Heat Transfer Coefficients Involving the Wall

If the bioreactor wall is not recognized explicitly as a separate phase in the model, then the heat transfer to the surroundings will be described as a direct transfer from the outer surface to the bed to the surroundings (Eq. (18.6) in Sect. 18.3.1). In this case, the various resistances, that is, for transfer from the bed to the wall, transfer across the wall and transfer from the wall to the surroundings, will be lumped together into an overall heat transfer coefficient hov. In cases where the bioreactor wall is recognized as a separate system, then separate heat transfer coefficients may be needed for terms describing heat transfer from the bed to the wall, from the headspace air to the wall and from the wall to the surroundings.

20.3.1. Bed-to-Wall Heat Transfer Coefficients

The efficiency of the heat transfer from the bed to the bioreactor wall will depend on whether the bed is agitated or static. Further, in an agitated bed, the heat transfer coefficient is likely to depend on the type of agitation.

For rotating drum bioreactors, Stuart and Mitchell (2003) used literature data for non-SSF applications of rotating drum bioreactors to estimate the bed to wall heat transfer coefficient (hbw, J s-1 m-2 °C-1) as:

u 77100

for the drum diameter (D) in meters.

The major source of resistance in transfer from the bed to the wall resides in the bed itself, so the heat transfer coefficient of the bed (hb, J s-1 m-2 °C-1) can be used as an approximation of the bed-to-wall heat transfer coefficient (Oostra et al. 2000). The value of hb will depend on whether the bed is mixed or not, with hb increasing as the intensity of mixing increases (Oostra et al. 2000). They estimated hb as:

where kb is the thermal conductivity of the bed (W m-1 °C-1), pb is the bed density (kg m-3), CPb is the bed heat capacity (J kg-bed °C-1), and tc is the time of contact between the solid particles and the wall (s), which is inversely proportional to the rotational speed of the agitator. In a static bed, a value of hb of 40 J s-1 m-2 °C-1 was determined experimentally for moist oats (Schutyser et al. 2004).

However, it is often not a simple matter to determine hb and, instead of doing this, the overall heat transfer coefficient for transfer from the bed to the outside is often determined experimentally (see Sect. 20.3.4).

20.3.2 Wall-to-Headspace Heat Transfer Coefficients

In most bioreactors heat transfer between the wall and the headspace gases is not described within mathematical models because the air has already left the bed and will leave the bioreactor without any further interaction with the bed. The situation is different in rotating drum bioreactors, in which the headspace air interacts with the bed as it travels along the drum (see Sect. 20.5). This headspace air also interacts with the drum wall. Once again, this transfer has received little attention in SSF bioreactors.

Stuart and Mitchell (2003) derived the following equation for the wall-to-headspace heat transfer coefficient (hwg, J s-1 m-2 °C-1) from the studies undertaken in a rotary kiln by Tscheng and Watkinson (1979):

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