# Insights from Mathematical Modelling into Energy Requirement and Process Design of Continuous and Batch Stirred Tank Aerobic Bioreactors

## Abstract

**:**

_{S}values, continuous mode of operation required a much smaller bioreactor volume, due to higher operating cell concentration, and this is a major advantage of continuous over batch.

## 1. Introduction

_{L}a) and agitator power and air flowrate is crucial in the design, operation and scale-up of bioreactors [1,6,7,8,9,10,11,12]. Oxygen transfer rate in a bioreactor is strongly influenced by the oxygen concentration in the broth, the broth’s physical chemical properties, agitator selection, and hydrodynamic conditions [1,13,14,15,16,17,18].

^{3}and larger using conventional agitators. Fitzpatrick et al. [2] carried out a mathematical simulation study for a batch system, which showed that the minimum or near-minimum total energy requirement for oxygen transfer occurred when operating at the onset of impeller flooding throughout the bioreaction, by continuously varying both impeller power and air flowrate over the bioreaction time. Operating at the onset of flooding may not be practical to implement in practice. However, the minimum energy can be approached by dividing the bioreaction time into a small number of time segments with appropriately chosen constant agitator powers and varying the air flowrate within each segment. This is potentially much more practical to implement.

## 2. Mathematical Modelling

_{R}) for both the continuous and batch bioreactors is given a value of 100 kg h

^{−1}. Details of the mathematical models are provided in the following sections and the calculations for the continuous mode were implemented using Microsoft Excel (Microsoft, Seattle, WA, USA) and those for batch mode were mainly implemented using Matlab (MathWorks, Natick, MA, USA).

#### 2.1. Bioreaction Kinetics

_{x}is the cell growth rate, X is the cell concentration and µ is the specific growth rate, which is modelled using the Monod model (Equation (2)):

_{OL}are the sugar and oxygen concentrations, respectively, µ

_{max}, K

_{S}and K

_{O}are constants.

_{p}) and sugar concentration (r

_{s}) due to microbial metabolism were modelled using the following Equations (3) and (4):

_{max}and K

_{S}can vary significantly where some reported values for µ

_{max}varied from 0.09–4.2 h

^{−1}[21,22,23,24]. K

_{S}values are typically in the mg·L

^{−1}range, with reported values typically varying from 0.07 to 200 mg·L

^{−1}[21,25,26], although Znad et al. [22] reported a value of 130,900 mg·L

^{−1}. K

_{O}can have significant variation but typically it is in the range of 0.1 to 1 mg·L

^{−1}[22,26,27]. The values of Y

_{XS}, Y

_{PS}and m

_{S}were obtained from van’t Riet and Tramper [21]. The values of α and β were obtained from Znad et al. [22].

_{R}is the dilution rate, S

_{f}and P

_{f}are the steady-state sugar and product concentrations, respectively in the CSTB, and P

_{0}is the concentration of any product in the feed.

_{0}= 150 g·L

^{−1}and P

_{0}= 0 g·L

^{−1}, respectively. For the batch bioreactor, the initial cell concentration was X

_{0}= 0.25 g·L

^{−1}, and the bioreaction was completed when sugar concentration was reduced to 0.1 g·L

^{−1}. For the CSTB, the kinetic modelling was applied to evaluate µ, P

_{f}and the steady-state cell concentration (X

_{f}) for given values of S

_{f}. For batch, it was applied to evaluate the evolution of substrate, product and cell concentration over time, and in particular the final product concentration and bioreaction time when the bioreaction was completed.

#### 2.2. Bioreactor Working Volume and Substrate Utilisation

_{R}= 100 kg·h

^{−1}). The feed flowrate requirement for the CSTB was also evaluated.

#### 2.2.1. Continuous Bioreactor

_{R}) by the continuous stirred tank bioreactor (CSTB) is obtained from Equation (7):

_{R}is the dilution rate and V

_{L}is the working volume of the CSTB. Equation (8) is used to calculate V

_{L}.

