A SIMPLIFIED MIXED-CULTURE BIOFILM MODEL WOLFGANG RAUCH 1 * * M , HENK VANHOOREN 2 and PETER A. VANROLLEGHEM 2* M 1 Department of Environmental Science and Engineering, Technical University of Denmark, Building 115, DK-2800 Lyngby, Denmark and 2 BIOMATH Department, University Gent, Coupure links 653, 9000 Gent, Belgium (First received December 1997; accepted in revised form September 1998) AbstractÐA simple dynamic model is presented for fast simulation of the removal of multiple substrates by dierent bacterial species growing in a bio®lm reactor. The model is an extension to the well-known half-order reaction concept that combines a zero-order kinetic dependency on substrate concentration with diusion limitation. The basic idea behind this model implementation is to decouple the calcu- lations of the two major processes in the bio®lm: substrate diusion and biochemical conversion. The separate assessment of substrate diusion allows to relate the penetration depth of substrates to a frac- tion of biomass that is active in conversion. The conversion is then calculated considering only the active fraction of the biomass. The model is compared to experimental data from the literature and is found to be able to closely replicate the overall dynamics in a bio®lm system. The major advantage of the pro- posed model is the simple structure which leads to a reduction of the computational eort as compared to state-of-the-art mixed-culture bio®lm models. # 1999 Elsevier Science Ltd. All rights reserved Key wordsÐbio®lm, diusion, modelling, mixed-culture, simulation NOMENCLATURE A F = surface of bio®lm (L 2 ) D = diusion coecient (M 2 T 1 ) J = total transport of substrate through the sur- face of the bio®lm (ML 2 T 1 ) L = bio®lm thickness (L) M = mass of particulate component (M) Q = ¯ow in the water phase (L 3 T 1 ) r = reaction rate (ML 3 T 1 ) r l = reaction rate in the lower part of the bio®lm (ML 3 T 1 ) r u = reaction rate in the upper part of the bio®lm (ML 3 T 1 ) S = concentration of soluble substrate (ML 3 ) t = time (T) u f = velocity by which the surface of the bio®lm moves relative to the substratum (MT 1 ) V = volume (L 3 ) X F = concentration of bacterial species in the bio- ®lm as biomass per total bio®lm volume (ML 3 ) X B = concentration of bacterial species in the bio- ®lm as biomass per bulk liquid volume (ML 3 ) Y = yield coecient (MM 1 ) fp = fraction of inert material in biomass (MM 1 ) i X = ammonia fraction in biomass (MM 1 ) m = maximum growth rate of biomass (T 1 ) b = decay rate (T 1 ) kh = hydrolysis rate (T 1 ) k d = detachment rate (MT 1 ) k dt = detachment coecient ( ) b = dimensionless penetration depth of a sub- strate ( ) e = fraction of volume occupied by the phase/ component ( ) f = active fraction of a species in the bio®lm ( ) r = density (ML 3 ) n = stoichiometric coecient ( ) Subscripts at = sux denotes attachment dt = sux denotes detachment i = sux denotes substrates j = sux denotes species l = liquid phase s = solid phase A = autotrophic bacteria B = bulk liquid F = bio®lm H = heterotrophic bacteria I = inert substance S = biodegradable substance INTRODUCTION The growth of microbial species inside attached bio- logical ®lms causes a signi®cant ¯ux of substrates from the bulk liquid, i.e. the water phase outside of Wat. Res. Vol. 33, No. 9, pp. 2148±2162, 1999 # 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0043-1354/99/$ - see front matter PII: S0043-1354(98)00415-1 *Author to whom all correspondence should be addressed. [Fax: +45 4525 2800; e-mail: wr@imt.dtu.dk]. 2148