Estimation of volumetric mass transfer coefficient (kLa)—Review of classical approaches and contribution of a novel methodology

Journal Pre-proof Estimation of volumetric mass transfer coefficient (kL a) - Review of classical approaches and contribution of a novel methodology Magdalini Aroniada, Sofia Maina, Apostolis Koutinas, Ioannis K. Kookos

PII:

S1369-703X(19)30397-3

DOI:

https://doi.org/10.1016/j.bej.2019.107458

Reference:

BEJ 107458

To appear in:

Biochemical Engineering Journal

Received Date:

24 April 2019

Revised Date:

17 October 2019

Accepted Date:

25 November 2019

Please cite this article as: Aroniada M, Maina S, Koutinas A, Kookos IK, Estimation of volumetric mass transfer coefficient (kL a) - Review of classical approaches and contribution of a novel methodology, Biochemical Engineering Journal (2019), doi: https://doi.org/10.1016/j.bej.2019.107458

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Estimation of volumetric mass transfer coefficient (kLa) - Review of classical approaches and contribution of a novel methodology

Magdalini Aroniadaa, Sofia Mainab, Apostolis Koutinasb,* [email protected] & Ioannis K.

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Kookosa,c,* [email protected]

a

Department of Chemical Engineering, University of Patras, 26504, Patras, Rio, Greece

b

Department of Food Science and Human Nutrition, Agricultural University of Athens, Iera Odos

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75, Athens 11855, Greece c

INVALOR: Research Infrastructure for Waste Valorization and Sustainable Management,

Equally contributed as corresponding authors: Apostolis Koutinas, Ioannis K. Kookos

Highlights

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*

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Caratheodory 1, University Campus, GR-26504, Patras, Greece

Various kLa estimation methodologies have been compared

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kLa is determined using experimental results only from gassing-in experiments Avoid oversimplified approaches in kLa estimation using fast oxygen (polarographic) probes

ABSTRACT The volumetric mass transfer coefficient, usually denoted by kLa, is an important parameter for both the design, scale-up and monitoring of aerated bioreactors. Significant research efforts have been 1

expended over the last decades in order to develop reliable and easy to apply methodologies for determining kLa using aeration experiment and fast oxygen (polarographic) probes. These methodologies have been reviewed in this study followed by the presentation of a new and promising methodology for kLa determination, which can be used when data from step experiments are not available. Step and aeration experiments have been carried out in laboratory scale bioreactors and the experimental data were used for the validation of the proposed methodology and

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also to critically assess all alternative methodologies for estimating kLa.

Keywords: aeration experiments, volumetric mass transfer coefficient, mathematical model, parameter estimation

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1. Introduction

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The volumetric mass transfer coefficient (kLa) is a parameter of paramount importance for the design, operation, scale-up and optimization of bioreactors [1-5]. Its reliable calculation has been

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the subject of extensive studies in the past and a number of different techniques are now available. As dissolved oxygen can become the limiting substrate and thus determine the efficiency of the

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operation of a bioreactor it is important to monitor a biotechnological process so as to ensure adequate supply of oxygen. The oxygen transfer rate (OTR) is influenced by the hydrodynamic conditions in a bioreactor that are determined be several parameters including bioreactor and impeller geometry, agitation speed, energy dissipation, physicochemical characteristics of the broth

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and the nature of the microorganism used.

Since their introduction in the mid 1950’s, DO electrodes or probes have found widespread

acceptance and wide range of applications [6-10]. As the DO probes are based on the use of an oxygen permeable membrane that covers the electrolyte, additional resistance to mass transfer of oxygen is introduced into the system. This needs to be taken into consideration in order to estimate reliably the volumetric mass transfer coefficient. Initial work was focused on the determination of

2

the diffusion coefficient of oxygen in polymeric membranes using step response experiments [11,12]. Aeration experiments were introduced latter and their mathematical analysis performed and elucidated [13-16]. The more comprehensive analysis of the membrane-covered polarographic probes is performed by Linek and Vacek [17] and Linek [18] that includes multi-region, multi-layer and non-uniform response analysis [19] and advanced analysis based on the method of moments [17,20]. In many cases, alternative but simplified methods for estimating kLa offer distinct advantages over the more elaborated methods (e.g. see ref. [21]).

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The aim of this work is to review the available modeling approaches and results for the estimation of the volumetric mass transfer coefficient using membrane covered probes and express the results on a consistent basis and notation. This is particularly important as different notations have been used in the literature making difficult to obtain a unified view of the scattered results.

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The conditions under which the more complicated models simplify to less complicated ones are

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discussed and expressed as a function of the parameters of the physical model (with the aim to stress further a unified view of the existing results). A novel approximation method for estimating

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the volumetric mass transfer coefficient is also proposed for the cases where experimental results from aeration (gassing-in) experiments are only available (a case commonly encountered when

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large bioreactors are studied). Both step response and aeration experiments were carried out and the data collected were used to investigate and compare the alternative methodologies. The conclusions drawn in this manuscript will be useful to the research community to avoid common misconceptions and oversimplified approaches when dealing with the problem of estimating

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reliably the volumetric mass transfer coefficient using fast oxygen (polarographic) probes. The new approximation methodology presented in this study can also be useful in the validation of kLa estimations obtained by alternative, more demanding and elaborated methodologies.

