It is apparent by now that a packed tower is a very involved biochemical operation that is difficult to model theoretically. Consequently, empirical and semiempirical approaches have been used to model it and a number of design equations have been developed. The NRC"' and Gollar-Gotaas1 equations are strictly empirical because they are based only on the application of regression analysis techniques to experimental observations. Consequently, they are valid only in the limited region over which the original data were available and can only be used for interpolation within that region. They cannot be used for extrapolation. Semiempirical models are also primarily interpolative, but can be successfully employed for extrapolation if the extrapolations do not extend them beyond the range over which their simplifying assumptions are valid. Finally, a truly mechanistic model can be used for extrapolation to new conditions, but care must still be exercised to be sure that extrapolations do not violate the basic assumptions of the model. In order to recognize the assumptions inherent in them we shall review some semiempirical and mechanistic models.

The model of Velz2' states that the substrate concentration decreases in an exponential manner with depth, which is a characteristic we observed in Figure 16.2:

where K is a constant. Comparison of Eq. 16.9 with Eq. 16.6 shows that the Velz equation is limited to the situation in which the removal of substrate by suspended microorganisms is negligible, the substrate consumption rate is first order with respect to the substrate concentration (Ssb « Ks), and the overall effectiveness factor is constant throughout the tower. Under these circumstances Eq. 16.6 can be integrated to yield:

|5£ = exp{-[(q„/Ks)aA-tlc..XB.„,Lf/F(l + ot)]L}

from which we see that:

K = (qH/Ks)aA-Ol(»X,,n,L,/F(l + «) = (q„asTicl)XI1,IfLr)/(KsAll) (16.11)

Thus, the Velz equation is limited to one THL rate.

16.4.2 Eckenfelder Model

Eckenfelder has explicitly accounted for the effects of flow rate:

where m and n are parameters that are dependent on the specific media used and K, is a rate coefficient. This equation is a modification of the Velz equation in which the effluent concentration is allowed to depend explicitly on F, a, and as. Therefore, we may look upon it as a limiting case of Eq. 16.6 in which the removal of substrate by suspended microorganisms is negligible, the substrate consumption rate is first order with respect to substrate (SSi, << Ks), and the overall effectiveness factor remains constant throughout the tower but varies with the THL, F(1 + a)/Av. The last point becomes clear if we rearrange Eq. 16.12:

and compare it to Eq. 16.10. Thus, if TqcC, is proportional to [A^/Fil -I- a)]" 1 they are equivalent for a given media. According to Liptak,1" most media have an n value of 0.7 to 0.8. Consequently, a 28 fold increase in THL would result in a 1.95 to 2.72-fold increase in [A,./F(l + en)]" '. Likewise, for the values of Bi, <|r< and 4>t likely to be found in a packed tower, a 28-fold increase in THL will increase the overall effectiveness factor by a factor of 2.5. Thus, it appears that the changes in [A^./F(l + a)]" 1 in the Eckenfelder model are similar to the changes in the overall effectiveness factor caused by flow.

16.4.3 Kornegay Model

The differential equation used by Kornegay'' is:

where K? depends on the substrate and type of media, and K,, is a pseudo halfsaturation coefficient which decreases asymptotically to Ks as the THL becomes very large. Equation 16.14 may be rewritten as:

where Ks is the intrinsic half-saturation coefficient which remains invariant for a particular microorganism and substrate. If there is no substrate removal by suspended microorganisms, Eq. 16.6 reduces to:

in which K, depends on the substrate and the type of media. Comparison of Eq. 16.15 with Eq. 16.16 reveals that they are equivalent for a given media if:

Let us examine the requirement on -qLl, in more detail. Inspection of Eq. 16.17 reveals that the overall effectiveness factor: approaches unity as the substrate concentration becomes very large; approaches unity when the THL becomes very large so that Kk approaches K.,; decreases as the substrate concentration decreases, approaching an asymptotic value of Ks/K,. as the substrate concentration approaches zero; and decreases as the THL is decreased, causing K,. to increase. These trends are consistent with those predicted from Figure 15.9. Thus, it appears that the Kornegay model can be interpreted as a limiting case of Eq. 16.6 in which substrate removal by suspended microorganisms is negligible and the overall effectiveness factor is given by Eq. 16.17.

16.4.4 Schroeder Model

Sehroeder's'" model states that:

where is the internal effectiveness factor. Comparison of Eq. 16.18 with Eq. 16.6 reveals that the Schroeder equation is a limiting case in which both the removal of substrate by suspended microorganisms and the external mass transfer resistance are assumed to be negligible. The assumption of negligible external mass transfer resistance may only be valid when the THL is very high. Furthermore, Schroeder assumed that the internal effectiveness factor is directly proportional to the bulk substrate concentration. Figures 16.2 and 16.6 suggest, however, that this assumption is a poor one.

Logan et al.'4 developed a model that attempts to account for the hydrodynamic properties of different packed tower media. The model assumes that the fluid flows in thin films that is generally laminar, allowing a parabolic velocity profile to be established. Transport of a single limiting substrate is assumed to occur across that liquid film in the absence of reaction in the film, i.e., there is no substrate removal by suspended microorganisms. The flux into the biofilm is given by an empirical expression that assumes first order kinetics and Brownian collisions between the substrate molecules and the cells." Although the rationale for the flux equation is not entirely clear, the model essentially assumes that substrate removal is limited by transport to the biofilm surface and not by the biofilm kinetics.1' In other words, only external mass transport is assumed to be limiting, which is consistent with the experimental observations of others." The unique feature of this model is that it considers the hydraulic characteristics of the media. In some types of media, the fluid is assumed to flow in thin films throughout the tower depth, allowing stable velocity and concentration profiles to be established. In others, however, intermixing is assumed to occur at regular intervals because of the geometry of the media, disrupting the velocity and concentration profiles at that point, thereby requiring them to be reestablished from the average concentration after intermixing. Although simultaneous transport of substrate and oxygen is not considered and only a single substrate is assumed to limit reaction, the capability is provided to allow computation of the maximum possible oxygen transport rate in the system. A computer code for the model suitable for microcomputers can be downloaded from the World Wide Web. Links to it can be found in the website of the Association of Environmental Engineering Professors, http://bigmac.civil.mtu.edu/aeep.html.

Hinton and Stensel" developed a model that considers transport of both substrate and oxygen into a biofilm, as well the effects of hydrodynamics in the tower. Consumption of substrate within the biofilm is conceptualized as a one-dimensional diffusion mass transport process with dual substrate limited kinetics. Rather than Monod kinetics, however, the model uses Blackman kinetics, which treats the reaction as being either first-order or zero-order with respect to the substrate and oxygen concentrations. The biofilm is assumed to be deep, with the active depth being determined by exhaustion of either the substrate or oxygen. The hydraulic characteristics of the tower are described as laminar liquid film flow which is interrupted at regular intervals by falling liquid drops. The interruptions provide mixing so that a uniform average concentration is established at the start of each laminar flow zone. Overall tower performance is determined by repeatedly solving the model equations for the short laminar flow sections and combining the results of many sections. The average section length is similar to the distance between mixing zones in the model of Logan et al.,'J so although the models are conceptually different, their portrayals of the system hydrodynamics are similar in effect. Substrate transport is all through the laminar liquid film, but oxygen transport occurs both through the liquid film and through direct contact of the biofilm with the bioreactor gas phase when flow is shifted between paths. Predictions from the model were in agreement with experimental observations in the lab and in pilot plants."

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