Page 313 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

e.g., q(raw water) ¼ 1.0 s, as well as Q. With these
assumptions, the calculated result, i.e., n ¼ 153 rev=s, is BOX 10.6 MEASUREMENT
too high from a practical standpoint, e.g., impeller power OF IMPELLER TORQUE
is proportional to n to the third power. A spreadsheet setup
would be useful to examine the trade-offs, e.g., q(raw For an experimental impeller mixing system, fabricated
water) and thus D(impeller), Q(impeller)=Q(raw water), by a machinist, the torque, T, imparted to an impeller
3
5
and n. The use of the power number, P ¼ P=(rn D ) and may be measured if the drive motor is mounted on a
the plot for the Rushton system power number, P, provide bearing plate (such as a ‘‘lazy susan’’ from a hardware
a basis for estimating the power required for the system store) above the basin. A lever rod is attached to the
and thus the reasonableness of any design trials, e.g., for n
motor and rotates with the motor; the motor as set up is
and D(impeller). As noted, changes in the geometric pro- free to rotate. A force gage is then mounted on the top of
portions of the system, i.e., deviations from the Rushton
the basin with a hook to restrain the lever rod (Figure
design, result in changes to Q and P, which are not neces-
sarily available in the literature. 3.5). When an impeller is attached to the shaft from the
motor, the associated force may be measured. The meas-
ured force in Newtons (or lb) times the lever arm distance
10.4.2.3.4 Power Dissipation in m (or ft) is the torque in N m (lb ft) generated by the
The power transferred to an impeller is the product of the motor to overcome the drag resistance of the impeller. If
torque applied and its rotational velocity, i.e., a luminescent tape strip is attached to the shaft, its rota-
tional velocity, n, may be measured by a strobe. Since
P ¼ T v (10:38) n 2p ¼ v, the power, P, dissipated by the impeller is,
P ¼ T (n 2p). The power dissipated by friction may be
where measured by removing the impeller and taking the same
P is the power dissipated by the impeller (W) measurements. Ordinarily, the measured friction torque
T is the torque exerted by impeller due to fluid drag (N m) should approach zero. To convert from kilograms-force
v is the rotational velocity (rad=s) to Newtons, N ¼ kg-force 9.80665; i.e., a kilogram of
force (2.2 lb force) is that force exerted by gravity on a
v (rad=s) ¼ n (rev=s) 2p mass of 1 kg.

The experimental procedure to measure T and w is
In other words, the energy dissipated as shear, i.e., turbu-
described in Box 10.6. 2
lence, is proportional to the impeller speed squared, n , and
2
impeller diameter squared, D(impeller) .
10.4.2.3.5 Shear
The power imparted to the water by an impeller causes a 10.4.2.3.6 Shear=Flow Ratio
pressure (head) increase if confined by a casing. In the For most mixing situations, high shear (i.e., turbulence) is
absence of a casing, the impeller energy is distributed between required, along with some level of advection. To obtain an
advection and turbulence, aggregated here as H(shear), since expression for the shear-to-flow ratio, divide Equation 10.40
the advective flow sheds eddies and ends up as turbulence. by Equation 10.37 to give (Myers et al., 1999, p. 35),
The power dissipated is thus (Myers et al., 1990, p. 35),

n
H(shear) P
P ¼ Q(impeller)g H(shear) (10:39) ¼
w Q(impeller) gQ 2 D
(10:41)

H(shear) P
n
where ¼ 2
3
g w is the specific weight of water, i.e., g w ¼ r w g (N=m ) Q(impeller) gQ D
H(shear) is the pseudo head due to impeller rotation dissi-
pated as shear (m) Equation 10.41 shows that for a given system, i.e., for unique
values of P and Q, the shear-to-flow ratio is proportional to
the n=D ratio, i.e., higher values of n and smaller values of
Equating the power terms of Equation 10.23, i.e., P ¼
3 5 D(impeller). Illustrative values are (Myers et al., 1999, p. 36).
P=(n D r), Section 10.3.3.1, and Equation 10.39, i.e., P ¼
Q(impeller)g w H, and substituting Equation 10.37, Q
3
(impeller)¼ QnD ,for Q, gives (Meyers et al., 1990, p. 35),
Shear Pumping n (rpm) D (m)
High Low 3500 0.1
P 2 2
n D(impeller) (10:40) Low High 100 1.0
Qg
H(shear) ¼
where P is the power number for impeller–tank system In other words, the distribution between shear and pumping
(dimensionless). flow may be controlled by the n=D ratio. Also, as seen, the