1. Introduction

Previous CFD studies have employed the assumption of a unified bubble diameter, which can generate reliable predictions if the bubble size distribution is very narrow. However, numerical modelling of gas-liquid two-phase flow behaviors should also take into account scenarios

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

where wide bubble size distributions and eddy/bubble-bubble interactions exist. These are very influential factors in the calculation of the gas-liquid interfacial area, which in turn affects the prediction of the mass and heat transfer between the two phases. By solving the population balance equations (PBEs) during the numerical simulation, the bubble size distribution can be derived directly, while the behaviour of the eddy/bubble-bubble interactions can be reflected within coalescence and breakup models.

2. Mathematical modelling

The bubble size distribution is determined by employing the population balance model with a consideration of bubble coalescence and breakup. Bubbles are divided into several size groups with different diameters specified by the parameter deq,i and an equivalent phase with the Sauter mean diameter to represent the bubble classes. In this study, 16 bubble classes with diameters ranging from 1 to 32 mm are applied based on the geometric discretization method

Modelling of Bubbly Flow in Bubble Column Reactors with an Improved Breakup Kernel Accounting for Bubble…

! <sup>i</sup> � ni

The source term for the i-th group, Si, can be thought of as the birth and death of bubbles due to coalescence and breakup, respectively. The expression for this particular term is given by Eq. (2)

where f <sup>i</sup> is the i-th class fraction of the total volume fraction and Vi is the volume for the i-th

To describe the coalescence between two bubbles, the coalescence kernel proposed by Luo [10] was utilized in this study. As this is not the main concern of this work, further details of the

The breakup model proposed in this study is based on the work of Luo and Svendsen [3]. Several improvements have been introduced in this study to produce a more realistic breakup model. In Luo and Svendsen's model, the shape of breakage bubbles was assumed to be spherical. However, the experimental studies and statistical results, such as Grace et al. [11] and Tomiyama [12], have found that bubbles exist in various shapes and the dynamics of bubble motion strongly depend on the shape of the bubbles. For example, Figure 1 shows the

Ω<sup>C</sup> deq,j : deq,i

� � <sup>þ</sup> <sup>X</sup>

dmax

dj¼di

αgf <sup>i</sup> ¼ niVi (3)

� � <sup>¼</sup> Si (1)

http://dx.doi.org/10.5772/intechopen.76448

! is the mass average velocity vector and Si is

Ω<sup>B</sup> deq,j : deq,i

� � � <sup>Ω</sup><sup>B</sup> deq,i

� �

(2)

67

where Vi ¼ 2Vi�1. The population balance equation is expressed by Eq. (1) ∂ni

where ni is the number density for i-th group, vi

Si ¼ Bcoalescence, <sup>i</sup> � Dcoalescence, <sup>i</sup> þ Bbreakup, <sup>i</sup> � Dbreakup, <sup>i</sup>

The local gas volume fraction can be calculated using Eq. (3),

Ω<sup>C</sup> deq,j : deq,i � deq,j � � �

coalescence kernel can be found in Luo's paper.

<sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>∇</sup> � <sup>v</sup>

deq,maxX� deq,i deq,j

2.1. Bubble size distribution

the source term.

d Xeq,i=2 deq,j¼deq,min

¼

class.

For the process of bubble breakup, Coulaloglou and Tavlarides [1] assumed that the breakup process would only occur if the energy from turbulent eddies acting on the fluid particle was more than the surface energy it contains. Prince and Blanch [2] acknowledged that bubble breakup is caused by eddy-bubble collision and proposed that bubble breakup can only be induced by eddies with approximately the same characteristic length. For instance, eddies at a much larger length scale transports the bubbles without causing any breakups. Luo and Svendsen [3] described the bubble breakup process by considering both the length scale and the amount of energy contained in the arriving eddies. The minimum length scale of eddies that are responsible for the breakup process is equivalent to 11.4 times the Kolmogorov scale. The critical probability of bubble breakup is related to the ratio of surface energy increase of bubbles after breakup to the mean turbulent kinetic energy of the colliding eddy. Therefore, very small eddies do not contain sufficient energy to cause the bubble breakup process. Lehr et al. [4] proposed a slightly different breakup mechanism from Luo and Svendsen [3] by considering the minimum length scale of eddies to be determined by the size of the smaller bubble after breakup. They also specified that the breakup process is dependent on the inertial force of the arriving eddy and the interfacial force of the bubble. Based on the results of Luo and Svendsen [3] and Lehr et al. [4], Wang et al. [5] proposed an energy constraint and capillary constraint criteria for the breakup model. The energy constraint requires the eddy energy to be greater than or equal to the increase of surface energy of bubbles after the breakage has occurred. The capillary constraint requires the dynamic pressure of the eddy to exceed the capillary pressure of the bubble. The use of these two breakup criteria has restricted the occurrence of breakage that generates unphysically small daughter bubbles and demonstrated more reliable results than that of Luo and Svendsen [3]. Similar ideas to those of Wang et al. [5] have also been adopted by Zhao and Ge [6], Andersson and Andersson [7] and Liao et al. [8]. A more concise breakup constraint of energy density increase was proposed by Han et al. [9]. The constraint of energy density increase involves only one term, which is the energy density itself, to represent what was originally expressed by two terms: capillary pressure and surface energy. It was shown that the energy density increase during the entire breakup process should not exceed the energy density of the parent bubble.

Incorporation of a bubble shape variation into the breakup model has rarely been documented in the open literature. Therefore, the aim of this study is to consider the influence of bubble shape variation on the bubble breakage process in bubble column flows. A breakage model accounting for the variation of bubble shapes, coupled with the breakage criterion of energy density increase, is proposed here.
