Gas phase energy balance (Tg)

g !Vbed

Solid phase (s)

Solid phase energy balance (Ts)

g !Vbed

Solid phase water balance (W)

Fig. 22.2. Summary of the model of a well-mixed, forcefully-aerated bioreactor. The variable shown in parentheses after the heading in each text box is the variable that is isolated in the differential term on the left hand side of the equation before the model is solved. Subscripts: s = solids phase, g = inter-particle gas phase, sat = saturation, a = surroundings, b = bioreactor wall. The meanings of the symbols representing key system variables are explained in Fig. 22.1

flow of dry air and water vapor (Fn = kg of dry air entering the bed per second)

Fig. 22.2. Summary of the model of a well-mixed, forcefully-aerated bioreactor. The variable shown in parentheses after the heading in each text box is the variable that is isolated in the differential term on the left hand side of the equation before the model is solved. Subscripts: s = solids phase, g = inter-particle gas phase, sat = saturation, a = surroundings, b = bioreactor wall. The meanings of the symbols representing key system variables are explained in Fig. 22.1

= A exp (-A /(R(TS - (Topt - 38) + 273.15))) Mft 1 + A3 exp (-A4 /(R(TS - (Topt - 38) + 273.15)))' (221)

where Topt is the optimum temperature for growth (°C). The curve described by Eq. (22.1) is shown in Fig. 22.3(a), along with the values of A1 to A4 that describe this curve.

To describe the effect of water activity, the following equation is used to calculate a fractional growth rate, iuFW (i.e., ¡uW/^opt):

where Di to D4 are fitting parameters. The model allows for the selection of two different types of water relations, namely either an Aspergillus-type water relation, in which growth is optimal at a water activity of 0.95, or a Rhizopus-type water relation, in which growth is optimal at a water activity of 1.0. Figure 22.3(b) shows these two relations and gives the values of D1 to D4 used to describe them.

The value of n for the logistic equation is then calculated as the product of the geometric mean of the fractional growth rates and the value of the specific growth rate constant under optimal conditions (^oph h-1):

Fig. 22.3. How the effects of temperature and water activity on growth are modeled. The Y-axis represents the fraction of the value of the specific growth rate for optimal conditions. (a) Effect of temperature. This curve is described by Eq. (22.1) with Al=8.31174x1011' A2=70225 J mol-1, A3=1.3x1047, and A4=283356 J mol-1. Adapted from Saucedo-Castaneda et al. (1990) with kind permission from John Wiley & Sons, Inc. (b) Effect of water activity. Both curves are described by Eq. (18.2). (—) Aspergillus-type water relation, for which D1=618.9218, D2=-1863.527, D3=1865.097, and D4=-620.6684; (- - -) Rhizopus-type water relation, for which D1=-131.600, D2=94.9959804, D3=214.219223, and D4=-177.66756. Based on experimental data of Glenn and Rogers (1988) (see Fig. 16.6)

Fig. 22.3. How the effects of temperature and water activity on growth are modeled. The Y-axis represents the fraction of the value of the specific growth rate for optimal conditions. (a) Effect of temperature. This curve is described by Eq. (22.1) with Al=8.31174x1011' A2=70225 J mol-1, A3=1.3x1047, and A4=283356 J mol-1. Adapted from Saucedo-Castaneda et al. (1990) with kind permission from John Wiley & Sons, Inc. (b) Effect of water activity. Both curves are described by Eq. (18.2). (—) Aspergillus-type water relation, for which D1=618.9218, D2=-1863.527, D3=1865.097, and D4=-620.6684; (- - -) Rhizopus-type water relation, for which D1=-131.600, D2=94.9959804, D3=214.219223, and D4=-177.66756. Based on experimental data of Glenn and Rogers (1988) (see Fig. 16.6)

The balance on the overall mass of dry solids (i.e., the sum of dry biomass and dry residual substrate) is necessary since not all the consumed substrate is converted into biomass; a proportion is lost in the form of CO2. This is Eq. (16.11), although without the maintenance term:

dM dt v yxs dX

In the gas phase water balance presented in Fig. 22.2, all terms have units of kg-H2O h-1 and:

• the left hand side represents the temporal variation in the amount of water vapor in the air phase within the bed;

• the first term on the right hand side represents the entry of water vapor with the inlet air and the leaving of water vapor with the outlet air;

• the second term on the right hand side represents the water exchange between the solid and gas phases.

In the gas phase energy balance presented in Fig. 22.2 all terms have units of J h-1 and:

• the left hand side represents the temporal variation in the sensible energy of the dry air and water vapor in the air within the bed;

• the first term on the right hand side represents the sensible energy of the dry air entering and leaving the bed in the process air stream;

• the second term on the right hand side represents the sensible energy of the water vapor entering and leaving the bed in the process air stream;

• the third term on the right hand side represents the sensible heat exchange between the solid phase and the gas phase.