_{S0}) is given in Equation (9):

_{SW}) is given in Equation (10):

#### 2.2.2. Batch Bioreactor

_{b}is the product concentration at the bioreaction time (t

_{b}) when the bioreaction is completed and the sugar concentration (S

_{b}) is 0.1 g·L

^{−1}. The average downtime between batches (t

_{d}) is given a value of 8 h.

_{SI}) and wasted substrate (M

_{SW}) is given in Equations (12) and (13), respectively:

#### 2.3. Oxygen Transfer and Detrmination of Agitator Power Requirement

^{−1}based on a range of values provided in an OUR review article by Garcia-Ochoa et al. [28].

_{L}a value required to supply the oxygen transfer rate (OTR) to satisfy the OUR at steady-state:

_{OG}is the oxygen concentration in the air bubbles and M is the Henry’s law equilibrium constant (=35). C

_{OG}varies from the concentration of oxygen in the ambient air (C

_{OGI}= 280 mg·L

^{−1}) to the concentration of oxygen in the air leaving the bioreactor (C

_{OGO}). Thus Equation (17) was used to evaluate an average equilibrium concentration of oxygen in the liquid:

_{G}is air volumetric flowrate and V

_{L}is the working volume of the bioreactor.

_{L}a and agitator mechanical power input in the gassed bioreactor (Pag) and air superficial velocity (v

_{s}) is given by Equation (19). This equation was used to calculate Pag in the mathematical simulations:

_{1}= 0.4 and n

_{2}= 0.5 and these were obtained from van’t Riet [29]. Equation (19) is an important correlation and the values of the constants in this equation should be experimentally evaluated in the application of this modelling approach to a real bioreaction. Furthermore, the size of the bioreactor volume may influence the values of constants in Equation (19). Even though the bioreactor volume changes in this study, a simplifying assumption is made that the values of the constants do not vary in Equation (19). However, in the application of the mathematical approach to a real bioreactor, some work should also be undertaken to investigate how scale-up influences these constants.

_{s}) is defined in Equation (20):

_{T}is the cross-sectional area of the bioreactor. The parameter vvm is used in this study, because the air flowrate is commonly expressed in terms of vvm in the bioprocess industry [15], and this is defined in Equation (21):

^{−1}.

_{G}and v

_{s}were evaluated from Equations (21) and (20), respectively. C

_{OL}is given a constant value, thus OTR = OUR. C

_{OGO}was calculated from Equation (18), after which ${C}_{OL}^{*}$ was calculated from Equation (17). Then, k

_{L}a was calculated from Equation (15) and Pag was calculated from Equation (19).

#### 2.4. Flooding and Phase Equilibrium Constraints

_{GF}) at the onset of flooding, for a fixed value of Pag, was evaluated by solving Equations (22)–(24). Equation (22) was obtained from Bakker et al. [18]:

^{−1}, N is the impeller rotational speed and N

_{PG}is the impeller power number. N

_{PG}varies with the air flow rate, and Equation (24) was applied to take this into account. This equation was obtained from Bakker et al. [18] along with values for the constants:

_{P}is the ungassed power number (=6), μ is the liquid viscosity (5 mPa·s), a = 0.72, b = 0.72, c = 24, d = 0.25 [flat-bladed turbine impeller].

_{OGO}) was limited by the equilibrium constraint (25):

#### 2.5. Aeration System Power Requirement

_{atm}is atmospheric pressure and Pi is the atmospheric pressure plus the static pressure acting on the bottom of the bioreactor due to weight of liquid, γ = 1.4. η

_{C}is the isentropic efficiency of the compressor (assumed to be constant at 0.7). The sum of the agitator and compressor electrical power requirements (P

_{tot}) was given by Equation (27):

_{m}is the electric motor efficiency (assumed to be constant at 0.9 for both the agitator and compressor). For the batch bioreactor, the power will vary over time, thus the average power requirement over the batch cycle time is evaluated. The average electrical power requirement was defined as the batch electrical energy requirement divided by the batch cycle time. This is useful for comparison with CSTB power requirement.

#### 2.6. Refrigeration Power Requirement for Cooling

_{T}) is given in Equation (28):

_{M}is the rate of microbial metabolic energy production, which is estimated from Equation (29):

_{HO}= 14.7 kJ·g

^{−1}of oxygen [21].

