The effect of probability of coalescence on the evolution of bubble sizes in a turbulent pipeline flow: A numerical study

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Computers and Chemical Engineering 32 (2008) 1249–1256

The effect of probability of coalescence on the evolution of bubble sizes in a turbulent pipeline flow: A numerical study Miguel Miura, Oleg Vinogradov ∗ Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4 Received 1 November 2006; received in revised form 2 May 2007; accepted 24 May 2007 Available online 29 May 2007

Abstract A population balance method in which continuum and discrete phases are integrated is developed to simulate the evolution of polydisperse population of bubbles in a turbulent pipeline flow. The investigation is focused on the effect of the coalescence efficiency on this evolution. A dilute system of bubbles under microgravity conditions is considered. It is found that if the initial coalescence efficiency is low, a slight increase produces a significant effect on the bubble coalescence rate, and thus on the evolution of the population. If, however, the initial coalescence efficiency is high, its increase results in a marginal effect on the way the population evolves. The results of simulations are validated against experimental data on the population mean. © 2007 Elsevier Ltd. All rights reserved. Keywords: Bubbles; Coalescence; Population evolution; Turbulent flow

1. Introduction Bubbles play a major role in many technological processes in chemical, metallurgical, and oil and gas industries. In many situations, knowledge of, and ability to control bubble size distribution, would enable optimizing the outcome of the process. In one specific application, the efficiency of a hydro-transport technology of bitumen extraction from the oil sands is based on the efficiency of coalescence of bitumen droplets and air bubbles, which, in addition to many other factors, depends on the size distributions of both droplets and bubbles and on the distance along the pipeline where the bubble population stabilizes. Here we concentrate on the evolution of the bubble population in a turbulent pipeline flow due to the effect of only one factor, the coalescence. The methodological component of the paper deals with further development of the approach suggested in Leonenko, Vinogradov, and Braverman (2004). In the latter paper the dynamic properties of the discrete phase were investigated separately based on the methodology of simulating granular multi-body systems by Vinogradov and Sun (1997), which, in general, can handle various kinds of particles, vari-

∗

Corresponding author. Tel.: +1 403 220 7187; fax: +1 403 282 6284. E-mail address: [email protected] (O. Vinogradov).

0098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2007.05.015

able concentrations and also interactions leading to formations of clusters. This approach of integrating a discrete phase comprising solid particles and a turbulent carrier in a pipeline flow was implemented and validated against experimental data in Leonenko and Vinogradov (2001). The advantage of the latter approach is that each particle is tracked individually and thus the events of breakage/coalescence can be correlated with particle collisions. The drawback, however, was that for the nonsteady-state regimes of the discrete phase, the representative volume (or the length of the pipeline in this case) had to be large enough to provide statistically stable results, which, in industrial type applications, was computationally expensive. The idea of the approach suggested in Leonenko et al. (2004) was to combine the method of population balance with the direct simulation of the discrete phase used in Leonenko and Vinogradov (2001). Namely, it was suggested finding collision frequencies for various combinations of species sizes, and then introducing a collision frequency function, which can be correlated with the sink/source terms in the population balance equation. This approach was validated in Leonenko et al. (2004) on simple models with analytical solutions. In the present paper the approach of Leonenko et al. (2004) is further extended to a polydisperse system of bubbles in a turbulent pipeline flow with possible coalescence between the bubbles. The latter limitation is imposed in order to compare the

