Generación del pipeline de la aplicación backend

For the evaluation of the diagrams presented in the next sections the turbulence eddy dissipation ε has been selected as the only variable parameter. The flow parameters presented in the formulas presented in the previous sections of this chapter for the evaluation of the interaction mechanisms frequency and efficiency have been kept constant and their values are:

𝛼 = 0.2  [−] 𝐷 = 0.003  [𝑚] 𝜌 = 997  [𝑘𝑔 𝑚⁄ ] 𝜌 = 1.185  [𝑘𝑔 𝑚⁄ ] 𝜎 = 0.0727  [𝑁 𝑚⁄ ]

7.6.1 Bubble Coalescence Due to Random Collision

In the case of the random collision coalescence terms, it is not possible to compare all the above reported models together. This is due to the fact that, on one side, Hibiki and Ishii [Hibiki and Ishii 2000a] and Yao and Morel [Yao and Morel 2004] are furnishing an expression for the calculation of the bubble coalescence efficiency and are treating it separately from the bubble collision frequency. On the other side, the model of Wu et al. [Wu et al. 1998] and the others derived from it, (Ishii and Kim [Ishii and Kim 2001] and Wang [Wang 2010]) do not consider the bubble coalescence frequency separately, as Wu et al. decided to use a constant coefficient for the collision frequency 𝜂 in (Eq.173). The value of this coefficient is not explicitly given in the work, but it is contained in the constant C .

For this reason it is not possible to separate the effects of the two components and it is not possible to perform a comparison.

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Figure 27: Trend of the Random Collision Frequency x Efficiency as a function of the Turbulence eddy dissipation

In Figure 27, it is possible to see how the different proportionality coefficients chosen by Wu et al. [Wu et al. 1998], Ishii and Kim [Ishii and Kim 2001] and Wang [Wang 2010], affect the trend of the random collision frequency multiplied by a constant coalescence efficiency as a function of the turbulence eddy dissipation. The highest values are obtained by the model proposed by Wu et al. [Wu et al. 1998]. The slope of the curve is very high for very low values of the energy dissipation. The lowest values are those obtained by Ishii and Kim [Ishii and Kim 2001].

Figure 28: Trend of the Random Collision Frequency as a function of the Turbulence eddy dissipation

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In Figure 28, the random collision frequency model of Hibiki and Ishii [Hibiki and Ishii 2000a] is compared to that of Yao and Morel [Yao and Morel 2004]. This last model delivers results in the order of two orders of magnitude higher than the Hibiki and Ishii model even at very high values of the energy dissipation.

Figure 29: Trend of the Coalescence Efficiency as a function of the Turbulence eddy dissipation

In Figure 29, the trend of the coalescence efficiency as a function of the turbulence eddy dissipation is shown. The model of Hibiki and Ishii [Hibiki and Ishii 2000a] and the model of Yao and Morel [Yao and Morel 2004] are delivering essentially the same results.

7.6.2

Bubble Breakup Due to Turbulence Impact

In case on the impact of turbulent eddies against the bubbles, it is possible to compare all the available models for the bubble-eddy collision frequency and for the efficiency since the expressions for the calculation and almost all coefficients have been given explicitly in the respective works. In case of a coefficient have not been given separately for each effect, it has been possible to separate it from other constants for the comparison. These coefficients have been given in the preceding chapter.

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Figure 30: Trend of the Turbulence Impact Frequency as a function of the Turbulence eddy dissipation:

whole frequency scale range (left), reduced frequency scale range (right)

In Figure 30, the turbulent impact frequency models have been compared. The highest values are obtained by using the Wu et al. [Wu et al. 1998], model as already seen for the coalescence frequency. The second highest values are obtained using the Yao and Morel [Yao and Morel 2004] model.

In order to appreciate the trend of the Wang 2010 and Hibiki and Ishii 2000 models, a scale range reduction is needed (Figure 30 right). In this figure it is possible to see that the Wang and Ishii and Kim models do not present breakup events at all up to a critical value of the turbulence eddy dissipation. This value in this special case lies between 6 and 7 [m2 _s3_].

The Hibiki and Ishii [Hibiki and Ishii 2000a] model delivers in general the lowest results of the collision frequency caused by the turbulent impact of the eddies against the bubbles.

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Figure 31: Trend of the Breakup Efficiency as a function of the Turbulence eddy dissipation

In Figure 31, the breakup efficiency models as a function of the turbulence eddy dissipation have been compared. The highest value is obtained again using the Wu et al. model. The lowest value of the breakup efficiency is obtained using the model of Wu et al. with coefficients from Wang.

It is worth of notice the fact that the same trend is appreciable both for the breakup and coalescence cases: Wu el al. model is delivering the highest coalescence and breakup rates. The model delivering the second highest rates is that proposed by Yao and Morel; both for bubble breakup and coalescence case. The model delivering the lowest bubble interaction rates is that proposed by Hibiki and Ishii. However, the absolute value assumed by the models is not so relevant to the evaluation of the capability of a model. In fact, the source and sink terms explained in the previous sections are applied together, for each author, to modify the transported value in the interfacial area transport equation. For this reason the important is the resultant value from the balance between creation and destruction of bubbles and not the values of the single interaction terms considered alone.

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Chapter 8