# Application of Mathematical Modelling to Reducing and Minimising Energy Requirement for Oxygen Transfer in Batch Stirred Tank Bioreactors

^{*}

## Abstract

**:**

## 1. Introduction

_{L}a) under direct operational control are agitator mechanical power input per unit volume (Pag/V

_{L}) and the superficial velocity (v

_{s}). Increasing v

_{s}will increase k

_{L}a simply because there is more air throughput but this is only up until the impeller starts to flood with air, after which the k

_{L}a will decrease. Consequently, the superficial air velocity must be less than the flooding value, which in turn is a function of the agitator Pag/V

_{L}. Measurement and evaluation k

_{L}a and how it is influenced by Pag/V

_{L}and v

_{s}is crucial in the design, operation and scale-up of bioprocesses [1,15,16,17,18,19,20].

- Air compressor power requirement
- Agitator power requirement

^{3}and larger using conventional agitators.

## 2. Mathematical Modelling

^{3}. A six bladed Rushton turbine impeller was used with a standard design configuration. The height to tank diameter ratio was 1. The tank diameter and cross-sectional area were 2.94 m and 6.8 m

^{2}, respectively. The impeller to tank diameter ratio was 0.35, giving an impeller diameter of 1.03 m.

#### 2.1. Bioprocess Kinetics and Oxygen Uptake

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

_{max}, K

_{S}and K

_{O}are constants.

_{max}and K

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

_{max}varied from 0.09–4.2 h

^{−1}[27,28,29,30]. K

_{S}values are typically in the mg·L

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

^{−1}[27,31,32], although Znad et al. [28] 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}[28,32,33]. The values of Y

_{XS}, Y

_{PS}and m

_{S}were obtained from van’t Riet and Tramper [27]. The values of α and β were obtained from Znad et al. [28]. Concentration values at the start of the bioprocess were X

_{0}= 0.1 g·L

^{−1}, S

_{0}= 150 g·L

^{−1}and P

_{0}= 0 g·L

^{−1}, and the bioprocess was completed when S was reduced to 0.1 g·L

^{−1}.

^{−1}were selected based on a range of values provided in an OUR review article by Garcia-Ochoa et al. [34].

#### 2.2. Oxygen Transfer and Agitator Mechanical Power

_{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 ambient air (C

_{OGI}= 280 mg·L

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

_{OGO}). Thus Equation (8) was used to evaluate an average equilibrium concentration of oxygen in the liquid.

_{G}is the air volumetric flowrate and V

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

_{L}a, agitator mechanical power input (Pag) and air superficial velocity (v

_{s}) is given by Equation (10).

_{1}= 0.4 and n

_{2}= 0.5, and these were obtained from van’t Riet [35]. For a particular bioprocess, the values for these constants need to be determined, as these will be influenced by the physicochemical properties of the bioprocess broth, including its viscosity.

_{T}is the cross-sectional area of the bioreactor.

^{−1}.

#### 2.3. Flooding and Phase Equilibrium Constraints

_{GF}) at the onset of flooding, for a fixed value of Pag, was evaluated by solving Equations (13) to (15). Equation (13) was obtained from Bakker et al. [14].

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

_{PG}is the impeller power number. N

_{PG}varies with the air flowrate, and Equation (15) was applied to take this into account. This equation was obtained from Bakker et al. [14] 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 and d = 0.25 (flat-bladed turbine impeller).

_{OGO}) was limited by the phase equilibrium constraint in Equation (16).

#### 2.4. Aeration System Energy Requirement

_{atm}is atmospheric pressure and Pi is the atmospheric pressure plus the static pressure acting on the bottom of the bioreactor due to the 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 energy requirements (E

_{tot}) for the duration time of the bioprocess (t

_{f}) is given by Equation (18).

_{m}is the electric motor efficiency (assumed to be constant at 0.9 for both the agitator and compressor).

