New Conceptual Formulation for Predicting Filter Performance Vlnod tare' and C. Venkobachar Environmental Engineering Division, Department of Civil Engineering, Indian Institute of Technology, Kanpur 2080 16, India
rn This paper describes the development of a new conceptual model for predicting head loss buildup and removal efficiency during filtration by packed beds based on the postulate that some retained particles can act as collectors and thereby improve the filtration efficiency. The concept of Unit Bed Element (UBE) and of the unit collector is used for the quantitative description of the various processes of interest which occur in the porous medium. The filtration efficiency is described by a set of three interdependent equations for monosized particles in suspension assuming the porosity of the bed to remain constant as filtration progresses. The first equation computes the single collector removal efficiency related to clean grain efficiency and retairied particles acting as collectors. The second equation estimates the number of retained particles acting as collectors as a function of depth, time, and local particle concentration in suspension. The third equation predicts suspended particle concentration with time over the entire filter depth. The head loss buildup is predicted by modifying the classical head loss equation for clean beds incorporatihg change in the surface area to volume ratio of the filter grains. The model is calibrated and verified with a set of laboratory data obtained from literature under varying conditions of operating and system variable?. The results show strong agreement between model predictions and experimental observations as indicated by low values of standard error of estimate associated with coefficient of correlation values approaching unity.
Introduction Filtration, a physicochemical process, is one of the most important unit operations employed for the final clarification of water and wastewater in treatment plants. Despite its wide application, the filtration process is not well understood. This is because conceptual mathematical formulations do not exist that can quantitatively describe the complete dynamics of deep bed filtration, taking into account the complex interaction between filter grains and liquid solution particles to be removed as well as the effects of operating and system variables. Only recently, OMelia and Ali (1)proposed a model for the development of head loss and removal efficiency during filtration by packed beds based on the postulate that some retained particles can act as collectors and thereby improve the filtration efficiency. Though the concept seems to be appropriate and the mathematical model very well agrees within the certain limits of experimental data, the estimated coefficients may not reflect or may not have any correlation with the actual properties of the suspension and filter media responsible for filtration. This is because the aforementioned model 0013-936X/85/0919-0497$01.50/0
does not incorporate a few important factors such as surface coverage of the filter media grains and that of retained particles acting as collectors, variation of retained particles acting as collectors along the filter depth, etc., which would influence the filter performance to a large extent. The present research attempts to modify the model proposed by previous workers (1) for predicting filter performance by incorporating the above-mentioned factors. The model is calibrated with a set of laboratory experimental data and tested with other experimental data.
Model Formulation According to O'Melia and Ali (1)the single collector removal efficiency (v,) of a filter grain and its associated retained particles can be written as
vr = IICY + N v P ~ , ( d , / d J 2
(1)
Here, v is the single collector efficiency of a clean collector, 71, is the collision efficiency of a retained particle, N is the number of retained particles acting as collectors, d, and d, are the diameters of media grain and suspended particles, respectively, and CY and CY, are the particle-to-filter grain and particle-to-particle attachment coefficients that can range from 0 to 1. The prediction of filter performance is dependent on estimation of the number of retained particles acting as collectors. OMedia and Ali (I) have proposed the equation for estimating N , neglecting the change in surface coverage that must occur with time. The present study takes into consideration the surface coverage of media grain and retained particles acting as collectors along with variation of N with time and depth. The development of a modified equation is described subsequently. Considering the number of retained particles that can act as collectors (N), which can vary with time and with location in the bed, the removal efficiency of a clean filter grain is CY. Consequently, the rate of removal of suspended particlee by the clean grain is 7cunuM:(7r/4). As the filtration progresses, the single collector removal efficiency increases by N times the rate a t which particles are removed by the retained particles acting as collectors; a t the same time there is a decrease in the single collector removal efficiency because of the decreased effective area of the media grain and retained particles acting as collectors (N)as a result of surface coverage. The reduction in effective area of the media grain will be proportional to the number of retained particles acting as collectors (N), while the reduction in effective area of the retained par-
0 1985 American Chemical Society
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ticles acting as collectors would be proportional to the square of the number of retained particles acting as collectors (W). This is based on a simple logic as follows. Suppose that on a retained particle two particles can be deposited a t any instance. Then the surface coverage of this particle would depend on the particles deposited on these two particles which are 4, Le., 2,. If we suppose that on a retained particle three particles can be deposited a t any instance, then the surface coverage of this particle would depend on the particles deposited on these three particles which are 9, i.e., 3,. Similarly, if N particles are deposited a t any instance, then the surface coverage would depend on the particles deposited on these N particles which are N X N, i.e., W . Thus, the change in the number of collector particles on a single filter grain can be expressed as follows: rate of change of the number of collector particles on a single filter grain equals rate of removal of particles by a single filter grain plus N times rate of removal of particles by retained particles on a filter grain acting as;collectors minus0 N times rate of particles removed which would contribute to the reduction in effective area of filter grain minus IP times rate of particles removed which would contribute to the reduction in effective area of the retained particles acting as collectors,
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Figure 1. Model calibration.
