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Predicting optimum backwash rates and expansion of multi-media filters
War. Res. Vol. 21, No. 9, pp. 107%1087, 1987 Printed in Great Britain. All rights reserved
0043-1354/87 $3.00+0.00 Copyright © 1987 Pergamon Journals Ltd
PREDICTING OPTIMUM BACKWASH RATES A N D EXPANSION OF MULTI-MEDIA FILTERS B. A. QUAYE Senior Lecturer, Sheffield City Polytechnic, Civil Engineering Department, Pond Street, Sheffield SI 1WB, U.K.
(Received May 1986) Abstract--A new model for predicting the optimal expansion of dual and tri-media beds is developed and tested experimentally. This is used to deterrqine the height of filter weirs in order to prevent the loss of filter media during backwash. For more accurate prediction of the expansion of the dual and tri-media filter beds it was assumed in an earlier work that the ratios of the porosities of tightly compacted beds and the porosities of the expanded beds are equal. This heuristic assumption for prediction of the exact expansion of the multi-layer filter beds is now verified theoretically.
Key words--settling velocity, fluidization, interstitial velocity, geometric mean hydraulic diameter, geometric mean porosity, optimum wash rate, porosity anthracite
NOMENCLATURE CD = D = d= dh = d~ = d~2 = dh3 = ds = Fb = Fd = Fi = f = fo = fc = f~ = f: = f3 = f,i = fit = f~2 = f~3 = f, = f* = G= g = h= K= K, = L = L, = L,~ = L,2 = Le3 = Let = n= P~ = P2 = Re =
Drag coefficient Diameter of filter column Grain diameter Hydraulic diameter of solid grains Geometric mean hydraulic diameter of solid grain Hydraulic diameter of the light solid Hydraulic diameter of the heavy solid grain Sieve diameter of solid grain "Buoyant" force Drag force Net impelling force Porosity of filter medium Geometric mean porosity Porosity at critical fluidization Initial porosity of anthracite Initial porosity of flintag (sand) Initial porosity of alundum Expanded porosity of anthracite Predicted expanded porosity of anthracite Expanded porosity of flintag (sand) Expanded porosity of alundum Expanded porosity Optimum expanded porosity Velocity of shear gradient (du/dy) Gravitational acceleration Head loss in filter media Constant Constant Depth of granular bed Expanded depth of the filter medium Expanded depth of anthracite Expanded depth of flintag (sand) Expanded depth of alundum Total expanded depth of the filter media Expansion coefficient Power input due to the shear forces Power dissipated by elemental volume Reynolds number (hydrodynamic mechanisms) V, = Stoke's settling velocity of a solid grain Vw= Upflow wash rate Vc = Critical interstitial velocity
V.* = Optimum upflow wash rate W2 = Fractional mass (per unit volume of bed) of light grains W3 = Fractional mass (per unit volume of bed) of heavy grains x = Exponent of the ratio of wash rate to settling velocity Xl = Exponent of the ratio of wash rate to settling velocity of anthracite with expanded coefficient, n~ x 2 = Exponent of the ratio of wash rate to settling velocity of sand with exponent coefficient, n 2 x3 = Exponent of the ratio of wash rate to settling velocity # = Dynamic viscosity of liquid Pl = Density of liquid Pm= Mass density of mixture pp = Density of solid grains p, = Effective density of solid grains Vd = Volume of mixture displaced ¥2 = Volume fraction of light grains with density, P2 V3 = Volume fraction of heavy grains with density, P3-
THEORY OF BACKWASHING
Derivation o f optimum wash rate at criticalfluidization In the study o f particle mechanics or general m e c h a n ics dealing with the relative m o t i o n between a particle (solid or liquid) a n d a s u r r o u n d i n g fluid (liquid or gas), it is immaterial w h e t h e r the particle m o v e s in a s t a t i o n a r y fluid or is suspended in a m o v i n g fluid. Typical examples are the experimental d e t e r m i n a t i o n o f the terminal settling velocity o f a discrete particle a n d the wash rate o f a filter m e d i u m . T h e o p t i m u m wash rate depends to a great extent o n the o p t i m u m e x p a n d e d porosity, f * w h i c h is a f u n c t i o n o f the t e r m i n a l settling velocity a n d the o p t i m u m wash rate. T h e general expression for the 1077
B.A. QUAYE
1078
expanded porosity, f,, will therefore be the basis for the derivation of the optimum wash rate at critical fluidization as shown in equation (18). The expression for the expanded porosity, f,, is given by equation (1)
Predicting the optimal expansion of tri-media filter beds during backwash PiLot f i l t e r plant I d e a l t r i - m e d i a filter
=FvwT f, Lv, J
"
•
/FiLter co(umn / 2 1 0 0 mm Long ~ 15omm o.d. J 140mm i.d.
(1) --4--
in which Vw= upflow wash rate, Vs = terminal settling velocity and x = exponent, its value depends upon the medium. Fair et al. (1968) determined the value of x, experimentally and recommended the exponential of 0.20 and 0.25 for free and hindered settling of sand medium. The mean unhindered terminal velocity, V,, in equation (1) can be calculated by timing--e.g. 20 individual particles settling through 1 m in a sedimentation tube containing a column of water--the velocity term in the transitional regime can be evaluated; and the equivalent (hydraulic) diameter, dh, of the solid grains calculated from the dimensionless relationships of Co/Re with Re, and with use of the "Camp curves", the optimum wash rates of multimedia filters can be calculated as demonstrated in the "model for computation of the optimum wash rate".
l
i
Depth of LooseLy compacted medm (m)
Depth of tightly compacted media (m)
0.97
j
Anthrocite (10/12 )
0.52 046
Computation of porosity at critical fluidization of Welsh anthracite /crushed flintag
(22/25)
In a fluidized bed, the grains of light filter medium will not settle into and mix with the grains of heavy material unless the density of the light grain exceeds the density of the mixture of heavy grains and water. Anthracite/Flintag. Consider a layer of anthracite with a density, Pl = 1400 kg m -3 and porosity, fl, on top of a flintag bed, density, P2 2650 kg m -3 and porosity, f2, density of water is p/. At critical fluidization when anthracite will no longer "float" the critical porosity,f~, of the mixed media is obtained by solving equation (2), which gives f~ = 0.760
0.21
0.20
undum (30/36) 0
0
=
p~ = (1
-
f~)P2 +f~ P:
Computation of porosity at critical fluidization of crushed flintag/alundum [fused alumina (A1203)] Flintag/Alundum. Consider a layer of flintag with density, P2 = 2650 kg m -3 and porosity, f : , on top of alundum bed, density, P3 = 3950 kg m -a and porosity f3. At critical fluidization when the flintag will no longer "float", the critical porosity, f~, of the mixed media is obtained by solving equation 3 which gives
p2= (l -L)p3 +f~p/
for mixing fluidization f~ > 0.440.
Fig. 1. Pilot plant--ideal tri-media filter beds.
(2)
for mixing fluidization f~ > 0.760. Since in equation (18), the geometric mean porosity of the anthracite/flintagfa 0.760, the anthracite will "float" on top of the fluidized flintag or heavy medium, without mixing during the backwashing of the filter.
f , = 0.440
morbl.es
Since in equation (18), the geometric mean porosity of the flintag/alundum, fo >f~, i.e. 0.485 > 0.0440, there will be intermixing at the interface during the backwashing of the filter as shown in Fig. I.
Derivation of the mass density of the mixture Consider a mixed media (flintag/alundum) bed fluidized by water, the mass density of the mixture, p,,, at a level where the porosity of the mixed media isf~ can be derived by re-arranging equations (6) and (5) and substituting them in equation (4) Pm =
~/'2P2 + V3P3 @ f c P f
(4)
V__2= W2P3 V3 W3P2
(3)
(5)
V 2 + V 3 = I - f~
in
which
Pm= mass
density
(6)
of
the
mixture,
1079
Predicting optimal expansion of dual and tri-media beds volume fraction of light grains with density P2, V3 = volume fraction of heavy grains with density P3, f~ = porosity at critical fluidization, p / = density of water, W2 = mass fraction (per unit volume of bed) of light grains and I4"3= mass fraction (per unit volume of bed) of heavy grains. The resultant mass density of the mixture is given by equation (7) V2 =
Pm= (1 - f~)P, + fcP/
(7)
in which p~ is the effective density of the solid grains given by equation (8)
Ps ~" P3
(8) W2p3 + 1 W3p~
Geometric mean hydraulic diameter The geometric mean hydraulic diameter, dh~, is given by equation (9) which is used in the derivation of the modified diameter term, CD Re 2, in equation (17) dh~ = (dh2 x dh3)1/2 (9) in which dh2 = hydraulic diameter of the light solid grains and dh3 = hydraulic diameter of the heavy solid grains. Geometric mean porosity of the granular bed The geometric mean porosity of the granular bed, fa, is given by equation (10) which is used in the derivation of the optimal upflow wash rate, V*, in equation (18) L = (A x A) '/2
(10)
in which f2 = porosity of the tightly compacted light solid grains of the granular fixed bed and f3 = porosity of the tightly compacted heavy solid grains of the granular fixed bed. In a fully fluidized graded bed (i.e. grains of various sizes), the smaller grains tend to accumulate in the upper layers and the larger grains in the lower layers. There is, therefore, a gradual increase in porosity from bottom to top which results in decreasing interstitial velocity from bottom to top. Since the BS sieve sizes are in geometric progression, the use of geometric mean for both hydraulic diameter and porosity is justified. Derivation of modified diameter term, Co Re 2,for dual and tri-media filter beds The modified diameter term, CoRe 2, is derived by assuming that at incipient fluidization, the upward buoyant force, F~, on the solid grains in a fluidized bed and the downward net impelling force are equal at equilibrium. The buoyant force, Fb, on the solid grains in a fluidized bed is equal to the weight of the mixture
displaced which is given by equation (11) Fb = ¥dgP,~
(11)
in which Yd = volume of the mixture displaced and Pm= mass density of the mixture. The floating solid grains of effective density, p,, within the mixture of volume, Va, must have a net impelling force downward equal to its weight less the drag force of the wash water past the solid particles. The net impelling force, F~, is given by equation (12), since the critical interstitial velocity, v c = v*/f~.
