Page 392 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Rapid Filtration 347

12.3.3.3.1 Collisions within Depth of Filter Media log C log C l Z
C o C o ¼ 2:3
As is evident in Figure 12.19, some of the particles will have
¼ (4:8 ln cycles=m) (1=2:3) 1:00 m
contact with a given media grain and some will not, depend-
ing upon the proximity of the particle to a grain and the ¼ 2:1 log cycles
magnitude of attachment forces. If a particle is not in a
C 2:1
streamline proximity for making contact with one collector, ¼ 10
C o
the particle has another chance at each new collector level. 2:1
C ¼ 5200 particles=mL 10
The probability is quite high that a contact will be made at
¼ 41 particles=mL
some level within the filter bed, depending upon velocity,
grain diameter, and other variables involved in transport
The difference between the two filter coefficients, that
efficiency. At the same time, since we are dealing with a 1
is, l ¼ 20 and 4.8 m , respectively, results in a difference
probability phenomenon, some particles, will escape colli- 7
of 10 in order of magnitude between the effluent
sion with a collector and leave the filter. The number escap-
concentrations.
ing depends on the filter coefficient and media depth.
Habibian and O’Melia (1975) refer to this as ‘‘contact oppor-
12.3.3.3.2 Particle Transport Equations
tunities’’; in other words, the larger values of h result in a
higher rate of contact opportunities. If h is low, then a The respective influences of interception, diffusion, and grav-
deeper filter bed can compensate to give the total number ity, are given in terms of the component coefficients, h I , h D ,
of contact opportunities needed to meet a specified filter and h G ,defined mathematically by Yao et al. (1971), as
effluent concentration.
2
3 d p
I (12:18)
h ¼
2 d c
Example 12.2 Calculation of Effluent 2=3
Concentration from Iwasaki’s Equation h ¼ 0:9 kT (12:19)
D
md p d c v o
Given (r r )gd 2
Suppose 5200 particles per mL enter a filter bed with P w P (12:20)
G
h ¼
length 1.00 m. The conditions are as given in Table 12.2, 18mv o
Eliassen’s data with l ¼ 0.20 cm 1 ¼ 20 log cycles=m.
where
Required
Calculate the concentration of particles leaving the filter h I is the collision frequency coefficient due to interception
bed. d P is the diameter of particle (m)
d c is the diameter of collector; same as grain diameter, d (m)
Solution
1. Apply Equation 12.16, h D is the collision frequency coefficient due to diffusion
23
k is the Boltzmann constant (1.38 10 J=K=molecule)
T is the absolute temperature (K)

C l v o is the interstitial velocity of water (m=s)
log ¼ Z 2
C o 2:3 m is the dynamic viscosity (Newton s=m )
h G is the collision frequency coefficient due to sedimenta-
tion
2. Substitute numerical data, 3
r P is the density of suspended particle (kg=m )
3
r w is the density of water (kg=m )

C l g is the acceleration of gravity (m=s )
2
log ¼ Z
C o 2:3
¼ (20 ln cycles=m) (1=2:3) 1:00 m The overall coefficient, h, is the sum of the three com-
¼ 8:7 log cycles ponents,
C 8:7 h ¼ h þ h þ h G (12:21)
I
D
¼ 10
C o
C ¼ 5200 particles=mL 10 8:7 where h is the overall transport coefficient, defined as ratio of
particles striking a collector to the particle flux approaching
¼ 0:000 010 4 particles=mL
(dimensionless).
The above equations, that is, 12.18 through 12.21 identify
Discussion the independent variables that affect particle-collector con-
Suppose the coefficient from Khan (1993, as given in tacts and permit calculation of h.
Table 12.2) is used, that is, l ¼ 0.048 cm 1 4.8 ln Table CD12.3 is a spreadsheet that calculates the transport
cycles=m. Then, coefficients h I , h D , h G , and h as defined in Equations 12.18