STNI Science and Technology of Nuclear Installations 1687-6083 1687-6075 Hindawi Publishing Corporation 890815 10.1155/2012/890815 890815 Research Article A Derivation of the Nonlocal Volume-Averaged Equations for Two-Phase Flow Transport Espinosa-Paredes Gilberto Končar Boštjan Área de Ingeniería en Recursos Energéticos Universidad Autónoma Metropolitana P.O. Box 55-535 Iztapalapa 09340 Mexico City DF 09340 Mexico uam.mx 2012 16 09 2012 2012 29 05 2012 07 08 2012 09 08 2012 2012 Copyright © 2012 Gilberto Espinosa-Paredes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper a detailed derivation of the general transport equations for two-phase systems using a method based on nonlocal volume averaging is presented. The local volume averaging equations are commonly applied in nuclear reactor system for optimal design and safe operation. Unfortunately, these equations are limited to length-scale restriction and according with the theory of the averaging volume method, these fail in transition of the flow patterns and boundaries between two-phase flow and solid, which produce rapid changes in the physical properties and void fraction. The non-local volume averaging equations derived in this work contain new terms related with non-local transport effects due to accumulation, convection diffusion and transport properties for two-phase flow; for instance, they can be applied in the boundary between a two-phase flow and a solid phase, or in the boundary of the transition region of two-phase flows where the local volume averaging equations fail.

1. Introduction

The technique of local volume averaging of microscopic conservation equations of motion and transport has received numerous research and analysis , in order to obtain macroscopic balance equations applicable to multiphase systems. The approximation of local volume-averaged conservation equation of two-phase flow is valid when the following length-scale restriction is fulfilled : (1)1, where is the characteristic length of the dispersed phases, and is the characteristic length of the global system: (2)1=MAXxVk|ψk(x_,t)|MAXxVk|ψk(x_,t)|. In this equation ψk is the intrinsic property and ψk represents the average and Vk is the volume of k-phase. Then, is associated where ψk varies significantly, and with the changes in ψk.

The imposition of the guarantees good behavior of the averaged variables. However, the most well-known multiphase flow systems where the length-scale restriction given by (1) are not true: geological systems , fractionation of hydrocarbons , transport of contaminants , elimination of contamination in aqueous streams , cuttings transport , and concentration of pharmaceutics , among others that include extraction and separation processes . Specifically, in nuclear systems of BWR type and other industrial applications that involve multiphase flow, the length-scale restriction is no longer satisfied following the transition to churn or slug flow regimes, where the number of bubbles is highly decreased and their size is increased to the length of the magnitude order of the averaging volume, including the whole system, that is, pipe diameter.

The local volume averaging of the conservation equations (mass, momentum, and energy) involves averaging the product of a volume-averaged variable ψk (in this paper it was sought to use the nomenclature defined by Lahey and Drew ), with the unaveraged variable (φk), that is, φkψk (here ψk and φk are intensive properties associated with the k phase). The conditions necessary to bring a volume-averaged variable outside the volume integral are the imposing of the length-scale restriction given by (1), that is, φkψk=φkψk, with the idea of obtaining manipulated variables associated with the processes of two-phase flow.

Another common case of the local volume averaging of the conservation equations is the average product of two unaveraged variables, that is, φkψk. The traditional representation is φkψk=αkφkψk+φ~kψ~k, where φ~k and ψ~k represent the spatial deviations around averaged values of the local variables and are defined by the decomposition φk=φk+φ~k and ψk=ψk+ψ~k . The removal of averaged quantities from the volume integrals is consistent with the length-scale restriction given by (1). The mathematical consequence of this type of inequality can be expressed as φ~k=0 and ψ~k=0.

However, for more realistic problems, this length-scale restriction given by (1) is not true. In general, this length-scale restriction is not valid within the boundary region (e.g., transition region in two-phase flows) due to significant spatial variations of the two-phase flow structure. The classical length-scale restriction which is implicit in the average transport equations are not satisfied.