F, =
3Vd PfV*2
V~gp,- co
2d~o V~
(12)
in which V * = o p t i m a l upflow wash rate, Vdgp~=weight of the solid grains, C o = d r a g coefficient of the solid grains, dh~ = geometric mean hydraulic diameter, 3Vd/2dha = cross-sectional area of the solid particles and p/V*2/2f~ = dynamic pressure. When the fluidized bed is in equilibrium. Upward force, Fb = downward force, F, V ~ p , , , = Vagp, -
3Vd pfV .2 CD 2dha ~ 2
3 n-V .2 Pm = P, -- CO --2dha ":-------~ 2gf~'
(13)
(14)
The Reynolds number for the friction drag past the particle is given by equation (15)
V* d~ Re = pf /1f~
(15)
from which V*w2/,f~ = #2 Re2/p}d~ substituting this value into equation (14) we get 3/12
Pm = Ps
4gpfd~ CDRe2
CDRe 2 = 4 gp/(p, - pm)d 3 3/12
(16)
(17)
Optimum wash rate at critical fluidization Knowing all the parameters in equation (17), the modified diameter term, CoRe s, can be evaluated and the corresponding Reynolds number, Re, obtained from the Camp curves. Although the Camp curves are derived from single spheres in a clean fluid, it was observed in an earlier work Quaye (1976) that the value of the modified diameter term, CoRe s, and the CoRe s for discrete particles was the same. The use of Camp curves in the present model is, therefore, justified. Considering the lowest two layers of a multi-layer filter, the optimal upflow wash rate at critical fluidization, V~*,is the lowest wash rate at which the upward buoyant force on the solid grains in a fluidized bed and the downward net impelling force are equal at equilibrium and the porosity at the
1080
B.A. QUAYE
critical fluidization is equal to the geometric mean porosity of the granular fixed bed, i.e. f~ -- fa. It can be obtained from equation (18) V* = ufoRe
(18)
pra~
When the fluidized bed is in equilibrium, upward force equals downward force. Upward force--apfg dh in which " a " is the area of the filter bed. Downward force = weight of the solid particles in the fluid = ag(pp - p:)(l - f~) dy
in which /~ = viscosity of the fluid, f~ = geometric mean porosity, Re = Reynolds number and d~ = geometric mean hydraulic diameter.
pfdh = (pp - py)(1 - f ~ ) dy
de
The present model has established that the optimum expanded porosity f * is given by equation (19)
:'e----IVy]
.9,
It is now proposed to obtain the value of the optimum expanded porosity, f * , by considering the hydrodynamic shear forces in a fluidized bed and identifying the velocity gradient, G, in a non-laminar flow. Camp and Stein developed an expression for the mean velocity gradient, G, by considering the shear forces on an elemental volume of liquid, as given by equation (20)
G
(20)
- L~VJ
in which G = d u / d y = m e a n velocity gradient, V = volume of liquid, # = dynamic viscosity and Pl = power input due to the shear force = loss in energy or head loss in the fluid which is dissipated as heat. Consider an elemental volume of fluid, height Ay and area A x A z , moving with a velocity u. If the hydraulic gradient in the y-direction or the head loss per unit length is dh/dy, the power dissipated by the elemental volume, A y A x A z is given by equation (21) dh P2 = ~y Ay A x A z p f g u P2
dh
(21) (22)
-~- = pfgu --~yy.
(Pv - P:) (1 - f~).
dh
Derivation of the optimum expanded porosity
(27)
p:
Substituting equation (27) in equation (26)
o =[: v.(p, -- Of)(1 q but from equation (1), Vw= Vsf~Ix= Vsf~ in which n = l/x f~)],/2
G=[g~(pp_p:)(l-
(28)
G = K,[f~-l(1 --fe)] 1/2
(29)
Ke =
(30)
in which V~(op - p:)]
.
Differentiating G with respect to f~ in equation (29) we get dG = ½K~[f~"-'(1 _ f~)]-,/2
x [(1 -f~)(n - 1)f7 - 2 - f ~ " - q ,
(31)
At maximum velocity gradient, G, dG -.'~=0'
and
d2G -d-~2=-ve.
Equation (36) becomes (n --
1)(1 - - f ~*) f , *n-2 __fe*n-I = 0 (n = 1)(l - - f ~ * ) = ~
fe -
(n -- 1)(1 - - f ~ * ) = f * n -- 1 - - n f * e + f * = f *
From equation (20), P, ~-- = #G 2.
n
(23)
Equating equations (22) and (23) (since Pl = / ' 2 ) dh laG 2 = pfgu ~
(24)
G - [ ",. dh 1"
-L~
n--1 f~* = - -
(25)
~YYJ
for a fluidized bed with a superficial velocity, Vw, where u ---- Vw/L.
(32)
The optimum porosity of the expanded medium, f~*, equals [(n - 1)/n] which is the same result obtained by Amirtharajah (1978). The values of the exponent, x, for anthracite and sand, determined experimentally in an earlier work Quaye (1976) were found to be 0.25 and 0.22 respectively. For anthracite, nl -- 1/Xm = 1/0.25 = 4. Optimum porosity of anthracite, f ~ , is given by equation (33) f~ = nl- 1 nl
[' /~ A d y ]
"
(26)
f ~ = 0.75.
(33)
Predicting optimal expansion of dual and tri-media beds For sand, n 2 = 1/x 2 = 1/0.22 = 4.5. Optimum porosity of sand, f * is given by equation (34) f,
_ n2 - 1
(34)
n2
f * = 0.78. For alundum, 1/0.22 = 4.5
n3 = 1/x 3 =
1081
(sand), f2 = initial porosity of the tightly compacted flintag (sand), f~3 = expanded porosity of alundum and f3 = initial porosity of the tightly compacted alundum. In predicting the optimum expansion of tri-media filter bed it must be noted that whereas the general expression for the expanded porosity given by equation (1) can be used to compute the expanded porosities of the flintag (sand) and alundum, the predicted porosity of anthracite must be obtained from either equation (37) or (38).
fe~ = n 3 - 1 = 0.78. n3
In laminar flow, n=5,
THEORETICAL VERIFICATION OF THE NEW MODEL FOR PREDICTING THE OPTIMAL EXPANSION OF DUAL AND TRI-MEDIA FILTER BEDS
f*=0.8;
in turbulent flow, n = 2.5, f * = 0.6. The optimum porosity of the expanded medium in the transitional regime therefore lies between 0.600 and 0.800, that is 0.600 < f * < 0.800.
Optimum expansion of dual and tri-media filter beds The expanded bed height at the optimal upflow wash rate, V*, is obtained from equation (35) Le L
-
-
=
-
1-f 1 -f* -
(35)
in which Le = expanded height of the medium, L = compacted height of the medium, f - - i n i t i a l porosity of the tightly compacted medium and fe* = optimum porosity of the expanded medium. A new mathematical model for predicting the optimum expansion of dual and tri-media beds developed and tested experimentally in an earlier work Quaye (1976) by the writer can be used to determine the optimum expansion of dual or tri-media filter bed and hence the height of the filter weirs in order to prevent the loss of filter media during backwash. An empirical relationship of the ratios of porosities of the tightly compacted beds and the porosities of the expanded beds was used to predict the exact expansion of the multilayer filter beds. This simple and efficient model for predicting expansion of dual and tri-media filter during backwash is given as follows: f~l
f~2 re3
fl
A
-- =
f~l__=__f,2
fl
A
f;1
f¢3
f,
A
-
A
It is proposed to verify equation (36) by computing the settling velocity in the transitional regime for the anthracite, flintag and alundum and to use the optimal wash rate (21 m m s -I) in equation (19) to determine the optimum expanded porosity. If the ratio of the optimum expanded porosity, fe*, and the initial porosity of the fixed bed, f, for the anthracite, flintag and alundum is the same in the transition and laminar zones then new model is verified theoretically.
Computation of the settling velocity of the flintag in the transition zone Consider a discrete particle of flintag, effective diameter 0.65 mm. The settling velocity of the flintag hydraulic diameter 0.44mm is derived from the general equation (39)
V22 = 4 g (Pp2 - Pf)dh2
(39)
3CDPf
The C D in the transition zone is given by Allen's approximation: 18.5 CD = Re06
(40)
/
~
= 18.5L~
\0.6
)
.
(41)
Substituting equation (41) in (39) we get
vh =
L
3pf
o,)e.2]
°.'
_] 18.5/1o. `
(42)
(36)
= LFO'O72g(PP2~--Pf)J d~26V~i6
(43)
(37) TO" ,41.6 1/0.6 ~tx2t*h2 • s2 1.4
(38)
in which f ~ = predicted expanded porosity of anthracite or the heuristic porosity of the expanded anthracite, f~ = initial porosity of the tightly compacted anthracite, f~2 = expanded porosity of flintag
Vs 2 ~ K-'0.71,41.14 "x2 "h2 "
From equation (43) /(2 = 0.072 g (Pp2 - Pl) p~.4#0.6
(44)
1082
B.A. QuAY~ The settling velocity of the anthracite hydraulic diameter 0.87 m m is given by equation (51)
At 20°C,
u=lx
1 0 - 3 k g m - l s -~,
where
Pp2 = 2650 kg m -3,
KI = 0.072 g (Ppl - Py)
g = 9.81 ms -~.
fl~.4//0.6
g 0'71 = (4639.34) °"71 = 401.04
(52)
At 20°C,
d TM = (0.44 x 10-3) TM = 1.5 X 10 -4.
/~ = 1 x 1 0 - a k g m - l s - l ,
F r o m equation (44)
p: = 1000 kg m - a
Vs2 = 401.04 x 1.5 x 10-4ms -1
Ppl = 1400 kg m -3,
= 60.2mm s -1 .
g = 9.81ms -2.
F r o m equation (19) _ ( 2 1 "]0.25 f * - \6--0~.2,] = 0.793
(51)
Vsl ----~t~.IF0"71"hl "41"141
p/= 1000 kg m - 3
F r o m equation (51) (45)
Vss = (1124.61) °71 (0.87 x 10-3) TM = 146.63 x 3.24 x 10-4ms -1
f*
0.793
f2
0.490
--=
= 1.62.
