WASTE MANAGEMENT, Vol. 9, pp. 141-149, 1989 Printed in the USA. All rights reserved.
0956-053X/89 $3.1)11 + .lit) Copyright © 1989 Maxwcll Pergamon Macmillan plc
MASS BALANCE FILTRATION EQUATION J. C. Agunwamba, N. Egbuniwe, and J. O. Ademiluyi Department of Civil Engineering, University of Nigeria, Nsukka, Nigeria
ABSTRACT. A new model filtration equation which incorporates the input variable, sludge concentration (C.,); the state variable, specific resistance (R); and the output variables, cake concentration (C) and filtrate concentration (C) was derived by material balance and regression analysis. On verification using experimental data from treatment plants, the theoretical predictions of the new equation agreed closely with the practical values, yielding a correlation coefficient of 0.98. For the experimental curve, the slope and intercept are 2.5 x 10 ~ m.h. and 0.02, respectively whereas the corresponding values for the theoretical curve are 2.75 x 10 ~5m.h. and 0.017, respectively. The graphical optimum dosage of ferric chloride for conditioning for both the Carman's and the new equation was found to be the same, 16% of the sludge solids by weight in each case.
experiments based on the sludge from the treatment plant of the University of Nigeria, Nsukka and at Enugu.
Sludge dewatering process reduces the volume of sludge before disposal. One of the ways of achieving this goal is through vacuum filtration which is a mechanical process. Many researchers (1-12, 14, 15) have proposed different models aimed at improving the performance of the vacuum filtration process. Carman (8) proposed the equation
Filtration Model The material balance equation for sludge filtration process can be stated as: Inflow - Outflow = Rate of Accumulations
P. A 2 -=
I~(RvV t + RmA)
which is the most generally used. V~.is the volume of the filtrate collected at the end of the filtration period (m3); t is the laboratory time of filtration (s); /~ is the absolute viscosity of the sludge (kN.s/m2); R is the specific resistance of the sludge to flow (m/kg); Rm is the specific resistance of the filtering medium (m/kg); A is the area of filtration (m2); and v is the instantaneous velocity (m/s). By mass balance analysis, this research improved on the existing models by introducing other filtration variables like the cake concentration, C (kg/m 3) and the filtrate concentration, (2/(kg/m 3) in addition to the specific resistance, R and the sludge concentration, Co (kg/m3). An equation of this nature would be useful in the process of optimization since it contains the most important filtration parameters. The new model is validated by carrying out laboratory
dC Q o C o - QCj = V/--~
where Qo is the rate of sludge inflow (m3/s); Q is the rate of volumetric increase of the filtrate (m3/s); Co and C• are the concentration of the sludge (kg/m 3) and the filtrate (kg/m3), respectively, while dC is a small change in the cake concentration (kg/m3). Assuming a constant rate of filtration,
V r = Qt
where t is the time taken to obtain the filtrate in seconds. If losses are ignored, the volume of sludge (Vo) filtered is distributed as the volume of cake (V~) formed and the volume of filtrate (Vr) i.e., vo
also, RECEIVED 2 MAY 1988; ACCEPTED 28 DECEMBER 1988.
Vo = Qot 141
A G U N W A M B A , N. E G B U N I W E , A N D J. O. A D E M I L U Y !
TABLE 1 Data for the Validation of the Filtration Equation at Constant Pressure
(mL10-6) (kg/m 3) (m/kgl0 '2) (kg/m ~) (kg/m 3)
400 380 Water 420 treatment 439 Plant Sludge 515
44 49 53 56 60
90 78 65 50 45
1.19 1.05 1.17 1.46 1.65
11 11.25 10 8.75 8.25
257.5 244.8 232
Pressure = 40.96 kN/m 2 Average Viscosity = 0.48 × 10 ~ kN.s/m 2 Average cake thickness = 0.050 m Min. cake thickness = 0.0115 m Aeff = 9.74 × I0 ~ m-'; Average, t, = 427.2 s Average Vj = 52.4 × 10 6 m3 By regression analysis the relationship between Cs and C is given C = 65.9 Cj - 428 where
65.9 and K2 = -428.
The coefficient of correlation is 0.9888, Integrating Eq. 3 for a case where Qo and Q are time independent,
CV~ = OoCot- OCtt.
T h e constant; K2/(1 + K1) does not affect the optimization of sludge filtration process, hence Eq. 10 reduces to
F r o m Eqs. 4, 6 and 7, and eliminating Qo and Q,
co(vj + v 3 =
C~- = 1 + K----~+ (1 + K,)V I
C~ : G + 1£2/(1 + K,).
