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War. Res. Vol. 28, No. 8, pp. 1851-1853, 1994

Pergamon

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RAPID COMMUNICATION

OPTIMIZING ROTATING BIOLOGICAL CONTACTOR DISC AREA IAN BUCHANANl and ROLAND LEDUC2*(~ ~Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Quebec, Canada H3A 2K6 and 2Department of Civil Engineering, University of Sherbrooke, Sherbrooke, Quebec, Canada JlK 2R1 (First received November 1993; accepted in revised form January 1994)

Abstraet--A method of minimizing the total active disc area required for soluble biochemical oxygen demand (SBOD) removal by a multi-stage rotating biological contactor (RBC) in which the substrate removal rate is not oxygen limited, is applied to two generalized steady-state SBOD removal models. The active disc area of each stage is optimized according to an analytically determined optimal relationship between SBOD concentrations of adjacent stages. Key words--rotating biological contactor, design, optimization, modelling, disc area

INTRODUCTION One of the major functions of a rotating biological contactor (RBC) is to remove soluble organic carbon (frequently expressed as soluble biochemical oxygen demand, or SBOD) from municipal wastewater. In order to determine the required disc surface area (or active disc area), this removal is modeled frequently as occurring according to Monod kinetics (Kornegay and Andrews, 1968; Kornegay, 1975; Clark et al., 1978) or first-order kinetics (Eckenfelder and Vandevenne, 1980; Fujie et al., 1983). Other kinetic orders (e.g. half-order) have also been proposed (Shieh, 1982; Arvin and Harremo~s, 1990). A procedure to minimize the active disc area required by an RBC in which SBOD removal occurs according to M o n o d or alternatively first-order kinetics is presented in Leduc and Buchanan (1993). In many cases, neither M o n o d nor first-order kinetics is applicable to all RBC stages. Hence there is a need for more flexible, yet easily calibrated, SBOD removal models intended for use in RBC design. Pano and Middlebrooks (1983) presented a carbon removal kinetic model in which SBOD removal in the first RBC stage occurs according to M o n o d kinetics, while the average removal in subsequent stages is proportional to the first stage SBOD concentration raised to a power n. As presented in Pano and Middlebrooks (1983), the model treats all but the first stage as one combined continuous-flow stirred tank

reactor. Average rate and order constants are applied at these later stages. A more flexible form of this kinetic model (termed the combined model) is adapted herein to RBC design. Additionally, a generalization of the firstorder model is considered. This generalized model is termed by the variable rate order (VRO) model. The magnitudes of the parameters of each model are assumed to differ from one RBC stage to the next. These parameters may be evaluated on a perstage basis during a treatability study of the wastewater in question, using flowrates and influent SBOD concentrations under which the proposed RBC is expected to operate. The optimization procedure outlined in Leduc and Buchanan (1993) is applied herein to these more general SBOD removal models.

METHODOLOGY The SBOD removal models include the following assumptions: (1) steady-state operation has been reached; (2) SBOD removal by agents other than the attached biomass may be neglected; (3) oxygen is not a limiting factor; (4) organism decay may be neglected due to the small decay rate relative to the growth rate; (5) complete mixing is achieved in each stage due to the action of the discs; and (6) the mass of attached-growth organisms is proportional to the active disc area (i.e. the active depth is constant). Taking the abovementioned assumptions into consideration, a mass balance of SBOD through the ith stage of a multi-stage RBC may be written as

*Author to whom all correspondence should be addressed.

Q( S,_ , - S,) = r~A,d,

1851

(1)

IAN BUCHANAN a n d ROLAND LEDUC

1852

For the combined model, expansion of equation (5) gives,

where Q = flow rate (m r d i); S,_ t = influent SBOD concentration (g m ~); S, = effluent SBOD concentration (g m 3); r , = r a t e of SBOD utilization per unit volume of attached-biomass (g m ~d ~); A, = area of active attached biomass (m:); d, = active depth of attached-biomass (m).

t/s,

K,\ s;,,

Variable rate order model

According to the VRO model, SBOD removal is proportional to the stage SBOD raised to the power v. That is, r, = k, ST', in which k, and v, are the rate coefficient (d l), and the reaction order applicable to the ith stage, respectively. Hence, equation (1) may be written as Q(S,

, - S,) = K,A,S~'

At= e,:,_

s:-',

PI S~ A O(So - S, ) = Ks, + Sl

I

(4)

in which K s and P represent the half-velocity saturation coefficient (mg 1-~), and the maximum specific SBOD removal rate (gm-Zd t), respectively. The mass balance applicable to each of the subsequent stages is given by equation (2). Thus AT is written as

(5) Total active disc area minimization

The total active disc area (A~.) given by either the VRO model or the combined model is first expanded and differentiated with respect to the stage SBOD concentrations (S~). The derivative is allowed to vanish and the expression is then solved for S v In the case of the VRO model, equation (3) may be expanded to (I

IS 0

,\

1 /S t

+ - -l/ - "{ S- ~ - S , - ? ,

g.

I

\

l

, \S;;'-?

_l F - , - s,.

