Page 301 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

Note: In Equation 10.23 for the power number the units include As evident in Equation 10.27, P is a function of R (for the
N (Newtons) on top, in the power term, P, and kg (kilograms) condition of no vortex); Rushton (1950, p. 401) gives empir-
on the bottom (in the density term). To reconcile this discrep- ical data to define m. When m ¼ 0, i.e., the viscous effect is
2
ancy, substitute the definition, N ¼ kgm=s , in the numerator to not dominant and Equation 10.27 becomes
eliminate N which results in cancellation of all units.
P ¼ K (10:28)
10.3.3.2 Variables of Impeller–Basin Mixing
Rushton et al. (1950a) grouped mixing variables as geometric,
This is the level part of the P vs. R experimental curve and
dynamic, and fluid properties. The functional relationship
occurs where inertia forces are dominant, i.e., high R; thus the
between dependent and independent variables was in terms
geometry of the system controls.
of dimensionless groups, i.e.,
10.3.3.3 Experimental Plots
t
l
c
h
s
T H C S L n
m n Figure 10.15 is a P=F vs. R plot from Rushton et al. (1950)
D D D D D for a radial-flow impeller–basin; proportions are as given in
P ¼ K d R F
w
j
b
W J B the plot. The plot as published showed several hundred
(10:25)
experimental plotted points (Box 10.5). Flow regimes illus-
D D R
trated are viscous, transition, gravity, and inertial; character-
where istics include the following:
K d is the constant, disaggregated from geometric variables
n
. For R 10, the slope of the log P=F vs. log R curve
(dimensionless)
P is the Power number is ‘‘ 1’’ and viscous forces govern.
. For a basin with no baffles, a ‘‘swirl’’ vortex
R is the Reynolds number
F is the Froude number (pronounced, fr-ud) formation is observed as R increases due to higher
D is the impeller diameter (m) rotational velocity, which is the bottom curve; Equa-
T is the tank diameter (m) tion 10.26 applies.
.
H is the depth of water in tank (m) For 10 R 300 all data points fall on the single
C is the distance of center of impeller from bottom of curve and viscous forces still dominate; Equation
tank (m) 10.27 applies.
. For tanks with baffles, and for R 10,000, the
S is the pitch of impeller (marine type) (m)
L is the length of impeller blade (flat blade type) (m) system geometry is the sole influence and Equation
W is the height of impeller (flat blade type) (m) 10.28 applies. To illustrate, for the plot of Figure
J is the baffle width (m) 10.15, as J=T (baffle width to tank diameter)
B is the number of blades on impeller increases, K increases (all other geometric variable
R is the number of baffles are constant). By the same token, for a given baffle
proportion, the K value will change with impeller
The exponents are empirical. shape, blade width, and blade length.
As seen in Equation 10.25, the reference dimension for
the length ratios is the impeller diameter, D, also called 10.3.3.3.1 Power Function Curves for Different
D(impeller). A given exponent indicates the quantitative Impellers and Tanks
effect of a particular ratio. Rushton et al. (1950a,b) actually
All impeller–tank combinations have characteristic curves
obtained such exponents by means of several thousand experi-
similar to those illustrated in Figure 10.15. Rushton et al.
ments for a wide variety of impeller types and basin geometries.
(1950, p. 472) have provided P and R data for R ¼ 5, 200,
and 100,000, which permit an approximate sketch of such
10.3.3.2.1 Simplifying Special Cases
characteristic curves for a given system. Table 10.3 provides
If the geometry of an experimental system remains fixed, then examples of such data for four impeller shapes with geometry
Equation 10.25 simplifies to ratios as given in the columns. The data show, in particular,
5
the effects of the J=T ratio and impeller type on P at R 10 .
m n
P ¼ KR F (10:26)
The presence of a vortex indicates that gravity forces influ- 10.3.3.4 Scale-Up by Fluid Similitude
n
ence P, and thus, n > 0. But with baffles or if the impeller To utilize one of the P=F vs. R curves generated by Rushton
axis is off-center with the basin or is tilted at an angle to et al. (1950a,b), e.g., Figure 10.15 or Table 10.3, the system
the vertical, then a vortex will not form; thus, n ¼ 0 and being considered must be geometrically similar to the one for
Equation 10.26 becomes which the respective curve was generated, i.e., a ‘‘Rushton’’
impeller–basin. An approach is to find a point on the level part
m 4
P ¼ KR (10:27) of the curve, i.e., P,at R 10 , which yields reasonable