A probability model for predicting the bubble size distribution in fully developed bubble flow in vertical pipes

Abstract

Bubble flow has good characteristics of mass and heat transfer, and is widely used in industrial practice. In bubble flow, the gas phase is mixed with the liquid phase in the form of bubbles. The gas-liquid two-phase structure of bubble flow is investigated by analyzing the motion and force states of bubbles. The most crucial bubble behaviors include the bubble collision and breakup, which continuously occur in the movement process of bubble flow. The bubble collision is usually brought by the fluid turbulence, buoyancy and laminar shear stress. The coalescence efficiency of bubble collisions is analyzed to describe the bubble coalescence behavior. A novel theory of bubble breakup is established according to the probability field analyses of turbulent eddies. Then, the generation and elimination rates of an entire bubble group can be obtained by calculating the collision and breakup probabilities of a single bubble. Combing with the investigations on the random motion rules of a single bubble and an entire bubble group, a theoretical model is first developed to predict the overall bubble size distribution in fully developed bubble flow. The predictions from the probability model generally agree with the experimental data of the bubble size distribution, validating the applicability and easier usage of the theoretical model.

Introduction

Bubble flow is widely used in industrial practice, such as nuclear reactors, rocket propellants, petroleum refining, metallurgy, wastewater treatment, and chemical industry. In ocean engineering, as the exploration of oil and gas enters into deep and ultra-deep waters, the exploitation and transportation of oil and gas are often conducted under extremely hostile conditions. The liquid-dominate flow patterns are commonly found when oil and gas are transported along deep water pipes. As a kind of liquid-dominate flow patterns, bubble flow has the characteristics of low friction, low fluid loss, strong backflow, strong carrying capacity of solid particles, water blocking and no oil blocking [1]. Moreover, the adoption of bubble flow can simplify the exploitation and transportation processes of oil and gas, short the construction time and reduce the engineering investment of oilfield development. With these good characteristics, bubble flow is widely used in the exploration and transportation of oil and gas. Bubble flow also possesses good characteristics of mass and heat transfer [2,3]. The bubble size and size distribution often determine the structure of bubble flow, and even affect the overall performance of the two-phase flow system. Under the condition of turbulent flow, bubbles in bubble flow usually have complicated random behaviors such as bubble collision and breakup. The bubble size distribution in a pipe has a great influence on the turbulent structure of the liquid phase and the mass and heat transfer characteristics in bubble flow. Meanwhile, the turbulent liquid phase in turn affects the bubble size, the volume fraction distribution and the mass and heat transfer characteristics of bubble flow. Thus, it is of great significance to investigate the mechanisms of bubble motion and bubble size distribution.

The characteristics of bubble flow, such as the fluid velocity, stress and pressure, vary randomly in a certain range. The structure of bubble flow is formed by the interactions between bubbles and the interactions between bubbles and turbulent eddies. Generally, in the fully developed bubble flow, the number and motion of bubbles are in a state of dynamic equilibrium. For example, in upward bubble flow, small bubbles collide with each other and coalesce into larger bubbles moving toward the centerline of the pipe, while larger bubbles break into smaller bubbles moving toward the wall [4]. The motion rules of bubbles in bubble flow are mainly reflected in the random movement direction and speed of a single bubble. These are all related to the collision and breakup of bubbles. The bubbles collide with each other and are broken up under the effects of the turbulence, drag force, buoyancy and wall force. The main mechanisms of bubble collision can be summarized as: (1) the effect of turbulence drives bubbles move to cause random collisions; and (2) the latter bubble is accelerated due to the wake action and collides with the previous bubble. The main mechanisms of bubble breakup can be summarized as: (1) the actions of turbulent eddies cause bubble breakup; (2) the large cap-bubbles are sheared into smaller bubbles by the laminar shear stress; and (3) the unstable interface causes the breakup of large bubbles. A new bubble is coalesced by bubble collisions. Initially, the models of collision and breakup were investigated through experimental methods. The experimental investigations of bubble coalescence can be divided into two categories: measuring the coalescence time in stagnant liquid [5,6], and determining the coalescence rate according to the bubble size data obtained in a turbulent system [6]. Ross et al. [7] and Valentas and Amundson [8] developed collision models. However, they were not examined for the lack of experimental data. Prince and Blanch [9] established a model for predicting the bubble breakup rate based on the void fraction and the time required for the breakup. Wu et al. [10] investigated the random collisions between bubbles under the effect of turbulence and the wake entrainment process due to the relative motions of bubbles. Hibiki and Ishii [11] established bubble collision and breakup models based on the random collision mechanisms of multiple bubbles and bubbles with turbulent eddies in the thermal insulation bubble flow. Hibiki and Ishii [12] developed the collision and breakup model, then they refined sink and source terms of the interfacial area concentration. Yao and Morel [13] proposed new time scales for bubble coalescence and breakup induced by turbulence, and developed the model of the nucleation of new bubbles in the volumetric interfacial area. Based on the theories of bubble collision and breakup, the bubble size distribution was investigated. The multi-bubble size class experimental solver was first introduced by Lucas et al. [14], which is necessary in investigating different velocity fields for bubbles with different sizes. Krepper et al. [15] developed the inhomogeneous multiple size group (MUSIG) model for collision and breakup based on the Eulerian modeling. Duan et al. [16] used numerical methods to simulate two population balance approaches, i.e., the average bubble number density (ABND) and inhomogeneous MUSIG models. The changes of gas volume fraction and bubble size distribution under complex flow conditions were estimated in their work. Cheung et al. [17,18] used a three-fluid model along with two-group ABND equations to simulate the classification of bubble interactions between sphere and cap bubbles. The local and axial distributions of void fractions as well as the volume equivalent bubble diameters were predicted in their work. Mukin [19] simulated isothermal mono and polydisperse bubbly flows based on the Diffusion Inertia Model (DIM) coupled with the method of function approximation. According to drag and non-drag forces, bubble breakup and coalescence processes and drag and lift coefficients on the bubble diameter, a new model of bubble breakup and coalescence was developed. Based on the baseline model, Liao et al. [20] simulated the evolution of bubble size distribution, gas volume fraction and velocity profiles along the pipe over a wide range of flow conditions. Montoya et al. [21] investigated benefits and limitations of the existent experimental approaches, discussed the availability for the development and validation of the approaches at high void fraction conditions, and proposed possible improvements. Vaidheeswaran and Hibiki [22] investigated the Eulerian two-fluid model and concluded that turbulence plays an important role in determining the void fraction distribution in a pipe.