In this paper a detailed derivation of the general transport equations for two-phase systems using a method based on nonlocal volume averaging, that is, without length-scale restriction is presented. The nonlocal volume averaging equations derived in this work contain new terms related to nonlocal transport effects due to accumulation, convection diffusion, and transport properties for two-phase flow heat transfer. The nonlocal terms were evaluated considering that these are a function of the local terms, which yield new coefficients or closure relationships.

2. Preliminaries

The two-phase flow is a system formed by a fluid mixture of l (liquid) and g (gas) phases flowing through a region V as is illustrated in Figure 1. Phase k (= l, g) has a variable volume Vk with a total interfacial area of Ak in the averaging volume V, which has an enveloping surface area (A) with a unit normal vector (n_) pointing outward. A portion of Ak is made of a liquid-gas interphase Alg and a fluid-solid interface Akw. The unit normal vector n_k of Ak is always drawn outwardly from phase k, regardless whether if it is associated with Alg or Akw.

Schematic figure of a two-phase flow with a transition region, showing the position vectors and the averaging volume.

Local averaging volume (3)V=Vl(t)+Vg(t). Volume fraction g phase in fluid mixture (4)αg=Vg(t)V. The method of volume averaging is a technique that can be used to rigorously derive continuum equations for multiphase systems. This means that the equations valid for a particular phase can be spatially smoothed to produce equations that are valid everywhere, except in the boundaries which contain the multiphase systems.

The volume average operator or superficial volume average ψks of some property ψk (scalar, vector, or tensor) associated with the k phase is given by (5)ψks|x_=1VVk(x_,t)ψk(x_+y_k,t)dV, where V is the averaging volume, Vk is the volume of the k phase (contained V), x_ is the position vector locating the centroid of the averaging volume, y_k is the position vector at any point in the k phase relative to the centroid, as is illustrated in Figure 1, and dV indicates that the integration is carried out with respect to the components of y_k. Then, (5) indicates that the volume-averaged quantities are associated with the centroid. In order to simplify the notation, we will avoid the precise nomenclature used and represent the superficial average of ψk as (6)ψks=1VVkψk|x_+y_kdV. The intrinsic average is expressed in the form (7)ψk=1VkVkψk|x_+y_kdV. These averages will be used in the theoretical development of the two-phase flow transport equations and are related by (8)ψks=αkψk. With ψk=1, the result leads to (9)1s=αk. As mentioned above V is a constant, which is invariant in both space and time as illustrated in Figure 1. In this case the volumes of each phase of the flow may change with the position and time, that is, Vk(x_,t). It should be clear that the volume fraction αk is a function of the position and time.

When the local instantaneous transport equations are averaged over the volume, terms arise which are averages of derivatives. In order to interchange differentiation and integration in the averaging transport equations, two special averaging theorems are needed. The first one is the spatial theorem [1, 28, 29] (10)ψks=ψks+1VAkψk|x_+y_kn_kdA, where ψk is a quantity associated with the k phase, n_k is the unit normal vector directed from the k phase towards the f phase, and Ak is the area of the k-f interface contained within V.

The second integral theorem is a special form of the Leibniz rule known as the transport theorem [10, 30]: (11)ψkts=ψkst-1VAkψk|x_+y_kW_k·n_kdA, where W_k is the velocity of the k-f interface in V,(12)Ak=Akw+Alg. If ψk=1, the previous theorems lead to (13)αk=-1VAkn_kdA,(14)αkt=1VAkW_k·n_kdA. In these theorems ψk should be continuous within the k phase.

It is important to note that these theorems are not restricted to the inequality given by (1).

In order to eliminate the point or local variable ψk in the spatial averaging theorem given by (10) we use the spatial decomposition (ψk=ψk+ψ~k) , (15)ψks=ψks+1VAk  ψk|x_+y_kn_kdA+1VAk  ψ~k|x_+y_n_kdA. In the homogeneous regions of the system, the following length-scale restriction given by (1) is usually satisfied, and the following simplification is considered: (16)ψk|x_+y_kψk,for1. Then, the second term on the right side of (15) can be written as (17)1VAkψk|x_+y_kn_kdA={1VAkn_kdA}ψk=-{αk}ψk,for1. In general, the averaged terms evaluated in the centroid can be removed from the integrals, where this result was obtained using the lemma given by (13). Then, (15) can be rewritten as follows:(18)ψks=αkψk+1VAk  ψ~k|x_+y_n_kdA,for1. The similar form, the theorem given by (11) can be rewritten as (19)ψkts=αkψkst-1VAkψ~k|x_+y_kW_k·n_kdA,for1, where this result was obtained using the lemma given by (14).

