Page 479 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

14.2.3.1 Hydraulics of Cake Filtration where
The water flow in filtration follows Poiseuille’s law, which W(pre-coat) is the mass loading application of pre-coat per
2
rearranged results in Darcy’s law (Hoffing and Lockhart, unit area of septum (kg pre-coat=m septum)
1951). The adaptation to diatomite filtration was taken from W(cake) is the mass loading application of body feed per
2
the work of Baumann et al., (1962). unit area of septum (kg body feed=m septum)
r(pre-coat) is the bulk density of pre-coat (kg pre-coat=m 3
14.2.3.1.1 Darcy’s Law pre-coat)
r(cake) is the bulk density of body-feed cake (kg body-
Flow of water through a porous medium cake is in accordance 3
feed cake=m body-feed cake)
with Darcy’s law (Equation E.2), that is,
As a last relation, the mass loading of body feed per unit
r g dh
w
v ¼ k (14:1) area of septum application is
m dz
W(cake) ¼ C(body feed) v t (14:7)
where
where
v is the filtration velocity (the same as hydraulic loading
C(body feed) is the concentration of body feed in raw-
rate, Q=A)(m=s) 2
2
k is the intrinsic permeability of medium (m ) water flow (kg body feed=m water)
3 t is the elapsed time since start of body feed (s)
r w is the mass density of water (kg=m )
2
g is the acceleration of gravity (9.806 650 m=s )
2 Substituting Equation 14.7 in Equation 14.6 and then insert-
m is the viscosity of water at a given temperature (N-s=m )
h is the hydraulic head, that is, p=g þ z (m) ing Equations 14.5 and 14.6 in Equation 14.4 gives
z is the coordinate in direction of velocity vector (m)
m W(pre-coat) C(body feed) v t
h L ¼ v þ
Applying Equation 14.1 to the pre-coat and the filter cake, r g k(pre-coat) r(pre-coat) k(cake) r(cake)
w
Baumann et al. (1962, p. 1114), developed the equations, (14:8)
r g h L (pre-coat) which is Equation 14.12 as given by Baumann et al. (1962,
w
v ¼ k(pre-coat) (14:2) p. 1114) and verified in over 500 experiments.
m Dz(pre-coat)
Equation 14.8 yields the following conclusions:
r g h L (cake)
w
v ¼ k(cake) (14:3) . The pre-coat headloss is proportional to the pre-coat
m Dz(cake)
diatomite load on the septum and the filtration
velocity.
where . The headloss caused by the body feed is proportional
2
k(pre-coat) is the intrinsic permeability of pre-coat (m ) to the body-feed concentration and the elapsed time
2
k(cake) is the intrinsic permeability of body-feed cake (m )
since the start of the body feed, and the square of the
h L (pre-coat) is the headloss across pre-coat (m)
filtration velocity.
h L (cake) is the headloss across body-feed cake (m)
Dz(pre-coat) is the thickness of pre-coat (m)
14.2.3.1.2 Calculations
Dz(cake) is the thickness of body-feed cake (m)
Table CD14.2 is a spreadsheet for the calculation of headloss
versus time in accordance with Equation 14.8; the final col-
Rearranging each equation in terms of the h L term and then
umns are assumed time, t, and the resulting headloss, h L ,
adding h L ¼ h L (pre-coat) þ h L (cake),
respectively. Figure CD14.10 is a plot of headloss versus
time calculations from the spreadsheet for five grades of

m Dz(pre-coat) Dz(cake) diatomite; also, the figure is embedded within the spreadsheet.
h L ¼ v þ (14:4)
r g k(pre-coat) k(cake)
w Figure CD14.10 shows that the rate of headloss increase varies
markedly with the grade of DE. A ‘‘nominal’’ limit to the
Now, Dz’s for the pre-coat or the body-feed cake are, headloss is shown as the horizontal line at 300 kPa (44 psi).
respectively,
Example 14.2 Calculation of Headloss
W(pre-coat) by Equation 14.8
(14:5)
r(pre-coat)
Dz(pre-coat) ¼
Given
W(cake) Assume Hyflo Super-Celt is being considered for the
(14:6)
Dz(cake) ¼
r(cake) treatment of a reservoir water. The pertinent data are