In order to develop a mathematical model for your bioreactor from scratch, you would need to undertake 7 steps (Fig. 12.6). These steps were followed in the development of the various mathematical models presented in Chaps. 22 to 25. Of course, with the availability of these models, it is currently possible to start in the middle of the process. For example, you could use model equations from the literature for the same type of bioreactor and start at Step 4, with the determination of the parameter values for your particular system. However, even it this is done, it is necessary to check the original development of steps 1 to 3 in the literature model, to make sure that you agree with the decisions made by the authors during these steps.

You should also note that even though the steps are presented as a linear sequence here, the modeling process does not necessarily occur in a simple linear fashion. Frequently it is necessary to return and revise earlier decisions as the model is refined.

This section covers the 7 steps of modeling an SSF bioreactor, highlighting the tasks and questions that arise at each of the steps. It does not offer answers to these questions. Chapter 13 discusses how several of the key questions have been answered in the past, for example, in the development of the various bioreactor models that are presented in Chaps. 22 to 25.

Thermodynamic parameters

• the saturation humidity of the air

• the heat capacity of the moist substrate particles and the air

• the enthalpy of vaporization of water

• the equilibrium concentration of O2 in the substrate

(each of these is a function of temperature)

State variables

• biomass concentration

• nutrient concentration?

• substrate bed, bioreactor wall, and headspace gas temperatures

• [O2] within the substrate particle?

• headspace O2 concentration

• substrate bed bulk density

• overall dry matter

Design variables bioreactor height, width, and depth

Design variables bioreactor height, width, and depth

Operating variables

(not all will appear in all models)

• cooling water flow rate and temperature

• inlet air flow rate, temperature, and humidity

• frequency, duration, and intensity of agitation

• setpoints that are used to activate control schemes

Biological parameters might include

• maximum specific growth rate

• fitting parameters in the equation used to describe how growth is affected by temperature

• yield coefficients (biomass/substrate, heat/biomass)

Transport parameters

• effective diffusivity of species such as O2, nutrients, enzymes and hydrolysis products within the substrate particle?

• effective diffusivity of O2 in the gas phase

• coefficient for heat transfer through the bioreactor wall

• thermal conductivity of the substrate bed

Independent variables

• space only - if we can make an assumption of a pseudo-steady state process, but there are spatial gradients

Fig. 12.5. Various parameters and variables that might be included in SSF bioreactor models. Not all these parameters and variables will appear within a particular model. Items marked with a question mark are typically not included within bioreactor models due to the complexity they would bring

12.4.1 Step 1: Know What You Want to Achieve and the Effort You Are Willing to Put into Achieving It

You will typically want to construct a model that can be used as a tool in the bioreactor design process or in the optimization of operation of a bioreactor that has already been built. Models that have already been constructed with this motivation are described in Chaps. 22 to 25.

At this stage it is necessary to decide on the appropriate balance between the effort required (i.e., the work involved in writing the model equations, determining the values of the model parameters, and solving the model) and the "power" of the model, where the power of a model is defined by its ability to describe the performance of the system under a range of operating conditions, including conditions outside of the experimental range on which the model development was based. The greater the degree to which a model describes mechanistically the many phenomena presented in Chap. 2, the more likely it is to be more flexible. However, the description of fundamental phenomena can greatly increase the complexity of the model, and can require significant experimental effort to determine the parameters. If, in the particular bioreactor being modeled, there are significant temperature, water, and gas gradients across the bed, then clearly the model needs to describe the heat and mass transfer processes within the bed and to include position as an independent variable. A choice must then be made as to whether to describe the intra-particle gradients that arise. Doing so will lead to a highly complex model, because it will be simultaneously describing heterogeneity at the macroscale and heterogeneity at the microscale. Chapter 13 addresses this question in some detail.

