Modeling of droplet–droplet interaction phenomena in gas–liquid systems for natural gas processing

Chemical Engineering Science 63 (2008) 3585 -- 3592

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s

Modeling of droplet--droplet interaction phenomena in gas--liquid systems for natural gas processing C.A. Dorao a,∗ , L.E. Patruno b , P.M. Dupuy b , H.A. Jakobsen b , H.F. Svendsen b a SINTEF

Materials and Chemistry, N-7465 Trondheim, Norway of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

b Department

A R T I C L E

I N F O

Article history: Received 6 September 2007 Received in revised form 10 March 2008 Accepted 25 March 2008 Available online 10 April 2008 Keywords: Population balance modeling Coalescence modeling Natural gas Least squares method

A B S T R A C T

The design of efficient gas liquid separation units for natural gas production lines depends on the accurate estimation of the droplet size distribution. The droplet size can be estimated by considering breakage and coalescence phenomena. In particular, off-shore separation units working at high pressure (100--200 bar) require special consideration of coalescence processes with multiple outcomes. This work discusses the introduction of multiple outcomes in the coalescence process. Numerical experiments are presented in order to highlight the effect of multiple coalescence behavior in the evolution of the droplet size distribution. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Gas liquid separation units play an important role in natural gas production lines because they may prevent breakdown of expanders, compressors and turbines, hydrate formation, and keep water or hydrocarbon dew point within sales gas or transport specifications, etc. A gas liquid separation unit, as a scrubber, most often consists of a vertical vessel with different internals or separation modules installed in series or in parallel, Fig. 1. The typical internals are the following: An inlet section pipe to distribute the flow uniformly in the scrubbers. Above the inlet arrangement, a removal or coalescing section is often installed consisting of a mesh pad or a vane pack. The mesh pad consists of layers of knitted wires while the vane packs consists of plates that are formed in a zigzag pattern. The final section of the scrubber normally consists of a battery of axial cyclones. In traditional production processes scrubbers are installed onshore in safe and stable areas. During the last years an increased interest in sub-sea separation facilities in remote and small off-shore gas fields has been observed (Devegowda, 2003; Austrheim, 2006; HelsZr, 2006). A particular aim is to enhance the efficiency of the sub-sea separation process and decrease the amount of water to be transported to the consumers. The estimation of the droplet size

∗ Corresponding author at: Department of Energy and Process Engineering, Norwegian University of Science and Technology, N7491 Trondheim, Norway. Tel.: +47 73 598462; fax: +47 73 593859. E-mail address: [email protected] (C.A. Dorao).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.03.042

distributions development inside the separator plays a major role in designing and optimizing separation units. The separation process is enhanced by coalescence as the bigger the liquid particle, the easier the separation is. Detailed experimental studies have been reported in the literature enabling the construction of so-called coalescence maps giving the outcome of droplet/droplet collisions. It is important to mention that coalescence is commonly implying the formation of one droplet due to the collision of two or more droplet. However, here the coalescence term is used wider and includes also the possibility of formation of multiple droplets when the new droplet breaks immediately after formation. This complex processes can be described by using the mentioned coalescence maps. These maps are defined in terms of plots of the impact parameter (B) as a function of the Weber number. The Weber number (We) and impact parameter (B) are defined as We = B=

2RU 2 

 2R

(1) (2)

with R being the droplet radius, U the relative velocity,  the projection of the separation distance between the droplet centers in the direction normal to that of U, and  and  the density and surface tension of the liquid and the gas--liquid interface, respectively. Thus, B = 0 designates head-on collisions while B = 1 designates grazing collisions. Fig. 2 shows as an example a schematic representation of the various collision regimes of hydrocarbon droplets at 1 atm in

3586

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

Population balance modeling (PBM) is a well established method for describing the evolution of populations of entities such as bubbles, droplets or particles. The main goal of this work is to discuss the modeling of the coalescence process including multiple outcomes within a population balance framework. Numerical examples are presented showing the effect in the density function by including multiple outcomes in the coalescence operator. In Section 2, the population balance equation is presented and the introduction of secondary breakage is discussed. Section 3 discusses how to incorporate the information of the coalescence maps in the modeling of the coalescence and droplet formation rate. Section 4 presents a few numerical examples in order to highlight the effects of including multiple collision outcomes. Finally, Section 5 presents the main conclusions drawn from this work.