#### 2.5. Modelling Scenarios

**Scenario K**: Both Pag and vvm are maintained as constant. This may occur in some bioprocesses which are operated under fixed conditions of agitator power input and air flowrate.**Scenario C**: The oxygen concentration in the liquid is kept constant throughout the bioprocess. This represents a practical control strategy by monitoring oxygen concentration in the liquid and controlling either agitator power input or air flowrate or both to maintain the oxygen concentration at a constant value. This was implemented for the following sub-scenarios:**Scenario C1**: Pag is kept constant throughout the bioprocess and vvm is varied. From an oxygen transfer energy perspective, it is later shown that dividing the bioprocess time up into two or more time segments with different constant Pag values and varying vvm in each time segment significant energy savings may be obtained (this is referred to as**Scenario C1-N**, where N is the number of time segments).**Scenario C2**: vvm is kept constant and Pag is varied.**Scenario C3 and Cmin**: Scenario C3 is where both Pag and vvm are varied. It will be shown later that a Pag-vvm combination profile over time can be evaluated that minimises the total electrical energy required for a specified constant oxygen concentration in the liquid. This is referred to as scenario Cmin.

#### 2.6. Model Implementation

_{OLf}):

_{OLi}is the oxygen concentration at the beginning of the time step.

## 3. Results and Discussion

#### 3.1. Bioprocess Progression

^{−1}for much of the bioprocess time. The bioprocess progresses slowly up to about 10 hours, after which there are significant changes in concentrations; the bioprocess ends at about 24 hours. The evolution of OUR is also presented in Figure 1a, where the OUR reaches a maximum of 4.7 g·L

^{−1}·h

^{−1}towards the end of the bioprocess. Figure 1b shows the influence of oxygen and sugar concentrations in the bioprocess liquid on the specific growth rate. The oxygen concentration has a major impact on specific growth rate. The specific growth rate is near to its maximum at the high oxygen concentration of 8 mg·L

^{−1}. It slowly decreases as the oxygen concentration is reduced to about 2 mg·L

^{−1}, after which there is a more precipitous decrease (it may be noted that K

_{O}is 0.363 mg·L

^{−1}). On the other hand, the sugar concentration during the bioprocess 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 1b.

#### 3.2. Scenario K

^{−1}. This constraint on oxygen concentration places a lower value constraint on vvm for a given Pag. The flooding and oxygen starvation constraints are presented in Figure 2. The region between the flooding curve and the oxygen starvation curve represents feasible combinations of vvm and Pag that will allow the bioprocess to achieve completion. From Figure 2, the minimum feasible Pag is around 15 kW.

^{−1}over the first 10 hours, which is expected as the OUR is low over this time period. For the lower vvm values, C

_{OL}decreases more rapidly to lower levels, which results in longer bioprocess times.

#### 3.3. Scenario C

_{OL}is maintained at a constant value throughout the bioprocess. This may be achieved by keeping vvm constant and varying Pag, or by keeping Pag constant and varying vvm, or by varying both Pag and vvm. This results in a number of corresponding sub-scenarios which are investigated in the following sections. Firstly, the constant values of C

_{OL}to be investigated must be stated, and this is highly influenced by the impact of C

_{OL}on bioprocess kinetics. C

_{OL}can vary between 0 and 8 mg·L

^{−1}and Figure 1b shows the impact of C

_{OL}on specific growth rate. Considering this, the following constant values of C

_{OL}were investigated: 5, 3, 2, 1, 0.4 and 0.1 mg·L

^{−1}.

#### 3.4. Scenario C1

_{OL}is maintained constant by varying vvm for a constant value of Pag throughout the bioprocess. Once Pag is fixed, the flooding velocity is also fixed. Below a certain critical value of Pag, the vvm required at maximum OUR would exceed the flooding velocity and C

_{OL}could not be maintained constant. At and above this critical value of Pag, vvm could be varied to maintain a constant C

_{OL}throughout the bioprocess. Consequently, the bioprocess needs to be operated at a Pag value equal to or above the critical Pag in order to maintain constant C

_{OL}throughout the bioprocess. At higher values of Pag above the critical value, the vvm required would decrease as expected. Furthermore, in the earlier part of the bioprocess when OUR is low, the vvm values required were low enough to break the constraint in Equation (16), giving rise to Cogo being less than M × C

_{OL}. This constraint cannot be broken and it thus limits the OTR that can be provided in Equation (9). To rectify this, vvm was increased so that Cogo = M × C

_{OL}.

