Applicability of a modified breakage and coalescence model based on the complete turbulence spectrum concept for CFD simulation of gas-liquid mass transfer in a stirred tank reactor

Journal Pre-proofs Applicability of a Modified Breakage and Coalescence Model Based on the Complete Turbulence Spectrum Concept for CFD Simulation of Gas–Liquid Mass Transfer in a Stirred Tank Reactor Lilibeth Niño, Ricardo Gelves, Haider Ali, Jannike Solsvik, Hugo Jakobsen PII: DOI: Reference:

S0009-2509(19)30762-6 https://doi.org/10.1016/j.ces.2019.115272 CES 115272

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Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

24 July 2019 30 September 2019 6 October 2019

Please cite this article as: L. Niño, R. Gelves, H. Ali, J. Solsvik, H. Jakobsen, Applicability of a Modified Breakage and Coalescence Model Based on the Complete Turbulence Spectrum Concept for CFD Simulation of Gas–Liquid Mass Transfer in a Stirred Tank Reactor, Chemical Engineering Science (2019), doi: https://doi.org/ 10.1016/j.ces.2019.115272

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Applicability of a Modified Breakage and Coalescence Model Based on the Complete Turbulence Spectrum Concept for CFD Simulation of Gas-Liquid Mass Transfer in a Stirred Tank Reactor Lilibeth Niño1, Ricardo Gelves2*, Haider Ali3, Jannike Solsvik3 and Hugo Jakobsen3 1

First author. Faculty of Exact and Natural and Sciences, University de Antioquia, Calle 67

53-108, Medellín, Colombia. 2*

Corresponding author. Department of Environment,

Santander,

Av.

Gran

Colombia

12E-96,

University Francisco de Paula

Cúcuta,

Colombia.

E-mail:

[email protected] 3

Department of Chemical Engineering, Norwegian University of Science and Technology,

7491 Trondheim, Norway.

Abstract

A generalized model for bubble breakage and coalescence is proposed using Computational Fluid Dynamics - CFD for considering the complete energy spectrum. An eulerian model and balance equations are simultaneously used to simulate the multiphase flow and bubble size distribution, respectively. The turbulent kinetic energy and its dissipation are calculated using the standard turbulence model ݇ െ ߝ. A semi-empirical model that solves the second-order longitudinal structure function based on an interpolation function is coupled to CFD via UDF (User Defined Functions) code. CFD results are compared with experimental data obtained from Sauter mean diameter measurements at different bioreactor positions and stirred by a Rushton turbine. A reasonable prediction is obtained in comparison with the original Coulaloglou and Tavlarides (Break up) and Prince and Blanch (Coalescence) model. Further, the generalized model was extended to other stirring and aeration geometries using the same 10 litter tank bioreactor. The latter for evaluating strategies for overcoming gas-liquid mass transfer problems commonly found in bioreactorsand a significant effect of the energy

spectrum is reached in the geometries studied. The above, explained by the ݇௅ ܽ oxygen transfer rate and bubble size determinations. It is numerically demonstrated that flow patterns and bubble size significantly influence the average ݇௅ ܽ mass transfer in a bioreactor.

Keywords: Bioreactor, ݇௅ ܽ Gas-Liquid Mass Transfer, Energy Spectrum. Population balance.

1. Introduction

One of the most common challenges in fermentation technology is to improve the stirring and aeration conditions for industrial biotechnology of microorganism cultures. The typical high cell density reached in a bioreactor considerably affects the ݇௅ ܽ mass transfer and stirring. Therefore, insufficient oxygen supply, limiting cell growth and productivity is expected. This type of limitation is prevalent in many applications related to pharmaceutical products through biotechnological routes such as cell, microorganisms and fungi culturing. Also, the latter is caused by problems associated with hydrodynamicssince bioreactors are often stirred by Rushton turbine devices. Rushtontype impeller design is well known for its disadvantages related to large-scale applications, such as high power requirements, inhomogeneous mixing zones, high local dissipation rates of turbulent kinetic energy, etc [1]. Detailed information from gas-liquid hydrodynamics in a stirred tank application would be suitable for identifying not only mass transfer limitations but also, the main criteria necessary to be implemented in a optimization bioprocess stage. Computational fluid dynamics (CFD) is a useful tool to simulate hydrodynamic phenomena and breakage process in stirred tank bioreactors.

The potentiality of CFD has been extended from the study of mixing in stirred systems, including single fluid rotating models [2] - [6], to the use of population balance models for particle breakage and coalescence phenomena [1], [7-11]. Breakage and coalescence rates

govern bubble size distribution in stirred tank bioreactors since bubble diameter results from its interaction with the resulting eddies. Hence, a mathematical model that captures bubble breakage and coalescence phenomena accurately are required to predict their impacts on ݇௅ ܽ.

A variety of kernels for describing bubble breakup and coalescence can be found in the literature[12]. However, all of them are limited to the inertial sub-range of turbulence [13]; consequently, current kernels are only valid for bubbles that have sizes similar to the inertial sub-range. This phenomenon is not entirely accurate since the continuous aeration in a stirred tank bioreactor leads to the formation of different bubble sizes from the order of micrometers to millimeters. One of the most promising solutions to address this problem and achieve more acceptable predictions would be the inclusion of a secondorder structural function. The latter, for considering not only the inertial sub-range but also the complete energy spectrum for improving bubble breakup and coalescence accurate and take into account the effect of the whole turbulence energy cascade in the bubble size distribution.Based on the above, [14] developed a mathematical model that takes into account the complete energy spectrum by obtaining an analytical solution of the second-order structural function [14] in a 0-Dimensional application. In this work, significant numerical evidence of the complete turbulence spectrum effect on drop breakup coalescence rates under different numbers of Reynolds is showed.

Motivated by the latter and considering implementation restrictions in the CFD code, a modified model is proposed in this research. A semi-empirical model of Sawford and Hunt [14] is used for including the complete energy spectrum. The results are compared with experimental data using a Rushton type radial turbine. Based on the improvement reached considering the entire energy spectrum, the proposed model is extended to other aeration-stirring designs. Therefore a significant effect of the energy spectrum was found in the bubble breakup and coalescence phenomena regarding the simulated geometries.

