Lu = av where L» is the film thickness, v is the withdrawal velocity, and a is a parameter that is dependent on the fluid properties. Since the withdrawal velocity of a point on a rotating disc depends on its radial position, some average velocity should be used, such as:

where r„ and r, are the outer and inner radii of the submerged sector, respectively, as shown in Figure 17.2. This suggests that the average film thickness is given by:

where a, is a coefficient whose value depends on the fluid properties, the size of the disc, and its degree of submergence. Hartman7 has reported a stagnant liquid film thickness of 40 p.m for a smooth rotating disc and Eq. 17.5 predicts a thickness of 60 p,m under the conditions studied. Grieves,4 on the other hand, reported that the thickness of the stagnant liquid layer on top of a rotating biofilm ranged from 50 to 200 p.m and was not reproducible. Such variability is due to the surface film depicted in Figure 15.1. Therefore, it is necessary to add an arbitrary amount to the thickness predicted by Eq. 17.5 to account for the retention of fluid by the surface biofilm. Hence, a more appropriate form might be:

where a2, a,, and a4 must be determined experimentally. This expression for the stagnant film thickness can be substituted into Eq. 15.3 to obtain an expression depicting the effect of rotational speed on the external mass transfer coefficient for the aerated sector, k, .,:

It is important to recognize that kL„ and kUl are influenced differently by the rotational speed of the disc. As Eq. 17.2 shows, the external mass transfer coefficient in the submerged sector will increase as the rotational speed is increased. However, as seen with Eq. 17.7, the external mass transfer coefficient in the aerated sector will decrease.

Once the external mass transfer coefficients have been estimated for each of the sectors, the overall effectiveness factors can be determined from Figure 15.9. They will not be the same, in part because of the differences between k,., and ku. Consequently, ti0Os and ti0O„ are used to denote the overall effectiveness factors for the submerged and aerated sectors, respectively. The substrate consumption rates per unit area of biofilm for each of the sectors are given by Eq. 15.16 with the appropriate values for the overall effectiveness factors.

First let us consider the case of a single bioreactor as shown in Figure 17.2. Taking the liquid volume in the tank as the control volume, V, we see that the substrate is brought in by two streams: the influent flow, F, and the liquid film on the aerated sector of each disc, F,. Two streams also comprise the output: the effluent flow, F,

and the liquid film entrained by the rotating discs, F,. Substrate is removed by the biofilm within the submerged sector and by suspended microorganisms. Thus, the steady-state mass balance equation on substrate in the tank is:

F'Sso + F[ • Ssi.R F • Ssh FL-Ssh "nc()s • X,( ,„ • L, qn - sSh

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