+ cv .
where T h e regression equation connecting the variables C and Ci is given as A g u n w a m b a (5) C = K1Cf + K2
where KI and K2 are constants. Substituting Eq. 9 in Eq, 8 and rearranging gives,
1 + K, + (1 + K1)Vf
1 + K,"
Since the specific resistance is an important par a m e t e r in sludge filtration, it is introduced in Eq. 11. F r o m Eq. 1, following C o a c k l e y ' s modification (10)
2PA2t V~ = RC~ TABLE 2 Computed Data Based on Table 1
0.116 0.150 0.174 0.123 0.105
12.4 12.2 13.2 8.9 7.8
4.5 3.75 3.5 2.25 1.75
0.05 0.048 0.054 0.045 0.039
K,, = ~CiVj by definition
after neglecting the septum resistance, Rm, where t is the laboratory time m e a s u r e d in seconds. The yield f r o m filter plant is given as Metcalf and E d d y , Inc. (13)
Y = VIC
A t,. where tc is the industrial filtration time expressed in hours. E q u a t i o n 13 can be further expressed as
C) = G - 6.5 from Eq. 12. Average K, = 0.1336 m/h. (Note that if the concentrations are expressed in percentage, the unit of K,, becomes kg/m:.h.)
V¢ ~ At,
_ 2PA x 3600 Rp
TABLE 3 Data for Validation of the Filtration Equation at Variable Pressure
56 58 60 63
50 50 50 50
Water 392 Treatment 400 Plant Sludge 432 472
(m/kg 10 ~2) (kg/m 3) 1.32 1.67 1.93 2.15
C Pressure (kg/m 3) (kN/m ~)
13.6 15.3 19 22
250 262 294 312.5
41.76 52.17 60.70 68.04
Average Viscosity = 0.89 x 10 -6 kN.s/m 2 Average cake thickness = 0.012 Acff = 9.74 x 10-' m 2 By regression analysis (5), C = 7.79 Cr + 143.5 where KI =
7.79, K2 = 143.5
and = 0.980
Computed values based on the above data (b) C] (Real values) (kg/m 3)
Theoretical Values C} (kg/m 3)
29.9 31.6 35.3 38.3
8.34 8.99 9.60 9.73
39.8 42.5 45.0 45.6
where C~ = G + 16.3. (5)
From Eqs. 14 and 15,
Equation 20 is the desired equation. It connects the output variable (CI), the state variable (R) with the input variable (Co). It can be rewritten as  =
The yield (Y) has been found to be proportional to the concentration of the sludge (5), i.e.,
where Y = CoKo
where Ko is the constant of proportionality. Substituting for Y, Eq. 16 becomes: Vj = 7200PA/RpCoKo.
K4 =,u d/7200P(1 + K1)
which are regarded as constants for constant pressure filtration. If however, this is not the case, for constant sludge concentration, Eq. 20 becomes
But V,, = A d, where d is the thickness of the cake. Equation 19 simplifies to Co C~oKodR C~ = 1 + K1 + 7200(1 + K1)"
Hence, Eq. 11 can be expressed as Co CoVoRl~CoKo C} = 1 + K-----~+ 7200(1 + KOPA"
K3 = 1/(K~ + 1)
where K~ = Co/(1 + K~)
K6 = ~pd/7200(1 + KO.
N. E G B U N I W E ,
A N D J. O. A D E M I L U Y I
TABLE 4 Data on the Variation of Sludge Concentration (Co) with the Resistance (R) for an Oxidation Pond Sludge (at Nsukka)
(kg/m 3) (kg/m 3) (kN/m 2) (m/kg.10 '2) (kg/m 3) (kg/m 3) 1.0 0.5 0.54 0.6 0.4
11.25 8.5 8.0 7.5 5.0
40.96 40.96 40.96 40.96 40.96
3.14 4.84 5.8 6.6 11.91
105 103.9 104 104 103
38.87 38.87 38.87 38.87 38.87
0.83 x 10 -6
d = 0.025 m. By regression analysis; (5), K~ = 1.71, K2 = 103
.'. C} = C/ + K~ + 1 - Cr + 37.87
Based on Eq. , the equation of regression analysis (5), becomes C) = ~
+ 0.0265 x 10 -1° C2°R P
The above constants, K1, K3,/(4,/(5 and K6 were evaluated using the data in Tables 1, 2, 3 and 4, (5). Substituting for the values of the constants in Eqs. 21 and 24, the result yields:
C)_ 1 + 1.16 x lO-15RCoKo Co 65.9
and C~ = 5.688 + 4.098 x IO-I°RKo/P.
A plot of C~/Co versus RCoKo in Eq. 27 gives a straight line while a plot of C~ versus RKo/P in Eq. 28 yields a straight line.
,~-~.~-~%,.X~Xv Experimental curve Slope: 2 . 5 x tO -iS m.h 8
Slope: 2.75x10 -15 m.h Intercept: O. 017
6 I 0
I 6 R Col'Co
x i d 2 (m. h )-I
FIGURE 1. Correlation between the theoretical and experimental plots of C]/Co versus RCo/Ko at constant pressure.
