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The interface between filtration and backwashing
Water Res. Vol. 19. No. 5, pp. 581-588, 1985 Printed in Great Britain. All rights rescued
0093-135.1 85 53.00+0.00 Copyright ~ 1985 Pergamon Press Ltd
THE INTERFACE BETWEEN FILTRATION AND BACKWASHING A . AMIRTHARAJAH
Professor of Civil Engineering and Engineering Mechanics, Montana State University. Bozeman, MT 59717, U.S.A.
(Received January 1984) Abstract--A conceptual model of the initial degradation phase of filtration is presented as an interface effect between backwashing and filtration. It is shown that the initial degradation of effluent quality is due to the backwash water remnants within the media and the backwash water abeve the media. The two peak characteristics of initial degradation due to the backwash water remnants within the media and above the media is established by an extensive experimental investigation. A mathematical model for the quality of backwash water as a function of backwash water volume is developed. Deductions made from the mathematical expression confirm the validity of some accepted facts on backwashing and also lay the basis for the peaks in initial degradation.
Key words--water treatment, filtration, backwashing, effluent quality, initial degradation, backwash water, modeling
INTRODUCTION
NOMENCLATURE
Ai2 = A~ = A(t) = a, b = c= =
Hamaker constant, J initial area (m-') surface renewal area at time t (m-" s -~) coet~cients mass concentration (kg m -j) Laplace transform with respect to time (kg m -3 s)
d = 2rn = diameter of collectors (m) D = particle diffusion coefficient (m-'s -1) Do = diameter of detached particle (m) k = ~4A° = coefficient (kg m- 3) V m --mass transfer rate (kg m - ' s -~) = time averaged mass transfer rate (kg m-'-s -~) M = total mass transferred from grain (kg) • /' -- ~ dt ~ total mass transfer per unit area (kg m--') r = radial distance (m) r 0 = radius of collector (m) R0 = radius of spherical shell (m) s = Laplace transform variable = renewal frequency
(s -~) v
Sc = ~ = Sehmidt number t = time (s) 7"8 = cumulative time of flow up to backwash water gutter (s) T.w = cumulative time of flow within and up to top of media (s) TR = cumulative filter ripening time (s) Tt = cumulative time of flow for lag period (s) v., = superficial velocity (m s -~) U * = friction velocity (m s -~) V = volume (m 3) Vi = interstitial velocity (m s -z) Z = clear distance between particle and collector (m)
Greek. letters :~ = Frossling number = porosity p i = fluid density (kg m -3) v = kinematic viscosity (m-"s-J).
The initial quality of effluent from a granular media filter that has been used for several runs is usually poor. It is one phase of a p h e n o m e n o n frequently referred to in quasi technical terms as "'filter ripening". A m i r t h a r a j a h a n d Wetstein (1980) have recently s h o w n that "filter ripening" always occurs at the initial stages of a filter run on old media, and that it is caused by a n interaction between the backwash stage a n d the initial stages o f filtration during which a degradative phase is followed by an improving phase. T h e d e g r a d a t i o n phase is characterized by two peaks in effluent quality. T h e two peaks are associated with the backwash water r e m n a n t s within the media a n d above the media. Thus modeling initial d e g r a d a t i o n becomes intrinsically intertwined with being able to model the quality o f the backwash water remnants. This p a p e r is in two parts: (1) The first reviews the basic causes for the poor quality effluent at the initial stages o f a filter run and confirms the initial two peak characteristics with experimental data. (2) The second part o f the p a p e r focuses o n the quality characteristics o f the b a c k w a s h water r e m n a n t s a n d presents a m a t h e m a t i c a l model to predict the variation of the quality o f backwash water. This model o f backwash water quality is a first a t t e m p t at providing a basis for modeling the characteristics o f the degradative phase d u r i n g filter ripening. CONCEPTUAL MODEL OF INITIAL DEGRADATION The following conceptualization of initial degradation shows that the p o o r quality of effluent during the initial stage o f a filter r u n is due to the b a c k w a s h water remaining in the filter system after
581
582
A. AMIRTHARAJAH
"FILTER RIPENING" _
,
T,.
'~..~.
OUTLET
~LEANFUNCTION OF FUNCTIQN OF ..'~ -'---'--~-EtACK W AS~ I INFLUENT lAiRI "&SH REMNRNTS I L RISING LIME I t / f
'
;
•
I [ i FILTER I ;SNEAKTNROUG~ I
RECEDING LIMB
/ ru
r,
rs
TR TIME
Fig. 1. Characteristics of initial effluent quality. backwash. At the end of the backwashing operation of a repeatedly used filter there would be remnants of the backwash water in the filter system. The backwash water remnants in the system can be subdivided into three types: (1) clean backwash water in the underdrain and connecting pipework from the backwash water supply system up to the bottom of the filter media, (2) backwash water remnants within the pores of the media and (3) backwash water remaining above the filter media up to the level of the wash water gutter. These three fractions of backwash water are shown in Fig. 1. The three fractions of backwash water remnants will have different characteristics. The first fraction up to the bottom of the media has the character of the clean backwash water. The remnant within the pores of the media would have a backwash water quality characterized by the last stages of the backwashing operation and the fact that the water remains within the pores of the media when the ffuidized bed during backwash is collapsed into a fixed bed at the end of backwashing. The collapse of the fluidized bed does not typically dislodge a significant number of particles from the surfaces of the media into the backwash water remnants within the pores, providing the backwashing operation has been effective. The third fraction of the backwash water above the media would have a quality which is poorer than the second backwash remnant since it preceded this backwash water during backwashing and hence would be removing more particles from the media. However, the remnant above the media is not subjected to the mechanism of collisions between media particles upon the collapse of the fluidized bed. Thus this fraction will exhibit characteristics unlike those of the other two backwash water remnants.
The peaks in initial degradation The early stages of filter ripening would be characterized by an initial period of relatively good water called the "lag" period in Fig. 1 and caused by the first fraction of remnant. The cumulative theoretical flow through times corresponding to each of the backwash water remnants are designated in Fig. 1.
The three fractions are noted as T~, (time up to underdrains), TM (time within and up to top of media) and 7"8 (time up to backwash water gutter). The total filter ripening period is TR. The "lag" period corresponds to TU and represents clean backwash water. It is followed by a "rising limb". During this period the effluent quality rapidly degrades up to the first peak at time T~, The first peak is due to the backwater remnant within the media and occurs at time T~t. Often the first peak is undetected since it occurs within 1 or 2 min of the start of a run at typical filtration rates of 2.7--4.0 mm s -~. Since the first peak is due to collisions at the end of the backwashing operation, such factors as the increased effectiveness and longer duration of backwash, the slow closure of the backwash valve and the increased strength of the adhesive forces between the filtered particles and the media may obscure the two independent peaks and a single plateau type response may be evident in the initial effluent quality. The backwash water remnant above the media (i.e. corresponding to time T B - TM) has particles which were removed from the filter grains during backwashing. The concentration of particles will be highest at level TB since it was at an earlier instant during backwashing, and lower at level TM. The author develops a new theory supported with experimental evidence in the second part of this paper that the concentration of particles in this backwash water remnant increases exponentially with height above the media. Thus the quality degrades between TM and TB reaching the second peak close to Ta. Oftentimes the second peak occurs at a time slightly ahead of the theoretical time Ta. This is due to mixing of the backwash remnant and the influent water. The second peak commonly occurs at times between 5-10rain at typical filtration rates. Experimental evidence substantiating the existence of the two peaks is presented in later sections of this paper. After the effluent quality has reached the second peak the quality of the product slowly improves.