2. Materials and methods

2.1 CFD model

In aerobic processes, the air is supplied using a sparger placed at the bottom of the bioreactor. The resulting gas-liquid fluid can be modeled by CFD using the Euler-Euler model in each phase. However, the previously mentioned model considers all phases as a continuum. It assumes that bubble sizes are constant and bubble dispersion effects are weakly addressed. In a bioreactor, the interaction between continuous airflow supplying and mixing induced by the stirring-aeration geometry generates different bubble sizes. Based on the above Euler-Euler method is limited to bioreactor applications. That is why the incorporation of population balance models to the Euler model to simulate different bubbles sizes resulting from interaction with turbulence effects is presented as an alternative to overcome this problem.

The commercial CFD code Fluent v 19 has been applied in this work. An Eulerian model was used to solve the system of n-equations of momentum and continuity for each phase. The motion induced by the stirring system is modeled by Multiple Reference Frame Model. The turbulence ݇ െ ߝ model is used since previous investigations [1], [15-17] have confirmed it accurate on gas-liquid hydrodynamics applications without affecting the computational resource.

In a stirred tank bioreactor, as mentioned before, different bubble sizes are formed as a consequence of bubble breakup and coalescence phenomena. That is why these are not homogeneously dispersed in the bioreactor. In this work, the discrete method is used [1820] to solve the population balance equations since the bubble population is discretized in a finite number of intervals. The population balance equations for the bubble bins can be written as [21- 23]:

డ డ௧

ሬԦீ ݊௜ ൯  ൌ  ߩீ ቀΓ஻ െ Γ஽ ൅ Γ஻ െ Γ஽ ቁ ሺߩீ ݊௜ ሻ ൅ ߘ ή ൫ߩீ ܷ ೔ ೔ ೔ ೔ ಴

಴

ಳ

ಳ

(1)

Where ݊௜ is the number of bubble group i, ‫ܤ‬௜಴ ƒ†‫ܤ‬௜ಳ are rates of birth due to coalescence and breakup, respectively, ‫ܦ‬௜಴ and ‫ܦ‬௜ಳ are death rates.

Equations (2)-(5) are the global models that describe general breakup and coalescence phenomena in terms of integral equations. Bubbles of ‫ ʹ ݒ‬formed in a stirred tank reactor will break into bubbles of size ‫ ݒ‬and these particles may coalesce with another bubble of size ‫ݒ‬. ଵ

௩

Γ஻೔ ൌ ଶ ‫׬‬଴ ܽሺ‫ ݒ‬െ ‫ ʹ ݒ‬ǡ ‫ݒ‬ሻ݊ሺ‫ ݒ‬െ ‫ ʹ ݒ‬ǡ ‫ݐ‬ሻ݊ሺ‫ ʹ ݒ‬ǡ ‫ݐ‬ሻ ݀‫ʹ ݒ‬ ಴

‫ן‬

(2)

Γ஽೔ ൌ ݊ሺ‫ݒ‬ሻ ‫׬‬଴ ܽሺ‫ݒ‬ǡ ‫ ʹ ݒ‬ሻ݊ሺ‫ ʹ ݒ‬ǡ ‫ݐ‬ሻ݊ሺ‫ ʹ ݒ‬ǡ ‫ݐ‬ሻ ݀‫ʹ ݒ‬

(3)

Γ஻೔ ൌ ‫׬‬ఆ ‫݃݌‬ሺ‫ ʹ ݒ‬ሻߚሺ‫ݒ‬ห‫ ʹ ݒ‬ሻ݊ሺ‫ ʹ ݒ‬ǡ ‫ݐ‬ሻ݀‫ʹ ݒ‬ ಳ

(4)

Γ஽೔ ൌ ݃ሺ‫ݒ‬ሻ݊ሺ‫ݒ‬ǡ ‫ݐ‬ሻ

(5)

಴

ೡ

ಳ

ܽሺ‫ݒ‬ǡ ‫ ʹ ݒ‬ሻ is the coalescence rate between bubbles of size ‫ ݒ‬and ‫݃ ; ʹ ݒ‬ሺ‫ݒ‬ሻis the breakup rate of bubbles of size ‫݃ ;ݒ‬ሺ‫ ʹ ݒ‬ሻ, is the frequency of bubbles breakup ‫ ʹ ݒ‬andߚሺ‫ݒ‬ห‫ ʹ ݒ‬ሻ is the probability of the density function of the bubbles broken from the volume ‫ ʹ ݒ‬to a bubble of volume ‫ݒ‬, ‫ ݌‬is the number of bubbles formed by each break; ݊ is the number of bubble classes. As it is described in equation (2) bubbles in a stirred tank reactor are continuously formed from larger bubbles breakup or coalescence of smaller ones (equation 3).

Also, bubbles will be destroyed by the breakup into smaller bubbles as modeled in the mathematical formulation (4) and by coalescence into larger ones as it is represented in expression (5).

Expressions for breakup and coalescence rate (kernels) are required for solving the population balance equations mentioned before. The breakup and coalescence phenomena have been studied extensively and are described by several authors [24-29]. A variety of kernels for describing bubble breakup and coalescence can be found in the literature[30]. However, those are limited to inertial sub-range of turbulence; thus, only the kinetic energy dissipation is dominated by the micro scales [13]. At this point, the average velocity of the bubbles ‫ݑ‬ത௜ଶ found in the sub-range of scales is defined as: ସ

∞

‫ݑ‬ത௜ଶ ൌ ଷ ‫׬‬଴ ‫ܧ‬ሾ݇ሿ ൅ ͵‫ܧ‬ሾ݇ሿ ቂ

ୡ୭ୱሾ௞௥ሻ ሾ௞௥ሿమ

െ

ୱ୧୬ሾ௞௥ሿ ቃ ݀݇ ሾ௞௥ య ሿ

(6)

The equation (6) to be solved, requires information of the energy spectrum ‫ܧ‬ሾ݇ሿ where ߢ represents the wave number and ‫ ݎ‬is here interpreted the eddies size ‫ݎ‬. In this case, it is defined by ߢolmogorov [14, 31] in the inertial sub-range scale:

‫ܧ‬ሾߢሿ ൌ ‫ ߝܥ‬ଶȀଷ ߢ ିହȀଷ

(7)

Where ‫ ܥ‬is a ߢolmogorov constant and ߝ denotes the turbulent kinetic energy dissipation rate. Substituting equation (7) into (6), an analytical solution is obtained to define the second-order structural function:

ସ

∞

‫ݑ‬ത௜ଶ ൌ ଷ ‫׬‬଴ ‫ܧ‬ሾߢሿ ൅ ͵‫ܧ‬ሾߢሿ ቂ

ୡ୭ୱሾ఑௥ሻ ሾ௞௥ሿమ

െ

ୱ୧୬ሾ఑௥ሿ ቃ ݀ߢ ሾ఑௥ య ሿ

ൌ ߙଵ ߝ ଶȀଷ ݀ଶȀଷ

(8)

The previous model is related to the turbulent kinetic energy in the bubble breakupkernels and simultaneously, the equation (8) is used in the determination of the bubble coalescence rate. The current breakup and coalescence kernels from the literature are only valid for bubbles that have similar intervals to the inertial sub-range. This phenomenon is not entirely accurate since the continuous aeration in a stirred tank bioreactor leads to a formation of different bubble sizes from the order of micrometers to millimeters. The above can generate inaccurate determinations of both bubble size and mass transfer rates in bioreactors. Consequently, the full effect of the spectrum energy cascade is not considered in breakup and coalescence rates.

