Bubble population balance

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. 

Interaction parameters, a(x,t), calculated from the bubble population balance itself, e.g., the total bubble density in flowing and stationary foam, the higher moments of the bubble number density distribution, etc. 

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. 

A meaningful simplification of the bubble population balance can be achieved by expanding the bubble number density into the moments of bubble mass. 

Some knowledge of the higher (>2) order moments is required to solve the zeroth moments of the bubble population balances. 

Bove S (2005) Computational Fluid Dynamics of Gas-Liquid Flows including Bubble Population Balances. PhD Thesis, Aalborg University, Esbjerg... 

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. 

To predict foam behavior mechanistically and quantitatively, it is necessary to account for bubble-size evolution. A foam bubble population-balance provides the necessary framework by including foam as a simple component in a standard reservoir simulator. Reduced gas mobility is modeled by a lowered gas relative permeability and a raised effective viscosity. A Stone-type model for relative permeability provides the requisite rules for modeling the relative permeability of both wetting liquid... 

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). 

The reaction engineering model links the penetration theory to a population balance that includes particle formation and growth with the aim of predicting the average particle size. The model was then applied to the precipitation of CaC03 via CO2 absorption into Ca(OH)2aq in a draft tube bubble column and draws insight into the phenomena underlying the crystal size evolution. 

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 )... 

Just like in the context of simulating solids suspension, one may wonder whether much may be expected from just sticking to the two-fluid approach combined with population balances. A better way ahead might rather be to combine population balances with LES, while proper relations for the various kernels used for describing coalescence and break-up processes could be determined from DNS of periodic boxes comprising a certain number of bubbles (or drops). The latter simulations would serve to study the detailed response of bubbles or drops to the ambient turbulent flow. 

External variables - the bubble position, x, as well as parameters, Tr(x,t), computed from the conservation equations and relationships external to the population balance, e.g., pressure, temperature, capillary pressure, etc. 

As shown in Appendix A, Equation (1) can be averaged over the volume of the porous medium to yield the population balances of bubbles in flowing foam... 

Finally, all the bubble generation and destruction functionals in Equations (2) and (3) must be known in advance to calculate their moments. It is vastly more difficult to find these functionals than to construct approximate generation and destruction functions in Equations (5) and (6). If we start from the zeroth moment equations, however, we forfeit the ability to calculate the higher order moments of the generation and destruction functionals that in turn are necessary to solve (5) and (6). To break this vicious circle without solving the full-blown population balances (2) and (3), we need to make guesses about shape of the bubble size distribution, and then iteratively solve Equations (5) and (6) until some specified criteria are met. 

The transport equation for the bubble number density distribution can be derived by use of the population balance method ... 

The zeroth order moments of the volume averaged bubble population equations, i.e., the balances on the total bubble density in flowing and stationary foam, have the form of the usual transport equations and can be readily incorporated into a suitable reservoir simulator. 

The purpose of this Appendix is to volume-average the population balance of bubble number density... 

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). 

Instead of arbitrarily considering two bubble classes, it may be useful to incorporate a coalescence break-up model based on the population balance framework in the CFD model (see for example, Carrica et al., 1999). Such a model will simulate the evolution of bubble size distribution within the column and will be a logical extension of previously discussed models to simulate flow in bubble columns with wide bubble size distribution. Incorporation of coalescence break-up models, however, increases computational requirements by an order of magnitude. For example, a two-fluid model with a single bubble size generally requires solution of ten equations (six momentum, pressure, dispersed phase continuity and two turbulence characteristics). A ten-bubble class model requires solution of 46 (33 momentum, pressure. 

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