MASS BALANCE FILTRATION EQUATION
TABLE 5 Variation of the Filtration Parameters at Different Dosages of Ferric Chloride. (a)
Domestic and industrial sludges were collected from the oxidation pond at the University of Nigeria, Nsukka and the water treatment plant at Enugu, respectively. Sludge, cake and filtrate concentrations were determined according to procedures described in "Standard Methods" (7). A moistened filter paper was placed on the perforated base of the Buchner funnel, and vacuum pressure applied for a few seconds to smoothen the paper and drain off the moisture. Then, the vacuum was put off. About 150 mL of well stirred sludge was poured into the funnel and the vacuum applied again. The volume of filtrate collected at the cylinder and the time it was collected were noted. For each filtration, the specific resistance was determined using the Buchner funnel technique of Coackley (10). To investigate the effect of chemical conditioning on the specific resistance based on both the new equation and Carman's equation, the domestic sludge was dosed with varying amounts of ferric chloride. Before each filtration cycle, the sludge was homogenously mixed using a magnetic stirrer for 60 sec. The specific resistance of these differently dosed sludges were then found.
% Ferric chloride (FeC13) of sludge solid content Parameters
Co ( k g / m 3) C ( k g / m 3) C/-(kg/m 3) t (s) Vf(ml) d(mm)
20 47 0.5 1080 96 2.0
20.3 60 0.6 1080 100 2.6
20.7 60 0.8 1080 102 2.1
23.3 141 1.3 1080 143 3.0
24.7 101 1.2 1080 115 2.4
P = 27.99 kN/m2; Ao, = 1.98 x 10 4 m 2 /~ = 0.89 x 10 -~ k N . s / m 2
(b) Specific resistance, R ( m / k g ) Dosage (FeCI3)
C a r m a n ' s equation
0 1.7 3.30 16.67 23.33
1.44 1.312 1.23 5.589 8.152
× x x x x
10 l° 10 I° 10 ~° 109 109
1.92 1.675 1.521 1.231 1.422
x x x x ×
10 ~4 10 TM 10 ~4 1014 10 TM
Slope: 4 . 0 9 8 x I0 -~ h / k g Intercept: 3 8 . 5 kg/m 3
Slope: 4 x l O - = O h / k g Intercept: 27" 5 k g / m 3
F I G U R E 2. Theoretical and experimental plots of (7} versus
--'- x I0 I0 m 3 h / k g
RKo/P at variable pressure.
J . C. A G U N W A M B A , x---
- - Carman~
N. E G B U N I W E ,
A N D J . O. A D E M I L U Y !
'- 1.5 u
4 '0 concentration
FIGURE 3. Variation of Co with R based on the two equations for water treatment plant sludge.
RESULTS The results of the experiments are shown in Tables 1, 2, 3, 4 and 5. Table 1 shows the data for the validation of the filtration equation at constant pressure which was used to compute Ko, KoCoR, C~ and C~/Co as recorded in Table 2. Table 3a records the values of the filtration parameters at variable pressure which was used in computing R.Ko/P and the theoretical values of C~ as shown in Table 3b. Recorded in Table 4 is the variation of the sludge concentration (Co) with the Resistance (R) at constant pressure. Table 5 shows the values of Co, C, Cr, t, Vr and d at various dosages of ferric chloride.
Model Validation Figure 1 shows the theoretical and experimental plots of C}/Co versus RCo/Ko at constant pressure for Eq. 27. The two plots are linear as earlier predicted in this study. For the experimental plot, the slope and intercept are 2.5 × 10-15 m.h and 0.02, respectively while the corresponding values for the theoretical plot are 2.75 × 10 -15 m.h and 0.017, respectively.
The coefficient of correlation was found to be 0.98. The close agreement between the theoretical predictions and experimental values for constant pressure filtration are shown in Fig. 1. A similar graph for variable pressure filtration is shown in Fig. 2 based on Eq. 28. Although the slopes agree, the intercepts differ by a wide margin. The experimental intercept was 55% higher than the theoretical intercept. This difference may be attributed to the fact that more solids are dislodged through the interstices of the septum at higher pressures than at lower pressure, leading to the rise in the filtrate concentration. However, the foregoing factor was not considered in the derivation of the new equation. The equation for constant pressure may be corrected for variable pressure by adding a term, 'blp' where 'b1' is a constant of proportionality and 'p' is the pressure. This, however, will be an approximate value since the two lines shown in Fig. 2 are not perfectly parallel lines. Figures 3 and 4 show the variation of the sludge concentration (Co) and the specific resistance (R) based on the new equation and Carman's equation
MASS BALANCE FILTRATION EQUATION
147 Carman's equation
- - I _ _
1 E oi x
G) U C u~ ¢)
.~_ o ~) Q. ~n
FIGURE 4. Variation of Co with R based on the two equations for an oxidation pond sludge.