The improving phase of filter ripening The improving phase of "filter ripening" termed
The interface between filtration and backwashing 0.4
RUNS 17 TO 19. A AND B
~)
r
0.5~-
FILTER 1. 2. A N n 3
RUNS
95,
W 15 ,
t
W 1E *
CLEAR WATER
0.4
MEAN AND
583
C.,.
/
~0.3 = i ~-
W 22
AFTE R T -
0.2
.
F,LTER A
t
CLEAR WATER
i ~0.2 .J
_1
0
v
T"
Tu
T:,
7, 1
2
,
,
Ts
T'
,
,
3
4
5
6
7
:
!
8
9
-
TIME - rain
~0.1 w
Fig. 4, Initial effluent quality with influent water variations. Filtration rate = 3.4 mm s-*, TuT,J
[I
TI
,
[,
5
, 15
10 TIME - MIN
Fig. 2. Initial effluent quality for uniform sand. Filtration rate = 2.4 mm s-L
the "'receding limb" in Fig. 1 is a function of the influent water characteristics. This long recessional limb has varying degrees of duration depending on filtration rate, influent concentration, particle size and the physicochemical character of the influent particles. This phase of filter ripening has been studied by O'Melia and All (1978) who associated this phase with the formation of particle chains or dendrites on the media during the initial stages of filtration shown to occur by Tien et al. (1977). Recently, Payatakes et al. (1981) have shown with direct visual experimental data, that the main mechanism causing alteration of the geometry of flow channels within filters was throat clogging. For ag-
RUNS 7 TO 12, A AND B FILTERS 1 TO 3
O.E
~ i =
~ ..R.NO.,- C,
0.(
i
~
0.4
0.2
M TI
°Tu I,
,
,
~o
'
,'o
TIME - mlft
Fig. 3. Initial effluent quality for uniform sand. Filtration rate = 4.7 mm s-L
glomerated aquasols this throat clogging phenomenon resulted in an increase of local capture efficiency and explained the initial increase of deep bed filtration efficiency which is shown as the improving phase in Fig. 1. The above summarizes the conceptual characteristics of initial degradation and improvement. EXPERIMENTAL VALIDATION CONCEPTUAL MODEL
OF
The above conceptual model has been validated with extensive experimental results by Amirtharajah and Wetstein (1980). The results are from two independent studies by Amirtharajah (1978) and Wetstein (1978), and details of the studies are reported in these references. All of the experiments consisted of a standard dirtying run, a backwash and a subsequent variable filtration run during which samples were taken at 1 rain intervals to monitor the quality of the initial effluent. The two studies were conducted on pilot plant filters and the influent suspension filtered was prepared by adding stock solutions of ferrous sulfate (FeSO4 • 7H20) or ferric chloride (FeCI~ • 6H20) to tap water. The iron in the effluent was measured spectrophotometrically. A sample of the results from the two studies are shown in Figs 2, 3 and 4. Figures 2 and 3 show the initial degradation from a uniform sand filter at filtration rates of 2.4 and 4.7 mm s -~. The results in Fig. 2 show the mean and 95% confidence intervals of 378 data points. At each minute the mean and error bar represents 18 data points. Figure 3 shows similar experimental data at a filtration rate of 4.7 mm s -*. On the abscissae of Figs 2 and 3 are shown the theoretically calculated flow through times Tu, Tu and TB. The figures clearly indicate the existence of the two peaks and the slow recessional improvement. It is also seen that the peaks correspond to the backwash remnants within the media (at time Tw) and that above the media (at time TB). Figure 4 shows the effect of influent water variations on initial effluent quality. It dramatically illustrates the correspondence between the backwash
584
A. AMIR'H-IARAJAH
FILTER RUNS. R 7.1 - R 7.6 (NFLUENT IRON. 14.5 rag. ) AVERAGE BACKWASH REMNANT IRON. 1.35 mg/I ~0.2C
e~" . . . . . . . . . . . .
-i,ooo
'-~"TRACE SOLUTION ABOVE MEDIA ONLY ONLY
E i
4 5000
30DD !
~TRACE
~ 0.1C
SOLUTION WITHIN VE MEDIA
-i~DDO r. T QUALITY
+ODO 0
u IT,,, 10
5
15
0
TIME - rain
Fig. 5. Initial effluent quality and dispersion characteristics of filter. Filtration rate = 4.7 mm s -~.
remnants and the initial degradation peaks. Run W16 with coagulated water is similar to the data shown in Figs 2 and 3, and confirms the two peak characteristics at times TM and Ta, the first peak being rather flat and the second occurring slightly before Ta. When clear tap water was filtered after T~t by removing the backwash water remnant above the media in Run W22, it is clearly seen that the second peak does not occur. When clear tap water was filtered in the subsequent run as in WI5 (after time 7"8) it is seen that the second peak is considerably suppressed even though it does exist like a plateau. The association of the backwash remnants with the peaks in the initial degradation shown in Figs 2, 3 and 4 are consistent with theoretically calculated flow through times T~,, TM and TB. In order to show this correspondence with direct experimental measurements and to evaluate the effect of dispersion of the remnants due to molecular and eddy diffusion, a series of 24 filter runs with six replications at each rate were completed at Montana State University. The runs noted as R 3.1-R 3.6 to R 9 . I - R 9.2 were made at filtration rates of 2.1, 3.4,
4.7 and 6.0ram s -t. Using the identical filter pilot plant and at the same filtration rates sodium salt tracer studies were made by replacing (a) the water above the media with tracer solution (for time T a - r~f), (b) the water within the media with tracer solution (for time T,~t- Tu) and (c) the water within and above the media with tracer solution (for time T8 - T6,). Typical results of initial degradation and sodium salt tracer studies are shown in Figs 5 and 6. The results confirm the two peak characteristics of initial degradation, and more importantly show the correspondence of the backwash water remnant volumes with the initial degradation. The sodium salt tracer studies clearly indicate the flow through times TM and Ta and prove that dispersion effects of the backwash water remnants are negligible. The predominant peak in initial degradation is due to the backwash remnant above the media. An attempt at explaining why this peak occurs necessitates a detailed analysis of the quality of backwash water. The following section presents a new theoretical model with experimental support for predicting the quality of backwash water.
FILTER RUNS. R 5.1 * R 5.6 INFLUENT IRON. 15.0 rag, I AVERAGE BACKWASH REMNANT IRON. 1,03 mg/I
so,o.o w..