The motion characteristics and bubble size distribution are different in different locations of a flow field. In the process of bubble breakup, the turbulent eddies appear randomly, and the sizes and energies of turbulent eddies are different in different locations. Thus, it is no longer appropriate using the average bubble method to investigate the bubble breakup behaviors in bubble flow. A more appropriate method is needed to describe the large-scale turbulent eddies.

Probability analysis is an appropriate method for investigating the occurrence possibilities of the statistic results of characteristic parameters and relevant events. Zhang et al. [23,24] used a probability model to predict the phase distributions for fully developed annular flow in vertical pipes. In their investigations, the probability model was used in describing three mechanisms: (1) the generation theory of turbulent eddies on the energy transferred from turbulent eddies to droplets; (2) the birth and death processes of liquid droplets; and (3) the interactions between turbulent eddies and droplets. The atomization and deposition rates were considered to be related to the probabilities of droplet generation and elimination, respectively. Then, a balance equation was established based on the dynamic equilibrium between atomization and deposition processes. The probability model is a statistical method with universality and practicability. In bubble flow, the liquid phase occupies a large proportion, and the gas phase exists in the liquid phase in the form of bubbles. The speeds and directions of bubbles located at different locations within the pipe are different under the effect of turbulence, and the bubble sizes are also different. For each bubble, the effects of turbulence, drag force, lift force and wall force on the bubble are different because of the bubble size. The bubble flow shows an irregular chaotic state in the whole pipe, and a certain state of the flow field appears randomly. Generally speaking, although the state of a physical system which is consisted of a bubble and its surrounding environments at a moment can be obtained from the previous state of the system through a series of analyses and calculations, many environmental details, such as the turbulent eddies, forces and local temperature caused by the energy dissipation, are always changing. Therefore, only statistical predictions rather than transient accurate ones are meaningful, which can be implemented by the probability analysis.

The brightest novelty of the present work is that an original probability model is derived from probability analyses. In contrast, the previous investigations usually adopted numerical models to predict the bubble size distribution in bubble flow. There is indeed a lack of theoretical investigations. The bubble movement in bubble flow is affected by a variety of complex forces, resulting in a sort of irregular movements. The probability model emphases on investigating this sort of irregular movements through probability analyses. Overall, according to analyzing the probability of the single bubble movement, the probability analysis of large-scale groups can be conducted, and the movements of bubbles in bubble flow can be finally obtained.

In the present work, the transient state of the bubble flow field is considered to be a random event. Based on three characteristic parameters, i.e., the liquid phase velocity, the gas phase velocity and the void fraction, the probabilistic field of bubble flow is established by analyzing the probability of the random event. The present work can be divided into four parts: (1) based on the rule of bubble collision, a bubble collision model is established through probability analyses; (2) based on the analyses of turbulent eddies, a bubble breakup model is established through probability analyses; (3) the bubble size distribution in fully developed bubble flow is regarded as a dynamic equilibrium, and then the equilibrium relationship can be established between bubble generation and elimination rates; and (4) according to the equilibrium relationship above, the bubble size distribution can be derived.