(46)
= 47.5 mm s -~ . F r o m equation (19)
Computation of the settling velocity of the alundum in the transition zone Consider a discrete particle of alundum, effective diameter 0.46 mm. The settling velocity of alundum hydraulic diameter 0.34 mm is given by equation (47)
Vs3 = K °71 d ~ 14
(47)
where K3 = 0.072 g (Pp3 - Pf)
p~.4#0.6
(48)
At 20°C, /~ =
1 ×
1 0 - 3 k g m - l s -l
g = 9.81ms -2.
Vs3 = (8294.02) °'71 (0.34 × 10-3) TM
V,3 = 67.2 m m s-1.
1.77.
(54)
The ratio o f f * / f t is rather high. The predicted expanded porosity of anthracite, f e*' is, therefore, obtained from equation (36) 0.460 x 0.793 0.490 = 0.744.
1.62.
(55)
(56)
fe*{__=__=__=f* f~ 1.62.
F r o m equation (19)
f, (49)
(50)
Computation of the settling velocity of the anthracite in the transition zone Consider a discrete particle of the anthracite, 1 . 2 4 ram.
effective diameter
f* 0.815 . . . . f~ 0.460
Using the hydraulic diameter in the computation of the settling velocity in the transition zone, equation (36) is verified theoretically as shown in equation (57)
= 605.79 × 1.11 × 10-4ms -1
= 1.61.
In equation (32), for laminar flow, n = 5 , f * = 0.8, the value of f * = 0.815 in the transition zone is, therefore not acceptable
f e*' 0.744 . . . . fl 0.460
F r o m equation (47)
0.774 0.480
(53)
In equation (32), for turbulent flow, n = 2 . 5 , f * = 0.6, the value o f f * ' = 0 . 7 4 4 in the transition zone is therefore acceptable
Pp3 = 3950 kg m -3
f~ --= f3
= ( 2 1 "]0.2, \4-~.5,] = 0.815.
f*' =
p = 1000 kg m - 3
_ ( 2 1 "]0.22 f ~ -- \6--~.2 ] = 0.774
f*
f2
(57)
f3
It will be noted from equations (55), (45) and (49) that each calculated optimum porosity of the expanded media, in the transition zone lies between 0.600 and 0.800, that is 0.600 < f * < 0.800; the calculated optimal upflow rate of 21 m m s -1 is therefore reasonable. In predicting the expansion of dual and tri-media filter beds, the writer has demonstrated that the new model is applicable in the transition and laminar zones as shown in equations (57) and (63).
1083
Predicting optimal expansion of dual and tri-media beds Table 1. Pilot plant fluidization experimental data 1. Wash rate, Vw,: 10 mm s-' Theoretical expansion Tightly compacted media
Filter media Anthracite (10/12) Flintag (22/25) Alundum (30/36) Total expansion (m)
Practical expansion
Loosely compacted media
Porosity of expanded media f,
Initial depth, L (m)
Expanded height, L~ (m)
Porosity of expanded media fe
Initial depth, L (m)
Expanded height, L, (m)
Expanded height, L, (m)
0.602 0,642 0.628
0.42 0.26 0.20
0.57 0.37* 0.28* 1.22
0.602 0.642 0.628
0.45 0.31 0.21
0,58 0.37 0.27 1.22
0.56 0.34* 0.31" 1.21
Note: 0.8% discrepancy in total expansion. The theoretical total expansion, Let, using the value of 0.25 of the exponent x~ in the calculation of the porosity of the expanded anthracite, fea, is given by: LeT =
L~ + Le2+ Le3
= (0.605 + 0.37 + 0.28) m = 1.26 m .
The 3.3% discrepancy in the theoretical and practical total expansion is high compared to the 0.8% discrepancy obtained in Table I by equating the ratio of the expanded and initial porosities of the anthracite to the expanded and initial porosities of either the flintag (sand) or alundum as shown in equations (37) and (38). *There is intermixing at the interface of flintag/alundum since for mixing fluidization, f~ > 0.440, i.e. 0.485 > 0.440. The discrepancy is due to the sinking of the flintag into ahindum, The total theoretical expansion is, however, equal to the total practical expansion.
C o m p u t a t i o n o f t h e t o t a l e x p a n d e d h e i g h t using either t h e initial p o r o s i t i e s o f t h e lightly c o m p a c t e d m e d i a o r t h e p o r o s i t i e s o f t h e loosely c o m p a c t e d m e d i a a n d t h e c o r r e s p o n d i n g d e p t h s o f t h e multilayer filter b e d s r e s u l t e d in a c c u r a t e p r e d i c t i o n o f the e x p a n s i o n o f t h e filter m e d i a as s h o w n in T a b l e s 1 a n d 2. APPLICATION OF THEORY
Pilot filter bed details Detail of ideal tri-media filter bed Pilot filter column 2.10 m x 150 mm o.d.; i,d.: 140 mm Top layer: 0.42 m deep, 10-12 anthracite ] Added by weight Middle layer: 0.26m deep, 22-25 flintag ~. and levels checked Bottom layer: 0.20m deep, 30-36 alundum .J after consolidation Base: 0.15 m deep, 7-10.5 nun glass marbles Total depth of tightly compacted media:
0.88 m
Total depth of loosely compacted media:
0.97 m
MODEL FOR COMPUTATION O F THE OPTIMUM WASH RATE
Design problem U s i n g the ideal t r i - m e d i a filter b e d d e s i g n p a r a m e t e r s listed a b o v e , calculate the flow rate for optimum backwashing of graded tri-media ( a n t h r a c i t e / f l i n t a g / a l u n d u m ) filter bed. T h e b a c k w a s h w a t e r is at a t e m p e r a t u r e o f 20°C a n d the t r i - m e d i a filter b e d is fluidized in a filter c o l u m n o f i.d. 1 4 0 m m . T h e d y n a m i c viscosity a n d d e n s i t y o f w a t e r at 20°C is 1 x 10- 3 kg m - ~s - ~ a n d 1000 kg m - 3 respectively.
Solution First, calculate the flintag (sand) d i m e n s i o n l e s s velocity t e r m , CD/Re in the t r a n s i t i o n a l r e g i m e by u s i n g e q u a t i o n (58)
Top surface 0.00 m above first sampling point Diameters of materials: anthracite: 1.24 mm fiintag: 0.65 mm ahindum: 0.46 mm Densities of materials: anthracite: 1400 kg m-3 flintag: 2650 kg m -3 ahindum: 3950 kg m - 3
Ct> = 4 g (P2 - Pf)# Re 3 3p f2 Vs2
Porosities of tri-media filter bed: (I) Porosity of anthracite: granular fixed bed flintag: (2) Porosity of media ahindum: in water Mass of materials: anthracite: flintag: ahindum:
C----~D----5.1 X 10 -2. Re
Settling velocities of materials: (1) Settling velocity in anthracite: the transition zone flintag: (2) Settling velocity in alundum: the laminar zone Exponent x values of the ratio of wash anthracite: flintag: alundum:
(1) 0.460 0.490
0.480
(2) 0.490 0.575 0.520
4.950 kg 5.410 kg 6.180 kg (1) 47.5 mm s-1 60.2 mm s - ~ 67.2 mm s - ~
(2) 65 mm s- ] 75 m m s 83 m m s - '
rate to settling velocity: 0.25 0.22 0.22
(58)
4 × 9 . 8 1 ( 2 6 5 0 - - 1000) x I x 10 -3 3 × 106 × (0.075) 3
U s i n g t h e empirical r e l a t i o n b e t w e e n C D / R e a n d R e (i.e. C a m p curves as s h o w n in Fig. 2), at CD/Re=5.1x 10 -2 , R e = 3 3 . S e c o n d , calculate t h e e q u i v a l e n t (hydraulic) dia m e t e r , dh2, o f flintag ( s a n d ) f r o m R e y n o l d s N o . o b t a i n e d in Step 1 b y u s i n g e q u a t i o n (59)
dh2 = ~'R---L p/V~2
(59)
# R e = 1 x 10 -3 x 33 -- 0 . 4 4 m m .
dh2 = p/Vs2
1000 x 0.075
B. A. QUAVE
1084
Table 2. Pilot plant fluidization experimental data 2. Wash rate, Vw2:20mm s-~ Theoretical expansion
Porosity of expanded media f, 0.700 0.747 0.732
Initial depth, L (m) 0.42 0.26 0.20
Practical expansion
Loosely compacted media
Tighly compacted media Expanded height, L, (m) 0.76 0.52 0.39 1.67
Porosity of expanded media f, 0.700 0.747 0.732
Initial depth, L (m) 0.45 0.31 0.21
Expanded height, L, (m) 0.77 0.52 0.38 1.67
Expanded height, L, (m) 0.78 0.53 0.37 1.68
Filter media Anthracite (10/12) Flintag (22/25) Alundum (30/36) Total expansion (m) Note: 0.5% discrepancy in total expansion. The theoretical total expansion, LeT, using the value of 0.25 of the exponent xt in the calculation of the porosity of the expanded anthracite. fel, is given by: LeT = Lel + Le2 + Le3 = ( 0 . 8 9 5 + 0.521 + 0 . 3 8 ) m = 1.80 m.
The 7.2% discrepancy in the theoretical and practical total expansion is high compared to 0.5% discrepancy obtained in Table 2 by equating the ratio of the expanded and initial porosities of the anthracite to the expanded and initial porosities of either the flintag (sand) or alundum as shown in equations (37) and (38). It is apparent from Table 2 that in the region of the optimal upflow wash rate (21 nun s -I) the theoretical expansion is equal to the practical expansion of each medium. The new model is therefore verified experimentally.
Third, calculate the a l u n d u m dimensionless velocity term, Co/Re in the transitional regime by using e q u a t i o n (60)
Co Re
4g(p3--'pf)# 3 3pf2 Vs3
F o u r t h , calculate the equivalent (hydraulic) diameter, dh3, o f a l u n d u m from Reynolds No. o b t a i n e d in Step 3 using e q u a t i o n (61) #Re
(60)
p:v,3
4 x 9 . 8 1 ( 3 9 5 0 - 1000) x l x 10 -3 dh3 ~
3 x 106 x (0.083) 3
Co
(61)
dh3 ~ - -
#Re
1 ×
pfVs3
1000 x 0.083
10 - 3 × 2 8
dh3 = 0.34 m m .
6.75 × 10 -3.