A more precise model should consider a second-order structural function that involves not only the sub-range scales but also the entire energy spectrum in a stirred tank bioreactor. Based on the above, Pope [14] proposes the following energy spectrum model to relate the kinetic energy distribution transported in the different eddy sizes:

‫ܧ‬ሾߢሿ ൌ ‫ߝܥ‬

ଶȀଷ ିହȀଷ

ߢ

఑௅ ቂሺሾ఑௅ሿమ ା஼ ሻቃ ಽ

ଵଵൗ ଷ

݁‫ ݌ݔ‬൤െߚ ൜ൣሺߢߟሻସ ൅ ‫ܥ‬ఎସ ൧

ଵൗ ସ

െ ‫ܥ‬ఎ ൠ൨

(9)

‫ ܮ‬is the eddy integral length scale, ߟ denotes the ߢolmogorov microscale (smallest eddy size). ߚ is a constant. The parameters ‫ܥ‬௅ and ‫ܥ‬ఎ are solved, accounting the restriction [30]:

∞

݇ ൌ ‫׬‬଴ ‫ܧ‬ሺߢሻ ݀ߢ ∞

ߝ ൌ ‫׬‬଴ ʹ‫ ߢݒ‬ଶ ‫ܧ‬ሺߢሻ ݀ߢ

(10)

(11)

The main difference between the ߢolmogorov energy spectrum and that of Pope is that the latter considers the energy-containing range, inertial and dissipation sub-ranges. The model mentioned here can be solved numerically using the numerical quadrature to be

coupled to CFD and PBM equations via UFD (User Defined Functions), which would involve a tremendous computational resource. However, it also can be solved by the implementation of analytical solutions, as previously investigated by Solsvik and Jakobsen [30]. The authors mentioneddeveloped analytical solutions of the second-order structural function from gamma, geometric and Bassel-type functions applied to a 0-Dimensional process.

Alternatively, due to implementation restrictions in the commercial code used a solution of the second-order structural function that includes the complete energy spectrum can be coupled to the CFD codes by implementing semi-empirical functions such as the one proposed by Sawfordand Hunt [14]. In the latter, the Second-order structural function is reasonably approximated by an interpolation function [14] to obtain:

‫ݑ‬ത௜ଶ

ൌ

ସ

∞ ‫ܧ ׬‬ሾߢሿ ଷ ଴

൅

ୡ୭ୱሾ఑௥ሻ ͵‫ܧ‬ሾߢሿ ቂ ሾ఑௥ሿమ

െ

ୱ୧୬ሾ఑௥ሿ ቃ ݀ߢ ሾ఑௥ య ሿ

௥మ

The effects of the integral scales ൤௥ మ ା௥మ ൨

ଶ

௥మ

ൌ ʹ‫ ݒ‬൤௥ మ ା௥మ ൨

ଵൗ ଷ

೏

ଶൗ ଷ

௥మ

൤௥ మ ା௥మ ൨

ଵൗ ଷ

(12)

ಽ

were introduced by Durbin [14] using an

ಽ

interpolation model. Later, Sawford and Hunt modified the Durbin model to develop a more precise expression by including the dissipation range and inertial sub-range ௥మ

contribution by adding the term ൤௥ మ ା௥మ ൨ ೏

ଶൗ ଷ

. The length scale ‫ݎ‬ௗ determines the boundary

from the viscous subrange to the inertial subrange and is calculated as:

‫ݎ‬ௗ ൌ ሺͳͷܿଵ ሻ

ଷൗ ସߟ

(13)

According to the ߢolmogorov second-order structure-function, ܿଵ = 2.0. The boundary ‫ݎ‬௅ from the inertial scales to the integral scales based on the Sawford and Hunt model is determined:

ଶ

‫ݎ‬௅ ൌ ቀଷቁ

ଷൗ ଶ

‫ܮ‬

(14)

Expression (13) and (14) require information regarding ߢolmogorov length scales (ߟ) and an integral length scale of the largest eddies (‫)ܮ‬. These are calculated in Sawford and Hunt model using the equations:

‫ܮ‬ൌ

య ௞ ൗమ

ఌ

ߟൌ‫ݒ‬

ଷൗ ିଵൗ ସߝ ସ

(15)

(16)

The Sawford and Hunt model [14] has been previously evaluated and compared together with the numerical and analytical solutions and similar results were found. This indicates that the model proposed here may be promising to be implemented in the CFD codes, taking into account its computational economy. The latter based on the numerical solution related to bubble interactions and the entire turbulence spectrum.

A variety of models (kernels) have been developed for the bubble breaking rate, based on different breakup criteria: in terms of critical turbulent kinetic energy [25], [26], [32], critical inertial force [27], [28] and critical velocity fluctuations [33]. The bubble breakup is analyzed in terms of bubble interaction with turbulent eddies generated by the stirring device. These turbulent eddies increase the surface energy of the bubbles until they cause deformation. The break occurs if the increase in surface energy reaches a critical value. The breakup kernel developed by Coulaloglou and Tavlarides [34] has been widely studied in various industrial applications. Even its low computational cost makes it attractive to be included in CFD codes to simulate multiple industrial applications involving bubble

breakagein stirred tank reactors. For that reason, according to the above, the breakup rate is defined as [35]:

ଵ οேሺௗሻ

݃ሺ‫ ʹ ݒ‬ሻ ൌ ఛ

್

(17)

ேሺௗ

Considering the above equation, the bubble breakup based on the inertial sub-range is defined in a stirred tank reactor as:

ௗ

߬௕ ൌ ത ൌ ௩ሺௗሻ

ௗ ට௨ ഥ೔మ

ൌ ܿଵ ݀

ଶൗ ିଵൗ ଷ ଷߝ

(18)

In which ‫ݒ‬ҧ ሺ݀ሻ represents the bubble translational velocity with diameter ݀. It is calculated by the second-order structural function according to the sub-range scales of turbulence.