2. Material and methods 2.1. Experimental equipment

3

The experiments were conducted in a 6.7-L bench top bioreactor (Ralf Advanced, Bioengineering, Switzerland) and in a 3.6-L bench top bioreactor (Labfors 4, Infors HT, Switzerland). Agitation was provided by two Rushton impellers with six blades equally spaced around the disc of the impeller. Both vessels were equipped with four baffles placed symmetrically on the bioreactor wall. Fig. 1 and Table 1 show the schematic and dimensions of bioreactors and impellers used in this study. The experiments were performed with deionized water, at 28 ˚C and atmospheric pressure. Gas was supplied with a flow rate of 1 vvm through a ring sparger located below the lower impeller.

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The step response and the aeration response experiments were conducted for agitation speeds of 150, 250, 300, 400 and 600 rpm. The experiments were repeated 2 times at each process

condition and the mean values for all measurements were presented. The step change response was measured by transferring the oxygen probe from a nitrogen saturated vessel into the air saturated

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bioreactor. The dissolved oxygen (DO) concentration was recorded at 5 second intervals until DO

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saturation was reached. For aeration response experiments, the measurements were initiated by sparging the bioreactor with nitrogen the DO concentration was reduced to 0% of saturation. Then,

approached saturation.

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aeration was initiated and the DO concentration was recorded at 5 second intervals until it

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DO was measured using a polarographic Mettler Toledo electrode InPro 6000 with polytetrafluoroethylene (PTFE) coated membrane. The electrode consisted of a platinum (Pt) cathode and a silver (Ag) anode. Two-points calibration of the electrode was carried out prior to the experimental runs. Nitrogen was initially supplied into the bioreactor and the zero point was set

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when a zero steady-state level of DO concentration was achieved. Subsequently, the saturated value was adjusted by supplying air until a steady-state saturation concentration of DO was reached.

2.2. Oxygen dynamics in the liquid phase during an aeration (gassing-in) experiment When aeration of the well stirred liquid in a vessel (which has been kept free of oxygen by bubbling an inert gas) is initiated, oxygen diffuses from the air into the liquid. The model adopted is that of

4

the mass transfer between different phases and two resistances to oxygen transfer are considered: the resistance in the gas film and the resistance in the liquid film, as shown schematically in Fig. 2. There is a concentration gradient developed in both films and simple, order of magnitude calculations indicate that the resistance in the liquid film is the only significant resistance. Under the assumption of well mixed phases the concentration of the oxygen is uniform in each phase. The flux of oxygen from the gas phase to the liquid phase is proportional to the liquid mass transfer coefficient kL and the difference in the oxygen concentration between the bulk of the liquid (CO2)

kL (CO2,s CO2 )

(1)

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N O2

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and the oxygen concentration in equilibrium with the bulk of the gas phase (CO2,s):

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The mass balance of oxygen in the liquid phase can be written as

Ai N O2

(2)

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d (VLCO2 ) dt

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where Ai is the interfacial area available for mass transfer between the gas and the liquid. The liquid hold-up is constant and combining Eqs (1) and (2) the following equation is obtained:

kL a (CO2,s CO2 )

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dCO2 dt

(3)

where a=Ai/VL is the interfacial area for mass transfer per unit volume of the liquid into the vessel. However, the oxygen that is accumulated into the liquid is equal to the imbalance of the oxygen between the input and the output of the gas stream:

5

dCO2 dt

NG VL

pO2 ,in

pO2

NG H CO2,in, s VL P

P

CO2, s

(4)

where NG is the molar flowrate of air, P is the total pressure, pO2 (=HCO2,s)is the partial pressure of oxygen and H is the Henry’s law constant (for ideal gases the ratio of the partial pressure to the total pressure equals the mole fraction of a component). It is important to note that CO2,in,s is the concentration of oxygen in the liquid in equilibrium conditions with the incoming air stream. If the

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dynamics of the gas phase is considered important then an equation similar to Eq. (4) can be written for the gas phase. Adding Eq. (3) and (4) results in an equation that describes the oxygen dynamics

CO2

CO2,in, s

(5)

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1 dCO2 β dt

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in the liquid phase expressed in convenient quantities:

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where

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1 β

1 kLa

VL P NG H

(6)

It can be shown that (1/kLa)>>(VL/NG)(P/H) and it is usually the case that β kLa. When the gas

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stream is switched from an inert gas to air then CO2,in,s undertakes a step change and the solution of Eq. (5) gives the response of the concentration of oxygen in the liquid phase:

CO2,s (t ) CO2,in,s (1 e

βt

)

Eq. (7) predicts an exponential change of the oxygen concentration in the liquid during an aeration experiment. The concentration of oxygen in the liquid increases immediately after the 6

(7)

aeration is commenced. The rate of increase is maximum at the beginning and gradually slows down and finally becomes practically zero when time is greater than about 6/β. However, when an oxygen probe that features an oxygen permeable polymeric membrane (such as a polarographic probe) is used to monitor the oxygen dynamics a fundamentally different response is obtained. The response is characterized by a significant dead time (time interval at which no detectable change in the oxygen concentration is observed) followed by a rate of change in the measured oxygen concentration that is initially slow, then fast and finally decreases to zero. The reason for the

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observed dynamics is the additional and significant resistance to mass transfer introduced by the membrane and possibly by a hydrodynamic liquid film formed next to the membrane. As the

oxygen needs to overcome the liquid film, dissolve in the polymeric material and then diffuse

across to the membrane to reach the electrolyte, significant additional dynamics is introduced. The

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dynamics of the mass transfer phenomena in an oxygen permeable membrane needs first to be

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deconvoluted from the sensor readings (in order to obtain the dynamics of the mass transfer phenomena involved in the transfer of oxygen from the gas to the liquid phase) and then determine

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the volumetric mass transfer coefficient.