The local average volume in principle cannot describe significant variations or sudden changes where the characteristic length can be of the order of . Then, it is necessary to extend the scope of the theorems given by (18) and (19), which is the goal of the next section.

3. <italic>Nonlocal</italic> Averaged Volume

The spatial decomposition given by ψk=ψk+ψ~k represents a decomposition of length scales, that is, the average ψk undergoes significant change only over the large length scale, while that the spatial deviation ψ~k is dominated by the small length scale . However, this idea considered that the nonlocal effects are negligible.

Returning to (10) and (11), it clearly indicates that is a nonlocal spatial averaging theorem since the dependent variable ψk is evaluated at other points than the centroid (which is indicated by   ψk|x_+y_k). In this context, we use nonlocal in the sense that it does not involve the use of length-scale restriction in its derivation [31, 32].

3.1. Nonlocal Averaged Volume Approximation

The area integral of   ψk|x_+y_kn_k is evaluated in the k phase indicated by position vector y_k shown in Figure 1. Then, (20)1VAkψk|x_+y_kn_kdA, which is essentially a nonlocal term since that the dependent variable ψk is not evaluated at the centroid, x_ (Figure 1). The nature of the volume-averaged variable ψk|x_+y_k can be known applying a Taylor series expansion about the centroid of the averaging volume : (21)ψk|x_+y_k=ψk+y_k·  ψk+12y_ky_k:ψk+. The second, third, and following terms on the left side correspond to nonlocal effects. Then, this equation can be approximate by (22)ψk|x_+y_k=ψk|x_+ψkNL, where ψkNL=y_k·ψk+(1/2)y_ky_k:ψk+. Inclusive can be treated as source term formed by ψk|x_+y_k-ψk. It is important to emphasize that the presence of the term ψkNL involves that the nonlocal representation avoids imposing length-scale restrictions. Then, the general representation of nonlocal term is (23)ψkNL={ψk|x_+y_k-ψk,withoutlength-scalerestrictions,0,for1. The physical interpretation of (23) indicates that the nonlocal contribution is negligible in the homogeneous region, that is, those portions of the two-phase flow that are not influenced by the rapid changes in the structure which occur in the boundary region. Therefore, nonlocal term can be important in the boundary region, where ψk|x_+y_k-ψk is important and the length-scale constraints given by (1) are not valid.

Applying these ideas the theorems can be expressed in nonlocal terms: (24)ψks=αkψk+1VAkψkNLn_kdA+1VAkψ~k|x_+y_n_kdA,(25)ψkts=αkψkt-1VAkψkNLW_k·n_kdA-1VAkψ~k|x_+y_kW_k·n_kdA. The forms of these integral theorems are applied in this work to obtain nonlocal volume-averaged conservation equation for two-phase flow, that is, without restriction of the length scale.

3.2. Average Volume of the Product of Two Local Variables <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M103"><mml:msub><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

The explicit representation of the average volume of the product of two local variables is given by (26)φkψks=1VVk(φkψk)|x_+y_kdV. Substituting the correspondent spatial deviations for the local variables φk and ψk leads to (27)φkψks=αkφkψk+ψkφ~ks+φkψ~ks+φ~kψ~ks. This is rewritten as (28)φkψks=φkψkNL+αkφkψk+φ~kψ~ks, where φkψkNL is a nonlocal term, since it involves, indirectly, values of φk and ψk that are not associated with the centroid of the averaging volume illustrated in Figure 1. The nonlocal contribution is given by (29)φkψkNL=φkψks+φ~kψks+φkψ~ks-αkφkψk. It can be demonstrated that (30)φkψkNL=0,for1.