_{OL}, the agitator, compressor and total energies were evaluated for agitator power values at and above the critical Pag value. Figure 6 presents total energy data as a function of constant Pag for three different values of C

_{OL}. It shows that the constant value of Pag chosen has a major impact on total energy requirement and thus care needs to be taken in choosing this value. For a fixed C

_{OL}, the minimum total energy occurs at the critical Pag, which is the value of Pag where flooding will start to occur at the time when OUR is a maximum. This is referred to as C1min. Considering this, it may be more prudent to operate at Pag values above the critical Pag due to uncertainties associated with flooding correlations.

_{OL}decreases. Less vvm is required at lower C

_{OL}values because the maximum OUR is less and the mass transfer driver is greater, meaning the critical Pag is lower. Figure 7 presents the minimum total energy at discrete values of C

_{OL}between 0.1 and 5 mg·L

^{−1}. The lowest minimum total energy was about 3.8 GJ, which occurred at 0.1 mg·L

^{−1}and after a greatly increased bioprocess time of 98 hours. Thus, there is a trade-off between lower minimum total energy values and higher bioprocess times as C

_{OL}is reduced.

#### 3.5. Scenario C2

_{OL}is maintained constant by varying Pag for a constant value of vvm throughout the bioprocess. There exists a minimum vvm that must be supplied to meet the maximum OUR and not break the constraint given in Equation (16), i.e., ${C}_{OGO}\ge M\left({C}_{OL}\right)$. The maximum OUR was evaluated previously and the minimum v

_{S}(v

_{S_min}) was estimated, using Equations (9), (11) and (16), to formulate Equation (20), noting that OTR equals OUR when C

_{OL}is constant. This was then used to evaluate the minimum vvm.

_{OL}at a constant value in this scenario was shown to be impossible and thus this scenario was not investigated further. At vvm values higher than the minimum, it was found that the Pag required, for much of the bioprocess, was so low that the flooding vvm was less than vvm, meaning the impeller was flooding. To overcome this flooding problem, the Pag had to be increased to the onset of the flooding value, which increased OTR and caused C

_{OL}to deviate from its constant value for much of the bioprocess.

#### 3.6. Scenario C3

_{OL}is maintained constant by varying both vvm and Pag throughout the bioprocess. At a specific time during the bioprocess, many different combinations of vvm and Pag can be evaluated to provide an OTR that satisfies the OUR at that time in order to maintain C

_{OL}at a constant value [22,23]. However, the values of vvm and Pag are subject to the constraints of phase equilibrium and agitator flooding outlined in Section 2.3. A typical evolution of OUR over the bioprocess time is given in Figure 1a. For a constant C

_{OL}of 2 mg·L

^{−1}, the OUR varies from around 0 up to 4.55 g·L

^{−1}·h

^{−1}. Simulations were performed to evaluate the effect of vvm on the power requirement of the compressor and agitator that could supply the OTR required to satisfy the OUR in the range of 0 up to 4.55 g·L

^{−1}·h

^{−1}. Figure 8 presents data for an OUR value of 4.5 g·L

^{−1}·h

^{−1}. As expected, compressor power increases as vvm increases and there is a corresponding decrease in agitator power to supply the OTR required to meet the OUR. However, compressor power increases linearly with vvm while agitator power decreases in an exponential fashion. The trade-off between compressor and agitator power results in a value of vvm that minimises total power. In the simulations for OUR values in the range of 0 up to 4.55 g·L

^{−1}·h

^{−1}, this minimum tended to be located at a value of vvm that was beyond the flooding value. Consequently, the minimum total power that also satisfies the constraints was located at the onset of flooding. Figure 8 also highlights that care needs to be taken in the selection of vvm, so as to remain within constraints, especially the flooding constraint, and to avoid excessive total power requirement at lower vvm, due to excessive agitator power. Consequently, major savings in aeration system energy requirement can be made by careful selection of vvm and agitator power input requirement throughout the bioprocess, and mathematical modelling can assist in this selection.