A well documented methodology for achieving this objective is to build an appropriate

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mathematical model for the transport phenomena taking place across the membrane and then use this model in order to reconstruct the liquid phase dynamics. This is presented in Fig. 3 where the basic elements of the problem under study in this work are presented. The system is interrogated using an appropriate input excitation (air is fed to the system following that of an inert gas such as

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nitrogen) and measurements are recorded (output of the system) using an appropriate fast oxygen response probe. As the real concentration of the oxygen in the liquid is unattainable the measurements are used together with the mathematical model of the probe so as to determine the actual input to the probe. At the final step the actual dynamics of the oxygen concentration in the liquid is used together with Eq. (7) to determine the volumetric mass transfer coefficient. The

7

mathematical models of the probe that have been proposed in the literature are presented in the section that follows.

2.3. Fast Oxygen Probe mathematical models and dynamics The main layers considered in the formulation of the mathematical model of the fast oxygen probe are shown in Fig. 4. The main part of the model consists of the mass balance of the diffusing oxygen in the oxygen permeable membrane of thickness L. The material balance is expressed by

2

C ( x, t ) x2

D

(8)

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C ( x, t ) t

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Fick's 2nd "law" of diffusion [6,7,22]:

where C(x,t) is the oxygen concentration in the membrane at distance x from the electrolyte and at

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time t. D is the (constant) diffusion coefficient of the oxygen in the membrane. Fick's 2nd "law" of

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diffusion has been studied extensively in polymeric membranes and has been found to represent the process of dissolving gas diffusion with acceptable accuracy. The mathematical model is complete

I.C.

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when the initial and appropriate boundary conditions are considered. The initial conditions are

C(x,0)=0

(9)

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i.e. the membrane is initially free of oxygen. The boundary condition at the interface between the membrane and the electrolyte is

B.C.1

C(0,t)=0

(10)

which states that the electrochemical reaction is instantaneous and consumes all oxygen reaching the interface between the membrane and the electrolyte [17]. 8

2.4. Step response experiments – probe immersed in a liquid saturated in oxygen The second boundary condition at x=L is the complicating factor to the problem under study. The first case considered is when the probe is immersed instantly into a liquid that is saturated in oxygen and the oxygen concentration at the membrane interface becomes instantly equal to the equilibrium concentration Cs:

C(L,t)=Cs

(11)

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B.C.2.1

This case, known as step change response, has been studied by Aiba et al. [11] and the following equation has been derived for the oxygen distribution as a function of the distance from the

x L

2 ( 1) n x sin nπ e πn1 n L

( nπ )2 at

(12)

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C ( x, t ) Cs

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interface between the membrane and electrolyte and time:

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where a=D/L2 is a constant parameter characteristic to the probe used. The signal recorded by the electrochemical probe is given by the following equation [7]:

C x

ne FAD

(13) x 0

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i(t )

where ne is the number of electrons involved in the electrochemical reaction taking place at the interface between the membrane and the electrolyte, F is the Faraday's constant and A is the active area of the membrane. By using Eq. (12) to obtain the derivative at x=0 and then replace the result into Eq. (13) the following is obtained:

9

i(t ) i

1 2

( 1) n e

( nπ )2 at

(14)

n 1

where i =neFAD/L. A typical response predicted by Eq. (14) is presented in Fig. 5. As it has been assumed that the liquid is saturated in oxygen, transfer of oxygen from the air to the liquid does not take place and kLa does not appear in Eq. (14). However, Eq. (14) is used to determine the parameter a, which is characteristic of the probe. Aiba et al. [11] have presented two methods for determining a. The first is based on the fact that for t sufficiently large only the first

i(t ) i

ln 2 π 2a t

(15)

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ln 1

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term in the sum appearing in Eq. (14) is significant and the equation simplifies to

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This result shows that by plotting ln(1 i(t)/i ) vs time, for sufficiently large t, a straight line is

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obtained. If that is the case then the model is validated and a can be obtained by the slope of the straight line. The one term approximation is compared with the exact solution in Fig. 5. It can be

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observed that for i(t)/i∞>0.4 or at>0.12 the one term solution is an exact approximation of the solution. The one term solution can also be used to determine α from a single experimental point by solving Eq. (15) for α and substituting a pair of experimental data (t, i(t)/i ). The second method, known a the method of moments, is obtained by calculating the integral

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of (1 i(t)/i ):

μ

1 0

i(t ) dt i

10

1 6a

(16)

