Fig. 23.2. Summary of the model of a well-mixed rotating drum, which is a slightly simplified version of the model of Stuart and Mitchell (2003). In each text box, the variable shown in parentheses after the heading is the variable that is isolated in the differential term on the left hand side of the differential equation before the equation set is solved. Subscripts: h = headspace, b = bed, w = wall of bioreactor, s = surroundings, in = inlet, sat = saturation

Fig. 23.2. Summary of the model of a well-mixed rotating drum, which is a slightly simplified version of the model of Stuart and Mitchell (2003). In each text box, the variable shown in parentheses after the heading is the variable that is isolated in the differential term on the left hand side of the differential equation before the equation set is solved. Subscripts: h = headspace, b = bed, w = wall of bioreactor, s = surroundings, in = inlet, sat = saturation

Growth occurs according to the logistic equation, where the specific growth rate constant («) is affected by the temperature and water activity of the solid in a manner identical to that described by Eqs. (22.1), (22.2), and (22.3) (see Chap. 22.2 and Fig. 22.3). The value of fxopt chosen should be valid for growth of the process organism in a continuously-mixed system since the model itself does not include any equations to describe the effect of the rotation rate on the specific growth rate constant. The substrate is assumed to be corn, characterized by the isotherm given by Eqs. (19.8) and (19.9).

The balance on the overall mass of dry solids (i.e., the sum of dry biomass and dry residual substrate), shown in the lower right of the bed in Fig. 23.2, 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), which was deduced in Sect. 16.2.

The mass balance on water in the substrate bed, shown in the upper right of the bed in Fig. 23.2, has terms on the right hand side to describe, respectively, metabolic water production due to growth and maintenance and the evaporation of water to the headspace.

The mass balance on water in the headspace, shown on the right of the head-space in Fig. 23.2, has terms on the right hand side to describe, respectively, the entry and leaving of water with the gas flow through the headspace and the evaporation of water from the bed.

On the right hand side of the energy balance over the substrate bed, shown in the upper left of the bed in Fig. 23.2, the four terms describe, respectively:

• metabolic heat production due to growth and maintenance;

• sensible heat transfer between the bed and the headspace gases;

• sensible energy transfer between the bed and the bioreactor wall;

• removal of energy from the bed by evaporation of water into the headspace gases.

On the right hand side of the energy balance over the headspace gases, shown on the left of the headspace in Fig. 23.2, the four terms describe, respectively:

• sensible energy of the dry air entering and leaving the headspace;

• sensible energy of the water vapor entering and leaving the headspace;

• sensible energy transfer between the headspace gases and the bioreactor wall;

• sensible heat transfer between the bed and the headspace gases.

On the right hand side of the energy balance over the bioreactor wall, shown at the top of Fig. 23.2, the three terms describe, respectively:

• sensible energy transfer between the wall and the surroundings of the bioreac-tor;

• sensible energy transfer between the bed and the bioreactor wall;

• sensible energy transfer between the headspace gases and the bioreactor wall.

Within these energy balance equations, there are several heat and mass transfer coefficients. The bed-to-wall heat transfer coefficient (hbw) is calculated using Eq. (20.3). Before the other heat transfer coefficients are calculated, the value of the air flow rate in vvm (volumes of air per total bioreactor volume per minute) is used to calculate the air flow rate, F (kg-dry-air s-1). This and the cross sectional area of the headspace normal to the gas flow (Ag, m2) are used to calculate the two heat transfer coefficients involving the headspace gases (W m-2 °C-1), namely the bed-to-headspace coefficient (hbh) and the headspace-to-wall coefficient (hhw), according to the relationship given by Geankoplis (1993) (see Eq. (20.6)). The psy-chrometric ratio is then used to calculate the bed-to-headspace mass transfer coefficient:

Note that the denominator also contains a conversion factor due to the units used for the driving force in the term that describes evaporation. This equation uses hbh in W m-2 °C-1 and gives K in kg-^O s-1 m-2 (kg-^O kg-dry-solids-1)-1 Note that the units within the parentheses are the units of the driving force. After simplification, the units of K are kg-dry-solids s-1 m-2. Note that the driving force for evaporation is the difference between the water content of the solids (W, kg-H2O kg-dry-solids-1) and the water content that the solids would have if they were in equilibrium with the gas in the headspace phase (Wsat). Wsat is calculated using Eq. (22.5). In this case, the equation uses the headspace temperature and water activity, the latter calculated as explained in Sect. 19.4.1 (see Eq. (19.16)).