‫ן‬

ಶ೎

ି ‫׬‬ா ܲሺ‫ܧ‬ሻ݀‫ ܧ‬ൌ ݁‫ ݌ݔ‬ಶ ൌ ೎

οேሺௗሻ

(19)

ேሺௗሻ

‫ܧ‬௖ ൌ ܿଶ ߪ݀ ଶ

(20)

ଶ ଵ ‫ܧ‬ത ൌ ܿଷ ߩௗ ݀ଷ ‫ݑ‬ത௜ଶ ൌ ܿଷ ߩௗ ߝ ൗଷ ݀ ൗଷ

(21)

By combining the above equations, the original model is obtained to simulate the breakage rate according to Coulaloglou and Tavlarides [34]:

݃ሺ‫ ʹ ݒ‬ሻ ൌ ‫ܥ‬ଵǡ௕ ݀

ିଶൗ ଵൗ ଷ ߝ ଷ ݁‫ ݌ݔ‬൤െ‫ܥ‬ଶǡ௕

ఙ ఱ మ ఘ೏ ఌ ൗయ ௗ ൗయ

൨

(22)

In which ‫ܥ‬ଵǡ௕ , and ‫ܥ‬ଶǡ௕ are constants of the model with values of 0.08 and 0.001, respectively. As explained above, the exposed model is limited to the inertial sub-range. Therefore it can generate inaccuracies in the definition of bubble sizes in a bioreactor.

Considering the complete energy spectrum defined by Sawford and Hunt [14] to develop a more precise model, the bubble breakup time can be corrected to give:

ௗ

߬௕ ൌ

ට௨ ഥ೔మ

ൌ ܿସ

ௗ

(23)

మൗ భൗ య య మ మ ඨଶ௩ మ ቈ ೏ ቉ ቈ ೏ ቉ ೝమ శ೏మ ೝమ శ೏మ ಽ

೏

The corrected turbulent kinetic energy for the complete energy spectrum results in this case:

‫ܧ‬ത ൌ

ܿହ ߩௗ ݀ଷ ‫ݑ‬ത௜ଶ

ଷ

ଶ

ௗమ

ൌ ܿହ ߩௗ ݀ ʹߴ ൤௥ మ ାௗమ ൨ ೏

ଶൗ ଷ

ௗమ

൤௥ మ ାௗమ ൨

ଵൗ ଷ

(24)

ಽ

Combining the reformulated equations for considering the influence of the complete energy spectrum in the breakup kernel, the following model proposed in this investigation is developed :

݃ሺ݀ሻ ൌ ‫ܥ‬ଵǡ௕

మൗ భൗ య య మ మ ඨଶణ మ ቈ ೏ ቉ ቈ ೏ ቉ మ మ మ మ ೝ೏ శ೏ ೝಽ శ೏

ௗ

‡š’ ൮െ‫ܥ‬ଶǡ௕

ఙ మൗ భൗ య య ೏మ ೏మ మ ఘ೏ ௗଶణ ቈ మ మ ቉ ቈ మ మ ቉ ೝ೏ శ೏ ೝಽ శ೏

൲

(25)

Coalescence in a stirred tank reactor is generally divided into three sub-processes: (i) two bubbles collide, trapping a small amount of liquid between them; (ii) the bubbles are kept in contact until the liquid film is drained to a critical thickness; (iii) The film breaks resulting in coalescence. In all cases, contact and collision is the premise of coalescence.

The collision between bubbles is usually caused by their relative velocity. Most of the current coalescence models available in the literature are formulated according to the inertial effects in the sub-range of scales [15].

However, bubbles are influenced not only by the inertial subrange addressed by the kinetic energy dissipation but also the complete cascade of turbulence is essential to transport and define bubble sizes. That is why coalescence is very important in terms of modeling during a design stage of devices for the aim purpose of improving mass transfer. These bubble mechanisms trigger the formation of weakly aerated regions which can affect the metabolic activities for microorganism growth [1].

Prince and Blanch model have been used in various CFD applications. Therefore, this mathematical expression is promising to be coupled in CFD codes together to the entire spectrum due to its minimal computational effort to simulate multiple industrial processes involving bubble coalescence in stirred tank reactors. Prince and Blanch model define the coalescence between bubbles in the sub-range of the turbulence, therefore [36]:

ܽሺ‫ݒ‬ǡ ‫ ʹ ݒ‬ሻ ൌ ߱௖ ൫݀௜ ǡ ݀௝ ൯ܲ௖ ൫݀௜ ǡ ݀௝ ൯

(26)

߱௖ ൫݀௜ ǡ ݀௝ ൯ refers to the bubble collision frequency and ܲ௖ ൫݀௜ ǡ ݀௝ ൯ is the probability of coalescence. In the current work, it is assumed that the coalescence rate results mainly from the turbulence in a bioreactor and is expressed as [14]:

߱௖ ൫݀௜ ǡ ݀௝ ൯ ൌ ݊௜ ݊௝ ܵ௜௝ ‫ݑ‬௥௘௟ǡ௜௝

ܵ௜௝ is the transverse bubble collision area and is calculated from the equation:

(27)

గ ௗ

ௗೕ ଶ

ܵ௜௝ ൌ ቂ ଶ೔ ൅ ଶ ቃ

(28)

ସ

‫ݑ‬௥௘௟ǡ௜௝ is the relative average velocity between bubbles i and j and is represented as:

ଵȀଶ

‫ݑ‬௥௘௟ǡ௜௝ ൌ ൣ‫ݑ‬ത௜ଶ ൅  ‫ݑ‬ത௝ଶ ൧

(29)

Where ‫ݑ‬ത௜ଶ is the average velocity of bubbles i previously defined in the original model based on the inertial sub-range and results:

భ

మ

ଶ

మ

ଵȀଷ

߱௖ ൫݀௜ ǡ ݀௝ ൯ ൌ ܿଵǡ௖ ߝ య ൫݀௜ ൅ ݀௝ ൯ ቀ݀௜ య ൅ ݀௝ య ቁ

(30)

The probability of bubble coalescence ܲ௖ ൫݀௜ ǡ ݀௝ ൯ can be related to physical phenomena acting in a stirred tank reactor. In such a way, the force compresses the bubbles would be effective in sufficient time so that it is possible to drain the thinning film that covers the bubbles until a critical value, thus triggering the coalescence. In The Prince and Blanch kernel [26] the probability of coalescence is evaluated based on the process of film drainage between two bubbles and is calculated as [36]:

೏೔ೕ య ቆ ‫ ۇ‬మ ቇ ഐ‫ۊ‬

೓బ

‫ ۈ‬భల഑ ‫ۋ‬௟௡ቆ೓೑ ቇ

ܲ௖ ൫݀௜ ǡ ݀௝ ൯ ൌ ‡š’ െ ‫ۉ‬

‫ی‬

మ ೏೔ೕ ൗయ ቆ ቇ మ భ ഄ ൗయ

(31)

Finally grouping the equationswe obtain the original model proposed by Prince and Blanch to simulate the bubble coalescence rate:

‫ۇ‬ ௖మǡ೎ ‫ۈ‬ భ య

గ

మ య

ଶ

మ య

ଵȀଷ

ܽሺ‫ݒ‬ǡ ‫ ʹ ݒ‬ሻ ൌ ܿଵǡ௖ ସ ߝ ൫݀௜ ൅ ݀௝ ൯ ቀ݀௜ ൅ ݀௝ ቁ

‡š’ െ

ቆ

೏೔ೕ య ቇ ഐ మ ‫ۊ‬ భల഑

‫ۉ‬

మ ೏೔ೕ ൗయ ቆ ቇ మ భ ഄ ൗయ

‫ۋ‬ ‫ی‬

(32)

The constant ܿଵǡ௖ was adjusted to 2.5 and ܿଶǡ௖ was defined to a value of 2.3 [15].

Prince and Blanch model can be reformulated using the second-order structural function for including the effects of the complete spectrum to give:

೏೔ೕ య ቆ ‫ ۇ‬మ ቇ ഐ‫ۊ‬

௖మǡ೎ ‫ۈ‬

ܽሺ‫ݒ‬ǡ ‫ ݒ‬ᇱ ሻ=ܿଵǡ௖ ൫݀௜ ൅ ݀௝ ൯

ଶ

ଵȀଶ ൣ‫ݑ‬ത௜ଶ ൅  ‫ݑ‬ത௝ଶ ൧ ‡š’ െ

‫ۉ‬

భల഑

మ ೏೔ೕ ൗయ ቆ ቇ మ భ ഄ ൗయ

‫ۋ‬ ‫ی‬

(33)

In which ‫ݑ‬ത௜ଶ and ‫ݑ‬ത௝ଶ are calculated from the second-order structure function previously defined for the complete energy spectrum:

ௗమ

೔ ‫ݑ‬ത௜ଶ ൌ ቆʹߴ ଶ ൤௥మ ାௗ మ൨ ೏

‫ݑ‬ത௝ଶ

ଶ

ଶൗ ଷ

ಽ

೔

ௗೕమ

ൌ ൭ʹߴ ൤ మ మ ൨ ௥ ାௗ ೏

ೕ

ௗమ

೔ ൤௥మ ାௗ మ൨

ଶൗ ଷ

ଵൗ ଷ

ௗೕమ

൤௥మ ାௗమ ൨ ಽ

ቇ

(34)

൱

(35)

೔

ଵൗ ଷ

ೕ

The equations previously shown above are the modified breakup and coalescence models proposed in this research to evaluate the effects of the energy spectrum in bubble size and ݇௅ ܽ mass transfer coefficient. The results obtained are compared with experimental data taken at different positions of the bioreactor. Finally, the model proposed here is

extended to other geometries proposed by the author aimed at improving the mass transfer in bioreactors from CFD focused on a numerical approach.

However, the

validation of this last stage is out of the scope of this work.

2.2 Reactor characteristics

For CFD modeling and experimentation, a laboratory-scale stirred tank reactor made of plexiglass was used with a working volume of 10 liters. The tank has a diameter (D T) of 0.24 m, liquid height (HL) of 0.24 m, and four equally spaced baffles (angle = 90°) with a width (Wb) of 0.1 DT. A six-bladed Rushton turbine impeller with a diameter (Di) of 0.07 m was positioned at a height of 0.079 m from the bottom of the tank (Fig. 1). Air is supplied to the tank through a ring-type diffuser of 0.06 m in diameter. The diffuser contained 32 holes and the diameter of each hole was 0.0005 m. The stirring speed was set at 700 rpm and the aeration flow rate was adjusted to 6 LPM for all cases. Based on experimental data (see Fig. 6), bubbles sizes are estimated in a range between 1.0-1.8 mm.

Figure 1 Rushton 6-blade turbine. (a) Front View. (b) Diagonal view. (c) Cross-section.

2.3 Proposed geometries to improve mass transfer: Design I and Design II

Based on the improvement reached considering the entire energy spectrum, the proposed model is extended to other aeration-stirring designs. Therefore a significant effect of the energy spectrum was found in the bubble breakup and coalescence phenomena regarding the simulated geometries.

Previous studies [1] have confirmed the radial and axial pumping downward and small bubble sizes formation as the main criteria for improving mass transfer in stirred tank bioreactors. To meet these criteria, a stirring system (Di=0.17 m) with a 45° impeller blades inclination in the downwardspumping direction was proposed for designs I and II,

respectively (see Fig 2). Simultaneously, the addition of radial blades inclined at 90 ° was proposed at the impeller tip (see Fig 2) to achieve a radial flow pattern and improve air dispersion.The tank diameter (DT) and liquid height (HL) were kept the same as previously mentioned in the reactor characteristics section.

Figure 2 Proposed Stirring Device in both simulation designs: Design I and Design II. (a) Front View. (b) Diagonal view. (c) Cross-section. A modified hybrid radial-axial pumping impeller is proposed for increasing air dispersion.

The rotating sparger effects the increase of oxygen transfer rate ݇௅ ܽ has been previously reported in several studies [1], [42] - [45]. Based on what was said before, two kinds of spargers are proposed capable of generating a rotational motion tangent to the impeller discharge region to increase the bubble breakup phenomenon and bubble dispersion. The design I includes a rotating sparger composed of four cylinders and integrated curved blades for supplying and dispersing axially small bubbles. Each cylinder rotates on its x or y-axis, respectively (see Fig. 3). However, Design II is set up with sixteen cylinders and integrated radial paddles for increasing the pumping environment. Also, every four cylinders are coupled in a rotary base and the latter rotates on its x or y-axis respectively (see Fig 4). Both Designs (I and I) were configured at 0.17 m of sparger diameter.

The rotating type sparger concept has been previously studied for animal cell culturing[1]. However, that proposed design is limited to the bioreactor stirring velocity. Consequently, that aeration device is mounted on the stirring shaft generating a rotational field in the same pumping orientation of the discharge stirring device. Contrary to those mentioned above, the proposed designs here can be operated under different setup conditions independently of the mixing system according to biological process requirements.