#### 3.7. Scenario Cmin

_{OL}= constant), which is referred to as Scenario Cmin. This analysis was applied to a bioprocess where C

_{OL}was kept constant at 2 mg·L

^{−}

^{1}. Figure 9 shows the variation in vvm, compressor power, agitator power and total power over the course of the bioprocess. The values of vvm and Pag that minimised total power were at the onset of flooding throughout all of the bioprocess. The oxygen concentration in the air leaving the bioreactor (C

_{OGO}) was consistently high throughout the bioprocess, ranging from about 225 to 250 mg·L

^{−}

^{1}, which is not too far below the inlet oxygen concentration of 280 mg·L

^{−}

^{1}.

_{L}a correlation and the constraints (in particular the flooding constraint). Less vigorous aerobic bioprocesses have maximum OUR values of about 1 g·L

^{−}

^{1}·h

^{−}

^{1}while vigorous aerobic bioprocesses have OUR values of around 7 g·L

^{−}

^{1}·h

^{−}

^{1}[26]. Kreyenschulte et al. [26] have shown that the minimum total power requirement also occurs at the onset of flooding, but only for bioreactor sizes of around 20 m

^{3}and larger (the flooding constraint is irrelevant for smaller bioreactors using conventional agitators). Consequently, simulations were also performed at 2.5 m

^{3}and 100 m

^{3}for OUR values in the range of 1–7 g·L

^{−}

^{1}·h

^{−}

^{1}for the k

_{L}a correlation constants used in the current study. The results are presented in the first line of Table 3 and show that the minimum total power requirement was constrained by the onset of flooding for eight of the nine simulations presented. For the other simulation, the total power requirement at flooding was about 0.5% greater than the minimum. These results suggest that the minimum (or near minimum) total energy requirement during a bioprocess can be obtained by operating at the onset of flooding.

_{L}a correlation constants used in the current study (correlation 1 in Table 2). Consequently, further simulations were performed using four other k

_{L}a correlations presented in Table 2 and obtained from Benz [20], van’t Riet [35] and Bakker [14]. This was to further evaluate if the minimum (or near minimum) total power occurred at the onset of flooding. The results showed that three of the other correlations behaved similarly to the first correlation (Table 3), with minimum total power being constrained by flooding or having a value within 0.5% of that at flooding. However, for correlation 2 (Table 3), the minimum total power occurred before flooding for all the simulations. The total power at onset of flooding was about 1–25% greater than the minimum total power value, with higher values occurring at the smaller volume and bigger OUR values, which is consistent with the work of Keyenschulte et al. [26]. Overall, the data presented for minimising total energy in Scenario C suggest that the vvm-Pag combinations should be controlled so that they are at or are close to the onset of flooding to supply an OTR that equals the OUR throughout the bioprocess, although this does depend on the k

_{L}a correlation constants and the factors that influence the onset of flooding.

_{OL}will influence the energies, as highlighted earlier, but there is a trade-off. Lower C

_{OL}values require lower k

_{L}a values and thus lower electrical power while on the other hand giving rise to longer bioprocess times which could require more energy. Cmin simulations were performed for a number of values of C

_{OL}and the results are presented in Figure 10. This shows that reducing C

_{OL}has a major impact on reducing energy; there is a minimum energy but it occurs at a very low C

_{OL}value around 0.05 mg·L

^{–1}. It is beneficial for the simulated bioprocess to operate at this low C

_{OL}value from an oxygen transfer energy perspective. However the bioprocess time was 166 hours, which is much greater than the 26.6 hours required for C

_{OL}= 2 mg·L

^{−1}, and this may be very disadvantageous from other perspectives such as the size of a bioreactor required to provide a specified productivity.

#### 3.8. Scenario C1-N

_{OL}value of 2 mg·L

^{−}

^{1}. It can be seen that there is a minimum total energy of 2.43 GJ, which occurred when the first time segment had a duration of around 19 hours and the corresponding Pag at the onset of flooding was 8 kW. A comparison is presented in Table 6 of this time segment configuration (C1-2min) with the two C1-2 configurations presented in Table 4 and Table 5. It shows that the duration of the first-time step and size of the Pag step increase is in between those of the other two configurations. It highlights that the total energy requirement is significantly lower. However, this consequential energy saving (due to segment configuration) does diminish as the number of time segments increases.

