Modeling of Oxygen Transfer Correlations for Stirred Tank Bioreactor Agitated with Atypical Helical Ribbon Impeller

Problem Statement: Mixing filamentous fungi entails delicate balance between promoting high volumetric oxygen transfer coefficient (kLa) while keeping low hydrodynamic stresses in the microenvironment of cultures. This study examined the oxygen transfer capability of a prototype low shear helical impeller as a potential replacement for the standard Rushton turbine. Approach: The dependence of kLa upon specific power uptake, superficial gas velocity and apparent viscosity were examined under coalescent, non-coalescent and viscous pseudoplastic fluids scenario using dynamic gassing out technique. Subsequently, collected data were treated under historical data design of response surface methodology. Results: Three empirical power law kLa correlations were developed for each fluid. Correlation comparison with literature models for single turbine agitation suggest higher kLa augmentation within 1.5-3.6 folds for helical impeller in distilled water and as high as 78% improvement in electrolytes fluid. However impeller performance is comparatively 10% lower against theoretical kLa of triple turbines arrangement for power uptake between 0-1600 W m . In case of carboxymethylcellulose, better oxygen transfer is expected for design with higher proportionality of impeller-to-tank internal diameter. Conclusions: Helical impeller performance as gas-liquid contactor is comparable to single and triple Rushtons turbines, with superficial gas velocity proved to be more influential than power uptake in non-viscous liquids and vice versa in viscous pseudoplastic fluids.


INTRODUCTION
Solubilization of oxygen in liquid medium constitutes one of the decisive factors leading to the success of fermentation processes. More often than not, wholly oxygenation of vessel would be impossible without some form of mechanical agitation [1] . Stirred tank impellers are typically classified based on the discharged flow from the blade tips, i.e., radial and axial or as indicated by the physical size with respect to tank internal diameter. Open type impellers have from one third to half of D i /D T ratio. Else, if the diameter ratio is higher or as found in some setup, quite close to the vessel wall to effect a scraping action, impellers are otherwise generalized as proximity type agitators [2] .
Open type are primarily employed in aerobic fermentation whereas proximity types are mostly associated with the bulk blending of viscous non-Newtonian fluids [3] . As far aerobic fermentation is concerned, k L a is the yardstick for evaluating impeller performance. Extensive survey [4] on turbines, paddles, propellers and self aeration agitators for P g /V in the range of 500-10,000 W m -3 had dictated similar value of volumetric oxygen transfer coefficient (k L a) correlation indices; a, b and c in power law equation (1). Concluding that, under complete gas dispersion, open type impellers at equal D i /D T can achieve the same oxygen transfer rate regardless of the different geometric designs.
The correlation is still relevant considering studies observed a mere 20% deviation of predicted k L a against actual data for coalescing and noncoalescing fluid systems from investigations on newer open type turbine having blades of different concavity [5,6] . Hence, no distinctive connection could be established between open type designs with regards to their effect on k L a. Significant differences in correlation indices may come from the addition of extra turbines on the shaft [7][8][9] while Vasconcelos et al. [10] commented that retrofitting of Rushton turbine with others at equal power uptake should imply the use of larger diameter impeller to enhance the gas handling capacity.
The possibility of employing a half-pitched double blades Helical Ribbon Impeller (HRI), a novel design under close clearance category for the improvement of k L a in stirred tank is the focus of this study. This being regardless of its prevalent mixing reputation in chemical industries, little are known of HRI gas-liquid oxygen transport characteristics in fermentation processes. Previous implementations were rationalized on the basis of maintaining low shear conditions for culturing fragile higher eukaryotic cells [11][12][13][14] or capitalizing on the agitators' bulk handling and pumping capacity to homogenize highly dense and viscous broth, as reported in the quality improvement of extracellular microbial polysaccharides [15,16] .
Typical factorial experiments require changing the level of one test variable at a time whilst keeping the rest constant. Varying the impeller speed or gassing rate in non-viscous Newtonian fluid may have given satisfactory model exponent indices in Eq. 1, since the dynamic viscosity would be approximately constant at different speeds or aeration conditions. Nevertheless, it is problematic for shear-dependent fluid due to interdependency of correlator. The changes in one's parameter level will inevitable have an impact on others as well. Varied responses may require the flexibility of Response Surface Methodology (RSM) statistical tool in manipulating k L a results to arrive to the best multivariate equations. Correlations were developed for coalescent distilled water, non-coalescent electrolytes and viscous biopolymer fluids of varying concentrations. For design justification, models derived would be mutually compared to literature values of turbine impellers.

