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Figure 15.7 Traditional conceptualization of a base biofilm showing an idealized concentration profile for a single limiting nutrient.

Two approaches are commonly used to model external mass transfer. One assumes that transport across the liquid layer is by molecular diffusion, with diffusivity In that case, the thickness L„ is defined as the equivalent depth of liquid through which the actual mass transfer can be described by molecular diffusion alone. Consequently, the flux, Js, or mass of substrate transported per unit area per unit time, is given by:

Because the diffusivity is an intrinsic characteristic of the material being transported (the fluid is assumed to be water), L„ becomes the parameter that must be evaluated before Eq. I5.l can be used to depict the rate of transport of the substrate to the biofilm. Its value must be deduced from Eq. 15.1 using measured fluxes coupled with known diffusivities and concentration gradients. The second approach employs a liquid phase mass transfer coefficient, k,, that incorporates all of the effects of diffusive and advective mass transfer into one parameter. In that approach:

The value of kL must also be deduced from measured fluxes and concentration gradients. It is apparent from comparison of Eqs. 15.1 and 15.2 that:

Thus, measured values of k, may be used to estimate and vice versa. The value of k, (and L„) will depend on the properties of the fluid (such as its viscosity, and its density, pw), the diffusivity of the substrate in the fluid, and the nature of the turbulence, which can be represented in part by the bulk fluid velocity past the biofilm, v. Figure 15.817 illustrates how that velocity influences the gradient in the

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