Re U s i n g the empirical relation between Co/Re a n d Re (i.e. C a m p curves), at CD/Re = 6.75 x 10 -2, Re = 28.
Fifth, calculate the geometric m e a n hydraulic diameter, dh=, by using e q u a t i o n (9)
dha
=
(dh2 X dh3) 1/2
dha = (0.44 x 0.34) 1/2 = 0.39 m m .
Comp curves 10 6
10
Sixth, calculate the geometric m e a n porosity, f , , by using e q u a t i o n (10) f~ = (f2 x A ) '/2
10 5
fa = (0.490 x 0.480) 1/2 = 0.485. Seventh, calculate the effective density, p,, o f the solid grains (flintag/alundum) by using e q u a t i o n (8)
lo"
10-1
w2 Ps P3 =
10 - 2
10 3
10 - 3
10 2
= 3950 I 10
L 10 2
W2P3
~--~ o~ +1
55.410410 6.18-----9+ 1 × 3950 L6.180 × 2650 + 1
103
Re
Fig. 2. Camp curves.
Ps = 3200 kg m -3 .
Predicting optimal expansion of dual and tri-media beds Eighth, calculate the density of mixture, p,,, by using equations (7) and (62) p,, = (I -- fa)P,+
f,,Pf
(62)
in the laminar zone lies between 0.600 and 0.800, that is, 0.600 < f * < 0.800; the calculated optimal upflow wash rate of 21 mm s -~ is therefore reasonable.
in which p, = effective density of the solid grains, f~ = f~ = geometric mean porosity of the granular bed at the incipient fluidization and p: = density of water p~, = (1 - 0.485)3200 + (0.485 x 1000) = 2133 kg m -3. Ninth, calculate the modified diameter term, Co Re 2, for flintag/alundum in the transitional regime by using equation (17) Co Re 2 = 4gp/(p~ - - p,,)dL 3# 2 4 x 9.81 x 1000(3200-2133) x (0.39) 3 x 10 -9 3 x l x l 0 -6 CoRe 2 = 8.28 x 102. Using the empirical relation between CoRe 2 and Re (i.e. Camp curves) at C o R e 2 = 8 . 2 8 x 102, Re = 17. Tenth, calculate the optimum wash rate, V~*, from the Reynolds No. obtained in Step 9 by using equation (18) #faRe
V*=--
p:d~
I x 10 -3 x 0.485 x 17 1000 x 0.39 x 10 -3
V* = 2 1 m m s -1. Eleventh, calculate the optimum porosity of the expanded alundum, f,'~, by using equation (19) f* =
~
J
=
0.739.
Twelfth, calculate the optimum porosity of the expanded flintag, f ~ , by using equation (19)
rL V- sT2 j = r=,T L 3j
=
0.756.
PREDICTING EXPANSION OF DUAL OR TRI-MEDIA FILTERS
Alundum : exponent x 3 value Using equations (1) and (35) and assuming that the exponent x3 is 0.22, the porosity of the expanded alundum, fe3, and the expanded height, Le3, can be calculated. Since the theoretical expanded height is equal to the practical expanded height, as shown in Table 2, the assumed value of x 3 for the prediction of the expansion of alundum is justified. Flintag (sand): exponent x2 value For a tri-media filter, the value of 0.22 of the exponent, x2, determined experimentally in an earlier investigation Quaye (1976) as shown in Fig. 3 can be used in the calculation of the porosity of the expanded flintag, f,2, and the expanded height, Le2. Anthracite: exponent x~ value The value of 0.25 of the exponent Xl determined experimentally, as shown in Fig. 3, resulted in the calculation of a high value of the expanded height, L,~. It is therefore recommended that the predicted porosity of expanded anthracite must be obtained from either equation (37) or (38) and used in the computation of the expanded height, L~, of the anthracite. Predicting the expansion of tri-media filter bed Using equations (1), (35) and (37) or (38) and the exponents, x3 = 0.22, x2 =0.22 and the predicted porosity of expanded anthracite, f ~ , the total expanGraph of log
f*
Y,
A
(Yw/~)
vs
log[1-(L/Le)(1-f)]
Log ( v . / ~ -1.4 I
-1.2 I
-1.0 1
-0.8 I
)
-0.6 I
-0.4 I
-0.2 I ///~/
Thirteenth, calculate the predicted optimum porosity of the expanded anthracite, f ; , by using either equation (37) or (38) f;*
1085
//_o.,o
x/~/
-o.1~ "
nd (18/301
t$
W.R, 21/9--F
- 0 . 2 0 ~1
T
-0.25
f~l = 0.710.
f~__=__ = _f'e3 _ 1.54. (63) f, f2 A It will be noted in Steps 11, 12 and 13 that each calculated optimum porosity of the expanded media
-0.05
Anthracite ( 7 / 1 4 ) ~ / x " /
f ; * = f ~ _ _ = x f ~ 0.460 x 0.756 f2 0.490 Using equation (36) we get
0
- -0.30
o - -0.35 0 . 3 5 - 0.10
Slope of anthracite, x 1 • ~
,
0.2.5
Slope of sond,x 2 • (~55-0.10 • 0.22 1.32- 0.20
Fig. 3. Exponent graphs of antracite and sand.
1086
B.A. QUA'rE
sion of a tri-media filter was computed when the wash rate, Vw~= 10mm s -1. Each theoretical total expansion of a tightly compacted media and a loosely compacted media was computed and compared with the practical total expansion. The tabulated experimental results are shown in Table 1. The discrepancy in the computation of the total expansion is 0.8%. Using a wash rate, Vw2= 20 mm s -1, the experimental investigation was repeated and the experimental results tabulated as shown in Table 2. The discrepancy in the computation of the total expansion is 0.5%. Predicting the expansion of a dual-media filter bed In an earlier investigation Quaye (1976), the writer predicted the expansion of a commercial dual-media filter [anthracite (7/14)-flintag (18/30) supported by coarse sand] during backwash, by considering the mass density of mixture of flintag/coarse sand (14/16). The coarse sand (14/16) is supported by coarse sand (10/14). The effective size of anthracite and flintag (sand) is 1.24 and 0.65 mm respectively, and the uniformity coefficient is 1.56 and 1.25 respectively. Using the model for computation of the optimum wash rate and the optimal expansion of tri-media filter, the writer predicted the exact expansion of a dual-media filter for the wash rate, Vw2= 20 mm s- 1, as shown in the tabulated experimental results. Total expansion
Anthracite (7/14) Flintag (18/30) Coarse sand (14/16) Coarse sand (10/14) Total
Theoretical (m) 0.57 0.94* 0.16" 0.05 1.72
Practical (m) 0.65 0.87* 0.15" 0.05 1.72
*There is intermixing at the interface of flintag/coarse sand (14/16) since at the point of incipient fluidization fc = fa and fa > 0.422, the porosity of the coarse sand (14/16), i.e. 0.455 > 0.422. The discrepancy is due to the sinking of the flintag into coarse sand (14/16). It is apparent from this investigation that the new model for predicting the optimal expansion of a dual or tri-media filter during backwash is a useful tool for designing the height of filter weirs in order to prevent the loss of filter media. CONCLUSIONS A study of the works of other researchers involved in the prediction of expansion of granular media during fluidization caused by backwashing processes confirms that in all cases, the expressions derived for the upflow wash rates consist of empirical constants which must be determined in order to evaluate the formulae for upflow wash rates. In the present model, the only constant required to be determined experimentally is the exponent x of the ratio of the upflow
wash rate and the settling velocity of sand. By considering the physical parameters such as the effective sizes, porosities and the spbericities of dual and tri-media filters the optimal upflow wash rates for various water temperatures were determined in an earlier work Quaye (1976) and the expansion of the filter beds predicted. Whilst the United Kingdom system of backwashing filter beds uses an upward flush of air first (air scour) followed by an upward water flush at velocities barely sufficient, and sometimes insufficient, to fluidize the bed; the United States system uses an upflow wash of sufficient velocity to fluidize the bed fully and provide substantial bed expansion. Both methods have resulted in dirt problems such as mud balls and filter cracks Cleasby (1976), Cleasby et al. (1975, 1977). It is now, generally accepted that the upflow wash rate should reach incipient fluidization as confirmed by the writer. From the theoretical and experimental study of the optimum backwash of sand and multilayer filters, the following conclusions can be drawn: (1) It has been calculated that the maximum shear in the fluid for optimum cleaning of anthracite-flintag-alundum media occurs at the optimum expanded porosity of 0.750, 0.780 and 0.780 respectively. These may be used for predicting the optimal expansion of tri-media filter beds. (2) The optimum porosity of the expanded medium in the transitional regime lies between 0.600 and 0.800, that is 0.600 < f * < 0.800. (3) In the lowest two layers of a multi-layer filter, the optimal upflow wash rate at critical fluidization is the lowest wash rate at which the upward buoyant force on the solid grains in a fluidized bed and the downward net impelling force are equal at equilibrium and the porosity at the critical fluidization is equal to the geometric mean porosity of the granular fixed bed. (4) Backwashing of deep granular filter beds with an upflow wash rate less than the optimum, resulted in the accumulation of suspended solids in the filter pores after a period of filter operation. This is an inefficient process of backwashing a filter with water only. Comparisons made by other research workers based upon this inefficient backwashing process with air and water backwashing are therefore not justifiable. For an efficient backwashing of the filter bed, the optimum expanded porosity should be used in the computation of the filter bed expansion. (5) A new model for predicting the optimal expansion of dual or tri-media during backwash is a useful tool for designing the height of filter weirs in order to prevent the loss of filter media. Acknowledgements--The writer wishes to express his sincere appreciation to Professor K. J. Ives for his guidance and encouragement during the period of investigation at the Kempton Park Experimental Treatment Works and in the Chadwick Public Health Laboratory, University College,
Predicting optimal expansion of dual and tri-media beds London. Financial support provided by the Thames Water Authority (formerly Metropolitan Water Board) is gratefully acknowledged.
REFERENCES
Amirtharajah A. (1978) Optimum backwashing of sand filters. J. envir. Engng Div. Am. Soc. cir. Engrs 104, 917-832. Cleasby J. L. (1976) Filtration with granular beds. Chem. Engr 314, 663-667, 682.