MATERIALS AND METHODS
Mixing apparatus: Experiments were performed in a 2 L stirred tank bioreactor (Biostat DCU-200, B. Braun, Germany) retrofitted with a contemporary HRI [17] . Agitator was originally developed in-house to address the complexity of mixing M. purpureus cultures in stirred tank bioreactor. Design can be described as a hybrid having the combinatorial feature of a halfpitched helical ribbon jointed to a bottom anchor agitator. The equipment setup is shown in Fig. 1 with the tank-impeller geometries as follows: impeller diameter (D i ) = 0.088 m; pitch or vertical length for ribbon to complete a 360° spiral turn (s) = 0.114 m; ribbon width (w) = 0.008 m; impeller height (H i ) = 0.94 D i ; liquid height (H L ) = 1.5 D i ; tank diameter (D T ) = 1.3 D i ; no. of baffles = 4; baffle width, W b = 0.10 D T . Figure 2 provides an immediate perspective on the size and looks of the design compared to Rushton turbine. A variable speed A/C motor (Heidolph, Germany) drove the impeller shaft via an in-line miniature torque transducer (TP-20KCD, Kyowa, Japan). Torque (M) was sampled at 1 Hz with a multichannel amplifier (PCD 300A, Kyowa, Japan) prior to recorded in a PC-based data logger.
Volume was fixed at 1.5 L, ensuring complete immersion of HRI blades at all time. Adjustment in stirring rate was made between 150-400 rpm (complete particles suspension to near vortexing condition) with an increment of 50 rpm in successive run. Compressed air from 0.0-3.0 L min −1 was introduced at 0.012 m underneath the impeller via stainless steel ring type sparger.

Liquid phases:
The non-viscous aqueous phases consisted of distilled water, 0.5M Na 2 SO 4 and synthetic fermentation medium formulated for red biopigment producing fungus M. purpureus FTC5391 [18] was used in this study. This fermentation medium contained (g L −1 ): glucose, 50; monosodium glutamate, 12 with fractions of 0.2%, 0.4%, 0.6 and 0.8% w/v were prepared through dilution in pre-heated distilled water to mimic the shear thinning fungal broths.

Rheology analysis:
The relationship between shear stress, τ and shear rate, γ was regressed to Ostwald de Waele power law model, as in Eq. 3 with the apparent viscosity of fluid, η a estimated through Eq. 4: Rheological profiles of CMCNa solutions were determined with R/S-CC controlled shear rates rheometer (Brookfield, Middleboro, United States). Measuring bob is a standard CC48 DIN cylindrical spindle with rotational speed adjustable to 1000 rpm, effectively providing test shear rate from 0-5143 sec −1 and maximum shear stress of 206 Pa. Measurement cup requires 50 mL of sample withdrawn from bioreactor after each k L a determination made.
Mass transfer measurement: k L a was determined at 30±1.0°C (XR440 Thermistors, PACE Scientific, USA) through dynamic gassing out technique. DO was measured with a sterilizable polarographic electrode (InPro 6900, Mettler Toledo, Switzerland) fitted with a 25.4 µm thick oxygen-selective semi permeable FEP (fluorinated ethylene propylene) membrane. To overcome erroneous k L a interpretation largely attributed to the membrane diffusional resistance, mathematical correction to the original oxygen balance model [19] was made through the incorporation of delayed response constant, k e , yielding a non-linear two "first order in series" model Eq. 5: The response time of probe, τ p = 1/k e , is the time for the signal output, C p (t), to reach 63.3% of final value following an instantaneous stepwise change from completely deoxygenated to fully saturated solution. τ p was estimated at an average of 16.8 s. The experimental DO saturation curve measured from 0% to 100% (C L * ) was then fitted to Eq. 5 to extract the best k L a estimate from each data set through MATLAB curve fitting module (Matlab Release 14.0, MathWorks, USA) using Gauss-Newton non-linear least square regression.

Modeling of correlations:
Many articles constantly correlated k L a with the specific impeller power uptake (P g /V) and superficial gas velocity (U g ) expressed in an oft-cited power law form in which later reassessed for the accuracy in coefficient prediction for viscous solutions. Improvement was made through the inclusion of apparent viscosity (η a ) term into Eq. 1 [20] . Design Expert software (version 6.0.6, Stat Ease, USA) was used to simulate the predictive k L a model for the three different fluid systems. Mathematical representation of three process variables is written according to the general first order response surface model (6): Where: Y = The estimate response β i = The coefficients indicating the relative importance of the associated X correlator values.
Upon examining the general form of k L a correlation, conformation to the linear polynomial function as in Eq. 6 was made possible by simply converting beforehand all quantitative values of test variables and the corresponding responses to natural logarithmic form leading to Eq. 7. The minimum and maximum values of variables in actual, logarithmic and coded factors are shown in Table 1.
where, constant A = ln(a) taken from Eq. 1.