Gas

cyclone

mesh pad Gas/Liquid

inlet vane

Liquid

2. Population balance equation

Fig. 1. Sketch of a separation unit.

B 1

time time

time 0

We Wemax time

Fig. 2. Schematic of various collision regimes of hydrocarbon droplets in 1 atm air (Qian and Law, 1997).

air (Qian and Law, 1997). As seen, the outcome of binary collisions is not limited to the case of the formation of a single new droplet by coalescence. Instead multiple outcomes can be observed which constitute a severe modeling problem. An overview of the droplet--droplet interaction phenomena can be found in Orme (1997), Simon and Bart (2002). Ashgriz and Poo (1990) presented results of the coalescence process for water drops, Qian and Law (1997) discussed the coalescence process for fuel drops and Gotaas (2007) presented measurements for n-decane, monoethyleneglycol (MEG), diethyleneglycol (DEG) and triethyleneglycol (TEG), to mention some examples found in the literate. The coalescence process can also involve two droplets of different liquids (Chen and Chen, 2006; Gao et al., 2005). In general the coalescence maps are related to two droplet of similar size, the difference in the size can be introduced by including an extra dimension representing the droplet size ratio (Post and Abraham, 2002). The coalescence outcomes after the breaking of the new droplet was discussed by Brenn et al. (2001) and Brenn and Kolobaric (2006). The simulation of dispersed systems and in particular liquid--gas system and sprays is still a challenging area of research (Gouesbet and Berlemont, 1998; Nijdam et al., 2004; Guo et al., 2004). In particular, the high computational cost required for describing the behavior of the dispersed phase can be reduced by using a statistical approach using a Boltzmann-type equation also referred to as a population balance equation (Reyes, 1989; Laurent et al., 2004).

Based on the population balance approach the dispersed phase is described by a density function for instance f(r, , t) where r is the spatial vector position,  is the dispersed phase property of interest, and t the time. Thus, the density function f(r, , t) d, can represent the average number of particles per unit volume around the point r at the instant t with the property between  and  + d. On the other hand, the density function f can present any property of interest. For example f(r, , t) d can represent the mass of liquid contained in droplets with the property between  and  + d per unit volume around the point r at the instant t. The evolution of this density function must take into account the different processes that control the density function such as breakage, coalescence, growth and advective transport of the particles. The resulting equation is a non-linear partial integro-differential equation to be solved by a suitable numerical method. The population balance equation considering one property and time can be written as Lf(, t) = g(, t) in 

(3)

B0 f(, t) = f0 ()

(4)

on 0

with  = [min , max ] × [0, T ] where min and max are for instance the minimum and maximum particle sizes, and T the final simulation time. The right-hand side (RHS) of Eq. (3) is a source or sink term, representing the external mechanism of adding or removing particles from the system. Eq. (4) contains the initial condition f0 () of the problem which is applied on 0 = {(, t) ∈ j : t = 0} and where B0 is the identity operator, i.e. B0 f(, t) = f(, t). The operator L is a non-linear first order partial integrodifferential operator defined as Lf(, t) ≡

jf(, t) + Lc f(, t) jt

(5)

where Lc f(, t) is the coalescence operator. Normally, the coalescence operator is defined as Lc f(, t) = f(, t) −

  max +min −

 

min

min

c(, s)f(s, t) ds

c( − s, s)f( − s, t)f(s, t) ds

(6)

The first term on the RHS of Eq. (6) represents the change in the population due to the loss of droplets by pair interaction processes such as coalescence. Thus, c(, s) is the coalescence rate between particles of type  and s. It is important to note that the upper limit of the integral is defined such that the coalescence process cannot produce particles exceeding the maximum physical allowable size