1087
Cleasby J. L., Stangl E. W. and Rice G. A. (1975) Developments in backwashing of granular filters, d. emir. Engng Div. Am. Soc. cir. Engrs 101, 713-727. Cleasby J. L. et al. (1977) Backwashing of granular filters. J. Am. Wat. Wks Ass. 69, 115-126. Fair G. M., Geyer J. C. and Okun D. A. (1968) Water and Wastewater Engineering, Vol. 2. Wiley, New York. Ives K. J. (1977) In Deep Bed Filtration, Solid-Liquid Separation (Edited by Svarovsky), Chemical Engineering Series, pp. 199-208. Butterworths, London. Quaye B. A. (1976) Contact filtration of reservoir water. Ph.D thesis, University of London.
0043-1354/87 $3.00+0.00 Copyright © 1987 Pergamon Journals Ltd
PREDICTING OPTIMUM BACKWASH RATES A N D EXPANSION OF MULTI-MEDIA FILTERS B. A. QUAYE Senior Lecturer, Sheffield City Polytechnic, Civil Engineering Department, Pond Street, Sheffield SI 1WB, U.K.
(Received May 1986) Abstract--A new model for predicting the optimal expansion of dual and tri-media beds is developed and tested experimentally. This is used to deterrqine the height of filter weirs in order to prevent the loss of filter media during backwash. For more accurate prediction of the expansion of the dual and tri-media filter beds it was assumed in an earlier work that the ratios of the porosities of tightly compacted beds and the porosities of the expanded beds are equal. This heuristic assumption for prediction of the exact expansion of the multi-layer filter beds is now verified theoretically.
Key words--settling velocity, fluidization, interstitial velocity, geometric mean hydraulic diameter, geometric mean porosity, optimum wash rate, porosity anthracite
NOMENCLATURE CD = D = d= dh = d~ = d~2 = dh3 = ds = Fb = Fd = Fi = f = fo = fc = f~ = f: = f3 = f,i = fit = f~2 = f~3 = f, = f* = G= g = h= K= K, = L = L, = L,~ = L,2 = Le3 = Let = n= P~ = P2 = Re =
Drag coefficient Diameter of filter column Grain diameter Hydraulic diameter of solid grains Geometric mean hydraulic diameter of solid grain Hydraulic diameter of the light solid Hydraulic diameter of the heavy solid grain Sieve diameter of solid grain "Buoyant" force Drag force Net impelling force Porosity of filter medium Geometric mean porosity Porosity at critical fluidization Initial porosity of anthracite Initial porosity of flintag (sand) Initial porosity of alundum Expanded porosity of anthracite Predicted expanded porosity of anthracite Expanded porosity of flintag (sand) Expanded porosity of alundum Expanded porosity Optimum expanded porosity Velocity of shear gradient (du/dy) Gravitational acceleration Head loss in filter media Constant Constant Depth of granular bed Expanded depth of the filter medium Expanded depth of anthracite Expanded depth of flintag (sand) Expanded depth of alundum Total expanded depth of the filter media Expansion coefficient Power input due to the shear forces Power dissipated by elemental volume Reynolds number (hydrodynamic mechanisms) V, = Stoke's settling velocity of a solid grain Vw= Upflow wash rate Vc = Critical interstitial velocity
V.* = Optimum upflow wash rate W2 = Fractional mass (per unit volume of bed) of light grains W3 = Fractional mass (per unit volume of bed) of heavy grains x = Exponent of the ratio of wash rate to settling velocity Xl = Exponent of the ratio of wash rate to settling velocity of anthracite with expanded coefficient, n~ x 2 = Exponent of the ratio of wash rate to settling velocity of sand with exponent coefficient, n 2 x3 = Exponent of the ratio of wash rate to settling velocity # = Dynamic viscosity of liquid Pl = Density of liquid Pm= Mass density of mixture pp = Density of solid grains p, = Effective density of solid grains Vd = Volume of mixture displaced ¥2 = Volume fraction of light grains with density, P2 V3 = Volume fraction of heavy grains with density, P3-
THEORY OF BACKWASHING
Derivation o f optimum wash rate at criticalfluidization In the study o f particle mechanics or general m e c h a n ics dealing with the relative m o t i o n between a particle (solid or liquid) a n d a s u r r o u n d i n g fluid (liquid or gas), it is immaterial w h e t h e r the particle m o v e s in a s t a t i o n a r y fluid or is suspended in a m o v i n g fluid. Typical examples are the experimental d e t e r m i n a t i o n o f the terminal settling velocity o f a discrete particle a n d the wash rate o f a filter m e d i u m . T h e o p t i m u m wash rate depends to a great extent o n the o p t i m u m e x p a n d e d porosity, f * w h i c h is a f u n c t i o n o f the t e r m i n a l settling velocity a n d the o p t i m u m wash rate. T h e general expression for the 1077
B.A. QUAYE
1078
expanded porosity, f,, will therefore be the basis for the derivation of the optimum wash rate at critical fluidization as shown in equation (18). The expression for the expanded porosity, f,, is given by equation (1)
Predicting the optimal expansion of tri-media filter beds during backwash PiLot f i l t e r plant I d e a l t r i - m e d i a filter
=FvwT f, Lv, J
"
•
/FiLter co(umn / 2 1 0 0 mm Long ~ 15omm o.d. J 140mm i.d.
(1) --4--
in which Vw= upflow wash rate, Vs = terminal settling velocity and x = exponent, its value depends upon the medium. Fair et al. (1968) determined the value of x, experimentally and recommended the exponential of 0.20 and 0.25 for free and hindered settling of sand medium. The mean unhindered terminal velocity, V,, in equation (1) can be calculated by timing--e.g. 20 individual particles settling through 1 m in a sedimentation tube containing a column of water--the velocity term in the transitional regime can be evaluated; and the equivalent (hydraulic) diameter, dh, of the solid grains calculated from the dimensionless relationships of Co/Re with Re, and with use of the "Camp curves", the optimum wash rates of multimedia filters can be calculated as demonstrated in the "model for computation of the optimum wash rate".
l
i
Depth of LooseLy compacted medm (m)
Depth of tightly compacted media (m)
0.97
j
Anthrocite (10/12 )
0.52 046
Computation of porosity at critical fluidization of Welsh anthracite /crushed flintag
(22/25)
In a fluidized bed, the grains of light filter medium will not settle into and mix with the grains of heavy material unless the density of the light grain exceeds the density of the mixture of heavy grains and water. Anthracite/Flintag. Consider a layer of anthracite with a density, Pl = 1400 kg m -3 and porosity, fl, on top of a flintag bed, density, P2 2650 kg m -3 and porosity, f2, density of water is p/. At critical fluidization when anthracite will no longer "float" the critical porosity,f~, of the mixed media is obtained by solving equation (2), which gives f~ = 0.760
0.21
0.20
undum (30/36) 0
0
=
p~ = (1
-
f~)P2 +f~ P:
Computation of porosity at critical fluidization of crushed flintag/alundum [fused alumina (A1203)] Flintag/Alundum. Consider a layer of flintag with density, P2 = 2650 kg m -3 and porosity, f : , on top of alundum bed, density, P3 = 3950 kg m -a and porosity f3. At critical fluidization when the flintag will no longer "float", the critical porosity, f~, of the mixed media is obtained by solving equation 3 which gives
p2= (l -L)p3 +f~p/
for mixing fluidization f~ > 0.440.
Fig. 1. Pilot plant--ideal tri-media filter beds.
(2)
for mixing fluidization f~ > 0.760. Since in equation (18), the geometric mean porosity of the anthracite/flintagfa
f , = 0.440
morbl.es
Since in equation (18), the geometric mean porosity of the flintag/alundum, fo >f~, i.e. 0.485 > 0.0440, there will be intermixing at the interface during the backwashing of the filter as shown in Fig. I.
Derivation of the mass density of the mixture Consider a mixed media (flintag/alundum) bed fluidized by water, the mass density of the mixture, p,,, at a level where the porosity of the mixed media isf~ can be derived by re-arranging equations (6) and (5) and substituting them in equation (4) Pm =
~/'2P2 + V3P3 @ f c P f
(4)
V__2= W2P3 V3 W3P2
(3)
(5)
V 2 + V 3 = I - f~
in
which
Pm= mass
density
(6)
of
the
mixture,
1079
Predicting optimal expansion of dual and tri-media beds volume fraction of light grains with density P2, V3 = volume fraction of heavy grains with density P3, f~ = porosity at critical fluidization, p / = density of water, W2 = mass fraction (per unit volume of bed) of light grains and I4"3= mass fraction (per unit volume of bed) of heavy grains. The resultant mass density of the mixture is given by equation (7) V2 =
Pm= (1 - f~)P, + fcP/
(7)
in which p~ is the effective density of the solid grains given by equation (8)
Ps ~" P3
(8) W2p3 + 1 W3p~
Geometric mean hydraulic diameter The geometric mean hydraulic diameter, dh~, is given by equation (9) which is used in the derivation of the modified diameter term, CD Re 2, in equation (17) dh~ = (dh2 x dh3)1/2 (9) in which dh2 = hydraulic diameter of the light solid grains and dh3 = hydraulic diameter of the heavy solid grains. Geometric mean porosity of the granular bed The geometric mean porosity of the granular bed, fa, is given by equation (10) which is used in the derivation of the optimal upflow wash rate, V*, in equation (18) L = (A x A) '/2
(10)
in which f2 = porosity of the tightly compacted light solid grains of the granular fixed bed and f3 = porosity of the tightly compacted heavy solid grains of the granular fixed bed. In a fully fluidized graded bed (i.e. grains of various sizes), the smaller grains tend to accumulate in the upper layers and the larger grains in the lower layers. There is, therefore, a gradual increase in porosity from bottom to top which results in decreasing interstitial velocity from bottom to top. Since the BS sieve sizes are in geometric progression, the use of geometric mean for both hydraulic diameter and porosity is justified. Derivation of modified diameter term, Co Re 2,for dual and tri-media filter beds The modified diameter term, CoRe 2, is derived by assuming that at incipient fluidization, the upward buoyant force, F~, on the solid grains in a fluidized bed and the downward net impelling force are equal at equilibrium. The buoyant force, Fb, on the solid grains in a fluidized bed is equal to the weight of the mixture
displaced which is given by equation (11) Fb = ¥dgP,~
(11)
in which Yd = volume of the mixture displaced and Pm= mass density of the mixture. The floating solid grains of effective density, p,, within the mixture of volume, Va, must have a net impelling force downward equal to its weight less the drag force of the wash water past the solid particles. The net impelling force, F~, is given by equation (12), since the critical interstitial velocity, v c = v*/f~.