The integral is the cross-hatched area shown in Fig. 5. The integral can be calculated easily using a numerical integration technique directly on the experimental data and then a can be calculated from Eq. (16). If the resistance of the liquid film adjacent to the liquid-membrane interface is taken into consideration then the second boundary condition needs to be modified. The appropriate form of the boundary condition is based on the assumption that no mass accumulation occurs at the interface and as a result the mass of oxygen that reaches the interface by convection from the liquid must be

B.C.2.2

C x

hK (C Cs ) x L

(17)

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D

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removed by diffusion in the membrane [17]:

where h is the convective mass transfer coefficient and K the solubility parameter of oxygen into the

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membrane (a parameter similar to Henry’s constant for gas-liquid systems but applied to fluid-

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membrane systems). This case, known as step change response with significant resistance to the liquid film, is studied by Linek and Vacek [17] and the following equation has been derived for the

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oxygen distribution:

Bim x x 2 Λn sin λn 1 Bim L L n 1

e

λ2n at

(18)

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C ( x, t ) Cs

where Λn=λncosλn/(λn sinλncosλn), λn are obtained by solving the equations λn+Bimtanλn=0 (or λncotλn+Bim=0) and Bim=hcLK/D is the dimensionless Biot number for mass transfer. The response of the probe is given by

i(t ) i

1 2 1

1 Bim

Λn e n 1

11

λ2n at

(19)

When the resistance in the liquid film is negligible then Bim ∞, tanλn=0 or λn=nπ, Λn=( 1)n and Eq. (19) simplifies to Eq. (14). The one term approximation and the first moment are given by

i(t ) i

μ

ln 2

1 0

tan λ1 λ1

i(t ) dt i

1 Λ1

λ21 αt

1 Bim 3 6a Bim 1

(20)

(21)

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ln 1

In contrast to Eqs (15) and (16) that involve one parameter only, Eqs (20) and (21) involve two

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unknown parameters and cannot be used to determine parameter a graphically.

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2.5. Exponential response experiments – probe immersed in liquid saturated in inert gas and

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aeration commences

When aeration begins in a liquid that is initially free of oxygen and assuming that the oxygen at the

B.C.2.3

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membrane-liquid interface is always at equilibrium then the second boundary condition becomes:

C ( L, t ) Cs (1 e

βt

)

(22)

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Heineken [13,14] and Benedek and Heideger [15] have presented the solution for this case which is given by the equations:

oxygen distribution: C ( x, t ) Cs

x L

1 e

βt

2 n ( 1) n β e 2 π n 1 n nπ a β

12

βt

e

nπ 2 at

sin nπ

x L

(23)

probe response:

i(t ) i

1 e

β a

βt

sin β a

2

n

( 1) n

n 1

β a e 2 nπ β a

nπ 2 at

(24)

one term approximation: i(t ) i

ln

β a

βt

sin β a

first moment:

μ

1

1 β

1 6a

(26)

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0

i(t ) dt i

(25)

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ln 1

Eq. (25) shows that the plot of ln(1 i(t)/i ) vs time, for sufficiently large t, is a straight line

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and β can be determined from the slope of this straight line. A more interesting and potentially

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useful result is obtained if we compare Eqs (16) and (26). If all experimental conditions are held constant and we then perform a step experiment followed by an aeration experiment then β can be

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obtained by the following equation [17]:

μaeration μstep

0

i(t ) i step

i(t ) dt i aeration

(27)

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1 β

i.e. the area of the region between the two experimental responses equals to the inverse of the parameter β (see also Fig. 6) which is, in most cases, equal to kLa. This simple idea is also shown in Fig. 6. This is a very attractive and easy to apply methodology for estimating kLa. However, it is based on the availability of data from a step experiment. If data from a step experiment are difficult to obtain or they are unreliable then we need to rely on the slope of the transformed response (ln(1 i(t)/i ) vs time) for sufficiently large times. 13

2.6. Novel methodology for kLa estimation An alternative novel method, which can be used when data from step experiments are not available, is proposed in this work based on the observation that the exponential response can be approximated by the following equation:

~ i (t ) i

1 e 0, t

β a

βt

sin β a t*

, t

β a

2

sin β a

n

( 1) n

n 1

β a e 2 nπ β a

nπ 2 at

(29)

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βt

(28)

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This is based on the observation that when t
1 e

t*

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i(t ) i

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Use is then made of the following results in order to obtain an approximation of β:

1

t

*

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1

0

~ i (t ) dt i

~ i (t ) dt i

1 β

(30)

1 * t β

(31)

t* can be estimated by using data at sufficiently large times (when Eq. (25) is valid) by fitting Eq. (25) to experimental data and then solve for t. To the best of our knowledge, the proposed methodology has been proposed for the first time in the open literature.