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

3587

f ()

Δf ()  Original distribution of droplets Unstable droplets

ξ Droplets for secondary breakage Redistribution step ∫(, ) Δf ( )d redistribution function  (, )

 (, ) 

ξ droplets created Fig. 3. Schematic of coalescence process ending in a secondary breakage.

max . The second term on the RHS represents the arrivals of new individuals due to droplet pair interaction, one particle of type 1 that coalesces with a particle of type 2 will produce a particle of type  = 1 + 2 . Depending on the property  chosen, i.e. volume, radio, etc., the previous relationships should be modified accordingly. As mentioned before the outcome of binary collisions is not limited to the creation of a new droplet with property  = 1 + 2 due to coalescence. Hence, the standard coalescence model needs to be modified in order to handle more complex situations. Considering the general case shown in Fig. 2, the coalescence operator can be expressed like the linear combination of four different coalescence processes Lc • =(Lc1 + Lc2 + Lc3 + Lc4 )•

(7)

The first operator corresponds to the standard coalescence process, i.e. only one particle is created after the collision, Lc1 f(, t) = f(, t) −

  max +min −

 

min

min

c1 (, s)f(s, t) ds

c1 ( − s, s)f( − s, t)f(s, t) ds

Lc2 f(, t) = f(, t) −

min

  max 

(8)

c2 (, s)f(s, t) ds

(, ∗ )f(∗ , t) d∗

(9)

where c2 (1 , 2 ) is the coalescence rate of particles with property values 1 and 2 which produce unstable droplets with a droplet size distribution given by f(∗ , t) =

 ∗ min

c2 (∗ − s, s)f(∗ − s, t)f(s, t) ds

k (s) =

  min

k (, s) d = sk

(11)

where k (s) is the moment of the new particles that appear after the secondary breakage, provided that k is the moment which is conserved in the breakage process. For instance, assuming that  represents the volume of the particle, then the 1st moment is conserved (k = 1) when the sum of the volumes of the particles that are produced due to breakage equals the parent particle volume s. The last operator corresponds to the bouncing process which does not affect the droplet size distribution, i.e. Lc4 ≡ 0. 3. Coalescence rate modeling

where c1 (1 , 2 ) is the coalescence rate for particles with property values 1 and 2 . The second and third operator correspond to coalescence processes with secondary breakage, i.e. one particle is created but almost instantaneously the droplet breaks giving a spectrum of small droplets, Fig. 3. In this case, the process can be expressed as   max +min −

These unstable droplets go into a secondary breakage process producing droplets with a size distribution function dictated by the secondary breakage yield function (SBYF) (, ∗ ). This function is equivalent to the breakage yield function of the breakage kernel. Hence, this function must satisfy the property that

(10)

The coalescence rate determines the fraction of particles that coalesce per unit time. A simple modeling approach consists of considering the coalescence process as a two stage process. The first stage determines the collision rate, , between the droplets and the second stage the probability or efficiency of coalescing, P, i.e. c(1 , 2 ) = (1 , 2 )P(1 , 2 )

(12)

The collision rate ( ) is related to the hydrodynamic properties of the medium such as turbulence, shear, density difference, etc. The coalescence efficiency (P), also referred to as the coalescence probability, gives a measure of the efficiency of the coalescence process, i.e. P=

Number of particles that coalesce Number of particles that collide

(13)

In this section, we discuss alternative modeling approaches of P based on the information available in the coalescence maps. The coalescence efficiency is related to the fluid properties such as density ratio, viscosity, etc. which determine the local phenomena of two

3588

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

B

B

B

We 0.7atm

We

We

2.4atm

4.4atm

Pressure Fig. 4. Effect of the pressure change in the coalescence map (Qian and Law, 1997).

B

1

0

P

We

1

We Fig. 5. Schematic of coalescence process and derivation of the coalescence probability.