Figure 3 Proposed Aeration Device: Design I. (a) Front View. (b) Diagonal view. (c) Crosssection.A rotating sparger composed of four cylinders and integrated curved blades is proposed for dispersing small bubbles. Each cylinder rotates on its x or y-axis respectively.

Figure 4 Proposed Aeration Device. Design II. (a) Front View. (b) Diagonal view. (c) Crosssection.A rotating sparger composed of sixteen cylinders and integrated paddles is proposed for dispersing small bubbles. Every four cylinders are integrated into a rotary base and the latter rotates on its x or y-axis respectively.

2.4 Experimental techniques

The ݇௅ ܽ value was determined experimentally using water to verify the CFD model. The bioreactor is gasified with nitrogen (N2) until reaching a minimum concentration of dissolved oxygen. In this stage, the air is supplied to the bioreactor through the sparger and temporary measurements of dissolved oxygen ‫ܥ‬௅ are taken until saturation is reached. The concentration of dissolved oxygen is measured using a dissolved oxygen sensor. The rate of change of oxygen in the liquid phase is determined using the following equation:

ௗ஼ಽ ௗ௧

ൌ ݇௅ ܽሺ‫ כ ܥ‬െ ‫ܥ‬௅ ሻ

(36)

Where ‫ כ ܥ‬is the oxygen saturation concentration and ‫ܥ‬௅ is the measured dissolved oxygen concentration. By integrating the equation (36), the following equation results: ஼ ‫ି כ‬஼ಽ ቁ ஼‫כ‬

݈݊ ቀ

ൌ  ݇௅ ܽǤ ‫ݐ‬

(37)

The bubble size distributionwas measured using a photo-optical probe (VI Kr SOPAT, Germany). The bubbles motion was recorded using four sets of images (total of 500

images) with a frame rate of 5 Hz and an acquisition time of 10 minutes. The recorded images were analyzed and processed in the SOPAT's dedicated software to obtain the bubble size distributions and the Sauter mean diameter. The total number of bubbles analyzed for every measurement was above 2000. The Sauter mean diameter was measured at four z liquid heights (0.055 m apart) in the stirred tank by keeping dimensionless radial position r/R constant. Three sets of experiments were repeated with the same operating conditions to minimize the standard deviation between the values of ݇௅ ܽ and the Sauter mean diameter.

The d index proposed by [37] was evaluated to quantify the similarity degree between the experimental data of the bubble diameters and the values of ݇௅ ܽ compared with the simulated data. The mentioned statistics consider the use of the parameter specified in place of the correlation coefficient, which is unable to distinguish the nature of the error [38]. The d indexes use information from the variance of observations and predictions. Its range can vary between 0.0 and 1.0, where 1.0 indicates a perfect concordance between the observations and the simulated data. Similarly, the d indexes can be easily interpreted and given their dimensionless nature, it can be compared with the values obtained from other models [39].

In this research, the d index is calculated with the following expression:

ே ᇱ ᇱ ଶ ଶ ݀ ൌ ͳ െ σே ௜ୀଵሾܲ௜ െ ܱ௜ ሿ Ȁ σ௜ୀଵሾȁܲ௜ ȁ െ ȁܱ௜ ȁሿ

(38)

ܲ௜ᇱ ൌ ܲ௜ െ ܱത

(39)

ܱ௜ᇱ ൌ ܱ௜ െ ܱത

(40)

ܲ௜ is the data simulated by CFD, ܱ௜ is the experimental values and ܱത is the average of the experimental data evaluated in each condition.

For ݇௅ mass transfer coefficient calculation from CFD simulations, the expression (41) was used based on the Higbies penetration theory [16]:

ఌఘ ଵൗ ସ

݇௅ ൌ ‫ܥ‬௦ ඥ‫ܦ‬௅ ሾ ఓ ೔ሿ

(41)

ಽ

The interfacial area of the bubbles is calculated as:

ܽൌ

଺‫ן‬ಸ ௗయమ

(42)

݇௅ is the transfer coefficient of the liquid phase, ‫ܦ‬௅ the oxygen diffusivity, and ε is the

dissipation of the turbulent energy, ‫ ீן‬is the air volume fraction, ݀ଷଶ is the Sauter diameter and ߤ௅ is the liquid phase molecular viscosity. The values of ݇௅ ܽ obtained by CFD are calculated as the product between equations (41) and (42).

2.5 Meshing procedure and numerical techniques

Meshing is composed of more than 1000 k computational cells. To solve the partial differential equations, a multiphase coupled algorithm that couples the pressure and velocity was used. A second-order Upwind scheme for spatial terms was also used. The solution converges when the scaled residuals remain with values less than 10-5 and when the pseudo-stable state is reached [40]. The initial bubble size was calculated according to a discretization scheme based on the geometric radius with 21 bubble classes based on the reports of [1], [8].

3. Results and Discussion

3.1 Prediction capacity of the Modified Break and Coalescence Model in a Rushton Turbine

The main objective of this research was to evaluate a modified mathematical model to calculate the bubble breakup and coalescence rates in a stirred tank bioreactor considering the complete energy spectrum. The model was compared with experimental data obtained in a stirred tank reactor mixed by a radial turbine (Rushton). Also, kernels developed here were extended to two geometries proposed by the author aimed at improving mass transfer in stirred tank bioreactors.

The breakup model of Coulaloglou and Tavlarides[34] and Prince and Blanch [88] coalescence model was used as the basis to investigate mainly the mass transfer and bubble diameter in a stirred tank bioreactor. The results were evaluated experimentally by determining the mass transfer coefficient ݇௅ ܽ and the mean bubble diameter ݀ଷଶ in four different positions of the bioreactor previously exposed in the materials and methods section.

Fig. 5 shows the behavior of the average bubble diameter determined experimentally and by CFD simulations. Calculations are based on the original model (sub-range of scales) and the modified model valid for the entire turbulence spectrum.

Figure 5 Comparison of (a) Sauter mean diameter and (b) ݇௅ ܽ mass transfer coefficient by experimental methods (EXP) and CFD simulations (Spectrum Model vs. Original Model).

Discrepancies showed in each model can be attributed to the effects of the energy spectrum. This is because of the original model, according to the results shown in Fig. 5, only considers the inertial sub-range of turbulence. In contrast, the results obtained with

the modified model seem to be closed to the experimental data. These findings obtained in this research reveal the beneficial effect of the inclusion using the entire spectrum.