#### 3.9. Comparison of Scenario Minimum Total Energy Requirements

_{OL}. It also provides the percentage energy savings associated with them when compared to minimum requirement in scenario C1 (C1min). Table 7 shows significant variation between the minimum total energies for each scenario and thus appropriate selection of Pag and vvm can make major energy savings. Cmin requires only 32% of the total energy of C1min (C

_{OL}= 2 mg·L

^{−}

^{1}). The use of two appropriately chosen constant Pag time segments, as in C1-2min, provides a 46% energy saving over C1min. Furthermore, this saving represents 67% of the energy difference between C1min and Cmin. As expected, reducing C

_{OL}can have a significant impact in reducing energy.

_{OL}= 2 mg·L

^{−}

^{1}). The reason for this is somewhat complex but it is essentially because the constant Pag is much higher at 39 kW for C1min than 15 kW for Kmin. This is because the C

_{OL}value is constrained in Cmin at 2 mg·L

^{−}

^{1}, while it reduces to the oxygen starvation constraint value of 0.01 mg·L

^{−}

^{1}in Kmin, as highlighted in Figure 4, where OUR increases to is highest values. The lower oxygen concentration provides a higher mass transfer driver which reduces the power requirement to provide the same OTR. Furthermore, the very low oxygen concentration did not negatively impact bioprocess time because it only occurred at the end of the bioprocess.

## 4. Conclusions

_{OL}in Scenario C) can result in much lower total energy requirements while carrying out the same bioprocess. For example, in Scenario K the minimum total energy was 3.2 GJ at Pag = 15 kW, while it was 5.4 GJ at Pag = 45 kW. This is a large difference depending on the selection of Pag. Consequently, mathematical modelling of microbial kinetics, oxygen transfer and energy analysis can assist in this selection process. The modelling also highlights sensitivities of the operating conditions to oxygen concentration in the liquid, oxygen starvation, impeller flooding and phase equilibrium constraints. For Scenario K, using the k

_{L}a correlations in this study, the minimum total energy occurred at the lowest feasible Pag value subject to the constraints. However, this occurred at a very sensitive point, being at the intersection of the flooding and oxygen starvation constraints (Figure 2).

_{OL}) the minimum total energy occurred at the lowest value of Pag such that the onset of flooding only occurred at maximum OUR. Overall, for Scenario C, the minimum total energy requirement was obtained when both Pag and vvm could be varied over time. For typical OUR values, it was shown that the minimum (or near minimum) total energy requirement tended to occur when operating at the flooding constraint throughout the bioprocess.

_{OL}has a major impact on total energy requirement, although there is a potential trade-off between oxygen mass transfer rate and microbial kinetics. In the bioprocess studied, the optimum C

_{OL}value in Scenario C, which minimised total energy, was very low, but this may be problematic from other perspectives such as longer bioprocess time and the bioreactor size required to produce a given amount of product. However, it may be beneficial to operate at higher C

_{OL}values of, say, 2 mg·L

^{−}

^{1}when OUR and power requirements are relatively low, and then to operate at lower C

_{OL}values when OUR increases towards its highest values to reduce the much higher power requirement (and overall total energy requirement) without having significant negative impacts on bioprocess time.

_{OL}, can help minimise the energy required, and mathematical modelling can assist in this process.

## Author Contributions

## Funding

## Conflicts of Interest

## List of Symbols

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 bioprocess broth (mg·L^{−1}) |

D | impeller diameter (m) |

Eag | agitator electrical energy (GJ) |

Ec | compressor electrical energy (GJ) |

E_{tot} | sum of agitator and compressor electrical energy (GJ) |

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

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

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

K_{R} | Monod kinetic constant (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}) |

P | production concentration (g·L^{−1}) |

Pag | agitator mechanical power input (kW) |

P_{atm} | atmospheric pressure (Pa) |

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

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

S | sugar concentration (g·L^{−1}) |

S_{R} | rate-limiting substrate concentration (g·L^{−1}) |

t | time (h) |

t_{f} | bioprocess time (h) |

T | bioreactor diameter (m) |

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

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

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

X | cell concentration (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 | product production rate model constants |

δ, Φ | OUR model constants |

µ | specific growth rate (h^{−1}) |

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

γ | isentropic exponent of compression |

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**Figure 1.**Microbial kinetics: (

**a**) cell, sugar and product concentrations and oxygen uptake rate (OUR) (Scenario K: Pag = 45 kW; vvm = 1); (

**b**) effect of oxygen and sugar concentrations on specific growth rate.

**Figure 2.**Scenario K: Limiting constraints on vvm and agitator mechanical power: oxygen starvation and impeller flooding.