The balance between model power and required effort may be decided from the outset, but it may also be decided later. Once the understanding of how the system functions is outlined in Step 2, the degree of complexity involved in a fully mechanistic approach becomes clearer, as do possible ways in which the mathematical description of the system can be simplified.

12.4.2 Step 2: Draw the System at the Appropriate Level of Detail and Explicitly State Assumptions

Once the aim of the modeling project is clear, the next step is to draw a diagram that summarizes the system and the important phenomena occurring within it. It is probably best to do this in two steps. Firstly, a detailed diagram should be drawn to include all the phenomena occurring within the system. Such a diagram might be similar to Fig. 2.6. Secondly, a simplified version should be drawn that includes only those phases and phenomena that have been selected as being sufficiently important to include in the model. For example, Fig. 12.3 shows a simplified diagram for a well-mixed bioreactor. An especially important question is as to whether the solid and air phases within the bed will be treated as separate subsystems, or whether the whole bed will be treated as a single pseudo-homogeneous subsystem that has the average properties of the solid and inter-particle air phases.

It will also need to be decided whether the bioreactor wall will be recognized as a separate subsystem. The diagram should clearly indicate the boundaries of the overall system and the various subsystems within it, the processes occurring within each subsystem, and the processes of exchange between different subsystems and between these subsystems and the surroundings of the bioreactor. It should be clearly annotated with the following information

• the state variables. In Fig. 12.3, these are the bed temperature and the biomass. Each of these should be given a symbol, which will be used in the equations;

• the interaction between the parameters and the state variables. For example, it should be noted that the growth rate of the organism will be modeled as depending on the bed temperature;

At the time of drawing these diagrams, the process of organizing the related information of assumptions, symbol definitions, and units should be started. All the symbols used to label the variables and parameters in the diagram should be listed and described, with their units. Also, all the assumptions and simplifications made should be carefully written down. As an example, for the well-mixed bioreactor in Fig. 12.3, it is assumed that:

• the substrate bed is well-mixed such that the whole bed can be represented by a single temperature, and the heat generation is uniform throughout the bed;

• the gas and solid phases are at temperature and moisture equilibrium, such that the air is saturated at the air outlet at the temperature of the bed;

• saturated air is used to aerate the bed;

• the loss of bed mass as CO2 during the process is not significant, allowing the bed mass to be represented by a constant (M, kg);

• the water lost during the fermentation is replaced by a spray, such that the water content of the bed does not change during the fermentation;

• growth follows logistic growth kinetics;

• the specific growth rate constant depends only on the biomass concentration and the temperature and therefore growth is not limited by the supply of O2 or nutrients;

• the thermal properties of the bed remain constant, even as the bed is modified by the growth process.

Of course many other assumptions are possible in order to reduce the complexity of models. Note that final decisions on the necessary variables and parameters and their appropriate units and the necessary assumptions might be made only at the stage of writing the equations.

12.4.3 Step 3: Write the Equations

This step builds on the foundation provided by the first two steps. The qualitative description of the system produced in Step 2 shows what equations need to be written and what terms should be included within these equations. The importance of the diagram drawn in Step 2 cannot be overstated. Lack of clarity in this diagram will lead to great difficulty in writing a coherent set of equations. The basic approach is to write:

• material and energy balance equations, usually in dynamic form (i.e., differential equations), with the state variables each expressed as:

d(variable) . , • , • , -= system inputs - system outputs + / - changes within the system dt

• These balances must originally be written in terms of quantities that are conserved, although the equations can be rearranged later. For example, a balance on water would originally be written with each term having units of the mass of water per unit volume of bioreactor (i.e., kg-H2O m-3) and not the water content (kg-water kg-dry-substrate-1). The differential term would therefore be d(WS)/dt and not dW/dt, where W is the water content and S is the kg of dry substrate per m3 of bioreactor. If it were desired to predict the water content, then the differential terms in W and S would be separated using the product rule, such that in the final equation only dW/dt appeared on the left-hand-side;

• relevant thermodynamic relationships for important parameters of the equations (e.g., the saturation water content of the gas phase as a function of temperature, using the Antoine equation);

• relationships for other parameters that are functions of the state of the system (e.g., the specific growth rate may be expressed as depending on the temperature);

• other intrinsic relationships.