.ooo ,
AND ABOVE MEDIA
~-
0.1C
~_
D.O..
o T-
O.D
I . . . . . . 5
I.
:
10
1~5
D
20
TIME - rain
Fig. 6. Initial effluent quality and dispersion characteristics o f filter. F i l t r a t i o n rate = 3.4 m m s -~.
The interface between filtration and backwashing
585
A THEORY FOR BACKWASHWATERQUALITY
c(r. O) = Co
(3)
Amirtharajah and Giourgas (1981) have recently presented a theory for particle detachment from sand grains during backwashing of filters with water alone by fuidization, typical of U.S. practice. Amirtharajah (1984) has recently dealt with theoretical aspects of air scour backwashing in which particle collisions and abrasion play a much greater role. The following theory assumes that particle detachment from the collectors in a water fluidized bed occurs due to the
c(ro, t) = c'o
(4)
~c(Ro, t)
= o
~r
(5)
in which Ro = radius of spherical shell. Equation (2) can be solved for the above boundary conditions by taking the Laplace transform with respect to time ?. Nelson and Galloway's (1975) solution to equation (2) is,
ro(Co-Co)N/~c°sh[(Ro-r)~/~]--~osinh[(Ro-r)X/~D
]
CO
+ --
"0 =
(6)
S
rs / ~ c ° s h [ ( R o - r o ) 3 ] - ~ o o S i n h [ ( R o - r o ) 3 ] hydrodynamic forces exerted by turbulent bursts. The turbulent bursts cross the viscous sub layer and exert lift forces which detach particles attached to the media. The detachment of particles is resisted by the London-Van der Waals adhesive forces. Amirtharajah and Giourgas (1981) considered the dynamics of a single particle cluster as it is removed from a collector. From the momentum equation, Rate of change of momentum = Lift Force - Adhesive Force '\dt
J
(oo .y 12Z2 :o0
0.076p/v: - -
in which s = Laplace transform variable. The radius of the spherical shell Ro is so chosen that the porosity of the sphere and its surroundings are equal to the porosity of the system. r0
R° = (1
-
,3
e)
(7)
in which e = porosity. Following Danckwert's (1951) analysis, the time averaged mass transfer rate is assumed as an exponential distribution of surface ages given by,
= - sD
(1)
(8) rzt 0
in which V,= interstitial velocity, dM/dt = t o t a l mass transfer rate from grain, p/=fluid density, v = kinematic viscosity, Do = diameter of detached particle, U* = friction velocity, A~ = Hamaker constant and Z = clear distance between particle and collector. The rate of change of momentum V~(dM/dt) was calculated by using Nelson and Galloway's (1975) mass transfer theory for fluidized beds. This model was formulated by considering mass transfer through particle diffusion across a film around a collector. The mechanism of mass transfer by diffusion from the surface was formulated in terms of Danckwerts' (1951) surface renewal theory. Consider the flux of particle clusters leaving the surface of the media grains as shown in Fig. 7. Due to diffusion the particle clusters would cause a concentration gradient ~c/Or from the surface of the media grain to the surface of the spherical shell, where it is assumed to be zero. The continuity equation in radial coordinates
in which ~ = time averaged mass transfer rate per unit area per unit time. Nelson and Galloway (1975) interpreted the Laplace transform variable s as proportional to the surface renewal frequency and assumed,
s = ~2 Vi
1 2r0 Sc t/3
(9)
in which Sc = v/D = Schmidt number, ~t was the only free parameter in Nelson and Galloway's model and by its judicious selection the model transformed exactly into accepted theories for single spheres at infinite dilution (~ = 1). Using equations (6), (7) and (9) it is possible to solve equation (8) to obtain the time averaged mass transfer fux, ffz.
/
s --~--~"~ 5"o'~-~ ~ °°°°ooooo~
/
CONCENTRATION GRADIENT
\
*o
//
Clr,o)=C,
gives, l[" R~
=z)pN\
aU
(2)
in which c =concentration, D = particle diffusion coefficient and r = radial distance. The boundary conditions as indicated in Fig. 7 are
\\
r ~
\
SPHERICAL_~..//¢%'. SHELL ~-
Clro, ' o=r C )
, / ~"
./~'~...
~ C(R.,t) =0
Fig. 7. Single collector mass transfer model.
586
A. AMIRTHARAJAH
Amirtharajah and Giourgas (1981) estimated the time averaged mass transfer rate as 8.7 x 10 -5 mg m m - : s -t. Assuming an average size for particle clusters detached from the collectors as 7,am, the rate of change of momentum of a particle cluster was calculated to be 8.1 x 10-3pN (pico Newtons). In contrast to this small value of the momentum change the lift force due to turbulent bursts was 4.5 x 10-:pN. It was also shown that the lift force is greater than the adhesive London-Van der Waals" forces only up to a distance of several nanometers. Thus a coating layer will remain on the collectors after backwashing. Thus total removal of
attached particles from the collectors is impossible by fluidi:ation during backwashing. This central result has been macroscopically confirmed with extensive experimental data by Amirtharajah (1978).
CONCETRAT,OH GRADIENT
-(--_z~
C = * Ke '
MASS TRANSFER FROM //SINGLE COLLECTORINTO VOLUME V AT RATE
BACKWASH REMNANT ABOVE MEDIA
~V~'~r
BACKWASHED.- - ~ " FILTER MEDIA
J
SURFACE RENEWAL MECHANISM
'
JJ
MOVEMENT OF PARTICLE /REMOVAL FRONT DURING SACKWASHING - CAUSES EXPONENTIAL DECLINE IN
"--
r t~, SUPERFICIAL - " " ~ i i
VELOCI,*"" ,i
SURFACE
RENEWAL AREA
~).....< AN ARSITRARY VOLUME V OF LIQUID
Fig. 8. Schematic diagram for backwash water quality theory.
Backwash water quality model The mass transfer model presented above can be used to formulate a theory for predicting backwash water quality. Consider the schematic diagram of a filter being backwashed as in Fig. 8. As an arbitrary volume of liquid V traverses the fluidized bed it would accumulate particles detached from the media surfaces. The mass transfer into the volume V would be at a rate ASI-- t~ dt as shown in equation (8). The total mass transfer into the volume V will depend on the area A (t) across which the surface renewal mechanism will transport material. The area A (t) would be a function of time. By a mass balance,
At a certain finite but large time t the particle removal front would have moved completely through the bed and no more particles can be removed from the collectors. As t becomes large, e x p ( - s t ) - - , 0 and c---,0, therefore the integration constant = 0. Hence equation (13) becomes,
V dc = -ff4A(t). dt
While equation (9) could be used for renewal frequency, at this preliminary modeling stage it wilt be assumed that the renewal frequency is the ratio of the superficial velocity v, to the diameter of the collectors d = 2r0. Therefore, equation (14) transforms to,
(10)
The - r e sign exists because the rate of transfer into the volume declines with time. As particles are removed from the surfaces of the collectors, some of the collectors will reach the non erodible layer and will no longer supply particles into the volume V. Thus a front at which no further particle detachment takes place will move up the filter bed as shown in Fig. 8. Since the surface renewal mechanism controls particle transfer, it is rational to assume that the surface renewal area is an exponential function of renewal frequency s,
A(t)---. ADs exp(--st)
(11)
in which A0 = initial area. From equations (10) and (11)
v ~ = - ~AoS exp ( - st) dt
(14)
in which
c = k exp - (vxt/d).