14

If the resistance of the liquid film adjacent to the liquid-membrane interface is taken into consideration, then the second boundary condition is analogous to Eq. (17) but C(L,t) is now time varying and is given by Eq. (22):

B.C.2.4

D

C x

hK (C ( L, t ) Cs )

(32)

x L

follows [17-20]:

ln 1

βt

i(t ) i

Bi m 2(1 Bi m )

Λn n 1

1

n 1

β a e β a λ2n

λ2n αt

(33)

i(t ) dt i

1 β

1 Bim 3 6a Bim 1

βt

(34)

(35)

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0

Λn

1 β a ) Λn 2 Bim n 1 λn β a

ln 1 2(1

μ

β a β a λ2n

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1 ) e Bim

re

1 2(1

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i(t ) i

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The equations for the probe response, one term approximation and moment calculation are as

When the resistance in the liquid film is negligible then Bim ∞, tanλn=0 or λn=nπ, Λn=( 1)n and

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Eqs (33), (34) and (35) simplify to Eqs (24), (25) and (26). To prove that this is actually true use is made of the following results

β a

1 2

sin β a

( 1) n n 1

λn n 1

( Bi m

Bi

2 m

λ ) sin λn 2 n

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β a (nπ )2 β a

1 2( Bi m 1)

(36)

(37)

n 1

λn ( Bim

1 Bim2 λ2n ) sin λn

Bim 3 12( Bim 1)2

(38)

Several important facts can be noted from Eqs (33)-(35). First, the plot of ln(1 i(t)/i ) vs time, for sufficiently large t, is a straight line and β can be determined from the slope of the straight line. This is, therefore, a general method for determining β irrespectively of whether resistance to the liquid film is important or not. By comparing Eqs (35) and (21) we note that Eq. (27) is also valid

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when resistance to the liquid film is deemed important. Thus, the calculation of β from experimental data using the area between the step and the aeration responses is a valid methodology

irrespectively of the particular assumption made for resistances to mass transfer phenomena in the

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probe. Therefore, if a step experimental can be made by keeping all other experimental conditions

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constant then this method should be the method of choice for determining kLa. The value obtained can be validated from the slope of the transformed response for large t.

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Least squares is a general optimization-based technique that can be used to determine the parameters appearing in any of the equations presented so far. The method consists of determining

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the parameters appearing in the mathematical model so as to minimize the sum of the squared (scaled or unscaled) deviations between the experimental measurements and the model predictions:

np

min

wk

k 1

e

i(tk ) i

(39) m

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a , β , Bi m

2

i (t k ) i

where subscript (m) denotes experimental values (model predictions) and wk are selected weights. As the methodology has a strong theoretical basis [23] and makes use of all experimental points, it is arguably a method that could be used in all cases and has the potential to offer superior results.

16

However, it relies on solving fairly complex equations, such as Eq. (33), and the underlying numerical optimization may suffer from convergence problems.

3. Results and discussion 3.1. Experiments with step change in the oxygen concentration When the oxygen probe is instantly immersed in liquid that is saturated in oxygen then the boundary conditions that are applicable at the membrane-liquid interface are the B.C.2.1 and

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B.C.2.2 giving rise to the probe dynamics described by Eqs (14) and (19). The response is obtained experimentally for agitation speeds of 150, 250, 300, 400 and 600 rpm. In what follows, all results will be reported for the Ralf Advanced bioreactor while the results for the Infors HT bioreactor will be summarized in the Supporting data. In Fig. 7, the responses obtained for 150, 300 and 600 rpm

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are compared. If B.C.2.1 is valid then the only parameter appearing in the solution (Eq. (14)) is the

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membrane parameter α=D/L2 and the predicted response does not depend on hydrodynamic conditions. However, the actual experimental results indicate that the response becomes

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significantly faster when the speed of agitation is increased, as shown in Fig. 7. This shortcoming of the basic mathematical model is also reflected in the estimated values for all alternative methods

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presented in Table 2. Parameter α increases as agitation speed increases indicating a decrease in the mass transfer resistance that can be attributed to a diminishing contribution of the hydrodynamic layer formed adjacent to the membrane. To test this hypothesis we need to apply the model given by Eq. (19) that takes the contribution of the hydrodynamics into consideration. As far as the validity

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of the three alternative techniques for estimating the parameter α is concerned it can be said that, based on the values given in Table 2, the least squares method agrees well with the moment (integral) method. The results obtained when the slope of the transformed response at large t method is used deviate significantly from the other two methods. As shown in the supplementary material the method also fails to predict the observed response.

17

When B.C.2.2 is used to complete the mathematical description of the problem of oxygen diffusion in the probe then Eq. (19) is obtained. As two parameters (α and Bim) appear in Eq. (19) the use of either the method of slope (Eq. (20)) or the moment method (Eq. (21)) is not enough in order to determine both parameters. The least squares method is, therefore, used to each set of experimental data and the results are also summarized in Table 2. The results are not satisfactory as the values of the parameter α vary significantly, while the parameter Bim is decreasing while the agitation speed increases, which contradicts its physical meaning. Despite these alarming

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indications implying failure of the approach, the fit of the theoretical model to the experimental data is almost perfect. This is an indication that a more involved approach needs to be devised. This

approach can be based on the least squares method but considers all experiments simultaneously. Parameter α is considered common to all experiments and only Bim is allowed to vary in each

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experiment. In this way, consistent results are obtained that do not suffer from the shortcomings

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discussed above (see Table 2). Parameter α is estimated to be α=0.556, a value that is larger that all values calculated from Eq. (14) which is consistent with the underlying theory of mass transfer.

