Page 141 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

96 Fundamentals of Water Treatment Unit Processes: Physical, Chemical, and Biological

After substituting the projected area for a sphere, i.e.,
TABLE 6.1 2 3
A ¼ pd =4, and the volume, V ¼ pd =6, Equation 6.6 becomes
Suspensions in Water and Wastewater Treatment
Stokes’ law:
Suspension Occurrence Settling Unit
1 g 2
Mineral particles Raw water supply Plain sedimentation v s ¼ (SG s SG f )d (6:7)
18 v
Oil Refinery wastes Separators
Floc particles with Air flotation Flotation thickening where
2
2
bubbles attached v is the kinematic viscosity of fluid (m =s) or (ft =s)
Organic particles Raw sewage Primary settling SG s is the specific gravity of particle
Biological floc Biologically treated Final settling SG f is the specific gravity of fluid
sewage
Chemical floc Chemically treated Flocculent settling
While Stokes’ law is useful, its important to keep in
water=sewage
mind that its merely an equality of forces for the special
Sludge’s Settled chemical Thickening compartments
case in which R 1, i.e., the viscous range for which
and biological flocs of settling basins
C D ¼ 24=R. Example 6.1 illustrates numerical calculations;
and settled organics
Table CDEx6.1 is set up as an algorithm for computations.
A relationship for C D for spheres that includes the range,
5
1 R < 10 was given by Fair et al. (1968, p. 25-3), i.e.,
0.5
C D ¼ 24=R þ 3=R þ 0.34.
BOX 6.1 THE COMPLETE
MATHEMATICAL MODEL OF FLUID FLOW
Example 6.1 Application of Stokes’ Law
As seen in fluid mechanics texts, the mathematical
description of any fluid flow is provided by the classical Illustrate the application of Stokes’ law for different situ-
Navier–Stokes equations. They were formulated by ations of fall of a quartz sand particle in water.
Navier, Cauchy, and Poisson, early in the nineteenth
1. Fall velocity: Calculate from Equation 6.7 the fall vel-
century, and by Saint-Venant and Stokes in the mid-
ocity, v s , for a quartz sand particle of 0.1 mm with equiva-
nineteenth century. Equations are named as a matter of lent diameter at 208C (688F),
custom after the first and last of these investigators
(Rouse, 1959, p. 208), hence Navier–Stokes. The 1 1 (g g )d 2 (Ex6:1:1)(6:7)
Navier–Stokes equation was merely an expansion, in v s ¼ 18 m s f
differential form, of Newton’s second law, i.e., the !
1 1
3
familiar, F ¼ ma. In the expansion, expressed as a dif- (998:21 kg=m )
v s ¼ 18 1:002 10 3 2
ferential equation, the left-hand side included all of the Ns=m
2
forces that act on an infinitesimal volume of fluid, e.g., (9:81 m=s )(2:65 1:00)(0:1 10 3 m) 2 (Ex6:1:2)
pressure, gravity, viscous, surface energy. The ma
side is the dynamic response to the forces. The ¼ 0:0090 m s ¼ 0:90 cm s == (Ex6:1:3)
Navier–Stokes expression has several dependent vari-
Note that g s ¼ r s g, and g f ¼ r f g, where g s is the specific
ables and so has defied solution, i.e., until the advent of weight of particle (N=m ) and g f is the specific weight of
3
the computer, which provided the means for a numer- fluid (N=m ).
3
ical solution, done in the 1960s by Fortran program-
ming in the 1990s by ‘‘computational fluid dynamics’’ 2. Largest particle diameter at 108C: Calculate the largest
diameter quartz sand particle (SG ¼ 2.65) for which
(CFD) software.
Stokes’ law is applicable at 08C, i.e., R ¼ 1.
a. Trial-and-error solution: The solution is by trial and
error and involves the following steps: (1) assume a
F D value for d; (2) using the assumed d, calculate v s
from Stokes’ law; and (3) from the calculated v s and
assumed d, calculate R. Since Stokes’ law is valid for
R 1.0, the largest d is for R ¼ 1.0. Therefore, if the
assumed d, gives R < 1.0, increase d for the next
trial; if the assumed d, gives R > 1.0, then decrease d
for the next trial.
v S
b. Spreadsheet: The easiest way to execute the fore-
going algorithm is by means of a spreadsheet (Table
W B CDEx6.1). Several trials are shown, i.e., d ¼ 0.010,
0.050, 0.10, 0.15, ..., 0.124 mm, with the last
FIGURE 6.1 Forces acting on a falling particle. being the size that meets the criterion R ¼ 1.00.