**Figure 3.**Scenario K: Influence of vvm on agitator (Eag), compressor (Ec) and total (Etot) electric energies and bioprocess time for Pag = 45 kW.

**Figure 4.**Scenario K: Evolution of oxygen concentration in bioprocess liquid when operating at Pag = 45 kW at vvm = 0.55, 1 and 4.3.

**Figure 5.**Scenario K: (

**a**) Influence of vvm on total electric power for constant values of Pag; (

**b**) Influence of Pag on minimum total electric energy requirement (Etot) and corresponding agitator (Eag) and compressor (Ec) electric energy inputs.

**Figure 6.**Scenario C1: Influence of constant values of Pag and oxygen concentration in the bioprocess liquid (C

_{OL}) on total electric energy requirement.

**Figure 7.**Scenario C1: Influence of oxygen concentration in the liquid on minimum total electric energy requirement.

**Figure 8.**Scenario C3: Influence of vvm on compressor, agitator and total electrical power requirements for OUR = 4.5 g·L

^{−1}·h

^{−1}(C

_{OL}= 2 mg·L

^{−}

^{1}).

**Figure 9.**Scenario Cmin: Evolution of vvm and compressor, agitator and total electrical power inputs (C

_{OL}= 2 mg·L

^{−}

^{1}).

**Figure 10.**Scenario Cmin: Influence of oxygen concentration in the liquid on bioprocess time (tf), and agitator (Eag), compressor (Ec) and total (Etot) electric energies (C

_{OL}= 2 mg·L

^{−}

^{1}).

**Figure 11.**Scenario C1-5: (

**a**) Segments with equal time durations: evolution of Pag and vvm (inset) and (

**b**) constant Pag step increase of 7.7 kW between time segments: evolution of Pag (onset of flooding occurs at end of each time segment; C

_{OL}= 2 mg·L

^{−}

^{1}).

**Figure 12.**Scenario C1-2: Influence of the duration of the first time segment on the total electric energy requirement and Pag (C

_{OL}= 2 mg·L

^{−}

^{1}).

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 |

k_{L}a Correlations and References | Constants | ||
---|---|---|---|

K | n1 | n2 | |

1—van’t Riet non-coalesce [van’t Riet, 1979] | 0.026 | 0.4 | 0.5 |

2—van’t Riet coalesce [van’t Riet, 1979] | 0.002 | 0.7 | 0.2 |

3—Benz broth 1 [Benz, 2013] | 0.015 | 0.55 | 0.6 |

4—Benz broth 2 [Benz, 2013] | 0.088 | 0.542 | 0.741 |

5—Bakker [Bakker, 1994] | 0.015 | 0.6 | 0.6 |

**Table 3.**Influence of k

_{L}a correlation (Table 2), working volume (V

_{L}) and OUR on whether or not minimum total power occurs at the onset of flooding in Scenario Cmin—Y: yes, minimum total power occurs at onset of flooding; N: no, it does not (C

_{OL}= 2 mg·L

^{−1}).

k_{L}a Correlation | V_{L} = 2.5 m^{3} | V_{L} = 20 m^{3} | V_{L} = 100 m^{3} | ||||||
---|---|---|---|---|---|---|---|---|---|

OUR (g·L^{−1}·h^{−1}) | OUR (g·L^{−1}·h^{−1}) | OUR (g·L^{−1}·h^{−1}) | |||||||

1 | 4 | 7 | 1 | 4 | 7 | 1 | 4 | 7 | |

1 | Y | Y | N | Y | Y | Y | Y | Y | Y |

2 | N | N | N | N | N | N | N | N | N |

3 | Y | Y | N | Y | Y | Y | Y | Y | Y |

4 | Y | Y | Y | Y | Y | Y | Y | Y | Y |

5 | Y | Y | N | Y | Y | Y | Y | Y | Y |

**Table 4.**Energy requirements (MJ) for the C1-N scenario with equal time segment durations (onset of flooding occurs at end of each time segment) and comparison with C1min and Cmin (C

_{OL}= 2 mg·L

^{−}

^{1}; % energy savings are relative to Scenario C1min).

