertia forces to gravity forces, and inertia forces to viscous
Dimensionless numbers are used frequently in the literature forces. They provide an empirical means to characterize fluid
and often without definition or explanation. The intent of this flow phenomena.
appendix was to provide definitions for the commonly used The Navier–Stokes equation has long been recognized as
dimensionless numbers and to review the topic in general. the most comprehensive mathematical description of bulk
fluid flow. But as a ‘‘second-order non-linear partial differen-
tial equation’’ (White, 1979), it has been considered too
G.1 THE WORLD OF DIMENSIONLESS complex for traditional mathematical solutions. The task of
NUMBERS computational fluid mechanics is to solve the Navier–Stokes
equation numerically for the particular boundary conditions of
The world of dimensionless numbers is actually much more
interest. The Navier–Stokes equation can be understood more
extensive than might be imagined, i.e., based on the usual
easily if looked at as merely an expansion of Newton’s second
exposures in textbooks. This is illustrated by the Land Chart
law, F ¼ ma (see, for example, Einstein, 1963). The force term
of Dimensionless Numbers (Omega Engineering, Inc., 1997;
on the left side incorporates terms for pressure, gravity, and
http:==www.omega.com=literature=posters, 2009), which lists
viscous forces. The right side is the inertia term. If one force
some 154 dimensionless numbers. Table G.1 also lists these
term is dominant, such as gravity, then the Froude number can
numbers; as seen, most are not recognizable from the literature.
serve as means to characterize the dynamics of the system,
The purpose of Table G.1 is merely to give an indication of the
which is the ratio of gravity forces to the inertia forces.
extent to which dimensionless numbers have been proposed.
Two kinds of forces may be important, however, especially
Table G.2 lists dimensionless numbers that could be applic-
in certain ranges. Consider, for example, pressure forces and
able to treatment processes as compiled from the given refer-
viscous forces, as characterized by E, are a function of R.
ences. Table G.2 gives the name, grouping of variables,
An example is shown in Figure G.1 in which E declines
definitions of variables, and nature of the ratios involved.
rapidly with increasing R and then levels. E is influenced
Table CDG.3 is a matrix of physical phenomena and
strongly by viscous forces at low R. Then as R increases,
associated dimensionless numbers (Omega Engineering,
the inertia forces predominate over viscous forces and E is no
Inc., 1997). Table CDG.3 shows the 45 phenomena as col-
longer affected by viscosity.
umns and 154 dimensionless numbers as rows. The dimen-
Such a curve as shown in Figure G.1 must be generated
sionless numbers applicable to a given phenomena could be
empirically by means of a laboratory setup. Discharge of a
indicated by ‘‘x’’ in the appropriate columns (not done as
fluid through an orifice is a case in which the Euler number is
given). The table is in the form of a spreadsheet on a CD disk.
important.
In this case, the discharge coefficient, C d , is a mathematical
G.2 UNDERSTANDING DIMENSIONLESS identity with the Euler number and Figure G.1 is seen more
commonly as C d versus R. At high R, the Euler number (or
NUMBERS
C d ) is a function of the geometry only. As another example,
Dimensionless numbers provide a means to group variables the Euler number may be a function of the Weber number at
such that a large amount of data may be condensed into a low values of W.
single set of plots. For example, Reynolds number, R,defined In the same fashion, the Froude number, F, is a function of
as R ¼ rvD=m, can be applied to a wide range of combinations R and a plot would be similar to Figure G.1. As a matter of
of fluid densities, velocities, diameters, and viscosity. Instead practical interest, the Froude number is an identity mathemat-
of conducting laboratory tests for an unknown condition such ically with the discharge coefficient for a weir.
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