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Furthermore, as shown in Table 2, Bim increases as agitation speed increases and its variation is almost linear with respect to the agitation speed in rpm (Bi 0.9(rpm/150)). Table 2 also presents the

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ratio of the coefficients that multiply the parameter α in the one term approximations that are valid for large t given by Eqs (15) and (20). The ratio is significantly different from 1 and this explains partially the failure of Eq. (15) to predict reliably parameter α. Table 2 also presents the first moment as calculated from the experimental data and the results of the least squares method. These

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values are close enough and validate the consistency of the approach proposed in this study to consider all experimental data simultaneously. Concluding this section, it should be stressed further the fact that one of the conclusions is that

achieving a fit to experimental data that appears satisfactory to the eye is not a strong argument to support the validity of the approach taken or of the theoretical model developed. Consistency of the estimated numerical values of the parameters involved in the mathematical model with the

18

underlying assumptions used to develop the model are also important and careful investigation needs to be performed. As far as the estimation of the parameter α from step response experiments is concerned, it could be said that it is not a straightforward exercise and the simultaneous consideration of experiments undertaken under different hydrodynamic conditions appears to be a promising strategy. The commonly employed method of fitting the response for large values of t gives estimations that are significantly different from the least squares method or the method of

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moments.

3.2. Aeration experiments

In Table 3, the results of ignoring the potential resistance to the liquid film and applying Eq. (24) to estimate α and kLa are summarized. It is interesting to note that the values of α estimated by aeration

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experiments and the least squares method are less scattered and that all methods offer comparable

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values for the volumetric mass transfer coefficient kLa. The new methodology given by Eqs (30) and (31), despite its simplicity, offers predictions that are close to the more demanding or

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elaborated techniques.

When the model given by Eq. (33) (that accounts for potential resistances to mass transfer due

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to a liquid film) is used to determine the three parameters of the process (α, kLa and Bim) numerical problems are encountered when a derivative based numerical optimization scheme was used. Then a derivative-free optimization routine was used [24] and the results obtained are summarized in Table 4. The results obtained for the agitation speeds of 400 and 600 rpm suffer from the same

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shortcoming as the ones reported in Table 3. The reason for the unreliable results have been investigated and it was found that severe correlation exists between the parameters kLa and Bim as contour plot are straight lines almost parallel to the kLa axis (see Fig. S4 in the supplementary data). This is also to be expected as Eq. (35) is a constraint that needs to be satisfied by any set of valid values of the three parameters (α, kLa and Bim). This fact indicates that the three parameters are

19

correlated in the optimal solution of the lest squares problem leading to the failure of the derivativebased approaches to least squares optimization.

4. Conclusions In this work the problem of determining the volumetric mass transfer coefficient using aeration experiments and fast-oxygen (polarographic) probes has been investigated. A critical review of the available mathematical models and the underlying assumptions has been presented. The models are

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then applied to experimental data obtained using step and aeration experiments in two bench-top bioreactors. In all cases, ignoring the resistance to the mass transfer of oxygen through the oxygenpermeable membrane introduces an error in kLa estimation and we recommend against such

oversimplifying approaches. Based on the results obtained by applying all available methodologies

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to the experimental results, the following recommendations are made.

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If reliable step and aerations experiments can be performed under exactly the same experimental (mainly hydrodynamic) conditions then Eq. (27), whose physical meaning is depicted

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in Fig. 6, can be used to obtain an accurate estimation of kLa. It is important to note that Eq. (27) was shown to hold irrespectively of the existence (or not) of a hydrodynamic layer next to the

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membrane of the electrochemical probe.

If only aeration experiments can be performed then numerical problems will be encountered when the least squares methodology is applied. As it has been pointed out, this is due to the fact that in the more elaborated mathematical models the parameters related to the hydrodynamic layer are

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correlated with the kLa (this is noted for the first time in the open literature). A relative accurate estimation can be obtained by fitting the response to a linear model for relatively large times. However, this estimation can be erroneous as it was demonstrated in this work for the case of step experiment and we recommend against its use. An alternative integral method, that is proposed for first time in this work, appears to offer accurate and reliable estimations of the volumetric mass

20

transfer coefficient. The proposed method, despite its simplicity, has many advantages over the more elaborated methods. Finally, a note of caution needs to be stressed when applying the least squares method for determining the model parameters (including the kLa) for different hydrodynamic conditions. It has been observed in this work that when each experiment, that corresponds to different hydrodynamic conditions, is considered in isolation from the other experiments there is no guarantee that the values obtained for the parameters are consistent with the underlying modeling assumptions. The

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simultaneous consideration of all experiments when applying the least squares methods appears to be a more appropriate methodology for estimating the model parameters. In any case, one must be prepared for numerical problems that can be attributed to either the complexity of the analytical solutions or the potential correlation of the parameters. This is the reason why simplified methods,

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as the one proposed in this work, may offer estimations that are reliable at a minimum

Declaration of interests

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computational cost.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Acknowledgments

This work was supported by the project “INVALOR: Research Infrastructure for Waste Valorization and Sustainable Management” (MIS 5002495) which is implemented under the Action “Reinforcement of the Research and Innovation Infrastructure”, funded by the Operational Programme “Competitiveness, Entrepreneurship and Innovation” (NSRF 2014–2020) and cofinanced by Greece and the European Union (European Regional Development Fund).