3.3. Operators Applied to Two Local Variables <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M114"><mml:msub><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

The typical expressions in the transport phenomena in a two-phase flow involve an average differential operator with two local variables, (31)φkψks=φkψks+1VAkφkψk|x_+y_kn_kdA,φkψkts=φkψkst-1VAk  φkψk|x_+y_kW_k·n_kdA. With the previous ideas we obtain expanded form of the theorems for the product of two-local variables (32)φkψks=φkψkNL+k(φkψk)+φ~kψ~ks+{1VAk(φkψk)n_kdA}NLnonlocal+{1VAk(φ~kψ~k)|x_+y_kn_kdA}Ddispersion,(33)φkψkts=αk(φkψk)t+φkψkNLt+φ~kψ~kst+{1VAk(φkψk)W_k·n_kdA}NLnonlocal+{1VAk(φ~kψ~k)|x_+y_kW_k·n_kdA}Ddispersion.

4. <italic>Nonlocal</italic> Volume-Averaged General Balance Equations for Two-Phase Flow

The starting point for the development of the nonlocal volume-averaged conservation equations is the point conservation equations. In order to illustrate the application of the nonlocal theorems and related definitions, we considered the general balance equation for some ψ properties in the k phase: (34)(ρkψk)t+·(ρkU_kψk)+·D_k=ρkf, where ψk is the quantity conserved, D_k is the molecular flux, and f is a volumetric source. As summarized in Table 1, depending on the choice of the quantity to be conserved, either of these equations can be used to quantify the mass, momentum, and energy conservation of each phase.

Conservation terms.

Conservation principle ψ k D _ k f
Mass 1 0 0
Momentum U _ k p k I _ _ - τ _ _ k g _ k
Energy e k - p k / ρ k q _ k - ( p k I _ _ - τ _ _ k ) · U _ k g _ k · U _ k + q k ' ' ' / ρ k

Note: ek=hk+Uk2/2 is the total specific energy for the k phase, where hk is the enthalpy.

q _ k denotes the heat flux vector for the k phase.

The volume averaged of the general balance equation can be expressed as (35)(ρkψk)ts+·ρkU_kψks+·D_ks=ρkfks. The nonlocal transport theorem of the product of two variables derived in this work given by (33) with φk=ρk is used in order to express the first term of this equation: (36)ρkψkts=αk(ρkψk)t+ρkψkNLt+ρ~kψ~kst-{1VAkρkψkW_k·n_kdA}NL-{1VAk(ρ~kψ~k)|x_+y_kW_k·n_kdA}D. The nonlocal averaging theorem for the product of three variables can be developed following the same procedure given by (32). Then, the second term in (35) is given by (37)·ρkU_kψks=αk·(ρkU_kψk)+·ρkU_kψkNL+·ρkU_kψkD+{1υAk(ρkU_kψk)·n_kdA}NL+{1υAk(ρ~kU_~kψ~k)|x_+y_k·n_kdA}D, where the dispersion term is given by (38)·ρkU_kψkD=·ρkψ~kU_~ks+·U_kρ~kψ~ks+·ψkρ~kU_~ks+·ρ~kU_~kψ~ks. The nonlocal averaging theorem given by (24) with ψk=D_k is used in order to obtain the diffusive term (39)·D_ks=αk·D_k+1VAkD_kNL·n_kdA+1VAkD_~k|x_+y_·n_kdA. The terms ρkfks are obtained with the application of (28) (40)ρkfks=ρkfkNL+αkρkfk+ρ~kf~ks. In order to simplify the previous equations, the following representations are proposed: (41)ρkψkNLt=ηaαk(ρkψk)t,nonlocalaccumulation,(42)ρ~kψ~ks|x_t=δaαk(ρkψk)t,dispersionforaccumulation,(43)·ρkU_kψkNL=ηbαk·(ρkU_kψk),nonlocalconvection,(44)·ρkU_kψkD=δbαk·(ρkU_kψk),dispersionforconvection,(45)ρkfkNL=ηcαkρkfk,nonlocalsource,(46)ρ~kf~ks=δcαkρkfk,dispersionforsource, where η and δ are dimensionless parameters. The parameter η is the nonlocal nature, while the δ parameter agglutinates the dispersion effects. Now, the diffusive flux of (39) can be expressed as (47)·D_ks=αk·D_k+MkNL+M~k. In this equation the following definitions were used: (48)MkNL=1VAkD_kNL·n_kdA,interfacialNonlocaldiffusion,(49)M~k=1VAkD_~k|x_+y_·n_kdA,interfacialDiffusionduetodispersion. Finally, substituting (41)–(49), the nonlocal volume-averaged of the general balance equation (without length-scale restriction) finally is obtained: (50)λaαk(ρkψk)t+λbαk·(ρkU_kψk)+αk·D_k=λcαkρkfk-MkNL-M~k-MkDΓ-MkNLΓ, where (51)λ=η+δ+1,(52)MkDΓ={1VAk(ρ~kU_~kψ~k)|x_+y_k·n_kdA}D-{1VAk(ρ~kψ~k)|x_+y_kW_k·n_kdA}D,(53)MkNLΓ={1VAk(ρkU_kψk)·n_kdA}NL-{1VAkρkψkW_k·n_kdA}NL. Recalling that a portion of Ak is made of a liquid-gas interphase and a fluid-solid interphase Akw. Then, MkDΓ (and MkNLΓ) consider the transport phenomena related with interfacial mass transfer between fluid-fluid and fluid-solid interphase, that is, MkΓ=MkfEΓ+MwkEΓ (with E=D, NL).