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results against experimental data (Colin, Riou, & Fabre, 2004; Kamp, Chesters, Colin, & Fabre, 2001) collected on turbulent bubbly pipe flow under microgravity conditions. Microgravity provides a unique situation in which it is possible to isolate turbulence-induced coalescence, since there are no other relevant body forces and the contribution of the mean shear plays a small role due to the flatness of the velocity profile in the center region of the pipe, where most of the bubbles can be found (Kamp, Colin, & Fabre, 1995). Additionally, the experimental investigations (Colin et al., 2004; Kamp et al., 2001) reported no bubble break-up for the whole range of operating conditions studied (in the turbulent regime). Since both coalescence and break-up phenomena are associated with uncertainties on their prediction, performing the simulation of the problem under microgravity conditions allows eliminating one of the uncertainties and thus provides a more reliable validation of the methodology of the direct simulations of collisions. We use the population balance (PB) technique to simulate the evolution of the polydisperse bubble population in a pipeline. For the general theory of the PB the reader is referred to Ramkrishna (2000). In this paper, the polydisperse system is approximated by a discrete set of bubbles of equal sizes using a geometric grid discretization technique (note that the problem of discretizing a polydisperse system is a subject of research by itself; see, e.g. Kumar, Peglow, Warnecke, Heinrich, & Morl, 2006). Then the collisions between the bubbles with all different classes are investigated by direct numerical simulations of a representative volume of the discrete system. Such simulations are conducted a sufficient number of times as to obtain statistically stable mean values for the collision frequencies for any given state of bubbles sizes. This information is then incorporated into a continuous population balance model, where the event of coalescence is introduced phenomenologically through the probability of coalescence. The following numerical investigation, in addition to demonstrating the methodology, concentrates on the sensitivity of the population evolution to the probability of coalescence. 2. The two-phase model The physical model is comprised of a polydisperse system of bubbles (air) flowing together with a continuum phase (water) through a horizontal pipe. The description of the carrier phase (water) is based on Kolmogorov’s concepts of fully developed turbulence (Frisch, 1995; Landau & Lifshitz, 1987). Its velocity field is considered to be independent of the presence of the dispersed phase (assumption of a dilute two-phase system), and the usual convention that the true velocity v at any point in the fluid is composed of the aver and the fluctuating one v (v = U  + v ) is also age component U applied, as assumed and validated in Leonenko and Vinogradov (2001) for fully developed turbulence. For pipeline flows, the mean flow velocity has an axial component only and can be assumed to be uniform along the cross-section (Massey, 1989), whereas the fluctuating velocity has components in both axial and radial directions. The corresponding vector is described by

the following expression (Landau & Lifshitz, 1987): v =

N 

Ai [cos θi , sin θi ]T

(1)

i=1

where Ai is the velocity fluctuation associated with the ith eddy, and θ i is a random direction. The bubbles are assumed to be “small”, in the sense that they are smaller than the characteristic length scale of turbulence. A more formal description of this assumption is that bubble sizes lie in the inertial sub-range of isotropic turbulence, which is a customary assumption (Prince & Blanch, 1990). This inertial region includes all the length scales located within the large energy containing eddies and the viscous dissipation eddies. The justification for restricting the turbulence spectrum is that very small eddies do not contain sufficient energy to affect the bubble motion, while eddies much larger than the bubble size transport groups of bubbles together without leading to significant relative motion (Prince & Blanch, 1990). Consequently, the size of the smallest eddy lmin is assumed to be greater than the diameter of the largest bubble in the system dmax . A limit on the size of the largest eddy lmax was found by correlating numerical experiments in Leonenko and Vinogradov (2001). The results showed a marginal variation when the largest eddy size was between H/2 and H/7, where H is the diameter of the pipe. The number of eddies N in the range lmin to lmax was found to have a marginal effect on the results when its value was greater than 100, using systems with low volumetric fractions of the particulate phase. Assuming that lmin equals the diameter of the largest bubble D, the uniformly distributed series of eddy sizes is calculated as: lj = D + j

lmax − D , N

j = 1, . . . , N

(2)