_{E}) of 15 °C and a condensation temperature (T

_{C}) of 35 °C. The refrigeration electrical power requirement (P

_{ref}) was estimated using the co-efficient of performance (COP) in Equation (30):

_{r}is the refrigeration efficiency which is typically around 0.6.

## 3. Results and Discussion

#### 3.1. Bioreaction Kinetics

^{−1}down to about 2 mg·L

^{−1}. There is a more precipitous decrease at lower oxygen concentrations because the value of K

_{O}is 0.363 mg·L

^{−1}. On the other hand, the sugar concentration during bioreaction varies from 150 g·L

^{−1}down to 0.1 g·L

^{−1}and this does not have a major influence on specific growth rate, as illustrated in Figure 1a.

_{f}) on the CSTB bioreaction kinetics (at a steady-state oxygen concentration of 2 mg·L

^{−1}) is illustrated in Figure 1b. As S

_{f}decreases, product concentration increases as more sugar is converted, and cell concentration increases as the average residence time increases. The batch bioreaction kinetics (at C

_{OL}= 2 mg·L

^{−1}) is illustrated in Figure 1c. The batch bioreaction progresses slowly up to about 10 h after which there are significant changes in concentrations and the bioreaction time is about 23 h. The evolution of OUR is also presented in Figure 1c, where the OUR reaches a maximum of 4.7 g·L

^{−1}·h

^{−1}towards the end of the bioreaction.

#### 3.2. Bioreactor Volume and Feed/Wasted Sugar Substrate

#### 3.2.1. CSTB

_{f}is very important in the process design of a CSTB to produce a specified rate of product leaving the bioreactor, as it impacts on the size of the bioreactor, the amount of feed sugar required and wasted sugar leaving the bioreactor. Figure 2a shows that operating at higher values of S

_{f}results in both a larger feed sugar requirement and more unused or wasted sugar leaving the bioreactor. This is because more wasted sugar requires more feed sugar to meet the product production specification.

_{f}results in higher feed flowrate, which is to be expected as more feed sugar is required because less of the feed substrate is being metabolised when operating at higher values of S

_{f}. The bioreactor working volume is a key design variable. Re-arranging Equation (8) gives Equation (33) which is used to evaluate the working volume (V

_{L}):

_{f}, as illustrated in Figure 2b. Furthermore, there is a trade-off between F and μ, as can be seen from Equation (33), because they both decrease as S

_{f}decreases and thus trade-off against each other in the determination of V

_{L}, which produces the minimum. However, for this bioreaction, the minimum V

_{L}of about 4.9 m

^{3}occurs at a low value of S

_{f}at around 1 g·L

^{−1}and V

_{L}is only slightly larger at 5.1 m

^{3}at a very low S

_{f}value of 0.1 g·L

^{−1}. Consequently, for this bioreaction, it is desirable to operate at low steady-state sugar concentration (e.g., 0.1–1 g·L

^{−1}) because this reduces bioreactor working volume, feed sugar requirement and wasted sugar leaving the bioreactor.

_{OL}has a significant impact on the specific growth rate at lower values of around 1 mg·L

^{−1}and less, as illustrated in Figure 1a. Consequently, simulations were run to investigate the effect of C

_{OL}on the feed and wasted sugar and on the bioreactor working volume. Figure 3a shows the effect of C

_{OL}on the amount of wasted sugar, where it is shown that there were only very small differences. Similar trends were obtained for the feed sugar requirement and feed volumetric flowrate. Overall, C

_{OL}has very little impact on these variables.

_{OL}on the working volume of bioreactor, which shows a major impact, especially at low concentrations, e.g., at less than 0.5 mg·L

^{−1}. The specific growth rate is reduced by lowering C

_{OL}, which results in a larger bioreactor working volume, considering Equation (33), as the feed flowrate is not much affected. Consequently, the influence of C

_{OL}on the specific growth rate is directly impacting on the bioreactor working volume.