(12)
f dc = ~°S f exp(-st) dt (13)
(15)
The expression will not be valid at extreme values of t, i.e. as t ~ 0 or ~ . The equation is the final result of the theory.
Deductions from the backwash water quality model Several important deductions can be made from the final result of the theory, equation (15). (1) Transform the equation by taking logs lnc = l n k - ( ~ )
Integrating,
c = [--~--2] exp ( - st) + constant.
c = k exp(-st)
\
(16) /
thus a plot of In c vs (v~t) would show a linear variation with a - v e slope. Experimental evidence to validate this result is presented in a following section. (2) The term (vxt) is superficial velocity x time. This implies that backwashing a filter to a given terminal backwash water quality is dependent only on the total washwater volume used as long as the particle removal mechanism remains unaltered. Thus
The interface between filtration and backwashing
587
500.0
"''-
,
.
I ~
2,0
,oo
~
1.5
10.0
~ 3 ~
1.0
AMiRTHARAJAH'S DaTA ~" DeFFEI~NTFILTERSWITH
100.0 ,
5.0
MEANAN0 STANOARO 01[VIATtON" ~.OTTED POINTS: 25 .
i
0.1
FILTRATIONRATE~
I
L
mmll
Fig. 11. Peak of initial degradation vs filtration rate.
1.0 0.50
o.,o,
~
~
;
Empirical evidence for backwash water quality model
,
WASHWATERVOLUME- m3/m2
Fig. 9. Backwash water quality as a function of backwash water volume.
backwashing at 12 mm s -~ for 6 min will produce the same cleanliness as 9 m m s -~ for 8min, since (v.J) = (12 x 6 x 60)/1000 or (9 x 8 x 60)/1000 which are both equal to 4.32 m s m--'. (3) After the removal front has passed through the bed and all collectors have reached their permanent coating layer, very few particles will be detached into the washwater. This would result in a constant but low concentration of particles in the washwater after a given total volume of washwater is used. A large body of empirical evidence exists validating all three of the above deductions. A sample of this evidence is presented in the next section.
5U
500
~ ~
HUDSON'S ('1981)
40
~
3o ~ 20
~ 0
100
J o o~
DATA
' ~
B
50
u~
10
=
a
-
b (c.j)
(17)
in which a and b are coefficients dependent on the units c and v,t. The data shown in Fig. 9 has a correlation coefficient of 0.93. It is seen that the equation is identical in form to equation (16). The data also shows that a constant backwash water quality is reached at a total backwash volume of approx. 6.0 m 3 m -2. Two examples of data from the literature are presented in Fig. 10. The data again confirms the general validity of equations (15) and (16) and the deductions that are derived from it. INTERACTION OF BACKWASH WATER QUALITY A N D INITIAL D E G R A D A T I O N
,=,,
z
log c
100
'1000
I
Results illustrating the model developed above are shown in Figs 9 and 10. Figure 9 presents data collected by Amirtharajah (1971) from 16 runs at expanded porosities ranging from 0.55 to 0.78. It is seen that the data plots as a straight line on the semi-logarithmic plot. A regression analysis of the data gave an empirical equation of the form.
I
i
I
i
I
1 2 3 4 5 6 TOTAL WASHWATERVOLUME-m3/mz
Fig. 10. Backwash water quality vs washwater volume-literature data.
The theory and experimental results presented above indicate that the concentration of particles in the backwash water remnant above the media varies as an exponential function with backwash water volume. As can be seen in Fig. 8, the highest concentration of detached particles would therefore be at the top of the backwash remnant above the media. As filtration occurs in the following run the second peak in initial degradation occurs close to the time that the backwash remnant volume above the media which corresponds to a time T s as shown in Fig. 1 moves through the filter. Filtration efficiency would be an inverse function of filtration rate at the initial stages of filtration, when all other parameters remain unchanged. This would imply that the magnitude of the second peak in initial degradation would be a direct function of filtration rate. Since the concentration of particles in the
588
A. AMIRTHARAJAH
backwash remnant varies exponentially, it could be expected that the magnitude of the second peak during initial degradation would be a power function of filtration rate. Preliminary results illustrating this variation are shown in Fig. I1. The power function shown in Fig. I 1 had a correlation coefficient of 0.91. The previous theoretical analysis and the results shown provide a reasonably satisfying model for the existence and characteristics of the initial degradation of effluent quality. The analysis applies directly to filters with high level collecting launders typical in the American continent, which are often backwashed with water alone. For other backwash systems or operations the analysis should be suitably modified. A CONCLUDING SUMMARY A conceptual model of the initial degradation phase of filtration is presented as an interface between backwashing and filtration. It is shown that the initial degradation of effluent quality is due to the backwash water remnants within the media and the backwash water above the media. The two peak characteristics of initial degradation due to the backwash water remnants within the media and above the media is established by an extensive experimental investigation. A mathematical model for the quality of backwash water as a function of backwash water volume is developed. Deductions made from the mathe.natical expression confirm the validity of some accepted facts on backwash and also lay the basis for the peaks in initial degradation. Acknowledgements--This paper was presented to the Symposium on Water Filtration organized by K V I V and the
Belgian chapter of the Filtration Society, in Antwerp, Belgium, April 1982 entitled "Modeling Initial Degradation of Effluent Quality. The paper incorporates experimental data collected by Daniel P. Wetstein and Mohamad R. Farvardin, former graduate research assistants at Montana State University, whose contribution is gratefully acknowledged. The study reported was partially supported by the Engineering Experi-
ment Station at Montana State University and the National Science Foundation under Grant No. ISP-80411449. REFERENCES
Amirtharajah A. (1971) Optimum expansion of sand filters during backwash. Dissertation presented to Iowa State University at Ames, Iowa in partial fulfilment of the requirements for the degree of Doctor of Philosophy. Amirtharajah A. (1978) Optimum backwashing of sand filters. J. em'ir. Engng Die. Am. Soc. cir. Engrs 104, 917-932. Amirtharajah A. (1984) Fundamentals and theory of air scour. J. enrir. Engng Am. Soc. cit'. Engrs 110, 573-590. Amirtharajah A. and Giourgas A. J. (1981) Theory for particle detachment from sand grains and during backwashing of filters. International Symposium on Applications of Fluid Mechanics and Heat Transfer to Energy and Environmental Problems. University of Patras, Patras, Greece. Amirtharajah A. and Wetstein D. P. (1980) Initial degradation ofeffiuent quality during filtration. J. Am. Wat. Wks Ass. 72, 518-524. Cleasby J. L. and Baumann E. R. (1977) Backwash of Granular Filters Used in Wastewater Filtration. EPA600/2-77 MERL, U,S. Environmental Protection Agency. Danckwerts P. V. (1951) Significance of liquid film coefficients in gas absorption. Ind Engng Chem. 43, 1460--1467. Hudson H. E. Jr (1981) Water Clarification Processes-Practical Design and Evaluation, p. 215. Van Nostrand Reinhold, New York. Nelson P. A. and Galloway T. R. (1975) Particle-to-fluid heat and mass transfer in dense systems of fine particles. Chem. Engng Sci. 30, 1-6. O'Melia C. R. and Ali W. (1978) The role of retained particles in deep bed filtration. Prog. Wat. Technol. 10, 167-182. Payatakes A. C., Park H. Y. and Petrie J. (1981) A visual study of particle deposition and reentrainment during depth filtration of hydrosols with a polyelectrolyte. Chem. Engng Sci. 36, 1319-1335. Tien C., Wang C. S. and Barot D. T. (1977) Chain-like formation of particle deposits in fluid-particle separation. Science 196, 983-985. Wetstein D. P. (1978) The initial quality of effluent from granular media water filters. Report presented to Montana State University in partial fulfillment of the requirements for the requirements for the degree of Master of Science.