The velocity fluctuation associated with each eddy vj is related to the eddy size by (Leonenko & Vinogradov, 2001):   lj 1/3 Aj = vj = α (3) lmax where α is an empirical parameter whose correct value must be found experimentally. However, an estimate can be made by considering that the root mean square of the velocity fluctuations is approximately equal to the shear velocity in the viscous sublayer (Oroskar & Turian, 1980). It was shown in Leonenko and Vinogradov (2001) that an estimate for α can be found from:    −0.5 √  li 2/3 α = υ∗ 2 (4) lmax i

where υ* is the shear velocity in the viscous sub-layer, which is a function of the wall friction coefficient and the mean fluid velocity. It is assumed here that coalescence occurs due to binary collisions between the bubbles (Prince & Blanch, 1990; Wu, Kim, Ishii, & Beus, 1998), which, in turn, are caused by turbulent eddies. The latter assumption was validated in Kamp et al. (2001) for microgravity conditions. Thus, the above Kolmogorov model provides a convenient tool to assess the collisions between the

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bubbles. The latter is done by direct numerical simulation of flow in a 2D representative volume (RV), which, for a homogeneous flow regime in a pipe, reflects the properties of a 3D system. The main objective of the direct simulations is to obtain collision frequencies between various size bubbles as functions of the populations of these sizes. Since collision frequencies and coalescence rates are correlated, the former information allows using the population balance technique to calculate the bubble number density for each class (Leonenko et al., 2004). 3. Population balance equations for a polydisperse system For polydisperse systems, the bubble size distribution (BSD) function can be divided into discrete systems of classes in such a way that in each class the size is assumed constant while the overall mass (or volume in this case) of bubbles is conserved. The system of PB equations reduces to the system of coupled ordinary differential equations shown in Eq. (5) once the source term associated with the break-up of bubbles is neglected, the dispersive component of the PB formulation is neglected (as numerically justified in Miura, 2005), and the bulk velocity is assumed one-dimensional. ∂Ct ∂Ci i , = Sco +U ∂x ∂t

for i = 1, . . . , m

(5)

where Ci is a measure of bubble concentration in class i, U the local mean velocity of the carrier in the direction along the pipe, i is the coalescence rate term. and Sco For a polydisperse system of bubbles, the source/sink term due to coalescence in any class i becomes a function of coalesi term in cence in class i and various other classes, i.e. the Sco Eq. (5) takes into account the coalescence between all combinations of particles of other classes that lead to the formation of particles of class i, and the coalescence of the particles of class i itself, which results in the vanishing of particles of this class. i term can be expressed in the form: Accordingly, the Sco i Sco = Si+ − Si−

Si+ =

m−1 

(6)

c+ (k, j),

for i = 2, . . . , m

(7)

c− (i, k),

for i = 1, . . . , m − 1

(8)

k=1

Si− =

m−1  k=1

where c+ (k, j) represents the coalescence rate between bubbles of certain pairs of classes k and j such that their union would produce particles of class i, and c− (i, k) represents the coalescence rate between bubbles of class i and bubbles of any other class except for the largest one, which is not allowed to coalesce in order to keep the number of classes finite. The parameter m defines the total number of classes in the system and is also the index for the class with the largest bubbles. One way to express the coalescence rate c(i, j) between the bubbles from two generic classes i and j is by presenting them as a product of the coalescence efficiency Ωi,j and the bubble–bubble

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collision frequency per dispersion volume λi,j (Prince & Blanch, 1990): c(i, j) = Ωi,j · λi,j

(9)

The coalescence efficiencies for all possible combinations between the classes were found using the following equation (Kirkpatrick and Lockett, 1974):  coal  ti,j (10) Ωi,j = exp − cont ti,j where tcoal is the film drainage time between two bubbles, and tcont is the bubble interaction/contact time. These times were estimated using the following expressions (Prince & Blanch, 1990): (di,j /2)2/3 ε1/3 1/2    (di,j /2)3 ρc h0 =c ln 16σ hf

cont ti,j =

(11)

coal ti,j

(12)

