## 5 Results and Discussion
### Mass Transfer Coefficient Estimation Approach Based on Developed Methodology  
Now when we are all set to handle the crucial parameter of the fermenter's efficiency mentioned in Section 2, namely to provide the phase contact area calculation methodology, it can be assumed again for CCV that bubbles have ellipsoidal form. Since the two-dimensional image gives us only an estimate of the cross-sectional area of the ellipsoid's projection, we can assume that the value of the missing semi-axis along the line of sight is equal in order of magnitude to \(c=\sqrt{ab}\). Using the approximate formula [49] for estimating the surface area of an ellipsoid, we estimate the contact area for one bubble of the gas phase and liquid as \(S=\gamma 4\pi\left(\frac{a^{p}p+\mu\sqrt{p}+a^{p}c^{p}}{3}\right)^{1/p}\), where \(p\approx 1.6075\) (for this parameter  
Figure 33: Ratio between the arithmetical mean of (**a**) number and (**b**) total area of bubbles per frame and bioreactor modes for the classical computer vision (blue) and neural network (orange) methods. The NN method produces results with less variance for the bubble number estimation, whereas the CCV method does the same for the total area estimation.  
Figure 32: Comparison of the results of the classical computer vision (blue) and neural network (orange) methods for (**a**) number and (**b**) total area of bubbles. The number estimations seems to be close but the NN method results have lower variance. There is also a gap between the total area estimation but the CCV method shows in its turn lower variance results for this case.  
the relative error of the formula is ca. one percent), \(\gamma\) -- conversion factor of pixel area to physical, \((a,b,c)\) -- semiaxes of the ellipsoid. In this case the total observed phase area is the sum of the bubbles areas.  
For irregular-shaped bubbles in the NN method we use another formula, \(S\approx\gamma(V)^{2/3}\), where \(V\approx RS_{projection}\)--the 3D volume of cluster, \(S_{projection}\)--the total sum of the cluster's pixels and we assume \(R=\sqrt{\frac{S_{projection}}{4\pi}}\). Since the capabilities of optical methods for bubble detection have certain limitations on the minimum sizes of the objects under consideration, it is necessary to use methods other than a simple summation of all the phase contact areas in order to evaluate the real mass transfer value. Indeed, a significant contribution to mass transfer might be made by submillimeter bubbles and even smaller ones, not caught by conventional camera, as also discussed in Sections 3 and 4, as well as in [45; 50].  
As seen in Figure 34 applying a mathematical function to random variables increases the final variances of estimates. In addition, the calibration coefficients are expected to have much more variance than estimated standard deviations for the obtained total area estimates since there is more uncertainty in calibration estimating. Despite the fact that the methods used made it possible to determine the type of bubble size distribution functions, the number of bubbles which are invisible to the eye and their characteristics remain unknown. In order to overcome this barrier it is possible to compare the observed bubbles with the final phase contact area calculated using other methods.  
Thus, for \(k_{L}a\) calculations the methods presented in [8; 51; 52] might be used. The _observed_ and _real_ areas of phase contact should be compared based on experimental data, then the coefficient \(k_{L}\) can also be found experimentally or theoretically by analyzing the chemical components of the liquid. Summarizing, the final (real) phase contact area is represented as an implicit function of the observed one \(a_{real}=f(a_{observed})\) and \(k_{L}a=k_{L}f(a_{observed})\). The represented approach defines trends for the further development of the experimental measuring and analytical part of the methodology assessment of the mass transfer coefficient for highly intensive gas-liquid flows in bioreactor loops.