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Ind. Eng. Chem. Res. 2005, 44, 7540-7549
Population Balance Model for Gas-Liquid Flows: Influence of Bubble Coalescence and Breakup Models Tiefeng Wang, Jinfu Wang,* and Yong Jin Department of Chemical Engineering, Tsinghua University, Beijing 100084, People’s Republic of China
In dispersed gas-liquid flows, the bubble size distribution plays an important role in the phase structure and the interphase forces, which, in turn, determine the multiphase hydrodynamic behaviors, including the spatial profiles of the gas fraction, gas and liquid velocities, and mixing and mass-transfer behaviors. The population balance model (PBM) is an effective method to simulate the bubble size distribution. The bubble coalescence and breakup models have a distinct influence on the prediction ability of the PBM. This work compares several typical bubble coalescence and breakup models. The results show that the bubble size distributions predicted by the PBM are quite different when different bubble coalescence and breakup models are used. By using proper bubble coalescence and breakup models, the bubble size distribution and regime transition can be reasonably predicted. The results also show that it is necessary to take into account bubble coalescence and breakup due to different mechanisms. 1. Introduction
2. Population Balance Model
In dispersed gas-liquid flows, the bubble size distribution plays an important role in the phase structure and interphase forces, which, in turn, determine the multiphase hydrodynamic behaviors, including the spatial profiles of the gas fraction, gas and liquid velocities, and mixing and mass-transfer behaviors. It is therefore necessary to take into account these influences to get good predictions in wide conditions by the computational fluid dynamics (CFD) simulation. The population balance model (PBM) is an effective method to simulate the bubble size distribution. The PBM was first formulated for chemical engineering purposes by Hulburt and Katz1 and has drawn much attention from both academic and industrial researches because it can describe the size distribution of the dispersed phase in a wide variety of particulate processes.2 In recent years, much work has been done using the PBM to simulate the bubble size distribution in gas-liquid flows, which shows the great potential of the PBM in such systems.3-7 In the PBM, the bubble coalescence and breakup models are very important for reasonable predictions of the bubble size distribution. Many bubble coalescence and breakup models have been proposed.8-17 However, some obvious discrepancies exist in these models; for example, the daughter bubble size distributions are greatly different for different bubble breakup models, as reviewed in our previous publication.18 Therefore, it is necessary to compare the typical bubble coalescence and breakup models that have been commonly used in the literature. This comparison is also valuable for further coupling of the PBM into a CFD framework to get a better understanding of the hydrodynamic behaviors in a gas-liquid flow. This work aims to make a comparison of several typical bubble coalescence and breakup models and to discuss in detail the ability of the PBM to predict the bubble size distribution.
The PBM is a statistical formulation to describe the size distribution of the dispersed phase in a multiphase flow. Generally speaking, the PBM equation for a gasliquid flow can be expressed as follows:19
* To whom correspondence should be addressed. Fax: 8610-62772051. E-mail:
[email protected].
∂n(v,t) + ∇‚[ubn(v,t)] ) ∂t 1 v n(v-v′,t) n(v′,t) c(v-v′,v′) dv′ 2 0
∫
c(v,v′) dv′ +
∫0∞n(v,t) n(v′,t) ×
∫v∞β(v,v′) b(v′) n(v′,t) dv′ - b(v) n(v,t) (1)
Equation 1 is an integrodifferential equation and can only be solved numerically. Different approaches have been proposed in the literature to solve the PBM equation, as reviewed by Ramkrishna.19 The discretization method developed by Kumar and Ramkrishna20 was used in this work. In this method, the bubble size is divided into different size subregions, and eq 1 is integrated over each discrete bubble size region. The bubbles with volume unequal to any pivot are redistributed to the two nearest pivots using an approach ensuring the conservation of bubble mass and number. The final discrete PBM is in terms of Ni and reads
dNi(t) dt
+ ∇‚[ubNi(t)] ) jgk
∑ j,k
gi-1egj+gkegi+1 M
Ni(t)
∑
(
k)1
)
1 1 - δj,k ηi,j,kcj,kNj(t) Nk(t) 2 M
ci,jNk(t) +
ζi,kbkNk(t) - biNi(t) ∑ k)i
(2)
where cj,k ) c(vi,vj), bi ) b(vi), M is the number of size subregions, and ζi,k and ηi,j,k are as follows:
ζi,k )
∫gg i
gi+1 -v β(v,gk) dv+ gi+1 -gi
i+1
∫gg
10.1021/ie0489002 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/13/2005
i
v-gi-1 β(v,gk) dv (3) gi - gi-1
i-1
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7541
ηi,j,k )
{
gi