Control Strategy | Agitator Eag | Compressor Ec | Total E_{tot} | Energy Savings (%) |
---|---|---|---|---|

C1min | 4097 | 460 | 4557 | - |

C1-N | ||||

C1-2 | 2166 | 464 | 2630 | 42.3 |

C1-5 | 1220 | 529 | 1749 | 61.6 |

C1-10 | 972 | 588 | 1560 | 65.7 |

Cmin | 746 | 705 | 1415 | 68.1 |

**Table 5.**Energy requirements (MJ) for C1-N scenario with constant Pag step increase from one time segment to the next. (C

_{OL}= 2 mg·L

^{−}

^{1}; onset of flooding occurs at end of each time segment).

Control Strategy | Agitator Eag | Compressor Ec | Total E_{tot} |
---|---|---|---|

C1-2 | 2279 | 519 | 2798 |

C1-5 | 1298 | 594 | 1892 |

C1-10 | 996 | 636 | 1632 |

**Table 6.**Energy requirements (MJ) for time segment configurations in C1-2 (the minimum energy requirement (C1-2min), the two time segment durations are the same (Δt

_{SEG}= constant), and the two Pag step increases are the same (ΔPag = constant)).

Control Strategy | Agitator Eag | Compressor Ec | Total E_{tot} | First Time Segment | |
---|---|---|---|---|---|

Pag (kW) | Duration (h) | ||||

C1-2min | 1787 | 642 | 2429 | 8 | 19 |

C1-2 (Δt_{SEG} = constant) | 2166 | 460 | 2630 | 2.4 | 13.3 |

C1-2 (ΔPag = constant) | 2279 | 519 | 2798 | 19.2 | 23.1 |

**Table 7.**Comparison of energy requirements (MJ) for the different scenarios at minimum total energy requirement within a scenario (C

_{OL}= 2 mg·L

^{−}

^{1}; % energy savings are relative to Scenario C1min).

Scenario | Bioprocess Time (h) | Agitator Eag | Compressor Ec | Total E_{tot} | % Energy Savings |
---|---|---|---|---|---|

Scenario K | |||||

Kmin | 28.8 | 1730 | 1475 | 3205 | 30 |

Scenario C | |||||

(C_{OL} = 2 mg·L^{−}^{1}) | |||||

C1min | 26.6 | 4097 | 460 | 4557 | - |

C1-2min | 26.6 | 1787 | 642 | 2429 | 46 |

Cmin | 26.6 | 746 | 705 | 1451 | 68 |

(C_{OL} = 0.4 mg·L^{−}^{1}) | |||||

C1min | 42.3 | 3560 | 320 | 3880 | 15 |

Cmin | 42.3 | 610 | 510 | 1120 | 75 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fitzpatrick, J.J.; Gloanec, F.; Michel, E.; Blondy, J.; Lauzeral, A. Application of Mathematical Modelling to Reducing and Minimising Energy Requirement for Oxygen Transfer in Batch Stirred Tank Bioreactors. *ChemEngineering* **2019**, *3*, 14.
https://doi.org/10.3390/chemengineering3010014

**AMA Style**

Fitzpatrick JJ, Gloanec F, Michel E, Blondy J, Lauzeral A. Application of Mathematical Modelling to Reducing and Minimising Energy Requirement for Oxygen Transfer in Batch Stirred Tank Bioreactors. *ChemEngineering*. 2019; 3(1):14.
https://doi.org/10.3390/chemengineering3010014

**Chicago/Turabian Style**

Fitzpatrick, John J., Franck Gloanec, Elisa Michel, Johanna Blondy, and Anais Lauzeral. 2019. "Application of Mathematical Modelling to Reducing and Minimising Energy Requirement for Oxygen Transfer in Batch Stirred Tank Bioreactors" *ChemEngineering* 3, no. 1: 14.
https://doi.org/10.3390/chemengineering3010014