0093-135.1 85 53.00+0.00 Copyright ~ 1985 Pergamon Press Ltd
THE INTERFACE BETWEEN FILTRATION AND BACKWASHING A . AMIRTHARAJAH
Professor of Civil Engineering and Engineering Mechanics, Montana State University. Bozeman, MT 59717, U.S.A.
(Received January 1984) Abstract--A conceptual model of the initial degradation phase of filtration is presented as an interface effect between backwashing and filtration. It is shown that the initial degradation of effluent quality is due to the backwash water remnants within the media and the backwash water abeve the media. The two peak characteristics of initial degradation due to the backwash water remnants within the media and above the media is established by an extensive experimental investigation. A mathematical model for the quality of backwash water as a function of backwash water volume is developed. Deductions made from the mathematical expression confirm the validity of some accepted facts on backwashing and also lay the basis for the peaks in initial degradation.
Key words--water treatment, filtration, backwashing, effluent quality, initial degradation, backwash water, modeling
INTRODUCTION
NOMENCLATURE
Ai2 = A~ = A(t) = a, b = c= =
Hamaker constant, J initial area (m-') surface renewal area at time t (m-" s -~) coet~cients mass concentration (kg m -j) Laplace transform with respect to time (kg m -3 s)
d = 2rn = diameter of collectors (m) D = particle diffusion coefficient (m-'s -1) Do = diameter of detached particle (m) k = ~4A° = coefficient (kg m- 3) V m --mass transfer rate (kg m - ' s -~) = time averaged mass transfer rate (kg m-'-s -~) M = total mass transferred from grain (kg) • /' -- ~ dt ~ total mass transfer per unit area (kg m--') r = radial distance (m) r 0 = radius of collector (m) R0 = radius of spherical shell (m) s = Laplace transform variable = renewal frequency
(s -~) v
Sc = ~ = Sehmidt number t = time (s) 7"8 = cumulative time of flow up to backwash water gutter (s) T.w = cumulative time of flow within and up to top of media (s) TR = cumulative filter ripening time (s) Tt = cumulative time of flow for lag period (s) v., = superficial velocity (m s -~) U * = friction velocity (m s -~) V = volume (m 3) Vi = interstitial velocity (m s -z) Z = clear distance between particle and collector (m)
Greek. letters :~ = Frossling number = porosity p i = fluid density (kg m -3) v = kinematic viscosity (m-"s-J).
The initial quality of effluent from a granular media filter that has been used for several runs is usually poor. It is one phase of a p h e n o m e n o n frequently referred to in quasi technical terms as "'filter ripening". A m i r t h a r a j a h a n d Wetstein (1980) have recently s h o w n that "filter ripening" always occurs at the initial stages of a filter run on old media, and that it is caused by a n interaction between the backwash stage a n d the initial stages o f filtration during which a degradative phase is followed by an improving phase. T h e d e g r a d a t i o n phase is characterized by two peaks in effluent quality. T h e two peaks are associated with the backwash water r e m n a n t s within the media a n d above the media. Thus modeling initial d e g r a d a t i o n becomes intrinsically intertwined with being able to model the quality o f the backwash water remnants. This p a p e r is in two parts: (1) The first reviews the basic causes for the poor quality effluent at the initial stages o f a filter run and confirms the initial two peak characteristics with experimental data. (2) The second part o f the p a p e r focuses o n the quality characteristics o f the b a c k w a s h water r e m n a n t s a n d presents a m a t h e m a t i c a l model to predict the variation of the quality o f backwash water. This model o f backwash water quality is a first a t t e m p t at providing a basis for modeling the characteristics o f the degradative phase d u r i n g filter ripening. CONCEPTUAL MODEL OF INITIAL DEGRADATION The following conceptualization of initial degradation shows that the p o o r quality of effluent during the initial stage o f a filter r u n is due to the b a c k w a s h water remaining in the filter system after
581
582
A. AMIRTHARAJAH
"FILTER RIPENING" _
,
T,.
'~..~.
OUTLET
~LEANFUNCTION OF FUNCTIQN OF ..'~ -'---'--~-EtACK W AS~ I INFLUENT lAiRI "&SH REMNRNTS I L RISING LIME I t / f
'
;
•
I [ i FILTER I ;SNEAKTNROUG~ I
RECEDING LIMB
/ ru
r,
rs
TR TIME
Fig. 1. Characteristics of initial effluent quality. backwash. At the end of the backwashing operation of a repeatedly used filter there would be remnants of the backwash water in the filter system. The backwash water remnants in the system can be subdivided into three types: (1) clean backwash water in the underdrain and connecting pipework from the backwash water supply system up to the bottom of the filter media, (2) backwash water remnants within the pores of the media and (3) backwash water remaining above the filter media up to the level of the wash water gutter. These three fractions of backwash water are shown in Fig. 1. The three fractions of backwash water remnants will have different characteristics. The first fraction up to the bottom of the media has the character of the clean backwash water. The remnant within the pores of the media would have a backwash water quality characterized by the last stages of the backwashing operation and the fact that the water remains within the pores of the media when the ffuidized bed during backwash is collapsed into a fixed bed at the end of backwashing. The collapse of the fluidized bed does not typically dislodge a significant number of particles from the surfaces of the media into the backwash water remnants within the pores, providing the backwashing operation has been effective. The third fraction of the backwash water above the media would have a quality which is poorer than the second backwash remnant since it preceded this backwash water during backwashing and hence would be removing more particles from the media. However, the remnant above the media is not subjected to the mechanism of collisions between media particles upon the collapse of the fluidized bed. Thus this fraction will exhibit characteristics unlike those of the other two backwash water remnants.
The peaks in initial degradation The early stages of filter ripening would be characterized by an initial period of relatively good water called the "lag" period in Fig. 1 and caused by the first fraction of remnant. The cumulative theoretical flow through times corresponding to each of the backwash water remnants are designated in Fig. 1.