In writing the equations, it is necessary to know the mathematical forms appropriate for describing the various phenomena. These mathematical forms are presented in Chaps. 14 to 17 for empirical growth kinetic equations and in Chaps. 18 to 20 for the processes described in balance/transport equations. As an example from Fig. 12.4, convection of heat to the surroundings appears within the energy balance as "h.a.(T-Tsurr)", or, in other words: "the rate of heat removal through the bioreactor wall is equal to a heat transfer coefficient times the area for heat transfer times the difference in temperature between the bed and the surroundings".

Attention to detail is paramount in the writing of equations. All terms of an equation must have the same units. For example, each term in a material balance would have units of kg h-1 (or kg m-3 h-1), while each term in an energy balance would have units of J h-1 (or J m-3 h-1). In fact, the necessity for terms to have certain units can help to give insights into how a particular term is to be constructed. Careful attention must be given as to whether terms are to be added or subtracted within an equation.

The number of dependent state variables selected will depend on the decisions made in Steps 1 and 2. For example, in the simple model in Fig. 12.4, equations are not written to describe the change in either the total mass of dry solids in the bed or the total mass of water in the bed. If the aim were to describe product formation then an extra equation would be written for the product.

For systems that are both temporally and spatially heterogeneous, and which therefore involve partial differential equations, it is necessary to write equations to describe the "boundary conditions". For example, it may be necessary to write that the temperature at the inlet of the bed is maintained at a particular temperature, and it may be necessary to write an equation that says that the rate at which heat is removed from the side walls of the bioreactor by convection to the cooling water is equal to the rate at which heat reaches the wall by conduction from the bed.

12.4.4 Step 4: Estimate the Parameters and Decide on Values for the Operating Variables

In order to solve a set of differential equations, you must have values for all of the parameters of the model, and, in addition to this, initial values must be given for the dependent state variables for which the differential equations are written.

Parameters. The types of parameters that appear in the model depend on the particular bioreactor and the phenomena that the model is describing. The values of the parameters may be determined in separate experiments, although at times values from the literature may be used. Note that some parameters might be constants, in which case only a single value is required, or they may vary as the state of the system varies, in which case an equation is needed that relates the parameter value to the state of the system. In the model presented in Fig. 12.4, the parameters were determined as follows:

• The parameters in the equation describing the dependence of the specific growth rate constant on temperature were determined by Saucedo-Castaneda et al. (1990) on the basis of experimental results for the growth of Aspergillus niger, obtained by Raimbault and Alazard (1980), by non-linear regression of the equation against these experimental results.

• The heat transfer coefficient (h) was obtained from Perry's Chemical Engineer's Handbook (Perry et al. 1984), as a typical value for the transfer of heat across steel. However, it could also be determined experimentally for a particular bioreactor.

• The design parameter A (area for heat transfer to the water jacket) was calculated assuming that the water jacket is in contact with the curved outer surface of the cylindrical bioreactor.

• Thermodynamic parameters were obtained from reference books (e.g., heat capacities of water and water vapor, coefficients of the Antoine equation used in the calculation of humidities, the enthalpy of evaporation of water).

• The heat capacity of the bed (CPB) was calculated on the basis of a starchy substrate of 50% moisture content.

Sometimes it is difficult to determine the value of a parameter in independent experiments. Although it is not particularly desirable, it is possible to allow this parameter to vary in the solving of the model, using an optimization routine to find the value of the parameter that allows the model to fit the data most closely.