#### 3.2.2. Batch

^{−1}to a final concentration of 0.1 g·L

^{−1}. The low final concentration ensures that there is little wasted sugar (M

_{SW}= 0.1 kg·h

^{−1}) which consequently reduces the feed sugar requirement (M

_{SI}= 155 kg·h

^{−1}). Comparing this to the CSTB operated at S

_{f}= 1 g·L

^{−1}(and at the same C

_{OL}of 2 mg·L

^{−1}) shows that wasted and feed sugar is slightly greater for the CSTB at M

_{SW}= 1 kg·h

^{−1}and M

_{SI}= 155.7 kg·h

^{−1}(C

_{OL}= 2 mg·L

^{−1}). However, the working volume of the batch bioreactor is much greater (batch V

_{L}= 31 m

^{3}and CSTB V

_{L}= 5 m

^{3}). This is mainly due to the differences in cell concentration and in-part due to the down-time between batches. For the CSTB, the steady-state cell concentration is about 27 g·L

^{−1}while cell concentration increases slowly during the batch bioreaction from a low value of 0.25 up to around 27 g·L

^{−1}. Consequently, this lower bioreactor working volume is a major advantage for the CSTB, especially when it is operated at lower values of S

_{f}, as can be seen from Figure 2b.

_{OL}influences the bioreaction kinetics, which in turn influences the bioreaction time and the working volume of the batch bioreactor to produce a given amount of product. This is illustrated in Figure 4 and shows that the C

_{OL}only starts to significantly influence V

_{L}when it is reduced below around 1 mg·L

^{−1}, which is to be expected considering that K

_{O}is 0.363 mg·L

^{−1}.

#### 3.2.3. Effect of K_{S} and Sugar Concentration

_{f}. On one hand, a CSTB operated at S

_{f}values higher than the final batch concentration leads to higher feed sugar requirement and wasted sugar. On the other hand, operating at lower S

_{f}values near to the final batch value may slow the bioreaction kinetics and increase V

_{L}of the CSTB in comparison to the batch bioreactor, however this depends on K

_{S}. In this work, K

_{S}is given a value of 0.005 g·L

^{−1}and Figure 2a shows that S

_{f}can be operated at low values before V

_{L}minimises. Consequently, the CSTB can be operated at a low S

_{f}value providing the benefits of both lower V

_{L}and lower feed/wasted sugar. In practice, a wide range of K

_{S}values have been reported typically varying from 0.00007 to 0.2 g·L

^{−1}(as highlighted in Section 2.1), although Znad et al. [22] reported an extremely high value of 130 g·L

^{−1}for a fungal bioreaction. Simulations were performed at three K

_{S}values of 0.005, 1 and the extreme value of 130 g·L

^{−1}to investigate the effect of K

_{S}and S

_{f}on the bioreactor working volume. These results are presented in Figure 5 and it can be seen that for typical K

_{S}values, the CSTB can be operated at low S

_{f}values where V

_{L}is maintained at or close to its minimum. This results in a V

_{L}lower than that of batch mainly because of the higher cell concentration in the CSTB and also partly because it has no downtime between batches. This has also the benefit of greatly reducing the wasted sugar and the consequential feed sugar requirement.

_{S}(= 130 g·L

^{−1}) shows that the minimum V

_{L}occurs at a S

_{f}value of about 50 g·L

^{−1}, and thus the selection of S

_{f}represents a trade-off between reducing V

_{L}and reducing feed/wasted sugar, as highlighted by Fitzpatrick et al. [30]. This is an extreme case, but may occur in some bioreactions, such as pelleted fungal bioreactions, possibly due to sugar mass transfer resistance.

#### 3.3. Electrical Power Requirement for Oxygen Transfer

_{OL}) influences cell growth and the oxygen mass transfer driver, which in turn influence the OTR and the electrical power requirement. Thus, the initial part of the analysis on power requirement is conducted at a constant C

_{OL}value of 2 mg·L

^{−1}, and the effect of varying C

_{OL}is presented later.

