Chinese Journal of Chemical Engineering, 16(2) 277 286 (2008) A New Thiele’s Modulus for the Monod Biofilm Model FANG Yuanxiang ( ) 1, * and GOVIND Rakesh 2 1 National Risk Management and Research Laboratory, U.S. Environmental Protection Agency, 26 W. Martin Lu- ther King Dr. Cincinnati, OH45268, USA 2 Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, OH45221, USA Abstract A new Thiele’s modulus, F , was developed to provide a gradual transition between zero and the first order of kinetics, and to accurately calculate the mass transfer flux and the effectiveness factor for the Monod biofilm. Values of the effectiveness factor, calculated using the new Thiele’s modulus, were compared with those obtained from numerical solutions and from other published moduli and empirical formulae. The comparison indi- cated that the new Thiele’s modulus was the best modulus for the Monod biofilm model. In addition, another Thiele’s modulus, G , was developed for a Monod biofilm, covered with an external water layer. The overall effec- tiveness factor can also be calculated by using both moduli F and G . The criteria that were proposed for identifica- tion were based on the values of F and G , the limiting processes for biomass growth, and substrate conversion. Developed from F , a new parameter was related uniquely to such features as the depth and shallowness of the generalized substrate concentration profiles inside a Monod biofilm. Criteria were developed to identify the types of concentration distribution inside a Monod biofilm. These methods were used to estimate the substrate flux and the concentration distribution of the biofilms defined in the first benchmark problem (BM1), by a task group of the In- ternational Water Association on Biofilm Modeling. Keywords Monod biofilm, kinetic model, effectiveness factor, Thieles modulus, generalized concentration profiles 1 INTRODUCTION The biofilm is the core functional part of many microbial processes, such as, fermentation, biofiltra- tion, and biological wastewater treatment. A biofilm model, which is conceptualized to describe the essen- tial processes of mass transfer and reactions inside an idealized homogeneous biofilm, needs to be incorpo- rated into various process models, for estimating the kinetic parameters for mass transfer and biological reactions, and for designing and simulating processes in fermentators, biofilters, biotrickling filters, and wastewater treatment plants [1 4]. The Monod biofilm model conceptualizes the biofilm as a homogenous layer of immobilized, active biomass, where substrates transfer according to Fick’s second law, and mean- while they are consumed by the biomass according to Monod kinetics. The mass flux (or effectiveness factor), its values determined by the kinetics of mass transfer and reac- tions, into a Monod biofilm without an external water layer (such as those inside biofilters) can be estimated approximately by using moduli and empirical expres- sions [5 7]. The mass flux into a Monod biofilm with an external water layer, such as those in biotrickling filters and biological wastewater treatment plants, can be estimated using various models based on concepts, such as, the steady-state biofilm and the minimum substrate concentration for the biofilm, which have been developed for describing the biofilm processes [8 10]. In addition, simplified algebraic expressions and pseudo-analytical solutions have been developed for calculating the mass transfer flux to a biofilm [11 14]. Discussions of these biofilm models and their later developments are presented by Chaudhry and Grady [15, 16]. At present, there is no modulus that can be used to calculate the analytical values of the mass flux that is accurate over the entire range of the Monod biofilm model: its order of reaction varies between 0 and 1; the generalized modulus covers an nth ( 1 n ) order of reactions, but not the fractional order of reactions [17]. Values of the flux, calculated by using the current moduli, derived from asymptotic situations and the half-order modulus for soft transition (one-step transi- tion), suffer large errors in the transient region between the first and the zero order of reaction kinetics [18]. Of late, a task group of the International Water Association (IWA) on Biofilm Modeling presented results of applications of various biofilm models, to three benchmark (BM) problems [19]. The first benchmark problem (BM1) described a monospecies biofilm system on a flat substratum in contact with a completely mixed bulk phase [20]. For a flat biofilm, similar values of bulk phase substrate concentration (one-dimensional) and flux were calculated using three methods: the analytical solution, by combining the first-order and half-order solutions; the pseudo-analytical solution, by taking into account the Monod kinetics for the limiting substrate [14]; and numerical simulations [21 23]. When using the ana- lytical method, the modelers had to make decisions on how to combine and average the first and half-order solutions, based on the substrate concentration range [20]. Such decision-making required the expertise of the Monod kinetics and substrate concentration pro- files inside a biofilm. The concentration distribution inside a Monod biofilm can be obtained only from the numerical solu- tion of the Monod biofilm model. This type of a con- centration profile can be deep, shallow, or fully pene- trated [9]. Each of these types can occur with reaction kinetics of zero-, first-, or a variable-order [24, 25]. It is possible that both the zero-order biofilm and the first-order biofilm can be deep, shallow, or fully Received 2007-06-11, accepted 2007-11-19. * To whom correspondence should be addressed. E-mail: fangjames@hotmail.com