• the fourth term on the right hand side represents the sensible heat exchange between the gas phase and the bioreactor wall, using the void fraction (s) as an estimate of the fraction of the total wall area in contact with the gas phase;

In the solid phase water balance presented in Fig. 22.2 all terms have units of kg-H2O h-1 and:

• the left hand side represents the temporal variation in the water content of the solids phase;

• the first term on the right hand side represents metabolic water production;

• the second term on the right hand side represents the exchange of water between the solid and gas phases.

In the solid phase energy balance presented in Fig. 22.2 all terms have units of J h-1 and:

• The left hand side represents the temporal variation of the sensible energy within the solids phase;

• the first term on the right hand side represents the liberation of waste metabolic heat in the growth process

• the second term on the right hand side represents sensible energy exchange between the solids and the gas phase;

• the third term on the right hand side represents the removal of energy from the solid as the latent heat of evaporation or addition of energy to the solid due to condensation, depending on the direction of water transfer;

• the fourth term on the right hand side represents the sensible energy exchange with the bioreactor wall, using (1-£-) as an estimate of the fraction of the total wall area in contact with the solids phase;

In the energy balance over the bioreactor wall presented in Fig. 22.2 all terms have units of J h-1 and:

• The left hand side represents the temporal variation of the sensible energy within the wall;

• the first term on the right hand side represents sensible energy exchange between the wall and the gas phase within the bed, using the void fraction (£•) as an estimate of the fraction of the total wall area in contact with the gas phase;

• the second term on the right hand side represents sensible energy exchange between the wall and the solids phase within the bed, using (1-^) as an estimate of the fraction of the total wall area in contact with the solids phase;

• the third term on the right hand side represents sensible energy exchange between the wall and the water in the water jacket.

Several of the assumptions that this model makes are:

• the bed volume does not change during the fermentation. The effect of consumption of dry matter is to decrease the packing density of the bed.

• the fermenting solids have the same isotherm as the substrate itself.

• only the side walls of the bioreactor are available for heat transfer to the cooling water in the jacket. Further, the whole of this side wall has the same temperature.

• The bed porosity (s) does not change during the fermentation. Further, in choosing a value for s, it is assumed that the porosity is higher than that for a normal packed bed due to the continuous mixing action.

• there is no maintenance metabolism.

• there is no microbial death. The sole effect of high temperatures is to limit the growth rate.

Further, no special attempt is made to describe the deleterious effects of mixing on the growth of the organism. The value of the optimal specific growth rate constant (Mopt) used in the model therefore should be an experimental value obtained in a continuously-mixed system.

22.2.2 Values of Parameters and Variables

Tables 22.1 and 22.2 show the values used in the base case simulation for the various parameters and variables in the model.

The coefficients for heat transfer from (1) the gas and the solid phase to the bioreactor wall and (2) the bioreactor wall to the water in the water jacket were chosen as 30 W m-2 °C-1 in order to give an overall coefficient for heat transfer from the bed to the water in the water jacket (calculated on the basis of the law of resistances in series) of the order of magnitude of 15 W m-2 °C-1, a value calculated from data provided by Nagel et al. (2001a) for a glass-walled laboratory-scale bioreactor (see Sect. 20.3.1).

Note that the side wall area (A, m2) and the bed volume (Vbed, m3) are calculated on geometric principles for an upright cylinder of circular cross-section.

The mass balance part of the model calculates the water content of the solids (W, kg-H2O kg-dry-solids-1), whereas in the growth kinetic part of the model the specific growth rate constant is expressed as a function of the water activity of the solids phase and not its water content (see Eq. (22.2)). The isotherm determined for corn by Calgada (1998) is used to convert the water content into the corresponding water activity (see Eqs. (19.8) and (19.9) in Sect. 19.2.5).

This isotherm is also used in the calculation of the evaporation term. The driving force for evaporation is the difference between the water content of the solids (W, kg-H2O kg-dry-solids-1), given by Eq. (19.9), and the water content that the solids would have if they were in equilibrium with the gas in the headspace phase (Wsat). To calculate Wsat, Eq. (19.9) is again used, but with the gas phase water activity and temperature, giving:

The coefficients for heat transfer and water mass transfer between the solid and gas phases were those determined for corn (see Eqs. (20.8) and (20.9)). Note that, since these transfer coefficients were determined for a packed-bed and, considering that solids/gas transfer is potentially more efficient in a mixed bed, the model allows for manipulation of these transfer coefficients, through the input variable "fold", which is used to multiply the values calculated in Eqs. (20.8) and (20.9).

A simple control scheme is incorporated to control the temperature of the water in the water jacket.

Tw Tsetpoint J(Ts Tsetpoint)> (22.6)

where J is the proportional gain. In other words, the program calculates the temperature difference between the solids and the set point temperature. It then sets the cooling water temperature so that the difference between the cooling water temperature and the set point temperature is J-fold greater, but in the opposite direction, such that the cooling water will warm the bed if the bed temperature is below the set point and cool the bed if the bed temperature is above the set point.

Table 22.1. Values used for the base case simulation of those parameters and variables that can be changed in the accompanying model of a well-mixed forcefully-aerated bioreactor

Symbol Significance

Base case value and unitsa

Design and operating variables and initial values of state variables

^wg fold T

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