Convective cooling, that is, cooling by transfer of heat to a moving fluid, which then transports the heat away due to bulk flow, occurs in various situations in SSF bioreactors that we might like to describe within bioreactor models:

• at the bioreactor wall, the removal of heat to flowing water in a water jacket, or to flowing air, which might either be forcefully agitated or be undergoing natural convection;

• at a bed surface in which there is a cross-flow of air;

• within a forcefully aerated bed, in which heat is removed from the solid phase to the flowing air phase between the particles and then removed from that location by the flow of air through the bed.

18.4.1 Convection at the Bioreactor Wall

The rate of heat removal by convection (Qconv, J h-1) at a surface in contact with a fluid depends on (Fig. 18.3):

• the coefficient of convective heat transfer (h, J m-2 h-1 °C-1). This depends on the velocity of the fluid flow because there is a layer of stagnant fluid at the solid surface, and heat transfer through this stagnant layer is limited to conduction. The thickness of the stagnant layer decreases as the flow velocity of the bulk fluid increases; this decreases the resistance to heat transfer and therefore increases the coefficient;

• the area of contact between the surface and the fluid (A, m2);

• the difference in temperature between the surface and the bulk fluid (°C).

That is, for the case where heat is transferred from the outer surface of the bio-reactor wall to cooling water in a cooling jacket, we would write:

Qconv hA (Twall outer surface Twater)• (18.10)

This equation applies if we can assume that the fluid is well mixed and can therefore be represented by a single temperature. The equation will be more complicated if we want to describe how the temperature of a fluid increases as it flows in a unidirectional manner past the surface.

To increase heat removal from the bioreactor wall, it is necessary to increase one or more of the three terms. The heat transfer coefficient can often be increased by increasing the velocity of fluid flow, while the area of contact can be increased by using projections on the wall or a bioreactor geometry that increases the overall wall surface area (for a given bioreactor volume). The driving force for heat transfer (i.e., the temperature difference) can be increased by cooling the water before it is passed through the water jacket.

Fig. 18.3. Convective heat transfer from a surface to a well-mixed flowing phase. (a) The example shown here is for heat transfer from the bioreactor wall to surrounding air or the water in a water jacket. (b) Similar considerations apply for the transfer of heat from the surface of a bed to a passing gas phase

Fig. 18.3. Convective heat transfer from a surface to a well-mixed flowing phase. (a) The example shown here is for heat transfer from the bioreactor wall to surrounding air or the water in a water jacket. (b) Similar considerations apply for the transfer of heat from the surface of a bed to a passing gas phase

18.4.2 Convective Heat Removal from Solids to Air

The rate of heat removal from the solid phase to the gas phase by convection (Qconv, J h-1) depends on (Fig. 18.4):

• the coefficient for heat transfer between the solid particles and the air phase (h, J m-2 h-1 °C-1), the value of which depends on the velocity of the air flow;

• the superficial area of contact between the solids and the air phase (A, m2);

• the difference in temperature between the solids and the air phase (°C).

flowing air

Fig. 18.4. Heat and mass transfer between the solid and gas phases in the case where the solid and gas phases are treated as separate phases flowing air

Fig. 18.4. Heat and mass transfer between the solid and gas phases in the case where the solid and gas phases are treated as separate phases

To describe solid-to-gas heat transfer we therefore write:

Note that the area of contact between the solid and gas phases can be difficult to measure and therefore the product "hA " is often expressed and determined as a global heat transfer coefficient that combines the two quantities ("hA", J h-1 °C-1). It may even be expressed as the overall coefficient per m3 of bed volume (i.e., with units of J h-1 °C-1 m-3-bed).

The amount of heat removed from the solids by convective cooling can be increased by increasing the air flow rate or decreasing the air temperature at the air inlet. Either of these strategies should increase the average temperature difference between the air and solid phases. Also, the higher air flow rate will increase the value of the heat transfer coefficient.

At times the solids and air are assumed to be in thermal equilibrium (this is the assumption of a pseudo-homogeneous bed). Note that this does not necessarily mean that the bed has the same temperature at all positions. It means that the solid particles at any particular position within the bed are at the same temperature as the gas phase at that position. Therefore a single temperature variable can be used to represent the temperature at a given position in the bed. In this case it is not necessary to write an equation describing solids-to-air heat transfer, as this is subsumed in the term that describes the heat removal associated with the flow of gas through the bed.

18.4.3 Convective Heat Removal Due to Air Flow Through the Bed

The rate of heat removal by flow of the air through the bed (Qconv, J h-1) depends on (Fig. 18.5):

• the mass flux of dry air (G, kg-dry-air m-2 h-1), which is given by the superficial velocity of the air (VZ, m h-1) multiplied by the density of the air (pair, kg-dry-air m-3). Of course, the superficial velocity itself is simply equal to the volumetric flow rate (m3-dry-air h-1) divided by the total cross sectional area of the bed (note that this is the total area, not the area occupied by the void spaces);

• the heat capacity of the air (CPair, J kg-dry-air-1 °C-1);

• the difference between the air temperatures at two different locations (°C).

Applied over the whole bed (i.e., in a balance that considers the difference between the air inlet and the air outlet), the rate of heat removal by convection (J h-1) would be given by:

Qconv = Gair CPairAb (Toutlet ~ Tinlet ) > (18.12)

where in this case Ab is the cross sectional area of the bioreactor.

However, in static beds, in which the temperature is a function of height within the bed, it is often of more interest to write an equation that allows the calculation of the temperature as a function of height. In this case, the balance equation is initially written over a thin layer of the bed. Within this equation the convection term will appear as:

dz since the temperature difference (°C) between the inlet and outlet of this thin layer is simply the temperature gradient (dT/dz, °C m-1) multiplied by the thickness of the thin layer (Az, m).

Typically the energy balance equation will be divided through by the volume of the thin layer during later rearrangements, such that in the final equation this term will appear containing neither Ab nor Az. Note also that if the temperature at the outlet of the thin layer is higher than the temperature at the inlet of the thin layer, then convection will be reducing the sensible energy of the thin layer, and therefore this term will be preceded by a negative sign if it appears on the right hand side of an equation such as Eq. (18.1). In fact, it is often put on the left hand side of the balance equation.

Note also that it is often convenient to use the same term to express the contribution of the water vapor to the removal of sensible energy. Given the humidity (H, kg-water kg-dry-air-1) and the heat capacity of the vapor (CPvapor, J kg-vapor-1 °C-1), the term would simply become:

Again, Ab and Az may be cancelled out in the manipulations that are made to arrive at the final equation in the bioreactor model.

Note that convective cooling by the forced aeration of a static bed in which there is continual heat liberation by the growth process will cause temperature gradients in the bed. This phenomenon was explained in Fig. 4.3.

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