where di,j is the equivalent diameter for bubbles of classes i and j calculated as 2 (di ·dj )/(di + dj ), ρc the density of the carrier, ε the turbulent energy dissipation rate, σ the bubbles surface tension, h0 the initial film thickness (film thickness when the bubbles are assumed to start contact), and hf is the final film thickness (critical or rupture film thickness). Constant c has been proposed by Laari and Turunen (2003) as an adjustable parameter that should be determined experimentally for different liquids. A commonly accepted average value for the surface tension σ is 0.0725 N/m, as presented by Hesketh, Russell, and Etchells (1987) from experiments with air–water systems. Several approaches have been used by different researchers in the estimation of the initial and final film thicknesses. Some assume both are constant according to typical values (Prince & Blanch, 1990), while others propose ways to evaluate both thicknesses in terms of the size of the thinning disk, the equivalent diameter, the approach velocity, etc. (Venneker, Derksen, & Van den Akker, 2002). Once the contact and coalescence times have been estimated, the coalescence efficiency may be calculated using Eq. (10). As it can be seen from Eqs. (11) and (12), the ratio tcoal /tcont is 5/6 proportional to di,j , which implies that as the equivalent diameter di,j increases, the coalescence efficiency between the particles of the classes i and j decreases. It should be noted that not all combinations of particles smaller than class i will coalesce into particles of class i. If the discretization of bubble sizes is based on a geometrical scheme according to which the next largest size has a mass of two times that of the previous class, the coalescence source and sink terms can be formulated in such a way that both the number and the mass balances are conserved (Hagesaether, Jakobsen, & Svendsen, 2002; Laari & Turunen, 2003). This requires adding multiplicative factors to the coalescence rates in order to split the source/sink terms in the case of a “daughter” particle with a volume that falls between two of the initially specified class

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volumes. The coalescence rate is then split between the larger of the two classes colliding and the class immediately larger, defined by certain weight factors that depend on the volume of the new particle (see Hagesaether et al., 2002; Hounslow, Ryall, & Marshall, 1998 for more details). Since the source terms of the PB equation for the class i depend on the population mean, which, in addition, is a function of the number density of all particle classes, the PB equations for every class are coupled and must be solved simultaneously. The problem statement requires proper formulation of initial and boundary conditions, which are defined as follows: (a) the population will eventually reach a stable configuration along the pipeline, which is the focus of our interest, and, in this respect, any initial condition may be used as long as it is compatible with the Dirichlet boundary condition imposed; (b) the number density function along the pipe is known at the initial state and it remains constant at any cross-section of the pipe; (c) the space derivative of the concentration of each species will be zero at a sufficiently far position along the pipe, referred to as the extended length. Finding this extended length, or the value of the sufficiently long simulation time, can be done by performing numerical experiments. The collision frequency functions (CFFs) are found by numerical simulations of discrete systems using a discrete particle method. According to the latter, the equation of motion is solved for every bubble in the system, taking into account the liquid–bubble interaction forces, such as drag, added mass, and buoyancy. Adopting similar assumptions to those made in Leonenko et al. (2004), other forces, such as the Saffman lift force, caused by the local mean and turbulent velocity gradients, and the Magnus force, due to the bubble rotation, are neglected. The event of a collision between two bubbles is treated as that between two hard spheres, with the corresponding coefficients of restitution and friction. As it was pointed out in Hesketh et al. (1987) and Fan, Yang, Lee, Tsuchiya, and Luo (1999), and as experimentally confirmed in Holmes (1973), the bubble size distribution in a horizontal dispersed flow can be described by a log–normal function. To incorporate into the model the fact that collision frequencies may vary as the bubble population evolves due to coalescence, different stages (configurations) of each system studied were simulated. This was accomplished by describing the evolution of the bubble size distribution, or equivalently, its probability density function (PDF), as a succession of log–normal PDF’s with constant γ parameter and variable population mean, subject to the mass conservation principle. Each PDF was represented by an associated histogram, which defined the system configuration, i.e. how many particles of each class were present. Note that a system with N classes will have m = N(N + 1)/2 possible binary combinations between its classes, thus, a triangular matrix of order N is required to store all possible binary collision frequencies for each of the m discrete system configurations. After the simulations, the outcome is a set of m triangular matrices of order N, which can be correlated as functions of the population mean, thus converting the discrete results into continuum expressions. In other words, each element

