Because of the complex nature of an RDR, certain simplifying assumptions must be made to model it. The first is that steady-state conditions prevail so that microorganisms are sheared from the surface of the biofilm at a rate equal to their growth. Thus, the biofilm behaves as a steady-state biofilm of known thickness. That thickness is an input parameter and is not computed in the model. The second assumption is that the turbulence in the bioreactor fiuid is sufficient to keep the detached biomass in suspension so that it can be washed out with the effluent. The third is that both the attached and detached microorganisms contribute to substrate removal. The fourth is that oxygen and other nutrients are present in excess so that the organic substrate is the growth limiting nutrient. In other words, dual limitation by both the electron donor and the electron acceptor in the biofilm is not considered. The fifth is that the thickness of the liquid film is uniform over the aerated sector of the disc. The final assumption is that the substrate concentration in the liquid film on the aerated sector depends only on the circumferential angle 0 and not on the radial position. In other words, the liquid film in the aerated sector is treated as a plug flow reactor on top of the biofilm.

The effectiveness factor approach to modeling transport and reaction was used by Grady and Lim.' Consequently, they incorporated the effects of both internal and external mass transfer resistances into an overall effectiveness factor as discussed in Section 15.2.2. Although the surface of a disc is generally not flat, the diameter of the undulations is large in comparison to the thickness of the active biofilm so that the effectiveness factor for a flat biofilm should be applicable. However, because of the differences between the submerged and the aerated sectors of the disc, separate effectiveness factors are required for each.

As shown in Figure 17.2, each disc can be divided into two sectors, submerged and aerated. Because the biofilm is attached to the disc, it moves through the bulk fluid in the submerged sector, thereby making the external mass transfer coefficient, kI s, a function of the rotational speed, w. As a point on the surface of the disc leaves the submerged sector and enters the aerated sector, a thin film of liquid adheres to

it and is carried along with it. Although this film can be assumed to have no motion relative to the biofilm on the disc, its thickness is a function of the rotational speed of the disc and consequently, the mass transfer coefficient for the substrate within it, k, ā also depends upon the speed.

Mass transfer from a fluid in laminar flow to the surface of a submerged rotating disc was analyzed by von Karman and given by Levich:"'

where k,, is the submerged external mass transfer coefficient, r is the radius of the disc, Dā is the diffusivity of the substrate in water, is the fluid density, is the fluid viscosity, and w is the rotational speed of the rotating disc. Equation 17.1 indicates that the external mass transfer coefficient will increase with the square root of the rotational speed. In practice, however, both the proportionality constant and the power on the rotational speed may be different due to deviations from the assumptions made in deriving the equation. Nevertheless, it is possible to obtain correlations of the form:

where e and f are coefficients with e depending on the physical properties of the fluid and radius of the disc.

In the aerated sector, the entrained fluid forms a stagnant layer on top of the biofilm. Thus, it will be assumed that the external mass transfer coefficient in the aerated sector, k, ā is equal to the diffusivity divided by the thickness of the stagnant liquid film as given by Eq. 15.3. Thus, quantification of that mass transfer coefficient requires knowledge of the liquid film thickness. The thickness of the liquid film entrained on a fiat plate withdrawn vertically from a quiescent liquid has been analyzed by Landau and Levich" and found to be:

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