The three fractions are noted as T~, (time up to underdrains), TM (time within and up to top of media) and 7"8 (time up to backwash water gutter). The total filter ripening period is TR. The "lag" period corresponds to TU and represents clean backwash water. It is followed by a "rising limb". During this period the effluent quality rapidly degrades up to the first peak at time T~, The first peak is due to the backwater remnant within the media and occurs at time T~t. Often the first peak is undetected since it occurs within 1 or 2 min of the start of a run at typical filtration rates of 2.7--4.0 mm s -~. Since the first peak is due to collisions at the end of the backwashing operation, such factors as the increased effectiveness and longer duration of backwash, the slow closure of the backwash valve and the increased strength of the adhesive forces between the filtered particles and the media may obscure the two independent peaks and a single plateau type response may be evident in the initial effluent quality. The backwash water remnant above the media (i.e. corresponding to time T B - TM) has particles which were removed from the filter grains during backwashing. The concentration of particles will be highest at level TB since it was at an earlier instant during backwashing, and lower at level TM. The author develops a new theory supported with experimental evidence in the second part of this paper that the concentration of particles in this backwash water remnant increases exponentially with height above the media. Thus the quality degrades between TM and TB reaching the second peak close to Ta. Oftentimes the second peak occurs at a time slightly ahead of the theoretical time Ta. This is due to mixing of the backwash remnant and the influent water. The second peak commonly occurs at times between 5-10rain at typical filtration rates. Experimental evidence substantiating the existence of the two peaks is presented in later sections of this paper. After the effluent quality has reached the second peak the quality of the product slowly improves.
The improving phase of filter ripening The improving phase of "filter ripening" termed
The interface between filtration and backwashing 0.4
RUNS 17 TO 19. A AND B
~)
r
0.5~-
FILTER 1. 2. A N n 3
RUNS
95,
W 15 ,
t
W 1E *
CLEAR WATER
0.4
MEAN AND
583
C.,.
/
~0.3 = i ~-
W 22
AFTE R T -
0.2
.
F,LTER A
t
CLEAR WATER
i ~0.2 .J
_1
0
v
T"
Tu
T:,
7, 1
2
,
,
Ts
T'
,
,
3
4
5
6
7
:
!
8
9
-
TIME - rain
~0.1 w
Fig. 4, Initial effluent quality with influent water variations. Filtration rate = 3.4 mm s-*, TuT,J
[I
TI
,
[,
5
, 15
10 TIME - MIN
Fig. 2. Initial effluent quality for uniform sand. Filtration rate = 2.4 mm s-L
the "'receding limb" in Fig. 1 is a function of the influent water characteristics. This long recessional limb has varying degrees of duration depending on filtration rate, influent concentration, particle size and the physicochemical character of the influent particles. This phase of filter ripening has been studied by O'Melia and All (1978) who associated this phase with the formation of particle chains or dendrites on the media during the initial stages of filtration shown to occur by Tien et al. (1977). Recently, Payatakes et al. (1981) have shown with direct visual experimental data, that the main mechanism causing alteration of the geometry of flow channels within filters was throat clogging. For ag-
RUNS 7 TO 12, A AND B FILTERS 1 TO 3
O.E
~ i =
~ ..R.NO.,- C,
0.(
i
~
0.4
0.2
M TI
°Tu I,
,
,
~o
'
,'o
TIME - mlft
Fig. 3. Initial effluent quality for uniform sand. Filtration rate = 4.7 mm s-L
glomerated aquasols this throat clogging phenomenon resulted in an increase of local capture efficiency and explained the initial increase of deep bed filtration efficiency which is shown as the improving phase in Fig. 1. The above summarizes the conceptual characteristics of initial degradation and improvement. EXPERIMENTAL VALIDATION CONCEPTUAL MODEL
OF
The above conceptual model has been validated with extensive experimental results by Amirtharajah and Wetstein (1980). The results are from two independent studies by Amirtharajah (1978) and Wetstein (1978), and details of the studies are reported in these references. All of the experiments consisted of a standard dirtying run, a backwash and a subsequent variable filtration run during which samples were taken at 1 rain intervals to monitor the quality of the initial effluent. The two studies were conducted on pilot plant filters and the influent suspension filtered was prepared by adding stock solutions of ferrous sulfate (FeSO4 • 7H20) or ferric chloride (FeCI~ • 6H20) to tap water. The iron in the effluent was measured spectrophotometrically. A sample of the results from the two studies are shown in Figs 2, 3 and 4. Figures 2 and 3 show the initial degradation from a uniform sand filter at filtration rates of 2.4 and 4.7 mm s -~. The results in Fig. 2 show the mean and 95% confidence intervals of 378 data points. At each minute the mean and error bar represents 18 data points. Figure 3 shows similar experimental data at a filtration rate of 4.7 mm s -*. On the abscissae of Figs 2 and 3 are shown the theoretically calculated flow through times Tu, Tu and TB. The figures clearly indicate the existence of the two peaks and the slow recessional improvement. It is also seen that the peaks correspond to the backwash remnants within the media (at time Tw) and that above the media (at time TB). Figure 4 shows the effect of influent water variations on initial effluent quality. It dramatically illustrates the correspondence between the backwash
584
A. AMIR'H-IARAJAH
FILTER RUNS. R 7.1 - R 7.6 (NFLUENT IRON. 14.5 rag. ) AVERAGE BACKWASH REMNANT IRON. 1.35 mg/I ~0.2C
e~" . . . . . . . . . . . .
-i,ooo
'-~"TRACE SOLUTION ABOVE MEDIA ONLY ONLY
E i
4 5000
30DD !
~TRACE
~ 0.1C
SOLUTION WITHIN VE MEDIA
-i~DDO r. T QUALITY
+ODO 0
u IT,,, 10
5
15
0
TIME - rain
Fig. 5. Initial effluent quality and dispersion characteristics of filter. Filtration rate = 4.7 mm s -~.
remnants and the initial degradation peaks. Run W16 with coagulated water is similar to the data shown in Figs 2 and 3, and confirms the two peak characteristics at times TM and Ta, the first peak being rather flat and the second occurring slightly before Ta. When clear tap water was filtered after T~t by removing the backwash water remnant above the media in Run W22, it is clearly seen that the second peak does not occur. When clear tap water was filtered in the subsequent run as in WI5 (after time 7"8) it is seen that the second peak is considerably suppressed even though it does exist like a plateau. The association of the backwash remnants with the peaks in the initial degradation shown in Figs 2, 3 and 4 are consistent with theoretically calculated flow through times T~,, TM and TB. In order to show this correspondence with direct experimental measurements and to evaluate the effect of dispersion of the remnants due to molecular and eddy diffusion, a series of 24 filter runs with six replications at each rate were completed at Montana State University. The runs noted as R 3.1-R 3.6 to R 9 . I - R 9.2 were made at filtration rates of 2.1, 3.4,
4.7 and 6.0ram s -t. Using the identical filter pilot plant and at the same filtration rates sodium salt tracer studies were made by replacing (a) the water above the media with tracer solution (for time T a - r~f), (b) the water within the media with tracer solution (for time T,~t- Tu) and (c) the water within and above the media with tracer solution (for time T8 - T6,). Typical results of initial degradation and sodium salt tracer studies are shown in Figs 5 and 6. The results confirm the two peak characteristics of initial degradation, and more importantly show the correspondence of the backwash water remnant volumes with the initial degradation. The sodium salt tracer studies clearly indicate the flow through times TM and Ta and prove that dispersion effects of the backwash water remnants are negligible. The predominant peak in initial degradation is due to the backwash remnant above the media. An attempt at explaining why this peak occurs necessitates a detailed analysis of the quality of backwash water. The following section presents a new theoretical model with experimental support for predicting the quality of backwash water.
