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Rathbun and Shuler (1983) found temperature gradients as high as 1.7°C cm within a static bed of tempe, while Ikasari and Mitchell (1998) measured temperatures as high as 50°C at 5 cm depth during the cultivation of Rhizopus oligosporus on rice bran in a tray within a 37°C incubator.

Szewczyk (1993) derived a simplified equation that can be used to describe the temperature profile within a tray bioreactor, from the central plane (z = 0) to the surface (z = 1), when the top and bottom half of the tray are identical, that is, in the situation shown in Fig. 6.3(a):

t =(T + 273) + 0 + N»T + 273) {ts - ra + e> - z 20, (6 2)

where Ts and Ta are the temperatures of the bed surface and surrounding air (°C), respectively. The spatial coordinate z is expressed as a dimensionless fraction of the total bed height (Z, m). NBi is the Biot number, given by a.Z/k where a is the heat transfer coefficient for bed-to-air heat transfer at the top of the bed (W m-2 °C-1), and k is the thermal conductivity of the bed (W m-1 °C-1). Finally, the symbol © represents the temperature difference that would occur between the bottom of the solid bed and the tray surface if there were no heat transfer through the bottom of the tray. It is given by the following equation:

RQZ 2

2k where Rq is the volumetric heat production rate (W m-3). It is not simple to apply Eq. (6.2), since the surface temperature of the tray needs to be known. The surface temperature depends on the value of the heat transfer coefficient a, but a also appears in Eq. (6.2), within NBi. A more complex modeling approach is needed to relate Ts and a. Szewczyk (1993) used such a model to derive the relationship

(b)

bottom

top height within the tray

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