RESULTS
The stirred tank operating variables of P g /V, U g and η a functions are dependent upon the changes made on impeller speed, gas throughput and CMCNa solute concentrations. For the analysis, the model of interest first took the form of linearized regression as depicted in Eq. 7. Table 2 shows the result for analysis of variance (ANOVA), for fisher-test (F-test) that is significant at 5% level (i.e., p<0.05), there is evident that the model has some power to explain the variation in the response. If F-test lacks of its significant, usually a more complicated model would fit better. The F-test with a very low probability value (P>F below than 0.05) has been demonstrated as to be a very high significance for the regression model. Moreover, for models' "Lack Of Fit" (LOF) which F-value higher than 5% probability level is considered not significant relative to the pure error. The LOF is the variation of the experimental data scattered around the simulated model values. If the model values did not fit well to the actual data, correlations developed will not be significant. The goodness of fit for the multi-linear models was assessed by the determination coefficient (R 2 ). Each of the k L a model exhibits satisfactory R 2 from 0.91-0.95. For example, for non-coalescent distilled water, the determination coefficient (R 2 = 0.9532) indicates that only 4.68% of the total variations not explained by the model. The value of its adjusted determination coefficient (Adj R 2 ) is also very high, indicating a high significance of the model. The "Pred R 2 " are found to be 0.9475, 0.9078 and 0.8929 against the "Adj R 2 " of 0.9512, 0.9181 and 0.8996 for distilled water, noncoalescent electrolytes (sodium sulphate salt and M. purpureus medium) and the shear thinning CMCNa solutions, respectively. Response surface application resulted in the generation of contour and surface plots as shown by where, n is the number of data. Correlations are also compared as well with the fitting parameter of linear regression fit (the a 1 and a 2 ) of Eq. 12. Best fitting criteria are one which provide values of 'a 1 ' close to zero and 'a 2 ' and the corresponding linear regression R 2 close to unity: k L a (theory) = a 1 +a 2 ⋅ (k L a (exp) ) The fitting parameters for the three empirical correlations reverted to power law form of equation are compiled in Table 3. From the data shown, every equation provides less than 20% of mean total error and quite satisfactory coefficient of determination (R 2 ) of linear fit regression. Graphical evaluation between values of k L a calculated from Eq. 8-10 with the collection of experimental data are presented throughout Fig. 4-6, enclosed together with the upper and lower 20% error limits. Upon observation, around 88 to 90% of all k L a data are found to cluster within the bounded error range. Table 4 compares the RSM models for HRI with established literatures. Correlations related to nonviscous Newtonian liquids and 0.2% w/v CMCNa from previous reports would serve as immediate comparison with the ones developed by applying a scenario of equal specific impeller power uptake to each stirred tank system sparged at a constant 1.0 VVM. The hypothetical k L a values predicted for different set of impellers are presented in Fig. 7 and 8, respectively.