Accurately simulating the breakup and coalescence phenomena is the most critical challenge for a new device optimization since the latter controls the reaction rates in a biological process.

That is why in aerobic fermentations oxygen plays a significant role as an electron terminal acceptor in the respiration process. Also, it constitutes a limiting factor in bioprocess engineering since it depends on the bioreactor operating conditions [41]. Therefore, the study of gas-liquid mass transfer from the basis of breakup and coalescence phenomena is considered one of the most fundamental stages of Bioprocess Engineering.

According to Higbie's penetration theory [16], the mass transfer depends on turbulent kinetic energy dissipation, air volume fraction, and bubble sizes. Based on the latter, simulated averages kLa values were found ݇௅ ܽ to be 45.45 and 27.0 h-1 using the modified model and the original model, respectively. While the mean ݇௅ ܽ obtained from experimental techniques showed a value of 43.0 h-1 (see Fig. 5 b). It should be noted that simulations were performed at the same operating conditions and constants mentioned in the materials and methods section. Thus the differences found here can be attributed to the effect of the complete energy spectrum.

Fig. 6 shows the experimental measurements of Sauter mean diameter. Calculations were performed at different positions of the bioreactor and its comparison with data obtained by CFD is shown. Simulations were evaluated using the original and the modified model including the energy spectrum. It is observed that the average bubble diameter is higher than the boundary limit between the dissipation and the inertial sub-range. This latter is calculated according to the value of ‫ݎ‬ௗ obtained using equation (13)(428 μm) and based on the average turbulent kinetic energy dissipation from the entire bioreactor (1.0 m2s-3).

However, the different scenarios showed can be explained by the concept of energy cascades since (as seen in Fig. 6) the original model over-predicts the bubble sizes. As mentioned before, if only the sub-range scales are considered, the breakup and coalescence rates are inaccurately estimated. Contrarily, the modified model calculates bubble size not only from the inertial sub-range but also the complete cascade of turbulence from the dissipation sub-range until the energy-containing subrange. Thus bubble breakup and coalescence rates are enhanced. Similar results have been obtained by Solsvik and Jakobsen [30].

Figure 6. Comparison of the average Sauterdiameter at different z axial positions using experimental methods (EXP) and CFD Simulations CFD (Spectrum Model vs. Original Model). The dimensionless radial position r/R was kept constant by setting a level of 0.42; r denotes the tank radial position. R is the tank radius and H means tank height.

The d index developed by Willmott[37] was evaluated for quantifying the degree of similarity between the experimental and the simulated data. Its range can vary between 0.0 and 1.0, where 1.0 indicates a perfect concordance between the observations and the simulated data.

Likewise, the concordance indexes can be easily interpreted and given their dimensionless character. It can be compared with the values obtained from other models [39]. In Fig. 7, the concordance values calculated for the experimental and simulated data of the Sauter mean diameter are observed. It is shown accurate according to the observations analyzed.

Figure 7 d indexes [-] calculated for experimental and simulated bubble size diameter data using the original breakup and coalescence model and the modified model. The dimensionless radial position r/R was kept constant by setting a level of 0.42. r denotes the tank radial position, R is the tank radius and H means tank height.

Fig. 7 shows that the original model has the lowest d indexes from each axial bioreactor position. Therefore, the mentioned model only shows acceptable Sauter predictions in zones closed to the bottom of the tank (Position 1) (d value of 0.70). Contrarily, The modified model shows the highest concordance rates in all positions compared to the original model. However, the concordance values are lower in the area closed to the turbine(value d below 0.50).

The above could be attributed to the fact that the breakup models do not consider the bubble interactions with the walls of the blade. Therefore new researches carried out are required to verify the phenomenon mentioned above.

The break-up and coalescence model proposed in this research showed better approximation in comparison to the experimental data of ݇௅ ܽ and bubble diameter. Regarding bioreactor design, the mass transfer and bubble size predictions are of great importance in the improvement of stirring-aeration devices.

3.2 Extension of the Modified Breakup and Coalescence Model to other Geometries: Designs I and II

A new aeration-stirring system (Design I and Design II) is simulated in this work using CFD. Both designs are described before on The Materials and Methods section (2.3 Proposed geometries to improve mass transfer: Design I and Design II).

CFD would be considered a promising tool in terms of mass transfer optimization due to its applicability in biological processes. Likewise, the implementation of bubble breakup and coalescence kernel, considering the complete energy spectrum can address more realistic theoretical approaches of ݇௅ ܽ.

In a biotechnological process, the ݇௅ ܽ mass transfer is one of the most important factors to consider for optimizing a new device. Oxygen is essential for ATP generation since it functions as a substrate in oxidative phosphorylation. There is a wide variety of effects of dissolved oxygen levels in the metabolism of microorganism cultures. Noteworthy, oxygen can affect cell growth, cell morphology and the production of the metabolite of interest [47].

The suggested designs mentioned before are simulated using the modified model, including the complete energy spectrum effect discussed in the previous results section. Contours of air volume fraction simulated for the new stirring-aeration designs are shown in Fig. 8 (a-b). Two different rotating spargers(a) Design I and (b) Design II are used. Both can increase air dispersion to the most staked bioreactor areas. For all cases, the simulations were performedusing the same conditions as the Rushton turbine (700 rpm and 6 LPM).

Figure 8 Contours of air volume fraction [-] simulated for the Geometries proposed in this research: (a) Design I and (b) Design II. Air is dispersed from each rotating sparger on the liquid surface. In both cases, it is shown that air is well-dispersed trough tank zones and a minimum air volume fraction is cumulated in areas closed to the sparger and stirring devices. Red color interval means high air volume fraction, and blue ones account for reduced dispersed air levels.

Fig. 8 shows a minimum air volume fraction cumulated in zones close to the spargersand stirring devicesin both simulated designs (I and II). However, air dispersion would be better dispersed using any of these proposed geometries compared with traditional turbines type Rushton.

Figure 9 Velocity magnitude vectors v/vtip[-] calculated for: (a) Design I and (b) Design II. Velocity magnitude vectors v are normalized based on the velocity tip vtip impeller. Both designs show the typical pumping downward flow in zones below impellers.

The velocity vectors observed in the bioreactor influence the air volume fraction dispersion, as seen in Fig. 9. The pumping of the downward flow is typical of an axial flow geometry. In general, rising pumping impellers push gas faster to the surface (lower gas retention) while downstream pumping drives induce recirculation and longer air bubble residence times [1]. Consequently, bubble sizes will decrease compared to the Rushton turbine device.