of the so-called collision frequency matrix is a function of the population mean, and it is used as a binary collision frequency function in the source/sink terms of the PB equations. 4. Numerical methodology and simulation results A 0.104 m × 0.104 m simulation box was used to conduct numerical experiments. The bubble-area fraction η was always less than 6%. The flow parameters used for simulations were as follows: U = 2 m/s, τ c = t = 0.01 s, H = 104 mm, k ∼ 0.01 m2 /s2 , ε ∼ 1 m2 /s3 , with the corresponding flow Reynolds number of 2 × 105 . The simulations were conducted for 5 size distributions and for each distribution 8 discrete classes were identified (Table 1). The distributions in Table 1 were generated according to the form of the lognormal distribution function used by Kamp et al. (2001):   √ −{ln(d/dmean )}2 −1 (13) P(d) = ( 2πγd) exp 2γ 2 where variable d is the bubble diameter, dmean the population mean, and γ is the parameter characterizing the spread of the variable. In the following dmean was varied between 1 and 3 mm, while γ was kept constant (0.25), based on averages of experimental data available in the literature (Colin et al., 2004; Kamp et al., 2001). The population mean dmean was selected as the parameter defining various size distributions according to Eq. (13). For an eight-class system, there are 36 different CFFs, which represent λi,j in Eq. (9). The binary collision frequency functions are found by best-fitting curves to the sequence of “i, j”-elements of the collision frequency matrices, with the population mean as the independent variable. The outcome is a symmetrical matrix in which the “i, j”-element represents the collision frequency function between the classes i and j, as a function of the population mean. The initial condition at the pipe inlet was given by the first size distribution (Dist. 1) in Table 1. The MacCormack’s finite-difference method, as presented in Griffiths and Higham (1999), was used to solve the coupled system of equations in Eq. (5) numerically. The solution estimate was calculated in three steps using a predictor–corrector Table 1 Number of particles per class for five discrete size distributions used in simulations Class number

Distribution 1 Distribution 2 Distribution 3 Distribution 4 Distribution 5

1

2

3

4

5

6

7

8

309 31 1 0 0

160 72 9 1 0

35 71 27 5 1

3 30 33 14 4

0 5 17 17 9

0 0 4 8 9

0 0 0 2 4

0 0 0 0 1

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procedure as outlined below for each class:

Uδt k δt − nkj ) + Stotal (nkj+1 ) + Stotal (nkj ) (n δx j+1 2

Uδt k+1 δt k+1 k+1 Wjk+1 = nkj − − Zj−1 )+ Stotal (Zjk+1 ) + Stotal (Zj−1 ) (Z δx j 2 1 nk+1 = (Zjk+1 + Wjk+1 ) j 2 Zjk+1 = nkj −

(14)

where the predictor-step, Zjk+1 , and the corrector-step, Wjk+1 , are used to calculate the number density at the next step, nk+1 j , corresponding to the spatial coordinate jδx and time (k + 1)δt. The term Stotal equals (Si+ − Si− ) in the previously used terminology. The solutions for the number of bubbles of each class and their evolution in space and time are presented as surface plots in Fig. 1 for the first four classes. As it can be seen from Fig. 1, the number of bubbles of class 1 changes monotonically and asymptotically in both space and time. Since the bubble-class 1 contains the smallest particles in the system, its number can only remain constant or decrease; there are no bubbles that could coalesce to create new class 1 bubbles. In the present case, the reduction in the number of class 1 bubbles indicates that those bubbles are coalescing to create new higher-class bubbles. The number of bubbles of the