Initial values of the state variables. The state variables that appear in the model depend on the combination of differential equations that make up the model. Their initial values will be determined by the way in which the bioreactor and inoculum were prepared. In the case of the well-mixed SSF bioreactor, it is necessary to give the initial mass of dry biomass and the initial temperature of the bed. Of course it is also possible to choose hypothetical initial values in order to explore the effect of the starting conditions on the predicted performance of the bioreactor.

Operating variables. The operating variables appearing in the model depend on the type of bioreactor and what manipulations it allows. The available operating variables for each SSF bioreactor type were presented in Chaps. 6 to 11. The values used for these variables in solving the model will either be experimental values, in the case of model validation, or hypothetical values, in the case where the model is being used to explore the effect of the operating conditions on the predicted performance of the bioreactor.

12.4.5 Step 5: Solve the Model

This book does not provide detailed information on how mathematical models are solved. Typically, numerical techniques will be used for solving differential equations. The amount of work that must be done to solve a model depends on the sophistication of the computer software available. In some cases it is necessary to write a program in a computer code such as FORTRAN or MatLab®, using prewritten subroutines as appropriate. With more sophisticated software packages, it may be sufficient simply to enter the equations and initial values in the appropriate fields and ask the computer to solve the equations.

Well-mixed systems will lead to a set of ordinary differential equations (ODEs), that is, equations in which the differential terms are only expressed as functions of time. Such a set of equations can be solved with well-known subroutines, such as the FORTRAN subroutine DRKGS, which is based on the Runge-Kutta algorithm. The solution of such models will be a graph, plotted against time, of the system variables that were described by the differential equations. In the case of the well-mixed SSF bioreactor model, the solution of the model is represented by temporal biomass and bed temperature profiles, such as the predictions presented in Fig. 12.7(a).

Systems with both spatial and temporal heterogeneity will lead to partial differential equations (PDEs), that is, equations that contain a mixture of differential terms that contain time in the denominator and differential terms that contain a spatial coordinate in the denominator. The solution methods involve transforming the PDEs into sets of ODEs, and then using numerical integration to solve these ODEs. Typically the transformation of the PDEs into sets of ODEs must be done by hand, and is not simple to do. The solution of such a model will be a graph against time of each of the state variables, with multiple curves, each curve representing a different position within the bed (Fig. 12.7(b)).

Fig. 12.7. What is the result obtained by solving a model? (a) For a model containing only ordinary differential equations, it is a predicted fermentation profile, or, in other words, a set of curves against time for each of the system variables. (b) For a model containing partial differential equations, fermentation profiles are predicted for various positions in the bed. For example, if the bioreactor shown in Fig. 12.3 were not mixed and the temperatures at various heights within the bed were predicted with an appropriate mathematical model, typical predictions would be as shown

12.4.6 Step 6: Validate the Model

If a model has been solved using independent estimates of all of the parameters, then it is of great interest as to whether the model manages to predict reasonably well the behavior of the system that is observed experimentally (Fig. 12.8). If it does, then this can be taken as supporting evidence, but not proof, that the mechanisms and phenomena included in the model are indeed those that are most important in determining the bioreactor behavior. Unfortunately, the validation of bioreactor models has only rarely been done well in the area of SSF to date.

As mentioned within Step 4, in some cases one or more of the parameters are determined during the solution step, by doing several simulations with different values for these parameters and seeing which solution agrees best with the experimental data (this being done most effectively by using an optimization routine to find the value of the parameter that gives the best statistical fit). The danger of this approach is that it might be possible to adjust the model to the data even if the mechanisms included in the model are inadequate. When this approach is used for parameter estimation, it is not possible to claim that the model has been validated, even if very close agreement is obtained.

