Droplet population balance

In more realistic situations there is a certain probability of the emulsion droplets coalescing with the bulk oil phase or a part of the bulk oil becoming emulsified. The physics of such complex fiow conditions is not well understood at present. The starting point of describing such a fiow would be to treat it as a normal two-phase flow and use the concept of relative permeability and a model for the rheological properties of the emulsion phase. To account for the material exchange between the bulk phase and the emulsion phase, some form of droplet population balance model will be needed. 

Population Balance Approach. The use of mass and energy balances alone to model polymer reactors is inadequate to describe many cases of interest. Examples are suspension and emulsion polymerizations where drop size or particle distribution may be of interest. In such cases, an accounting for the change in number of droplets or particles of a given size range is often required. This is an example of a population balance. 

A population balance is a balance on a defined set of countable or identifiable entities in a given system as a result of all phenomena which add or remove entities from the set. If the set in question is the number of droplets between diameter D and (D + dD) in a vessel, the set may receive droplets by flow into the vessel, by coalescence or by growth from smaller droplets. The set may also lose droplets by outflow from the vessel, by coalescence or by growth out of the set s size range. 

Equation (A12) is widely used in RE, but it does not account for the specific interactions of the dispersed phase. In this respect current research is focused on drop population balance models, which account for the different rising velocities of the different-size droplets and their interactions, such as droplet breakup and coalescence (173-180). 

Tavlarides presents a sophisticated model for representing coalescence and breakage of droplets in liquid-liquid dispersions. The model relies on the population balance equation and still requires the adjustment of 6 parameters. The solution of such equations is difficult and requires the use of Monte-Carlo methods... 

When the particles are formed at high temperatures the particles are often liquid droplets. These droplets stick together when they collide, altering the particle size distribution produced. Accounting for aggregation in the population balance in gas phase reactors is performed in the following way ... 

For different values of n ing(i ) = i2", other kinetic expressions can be developed. Figure 8.10 [18] shows the type of powder produced on spray diydng a solution that consists of metal salts of barium and iron in the ratio 1 12 (i.e., barium ferrite). Here we see the remains of the spherical droplets with a surface that consists of the metal salt precipitates, which form a narrow size distribution of platelet crystals (see Figure 8.10(a) and (b)). This narrow crystal size distribution is predicted by the population balance model if nudeation takes place over a short period of time. When these particles are spray roasted (in a plasma gun), the particles are highly sintered into spherical particles (see Figure 8.10(c)). 

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. 

The population balance approach is employed for the description of droplet dynamics in various flow fields. A significant advantage of the method is that a vehicle is provided to include the details of the breakage and coalescence processes in terms of the physical parameters and conditions of operation. A predictive multidimensional particle distribution theory is at hand which, in the case of well-defined droplet processes, can be employed for a priori prediction of the form and the magnitude of the particle size distribution. The physical parameters which affect the form... 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

It is encouraging that substantial progress has been made in analyzing the hydrodynamics of droplet interactions in dispersions from fundamental considerations. Effects of flow field, viscosity, holdup fraction, and interfacial surface tension are somewhat delineated. With appropriate models of coalescence and breakage functions coupled with the drop population balance equations, a priori prediction of dynamics and steady behavior of liquid-liquid dispersions should be possible. Presently, one universal model is not available. The droplet interaction processes (and... 

The work discussed in this section clearly delineates the role of droplet size distribution and coalescence and breakage phenomena in mass transfer with reaction. The population balance equations are shown to be applicable to these problems. However, as the models attempt to be more inclusive, meaningful solutions through these formulations become more elusive. For example, no work exists employing the population balance equations which accounts for the simultaneous affects of coalescence and breakage and size distribution on solute depletion in the dispersed phase when mass transfer accompanied by second-order reaction occurs in a continuous-flow vessel. Nevertheless, the population balance equation approach provides a rational framework to permit analysis of the importance of these individual phenomena. 

Chen [74] derived with the help of a population balance equation a mathematical mechanistic model, which enables the description and prediction of the droplet size distribution arising from emulsion and suspension processes in the loop reactor. 

The fundamental derivation of the population balance equation is considered general and not limited to describe gas-liquid dispersions. However, to employ the general population balance framework to model other particulate systems like solid particles and droplets appropriate kernels are required for the particle growth, agglomeration/aggregation/coalescence and breakage processes. Many droplet and solid particle closures are presented elsewhere (e.g., [96, 122, 25, 117, 75, 76, 46]). 

In the case of droplets of volume so small that their content can be regarded as uniform, the process can be modeled by applying to each droplet a macroscopic population balance (72) in the form... 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

A final topic concerns two phase liquid-liquid reactors, where droplet breaking and coalescence is of great importance. This complicated area cannot be covered here for a recent useful reference, see Coulaloglou and Tavlarides [86]. The methods are based on the use of population balance techniques, to be discussed in Sec. 12.6. 

If a flow system is visualized as consisting of a large number of fluid elements that collide, coalesce, and then reform into two new elements a population balance model similar to that for immiscible droplet interaction can be formulated. The latter was devised by Curl [118] also see Rietema [119] for a review of this complex area. 

M. Mezhericher, M. Naumann, M. Peglow, A. Levy, E. Tsotsas, 1. Borde, Continuous species transport and population balance models for first drying stage of nanosuspension droplets. Chemical Engineering Journal, 210 (2012) 120-135. 

A. Buck, M. Peglow, M. Naumann, E. Tsotsas, Population balance model for drying of droplets containing aggregating nanoparticles, AIChE Journal, 58 (2012) 3318-3328. 

To predict the evolution of the droplet (floe) size distribution is the central problem in emulsion stability. It is possible, in principle, to predict the time dependence of the distribution of droplets (floes) if information concering the main subprocesses (flocculation, floe fragmentation, coalescence, creaming), constituting the whole phenomenon, is available. This prediction is based on consideration of the population balance equation (PBE). 

To follow the dynamic evolution of PSD in a particulate process, a population balance approach is commonly employed. The distribution of the droplets/particles is considered to be continuous in the volume domain and is usually described by a number density function, (v, t). Thus, n(v, f)dv represents the number of particles per unit volume in the differential volume size range (v, v + dv). For a dynamic particulate system, undergoing simultaneous particle breakage and coalescence, the rate of change of the number density function with respect to time and volume is given by the following non-linear integro-differential population balance equation (PBE) [36] ... 

For each compartment, a population balance is derived as basis for the further discretization according to the selected pivot elements along the Vp axis and using number density functions ii p(F,0 and ricac(y,t) for the impeller and circulation compartments. These density functions are defined such that iiisip(y,t)dV and ncac(V,t)dV are the numbers of droplets per unit volume of the impeller and circulation compartments with a volume in the interval [V, V + dV]. 

The first term of the population balance, PBjSip, relates to the formation of droplets with a volume in the interval [F, F+dV] by breakage of droplets with a larger volume (maximally F ax)- In this term, the mesoscale parameter g U) is the breakage coefficient for a droplet with a volume U, u(U) is the number of droplets formed upon breakage of a droplet with a volume U (typically two), and P U,V) reflects the probability that a droplet with a volume U breaks into a droplet with a volume F. The second term represents the formation of droplets in the volume interval [F,... 

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