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Water treatment

in which ∆C is the

in which ∆C is the concentration change in mass (or number of particles) per unit volume during the time ∆t; Q, constant filtration rate; ∆σ a , absolute specific deposit, volume of particles deposited per unit volume of the bed; ∆LA, volume of filtration element. ∆σ a = β∆σ, in which β is the bulk factor, and can be interpreted as a conversion factor to obtain the specific deposit σ (mass or number of particles per unit volume). Irrespective of the units of C, the bulk factor helps in obtaining estimates of σ (Sembi and Ives, 1982, quoted by Ohja and Graham, 1994). Rearranging equation 2.1 and using the differential form, ∂C A ∂σ ∂C β ∂σ ∂C ∂σ + β = 0 ⇒ + = 0 ⇒ V + β = 0 (2.2) ∂L Q ∂t ∂L V ∂t ∂L ∂t in which V is the approach velocity (LT -1 ). In eq. 2.2, the change of σ with time can be estimated if the change of C with L is known. On the basis of experimental data on SSF units, Iwasaki (1937) presented the eq 2.3, in which λ (L -1 ) is the impediment modulus also called filter coefficient. ∂ C = −λC (2.3) ∂L In clean filter conditions λ becomes λ 0 , and in the integrated form of the eq. 2.3 C 0 is the initial concentration of particles. Then, the profile of C in the liquid phase throughout the filter depth is logarithmic for a bed of uniform grain size, as shown by eq. 2.4 −λ L 0 0 C = C e (2.4) The parameter λ is important in filtration studies and occurs in all of the theories of filtration (Amirtharajah, 1988). Since σ, λ, and C are functions of time in the differential equations 2.2 and 2.3, a third equation is necessary to determine concentration as a function of time. Since theoretical considerations are of little help, λ has to be determined from filtration experiments, with which ∂c/∂L at various combinations of L and t can be estimated and plotted against σ, to obtain graphs as those shown in figure 2.5. With this approach the entire filtration cycle can be modelled but needs extensive experimentation to determine many of the parameters included in the models. Research papers present λ = f (λ 0 , σ), with empirical coefficients and because of differences in theoretical considerations or in experimental conditions, many have found different results. Some examples follow (based on Huisman, 1986), λ 1 = λo + k σ λ = λ + k σ − k o 1 2 σ p − σ 0 2 (Stein, 1940) (2.5) (Ives, 1960) (2.6) σ λ = λ0(1 + ) 1 − p 0 a σ b σ c (1 − ) (1 − ) p σ 0 u (Ives, 1969) (2.7) 24

in which p 0 is the clean media porosity; σ u is the ultimate or saturation value of the specific deposit ratio; and k 1 , k 2 , a, b, and c are coefficients or exponents to be determined experimentally. λ= λ + κ σ λ λ= λ + κ σ − κ Figure 2.5. A. Variation of filter coefficient λ with specific deposit ratio, σ (Fox and Cleasby, 1966). B. Variation of concentration C with filter length L and time t (Ives, 1982) The eq. 2.6 includes a second term to account for the increase in removal efficiency during the initial filter ripening period as a result of increased surface area caused by deposits collecting on the filter grains. The third term accounts for the subsequent decrease in removal efficiency in RF units with constant-filtration rate due to the increasing pore velocities caused by deposits. Fox and Cleasby (1966, quoted by Ginn et al, 1992) found that in conventional water treatment the alum and ferric flocs produced were not adequately modelled by eq. 2.6, mainly due to the third term accounting for the clogging process. Many of the equations proposed for λ can be derived from the general eq. 2.7, obtained after assuming that the changes in filter efficiency were due to changes in pore geometry, and the increase in interstitial velocity after the narrowing of the pore flow paths (Lebcir, 1992) Adin and Rebhun (1982), quoted by Amirtharajah (1988) presented a different approach incorporating attachment and detachment terms explicitly in the model equation. Besides eq. 2.2, they used eq. 2.8 for the removal efficiencies, ∂σ = k3VC( F −σ ) − k4σI ∂t (2.8) in which k 3 is the accumulation coefficient; F is the theoretical filter capacity or the mass retained per unit bed volume that could clog the pores completely; k 4 the detachment coefficient; and I the hydraulic gradient. The F and I values are obtained from the porosity of the filter and Darcy´s Law. The accumulation term, k 3 VC(F-σ) is similar to eq. 2.3. The third equation used by Adin and Rebhun to relate σ to F, was eq. 2.9, k ko 3 ⎡ 0.5 ⎢ 1 σ ⎤ − ( ) ⎥ ⎦ = ⎣ F (2.9) in which k and k 0 are hydraulic conductivity of the filtering bed with and without (clean bed) deposits. The parameters have to be determined experimentally for specific conditions. 25