F, =
3Vd PfV*2
V~gp,- co
2d~o V~
(12)
in which V * = o p t i m a l upflow wash rate, Vdgp~=weight of the solid grains, C o = d r a g coefficient of the solid grains, dh~ = geometric mean hydraulic diameter, 3Vd/2dha = cross-sectional area of the solid particles and p/V*2/2f~ = dynamic pressure. When the fluidized bed is in equilibrium. Upward force, Fb = downward force, F, V ~ p , , , = Vagp, -
3Vd pfV .2 CD 2dha ~ 2
3 n-V .2 Pm = P, -- CO --2dha ":-------~ 2gf~'
(13)
(14)
The Reynolds number for the friction drag past the particle is given by equation (15)
V* d~ Re = pf /1f~
(15)
from which V*w2/,f~ = #2 Re2/p}d~ substituting this value into equation (14) we get 3/12
Pm = Ps
4gpfd~ CDRe2
CDRe 2 = 4 gp/(p, - pm)d 3 3/12
(16)
(17)
Optimum wash rate at critical fluidization Knowing all the parameters in equation (17), the modified diameter term, CoRe s, can be evaluated and the corresponding Reynolds number, Re, obtained from the Camp curves. Although the Camp curves are derived from single spheres in a clean fluid, it was observed in an earlier work Quaye (1976) that the value of the modified diameter term, CoRe s, and the CoRe s for discrete particles was the same. The use of Camp curves in the present model is, therefore, justified. Considering the lowest two layers of a multi-layer filter, the optimal upflow wash rate at critical fluidization, V~*,is the lowest wash rate at which the upward buoyant force on the solid grains in a fluidized bed and the downward net impelling force are equal at equilibrium and the porosity at the
1080
B.A. QUAYE
critical fluidization is equal to the geometric mean porosity of the granular fixed bed, i.e. f~ -- fa. It can be obtained from equation (18) V* = ufoRe
(18)
pra~
When the fluidized bed is in equilibrium, upward force equals downward force. Upward force--apfg dh in which " a " is the area of the filter bed. Downward force = weight of the solid particles in the fluid = ag(pp - p:)(l - f~) dy
in which /~ = viscosity of the fluid, f~ = geometric mean porosity, Re = Reynolds number and d~ = geometric mean hydraulic diameter.
pfdh = (pp - py)(1 - f ~ ) dy
de
The present model has established that the optimum expanded porosity f * is given by equation (19)
:'e----IVy]
.9,
It is now proposed to obtain the value of the optimum expanded porosity, f * , by considering the hydrodynamic shear forces in a fluidized bed and identifying the velocity gradient, G, in a non-laminar flow. Camp and Stein developed an expression for the mean velocity gradient, G, by considering the shear forces on an elemental volume of liquid, as given by equation (20)
G
(20)
- L~VJ
in which G = d u / d y = m e a n velocity gradient, V = volume of liquid, # = dynamic viscosity and Pl = power input due to the shear force = loss in energy or head loss in the fluid which is dissipated as heat. Consider an elemental volume of fluid, height Ay and area A x A z , moving with a velocity u. If the hydraulic gradient in the y-direction or the head loss per unit length is dh/dy, the power dissipated by the elemental volume, A y A x A z is given by equation (21) dh P2 = ~y Ay A x A z p f g u P2
dh
(21) (22)
-~- = pfgu --~yy.
(Pv - P:) (1 - f~).
dh
Derivation of the optimum expanded porosity
(27)
p:
Substituting equation (27) in equation (26)
o =[: v.(p, -- Of)(1 q but from equation (1), Vw= Vsf~Ix= Vsf~ in which n = l/x f~)],/2
G=[g~(pp_p:)(l-
(28)
G = K,[f~-l(1 --fe)] 1/2
(29)
Ke =
(30)
in which V~(op - p:)]
.
Differentiating G with respect to f~ in equation (29) we get dG = ½K~[f~"-'(1 _ f~)]-,/2
x [(1 -f~)(n - 1)f7 - 2 - f ~ " - q ,
(31)
At maximum velocity gradient, G, dG -.'~=0'
and
d2G -d-~2=-ve.
Equation (36) becomes (n --
1)(1 - - f ~*) f , *n-2 __fe*n-I = 0 (n = 1)(l - - f ~ * ) = ~
fe -
(n -- 1)(1 - - f ~ * ) = f * n -- 1 - - n f * e + f * = f *
From equation (20), P, ~-- = #G 2.
n
(23)
Equating equations (22) and (23) (since Pl = / ' 2 ) dh laG 2 = pfgu ~
(24)
G - [ ",. dh 1"
-L~
n--1 f~* = - -
(25)
~YYJ
for a fluidized bed with a superficial velocity, Vw, where u ---- Vw/L.
(32)
The optimum porosity of the expanded medium, f~*, equals [(n - 1)/n] which is the same result obtained by Amirtharajah (1978). The values of the exponent, x, for anthracite and sand, determined experimentally in an earlier work Quaye (1976) were found to be 0.25 and 0.22 respectively. For anthracite, nl -- 1/Xm = 1/0.25 = 4. Optimum porosity of anthracite, f ~ , is given by equation (33) f~ = nl- 1 nl
[' /~ A d y ]
"
(26)
f ~ = 0.75.
(33)
Predicting optimal expansion of dual and tri-media beds For sand, n 2 = 1/x 2 = 1/0.22 = 4.5. Optimum porosity of sand, f * is given by equation (34) f,
_ n2 - 1
(34)
n2
f * = 0.78. For alundum, 1/0.22 = 4.5
n3 = 1/x 3 =
1081
(sand), f2 = initial porosity of the tightly compacted flintag (sand), f~3 = expanded porosity of alundum and f3 = initial porosity of the tightly compacted alundum. In predicting the optimum expansion of tri-media filter bed it must be noted that whereas the general expression for the expanded porosity given by equation (1) can be used to compute the expanded porosities of the flintag (sand) and alundum, the predicted porosity of anthracite must be obtained from either equation (37) or (38).
fe~ = n 3 - 1 = 0.78. n3
In laminar flow, n=5,
THEORETICAL VERIFICATION OF THE NEW MODEL FOR PREDICTING THE OPTIMAL EXPANSION OF DUAL AND TRI-MEDIA FILTER BEDS
f*=0.8;
in turbulent flow, n = 2.5, f * = 0.6. The optimum porosity of the expanded medium in the transitional regime therefore lies between 0.600 and 0.800, that is 0.600 < f * < 0.800.
Optimum expansion of dual and tri-media filter beds The expanded bed height at the optimal upflow wash rate, V*, is obtained from equation (35) Le L
-
-
=
-
1-f 1 -f* -
(35)
in which Le = expanded height of the medium, L = compacted height of the medium, f - - i n i t i a l porosity of the tightly compacted medium and fe* = optimum porosity of the expanded medium. A new mathematical model for predicting the optimum expansion of dual and tri-media beds developed and tested experimentally in an earlier work Quaye (1976) by the writer can be used to determine the optimum expansion of dual or tri-media filter bed and hence the height of the filter weirs in order to prevent the loss of filter media during backwash. An empirical relationship of the ratios of porosities of the tightly compacted beds and the porosities of the expanded beds was used to predict the exact expansion of the multilayer filter beds. This simple and efficient model for predicting expansion of dual and tri-media filter during backwash is given as follows: f~l
f~2 re3
fl
A
-- =
f~l__=__f,2
fl
A
f;1
f¢3
f,
A
-
A
It is proposed to verify equation (36) by computing the settling velocity in the transitional regime for the anthracite, flintag and alundum and to use the optimal wash rate (21 m m s -I) in equation (19) to determine the optimum expanded porosity. If the ratio of the optimum expanded porosity, fe*, and the initial porosity of the fixed bed, f, for the anthracite, flintag and alundum is the same in the transition and laminar zones then new model is verified theoretically.
Computation of the settling velocity of the flintag in the transition zone Consider a discrete particle of flintag, effective diameter 0.65 mm. The settling velocity of the flintag hydraulic diameter 0.44mm is derived from the general equation (39)
V22 = 4 g (Pp2 - Pf)dh2
(39)
3CDPf
The C D in the transition zone is given by Allen's approximation: 18.5 CD = Re06
(40)
/
~
= 18.5L~
\0.6
)
.
(41)
Substituting equation (41) in (39) we get
vh =
L
3pf
o,)e.2]
°.'
_] 18.5/1o. `
(42)
(36)
= LFO'O72g(PP2~--Pf)J d~26V~i6
(43)
(37) TO" ,41.6 1/0.6 ~tx2t*h2 • s2 1.4
(38)
in which f ~ = predicted expanded porosity of anthracite or the heuristic porosity of the expanded anthracite, f~ = initial porosity of the tightly compacted anthracite, f~2 = expanded porosity of flintag
Vs 2 ~ K-'0.71,41.14 "x2 "h2 "
From equation (43) /(2 = 0.072 g (Pp2 - Pl) p~.4#0.6
(44)
1082
B.A. QuAY~ The settling velocity of the anthracite hydraulic diameter 0.87 m m is given by equation (51)
At 20°C,
u=lx
1 0 - 3 k g m - l s -~,
where
Pp2 = 2650 kg m -3,
KI = 0.072 g (Ppl - Py)
g = 9.81 ms -~.
fl~.4//0.6
g 0'71 = (4639.34) °"71 = 401.04
(52)
At 20°C,
d TM = (0.44 x 10-3) TM = 1.5 X 10 -4.
/~ = 1 x 1 0 - a k g m - l s - l ,
F r o m equation (44)
p: = 1000 kg m - a
Vs2 = 401.04 x 1.5 x 10-4ms -1
Ppl = 1400 kg m -3,
= 60.2mm s -1 .
g = 9.81ms -2.
F r o m equation (19) _ ( 2 1 "]0.25 f * - \6--0~.2,] = 0.793
(51)
Vsl ----~t~.IF0"71"hl "41"141
p/= 1000 kg m - 3
F r o m equation (51) (45)
Vss = (1124.61) °71 (0.87 x 10-3) TM = 146.63 x 3.24 x 10-4ms -1
f*
0.793
f2
0.490
--=
= 1.62.
