A Suite of Mathematical Models Suitable as Tools for the Selection ...

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A Suite of Mathematical Models Suitable as Tools for the Selection ...A Su

    A Suite of Mathematical Models Suitable as Tools for the

    Design and Optimization of Solid-State Fermentation


     123David Alexander Mitchell, Nadia Krieger, Oscar Felipe von Meien e Luiz Fernando 3de Lima Luz Junior

     1 Universidade Federal do Paraná Depto de Bioquímica e Biologia Molecular

    Caixa Postal 19046 81530-990 Curitiba PR Email: 2 Universidade Federal do Paraná Departamento de Química

    Caixa Postal 19081 81531-990 Curitiba - PR 3 Universidade Federal do Paraná Departamento de Engenharia Química

    Caixa Postal 19011 81531-990 Curitiba - PR


     We present a case study that demonstrates how mathematical models can be used as tools to guide the design and optimization of solid-state fermentation bioreactors. We show how a model describing growth and heat and mass transfer phenomena within an intermittently-mixed stirred bioreactor can be used to optimize design variables such as bioreactor height and operating variables such as the air flow rate and the strategy for triggering mixing steps.


     Solid-state fermentation (SSF) can be used to produce a range of microbial products from solid organic residues. This technology is particularly relevant to Brazil, given its large agro-industrial base, and various groups are currently investigating processes at laboratory scale. However, the successful development of large-scale applications is hampered by the lack of rational scale-up criteria in the face of difficulties in removal of the metabolic heat generated during the fermentation, difficulties which become increasingly severe with increase in scale (Mitchell et al., 2003). Over the past three years, we have developed mathematical models for several of the bioreactor types used in SSF, namely packed-bed bioreactors (Mitchell & von Meien, 2000), intermittently-mixed bioreactors (von Meien & Mitchell, 2002) and rotating drum bioreactors (Mitchell et al, 2002a, Stuart & Mitchell, 2003). The intention is to provide a suite of tools that can be used, in conjunction with experimental work, to select an appropriate bioreactor type, design a large-scale version of the selected bioreactor and to optimize the operation of the selected bioreactor once it is built. We undertake a case study with one of these models, to illustrate how models can be used as tools in the design and optimization process.


     Key features of the bioreactors for which we have mathematical models are shown in Fig. 1: a packed-bed bioreactor without mixing but with internal heat transfer plates, an intermittently-mixed bioreactor, a well-mixed rotating drum and a rotating drum operated such that it is well-mixed in the radial direction but unmixed in the axial direction. Figure 1 also highlights the phenomena that are described by the model in each case. The models for each of these bioreactors are written in FORTRAN programs.

     The case study is undertaken with the intermittently-mixed stirred bioreactor. The full model equations are presented in von Meien & Mitchell (2002) and are not reproduced here.

    exhaust air

     Headspace heat substrate bed transfer held between the air outlet plate plates

     z=HB GAP cooling water (same heat transfer plates

    for each plate) Metabolic


     x=L x=0 Radial conduction repeating unit within

    the bioreactor substrate bed height Convection by cooling water z=0 Axial convection and evaporation Cooling inlet air air inlet water Heat removal across direction of rotation the bioreactor wall

     HEADSPACE Air PHASE Air Air out in tumbling convection to evaporation to substrate the headspace the headspace bed SUBSTRATE Added PHASE HO 2

     Heat removal across

    the bioreactor wall

     detection of the relative humidity of the air Gas Phase leaving the bioreactor.

    When the relative heat humidity falls below a transfer setpoint, an agitation Solid Phase event is triggered.

     metabolic heat liberated in the solid phase water heat transfer across transfer the bioreactor wall

    assumed to be negligible flow of air

    through the bed inlet air

Fig. 1 The various bioreactor types for which we have mathematical models, indicating the key phenomena

    described in the models. From top to bottom the bioreactors are: A packed-bed bioreactor with internal heat

    transfer plates, a rotating drum and an intermittently-mixed bioreactor.

     The key equations are water and energy balances on both the solid phase and the gas phase, plus a growth kinetic equation describing logistic growth kinetics. Simulations are done for two fungi, Aspergillus niger and Rhizopus oligosporus. The same maximum growth rate and

    influence of temperature on growth are assumed for both, but their responses to the water activity of the medium are modeled differently (Fig. 2). The temperature and water effects, expressed as fractions of the maximum possible specific growth rate, are combined by calculating the geometric mean.

     Initially packed-bed operation is modeled. When the air at the outlet falls below a relative humidity (RH) set point, a 15 minute mixing event is initiated. Growth is completely inhibited during this mixing event for the aseptate fungus Rhizopus oligosporus, being reestablished on

    the return to static operation, but continues normally for the septate fungus Aspergillus oryzae.