D=

 ∀i

Di ,

Di ∩ Dj = ∅, i  = j

(14)

 (B; E)

Hydrodynamic and fluid parameters E

Enviromnent weight function

droplets approaching each other and the breakage of the fluid bridge between them. The coalescence efficiency is commonly modeled by estimating the drainage time of the fluid between particles and a characteristic time of the collision (Chesters and Hofman, 1982). An alternative is to consider an energy balance in terms of the surface energy of the droplet created. This approach normally ends up in rather empirical expressions for these time scales due to the complexity of the processes involved and that the functionality relating the probability and the characteristic time scales is also very uncertain. In our laboratory, experimental information about the coalescence process is being gathered for different fluid properties in terms of coalescence maps. In this article, alternative methods for using the information contained in these maps to estimate the coalescence efficiency are examined. The first approach consists in determining an overall estimate of the coalescence efficiency. Considering the coalescence map shown in Fig. 2, the region D = [0, 1] × [0, Wemax ], with Wemax a maximum bound for the We number, represents all the possible collision interactions. Each coalescence process defines a region Di with i the type of coalescence process such that

0

1

B

Fig. 6. Environment weight function.

In addition, each coalescence process can be represented by a coalescence process indicator function (Ci ) such that  1 if (We, B) ∈ Di Ci (We, B) = (15) 0 otherwise

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

 (B, We; E1) B

 (B, We; E2)

B 1

3589

1

0

We

0

We

Fig. 7. Environment weight function.

Based on the previous definitions, expression (13) can written as Pi =

Area of Di Area of D

or equivalently as  C dD  i Pi = D D dD

(16)

(17)

Therefore, changes in the area of a region implies a change in the efficiency of this particular coalescence process. For example, changes in pressure will modify considerably the coalescence map, Fig. 4. In this case, it is easy to see that the overall effect of increasing the pressure is to reduce the coalescence efficiency. In particular, this conclusion is relevant for the design of new separation units working at high pressure. This novel approach presents a drawback that the coalescence efficiency does not include any information about the droplet size. A possible way to improve the modeling of the coalescence efficiency is to assume that the droplet relative velocity is independent of the droplet size, and that different We numbers represent droplets of different sizes. For a given We number, or equivalently for a droplet of a given size, a vertical line in the coalescence map may represent all the types of collision interactions, i.e. B = [0, 1]. Therefore, the efficiency of a certain type of coalescence can be defined as the length of the vertical segment in the coalescence map of the corresponding case, see Fig. 5. This corresponds to an integration in the impact parameter, i.e. 1 C (We, B) dB Pi (We) = 0 i 1 0 dB

(18)

In this manner a better representation of the droplet size is derived. However, the We is defined in terms of the relative velocity which was assumed to be independent of the droplet size. The dependency of the relative velocity is then lost during the general derivation of the population balance framework. This can be explained as follows. In the most complete setting, the density function f is a function of the droplet velocity, i.e. f(r, , v, t), similar to the Boltzmann equation increasing the complexity of the problem. In this case the Weber number is defined for two colliding droplets as

We =

 + 2 2 1 |v1 − v2 |2 2 

(19)

The velocity dependency is dropped in the population balance framework requiring an introduction of a closure relation for the relative velocity |v1 − v2 |. This normally ends up in expressions containing the fluid properties and the hydrodynamics condition of the system

under study. The same remark is valid for the projection of the separation distance in the impact parameter. The modeling of the coalescence process in the population balance framework requires closure expressions for the relative velocity defining the Weber number and the projection of the separation distance defining the impact parameter. These two parameters are strongly dependent on the system under consideration. The previous two derivations assume independence of the environment. In order to include explicitly this effect, an environment weight function, (B, We; E) with E being the hydrodynamic and fluid parameters, is introduced which weighs the effect of the environment in the computation of the coalescence efficiency, Fig. 6. In this manner, the influence of the environment can be included into the coalescence process by weighing the coalescence map using (B, We; E). Fig. 7 shows an example of the effect of changing the environment weighing. By using the environment weight function, the coalescence efficiency global can be computed as  C (We, B)(B, We; E) dWe dB (20) Pi = D i D (B, We; E) dWe dB while the local efficiency can be computed as 1