(46)
= 47.5 mm s -~ . F r o m equation (19)
Computation of the settling velocity of the alundum in the transition zone Consider a discrete particle of alundum, effective diameter 0.46 mm. The settling velocity of alundum hydraulic diameter 0.34 mm is given by equation (47)
Vs3 = K °71 d ~ 14
(47)
where K3 = 0.072 g (Pp3 - Pf)
p~.4#0.6
(48)
At 20°C, /~ =
1 ×
1 0 - 3 k g m - l s -l
g = 9.81ms -2.
Vs3 = (8294.02) °'71 (0.34 × 10-3) TM
V,3 = 67.2 m m s-1.
1.77.
(54)
The ratio o f f * / f t is rather high. The predicted expanded porosity of anthracite, f e*' is, therefore, obtained from equation (36) 0.460 x 0.793 0.490 = 0.744.
1.62.
(55)
(56)
fe*{__=__=__=f* f~ 1.62.
F r o m equation (19)
f, (49)
(50)
Computation of the settling velocity of the anthracite in the transition zone Consider a discrete particle of the anthracite, 1 . 2 4 ram.
effective diameter
f* 0.815 . . . . f~ 0.460
Using the hydraulic diameter in the computation of the settling velocity in the transition zone, equation (36) is verified theoretically as shown in equation (57)
= 605.79 × 1.11 × 10-4ms -1
= 1.61.
In equation (32), for laminar flow, n = 5 , f * = 0.8, the value of f * = 0.815 in the transition zone is, therefore not acceptable
f e*' 0.744 . . . . fl 0.460
F r o m equation (47)
0.774 0.480
(53)
In equation (32), for turbulent flow, n = 2 . 5 , f * = 0.6, the value o f f * ' = 0 . 7 4 4 in the transition zone is therefore acceptable
Pp3 = 3950 kg m -3
f~ --= f3
= ( 2 1 "]0.2, \4-~.5,] = 0.815.
f*' =
p = 1000 kg m - 3
_ ( 2 1 "]0.22 f ~ -- \6--~.2 ] = 0.774
f*
f2
(57)
f3
It will be noted from equations (55), (45) and (49) that each calculated optimum porosity of the expanded media, in the transition zone lies between 0.600 and 0.800, that is 0.600 < f * < 0.800; the calculated optimal upflow rate of 21 m m s -1 is therefore reasonable. In predicting the expansion of dual and tri-media filter beds, the writer has demonstrated that the new model is applicable in the transition and laminar zones as shown in equations (57) and (63).
1083
Predicting optimal expansion of dual and tri-media beds Table 1. Pilot plant fluidization experimental data 1. Wash rate, Vw,: 10 mm s-' Theoretical expansion Tightly compacted media
Filter media Anthracite (10/12) Flintag (22/25) Alundum (30/36) Total expansion (m)
Practical expansion
Loosely compacted media
Porosity of expanded media f,
Initial depth, L (m)
Expanded height, L~ (m)
Porosity of expanded media fe
Initial depth, L (m)
Expanded height, L, (m)
Expanded height, L, (m)
0.602 0,642 0.628
0.42 0.26 0.20
0.57 0.37* 0.28* 1.22
0.602 0.642 0.628
0.45 0.31 0.21
0,58 0.37 0.27 1.22
0.56 0.34* 0.31" 1.21
Note: 0.8% discrepancy in total expansion. The theoretical total expansion, Let, using the value of 0.25 of the exponent x~ in the calculation of the porosity of the expanded anthracite, fea, is given by: LeT =
L~ + Le2+ Le3
= (0.605 + 0.37 + 0.28) m = 1.26 m .
The 3.3% discrepancy in the theoretical and practical total expansion is high compared to the 0.8% discrepancy obtained in Table I by equating the ratio of the expanded and initial porosities of the anthracite to the expanded and initial porosities of either the flintag (sand) or alundum as shown in equations (37) and (38). *There is intermixing at the interface of flintag/alundum since for mixing fluidization, f~ > 0.440, i.e. 0.485 > 0.440. The discrepancy is due to the sinking of the flintag into ahindum, The total theoretical expansion is, however, equal to the total practical expansion.
C o m p u t a t i o n o f t h e t o t a l e x p a n d e d h e i g h t using either t h e initial p o r o s i t i e s o f t h e lightly c o m p a c t e d m e d i a o r t h e p o r o s i t i e s o f t h e loosely c o m p a c t e d m e d i a a n d t h e c o r r e s p o n d i n g d e p t h s o f t h e multilayer filter b e d s r e s u l t e d in a c c u r a t e p r e d i c t i o n o f the e x p a n s i o n o f t h e filter m e d i a as s h o w n in T a b l e s 1 a n d 2. APPLICATION OF THEORY
Pilot filter bed details Detail of ideal tri-media filter bed Pilot filter column 2.10 m x 150 mm o.d.; i,d.: 140 mm Top layer: 0.42 m deep, 10-12 anthracite ] Added by weight Middle layer: 0.26m deep, 22-25 flintag ~. and levels checked Bottom layer: 0.20m deep, 30-36 alundum .J after consolidation Base: 0.15 m deep, 7-10.5 nun glass marbles Total depth of tightly compacted media:
0.88 m
Total depth of loosely compacted media:
0.97 m
MODEL FOR COMPUTATION O F THE OPTIMUM WASH RATE
Design problem U s i n g the ideal t r i - m e d i a filter b e d d e s i g n p a r a m e t e r s listed a b o v e , calculate the flow rate for optimum backwashing of graded tri-media ( a n t h r a c i t e / f l i n t a g / a l u n d u m ) filter bed. T h e b a c k w a s h w a t e r is at a t e m p e r a t u r e o f 20°C a n d the t r i - m e d i a filter b e d is fluidized in a filter c o l u m n o f i.d. 1 4 0 m m . T h e d y n a m i c viscosity a n d d e n s i t y o f w a t e r at 20°C is 1 x 10- 3 kg m - ~s - ~ a n d 1000 kg m - 3 respectively.
Solution First, calculate the flintag (sand) d i m e n s i o n l e s s velocity t e r m , CD/Re in the t r a n s i t i o n a l r e g i m e by u s i n g e q u a t i o n (58)
Top surface 0.00 m above first sampling point Diameters of materials: anthracite: 1.24 mm fiintag: 0.65 mm ahindum: 0.46 mm Densities of materials: anthracite: 1400 kg m-3 flintag: 2650 kg m -3 ahindum: 3950 kg m - 3
Ct> = 4 g (P2 - Pf)# Re 3 3p f2 Vs2
Porosities of tri-media filter bed: (I) Porosity of anthracite: granular fixed bed flintag: (2) Porosity of media ahindum: in water Mass of materials: anthracite: flintag: ahindum:
C----~D----5.1 X 10 -2. Re
Settling velocities of materials: (1) Settling velocity in anthracite: the transition zone flintag: (2) Settling velocity in alundum: the laminar zone Exponent x values of the ratio of wash anthracite: flintag: alundum:
(1) 0.460 0.490
0.480
(2) 0.490 0.575 0.520
4.950 kg 5.410 kg 6.180 kg (1) 47.5 mm s-1 60.2 mm s - ~ 67.2 mm s - ~
(2) 65 mm s- ] 75 m m s 83 m m s - '
rate to settling velocity: 0.25 0.22 0.22
(58)
4 × 9 . 8 1 ( 2 6 5 0 - - 1000) x I x 10 -3 3 × 106 × (0.075) 3
U s i n g t h e empirical r e l a t i o n b e t w e e n C D / R e a n d R e (i.e. C a m p curves as s h o w n in Fig. 2), at CD/Re=5.1x 10 -2 , R e = 3 3 . S e c o n d , calculate t h e e q u i v a l e n t (hydraulic) dia m e t e r , dh2, o f flintag ( s a n d ) f r o m R e y n o l d s N o . o b t a i n e d in Step 1 b y u s i n g e q u a t i o n (59)
dh2 = ~'R---L p/V~2
(59)
# R e = 1 x 10 -3 x 33 -- 0 . 4 4 m m .
dh2 = p/Vs2
1000 x 0.075
B. A. QUAVE
1084
Table 2. Pilot plant fluidization experimental data 2. Wash rate, Vw2:20mm s-~ Theoretical expansion
Porosity of expanded media f, 0.700 0.747 0.732
Initial depth, L (m) 0.42 0.26 0.20
Practical expansion
Loosely compacted media
Tighly compacted media Expanded height, L, (m) 0.76 0.52 0.39 1.67
Porosity of expanded media f, 0.700 0.747 0.732
Initial depth, L (m) 0.45 0.31 0.21
Expanded height, L, (m) 0.77 0.52 0.38 1.67
Expanded height, L, (m) 0.78 0.53 0.37 1.68
Filter media Anthracite (10/12) Flintag (22/25) Alundum (30/36) Total expansion (m) Note: 0.5% discrepancy in total expansion. The theoretical total expansion, LeT, using the value of 0.25 of the exponent xt in the calculation of the porosity of the expanded anthracite. fel, is given by: LeT = Lel + Le2 + Le3 = ( 0 . 8 9 5 + 0.521 + 0 . 3 8 ) m = 1.80 m.
The 7.2% discrepancy in the theoretical and practical total expansion is high compared to 0.5% discrepancy obtained in Table 2 by equating the ratio of the expanded and initial porosities of the anthracite to the expanded and initial porosities of either the flintag (sand) or alundum as shown in equations (37) and (38). It is apparent from Table 2 that in the region of the optimal upflow wash rate (21 nun s -I) the theoretical expansion is equal to the practical expansion of each medium. The new model is therefore verified experimentally.
Third, calculate the a l u n d u m dimensionless velocity term, Co/Re in the transitional regime by using e q u a t i o n (60)
Co Re
4g(p3--'pf)# 3 3pf2 Vs3
F o u r t h , calculate the equivalent (hydraulic) diameter, dh3, o f a l u n d u m from Reynolds No. o b t a i n e d in Step 3 using e q u a t i o n (61) #Re
(60)
p:v,3
4 x 9 . 8 1 ( 3 9 5 0 - 1000) x l x 10 -3 dh3 ~
3 x 106 x (0.083) 3
Co
(61)
dh3 ~ - -
#Re
1 ×
pfVs3
1000 x 0.083
10 - 3 × 2 8
dh3 = 0.34 m m .
