Bubble population balance equation

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

A simplified one-dimensional transient solution of the bubble population balance equations, verified by experiments, has been presented elsewhere (5) for a special case of bubble generation by capillary snap-off. 

Bubble size is required to calculate, for example, the drag force imparted on a bubble. Most Eulerian-Eulerian CFD codes assume a single (average) bubble size, which is justified if one is modeling systems in which the bubble number density is small (e.g., bubbly flow in bubble columns). In this case, the bubble-bubble interactions are weak and bubble size tends to be narrowly distributed. However, most industrially relevant flows have a very large bubble number density where bubble-bubble interactions are significant and result in a wide bubble size distribution that may be substantially different from the average bubble size assumption. In these cases, a bubble population balance equation (BPBE) model may be implemented to describe the bubble size distribution (Chen et al., 2fX)5). 

Venneker et al. (2002) used as many as 20 bubble size classes in the bubble size range from 0.25 to some 20 mm. Just like GHOST , their in-house code named DA WN builds upon a liquid-only velocity field obtained with FLUENT, now with an anisotropic Reynolds Stress Model (RSM) for the turbulent momentum transport. To allow for the drastic increase in computational burden associated with using 20 population balance equations, the 3-D FLUENT flow field is averaged azimuthally into a 2-D flow field (Venneker, 1999, used a less elegant simplification )... 

As mentioned before. Equations (5) and (6) are the differential transport equations of average bubbles and could be written from scratch without the convoluted derivations invoked here. Unfortunately, modeling of foam flow in porous media is a lot more complicated than Equations (3) and (6) lead us to believe. Having started from a general bubble population balance, we discovered that flow of foams in porous media is governed by Equations (2) and (3), and that Equations (5) and (6) are but the first terms in an infinite series that approximates solutions of (2) and (3). 

The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

To close the present derivation of the continuum population balance equations, one needs to simplify the last two terms on the left-hand side of Equations (A-23) and (A-24). These terms describe various mechanisms of mass and/or bubble transfer among the regions defined by the characteristic functions (A-2)-(A-4). 

Sha et al [130, 131] developed a similar multifluid model for the simulation of gas-liquid bubbly flow. To guarantee the conservation of mass the population balance part of the model was solved by the discrete solution method presented by Hagesaether et al [52]. The 3D transient simulations of a rectangular column with dimensions 150 x 30 x 2000 (mm) and the gas evenly distributed at the bottom were run using the commercial software CFX4.4. For the same bubble size distribution and feed rate at the inlet, the simulations were carried out as two, three, six and eleven phase flows. The number of population balance equations solved was 10 in all the simulations. It was stated that the higher the number of phases used, the more accurate are the results. 

Venneker et al [118] made an off-line simulation of the underlying flow and the local gas fractions and bubble size distributions for turbulent gas dispersions in a stirred vessel. The transport of bubbles throughout the vessel was estimated from a single-phase steady-state flow fleld, whereas literature kernels for coalescence and breakage were adopted to close the population balance equation predicting the gas fractions and bubble size distributions. 

Several extensions of the two-fluid model have been developed and reported in the literature. Generally, the two-fluid model solve the continuity and momentum equations for the continuous liquid phase and one single dispersed gas phase. In order to describe the local size distribution of the bubbles, the population balance equations for the different size groups are solved. The coalescence and breakage processes are frequently modeled in accordance with the work of Luo and Svendsen [74] and Prince and Blanch [92]. 

Lehr and Mewes [67] included a model for a var3dng local bubble size in their 3D dynamic two-fluid calculations of bubble column flows performed by use of a commercial CFD code. A transport equation for the interfacial area density in bubbly flow was adopted from Millies and Mewes [82]. In deriving the simplified population balance equation it was assumed that a dynamic equilibrium between coalescence and breakage was reached, so that the relative volume fraction of large and small bubbles remain constant. The population balance was then integrated analytically in an approximate manner. 

Chen et al [12] and Bertola et al [8] simulated mixtures consisting of A1+1 phases by use of algebraic slip mixture models (ASMMs) which have been combined with a population balance equation. Each bubble size group did have individual local velocities which were calculated from appropriate algebraic slip velocity parameterizations. In order to close the system of equations, the mixture velocity was expressed in terms of the individual phase velocities. The average gas phase velocity was then determined from a volume weighted slip velocity superposed on the continuous phase velocity. Chen et al [12] also did run a few simulations with the ASMM model with the same velocity for all the bubble phases. 