A sensitivity analysis might be done at this stage (Fig. 12.9). This involves making changes one at a time to the various parameters in the model and seeing how large the effect is on the model predictions. The objective is to determine which parameters are most important in determining bioreactor performance:

time

Fig. 12.8. Validation of the model. The graph illustrates three possible situations for a comparison between experimentally measured temperatures, represented by the solid circles, and the bed temperatures predicted by the model, represented by one of the curves. (—) Ideally there should be minimal deviation between the model predictions and experimental data; (- - -) At times general features of the experimental curve are described but are offset in magnitude and time. Possibly more accurate determination of one or more parameters is necessary; ( ) At times the predicted results are very different from the experimental results. A key phenomenon may have been omitted in the model time

Fig. 12.8. Validation of the model. The graph illustrates three possible situations for a comparison between experimentally measured temperatures, represented by the solid circles, and the bed temperatures predicted by the model, represented by one of the curves. (—) Ideally there should be minimal deviation between the model predictions and experimental data; (- - -) At times general features of the experimental curve are described but are offset in magnitude and time. Possibly more accurate determination of one or more parameters is necessary; ( ) At times the predicted results are very different from the experimental results. A key phenomenon may have been omitted in the model

Fig. 12.9. Sensitivity analysis. In this example, the model presented in Fig. 12.4 is solved for various values of the heat transfer coefficient, h, associated with heat removal through the bioreactor wall. Key: (—) solution using h, the best estimate of the heat transfer coefficient; ( ) solution using 2.h; (---) solution using h/2. Two possible situations are shown.

(a) The variations in h have relatively little effect on the model predictions. Probably removal through the bioreactor wall makes only a relatively small contribution to overall heat removal. It might be appropriate to remove this term from the model. (b) The variations in h have a significant effect on the model predictions. The term describing heat removal through the bioreactor wall should be maintained in the model, since it is obviously an important contributor to overall heat removal, and it is important to have an accurate value for h if the model is to predict the experimental data well time time

Fig. 12.9. Sensitivity analysis. In this example, the model presented in Fig. 12.4 is solved for various values of the heat transfer coefficient, h, associated with heat removal through the bioreactor wall. Key: (—) solution using h, the best estimate of the heat transfer coefficient; ( ) solution using 2.h; (---) solution using h/2. Two possible situations are shown.

(a) The variations in h have relatively little effect on the model predictions. Probably removal through the bioreactor wall makes only a relatively small contribution to overall heat removal. It might be appropriate to remove this term from the model. (b) The variations in h have a significant effect on the model predictions. The term describing heat removal through the bioreactor wall should be maintained in the model, since it is obviously an important contributor to overall heat removal, and it is important to have an accurate value for h if the model is to predict the experimental data well

• If relatively small changes in the value of a parameter significantly affect model predictions, then quite probably the phenomenon with which the parameter is associated is quite important in determining the system behavior and, furthermore, it is quite important to obtain accurate values for the parameter;

• If relatively large changes in the value of a parameter have a relatively small effect on model predictions, then possibly the phenomenon with which the parameter is associated is not very important in determining the system behavior, at least under the particular set of operating conditions used (the phenomenon might become more important under another set of operating conditions). The degree of accuracy needed for estimation of this parameter is not so great and possibly the term describing this parameter can be eliminated in order to simplify the model.

12.4.7 Step 7: Use the Model

The use to which the model is put will of course depend largely on the original motivation of the modeling work. For example, the model might be used to explore:

• how the same bioreactor will perform under operational conditions other than those for which experimental results were collected;

• how a different bioreactor geometry affects performance;

• how the size of the bioreactor affects performance.

Chapters 22 to 25 will show examples of such explorations for various different bioreactor types.

Of course there is no guarantee that the model will work well for a situation other than that for which it was validated. Predictions of the model about how performance can be improved must be checked experimentally. However, clearly an experimental program guided by use of a mathematical model has a good chance of optimizing performance more rapidly than a purely experimental program.

Deviations of the performance from predictions will lead to work to improve either or both of the model structure (the equations) and the parameter values. That is, it may be necessary to return to Steps 3 and 4. Such revisions lead to continual refinements of the model and to a greater understanding about how the various phenomena interact to control bioreactor performance.

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