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References [1]

N. Patel, J. Thibault, Enhanced in situ dynamic method for measuring kLa in fermentation media, Biochem. Eng. J., 47 (2009) 48-54.

[2]

R. I. Carbajal, A. Tecante, On the applicability of the dynamic pressure step method for kLa determination in stirred Newtonian and non-Newtonian fluids, culture media and fermentation broths, Biochem. Eng. J., 18 (2004) 185-192.

[3]

A.C. Badino, M.C.R. Facciotti, W. Schmidell, Volumetric oxygen transfer coefficients (kLa)

[4]

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in batch cultivations involving non-Newtonian broths, Biochem. Eng. J., 8 (2001) 111-119. M. Michelin, A.M.O. Mota, M.L.T.M. Polizeli, D.P. Silva, A.A.Vicente, J.A. Teixeira,

Influence of volumetric oxygen transfer coefficient (kLa) on xylanases batch production by Aspergillus niger van Tieghem in stirred tank and internal-loop airlift bioreactors, Biochem.

M.G. Acedos, A. Hermida, E. Gomez, V.E. Santos, F. Garcia-Ochoa, Effects of fluid-

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[5]

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Eng. J., 80 (2013) 19-26.

dynamic conditions in Shimwellia blattae (p424IbPSO) cultures in stirred tank bioreactors:

J., 149 (2019) 107238.

E. Gnaiger, H. Forstner, Polarographic Oxygen Sensors, Aquatic and Physiological

ur na

[6]

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Hydrodynamic stress and change of metabolic routes by oxygen availability, Biochem. Eng.

Applications, Springer-Verlag, Berlin, 1983. [7]

Y.H. Lee, G.T. Tsao, Dissolved oxygen electrodes, Adv. Biochem. Eng., 13, (1979) 35-86

[8]

K. Van’t Riet, Review of measuring methods and results in mass transfer in stirred vessels,

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Ind. Eng. Chem. Process Des. Dev., 18(3) 1979 357-364.

[9]

M. Sobotka, A. Prokop, I.J. Dunn, A. Einsele, Review of methods for the measurement of oxygen transfer in microbial systems, Annu. Rep. Ferment. Process., 5 (1982) 127-210.

[10] F. Gargia-Ochoa, E. Gomez, Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview, Biotechnol. Adv., 27 (2009) 153-176.

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[11] S. Aiba, S.Y.Huang, Oxygen permeability and diffusivity in polymer membranes immersed in liquids, Chem. Eng. Sci., 24(7) (1969) 1149-1159. [12] V. Linek, T. Moucha, M. Kordač, M. Dubcová, F. Hovorka, J.F.Rejl, Liquid film effect on dynamics of optical oxygen probe. Comparison with polarographic oxygen probes. Diffusion coefficients measuring technique, Chem. Eng. Sci., 64(18) (2009) 4005-4015. [13] F.G. Heineken, On the use of fast-response dissolved oxygen probes for oxygen transfer studies, Biotechnol. Bioeng., 12 (1970) 145-154.

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[14] F.G. Heineken, Oxygen mass transfer and oxygen respiration rate measurements utilizing fast response oxygen electrodes, Biotechnol. Bioeng.,13 (1971) 599-618.

[15] A.A. Benedek, W.J. Heideger, Polarographic oxygen analyzer response: The effect of

instrument lag in the non-steady state reaeration test, Water Res., 4(9) (1970) 627-640.

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[16] W. C. Wernau, C.R. Wilke, New method for evaluation of dissolved oxygen probe response

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for kLa determination, Biotechnol. Bioeng., 15 (1973) 571-578.

[17] V. Linek, V. Vacek, Oxygen electrode response lag induced by liquid film resistance against

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oxygen transfer, Biotechnol. Bioeng.,18 (1976) 1537-1555 . [18] V. Linek, Dynamic measurement of the volumetric mass transfer coefficient in agitated

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vessels, Biotechnol. Bioeng. 19 (1977) 983-1008. [19] V. Linek, P. Benes, Multiregion, multilayer, nonuniform diffusion model of an oxygen electrode, Biotechnol. Bioeng., 19 (1977) 741-748. [20] V. Linek, P. Benes, Comparison of regression and moment methods for evaluation of oxygen

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probe response, Biotechnol. Bioeng., 20 (1978) 903-912.

[21] A.C. Badino, W. Schmidell, Improving kLa determination in fungal fermentation, taking into account electrode response time, J. Chem. Technol. Biotechnol., 75 (2000) 469-474.

[22] H.W. Blanch, D.S. Clark, Biochemical Engineering, Marcel Dekker Inc., New York, USA, 1996. [23] Y. Bard, Nonlinear Parameter Estimation, Academic Press, New York, USA, 1974.

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[24] I.K. Kookos, Optimization of batch and fed-batch bioreactors using simulated annealing,

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re

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Biotechnol. Progress, 20(4) (2004) 1285-1288.

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Fig. 1. Configuration of bioreactor and impeller

VG

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CO2

pO2,i=HCO2,i

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NG, yO2

pO2 pO2,i CO2,i

VL

CO2

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NG, yO2,in

Fig. 2. A bioreactor with well mixed gas and liquid phases at the aeration stage.