FILTER RUNS. R 5.1 * R 5.6 INFLUENT IRON. 15.0 rag, I AVERAGE BACKWASH REMNANT IRON. 1,03 mg/I
so,o.o w..
.ooo ,
AND ABOVE MEDIA
~-
0.1C
~_
D.O..
o T-
O.D
I . . . . . . 5
I.
:
10
1~5
D
20
TIME - rain
Fig. 6. Initial effluent quality and dispersion characteristics o f filter. F i l t r a t i o n rate = 3.4 m m s -~.
The interface between filtration and backwashing
585
A THEORY FOR BACKWASHWATERQUALITY
c(r. O) = Co
(3)
Amirtharajah and Giourgas (1981) have recently presented a theory for particle detachment from sand grains during backwashing of filters with water alone by fuidization, typical of U.S. practice. Amirtharajah (1984) has recently dealt with theoretical aspects of air scour backwashing in which particle collisions and abrasion play a much greater role. The following theory assumes that particle detachment from the collectors in a water fluidized bed occurs due to the
c(ro, t) = c'o
(4)
~c(Ro, t)
= o
~r
(5)
in which Ro = radius of spherical shell. Equation (2) can be solved for the above boundary conditions by taking the Laplace transform with respect to time ?. Nelson and Galloway's (1975) solution to equation (2) is,
ro(Co-Co)N/~c°sh[(Ro-r)~/~]--~osinh[(Ro-r)X/~D
]
CO
+ --
"0 =
(6)
S
rs / ~ c ° s h [ ( R o - r o ) 3 ] - ~ o o S i n h [ ( R o - r o ) 3 ] hydrodynamic forces exerted by turbulent bursts. The turbulent bursts cross the viscous sub layer and exert lift forces which detach particles attached to the media. The detachment of particles is resisted by the London-Van der Waals adhesive forces. Amirtharajah and Giourgas (1981) considered the dynamics of a single particle cluster as it is removed from a collector. From the momentum equation, Rate of change of momentum = Lift Force - Adhesive Force '\dt
J
(oo .y 12Z2 :o0
0.076p/v: - -
in which s = Laplace transform variable. The radius of the spherical shell Ro is so chosen that the porosity of the sphere and its surroundings are equal to the porosity of the system. r0
R° = (1
-
,3
e)
(7)
in which e = porosity. Following Danckwert's (1951) analysis, the time averaged mass transfer rate is assumed as an exponential distribution of surface ages given by,
= - sD
(1)
(8) rzt 0
in which V,= interstitial velocity, dM/dt = t o t a l mass transfer rate from grain, p/=fluid density, v = kinematic viscosity, Do = diameter of detached particle, U* = friction velocity, A~ = Hamaker constant and Z = clear distance between particle and collector. The rate of change of momentum V~(dM/dt) was calculated by using Nelson and Galloway's (1975) mass transfer theory for fluidized beds. This model was formulated by considering mass transfer through particle diffusion across a film around a collector. The mechanism of mass transfer by diffusion from the surface was formulated in terms of Danckwerts' (1951) surface renewal theory. Consider the flux of particle clusters leaving the surface of the media grains as shown in Fig. 7. Due to diffusion the particle clusters would cause a concentration gradient ~c/Or from the surface of the media grain to the surface of the spherical shell, where it is assumed to be zero. The continuity equation in radial coordinates
in which ~ = time averaged mass transfer rate per unit area per unit time. Nelson and Galloway (1975) interpreted the Laplace transform variable s as proportional to the surface renewal frequency and assumed,
s = ~2 Vi
1 2r0 Sc t/3
(9)
in which Sc = v/D = Schmidt number, ~t was the only free parameter in Nelson and Galloway's model and by its judicious selection the model transformed exactly into accepted theories for single spheres at infinite dilution (~ = 1). Using equations (6), (7) and (9) it is possible to solve equation (8) to obtain the time averaged mass transfer fux, ffz.
/
s --~--~"~ 5"o'~-~ ~ °°°°ooooo~
/
CONCENTRATION GRADIENT
\
*o
//
Clr,o)=C,
gives, l[" R~
=z)pN\
aU
(2)
in which c =concentration, D = particle diffusion coefficient and r = radial distance. The boundary conditions as indicated in Fig. 7 are
\\
r ~
\
SPHERICAL_~..//¢%'. SHELL ~-
Clro, ' o=r C )
, / ~"
./~'~...
~ C(R.,t) =0
Fig. 7. Single collector mass transfer model.
586
A. AMIRTHARAJAH
Amirtharajah and Giourgas (1981) estimated the time averaged mass transfer rate as 8.7 x 10 -5 mg m m - : s -t. Assuming an average size for particle clusters detached from the collectors as 7,am, the rate of change of momentum of a particle cluster was calculated to be 8.1 x 10-3pN (pico Newtons). In contrast to this small value of the momentum change the lift force due to turbulent bursts was 4.5 x 10-:pN. It was also shown that the lift force is greater than the adhesive London-Van der Waals" forces only up to a distance of several nanometers. Thus a coating layer will remain on the collectors after backwashing. Thus total removal of
attached particles from the collectors is impossible by fluidi:ation during backwashing. This central result has been macroscopically confirmed with extensive experimental data by Amirtharajah (1978).
CONCETRAT,OH GRADIENT
-(--_z~
C = * Ke '
MASS TRANSFER FROM //SINGLE COLLECTORINTO VOLUME V AT RATE
BACKWASH REMNANT ABOVE MEDIA
~V~'~r
BACKWASHED.- - ~ " FILTER MEDIA
J
SURFACE RENEWAL MECHANISM
'
JJ
MOVEMENT OF PARTICLE /REMOVAL FRONT DURING SACKWASHING - CAUSES EXPONENTIAL DECLINE IN
"--
r t~, SUPERFICIAL - " " ~ i i
VELOCI,*"" ,i
SURFACE
RENEWAL AREA
~).....< AN ARSITRARY VOLUME V OF LIQUID
Fig. 8. Schematic diagram for backwash water quality theory.
