## Conduction

Conduction occurs in several places within subsystems of SSF bioreactors:

• within the solid bed (both within the solid and gas phases of the bed);

• across the bioreactor wall, usually treated as occurring only directly from the inside surface to the outside surface of the wall and not along the wall.

The mathematical forms for describing these processes are presented below.

18.3.1 Conduction Across the Bioreactor Wall

The rate of heat transfer across the bioreactor wall (Qcond, J h-1) depends on:

• the difference in temperature between the bed in contact with the wall and the phase on the other side of the wall (°C);

• the area of the wall across which heat transfer is taking place (A, m2);

• the heat transfer coefficient for conduction through the wall, representing the Joules of energy that will be transferred per unit of time per area of wall per degree of temperature difference (i.e., J h-1 m-2 °C-1);

• the heat transfer coefficients for transfer from the bed to the inner surface of the wall and for transfer from the outer surface of the wall to the surroundings (i.e., J h-1 m-2 °C-1).

It is common to treat the three steps in heat removal (that is from the bed to the wall, through the wall, and from the wall to the surroundings) as a single overall process (Fig. 18.1). In this case, the rate of heat transfer is written as:

Qcond = hA (Tbed outer surface Tsurroundings), (18.6)

where h is the "overall heat transfer coefficient". The temperatures are self-explanatory. On the other hand, if the bioreactor wall is treated as a different subsystem, then for transfer from the bed to the inner surface of the wall we write:

Qcondl = hiAi (Tbed outer surface Twall inner surface), (18.7)

where h1 is the heat transfer coefficient between the bed and the inner surface of the wall and A1 is the area of contact between the bed and wall. For transfer across the bioreactor wall we can write:

Qcond2 - h2A2 (Twall inner surface Twall outer surface), (18.8)

where h2 is the heat transfer coefficient for transfer within the material of the bio-reactor wall and A2 is the area of the wall.

In order to describe transfer from the wall outer surface to the surroundings (Qcondi) we would use an term of similar form, but describing convective heat transfer from a surface to a cooling fluid (see Eq. (18.10) in Sect. 18.4.1).

Fig. 18.1. Conductive heat transfer across the bioreactor wall, highlighting that it can be treated as consisting of three individual steps or simply as one overall process. Steps: (1) Heat transfer from the outer surface of the substrate bed to the inner surface of the bioreactor wall; (2) Conduction across the bioreactor wall; (3) Convective heat removal from the outer surface of the bioreactor wall to a well-mixed cooling fluid (air or water)

Fig. 18.1. Conductive heat transfer across the bioreactor wall, highlighting that it can be treated as consisting of three individual steps or simply as one overall process. Steps: (1) Heat transfer from the outer surface of the substrate bed to the inner surface of the bioreactor wall; (2) Conduction across the bioreactor wall; (3) Convective heat removal from the outer surface of the bioreactor wall to a well-mixed cooling fluid (air or water)

### 18.3.2 Conduction Within a Phase

Conduction will also occur within a phase, such as the substrate bed, the head-space gas, or even the bioreactor wall, although the significance of the contribution that it makes to overall heat removal will depend on the presence of other heat removal mechanisms such as convection and evaporation. Conduction will be the dominant mechanism within static beds without forced aeration (Group I bioreac-tors), that is, within the bed within tray bioreactors. In other bioreactors its contribution to heat removal may be relatively minor.

The rate of transfer of heat by conduction within a static phase (Qcond, J h-1) is determined by:

• the temperature gradient in the phase (dT/dx, °C m-1);

• the thermal conductivity of the phase (k, J m-1 h-1 °C-1). This is a property of the material that characterizes how easily it conducts heat, and which will be significantly affected by its composition. In the case of beds of solid particles, it depends on the bed water content, being higher with higher water contents. Note that the bed may be treated as a single pseudo-homogenous phase in which the thermal conductivity is calculated as a weighted average of the thermal conductivities of the solid phase and the inter-particle gas phase;

• the area across which heat transfer is being considered (A, m2). Note that this area term may be cancelled out in the final equation after it is rearranged.

Therefore the term for conductive heat transfer within a phase is given by:

Depending on the design and operation of an aerated bed, conduction within the bed can occur: (1) co-linearly with the air flow (in which case the transfer by conduction will be in the opposite direction to the air flow); (2) normal to the air flow; or (3) in both the co-linear and normal directions (Fig. 18.2). In other words, an energy balance may contain a term that includes dT/dz, a term that includes dT/dx, or two terms, one including dT/dz and the other including dT/dz.

Once there is a temperature gradient, conductive heat transfer will occur. Conversely, if conductive cooling is the only heat transfer mechanism in the bed (i.e., in the case of a static unaerated bed) and the surface is being cooled by heat transfer to the surroundings, then temperature gradients will arise in the bed. As shown in Fig. 18.2, conduction occurs "down" the temperature gradient, hence the minus sign on the right hand side of Eq. (18.9). In other words, the flux of heat is positive in the direction in which the temperature gradient is negative.

During the rearrangements made in simplifying the energy balance for a static bed, Eq. (18.9) is often divided by the volume of the bioreactor (volume being given by an axial distance, z, multiplied by a cross-sectional area, A). This has two consequences: firstly, the area term cancels out and, secondly, the axial distance (z) that is left over combines with the term dz to make the derivative a second-order derivative. That is, the conductive term will often appear as "kd2T/dz2".

air flow conduction normal to the direction of airflow conduction co-linear with the airflow, although in the opposite direction