In contrast to the two examples discussed above, where mass balances were written for boundary areas, we now need to write the oxygen balance for a thin slice within the biofilm, which is assumed to be smooth and of equal thickness. The oxygen balance is now written as a second-order differential equation (see Problem 7.2):

The first part is the local change of the rate of diffusion inside a very thin plate of thickness dz; and the second part is the rate of oxygen consumption inside this slice.

We need to integrate twice in order to calculate c' = f (z), the oxygen concentration profile inside the biofilm. For each integral, one unknown constant appears and must be determined. Thus, we need two boundary conditions for the two sides of the film (Fig. 7.6, curve IV)

Now, we will use dimensionless numbers:

and:

as well as the Damkohler-II number (Damkohler 1937, Thiele 1939):

r'maxL2 reaction rate

K' D diffusion rate inside the biofilm

In literature, the name Thiele modulus i;Da„ is more common.

Equations (7.6), (7.7a) and (7.7b) can be written using these dimensionless parameters:

Only numerical solutions to Eqs. (7.10) and (7.11) are available. Atkinson and Daoud (1968) were two of the first who published a solution for:

However, these concentration profiles are of limited interest. More important is to know the part of the biofilm which is supplied with oxygen and which is therefore aerobically active as well as to compare this with the highest possible activity:

oxygen uptake rate r'eff

1 maximal oxygen cosumption rate rinax with r| i efficiency coefficient.

The oxygen uptake rate can be calculated using the oxygen concentration profile, Eq. (7.12), and its gradient at z = 0:

D dc' T dz where A is the film surface area and V is the film volume = AL; as well as: Mo'

After introducing Eqs. (7.14) and (7.15) into Eq. (7.13) and considering Eqs. (7.8) and (7.9), we obtain:

Double logarithmic plots are presented in Fig. 7.7, showing the influence of Da„ and Mo'. With increasing Da„ (reaction rate compared to diffusion rate) and decreasing Mo' (dissolved oxygen concentration at boundary liquid/solid), a decreasing part of the biofilm can be supplied with oxygen (Atkinson and Davies 1967; Atkinson and Daoud 1968).

For Mo'P 1, the reaction approximates a first order reaction with: 1 dC'

0.06

0.02

0.01

0.06

0.02

- ^ |
r 1 |
- | |

// | |||

/ y | |||

/ ^^ |
1 1 1 |
60 100 200 Fig. 7.7 Diffusion and reaction inside of a plane biofilm without the influence of mass transfer liquid/solid; Da,, = r'max L2/K' D, Mo' = c*/K' (Atkinson and Daoud 1967). 60 100 200 Fig. 7.7 Diffusion and reaction inside of a plane biofilm without the influence of mass transfer liquid/solid; Da,, = r'max L2/K' D, Mo' = c*/K' (Atkinson and Daoud 1967). r = r and an analytical solution exists (Damkohler 1937; Thiele 1939): tanh DaII DaII which is not influenced by the Monod number. 7.4.6 |

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