Multiphase Bioreactor Design

Diffusion-Reaction Model: Calculation of Substrate Consumption

Rate for a Single Substrate and a Single Microbial Species in Steady-

state Biofilms

The model will be described for the case of single limiting substrate and single microbial
species in the biofilm, and the latter will be supposed to be in “steady state conditions”.
By this, it is meant that the amount of attached biomass, its thickness and the rate of
substrate consumption in the biofilm are constant with respect to time. Moreover, the
equations will be applied to describe the case of a flat biofilm, that is, a microbial layer
attached to one side of a flat particle or thin biofilms attached to particles of other shapes;
here, the meaning of “thin” depends on the relative dimensions of the biolayer and the
support particle: a biofilm may be considered “thin” if its thickness is smaller than
roughly 30–50% of the radius of the support particle (supposing the latter is spherical).
The model is derived from the well-known heterogeneous catalysis approach in chemical
engineering (Froment and Bischoff, 1979), and its application to enzyme reactors and
wastewater treatment biofilm reactors has been fully reported by several authors in the
last decades (Harremöes, 1978; Cabral and Tramper, 1994; Harremöes and Henze, 1995).
Let rf be the reaction rate inside the biofilm, that is, the substrate consumption rate per
unit volume of wet biofilm (kg.m−^3 .s−^1 ), y the distance inside the biofilm, measured from
the liquid-biofilm interface, and J the substrate flux through the biofilm, referred to the
unit area of a microbial layer attached to a supposedly flat surface (kg.m−^2 .s−^1 ). A mass
balance to the substrate across an element of thickness dy inside the biofilm results in:

(1)

Assuming unidimensional mass transfer in the biofilm, Fick’s law will be written as:

(2)

where Sf is the substrate concentration inside the biofilm at a distance y from the biofilm-
liquid interface, and Df is the effective diffusion coefficient (also called “effective
diffusivity”) of the substrate in the microbial layer. This coefficient may not be equal to
the molecular diffusivity of the same compound in the liquid phase, on account of the
tortuosity and porosity of the biofilm and of the fact that convective flow (and not only
molecular diffusion) may also take part in the transport of substrates inside the microbial
matrix (Stoodley et al, 1994).
The steady-state diffusion-reaction differential equation is obtained from Equations (1)
and (2):

(3)

Once the expression for the biological reaction rate (rf) is known, the next problem is the
integration of Equation (3). In the present case, an equivalent of the traditional Monod
formula will be used to describe the kinetics of biomass production and substrate
consumption in the biofilm. It should be stressed that the Monod model was developed

Multiphase bioreactor design 302