C (We, B)(B, We; E) dB Pi (We; E) = 0 i 1 0 (B, We; E) dB

(21)

In summary, estimation of the coalescence efficiency can be improved by using the information gathered in the coalescence maps. This map contains parametrized information of the different outcomes of the coalescence process. In particular, an overall estimation of the coalescence probability can be approximated as the relative area of the coalescence process over the total area of the map. A more detailed efficiency estimate can be obtained by considering an integration over the impact parameter. These two approaches do not include information of the environment which can affect the frequency of a given process. For that reason, it is possible to assume the existence of an environment weighing function which contains the effects of a particular system. 4. Numerical examples In order to show the effects of the multiple outcomes in the coalescence process, numerical simulations are performed. Due to the limited information available, the numerical experiments should be considered as a parameter analysis study. The solver is based on the least squares method (LSM) (e.g. Jiang, 1998; Proot and Gerritsma, 2002; Pontaza and Reddy, 2003, 2004). The basic idea in the LSM is to minimize the integral of the square of the residual over the

3590

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

2

10 0.2

0.2

6

0.15

4

0.1

2

0.05

0.1 0 10 t

1

5 0 0

0.5 

1.5 k

8

t

0.3

0

0.2

0.4

0.6

0.8

0 −1 −1/3

0.5

0

0

1

0

1

0

2

4

0.2

0.2

6

0.15

4

0.1

2

0.05

t

5

1 0 0

0.5 

0

0.2

0.4

0.6

0.8

0

2

4

2

0.2

6

0.15

4

0.1

2

0.05

5 0

0

0.5 

0

0 0

0.2

0.4

0.6

0.8

6

8

10

0 −1 −1/3

1.5 k

8

t

0.3

t

10

t 0.2

1

8

0

1

10

0 10

6

0 −1 −1/3

0.5

0 0

1



0.1

10

1.5 k

8

t

0.3

0 10

8

2

10

0.1

6 t



1

1 0.5 0

0

2

4 t



Fig. 8. Evolution of f(, t) and its moments for each coalescence operator. First row standard coalescence Lc1 , second row Lc2 and third row Lc3 .

computational domain. An extended discussion about the application of LSM to PBE can be found in Dorao and Jakobsen (2005a,b). The model problem is defined as jf(, t) + Lc1 f(, t) + Lc2 f(, t) + Lc3 f(, t) = 0 jt in  = [0, 1] × [0, T ]

(22)

with f(, t) d being the total volume of droplets of size [,  + d] at the time t. Hence, the 0th moment of the distribution is the conserved moment and represents the total volume in the system, the −1th moment gives the total number of droplets in the system and the −1/3th moment is the droplet area density in the system. The moments are defined as  1 0 = f(, t) d (23) 0

−1 =

 1 0

−1/3 =

−1 f(, t) d

 1 0

−1/3 f(, t) d

(24)

(25)

The second term in (22) corresponds to the standard coalescence process given by expression (8), i.e. only one droplet is created after the collision. The third and fourth terms in (22) correspond to

coalescence processes with secondary breakage given by expression (9), i.e. one droplet is created but almost instantaneously the droplet breaks giving a spectrum of small droplets. Fig. 8 shows the evolution of the density function f(, t) for each coalescence operator acting individually. The first row in Fig. 8 shows the evolution of the density function f(, t) and its moments for the standard coalescence operator. It is noted that the value of both the moment −1 and −1/3 decrease because of the coalescence of droplets. As can be seen the effect is strongest on the droplet area density. The second and third row in Fig. 8 show the effect of the breakage of the formed particle after coalescence. The redistribution function in the operator Lc2 is defined as 2 (, p ) = (p − )

(26)

with p the parent droplet, and the constant of normalization according to expression (11). This redistribution function produces mainly small droplets. The redistribution function in the operator Lc3 is defined as 2 (, p ) = 

(27)