Backwash water quality model The mass transfer model presented above can be used to formulate a theory for predicting backwash water quality. Consider the schematic diagram of a filter being backwashed as in Fig. 8. As an arbitrary volume of liquid V traverses the fluidized bed it would accumulate particles detached from the media surfaces. The mass transfer into the volume V would be at a rate ASI-- t~ dt as shown in equation (8). The total mass transfer into the volume V will depend on the area A (t) across which the surface renewal mechanism will transport material. The area A (t) would be a function of time. By a mass balance,
At a certain finite but large time t the particle removal front would have moved completely through the bed and no more particles can be removed from the collectors. As t becomes large, e x p ( - s t ) - - , 0 and c---,0, therefore the integration constant = 0. Hence equation (13) becomes,
V dc = -ff4A(t). dt
While equation (9) could be used for renewal frequency, at this preliminary modeling stage it wilt be assumed that the renewal frequency is the ratio of the superficial velocity v, to the diameter of the collectors d = 2r0. Therefore, equation (14) transforms to,
(10)
The - r e sign exists because the rate of transfer into the volume declines with time. As particles are removed from the surfaces of the collectors, some of the collectors will reach the non erodible layer and will no longer supply particles into the volume V. Thus a front at which no further particle detachment takes place will move up the filter bed as shown in Fig. 8. Since the surface renewal mechanism controls particle transfer, it is rational to assume that the surface renewal area is an exponential function of renewal frequency s,
A(t)---. ADs exp(--st)
(11)
in which A0 = initial area. From equations (10) and (11)
v ~ = - ~AoS exp ( - st) dt
(14)
in which
c = k exp - (vxt/d).
(12)
f dc = ~°S f exp(-st) dt (13)
(15)
The expression will not be valid at extreme values of t, i.e. as t ~ 0 or ~ . The equation is the final result of the theory.
Deductions from the backwash water quality model Several important deductions can be made from the final result of the theory, equation (15). (1) Transform the equation by taking logs lnc = l n k - ( ~ )
Integrating,
c = [--~--2] exp ( - st) + constant.
c = k exp(-st)
\
(16) /
thus a plot of In c vs (v~t) would show a linear variation with a - v e slope. Experimental evidence to validate this result is presented in a following section. (2) The term (vxt) is superficial velocity x time. This implies that backwashing a filter to a given terminal backwash water quality is dependent only on the total washwater volume used as long as the particle removal mechanism remains unaltered. Thus
The interface between filtration and backwashing
587
500.0
"''-
,
.
I ~
2,0
,oo
~
1.5
10.0
~ 3 ~
1.0
AMiRTHARAJAH'S DaTA ~" DeFFEI~NTFILTERSWITH
100.0 ,
5.0
MEANAN0 STANOARO 01[VIATtON" ~.OTTED POINTS: 25 .
i
0.1
FILTRATIONRATE~
I
L
mmll
Fig. 11. Peak of initial degradation vs filtration rate.
1.0 0.50
o.,o,
~
~
;
Empirical evidence for backwash water quality model
,
WASHWATERVOLUME- m3/m2
Fig. 9. Backwash water quality as a function of backwash water volume.
backwashing at 12 mm s -~ for 6 min will produce the same cleanliness as 9 m m s -~ for 8min, since (v.J) = (12 x 6 x 60)/1000 or (9 x 8 x 60)/1000 which are both equal to 4.32 m s m--'. (3) After the removal front has passed through the bed and all collectors have reached their permanent coating layer, very few particles will be detached into the washwater. This would result in a constant but low concentration of particles in the washwater after a given total volume of washwater is used. A large body of empirical evidence exists validating all three of the above deductions. A sample of this evidence is presented in the next section.
5U
500
~ ~
HUDSON'S ('1981)
40
~
3o ~ 20
~ 0
100
J o o~
DATA
' ~
B
50
u~
10
=
a
-
b (c.j)
(17)
in which a and b are coefficients dependent on the units c and v,t. The data shown in Fig. 9 has a correlation coefficient of 0.93. It is seen that the equation is identical in form to equation (16). The data also shows that a constant backwash water quality is reached at a total backwash volume of approx. 6.0 m 3 m -2. Two examples of data from the literature are presented in Fig. 10. The data again confirms the general validity of equations (15) and (16) and the deductions that are derived from it. INTERACTION OF BACKWASH WATER QUALITY A N D INITIAL D E G R A D A T I O N
,=,,
z
log c
100
'1000
I
Results illustrating the model developed above are shown in Figs 9 and 10. Figure 9 presents data collected by Amirtharajah (1971) from 16 runs at expanded porosities ranging from 0.55 to 0.78. It is seen that the data plots as a straight line on the semi-logarithmic plot. A regression analysis of the data gave an empirical equation of the form.
I
i
I
i
I
1 2 3 4 5 6 TOTAL WASHWATERVOLUME-m3/mz
Fig. 10. Backwash water quality vs washwater volume-literature data.
The theory and experimental results presented above indicate that the concentration of particles in the backwash water remnant above the media varies as an exponential function with backwash water volume. As can be seen in Fig. 8, the highest concentration of detached particles would therefore be at the top of the backwash remnant above the media. As filtration occurs in the following run the second peak in initial degradation occurs close to the time that the backwash remnant volume above the media which corresponds to a time T s as shown in Fig. 1 moves through the filter. Filtration efficiency would be an inverse function of filtration rate at the initial stages of filtration, when all other parameters remain unchanged. This would imply that the magnitude of the second peak in initial degradation would be a direct function of filtration rate. Since the concentration of particles in the
588
A. AMIRTHARAJAH
backwash remnant varies exponentially, it could be expected that the magnitude of the second peak during initial degradation would be a power function of filtration rate. Preliminary results illustrating this variation are shown in Fig. I1. The power function shown in Fig. I 1 had a correlation coefficient of 0.91. The previous theoretical analysis and the results shown provide a reasonably satisfying model for the existence and characteristics of the initial degradation of effluent quality. The analysis applies directly to filters with high level collecting launders typical in the American continent, which are often backwashed with water alone. For other backwash systems or operations the analysis should be suitably modified. A CONCLUDING SUMMARY A conceptual model of the initial degradation phase of filtration is presented as an interface between backwashing and filtration. It is shown that the initial degradation of effluent quality is due to the backwash water remnants within the media and the backwash water above the media. The two peak characteristics of initial degradation due to the backwash water remnants within the media and above the media is established by an extensive experimental investigation. A mathematical model for the quality of backwash water as a function of backwash water volume is developed. Deductions made from the mathe.natical expression confirm the validity of some accepted facts on backwash and also lay the basis for the peaks in initial degradation. Acknowledgements--This paper was presented to the Symposium on Water Filtration organized by K V I V and the
Belgian chapter of the Filtration Society, in Antwerp, Belgium, April 1982 entitled "Modeling Initial Degradation of Effluent Quality. The paper incorporates experimental data collected by Daniel P. Wetstein and Mohamad R. Farvardin, former graduate research assistants at Montana State University, whose contribution is gratefully acknowledged. The study reported was partially supported by the Engineering Experi-
ment Station at Montana State University and the National Science Foundation under Grant No. ISP-80411449. REFERENCES
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