#### 3.3.1. CSTB—Effect of vvm, Agitator Mechanical Power and Steady-State Sugar Substrate Concentration

_{OL}at a constant value of 2 mg·L

^{−1}. This is achieved by a combination of air flow and mechanical agitation, subject to the constraints of agitator flooding and phase equilibrium outlined in Section 2.4. The OTR and the power requirements of the compressor and agitator will depend on the OUR, which in turn is influenced by S

_{f}, as illustrated in Figure 6. The OUR increases as S

_{f}decreases, because more of the feed substrate is being utilized resulting in higher cell concentrations, which results in higher OUR according to Equation (14). However, Figure 6 also shows that the total OUR in the bioreactor volume remains fairly constant, and this is because the bioreactor volume decreases as S

_{f}decreases, as illustrated in Figure 2b, which counteracts the effect of the increasing OUR.

_{f}. Data for S

_{f}= 5 g·L

^{−1}are presented in Figure 7. This shows there can be a major variation in total electric power requirement (i.e., compressor + agitator power), to supply the same OTR, depending on the selection of vvm. Consequently, care needs to be taken in selecting vvm, so as to avoid excessive total power requirement, and this can result in major savings in aeration system energy requirement.

_{L}a correlation Equation (19), the values of OUR and bioreactor working volume will all influence the minimum total power requirement and whether or not it is constrained by flooding. Fitzpatrick et al. [2] examined this by performing simulations for a typical range of values for OUR and bioreactor working volume and for five different sets of constants in the k

_{L}a correlation equation (which were originally presented by Benz (2013). These simulations were applied to a system with similar bioreaction kinetics as in this work. This analysis showed that the minimum total power requirement tended to be constrained by flooding for many of the scenarios, and was close to the value at flooding for those scenarios where the minimum occurred before the onset of flooding. Consequently, the modelling showed that the minimum or near-minimum total power requirement occurred when operating at the onset of impeller flooding.

#### 3.3.2. CSTB—Agitator and Compressor Power Requirement that Minimises Total Electrical Power for Aeration

_{f}. These were evaluated as a function of S

_{f}for C

_{OL}= 2 mg·L

^{−1}, and these data are presented in Figure 8. These data show that the minimum total power does not majorly change and there is a gradual reduction from 22.5 kW (at S

_{f}= 1 mg·L

^{−1}) to 19.5 kW (at S

_{f}= 120 g·L

^{−1}).

_{L}a correlation Equation (19) will have an impact on the calculated total power requirement for oxygen transfer. Fitzpatrick et al. [30] examined this by investigating the effect of five different sets of constants in the k

_{L}a correlation equation (which were originally presented by Benz (2013) and regarded as providing a somewhat typical variation). This was applied to a system with similar bioreaction kinetics as in this work. The analysis showed that the correlation requiring the highest power had a power requirement of just over double that of the lowest power requirement correlation. The correlation applied in this work is close to that displaying the highest power requirement.

#### 3.3.3. CSTB—Effect of Steady-State Oxygen Concentration (C_{OL})

_{OL}will influence bioreaction kinetics, which in-turn can influence the working volume of the bioreactor, feed flowrate to the bioreactor, amount of feed sugar required and wasted or unutilized sugar leaving the bioreactor, as discussed earlier. It will also influence the electrical power requirement for the aeration system, because lowering C

_{OL}will increase the oxygen mass transfer driving force which will reduce the k

_{L}a required to supply a required OTR. Consequently, this section investigates the influence of C

_{OL}on these aspects.

_{OL}has a beneficial effect of reducing the total power requirement. This is because of an increase in the oxygen mass transfer driver which results in a lower k

_{L}a required to the deliver the OTR to satisfy the OUR requirement. However on the other hand, Section 3.2 shows that reducing C

_{OL}can increase the working volume of the bioreactor. This results in a trade-off between aeration system power requirement and bioreactor working volume, and this is illustrated in Figure 10.

#### 3.3.4. Batch—Agitator and Compressor Power Requirement

_{OL}to be less than a value that causes the microbes to die due to oxygen starvation). For example, a batch bioreactor may be operated at constant values of Pag and vvm; it may be operated such that C

_{OL}is controlled at a constant value by maintaining Pag constant and varying vvm throughout the bioreaction.