    The mixing event returns the solid and gas phase temperatures and water activities at all locations within the bed to their original values. The average biomass and dry solids concentrations are then used as the starting point for another phase of packed-bed operation.


     In the operation of an intermittently-stirred bioreactor, the operating and design variables available are the height of the bioreactor, the temperature, flux and humidity of the inlet air, the agitation frequency, duration and intensity and the RH setpoint of the outlet gas used to trigger the mixing event. The current work explores only the effects of the bioreactor height, the flux of the inlet air, and the strategy for triggering the mixing event.

    Base case simulation

     -2-1 The base case involves a bioreactor height of 2.5 m, a flux of 0.06 kg m s, an inlet gas

    RH of 99% and a setpoint of 87% RH, with all other parameter values being identical with those of von Meien & Mitchell (2002). With these operating conditions, A. niger performs

    significantly better than R. oligosporus (Fig. 3), the differences in performance being almost

    entirely due to the differences in the effect of water activity on growth.

    1.01.0(b) (a)




    Relative growth rate Relative growth rate 0.20.2


    0.800.850.900.951.00253035404550oWater activityTemperature ( C)

    Fig. 2 (a) Effect of water activity on the specific growth rates of (?) A. niger and (?) R. oligosporus, as fractions -1of the maximum possible specific growth rate of 0.324 h (Glenn & Rogers, 1998). (b) Effect of temperature on

    the specific growth rate, expressed as a fraction of the maximum possible specific growth rate. The same relationship of specific growth rate to temperature is assumed for both fungi.

    (a) (b) 1.01.0




    Fraction of maximum specific growth rateBiomassFraction of maximum specific growth rateBiomass0.20.2

    Based on temperatureBased on temperature





    Based on water activityBased on water activity0.00.0





    01020304050600102030405060Time (h) Time (h)

    Fig. 3. Base case simulation for growth of (a) Aspergillus orzyae and (b) Rhizopus oligosporus as a function of

    time and position. Legend: (yellow) bottom of column; (green) quarter height; (blue) half height; (magenta)

    three-quarter height; (red) top of column. In the biomass plot, the volume-averaged biomass concentration

    coincides with the blue line while the solid black line represents the profile for growth at the maximum possible -1specific growth rate of 0.324 h throughout the whole growth phase. The arrows denote the mixing events.

     Figure 3 illustrates how the severity of the temperature and water limitations vary as a function of both time and position. In both cases, temperature limitations increase in severity with height, this being due to convective cooling (von Meien & Mitchell, 2002). On the other hand, water limitations decrease in severity from a quarter height upwards. Mixing returns the bed to the optimum conditions, but the conditions quickly diverge from the optimum during the following period of static operation.

     For A. niger, temperature limitations have the greater effect, especially in the upper regions of the column. The water limitations are relatively brief, being relieved by the next mixing event. In the case of R. oligosporus the water limitations are much more severe, affecting

    growth in the whole column for periods of about 10 hours before the next mixing event is triggered. During these periods growth slows. As a result, waste metabolic heat production

    falls and the temperature falls, alleviating the temperature limitation of growth. However, after the water activity is brought back to its original value, the high temperatures are quickly reestablished in the upper regions of the column, causing temperature to once again be the major limiting factor. For both fungi, as a results of these limitations, growth becomes increasingly poorer with height. Growth right at the base is predicted to be significantly better than in the other regions due to the good temperature control and the fact that this region is not predicted to dry out so rapidly.

     Although this performance is much better than it would be in the absence of mixing and water replenishment events (von Meien & Mitchell 2002), no attempt has yet been made to explore how manipulations of the design and operating conditions might optimize bioreactor performance. We undertake such an exploration here.

    Effect of the relative humidity setpoint used to trigger the mixing event

     In the base-case simulation, a mixing event is triggered when the outlet gas RH falls below a setpoint of 87%. This allows significant water limitation (Figs 3a and 3b), because a significant portion of the bed dries out before the outlet humidity falls to the setpoint. Fig 4a shows a simulation in which the outlet humidity setpoint is changed to 98% RH. For both fungi this minimizes the water limitations of growth. However, in the case of the growth of A.

    niger the improvement of growth is minimal, since temperature limitation is already the major factor limiting growth in the base case. In the case of R. oligosporus, which prefers higher

    water activities, the predicted improvement of growth is considerable, but growth is still limited, this time by temperature limitations.

    Effect of the flux of the inlet air

     -2-1With the flux of air at the inlet doubled to 0.12 kg m s, a slight improvement is predicted

    for the growth of both A. niger and R. oligosporus (Fig. 4b). This is directly associated with

    the improved heat removal that is obtained with higher superficial velocities, decreasing the temperature limitation of growth (Mitchell & von Meien, 2000).

