Page 257 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

where where e o ¼ permittivity in a vacuum (F=m) ¼ 8.854 187 817
EM is the EM (mm=s=V=cm) F=m (Lide, 1996, back cover).
v is the velocity of particle in electric field (cm=s) The difference is that the permittivity, that is, e (in which
dV is the voltage drop across electrode plates (V) e ¼ e o D), replaces the dielectric constant alone and the p and
dx is the distance of separation between electrode plates (m) numerical terms cancel. The equation is dimensionally homo-
geneous and gives a conversion factor, that is, 12.9 reported
The measurements are taken on individual particles that by several investigators (next section).
are visible and tracked visually in the cell with time noted
for traveling a set distance. Enough results are obtained to 9.7.2.2.3 Empirical Relation
delineate a histogram and the mean is taken as the result. According to Black and Willems (1961, p. 592), Hall (1965,
With the early instruments the measurements and calculations p. 198) the group (4pm=D) has the numerical value,
were laborious involving the use of a stopwatch, thermometer, (4pm=D) ¼ 13, giving
with associated calculations. Example 9.4 illustrates how to
calculate EM from basic cell measurements. With the advent
z 13 EM (9:15)
of solid state electronics in the 1970s, laser optics in the
1980s, combined with personal computers in the 1980s,
in which z units are mV and EM units are mm=s=V=cm.
most measurements are done automatically and data are com-
piled by computer with only a minimal amount of labor 9.7.2.2.4 Examples of EM and Zeta Potential
required.
Calculations
Example 9.4 illustrates the calculation of EM from measured
9.7.2.2 Zeta Potential
data. Example 9.5 shows the following: (a) in Equation 9.13
The zeta potential, z, is calculated from EM measured data
the group, (4pm=D) 6¼ 13; (b) Equation 9.13 is not dimen-
(Matijevic, 1967). The Helmholtz–Smoluchowski equation is
sionally homogeneous; (c) Equation 9.14 is dimensionally
common and is given for reference. Another relation by
homogeneous and the group, (m=e o D) ¼ 12.9. Example 9.6
Hunter (1981) is dimensionally homogeneous and also com-
illustrates the application of Equation 9.15.
putes the correct value of z, and is favored here.
9.7.2.2.1 Helmholtz–Smoluchowski Equation Example 9.4 Calculation of Electrophoretic
The Helmholtz–Smoluchowski equation is (Pilipovich et al., Mobility (Black and Smith, 1962, p. 934)
1958, p. 1474; Riddick, 1961, p. 1021; Black and Smith,
1962, p. 925), This example illustrates how to convert basic measure-
ments to an EM value.
4pm Given
EM (9:13)
z ¼ Black and Smith (1962, p. 934) obtained the following
D
data using a Briggs cell:
where 6
. d(field) ¼ 49 mm ¼ 49 10 m
z is the zeta potential (mV) . A(cell) ¼ 0.100 cm 1.73 cm ¼ 0.001 m 0.0173 m
2
m is the viscosity of water medium (N s=m ) ¼ 1.73 10 5 m 2
. t(avg) ¼ 7.8 s (average)
. For water m(258C) ¼ 0.89 10 3 Ns=m 2 . i ¼ 4.0 10 4 A
. In the literature, viscosity is given often in poises. . R ¼ 4450 ohm-cm ¼ 44.50 ohm-m
The definition is: poise ¼ g=cm=s ¼ dyne s=cm 2
. The conversion is 0.89 10 2 Required
poises at 258C ¼
0.89 10 3 Ns=m 2 Calculate EM, based on measurements using the Briggs cell.
Solution
D ¼ dielectric constant for medium (dimensionless) Equation Ex9.4.1, from Black and Smith (1962, p. 934), may
¼ 78.36 the dielectric constant for water at 258C be used to calculate EM (with the Black and Smith units
(Lide, 1996, pp. 6–18) converted to SI). Insertion of data from the Briggs cell gives
9.7.2.2.2 Hunter Equation d(field)A(cell)
A modification of the Helmholtz–Smoluchowski and Debye– EM ¼ t(particle)iR
Hückel equations for the EM to z conversion was given by
49 10 6 m (1:00 10 3 m 1:73 10 2 m)
Hunter (1981, pp. 61, 359), which is
¼ 4
7:8s 4:0 10 amps 44:5 ohm-m
m m 2
EM (9:14) ¼ 0:061 10 7
z ¼
e o D s amps ohms