_{OL}, where it can be applied over the whole of the bioreaction time to evaluate the combinations of vvm and Pag that minimise the total electrical power requirement at each time increment throughout the bioreaction. Consequently, continuously controlling the bioreactor at these optimal combinations of Pag and vvm throughout the entire bioreaction provides the minimum total energy requirement for oxygen transfer for the bioreactor.

_{OL}varying from 0.1 to 6 mg·L

^{−1}. From this the corresponding average electrical powers were calculated and these are presented in Figure 11. This shows that reducing C

_{OL}leads to lower total electrical power/energy requirement for oxygen transfer, which is due to the higher mass transfer driver. Like the CSTB, there is an inherent process design trade-off whereby lower C

_{OL}which leads to lower electrical power requirement on one hand but leads to larger bioreaction times and larger bioreactor working volumes on the other hand, and this is also illustrated in Figure 11.

#### 3.4. Refrigeration Electrical Power Requirement for Cooling

_{OL}. In Section 2.6, it was assumed that the main sources of bioreactor heat production were metabolic and agitation.

_{OL}. Figure 12b shows that the refrigeration power requirement is significant. There is little variation up to about C

_{OL}= 2 mg·L

^{−1}, where the refrigeration power requirement is around 13.3 kW, after which it gradually increases up to 17.8 kW at C

_{OL}= 6 mg·L

^{−1}, due to the increased agitator heat dissipation shown in Figure 12a.

_{OL}. At C

_{OL}= 2 mg·L

^{−1}, the refrigeration, agitator and air compressor power requirements represent about 40%, 28% and 32%, respectively of their combined total. It should be kept in mind that the agitator and air compressor powers presented in Figure 12b represent values that minimised the power requirement for oxygen transfer and that the COP of the refrigeration system was estimated as 8.6, as these factors will influence the comparison. Overall, the electrical power requirements (both for oxygen transfer and refrigeration) were very similar for both batch and CSTB.

#### 3.5. Process Design Optimisation/Trade-Offs and Comparison of Batch and CSTB

- Working volume of the bioreactor
- Amount of feed sugar required
- Amount of wasted or unutilised sugar leaving the bioreactor
- Electric energy requirement for oxygen transfer and cooling
- Greenhouse gas emissions associated with electric energy supply
- Cost

_{f}) needs to be selected. Here, there is a potential trade-off between lower S

_{f}values leading to lower feed and wasted substrate (and associated environmental impact) on one hand and the potential for higher bioreactor volumes on the other hand due to bioreactor volume displaying a minimum value as illustrated in Figure 2b. However the effect on bioreactor volume and the position of this minimum is greatly influenced by the value of K

_{S}, as illustrated in Figure 5. The result of these simulations showed that for typical values of K

_{S}, S

_{f}could be chosen as a low value, such as 1 g·L

^{−1}, and thus have the dual benefit of approaching both minimum volume and minimum feed/wasted substrate. In terms of comparing bioreactor volume requirement for batch and CSTB, the simulations showed that the CSTB required a much smaller volume, especially when operated at lower values of S

_{f}, and this is a major advantage of CSTB over batch.

_{OL}is crucial. Decreasing C

_{OL}causes a decrease in electrical energy requirement (and associated greenhouse gas emissions), however this tends to increase the working volume of the bioreactor for both batch and CSTB. This is especially true when values of C

_{OL}decrease below 1 mg·L

^{−1}, where there tended to be an exponential increase in the working volume. Consequently, there is no value of C

_{OL}that satisfies all the desired objectives and thus a compromise must be sought to zone in on a value that is considered satisfactory. Output from the simulations may help in the determination of such satisfactory values. For, example, selecting a C

_{OL}value in the range of 0.5 to 2 mg·L

^{−1}(Figure 10) appears to represent a good compromise between the working volume of the bioreactor and the electrical power requirement. A more structured optimisation approach could be applied to evaluate a value of C

_{OL}that minimises the economic cost subject to constraints (such as limits on greenhouse gas emissions).

## 4. Conclusions

_{OL}. It was shown that the minimum or near-to-minimum total energy requirement occurred when operating at the onset of flooding throughout the bioreaction.