6.75 × 10 -3.
Re U s i n g the empirical relation between Co/Re a n d Re (i.e. C a m p curves), at CD/Re = 6.75 x 10 -2, Re = 28.
Fifth, calculate the geometric m e a n hydraulic diameter, dh=, by using e q u a t i o n (9)
dha
=
(dh2 X dh3) 1/2
dha = (0.44 x 0.34) 1/2 = 0.39 m m .
Comp curves 10 6
10
Sixth, calculate the geometric m e a n porosity, f , , by using e q u a t i o n (10) f~ = (f2 x A ) '/2
10 5
fa = (0.490 x 0.480) 1/2 = 0.485. Seventh, calculate the effective density, p,, o f the solid grains (flintag/alundum) by using e q u a t i o n (8)
lo"
10-1
w2 Ps P3 =
10 - 2
10 3
10 - 3
10 2
= 3950 I 10
L 10 2
W2P3
~--~ o~ +1
55.410410 6.18-----9+ 1 × 3950 L6.180 × 2650 + 1
103
Re
Fig. 2. Camp curves.
Ps = 3200 kg m -3 .
Predicting optimal expansion of dual and tri-media beds Eighth, calculate the density of mixture, p,,, by using equations (7) and (62) p,, = (I -- fa)P,+
f,,Pf
(62)
in the laminar zone lies between 0.600 and 0.800, that is, 0.600 < f * < 0.800; the calculated optimal upflow wash rate of 21 mm s -~ is therefore reasonable.
in which p, = effective density of the solid grains, f~ = f~ = geometric mean porosity of the granular bed at the incipient fluidization and p: = density of water p~, = (1 - 0.485)3200 + (0.485 x 1000) = 2133 kg m -3. Ninth, calculate the modified diameter term, Co Re 2, for flintag/alundum in the transitional regime by using equation (17) Co Re 2 = 4gp/(p~ - - p,,)dL 3# 2 4 x 9.81 x 1000(3200-2133) x (0.39) 3 x 10 -9 3 x l x l 0 -6 CoRe 2 = 8.28 x 102. Using the empirical relation between CoRe 2 and Re (i.e. Camp curves) at C o R e 2 = 8 . 2 8 x 102, Re = 17. Tenth, calculate the optimum wash rate, V~*, from the Reynolds No. obtained in Step 9 by using equation (18) #faRe
V*=--
p:d~
I x 10 -3 x 0.485 x 17 1000 x 0.39 x 10 -3
V* = 2 1 m m s -1. Eleventh, calculate the optimum porosity of the expanded alundum, f,'~, by using equation (19) f* =
~
J
=
0.739.
Twelfth, calculate the optimum porosity of the expanded flintag, f ~ , by using equation (19)
rL V- sT2 j = r=,T L 3j
=
0.756.
PREDICTING EXPANSION OF DUAL OR TRI-MEDIA FILTERS
Alundum : exponent x 3 value Using equations (1) and (35) and assuming that the exponent x3 is 0.22, the porosity of the expanded alundum, fe3, and the expanded height, Le3, can be calculated. Since the theoretical expanded height is equal to the practical expanded height, as shown in Table 2, the assumed value of x 3 for the prediction of the expansion of alundum is justified. Flintag (sand): exponent x2 value For a tri-media filter, the value of 0.22 of the exponent, x2, determined experimentally in an earlier investigation Quaye (1976) as shown in Fig. 3 can be used in the calculation of the porosity of the expanded flintag, f,2, and the expanded height, Le2. Anthracite: exponent x~ value The value of 0.25 of the exponent Xl determined experimentally, as shown in Fig. 3, resulted in the calculation of a high value of the expanded height, L,~. It is therefore recommended that the predicted porosity of expanded anthracite must be obtained from either equation (37) or (38) and used in the computation of the expanded height, L~, of the anthracite. Predicting the expansion of tri-media filter bed Using equations (1), (35) and (37) or (38) and the exponents, x3 = 0.22, x2 =0.22 and the predicted porosity of expanded anthracite, f ~ , the total expanGraph of log
f*
Y,
A
(Yw/~)
vs
log[1-(L/Le)(1-f)]
Log ( v . / ~ -1.4 I
-1.2 I
-1.0 1
-0.8 I
)
-0.6 I
-0.4 I
-0.2 I ///~/
Thirteenth, calculate the predicted optimum porosity of the expanded anthracite, f ; , by using either equation (37) or (38) f;*
1085
//_o.,o
x/~/
-o.1~ "
nd (18/301
t$
W.R, 21/9--F
- 0 . 2 0 ~1
T
-0.25
f~l = 0.710.
f~__=__ = _f'e3 _ 1.54. (63) f, f2 A It will be noted in Steps 11, 12 and 13 that each calculated optimum porosity of the expanded media
-0.05
Anthracite ( 7 / 1 4 ) ~ / x " /
f ; * = f ~ _ _ = x f ~ 0.460 x 0.756 f2 0.490 Using equation (36) we get
0
- -0.30
o - -0.35 0 . 3 5 - 0.10
Slope of anthracite, x 1 • ~
,
0.2.5
Slope of sond,x 2 • (~55-0.10 • 0.22 1.32- 0.20
Fig. 3. Exponent graphs of antracite and sand.
1086
B.A. QUA'rE
sion of a tri-media filter was computed when the wash rate, Vw~= 10mm s -1. Each theoretical total expansion of a tightly compacted media and a loosely compacted media was computed and compared with the practical total expansion. The tabulated experimental results are shown in Table 1. The discrepancy in the computation of the total expansion is 0.8%. Using a wash rate, Vw2= 20 mm s -1, the experimental investigation was repeated and the experimental results tabulated as shown in Table 2. The discrepancy in the computation of the total expansion is 0.5%. Predicting the expansion of a dual-media filter bed In an earlier investigation Quaye (1976), the writer predicted the expansion of a commercial dual-media filter [anthracite (7/14)-flintag (18/30) supported by coarse sand] during backwash, by considering the mass density of mixture of flintag/coarse sand (14/16). The coarse sand (14/16) is supported by coarse sand (10/14). The effective size of anthracite and flintag (sand) is 1.24 and 0.65 mm respectively, and the uniformity coefficient is 1.56 and 1.25 respectively. Using the model for computation of the optimum wash rate and the optimal expansion of tri-media filter, the writer predicted the exact expansion of a dual-media filter for the wash rate, Vw2= 20 mm s- 1, as shown in the tabulated experimental results. Total expansion
Anthracite (7/14) Flintag (18/30) Coarse sand (14/16) Coarse sand (10/14) Total
Theoretical (m) 0.57 0.94* 0.16" 0.05 1.72
Practical (m) 0.65 0.87* 0.15" 0.05 1.72
*There is intermixing at the interface of flintag/coarse sand (14/16) since at the point of incipient fluidization fc = fa and fa > 0.422, the porosity of the coarse sand (14/16), i.e. 0.455 > 0.422. The discrepancy is due to the sinking of the flintag into coarse sand (14/16). It is apparent from this investigation that the new model for predicting the optimal expansion of a dual or tri-media filter during backwash is a useful tool for designing the height of filter weirs in order to prevent the loss of filter media. CONCLUSIONS A study of the works of other researchers involved in the prediction of expansion of granular media during fluidization caused by backwashing processes confirms that in all cases, the expressions derived for the upflow wash rates consist of empirical constants which must be determined in order to evaluate the formulae for upflow wash rates. In the present model, the only constant required to be determined experimentally is the exponent x of the ratio of the upflow
wash rate and the settling velocity of sand. By considering the physical parameters such as the effective sizes, porosities and the spbericities of dual and tri-media filters the optimal upflow wash rates for various water temperatures were determined in an earlier work Quaye (1976) and the expansion of the filter beds predicted. Whilst the United Kingdom system of backwashing filter beds uses an upward flush of air first (air scour) followed by an upward water flush at velocities barely sufficient, and sometimes insufficient, to fluidize the bed; the United States system uses an upflow wash of sufficient velocity to fluidize the bed fully and provide substantial bed expansion. Both methods have resulted in dirt problems such as mud balls and filter cracks Cleasby (1976), Cleasby et al. (1975, 1977). It is now, generally accepted that the upflow wash rate should reach incipient fluidization as confirmed by the writer. From the theoretical and experimental study of the optimum backwash of sand and multilayer filters, the following conclusions can be drawn: (1) It has been calculated that the maximum shear in the fluid for optimum cleaning of anthracite-flintag-alundum media occurs at the optimum expanded porosity of 0.750, 0.780 and 0.780 respectively. These may be used for predicting the optimal expansion of tri-media filter beds. (2) The optimum porosity of the expanded medium in the transitional regime lies between 0.600 and 0.800, that is 0.600 < f * < 0.800. (3) In the lowest two layers of a multi-layer filter, the optimal upflow wash rate at critical fluidization is the lowest wash rate at which the upward buoyant force on the solid grains in a fluidized bed and the downward net impelling force are equal at equilibrium and the porosity at the critical fluidization is equal to the geometric mean porosity of the granular fixed bed. (4) Backwashing of deep granular filter beds with an upflow wash rate less than the optimum, resulted in the accumulation of suspended solids in the filter pores after a period of filter operation. This is an inefficient process of backwashing a filter with water only. Comparisons made by other research workers based upon this inefficient backwashing process with air and water backwashing are therefore not justifiable. For an efficient backwashing of the filter bed, the optimum expanded porosity should be used in the computation of the filter bed expansion. (5) A new model for predicting the optimal expansion of dual or tri-media during backwash is a useful tool for designing the height of filter weirs in order to prevent the loss of filter media. Acknowledgements--The writer wishes to express his sincere appreciation to Professor K. J. Ives for his guidance and encouragement during the period of investigation at the Kempton Park Experimental Treatment Works and in the Chadwick Public Health Laboratory, University College,
Predicting optimal expansion of dual and tri-media beds London. Financial support provided by the Thames Water Authority (formerly Metropolitan Water Board) is gratefully acknowledged.
REFERENCES
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