It is noted that all the terms in the continuum population balance equation have common units, (l/m s[m]). By discretizing the continuous bubble number density in groups or classes, a PBE formulation equivalent to the discrete macroscopic framework is obtained. 

Dorao CA (2006) High Order Methods for the Solution of the Population Balance Equation with Applications to Bubbly Flows. Dr ing thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim... 

The population balance equation is a framework for the modeling of particulate systems. These include dispersions involving solid particles, liquid drops, and gas bubbles spanning a variety of systems of chemical engineering interest. The detailed derivation of the population balance equation and its applications can be found in Ramkrishna (1985, 2000). Publications pioneering the general application of population balance are by Hulburt and Katz (1964), Randolph and Larson (1964), and Frederickson et al. (1967). 

In equation 4, the subscripts f and t refer to flowing and trapped foam, respectively, and ni is the foam texture or bubble number density. Thus, nf and t are, respectively, the number of foam bubbles per unit volume of flowing and stationary gas. The total gas saturation is given by Sg = 1 — Sw = S + St, and Qb is a source—sink term for foam bubbles in units of number per unit volume per unit time. The first term of the time derivative is the rate at which flowing foam texture becomes finer or coarser per unit rock volume, and the second is the net rate at which foam bubbles trap. The spatial term tracks the convection of foam bubbles. The usefulness of a foam bubble population-balance, in large part, revolves around the convection of gas and aqueous phases. 

A general population balance equation for bubbles located at position vector with a bubble volume Vj, at time t, can be written as (Chen et al., 2005)... 

To write the population balance equation, the birth and death terms due to bubble coalescence remain to be identified. In what follows, we denote... 

By using CFD, the fluid flows can be taken into closer examination. Rigorous submodels can be implemented into commercial CFD codes to calculate local two-phase properties. These models are Population balance equations for bubble/droplet size distribution, mass transfer calculation, chemical kinetics and thermodynamics. Simulation of a two-phase stirred tank reactor proved to be a reasonable task. The results revealed details of the reactor operation that cannot be observed directly. It is clear that this methodology is applicable also for other multiphase process equipment than reactors. 

The source terms for the bubble numbers are due to breakage and coalescence of bubbles, and mass transfer induced size change. Other sources (such as formation of small bubbles through nucleation mechanisms) were neglected in this study. The discretized population balance equation can then be written in the following form... 

In the population balance equations, the number density of the bubbles is counted. This approach has been used in the simulation of two-phase processes in flowsheet simulators and in testing of the population balance models. However, in the CFD, the bubbles are divided into size categories according to mass fractions. Thus an additional interface code is needed between the user population balance subroutines used in a flowsheet simulator and that used in CFD. 

R < Rm shrink and those with R > R grow. Typically, a population balance equation is solved using Eq. (161) to obtain the evolution of the bubble size distribution. A pivotal assumption made in deriving Eq. (161), which has been made in all subsequent papers [25,71,72,73] dealing with this phenomenon, is that the pressure inside a bubble of radius R is given by pi = + 2a/R,... 

Another objective in the study of the application of CFD in crystallization is to simulate the particle size distribution in crystallization. In order to solve this problem, the simulation should take into account the population balance. The internal coordinates of the population balance make it difficult to utilize it in the CFD environment. In addition, differentsized particles have different hydrodynamics, which causes further complications. Wei and Garside [42] used the assumption of MSMPR and the moments of population balance to avoid the above difficulties in the simulation of precipitation. In the CFX commercial application, the MUSIC model offers a method for solving the population balance equation in CFD and defines the flow velocity of different-sized particles on the basis of the ratio of the velocity to a referencesized particle. The velocity of this reference-sized particle is then simulated using the multiphase-flow model. In this way, the problem of the different hydrodynamics of different-sized particles is solved. However, the accuracy of this model is not sufficient for it to be directly used for bubble-sized distribution only. More work is required before this model can be applied to crystallization. 

---