In this case, the redistribution function produces mainly big droplets. Fig. 8 shows that if the redistribution function is producing mainly small droplets the coalescence process behaves like a breakage process instead. In a separation unit, it is expected that the coalescence

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

2

10 0.2

8 6

0.15

4

0.1

2

0.05

t

0.2 0.1 0 10 5

t

0

0

1

0.5

0



1.5 k

0.3

0 0

0.2 0.4 0.6 0.8

1

0.2

8

t

5

0 0

0.5

1

4

0.1

2

0.05

0



k

t

0 10

0.2 0.4 0.6 0.8

2

4

6

8

10

6

8

10

6

8

10

t 0 −1 −1/3

1 0.5

0 0

0

1.5

0.15

6

0.1

0 −1 −1/3

2

10

0.2

1 0.5

 0.3

3591

0

1

0

2

4 t



Fig. 9. Evolution of f(, t) and its moments for Lc1 + Lc2 in the first row, and Lc1 + Lc3 in the second row.

8

0.2

6 t

0.3

0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

0.1

4

0 10

t

5

0 0

1

0.5

2 0



0

0.2 0.4 0.6 0.8

2 1.5 k

10

1 0 −1 −1/3

0.5 0

1

0

2

4 t

 Fig. 10. Evolution of f(, t) and its moments for Lc1 + Lc2 + Lc2 .

process will shift the droplet size distribution to bigger droplet sizes where the separation can be done optimally. It is noted that if the redistribution function is producing mainly small droplets the droplet size distribution will be shifted to the smaller sizes, thus negatively affecting the separation. If the redistribution function produces mainly big droplets, the time scale of coalescence is affected. In particular, a longer residence time will be required for shifting the droplet size to bigger droplets. This fact could indicate that in this case if the secondary breakage process is dominant in a separation unit, the volume of the unit should be increased in order to allow the coalescence process to shift the droplet size and thus enhance separation. Fig. 9 shows the evolution of the density function f(, t) and its moments for the case of Lc1 + Lc2 in the first row and Lc1 + Lc3 in the second row. It is noted that the effect of the redistribution function after coalescence plays a major role in the dynamics of the system. Hence, accurate estimations of the redistribution function is required in order to improve the modeling of high pressure separation units. Finally, Fig. 10 shows the effect of combining the three coalescence operators, i.e. Lc1 + Lc2 + Lc3 . The net effect in this case is close to a breakage process shifting the droplet size distribution to the smallest droplet size.

5. Conclusions In this work, the modeling of a coalescence process with multiple outcomes is discussed. The coalescence process is normally followed by a breakage process which gives a wide range of new droplets. This effect is commonly neglected, but at high pressure this breakage process is enhanced. For that reason, the accurate prediction of the droplet size distribution plays a major role in simulating the efficiency of gas--liquid separation units working at high pressure conditions. The least squares method was applied for solving the partial integro-differential model. Simulations showing the effects of the multiple outcomes were discussed. It is noted that the effect of the redistribution function after the coalescence plays a major role in the dynamics of the system. If, after coalescence the droplet breaks into small ones, the net effect of the coalescence is to produce small droplets which affect severely the separation process. In some cases, a longer residence time can be required for shifting the droplet size to bigger droplets. The modeling concept presented in this article is quite general and not only limited to gas--liquid systems. The application of this concept to real cases demands further research in order to improve the description of the coalescence process. In particular, modeling