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measured O2 concentration

O2 concentration in the liquid

O2 concentration in the gas

Time

Time

Membrane and liquid film (probe) dynamics

Liquid phase dynamics Aeration initiated (Input)

Unmeasurable dynamics

Time

Measurement (O2 probe) (Output)

Oxygen permeable membrane

Liquid film

Liquid bulk

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Cathode

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Elecrolyte

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Fig. 3. Graphical representation of the conceptual problem considered in this work.

x=0

h x=L

D

Fig. 4. Main parts and resistances to mass transfer in the fast oxygen probe (lines denote oxygen

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concentration at steady state).

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1 0.9 0.8 μ i(t)/i∞

0.7 0.6 0.5 0.4

0.2 0.1 0

20

π2αt

40

60

80 Time (t)

100

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0

1 2e

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0.3

120

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Fig. 5. Predicted response when probe is instantly placed in a liquid saturated in oxygen.

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1 Step response

1/β i(t)/i∞

Aeration response

0.5

0

100

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Time (s)

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Experimental responses are at 600 rpm

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Fig. 6. Graphical representation of the calculation indicated in Eq. (27).

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200

1 0.9 0.8 i(t)/i∞ 0.7 0.6 0.5

0.3 0.2 0.1 20

40

60 Time (s)

80

100

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0 0

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150 rpm 300 rpm 600 rpm

0.4

120

Fig. 7. Probe responses when immersed instantly to liquid saturated with oxygen for several values

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of the agitation speed

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Table 1. Characteristics of the bioreactors Infors

Unit

Value

Value

Volume capacity

L

6.7

3.6

Operating volume

L

3

1

Liquid height

cm

18

7.5

Internal diameter of vessel (Di)

cm

15

15

Internal height of vessel (Hi)

cm

40

22

Number of impellers

2

2

Type of impellers

Rushton

Impeller diameter (D)

mm

60

Distance between impellers

mm

36

Bottom impeller clearance

mm

50

Number of blades

6

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Ralf Descriptions

60 45 20

mm

20

Blade height (w)

mm

15

10

4

4

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Number of baffles Baffle width Type of sparger

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Holes sparger

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Baffle height

10

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Blade width (l)

31

cm

1

1

cm

28

16

Ring 13

10

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Table 2. Summary of the results of the parameter estimation for the experiments with step change in the oxygen concentration (instant probe immersion to a liquid

Eq. (19) (Resistance to membrane and to liquid film) α (1/min) α (1/min) Bim Bim

Moment (Eq. (16)) 0.268

Large t (Eq. (15)) 0.185

Least squares

150 rpm

Least squares 0.281

0.440

2.386

250 rpm

0.327

0.312

0.209

0.601

1.261

300 rpm

0.323

0.310

0.213

0.475

400 rpm

0.345

0.329

0.230

600 rpm

0.398

0.372

0.222

π2/ λ12

μLS

μexperimental

0.961

2.4293

0.6058

0.6232

1.698

1.9969

0.5222

0.5350

3.111

1.674

2.0071

0.5242

0.5377

0.534

2.487

2.121

1.8450

0.4921

0.5065

0.970

0.302

3.635

1.5383

0.4293

0.4481

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Agitation speed

First moment

e-

Parameter

Ratio of slopes

pr

Eq. (14) (Resistance only to membrane) a (1/min)

Pr

Mathematical model

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saturate in oxygen)

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0.556

ignored

Eq. (24) (Resistance to mass transfer attributed only to membrane)

Parameter

α (1/min)

Agitation speed

Least squares

150 rpm

0.315

250 rpm

pr

Mathematical model

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Table 3. Summary of results of the parameter estimation with aeration (gassing-in) experimental data when potential resistance to the liquid film is

kLa (1/min)

kLa (1/min)

α (1/min)

0.471

Moment (Eq. (27)) 0.501

Large t (Eqs (25) or (34)) 0.438

New method (Eq. (31)) 0.443

Least squares (all exp. data considered simultaneously) 0.458

0.344

0.721

0.748

0.737

0.760

300 rpm

0.383

1.084

1.253

1.123

1.237

1.139

400 rpm

0.323

1.534

1.511

1.437

1.560

1.442

600 rpm

0.426

2.305

2.641

2.101

2.291

2.908

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Pr

e-

kLa (1/min)

kLa (1/min)

33

0.350

kLa (1/min)

0.717

oo

f

Table 4. Summary of results of the parameter estimation with aeration (gassing-in) experimental data with resistance to the liquid film

Mathematical model

Eq. (33) (Resistance to mass transfer attributed to membrane and to the liquid film adjacent to the membrane)

Parameter

α (1/min)

Agitation speed

Least squares

150 rpm

0.3773

250 rpm

Bim

α (1/min

0.472

8.906

Least squares (all exp. data considered simultaneously) 0.4833 1.5775

0.385

0.722

15.641

300 rpm

0.418

1.085

20.584

400 rpm

0.4258

3.453

600 rpm

0.506

5.301

Bim

e-

pr

kLa (1/min)

0.7217

2.7144

1.1116

3.6086

0.604

1.6920

1.4681

1.127

2.3542

6.5510

Pr

0.5280

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kLa (1/min)

34