for water treatment plant sludge and oxidation pond sludge, respectively. In each case, the two equations yielded curves of the same shape, although the curves of the new equation are shifted to the right. The shift may be attributed to three factors: firstly, the different assumptions made in deriving the two equations, secondly, the presence of such interactive variables as CI, Co and C which are absent in Carman's equation; and thirdly, the omission of the term 'R,,~/ PA' in Carman's equation as expressed in Eq. 13. The variation of the specific resistance with ferric chloride conditioning calculated from the two equations is shown in Fig. 5. Both curves show the specific resistance decreasing with increase in dosage until after a certain dosage when further increase of dosage yielded higher resistances. The optimum dosage in each case is about 16%, and the shapes are essentially the same. Hence, the two equations model the filtration process similarly.
CONCLUSION A new model equation which relates the input variable, Co, the state variable, R and the output variables, C and Cr have been proposed. It was derived by material balance and regression analysis and verified using experimental data from the oxidation pond sludge at Nsukka and water treatment plant sludge from Enugu. There was a close agreement between the theoretical predictions and the experimental values for constant pressure filtration. For the experimental curve, the slope and intercept are 2.5 x 10 -15 m.h and 0.02, respectively. While the corresponding values for the theoretical curves are 2.75 x 10 -15 m.h and 0.017, respectively; the correlation coefficient is 0.98. When compared with Carman's equation the new model yielded similar curves with a shift to the right. The high coefficient of correlation between theoret-
AGUNWAMBA, N. EGBUNIWE, AND J. O. ADEMILUYI x -- -- -o
Carman's equation ( x l 0 9) New equation ( x 1013 )
E 11) i ~ U e-
•k . 8 ~J
FIGURE 5. The effect of chemical conditioning on the specific resistance based on the two equations.
ical and practical values and the incorporation of all the essential variables into the equation, makes it useful in optimization, since it recognises the due significance of the sludge and filtrate concentration process. Also, the resistance could be minimized by the mathematical manipulation of the output variable, Co. The new equation was found to model the filtration process similarly with Carman's equation. Coagulating the Nsukka sample with ferric chloride, the optimum dosage was 16% for each of the equations. Hence, the new model finds application in the determination of the effectiveness of coagulants. REFERENCES 1. Ademiluyi, J. O., Anazodo, U. G. N., and Egbuniwe, N. Filtrability and compressibility of sludges, Part 1. Effl. Water Treat. J. 22:428 (1982).
2. Ademiluyi, J. O., Anazodo, U. G. N., and Egbuniwe, N. Filtrability and compressibility of sludges, conclusion. Effl. Water Treat. J. 23: 25-3[) (1983). 3. Ademiluyi, J. O. and Egbuniwe, N. LMT dimensional equations for compressible sludge filtration. Nigerian J. Sci. 18: 138-142 (1984). 4. Ademiluyi, J. O., Egbuniwe, N., and Agunwamba, J. C. A dimensionless number as an index of sludge dewaterability. J. Eng. Dev. 1:1-11 (1987). 5. Agunwamba, J. C. Optimization of the filtration process. An M. Eng. Thesis Dept. of Civil Eng., University of Nigeria, Nsukka (1987). 6. Anazodo, U. G. N. Dimensional equation for sludge filtration. Effl. Water Treat. 14:517-523 (1974). 7. APHA, AWWA, WPCF Standard Methods for the Examination of Water Wastewater, 13th edition. American Public Health Association, Washington D.C. (1971). 8. Carman, P. C. A study of the mechanism of filtration part I. Soc. Chem. Ind. Tram-Comm. 53: 159T (1934a). 9. Carman, P. C. A study of the mechanism of filtration part III. Soc. Chem. Ind. Trans-Comrn. 53: 301T (1934b). 10. Coackley, P. Laboratory Scale Filtration. Experiments and
MASS BALANCE FILTRATION EQUATION their application to Sewage Sludge Dewatering. Reynold Publishing Corporation, New York (1958). 11. Grace, H. P. Resistance and compressibility of filter cakes. Chem. Eng. Progr. 49:427-436 (1953). 12. Halff, A. H. An investigation of the rotary vacuum filter cycle as applied to sewage sludges. Sewage Ind. Wastes 24: 962984 (1952).
149 13. Metcalf and Eddy. Wastewater Engineering: Treatment Dispo~al, Reuse, 2nd Ed. McGraw-Hill Book Company, New York (1979). 14. Ruth, B. E Studies in filtration-derivation of general filtration equation. Ind. Eng. Chem. 27:708 (1935). 15. White, J. M. and Gale, R. S. Comments on dimensional equations. Effl. Water Treat. J. 15:422-423 (1975).