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second class shows a slight increase near the pipe inlet and in the early stages of the simulation, after which it starts an asymptotic descent in both coordinates. The increase is due to the fact that the fraction of class 1 bubbles coalescing into class 2 is higher than the number of class 2 bubbles coalescing into higher classes. The third class behaves similarly to the previous class, but with a larger increase before the descent. The class 4 bubbles noticeably increase their number due to the different coalescence reactions taking place in the smaller classes, and its population reaches its maximum within the domain shown in the plot. The behaviour of the classes 5, 6 and 7 is very similar to that of class 4. After 3 s of evolution, the population of every class undergoes very little variation; therefore, the bubble size distribution after that period is assumed to have reached a steady-state composition. The main factor that leads to a stable mixture composition is that the collision frequency decreases and eventually tends to zero as the number of bubbles decreases; consequently, coalescence reactions are less likely to occur. Additionally, as it was mentioned earlier, the coalescence efficiency of the higher-class bubbles is lower than that of the smaller classes. The steadystate curves for all the classes are plotted in Fig. 2 as functions of the longitudinal position in the pipe. An interesting finding is that most of the coalescence takes place approximately within the first four meters; after this length of population rearrangement, a stable composition is reached. In general, bubbles will

Fig. 1. The number of particles per class as a function of time (t) and position (x) in the pipe; the domain ranges from 0 to 60δt (3 s) in time and from 0 to 30δx (3.75 m) in position.

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Fig. 2. Steady-state number of particles per class as a function of the position in the pipe.

Fig. 4. Number of bubbles vs. bubble size at various locations along the pipe.

continue interacting indefinitely; however, the rates of reaction become so small that eventually tend to zero. This transition to the stable composition was expected to be smoother; a factor that may have induced this rather sudden behaviour is that the collision frequency functions were approximated (best-fit) with polynomials, which accurately describe the shape of the functions when the number of bubbles is “away” from zero, but do not follow the truly asymptotic behaviour expected in the vicinities of zero. However, in practical terms the true description of the CFF in the vicinity of zero is not important since the number of bubbles is negligibly small. The size distributions obtained with the present model were compared against the continuous log–normal probability distributions as defined by Eq. (13), with dmean obtained from the numerical results and amplitudes consistent with the mass conservation principle. The plots in Fig. 3 show the evolution of the size distribution of bubbles as the mixture flows through the pipe, in terms of the number of bubbles per class versus the bubble diameter. Plot number 1 represents the distribution at the pipe inlet, and plots 2–4 represent different positions along

the pipe. A good agreement is observed between the results of the discretized model (histograms) and the PDF’s (curves), implying that the model is capable of describing the evolution of the size distribution of the bubbles as they interact during their passage through the pipe, maintaining the expected statistical behaviour. Fig. 3 also shows, as mentioned earlier, that for distances greater than approximately 4 m from the pipe inlet, the distribution remains practically constant. The continuous approximations of histograms for four distances along the pipe are superposed in Fig. 4. The evolution to a stable distribution is clearly seen in this figure. Experimental observations in microgravity conditions have shown that the spread parameter of the distribution describing the bubble sizes increases as the system reacts (Colin et al., 2004). For conditions similar to those used in the present study, after a longitudinal distance equivalent to 80 diameters, the spread parameter exhibits an increase of about 25%. This variation was not considered in the model due to its implications in the complexity of the determination of the matrix of collision frequency functions; surface plots would be required instead of

Fig. 3. Steady-state number of particles per class vs. bubble diameter (mm) for various positions along the pipe compared to the continuous distribution with the same mean.

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the 2D curves used here. A variable spread parameter should reduce the deviation between the maximum values of the discrete and continuous solutions. An increase in the number of classes considered (and the corresponding increase in the number of particles) is expected to have the same effect. It is recommended to use an intermediate (average) value for the spread parameter γ at the moment of defining the size distributions for the numerical simulations. Regarding the population mean, the experiments showed an average increase of 70% after a flow length of 80 diameters (using data from 23 experiments by Colin et al., 2004). The population mean associated with the initial conditions of the PB equations in simulations was 1.13 mm, and the final mean, associated with the stable mixture composition, was 1.9975 mm. Thus, the corresponding increase in the population mean was 77%, which compares very well with the average result from the experiments. The positive agreement of the model and the experimental results pertaining to the evolution of the population mean validates the present methodology. It is important to note that the methodology can be adjusted to specific applications in order to produce more accurate simulations, mainly through the adjusting parameters in the turbulence model and by refining the determination of the coalescence efficiency. The effect of coalescence efficiency on the evolution of the bubble population is of special interest since it gives an indication of when, in practical applications, the investment in