_{OL}) and CSTB steady-state sugar concentration (S

_{f}) are two important input process design variables that can impact on important design output variables, such as the working volume of the bioreactor, energy requirements and impacts on the environmental. Decreasing C

_{OL}has the beneficial effect of reducing the aeration system energy requirement and associated carbon footprint, however at the same time, it can slow down the bioreaction leading to the need for a larger sized bioreactor and associated cost. This shows that varying C

_{OL}and S

_{f}may be beneficial for some design output variables but may be detrimental to the values of others. Consequently, compromises and trade-offs are required to determine superior process designs. Mathematical modelling can assist in more precisely zoning in quantitatively on the selection of values for key input design variables, such as C

_{OL}and S

_{f}. This can be coupled with economic and environmental optimisation that can help produce a best compromise between conflicting design output variables. The mathematical modelling can also highlight design sensitivities to changes in input variable values which can cause undesirable adverse changes in key design variables. Finally it is important that the model equations and values of constants in the equations are appropriate and representative.

## Funding

## Conflicts of Interest

## Abbreviations

A_{T} | cross-sectional area of bioreactor (m^{2}) |

C_{OG} | oxygen concentration in air bubble (mg L^{−1}) |

C_{OGI} | oxygen concentration in air entering bioreactor (mg L^{−1}) |

C_{OGO} | oxygen concentration in air leaving bioreactor (mg L^{−1}) |

C_{OL} | oxygen concentration in the bioreaction liquid (mg L^{−1}) |

CSTB | continuous stirred tank bioreactor |

D | impeller diameter (m) |

D_{R} | dilution rate (h^{−1}) |

F | volumetric flowrate of feed entering bioreactor (m^{3} h^{−1}) |

F_{G} | inlet air volumetric flowrate (m^{3} h^{−1}) |

F_{S0} | mass flowrate of sugar entering bioreactor (kg h^{−1}) |

F_{SW} | mass flowrate of wasted sugar exiting bioreactor (kg h^{−1}) |

k_{L}a | volumetric oxygen mass transfer coefficient (h^{−1}) |

K_{O} | Monod kinetic constant for oxygen (g L^{−1}) |

K_{S} | Monod kinetic constant for sugar (g L^{−1}) |

M | Henry’s Law constant |

m_{S} | specific maintenance coefficient (h^{−1}) |

N | agitator rotational speed (s^{−1}) |

N_{A} | aeration number |

N_{Fr} | Froude number |

N_{P} | agitator power number (ungassed) |

N_{PG} | agitator power number (gassed) |

OUR | oxygen uptake rate (g L^{−1} h^{−1}) |

OTR | oxygen transfer rate (g L^{−1} h^{−1}) |

Pag | agitator mechanical power input in gassed bioreactor (kW) |

P_{atm} | atmospheric pressure (Pa) |

P_{b} | product concentration in batch bioreactor when bioreaction is completed (g L^{−1}) |

P_{C} | compressor mechanical power input (kW) |

P_{f} | steady-state product concentration in CSTB (g L^{−1}) |

P_{i} | atmospheric pressure + static head in bioreactor (Pa) |

P_{0} | concentration of any product in the feed (g L^{−1}) |

P_{R} | product production rate (kg h^{−1}) |

P_{tot} | sum of compressor and agitator electrical power inputs (kW) |

S_{b} | sugar concentration in batch bioreactor when bioreaction is completed (g L^{−1}) |

S_{f} | steady-state sugar concentration in CSTB (g L^{−1}) |

S_{0} | concentration of sugar in the feed (g L^{−1}) |

t_{b} | bioreaction time in batch bioreactor (h) |

t_{d} | down time between batches in batch operation (h) |

T | bioreactor diameter (m) |

V_{L} | bioreactor working volume (m^{−3}) |

v_{s} | air superficial velocity (m h^{−1}) |

vvm | volume of air per minute per unit bioreactor working volume (min^{−1}) |

X_{f} | steady-state cell concentration in CSTB (g L^{−1}) |

Y_{XS} | Yield coefficient for biomass (g dry cell weight per g sugar) |

Y_{PS} | Yield coefficient for product (g product per g sugar) |

α, β | bioreaction model kinetic constants |

δ, Φ | OUR model constants |

µ | specific growth rate (h^{−1}) |

µ_{max} | maximum specific growth rate (h^{−1}) |

η_{C} | compressor isentropic efficiency |

η_{m} | electric motor efficiency |

η_{r} | refrigeration efficiency |

γ | isentropic exponent of compression |

## References

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**Figure 1.**Bioreaction kinetics (

