A finite element bed modelling approach for biological fluidized S. Soyupak, H. Nakiboglu, and G. Siiriicii Department of Environmental Engineering, Middle East Technical University, Ankara, Turkey A numerical solution approach is presented for the mathematical model developed for the biological fluidized bed wastewater treatment process. The model includes equations describing the pollutant concentration change within the reactor and biofilm: a dispersed piston flow reactor model equation with a removal rate term and a nonlinear biofilm model equation. The spatial solution of the dispersed piston reactor model was achieved by using zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC a finite element technique, Galerkin’s method of weighed residuals, and its solution through time was performed by using an explicit backwardjinite difference technique. Solution of the model equations gave results comparable to some experimental data ob- tained from the literature. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Keywords: biological fluidized bed, dispersed plug flow, finite element method, fixed film reactors. Gaierkin’s residual method, modelling Introduction Biological fluidized bed reactors are being used to re- move some pollutants, such as N03-N and NH4-N, from wastewaters. Considerable research efforts have been made within the last decade to understand the basic mechanisms involved and to develop models for design purposes, because of their high efficiency, which leads to small volume requirements.‘-4 Figure 1 gives a schematic representation of an upflow biological flu- idized bed reactor. Flow is applied from the bottom of the column reactor, and its level is ensured to fluidize the carrier particles with biological film such as sand and activated carbon. The basic mechanisms for the pollutant transport within the reactor are convection, dispersion, and mass flux to the biotilm.5 The biode- gradable pollutants that diffuse into biological layers on fluidized particles are exhausted by microorga- nisms. Figure 2 illustrates the pollutant concentration profiles that develop through a biotilm. Each biofilm involves a stagnant liquid layer through which no re- action occurs; diffusion is the only mechanism of trans- port to the actual biotilm layer. The pollutants that penetrate into the biological layers are exhausted by microorganisms while their transport continues. The complicated nature of transport and reaction mechanisms does not allow the use of simple equations for reactor design. Most existing designs were based Address reprint requests to Dr. Soyupak at the Department of En- vironmental Engineering, Middle East Technical University, 06531, Ankara, Turkey. Received 7 November 1988; accepted 10 October 1989 258 Appl. Math. Modelling, 1990, Vol. 14, May on pilot or laboratory-scale experiments. Mathematical models can be considered as an alternative method in designing the biological fluidized beds and also a useful tool in understanding their performance characteris- tics. This study was started with the aims of selecting a suitable model for a biological fluidized bed reactor and providing a numerical solution for model equa- tions. A comparison of the numerical solution of the model equations with the experimentally observed data, that exists in the literature was also within the scope of the study. Mathematical model The basic assumptions used in developing model equa- tions were given by Nakiboglu.6 The pollutant con- centration change through any finite depth (AZ) of the reactor can be described by the following partial dif- ferential equation, which was derived by using mass balance for the pollutant over the tinite depth? _MaSb+D a*sb w zyxwvutsrqponmlkjihgfedcbaZY aSb(t) a2 ’ zyxwvutsrqponmlkjihgfedcbaZYXWVUT a2* - - R,(t) = at (1) where u is the superficial velocity of the fluid (L/T); t is time; sb is the bulk substrate concentration (M/L3); D, is the axial dispersion constant (L*/T); z is the axial distance in the reactor (L); R, is the substrate removal rate in the reactor (M/(L3T)); and E is the porosity of the bed (dimensionless). The initial and boundary conditions for equation (1) are given below: Initial condition: s,(t) = sbo at O<z<H for t = 0 0 1990 Butterworth Publishers