5. Discussion

The volume averaged of the balance equation with length-scale restriction can be obtained starting from the nonlocal averaging equation (50), which contains local and nonlocal terms of the averaged volume. When η0, ρkU_kψkNL0, ρkfkNL0, MkNL0, and MkNLΓ0, the local averaging volume equation is recovered. Then, (46) simplifies to (54)(δa+1)αk(ρkψk)t+(δb+1)αk·(ρkU_kψk)+αk·D_k=(δc+1)αkρkfk-M~k-MkDΓ,1. The fundamental difference between local and nonlocal equations is that (50) involves, indirectly, values of the variables that are not associated with the centroid of the averaged volume as illustrated in Figure 1, while in (54) all the values of the volume-averaged variables are associated with the centroid of the averaged volume. The physical interpretation indicates that (54) describes the homogeneous two-phase flow. In this work the homogeneous term is used to indicate that the two-phase flow system has a behavior close to that of a homogeneous system; then to ensure homogeneity the system under study is based in length-scale restriction used to perform the upscaling in the two-phase flow system. However, (50) has not length-scale restriction and in principle it can describe regions of a two-phase flow, where drastic changes occur in the void fraction and transport properties (e.g., diffusivity).

The nonlocal volume averaging equations derived in this work contain new terms related to nonlocal transport effects due to accumulation, convection diffusion, and transport properties for two-phase flow. In general, the nonlocal terms were evaluated considering them as a function of the local terms, yielding new coefficients (ηs) that can be called nonlocal coefficients due to its nature these coefficients were defined through (41), (43), and (45) along with (48) and (53). It is important to note that these last two equations can also be expressed in terms of the local terms.

The nonlocal coefficients (ηs) are new closure relationships of the present novel formulation. For the application in a two-phase flow it is necessary as a first approximation to perform an analysis of order of magnitude, with the idea of identifying the predominant effects where the coefficients are not negligible (i.e., the temporal and diffusive effects are negligible). Then, the significant nonlocal coefficients can be evaluated with new or existing procedures in the experimental field, theoretical deduction, or numerical simulation, for instance.

The physical meaning of the nonlocal coefficients is related to the scaling process, that is, in the transition region as it can be observed in Figure 1. These coefficients act as coupling elements among the phenomena occurring in at least two different length scales. Outside of the interregion the length scales are smaller compared with those near the interregion (Figure 1).

Some examples where nonlocal general equation (50) can be applied are where αk presents abrupt changes [33, 34], in particular transitions of flow patterns, interface with stratified or annular flow drops and bubbles, and others as in the boundary region of the two-phase flow and solid, where the length-scale restriction given by (1) is not valid.