+ r . \ s;"

'

/

/)

Differentiating equation (6) with respect to S~ (with i = 1 , . . . , n - 1), and equating this derivative to zero, results in S,

I;i Si - I

KS~'

(7) 1 +v,

i

)

/~

\K~S7

/

Similarly, differentiating equation (8) with respect to S, (for i = 2 . . . . . n - 1), equating the resulting derivative to zero and solving for Sj results in equation (7). Therefore, under the combined model, AT is optimized when both equations (9) and (7) are satisfied. In this case, however, the index i in equation (7) ranges from 2 to n - 1.

Combined model

In the combined model, SBOD removal is modeled as occurring according to Monod kinetics in the first RBC stage and according to the VRO kinetic expression in each of the subsequent stages. This approach recognizes the fact that environmental conditions within the first stage of an RBC are quite different from those in the subsequent stages. Thus a mass balance of SBOD through the first RBC stage may be written as

, 5

Differentiating equation (8) with respect to the first stage SBOD, equating this derivative to zero and solving for S~ yields

(2)

in which K,(m d ~) is a proportionality constant (K~ = k,d,). The parameters, K~ and vj, are evaluated during a treatability study of the wastewater in question. The total active disc area of an n-stage RBC, with a desired effluent S, and an influent concentration S 0, is then given by

",

L

DISCUSSION Equipped with the influent and desired effluent S B O D concentrations, as well as the values of the model parameters applicable to each stage (evaluated during a treatability study), the RBC designer can use e q u a t i o n (7), or equations (9) and (7), to obtain the S B O D concentration o f each but the final (n th) stage (S, having been fixed at the outset as being the desired effluent S B O D concentration). These values may then be used to calculate the optimal active disc area of each stage. To facilitate comparison between these a n d other R B C design models for S B O D removal, the data of Clark et al. (1978) will be used herein, as they were in Leduc a n d B u c h a n a n (1993). Based on these data, the values of the parameters of the V R O and the c o m b i n e d models were estimated. The coefficients of determination (r 2 values) associated with each of the p a r a m e t e r estimates indicate that the V R O model represents these data better t h a n M o n o d kinetics (as used in Clark et al., 1978), or the c o m b i n e d model. The R B C design m e t h o d used by Clark et al. (1978) calls for a two-stage RBC having a total active disc area (AT) of 248,000 m 2. Application of the methodology presented herein to the V R O model results in a two-stage R B C h a v i n g a total active disc area of 2 1 6 , 4 0 0 m 2 being specified (see Table 1). Hence a 12.8% reduction in AT is realized. In some cases, the assumption that oxygen is not rate limiting may not be valid, especially in the first R B C stage where organic loading is greatest. In this

Stage i I 2

Table I. Optimal RBC disc areas (VRO model) K, S, A~ Loading (rod 1) v~ (mgl i) (m2) (gm 2d I) 0.186 1.21 40.0 121,820 26.1 0.139 1.17 23.0 94,580 12.8

Rotating biological contactor optimization case, the actual S B O D removal would be less than that predicted by either the V R O or combined models. Hence the minimized active disc area would be inadequate to achieve the desired SBOD removal. U n d e r these conditions, the active disc area of the RBC stage, in which overload conditions occur, may be increased as indicated in Leduc and Buchanan (1993).

CONCLUSIONS Two generalized RBC design models, termed the variable rate order and the combined models, have been outlined. The method of determining the optimal active disc area of each RBC stage, presented in Leduc and Buchanan (1993), has been applied to these models. U n d e r heavy organic loadings, the active disc area of a stage determined according to this method may be inadequate to effect the desired S B O D removal due to oxygen transfer limitation. This situation may be remedied by increasing the active disc area in the overloaded stage, as indicated in Leduc and Buchanan (1993). Acknowledgement--The Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged for its financial assistance.

1853 REFERENCES

Arvin E. and Harremo~s P. (1990) Concepts and models for biofilm reactor performance. Wat. Sci. Technol. 22, 171-192. Clark J. H., Moseng E. M. and Asano T. (1978) Performance of a rotating biological contactor under varying wastewater flow. J. Wat. Pollut. Control Fed. 50, 896-911. Eckenfelder W. W. Jr and Vandevenne L. (1980) A design approach for rotating biological contactors treating industrial wastewaters. In Proceedings: First National Symposium/Workshop on Rotating Biological Contactor Technology. Edited by Smith E. D., Miller R. D. and Wu Y. C., Vol. II, pp. 1065-1075. Champion, Va. Fujie K., Bravo H. E. and Kubota H. (1983) Operation design and power economy of a rotating biological contactor. War. Res. 17, 1153-1162. Kornegay B. H. and Andrews J. F. (1968) Kinetics of fixed-film biological reactors. J. War. Pollut. Control Fed. 40, R460-R468. Kornegay B. H. (1975) Modelling and simulation of fixed-film reactors for carbonaceous waste treatment. In Mathematical Modelling for Water Pollution Control Processes (Edited by Keinath T. M. and Wanielista M.), pp. 271 306. Ann Arbor Science, Ann Arbor, Mich. Leduc R. and Buchanan I. (1993) Minimization of multistage RBC active disc area. J. envir. Engng Div. Am. Soc. cir. Engrs 119, 271 286. Pano A. and Middlebrooks E. J. (1983) Kinetics of carbon and ammonia nitrogen removal in RBCs. J. Wat. Pollut. Control Fed. 55, 956-965. Shiek W. K. (1982) Mass transfer in a rotating biological contactor. Wat. Res. 16, 1071 1074.

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