The values of the transfer coefficients hbh and kw calculated as described in the previous paragraph are for a drum that is not mixed. Mixing should increase bed-to-headspace heat and mass transfer. However, there is not sufficient information available to incorporate this mechanistically within the equation. Therefore, in this model, hbh and kw are simply multiplied by an empirical factor "n", which represents the fold-increase in transfer rates due to mixing.

The model incorporates two simple control schemes, a control of the inlet air humidity and a control of the bed water activity (Fig. 23.3). Control of the inlet air humidity is desirable since, although the use of dry air to promote evaporation is an effective cooling strategy, if dry air is used at the beginning of the fermentation when the rate of metabolic heat production is low, then the bed temperature can fall to values low enough to retard early growth. Therefore the temperature of the bed is monitored hourly. If it is less than the optimum temperature for growth (38°C) then high-humidity air is fed to the bioreactor, while if it exceeds this temperature then low-humidity air is supplied.

The promotion of evaporation by supplying low-humidity air could potentially dry the bed to water activities low enough to restrict growth. Therefore it is assumed that samples are removed from the bed every hour and their water activity rapidly determined. If the water activity falls below a set point, then sufficient water is added to the bed to bring the water activity back to the initial value. Note that it is assumed that the added water is at the temperature of the solids and therefore does not affect the bed temperature.

The model equations are ordinary differential equations, since time is the only independent variable. They are solved using the Runge-Kutta numerical integration algorithm.

Fig. 23.3. Control schemes incorporated into the mathematical model air from blower

Fig. 23.3. Control schemes incorporated into the mathematical model

23.2.2 Predictions about Operation at Laboratory Scale

Figure 23.4 shows the kind of information that can be obtained from the model. In this case the data from Table 23.1 and 23.2 were used, withn = 10. The output can be plotted so as to see temporal variations in:

• The growth of biomass and decrease in the overall mass of dry solids (Fig. 23.4(a)).

• The value of the specific growth rate parameter in the logistic equation and the relative effects of temperature and water activity on the value of this parameter (Fig. 23.4(b)). In this case the temperature has the greater influence on the value of this parameter.

• The driving force for evaporation (Fig. 23.4(c)).

• The degree of saturation of the headspace (Fig. 23.4(d)). In this case the head-space remains saturated throughout the fermentation.

Table 23.1. Values used for the base case simulation for those parameters and variables that can be changed in the accompanying model of a well-mixed rotating-drum bioreactor

Symbol Significance

Base case value and unitsa

Design and operating variables (can be varied in the input file for the model)

D Diameter of the bioreactor

%fill Percentage of the drum volume occupied by the solid bed Ts Temperature of the surroundings vvm Rate of air flow (dry basis) Tin Temperature of the inlet air a„ini Water activity of the inlet air when T < Topt awi„2 Water activity of the inlet air when T > Topt awSP Set point bed water activity (below which the addition of water to the bed is triggered)

Initial values (can be varied in the input file for the model) Tbo Initial bed temperature 30°C

Two Initial temperature of the bioreactor body 30°C Tho Initial temperature of the headspace gases 30°C

awbo Initial water activity of the solids 0.99d

Microbial parameters (can be varied in the input file for the model)

bo bm

Popt Topt

Type

Yq mQ

0.3 (used to calculate the initial dry mass of solids in the bed M<,) 30°C

0.01 m3-air (m3-bioreactor)-1 min-1b

0.15c

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