The residence time is defined as the time of a bubble that remains in the reactor, which leads to higher gas retention [46].

The fluid is pumped not only downwards axial

orientation, but it is also dragged towards the bioreactor walls due to the new hybrid impeller. Consequently, oxygen mass transfer is increased in both systems (Designs I and II). The modified model that includes the complete energy spectrum is evaluated in this researchby determining the bubble mean diameter and the ݇௅ ܽmass transfer. The results are shown in Fig. 10.

Figure 10 Comparison of (a) ݇௅ ܽ mass transfer coefficient and (b) Sauter mean diameter by CFD simulations (Spectrum Model vs. Original Model) for Design I and II, respectively.

Average bubble sizes with values of approximately 400 μm are observed in both designs (Design I and Design II) using the original Breakup-Coalescence model (Fig. 10 b). However, a decrease in bubbles size of 200 μm in diameter is observed by implementing the modified model valid for the complete energy spectrum. Similar behavior was previously shown in the radial turbine analysis demonstrating the global effect of the energy cascade in the breakup and coalescence phenomena already explained in this research.

Fig. 10 (a) shows the results of the ݇௅ ܽ mass transfer coefficient using CFD for the new devices applying the original and the modified model considering the entire energy spectrum. Values from 140 h-1 (Design I) to levels higher than 360 h-1 (Design II) are observed according to the proposed geometries, which confirms a high breakup rate phenomenon.

One of the conventional strategies (Turbine Rusthon) most used today to provide an adequate oxygen transfer is to increase the stirring velocity. However, the operating conditions of stirring-aeration devices are limited to their geometry design. According to [48], the required values of ݇௅ ܽ are closed to 300 h-1 for the submerged culture of the fungus Trichoderma reesei at maximum stirring speed up to 1500 rpm.

The findings found in this research suggest that these demands can be well supplied using Desing I, according to the ݇௅ ܽvalues simulated, applying the modified model. In such a way, that it is possible to reach values of ݇௅ ܽ higher than 300 h-1using stirring velocities that do not exceed 700 rpm. This is how it is observed in Fig. 5 (b) and Fig. 10 (a), the stirring-aeration systems were found to be up to 8 times higher in terms of ݇௅ ܽ using the Design I concerning a radial turbine impeller.

Comparing both geometries (Design I and Design II), it is observed similar performance in terms of air dispersion, so that small bubble sizes are formed. This is due to the effect of the geometry, especially, the design of the rotary spargers, which further favors the negative axial flow pattern.

Due to the importance of the flow pattern generated by the impellers, a comparison was made regarding the flow pattern dominated by these geometries, concerning the radial turbine. Fig. 11 (a-b) show axial ‫ݒ‬௭ and radial ‫ݒ‬௥ velocities simulated using CFD according to the stirred tank dimensionless radial position (r/ R). For all cases (Turbine, Design I and

Design II) the z height tank was kept constant by setting up a zone closed from impeller blades to the tank wall.

Fig. 11 (a) shows that the flow pattern generated by the new designs simulates negative downward pumping axial velocities, in all areas between the impeller and the bioreactor wall. These velocities are higher compared to those obtained using the Rushton device. This confirms the purpose of the new device since the basis for improving the mass transfer and bubble dispersion is aimed at the flow patterns, as well as on bubble diameters reached. As mentioned before, CFD results were performed using the modified model, including the complete energy spectrum.

That is why the device designs are focused on generating an axial flow pattern of downwards pumping and simultaneously a radial pattern in the areas near the bioreactor wall to improve the bubble dispersion. This increases the oxygen transfer in comparison with the radial geometry evaluated in the previous section.

It is also observed in Fig. 11 (a) that axial velocities tend to be favorably using the Rushton turbine in most discharge region from the impeller zone to the tank wall. The latter induces the upwards pumping and losses are expected regarding gas retention. Contrary, negative axial velocities are simulated for designs I and II, increasing downwards pumping and bubble residence time.

Radial velocities also play an important role. it is observed (see Fig. 11 b) that new designs show the highest radial velocities compared to the radial geometry. In other words, the importance of generating a hybrid flow pattern for increasing the ݇௅ ܽ and improve the gas dispersion is again confirmed to maximize the required oxygen supply during a biological process.

Figure 11 Comparison of ‫ݒ‬௭ Axial (a) and ‫ݒ‬௥ radial (b) velocities of the Rushton turbines, design I and design II, as a function of the tank radial position (r/R). r denotes the tank radial position and R represents the tank radius. For all cases, the z height tank was kept constant by setting up a zone closed from impeller blades to the tank wall.

As previously described in Fig. 10 (a-b), the improvement of air dispersion in critical areas of the bioreactor is observed using the devices proposed in this research. Based on these results, CFD simulations together with models including the complete turbulence spectrum are considered as a promising tool, especially useful for optimization studies related to industrial applications commonly characterized by oxygen transfer limitations.

The devices proposed in this research could overcome the limitations that currently exist in the cultivation of biological organisms that involve high oxygen transfer requirements. Based on the latter adequate supply of stirring and aeration conditions is one of the biggest challenges and concerns regarding new bioreactor designs.

4. Conclusions

A modified breakup and coalescence model that considers the entire cascade of turbulence was evaluated and comparedto experimental data by estimating Sauter diameter measurements in different z axial bioreactor positions stirred by a Rushton turbine. A semi-empirical model that solves the second-order structural function for including the complete energy spectrum was coupled to CFD via UDF (User Defined Functions). Considering the results, the modified model considerably improves the bubble diameter predictions. Based on the latter the model was extended to other geometries to evaluate strategies for overcoming gas-liquid mass transfer problems commonly found in bioreactors.

A significant effect of the energy spectrum was found in the breakup and coalescence phenomena. The latter also explained by the determination of oxygen transfer and bubble sizes. It was numerically demonstrated that the flow pattern generated by a stirring aeration system significantly influences the mass transfer in a bioreactor. However, experimental validations of the new Designs evaluated are required to confirm these findings.

5. Acknowledgments

The study is part of a cooperation project together with Norwegian University of Science and Technology, University Francisco de Paula Santander and the University of Antioquia.

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*Declaration of Interest Statement

Declaration of interests ‫܆‬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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Highlights

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The Original Break up and Coalescence model over-predicts bubble sizes since it considers only the sub-range of turbulence scales.

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The Complete Energy cascade improves the accurate in gas-liquid CFD-PBM models.

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Hybrid geometries (axial-radial flow patterns) increase simulated kLa in a stirred bioreactors.