6. Conclusions

In this paper a derivation of the general transport equations for two-phase systems using a method-based on nonlocal volume averaging was presented. The nonlocal volume averaging equations derived in this work (50) contain new terms related to nonlocal transport effects due to accumulation, convection diffusion, and transport properties for two-phase flow.

The nonlocal terms were evaluated as a first approximation considering that these are a function of the local terms (41), (43), and (45), given as result of the nonlocal volume averaging equations (50) for practical applications. The nonlocal coefficients (ηs) are new closure relationships of the present novel formulation. The significant nonlocal coefficients can be evaluated with new or existent procedures: theoretical, numerical, and experimental. These coefficients act as coupling elements among the phenomena occurring in at least two different length scales, during the scaling process for pragmatic applications.

To illustrate the application of the representations of the nonlocal theorems and related definitions, the general balance equation for some ψ property in the k phase was considered, where it was demonstrated that a nonlocal volume averaging balance equation was obtained with meaningful averages. This general balance equation can be applied generally where αk presents abrupt changes [34, 35], such as transitions of flow patterns, interfaces with stratified or annular flow drops and bubbles, and others such as in the boundary region of the multiphase system, where the length-scale restriction (1) are not valid.

The nonlocal averaging model derived in this work represents a novel proposal and its framework could be the beginning of extensive research, both theoretical and experimental, as well as numerical simulation.