**a**) effect of oxygen and sugar concentrations on specific growth rate, (

**b**) effect of CSTB steady-state sugar concentration on product concentration and cell growth. (C

_{OL}= 2 mg·L

^{−1}) and (

**c**) batch bioreaction kinetics (C

_{OL}= 2 mg·L

^{−1}).

**Figure 2.**Effect of CSTB steady-state sugar concentration on (

**a**) mass flowrates of feed and wasted sugar, and (

**b**) bioreactor working volume and feed flowrate [steady-state oxygen concentration is 2 mg·L

^{−1}].

**Figure 3.**Effect of CSTB steady-state oxygen concentration on (

**a**) wasted substrate and (

**b**) bioreactor working volume.

**Figure 4.**Effect of oxygen concentration on bioreaction time and bioreactor working volume for batch bioreactor.

**Figure 6.**Effect of CSTB steady-state sugar concentration on oxygen uptake rate (OUR) and total oxygen uptake rate in the bioreactor volume (OUR*V

_{L}) [steady-state oxygen concentration is 2 mg·L

^{−1}].

**Figure 7.**Effect of vvm on compressor, agitator and total electrical power requirements for a steady-state sugar concentration of 5 g·L

^{−1}[steady-state oxygen concentration is 2 mg·L

^{−1}].

**Figure 8.**Minimum total electrical power required (and corresponding compressor and agitator power requirements) at each CSTB steady-state sugar concentration value [steady-state oxygen concentration is 2 mg·L

^{−1}].

**Figure 9.**Effect of CSTB steady-state oxygen concentration on minimum total electrical power requirement. Each line is for a constant C

_{OL}value and each point represents the minimum total electrical power requirement at the specified steady-state sugar concentration.

**Figure 10.**Effect of CSTB steady-state oxygen concentration on minimum total electrical power requirement and bioreactor working volume [steady-state sugar concentration is 1 g·L

^{−1}].

**Figure 11.**Effect of oxygen concentration on average power requirement for oxygen transfer and bioreactor working volume in a batch bioreactor.

**Figure 12.**Cooling requirement in batch bioreactor: (

**a**) rate of metabolic heat production and agitator heat dissipation; (

**b**) comparison of refrigeration, agitator and air compressor average electrical power requirements.

μ_{max} (h^{−1}) | K_{S} (g·L^{−1}) | K_{O} (g·L^{−1}) | α | β (h^{−1}) | Y_{XS} | Y_{PS} | m_{S} (h^{−1}) |
---|---|---|---|---|---|---|---|

0.25 | 0.005 | 0.000363 | 2.9220 | 0.1314 | 0.55 | 1 | 0.025 |

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**MDPI and ACS Style**

Fitzpatrick, J.J.
Insights from Mathematical Modelling into Energy Requirement and Process Design of Continuous and Batch Stirred Tank Aerobic Bioreactors. *ChemEngineering* **2019**, *3*, 65.
https://doi.org/10.3390/chemengineering3030065

**AMA Style**

Fitzpatrick JJ.
Insights from Mathematical Modelling into Energy Requirement and Process Design of Continuous and Batch Stirred Tank Aerobic Bioreactors. *ChemEngineering*. 2019; 3(3):65.
https://doi.org/10.3390/chemengineering3030065

**Chicago/Turabian Style**

Fitzpatrick, John J.
2019. "Insights from Mathematical Modelling into Energy Requirement and Process Design of Continuous and Batch Stirred Tank Aerobic Bioreactors" *ChemEngineering* 3, no. 3: 65.
https://doi.org/10.3390/chemengineering3030065