    Effect of bioreactor height

There is also a slight improvement for both A. niger and R. oligosporus when the bioreactor

    height is reduced from 2.5 m to 1.0 m (Fig. 4c). Of course this is to be expected, since the temperature profile which establishes itself in the lower parts of the bioreactor during the static periods is independent of the height of the bioreactor, so the temperature profile in the 1 m bioreactor is identical with the temperature profile in the first meter of the 2.5 m high base case bioreactor (Fig. 3). In other words, the 1 m high bioreactor represents the better performing region at the bottom of the 2.5 m high bed. The disadvantage of reducing the

    height is that the bioreactor has to take up much more floor area to have the same capacity. Also, aeration costs are higher, since the simulation is done with a constant air flow rate per square meter of bed.

    Effect of combined manipulations

     A simulation done using all three changes to the base case simultaneously, namely a RH -2-1setpoint of 98%, an aeration rate of 0.12 kg m s and a bioreactor height of 1 m, predicts

    near optimal growth for both A. niger and R. oligosporus (Fig 4d). Both water and

    temperature limitations are minimized.

    0.250.25(a) (b)





    Biomass (kg-biomass/kg-substrate)Biomass (kg-biomass/kg-substrate)0.000.00


    Time (h)Time (h)

    0.250.25(c) (d)





    Biomass (kg-biomass/kg-substrate)Biomass (kg-biomass/kg-substrate)0.000.00


    Time (h)Time (h)

    Fig. 4. Effect of changes in the design and operating conditions on predicted performance. In all cases blue

    represents A. niger and red represents R. oligosporus. A dashed colored line represents the base case simulation

    while a solid colored line represents the predicted results with the changed design and operating variables. The solid black line represents the maximum possible growth. (a) Effect of the RH setpoint used for triggering the mixing event. The value was changed from the base case setpoint of 87% RH to a new setpoint of 98% RH. (b) -2-1-2-1Effect of the aeration rate. The value was changed from the base case value of 0.06 kg m s to 0.12 kg m s.

    (c) Effect of bed height. The value was changed from the base case value of 2.5 m to 1 m. (d) Effect of all three -2-1changes combined, namely a setpoint of 98%, an aeration rate of 0.12 kg m s and a bed height of 1 m.


     In the current work we have used a case study to demonstrate how mathematical models can be used to guide bioreactor design and operation. The case study was done with one-by-one variations in only a few operating variables. However, the model can be used to investigate other operating variables. Further, it is possible to apply least squares optimization routines to identify the optimal combination of operating variables or even to use the model for the development of control strategies such as PID control.

     Obviously, these same approaches can be applied with the mathematical models that we have for other bioreactors. The aim of course should be for the modeling work to guide the experimental work, and not to replace it. In fact, in a combined program of modeling and

    experimental work, the mathematical model will undergo continual improvements, in the light of any discrepancies that might occur between model predictions and actual bioreactor performance.

     Several improvements are essential in these models before they can truly become flexible and powerful tools. There is currently insufficient knowledge about the effects of varying temperatures on the growth and death kinetics of microorganisms, with current kinetic models being based on experiments in which various cultures are incubated at different temperatures, but in which any one culture is incubated at a constant temperature throughout the growth cycle. This does not reflect the situation in SSF, where the organism suffers temporal variations in the temperature. Further very little is known about the magnitudes of shear effects during the agitation of beds of solids substrates and their effects on the growth kinetics. This is a crucial point for future studies, since, at the moment, current models simply do not take shear effects into account. On the physical side, it is not a simple matter to determine transfer coefficients in the mass and energy balance equations. Solid and air flow patterns within the bioreactor may not be simple, and must be described accurately if the correct values of transfer coefficients are to be extracted from experiments based on overall mass and energy balances over the bioreactor, such experiments typically being undertaken in the absence of growth. In fact, the models could attempt to describe such flow patterns. Further, current models typically assume that the transfer and physical parameters remain constant during growth, whereas the significant changes that the bed undergoes during the fermentation may mean that this is not true.

     Despite the current limitations, it is clear that models have the potential to be powerful tools in the selection, design and optimization of solid-state fermentation bioreactors. Once improved models are available, their use in scale-up programs will reduce the amount of experimental work required and will lead to the construction of more efficient bioreactors, thereby enabling solid-state fermentation to fulfil its potential as a fermentation technology.


    This work was supported by the Brazilian National Council for Scientific and Technological Development (CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico). David Mitchell and Nadia Krieger thank CNPq for research scholarships.


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    Mitchell, D.A. e von Meien, O.F. (2000), Mathematical modeling as a tool to investigate the design and operation of the Zymotis packed-bed bioreactor for solid-state fermentation. Biotechnology and Bioengineering,

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    Mitchell, DA, von Meien OF e Krieger, N. (2003), Recent developments in modeling of solid-state fermentation: heat and mass transfer in bioreactors. Biochemical Engineering Journal, v. 13 n. 2-3, p. 137-147.

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