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19.4.2 Heat Capacity of the Substrate Bed

The heat capacity of the substrate bed will appear in energy balance equations since it relates the amount of energy stored within the bed to the temperature of the bed. Typically the model will be written in terms of the heat capacity at constant pressure, even though at times the heat capacity at constant volume is the more correct term. For a bed of solid particles there will be little difference between these two heat capacities.

The heat capacity of the bed depends on the heat capacities of its various components, namely the dry solid, the liquid water, the dry air, and the water vapor. However, it is not simply the arithmetic mean of these heat capacities. It must be calculated as a "mass-weighted average".

Firstly, the heat capacity of the moist solids (CPs, J kg-wet-solids-1 °C-1) is a weighted average of the heat capacities of the dry solids (CPd, J kg-dry-solids-1 °C-1) and the liquid water (CPw, J kg-water-1 °C-1):

CPd + WCPw

where W is the water content of the solids on a dry basis (kg-water kg-dry-

solids-1). Likewise, the heat capacity of the gas phase is a weighted average of the heat capacities of the dry air (CPa, J kg-dry-air-1 °C-1) and the water vapor (CPw, J kg-vapor-1 °C-1):

where H is the air humidity (kg-vapor kg-dry-air-1). The overall heat capacity of the bed (CPb, J kg-bed-1 °C-1) is a weighted average of these two heat capacities.

msCPs + mgCPg

mb where mb is the overall mass of the bed (kg), equal to the sum of ms and mg.

The heat capacities of liquid water, dry air, and water vapor can be found in various books (see the further reading section at the end of the chapter). All three are functions of temperature. The heat capacity of liquid water (J kg-1 °C-1) is given by (Himmelblau 1982):

CPw = 1015.56 + 26.206(7 + 273) - 0.0743117(T + 273)2 + 7.2946x10~5(T + 273)3 . (19.24)

The heat capacity of dry air (J kg-1 °C-1) is given by (Himmelblau 1982):

CPa = 997.9 + 0.143T - 0.00011T2 - 6.776 x10T3 . (19.25)

The heat capacity of water vapor is given by (Himmelblau 1982):

CPv = 1857 + 0.382T -0.0004221T2 -1.994x!0~7T3 . (19.26)

In all three cases the temperature is in °C. However, the influence of temperature is likely to be small over the temperature range experienced during a fermentation, and therefore heat capacities determined for a temperature in the middle of the expected temperature range can be used. Reasonable values are 1006 J kg-1 °C-1 for Cpa, 1880 J kg-1 °C-1 for Cpv, and 4184 J kg-1 °C-1 for Cpw.

Of the various heat capacities, that of the dry solids is the most difficult to obtain. You can use the following equation to estimate the heat capacity if you know the composition of the substrate and have available literature data for the heat capacities of the various components:

where wt is the fraction of the total mass of the substrate (or "mass fraction") contributed by component number "i" and CPi is the heat capacity of that component. Since the components present in solid substrates used in SSF will typically be similar to the components of foodstuffs, books that tabulate the heat capacities of food components may be used (e.g., Sweat 1986; Hallstrom et al. 1988).

It is also possible to estimate Cpb experimentally. In this case it would be necessary to transfer a known quantity of energy into the bed and then measure the temperature rise. It would be best to do this in a bomb calorimeter. Although this will give CVb, the difference between CVb and CPb will be negligible. Schutyser et al. (2004) determined the heat capacity of oats experimentally as 2300 J kg-dry-solids-1 °C-1.

Note that the heat capacity of the bed can also change as the microorganism grows, given that the microbial biomass has a different composition and water content than the substrate particle itself. Little attention has been paid to this in bioreactor models.

19.4.3 Enthalpy of Vaporization of Water

This is an important property because of the importance of evaporation as a heat removal mechanism. It describes the enthalpy change in the process:

The enthalpy of vaporization of water (2, J kg-H2O-1) depends on the temperature, but over the range of temperatures that might be expected in a fermentation, this variation is not large. For example, the enthalpy of vaporization of water is 2438.4 kJ kg-1 at 27°C and 2389.8 kJ kg-1 at 47°C (Himmelblau 1982), which represents a decrease of only 2%. An average of 2414 kJ kg-1 would be appropriate for most situations.