Page 895 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

850 Appendix G: Dimensionless Numbers

TABLE G.2 (continued)
Selection of Dimensionless Numbers Applicable to Water Treatment Process Design
Name Grouping Variables Ratio and Description
Specific speed (Rouse, 1946, p. 304, 305) nQ 0:5 N s ¼ specific speed N s 0.05, radial flow pump
(gDH) n ¼ rotational velocity (rev=s) N s 0.1, radial mixed flow
N s ¼ 0:75
3
Q ¼ flow (m =s) N s 0.2, mixed flow
2
g ¼ gravitational constant (9.18 m=s ) N s 0.4, axial mixed flow
DH ¼ head developed by pump (m) N s 0.8, axial flow
v inertia forces
Weber (Rouse, 1946; McCabe et al., 1993) D ¼ length of contact (m)
(s=rD) surface tension forces
W ¼
3
r ¼ density of fluid (kg=m )
v ¼ velocity (m=s)
2
s ¼ coefficient of surface energy (N=m) or (kg=s )

TABLE CDG.3
Matrix of Physical Phenomena and Associated Dimensionless Numbers
Source: Omega Engineering, Inc., The land chart of dimensionless numbers, U.S. Patent No. 5,465,838, Omega Engineering, Inc., Stamford, CT, 1997.)

range of conditions not investigated. The selection of the proper
E =f(R) E ≠f(R)
dimensionless numbers also identifies the key variables
involved in an experiment, e.g., viscosity, diffusion, and gravity.

E REFERENCES

Einstein, H. A., Engineering derivation of the Navier–Stokes equa-
tions, Journal of the Engineering Mechanics Division,
Proceedings of the American Society of Civil Engineers,
Proceedings Paper 3533, Vol. 89(EM3):1–7, June 1963.
R
McCabe, W. L., Smith, J. C., and Harriot, P., Unit Operations of
FIGURE G.1 Euler number versus Reynolds. Chemical Engineering, 5th edn., McGraw-Hill, New York,
1993, ISBN 0-07-112721-6.
Omega Engineering, Inc., The land chart of dimensionless numbers,
U.S. Patent No. 5,465,838, Omega Engineering, Inc.,
involving an immersed object (form drag), Sherwood num- Stamford, CT, 1997.
Rouse, H., Elementary Mechanics of Fluids, John Wiley & Sons,
ber versus Reynolds number, etc.
New York, 1946.
Dimensionless numbers may be used to express experimental
Weber, W. J. Jr. and DiGiano, G. A., Process Dynamics in Environ-
results in which many variables may be involved. The consoli- mental Systems, John Wiley & Sons, New York, 1996, ISBN
dation of variables into dimensionless numbers means that a 0-471-01711-6.
given set of experimental results may be applicable to a wide White, F. M., Fluid Mechanics, McGraw-Hill, New York, 1979.