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^{[79]} 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 halfpitched 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 gasliquid oxygen transport characteristics in fermentation processes. Previous implementations were rationalized on the basis of maintaining low shear conditions for culturing fragile higher eukaryotic cells^{[1114]} 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 nonviscous 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 sheardependent 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, noncoalescent 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 DCU200, B. Braun, Germany) retrofitted with a contemporary HRI^{[17]}. Agitator was originally developed inhouse 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 tankimpeller 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 inline miniature torque transducer (TP20KCD, Kyowa, Japan). Torque (M) was sampled at 1 Hz with a multichannel amplifier (PCD 300A, Kyowa, Japan) prior to recorded in a PCbased data logger.
 Fig. 1: 
Diagram of experimental setup for stirred tank bioreactor
fitted with HRI agitator 
 Fig. 2: 
Dimensional comparison between the novel doubleflight helical
ribbon impeller with the standard 0.333 D_{i}/D_{T} rushton
turbine 
Power draw was computed according to Eq. 2:
Volume was fixed at 1.5 L, ensuring complete immersion of HRI blades at all
time. Adjustment in stirring rate was made between 150400 rpm (complete particles
suspension to near vortexing condition) with an increment of 50 rpm in successive
run. Compressed air from 0.03.0 L min^{1} was introduced at 0.012
m underneath the impeller via stainless steel ring type sparger.
Liquid phases: The nonviscous 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; K_{2}HPO_{4}, 2.5; KH_{2}PO_{4}, 2.5; MgSO_{4}.7H_{2}O, 1.0; KCl, 0.5; ZnSO_{4}.7H_{2}O, 0.001; FeSO_{4}.7H_{2}O, 0.001 and MnSO_{4}.7H_{2}O, 0.003. pH was adjusted to 6.5 either with 1.0M HCl or NaOH prior to sterilization. Whereas four sets of carboxymethylcellulose sodium salt solutions, CMCNa (Prod no. 279294T, BDH, United Kingdom) with fractions of 0.2%, 0.4%, 0.6 and 0.8% w/v were prepared through dilution in preheated distilled water to mimic the shear thinning fungal broths. Rheology analysis: The relationship between shear stress, τand shear rate, γ was regressed to Ostwald de Waele power law model, as in Eq. 3 with the apparent viscosity of fluid, ŋ_{a} estimated through Eq. 4:
Rheological profiles of CMCNa solutions were determined with R/SCC 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 05143 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 oxygenselective 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 nonlinear 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 GaussNewton nonlinear 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 oftcited 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 (ŋ_{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):
Table 1: 
Factor levels expressed in actual, logarithmic and coded values 

Where: Y = The estimate response β_{i} = The coefficients indicating the relative importanceof 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 ŋ_{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 fishertest (Ftest) 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 Ftest lacks of its significant, usually a more complicated model would fit better. The Ftest 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 Fvalue 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.
Table 2: 
ANOVA table for historical response surface data design 

The goodness of fit for the multilinear models was assessed by the determination coefficient (R^{2}). Each of the k_{L}a model exhibits satisfactory R^{2} from 0.910.95. For example, for noncoalescent 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 Fig. 3. Ultimately, RSM regression would produced the following empirical polynomial relationships expressed in uncoded factor: Case A: Coalescent distilled water system:
Case B: Noncoalescent salt and M. purpureus medium system:
Case C: Shear thinning fluid system (0.20.8% w/v CMCNa):
Alternatively, statistical verification of model accuracy are made by calculating
the mean error (%) of model values tested against a group of experimental data
for each correlation as defined by Eq. 11:
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:
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.
810 with the collection of experimental data are presented
throughout Fig. 46, 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.

Fig. 3: 
Example of contour and surface plot of RSM model for (a):
Distilled water system and (b): CMCNa at apparent viscosity ŋ_{a}
= 0.0153 Pa.s 
 Fig. 4: 
Comparison of experimental k_{L}a data with theoretical
values for coalescent water. (—) simulated data, () error limit
and (◊) distilled water 
The hypothetical k_{L}a values predicted for different set of impellers are presented in Fig. 7 and 8, respectively.
Table 3: 
Correlation indices, mean errors and linear regression coefficients
of experimental k_{L}a data fitted to the predictive RSM correlations 

Table 4: 
k_{L}a correlations for water, electrolyte and viscous
CMC solutions 

RT = Rushton disc turbine 
 Fig. 5: 
Comparison of experimental k_{L}a data with theoretical
values for noncoalescent electrolytes. (—) simulated data, ()
error limit, ( ) 0.5 M Na_{2}SO_{4} and (δ) Monascus
sp. synthetic medium 
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 ŋ_{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 determination coefficient
(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.
 Fig. 6: 
Comparison of experimental k_{L}a data with theoretical
values for viscous pseudoplastic samples. (—) simulated data, ()
error limit, (•) 0.2% w/v, (■) 0.4% w/v, (▲) 0.6% w/v
and (♦) 0.8% w/v CMCNa 
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 ribbonanchor 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]}.
 Fig. 7: 
Predicted k_{L}a as a function of P_{g}/V
for water and electrolytes sparged at 1.0 VVM. (■): Van’t^{[4]}
model for water, (▲) Present study for distilled water, ( ): Van’t^{[4]}
model for electrolyte salt, (◊): Linek et al.^{[7]} model
for electrolyte salt, (◊): Present study for electrolyte salt and (ο):
Linek et al.^{[7]} model for three Rushton turbines in electrolytes
salt 
for open type stirrers, Van’t ^{[4]} generalized correlations are
well established for nonviscous 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 ŋ_{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 nonviscous 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 noncoalescent 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
 Fig. 8:  Predicted
k_{L}a as a function of P_{g}/V for 0.2% w/v CMCNa sample.
(⊗): Present study, (•): Tecante et al.^{[3]}
model for helical screw and (δ): 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 flowtype Rushton fixtures when the impeller power draw started to exceed 800 W m^{3}. Above this, the prototype HRI managed to produce 1520% higher k_{L}a than Arjunwadkar et al.^{[8]} two turbines model. However, correlation representing the classical full pitched innerouter helicalscrew 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 nonviscous 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 breakup 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 nonviscous 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 gasliquid 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 gasliquid phase and uniform bulk mixing in stirred tank than what was reported for the turbulent open type stirrer system. " target="_blank">View Fulltext
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