these equations by making use of numerical methods. The development of head loss (kf)is based on the rate of change = qauon?r(d,2/4) + Nqpapvon?r(d,2/4)Carman-Kozeney (2) equation modified for change in area to volume ratio of the media due to deposition N~,~,u0ndd,2/4)Pc- ~ ? l p ~ p ~ o n ~ ( d , 2 / 4 ) P surface p as filter run advances. The equation is as follows: rate of change = (C, C2N - C3N- C4W)n (2)
+
where 6, is the fraction of removed particles which contributes to the reduction in effective area of filter grain and Pp is the fraction of removed particles which contributes to the reduction in effective area of the retained particles acting as collectors.
c1 = ?pUo(?r/4)d>
(3)
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(4)
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(5)
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(6)
Considering an extremely thin layer of thickness “hL” of the filter bed and making the mass balance for N, the following equation can be obtained:
aN + u,,-aN = [C, + (C, - C3)N- C4Wln at aL
(7)
The removal efficiency of a single filter grain is related to the removal accomplished by a packed bed through a mass balance about a differential volume element of the filter. For the case considered here, in which retained particles act as collectors so that removal varies with time, the result of such a mass balance is given as follows:
Here, F is the bed porosity. Equations 1, 7, and 8 comprise a mathematical model to describe the removal efficiency of a filter bed in time and space. Their solution results in the removal efficiency of a filter bed. The exact solution of the above equations is not possible. Simplifying techniques are used to obtain a reasonably accurate solution. In this qr and n are considered as step functions rather than continuous functions of time and depth. A new algorithm is developed to solve 498
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Here, K is an empirical coefficient,p and p are the dyanmic viscosity and density of fluid, g is the acceleration due to gravity, uo is the undisturbed superficial velocity above the filter bed, Nc is the number of filter grains, Np is the number of retained particles in the filter bed estimated by using the modified equation (eq 8), and p’ is an empirical coefficient that represents the fraction of retained particles thqt is exposed to the flowing fluid and contributes to the additional surface area.
Model Calibration Use of eq 1 and 7-9 to describe filter performance requires evaluation of the terms q , qp, a,ap,&, Pp, and p’. Data obtained by Habibian (3) have been used to estimate certain of these coefficients in calibrating the model (Figure 1). The model is calibrated by using the procedure adopted by O’Melia and Ali ( I ) . In calibrating the model, the single collector removal efficiency (qa)is calculated from the experimental data for a “clean” filter. For example, n/no is 0.6 for the filters in Figure 1. Hence, by use of eq 8 a t t = 0, qa = 1.44298 X With this initial condition established, the constants qpap,&, and PP are determined by using nonlinear curve-fitting techniques. Morquardt (BSOLVE regression algorithm) is used to solve for the coefficients in multivariable nonlinear regression equations (4). The values of qpap,P,, and Pp were found to be 0.4897 X lo-,, 0.7582, and 0.2437 X lo-’, respectively. The values of q and q calculated by using Happle’s flow model (5-7) for packe8 and 1.0, respectively. Thus, a and beds are 8.4192 X apcan now be computed as 0.17139 and 0.4897 X lo-,. In a similar manner, p’ was estimated by using eq 9. Calcu-
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Flgure 3. Model verification.