3592

C.A. Dorao et al. / Chemical Engineering Science 63 (2008) 3585 -- 3592

and experimental investigations are being carried out for characterizing the coalescence process for high pressure systems. It should be noted that a similar methodology can be used for droplet/film interactions and for droplet entrainment processes. Acknowledgments The Post-Doc fellowship (C.A. Dorao) and the Ph.D. fellowship (L.E. Patruno and P.M. Dupuy) financed by the Research Council of Norway Petromak programme through the (HiPGLS) project are gratefully appreciated. References Ashgriz, N., Poo, J.Y., 1990. Coalescence and separation in binary collisions of droplets. Journal of Fluid Mechanics 221, 183--204. Austrheim, T., 2006. Experimental characterization of high-pressure natural gas scrubbers. Ph.D. Thesis, University of Bergen, Norway. Brenn, G., Kolobaric, V., 2006. Satellite droplet formation by unstable binary drop collisions. Physics of Fluids 18 (8), 087101--087101-18. Brenn, G., Valkovska, D., Danov, K.D., 2001. The formation of satellite droplets by unstable binary drop collisions. Physics of Fluids 13 (9), 2463--2477. Chen, R.-H., Chen, C.-T., 2006. Collision between immiscible drops with large surface tension difference: diesel oil and water. Experiments in Fluids 41 (3), 453--461. Chesters, A.K., Hofman, G., 1982. Bubble coalescence in pure liquids. Applied Scientific Research 38, 353361. Devegowda, D., 2003. An assessment of subsea production systems. M.Sc. Thesis, Texas A&M University. Dorao, C.A., Jakobsen, H.A., 2005a. Application of the least square method to population balance problems. Computer & Chemical Engineering 30 (3), 535--547. Dorao, C.A., Jakobsen, H.A., 2005b. Application of the least squares method for solving . Chemical Engineering Science 61 (15), population balance problems in R 5070--5081. Gao, T.-C., Chen, R.-H., Pu, J.-Y., Lin, T.-H., 2005. Collision between an ethanol drop and a water drop. Experiments in Fluids 38 (6), 731--738. d+1

Gotaas, C., 2007. Experimental and numerical investigation of droplet phenomena. Ph.D. Thesis, Norwegian University of Science and Technology, Norway. Gouesbet, G., Berlemont, A., 1998. Eulerian and Lagrangian approaches for predicting the behaviour of discrete particles in turbulent flows. Progress in Energy and Combustion Science 25 (2), 133--159. Guo, B., Fletcher, D.F., Langrish, T.A.G., 2004. Simulation of the agglomeration in a spray using Lagrangian particle tracking. Applied Mathematical Modeling 28 (3), 273--290. HelsZr, T., 2006. Experimental characterization of wire mesh demisters. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim, Norway Jiang, B., 1998. The Least-Square Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, Berlin. Laurent, F., Massot, M., Villedieu, P., 2004. Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays. Journal of Computational Physics 194, 505--543. Nijdam, J.J., Guo, B., Fletcher, D.F., Langrish, T.A.G., 2004. Challenges of simulating droplet coalescence within a spray. Drying Technology 22 (6), 1463--1488. Orme, M., 1997. Experiments on droplet collisions, bounce, coalescence and disruption. Progress in Energy Combustion Science 23, 65--79. Pontaza, J.P., Reddy, J.N., 2003. Spectral/hp least-squares finite element formulation for the incompressible Navier--Stokes equation. Journal of Computational Physics 190 (2), 523--549. Pontaza, J.P., Reddy, J.N., 2004. Space--time coupled spectral/hp least squares finite element formulation for the incompressible Navier--Stokes equation. Journal of Computational Physics 197 (2), 418--459. Post, S.L., Abraham, J., 2002. Modeling the outcome of drop--drop collisions in Diesel sprays. International Journal of Multiphase Flow 28, 997--1019. Proot, M.M.J., Gerritsma, M.I., 2002. A least-squares spectral element formulation for Stokes problem. Journal of Scientific Computing 17 (1--4), 285--296. Qian, J., Law, C.K., 1997. Regimes of coalescence and separation in droplet collision. Journal of Fluid Mechanics 331, 59--80. Reyes Jr., J.N., 1989. Statistically derived conservation equations for fluid particle flows. Transactions of the American Nuclear Society 60. Winter Meeting of the American Nuclear Society (ANS) and Nuclear Power and Technology Exhibit, 26--30 November 1989, San Francisco, CA, USA. Simon, M., Bart, H.J., 2002. Experimental studies of coalescence in liquid/liquid systems. Chemical Engineering and Technology 25 (5), 481--484.