increasing this efficiency is worth it. To study the effect of the coalescence efficiency on the population evolution, or, in other words, its sensitivity to the coalescence efficiency, a bidisperse system with particle classes A and B was simulated (bubbles are such that A + A → B). The system is initially composed of A-bubbles only, and as it evolves, B-bubbles start to form. For this system, several values of ΩAA were used. Results for various efficiencies are shown in Fig. 5. The different lines represent values of the coalescence efficiency between 0 and 1. Fig. 5 refers to species A, and Fig. 6 exhibits the behaviour of species B. As the reaction efficiency is increased, larger and more sudden changes appear in the early stages of reaction. Approach to reaction completion is also accelerated by increasing the efficiency. If, for instance, we compare the results for Ω = 0.75 with those for Ω = 0.25, we can see that the former reaches 90% reaction completion in about 30 m or 14 s (for extreme time and length, respectively), while the latter is close to 80% at the same position and time. Consequently, the relationship between the rate of reaction and the coalescence efficiency is not linear. Note, for example, that with an efficiency of 50%, a reaction completion of 90% is reached in about 14 s and 30 m; whereas with 100% efficiency, reaction completion increases only by an additional 5%. For low efficiencies, a small change in Ω will lead to considerable changes in the rate of reaction. On the other hand, changes in high efficiencies will have very little effect on the reaction rate, and efforts made to increase the efficiency might not produce significant improvement.

Fig. 5. Normalized number density distribution of A bubbles at the pipe outlet (1) and at tmax (2) for various coalescence efficiencies.

Fig. 6. Normalized number density distribution of B bubbles at the pipe outlet (1) and at tmax (2) for various coalescence efficiencies.

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and utilizing the principle of ergodicity to optimize the simulation time, are also important, but were outside the scope of this paper and the reader is referred to Miura (2005) for details. References

Fig. 7. Normalized B-concentration at xmax and tmax as a function of coalescence efficiency (0–100%).

The effect of changing the coalescence probability can be quantified by studying the concentration of product (class B) at the pipe outlet for different Ω’s. A plot of CB /CB max at the pipe outlet versus probability of coalescence is shown in Fig. 7. As it can be seen from the above figure, the effort to increase the coalescence efficiency will, after some level, result in diminishing returns. The identification of this level is very important in applications. 5. Conclusions A method for simulating multiphase systems with reactions based on direct simulations of collisions between the species previously developed by Leonenko et al. (2004) was implemented and tested for a two-phase polydisperse bubble system in which coalescence between bubbles of different sizes takes place in a turbulent pipeline flow in a microgravity field. The focus of the numerical investigation was on the evolution of the bubble population along the pipeline for a given distribution of the bubble sizes, and on the effect of the coalescence efficiency on this evolution. It was found that if the coalescence efficiency is low, a small increase of it will lead to considerable changes in the rate of reaction and thus the evolution of the bubble population. On the other hand, if the efficiency is high the effect on the reaction rate is small. The decisions on what is “small” or “high” coalescence efficiency should be specific to the particular application; this may be determined by considering a balance between the costs of further increasing the efficiency and the outcome of this increase. The presented methodology allows to asses the latter. The validity of the methodology was based on the comparison of the population mean at the end of the evolution process against the experimental data; a good agreement on the population average was achieved. It is important to note that the methodology can be adjusted to specific applications in order to produce more accurate simulations, mainly through the adjusting parameters in the turbulence model and by refining the determination of the coalescence efficiency and the binary collision frequency functions. Some issues related to the efficiency and validity of the simulation procedure such as the size of the representative volume,

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