Since in a tray bioreactor air is not blown forcefully through the trays, O2 and CO2 can only move within the bed by diffusion. Potentially, due to the temperature gradients that arise, there could be natural convection within the void spaces within the bed, although this has not been studied. This discussion will focus on O2. Similar considerations apply to CO2, although it will typically be diffusing in a direction opposite to that of O2.

The limitation of O2 movement in the bed to diffusion through the void spaces, coupled with its simultaneous uptake by the microorganisms at the particle surfaces, leads to the establishment of O2 concentration gradients within the void spaces (Fig. 6.2). Rathbun and Shuler (1983) noted O2 gradients within the gas phase of a bed of tempe (which involves the cultivation of the fungus Rhizopus oligosporus on cooked soybeans) of the order of 2% (v/v) cm-1. This represents a drop equal in magnitude to 10% of the gas phase O2 concentration in air (~21% (v/v)) over 1 cm. Of course, the exact shape of the spatial O2 concentration profile will depend on whether the bottom of the tray is perforated or not and the rate at which O2 is being consumed by the organism.

Ragheva Rao et al. (1993) proposed an equation to estimate the maximum depth that a tray could be in order for the O2 concentration not to fall to zero at any part in the tray during a fermentation. They referred to this depth as the critical depth (Dc, cm):

l2DCYxn dc =J(6J)

v rxm top open but bottom not perforated top open and bottom perforated

Fig. 6.2. O2 concentration gradients within trays. Note that the spatial gradients will change over time, depending on the rate of growth of the microorganism. In the bioreactor with an unperforated bottom, O2 limitation will be the factor that has the greatest influence on growth at the bottom of the bed (the area indicated by the dotted circle) since heat removal through the bottom of the bed will control the temperature in this region reasonably well

Fig. 6.2. O2 concentration gradients within trays. Note that the spatial gradients will change over time, depending on the rate of growth of the microorganism. In the bioreactor with an unperforated bottom, O2 limitation will be the factor that has the greatest influence on growth at the bottom of the bed (the area indicated by the dotted circle) since heat removal through the bottom of the bed will control the temperature in this region reasonably well

In Eq. (6.1) YXO is the yield coefficient of biomass from O2 (g-dry-biomass g-O2"1), C is the O2 concentration in the surrounding atmosphere (g cm-3), D is the effective diffusivity of O2 in the bed (cm2 h-1), and RXM is the maximum growth rate (g-dry-biomass cm-3-bed h-1). Ragheva Rao et al. (1993) estimated D as 0.03 cm2 s-1 and YXO as 1.07 g-dry-biomass g-O2-1. In dry air at 25°C and 1 atm pressure, C will be 2.7x10-4 g cm-3. Using experimental estimates for RXM, they concluded that the critical depth would be of the order of 2.4 cm. For a tray with a perforated bottom, oxygen can penetrate this distance from both the top and bottom surfaces, meaning that the bed depth in the tray can be twice the critical bed depth, namely 4.8 cm. This can be taken as a typical value for trays, although of course the exact value is influenced not only by the growth rate but also by the effective diffusivity of O2 within the bed, which will decrease as the biomass grows into the inter-particle spaces during the fermentation.

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