Slattery J. C. Flow of viscoelastic fluids through porous media AIChE 1967 13 1066 1071 Drew D. A. Averaged field equations for two- phase media Studies in Applied Mathematics 1971 50 2 133 166 2-s2.0-0015077923 Drew D. A. Segel L. A. Averaged equations for two- phase flows Studies in Applied Mathematics 1971 50 3 205 231 2-s2.0-0015124675 Bear J. Dynamics of Fluid in Porous Media 1972 New York, NY, USA Elsevier Gray W. G. A derivation of the equations for multi-phase transport Chemical Engineering Science 1975 30 2 229 233 2-s2.0-0016470063 Yadigaroglu G. Lahey R. T. On the various forms of the conservation equations in two-phase flow International Journal of Multiphase Flow 1976 2 5-6 477 494 2-s2.0-0016941435 Gray W. G. O'Neill K. On the general equations for flow in porous media and their reduction to Darcy's law Water Resources Research 1976 12 2 148 154 2-s2.0-0016941151 Delhaye J. M. Kakac S. Mayinger F. Instantaneous space-averaged equations Two-Phase Flows and Heat Transfer 1977 1 Washington, DC, USA Hemisphere 81 90 Gray W. G. Lee P. C. Y. On the theorems for local volume averaging of multiphase systems International Journal of Multiphase Flow 1977 3 4 333 340 2-s2.0-0017505662 Lahey R. T. Drew D. A. The three-dimensional time and volume averaged conservation equations of two-phase flow Advances in Nuclear Science & Technology 1989 20 1 69 Nigmatulin R. I. Spatial averaging in the mechanics of heterogeneous and dispersed systems International Journal of Multiphase Flow 1979 5 5 353 385 2-s2.0-0018533050 Hassanizadeh M. Gray W. G. General conservation equations for multi-phase systems: 1. Averaging procedure Advances in Water Resources 1979 2 131 144 2-s2.0-37249006592 Banerjee S. Chan A. M. C. Separated flow models-I. Analysis of the averaged and local instantaneous formulations International Journal of Multiphase Flow 1980 6 1-2 1 24 2-s2.0-0018949238 Delhaye J. M. Bergles A. E. Collier J. G. Delhaye J. M. Hewitt G. F. Mayinger F. Basic equations for two phase flow modeling Two-Phase Flow and Heat Transfer in the Power and Process Industries 1981 Washington, DC, USA Hemisphere 40 97 Sha W. T. Chao B. T. Soo S. I. Time averaging of volume averaged conservation equation of multiphase flow Proceedings of the 21st ASME/AIChE National Heat Transfer Conference 1983 Seattle, Wash, USA 420 426 Sha W. T. Chao B. T. Novel porous media formulation for multiphase flow conservation equations Nuclear Engineering and Design 2007 237 9 918 942 2-s2.0-33947669350 10.1016/j.nucengdes.2007.01.001 Dagan G. Statistical theory of groundwater flow and transport: poro to laboratory, laboratory to formation, and formation to regional scale Water Resources Research 1986 22 9 1205 1345 Gupta T. C. S. M. Goswami A. N. Rawat B. S. Mass transfer studies in liquid membrane hydrocarbon separations Journal of Membrane Science 1990 54 1-2 119 130 2-s2.0-0025514065 10.1016/S0376-7388(00)82074-8 Brusseau M. L. Transport of reactive contaminants in heterogeneous porous media Reviews of Geophysics 1994 32 3 285 313 2-s2.0-0028608541 Quintard M. Whitaker S. Convection, dispersion, and interfacial transport of contaminants: homogeneous porous media Advances in Water Resources 1994 17 4 221 239 2-s2.0-0028669433 Sen T. K. Khilar K. C. Review on subsurface colloids and colloid-associated contaminant transport in saturated porous media Advances in Colloid and Interface Science 2006 119 2-3 71 96 2-s2.0-33644536708 10.1016/j.cis.2005.09.001 Correia P. F. M. M. De Carvalho J. M. R. A comparison of models for 2-chlorophenol recovery from aqueous solutions by emulsion liquid membranes Chemical Engineering Science 2001 56 18 5317 5325 2-s2.0-0035921780 10.1016/S0009-2509(01)00240-8 Cho H. Shah S. N. Osisanya S. O. A three-segment hydraulic model for cuttings transport in coiled tubing horizontal and deviated drilling Journal of Canadian Petroleum Technology 2002 41 6 32 39 2-s2.0-0036591647 Espinosa-Paredes G. Cazarez-Candia O. Two-region average model for cuttings transport in horizontal wellbores I: transport equations Petroleum Science and Technology 2011 29 13 1366 1376 2-s2.0-79957846109 10.1080/10916460903567574 Espinosa-Paredes G. Cazarez-Candia O. Two-region average model for cuttings transport in horizontal wellbores II: interregion conditions Petroleum Science and Technology 2011 29 13 1377 1386 2-s2.0-79957830862 10.1080/10916460903551099 Lee S. C. Continuous extraction of penicillin G by emulsion liquid membranes with optimal surfactant compositions Chemical Engineering Journal 2000 79 1 61 67 2-s2.0-0034283080 10.1016/S1385-8947(00)00173-X Bayraktar E. Response surface optimization of the separation of DL-tryptophan using an emulsion liquid membrane Process Biochemistry 2001 37 2 169 175 2-s2.0-0034836520 10.1016/S0032-9592(01)00192-3 Anderson T. B. Jackson R. A fluid mechanical description of fluidized beds Industrial & Engineering Chemistry Fundamentals 1967 6 527 538 Marle C. M. Écoulements monophasique en milieu poreux Revue de l'Institut Francais du Petrole 1967 22 1471 1509 Truesdell C. A. Toupin R. A. Flugge S. The classical field theories Encyclopedia of Physics 1960 3/1 Springer 226 858 Cushman J. H. Multiphase transport equations I: general equation for macroscopic local space-time homogeneity Transport Theory and Statistical Physics 1983 12 1 35 71 2-s2.0-0020689543 Espinosa-Paredes G. Jump mass transfer for double emulsion systems International Mathematical Forum 2007 2 32 1553 1570 Whitaker S. The Method of Volume Averaging 1999 Dodrecht, The Netherlands Kluwer Academic Espinosa-Paredes G. Nuñez-Carrera A. SBWR model for steady state and transient analysis Science and Technology of Nuclear Installations 2008 2008 18 428168 10.1155/2008/428168 Morel C. Modeling approaches for strongly non-homogeneous two-phase flows Nuclear Engineering and Design 2007 237 11 1107 1127 2-s2.0-34248212109 10.1016/j.nucengdes.2007.01.005