\
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Figure 2. Model verlflclatlon.
lations of head loss as a function of time were compared with experimental results (Figure 1). A value of p' = 0.193 is obtained as a result of curve fitting. Very good correlation is obtained for the experimental and model pred i d values of removal efficiency and head loss. Removal efficiency and head loss as predicted by O'Melia and Ali (I) are also shown in Figure 1. Model Verification The model is tested by using data from other experiments by Habibian (3) and Tare (8) in which one or more filtration variables were significantly different from conditions in the runs used for calibration. Observed values of the particle removal by the clean filters were used to calculate r/a for each run. Coefficients pc, BP,and p' are kept constant as they are not expected to vary with surface properties of media and particle and concentration of particles. The value of qPapwas estimated for the best fit of the experimental results as it would vary with the surface properties of the media, suspended particles, and size of the suspended particles. Only the typical data from two other filtration experiments by Habibian (3))which is also used by O'Melia and Ali (I),employing different suspended particle sizes (dp = 0.1 and 1 pm) and influent concentrations (no= 9.7 and 37.2 mg/L) are presented in Figures 2 and 3. Proposed model predicted values along with those of O'Melia and Ali (I) are also plotted. The model values are in better agreement with the experimental data than those predicted by using the previous model. Conclusions The following conclusions may be drawn on the basis of the present work. (1) The model presented here has better agreement with results of laboratory experiments compared to the model proposed by previous researchers. This clearly shows that the surface coverage of media grains and retained particles acting as collectors and the variation of retained particles with depth are the important factors. (2) High values (-0.758) for the surface coverage coefficient for media grains indicate that as the filtration progresses, surface properties of media grains are not important, while particle to particle interaction dominates the filter performance.
1)aUo(r/4)d? sp.,uo(r/4)d,2 ~papuo(*/4)d,2& 4ppuo(r/4)d,2P diameter of me&a grain diameter of suspended particle bed porosity acceleration due to gravity head loss empirical coefficient depth number of retained particles acting as collectors number of filter grains number of retained particles undisturbed or bulk particle concentration influent particle concentration time fluid velocity particle-to-filter grain attachment coefficient particle-to-particle attachment coefficient fraction of retained particles that is exposed to the flowing fluid and contributes to the additional surface area fraction of removed particles which contributes for reduction in the effective area of filter grain fraction of removed particles which contributes for reduction in the effective area of retained particles acting as collectors single collector efficiency of clean collector contact efficiency of a retained particle single collector removal efficiency density of fluid dynamic viscosity Literature Cited O'Melia, C. R.; Ali, W. Prog. Water Technol. 1978, 10, 167-182.
Carman, P. C. Trans. Inst. Chem. Eng. 1937,15,150-166. Habibian, M. T. Ph.D. Dissertation, Univeristy of North Carolina, Chapel Hill, NC, 1971. Kuester, J. L.; Mize, J. H. Optimization Techniques with Fortran, McGraw-Hill: New York, NY, 1973. Eliassen, R. J.-Am. Water Works Assoc. 1941,33,128-134. Spielman, L. A.; Friedlander, S. K. J . Colloid Interface Sci. 1974,46, 22-31.
Spielman, L. A.; Goren, S. L. Environ. Sci. Technol. 1970, 4, 135-140. Tare, V. Ph.D. Dissertation, Indian Institute of Technology Kanpur, India, 1981.
Received for review April 4, 1983. Revised manuscript received October 25, 1984. Accepted December 28, 1984.
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