DISCUSSION
Modeling exercise shows that the linear type first order surface response model chosen would adequately explained the effect of two to three affecting factors, i.e., the P g /V, U g and η a on k L a only when the exact optimum point was not of particular interest. This is rather obvious looking at the flat profile of 3D contour and surface plots for distilled water and concentrated CMCNa solutions featured in Fig. 3. Cumbrous higher order functions are usually required in order to detect the presence of curvature or peak in response surface and hence, closer approximation to the general vicinity of precise k L a prediction. Nevertheless, it might not be necessary for these cases since the  (R 2 ) illustrates high significance, supported by close agreement between "Pred R 2 " and "Adj R 2 " where the differences in all models are less than 0.2. A higher differences could have pointed out to a possible problem in the model employed or data collection [21] .
To the best of present knowledge, no known k L a correlation currently existed which is specific to halfpitched helical ribbon-anchor configurations, thus rendering it impossible to directly compare the present correlations with others obtained from identical geometry. The only available correlation in literature with particular relevance to proximity type agitator were derived for CMCNa solutions stirred with a helical screw impeller [3] . As for open type stirrers, Van't [4] generalized correlations are well established for non-viscous fluid systems while Linek et al. [7] correlated k L a specifically for 0.5 M Na 2 SO 4 system. Most correlations on oxygen transfer for viscous pseudoplastic CMC lack the level variation of apparent viscosity due to anomalous rheological nature of simulant media with the agitator shearing rate [3,15,20] . As such, there is conspicuous omission of η a term in the collection of k L a models for Rushton turbine listed in Table 4. Most investigators would only fix the biopolymer to a single level of concentration assumed to have closest imitation of fungal viscosity [9] .
It would appear that the notion of prevailing hydrodynamics resulted from agitator geometries influencing difference k L a profile is very much corroborated from these comparative analyses. Fictitious mixing with novel HRI in theory tends to perform better than a single Rushton turbine on both types of non-viscous media. Van't [4] model has produced gross underestimation of Rushton performance against HRI agitator for water system, evident from the differences of 1.5 to 3.6 folds in k L a readings between both impellers at the lowest to highest specific power uptake range. For non-coalescent electrolyte solutions, close agreement are found between Linek and Van't [4] model for fluids mixed by a solitary Rushton turbine. Again, higher oxygen transfer performance by HRI mixing is registered for 0.5 M Na 2 SO 4 with an average of 78% increased in k L a  [3] model for helical screw and ( ): Arjunwadkar et al. [8] model for two Rushton turbines setup when shifting to the highest impeller power uptake scale. The measure to improve the k L a of open type impeller is achieved through additional numbers of Rushton turbine as indicated by Linek's model. Nonetheless in a promising outcome, three Rushton turbines affixed to a stirred tank bioreactor shaft would hypothetically provide only an average of 10% increased in k L a than values from a standalone half pitched HRI system.
Judging from the flow pattern, the novel HRI design and helical screw falls under axial dispersion category. Under shear thinning fluid scenario, the particular flow pattern was predicted to fare better in mixing viscous 0.2% w/v CMCNa than the two radial flow-type Rushton fixtures when the impeller power draw started to exceed 800 W m −3 . Above this, the prototype HRI managed to produce 15-20% higher k L a than Arjunwadkar et al. [8] two turbines model. However, correlation representing the classical full pitched inner-outer helical-screw agitator having a relatively larger diameter ratio (D i /D T = 0.88) can offers up to 37% more enhancement of oxygen transfer in the upper range of P g /V studied. Inferior k L a values below 800 W m −3 for helical screw compared to others may be attributed to the absence of horizontal turbine disc or baffles structure. The spinning disk helps to create lower pressure region in bioreactor to redirect the bubbles passage into the high shearing region of the blades while baffles can create additional breakup near the tank wall areas [2] . Nevertheless as the power input increases, the energy dissipated by the close clearance geometry could have created higher turbulence throughout the stirred tank environment and increases the liquid drag force. Thereby reducing the bubbles upward rise velocity and in turn, achieved more contact with the revolving screw.
Tecante et al. [3] had perceived that the effectiveness of axial flow close clearance impeller to be strongly dependent on gassing rate rather than the specific power uptake (agitation intensity) in viscous pseudoplastic system, which is an obvious inverse from the established correlations for radial flow Rushton turbine. By obtaining a slightly similar exponent index for P g /V variable with the aforementioned work and noting that the exponent for U g in this study to be 1.5 times more influential than value recorded for P g /V (Eq. 10) as opposed to correlations for non-viscous cases, the findings further lends credibility to the previous investigators' point of view. In the two turbines system, the scenario may have suggested that mixing mechanism of turbulent eddies primarily induced by the fluid entrained by its remote stirring within the P g /V range applied might still be insufficient to penetrate the viscous boundary layers affecting bubbles break-up and consequently, led to the low global oxygen transfer in mixing vessel, thus resulting in poorer overall k L a performance for Rushton turbine compared to the two cases utilising helical type close clearance impellers.

CONCLUSION
The RSM treatment on historical k L a dataset for correlation development was considered successful in providing good statistical fit between calculated and experimental data within ±20% error level. The correlations adequately describe the effect of operating variables and fluid properties with k L a augmentation to be primarily dependent more on impeller power uptake for non-viscous media. Nevertheless rheological modification towards viscous shear thinning conditions collectively caused the effect of superficial gas velocity, U g to be more influential than P g /V term. Upon comparison with literature models, HRI hypothetically fares better as gas-liquid contactor than a single Rushton turbine and also quite remarkable when compared against a triple arrangement. In pseudoplastic solutions, design having larger wall to blades clearance would perform better above a certain power uptake. On that basis, the k L a correlation obtained from the current system actually can be assumed to give more valid representative of a well mixed gas-liquid phase and uniform bulk mixing in stirred tank than what was reported for the turbulent open type stirrer system.