Page 352 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Flocculation 307

Now if the blade is connected to a rotating paddle-wheel
n (rps) arm with rotational velocity, v rad=s, the velocity of the blade
canbe expressedas v b ¼ r b v,inwhich r b is the radial distance
to the blade. Inserting this in Equation 11.14 gives
Axis Blade #1
r
w 3 3
P(blade) ¼ C D v r A b (11:15)
b
2
r b3
r b2 where
r b1
r b is the distance to center of blade from shaft (m)
v is the rotational velocity of shaft (rad=s)
(a)
Let there be a set of n(blades) per paddle-wheel arm and
N(arms) per paddle wheel. In practice, 2 N(arms) 4. Using
the subscript, i, to designate any blade, the power expended
by all blades per paddle wheel is
V b
F (drag force) n(blades)
D
r
X
v 3 3
P(paddle wheel) ¼ C D v N(arms) r A bi (11:16)
bi
2
1
Since the whole water mass is subject to being set in motion
by the paddle wheel, a slippage factor k is inserted into
Equation 11.16 to give
(b)
1 3 3
P(paddle wheel) ¼ C D r(1 k) v N(arms)
FIGURE 11.12 Sketch of paddle-wheel flocculator showing terms 2
used in mathematical development of design equations. (a) Side view n(blades)
X 3
sketch of paddle wheel. (b) Inset diagram showing drag force on a (11:17)
r A bi
bi

blade with velocity v b . 1
where
BOX 11.3 DRAG COEFFICIENT N(arms) is the number of arms, with blades that comprise
the paddle wheel (m)
For a flat plate with surface normal to the velocity, k is the slippage factor which is ratio of rotational velocity
3
C D ¼ 1.8 when R > 10 (Rouse, 1946, p. 247; Figure
of water mass to rotational velocity of paddle wheel,
D.1, Appendix D). Thus, a paddle-wheel blade would k ¼ 0.24–0.32
3
have this drag coefficient. For 1 < R < 10 , the C D
versus R experimental curves should be used (see, for
To explain the (1 k)v term, let the relative velocity of
example Rouse, 1946, p. 247). The following drag the blade past the water be designated v b=w . The velocity of
3
coefficients are given for R 10 by Rouse (1966,
the rotating water mass is v w . Therefore, v b=w ¼ v b v w .
p. 249) for flat plates of different proportions:
If v w ¼ kv b , then, v b=w ¼ v b kv b ¼ v b (1 k). Since v b ¼ r b v,
L=w ¼ 1, C D ¼ 1.16; L=w ¼ 5, C D ¼ 1.20; L=w ¼ 20,
the grouping in Equation 11.17 follows.
C D ¼ 1.50; L=w ¼ infinity, C D ¼ 1.90. For R 1,
The effect of the radial spokes can be calculated in the
Stoke’s law applies and C D ¼ 24=R.
same fashion, only an integral expression is involved. If the
spoke is a flat blade shape, its integral is
where 1=4 w(blade) r o 4 (11:18)
F D is the drag force due to motion of blade (N)
C D is the drag coefficient for a flat plate (dimensionless) and must be added to the summation, multiplied by the
3
r w is the density of water (kg=m ) number of spokes. Thus, the complete equation is
v b is the velocity of paddle (m=s)
2
A b is the area of paddle (m ) 1 3 3
P(paddle wheel) ¼ C D r(1 k) v
2
The power, P, expended by the blade is the drag force, F D , " n(blades)
X 3
times its velocity, v b , i.e., N(arms)
r A bi
bi
1
r
3 #
P(blade) ¼ C D v A b (11:14) N(spokes)
b
2 4
4
þ w(blade) r o (11:19)
where P(blade) is the power dissipated by blade (N m=s).