No modeling case study will be presented for trays in this book and therefore this section will discuss the insights that dynamic mathematical models of tray biore-actors have given into the relative importance of temperature and O2 limitations in controlling the performance of trays.

Rajagopalan and Modak (1994) developed a model to describe heat and mass transfer in trays, which included the various processes shown in Fig. 6.4. They used their model to investigate the relative importance of high temperatures and low O2 concentrations in determining the specific growth rate in a 6.4-cm-high tray. Since the tray was assumed to be symmetrical around the center plane it was only necessary to consider a depth of 3.2 cm from the surface to the central plane. Their results are shown in Fig. 6.5. In interpreting these results, it must be remembered that the overall growth rate (RX, kg-dry-biomass m-3 h-1) is a combination of the biomass density (X, kg-dry-biomass m-3) and the specific growth rate according to the following equation:

where Xmax (kg-dry-biomass m-3) is the maximum possible value of the biomass density. The specific growth rate constant n (h-1) is affected by both temperature and the biofilm O2 concentration according to the relationship:

In this equation, fu^ (h-1) is the maximum value that the specific growth rate constant can have, that is, its value under optimal conditions for growth. On other hand, ¡uFT and ¡uFO are dimensionless fractions, that is, they vary between 0 and 1. The parameter juFT describes the limitation of growth by deviations from the optimum temperature of 38°C while ¡uFO describes the limitation of growth at low O2 concentrations.

bulk flow of O2 and CO2 with the headspace gases

J i exchange of O2 and CO2 between the surrounding gas phase and the void spaces within the bed

A convective heat removal from the bed surface ro the organism a transfer of O2 a CO2 between t void spaces an particle surface oc diffusion of O and CO2 in th void spaces

A convective heat removal from the bed surface the organism a transfer of O2 a CO2 between t void spaces an particle surface diffusion of O and CO2 in th void spaces

no heat or mass transfer through the center plane of the bed, around which the model is symmetrical

Fig. 6.4. Heat and mass transfer processes described by the model of Rajagopalan and Mo-dak (1994). Note that for simplicity, it was assumed that the whole biofilm was at the same O2 concentration, although in reality there would be an O2 concentration gradient due to the simultaneous diffusion and consumption of O2

A key prediction of this modeling work of Rajagopalan and Modak (1994) is that limitation of growth due to lack of O2 occurs even though the gas phase O2 concentration never falls to very low values; in their simulations the gas phase O2 concentration was always two-thirds or more of the O2 concentration in the surrounding atmosphere, regardless of time or depth (Fig. 6.5(b)). Since the organism within the biofilm can consume O2 much faster than the rate at which O2 can transfer from the gas phase to the biofilm, biofilm O2 concentrations can fall to low levels (Fig. 6.5(d)). This occurs at the top of the bed where, due to the fact that the temperature remains near to the optimum for growth since it is effectively cooled by the surrounding atmosphere (Fig. 6.5(a)), the organism grows rapidly and consumes the O2 in the biofilm, reducing it to levels that significantly decrease the specific growth rate. In this case, the growth of the biomass is controlled by the rate at which O2 is transferred to the biofilm ((Figs. 6.5(e) and (f)).

The highest temperatures occur at the central plane of the bed, and these are sufficiently high to decrease the specific growth rate significantly (Fig. 6.5(c)). Indeed, due to the fact that the high temperatures cause low growth rates in this

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