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Science and Technology of Nuclear Installations
Volume 2012 (2012), Article ID 890815, 8 pages
http://dx.doi.org/10.1155/2012/890815
Research Article

A Derivation of the Nonlocal Volume-Averaged Equations for Two-Phase Flow Transport

Área de Ingeniería en Recursos Energéticos, Universidad Autónoma Metropolitana, P.O. Box 55-535, Iztapalapa, 09340 Mexico City, DF 09340, Mexico

Received 29 May 2012; Revised 7 August 2012; Accepted 9 August 2012

Academic Editor: Boštjan Končar

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.

Abstract

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 [116], 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 [10]: 1,(1) where is the characteristic length of the dispersed phases, and is the characteristic length of the global system: 1=MAX𝑥𝑉k||𝜓k𝑥,𝑡||MAX𝑥𝑉k||𝜓k𝑥,𝑡||.(2) 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 [17], fractionation of hydrocarbons [18], transport of contaminants [1921], elimination of contamination in aqueous streams [22], cuttings transport [2325], and concentration of pharmaceutics [26], among others that include extraction and separation processes [27]. 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 [10]), 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+𝜑k𝜓k, where 𝜑k and 𝜓k represent the spatial deviations around averaged values of the local variables and are defined by the decomposition 𝜑k=𝜑k+𝜑k and 𝜓k=𝜓k+𝜓k [5]. 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 𝜑k=0 and 𝜓k=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 𝑛k of 𝐴k is always drawn outwardly from phase k, regardless whether if it is associated with 𝐴lg or 𝐴kw.

890815.fig.001
Figure 1: Schematic figure of a two-phase flow with a transition region, showing the position vectors and the averaging volume.

Local averaging volume 𝑉=𝑉l(𝑡)+𝑉g(𝑡).(3) Volume fraction g phase in fluid mixture 𝛼g=𝑉g(𝑡)𝑉.(4) 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 𝜓ks||𝑥=1𝑉𝑉k(𝑥,𝑡)𝜓k𝑥+𝑦k,𝑡𝑑𝑉,(5) 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, 𝑦k 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 𝑦k. 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 𝜓ks=1𝑉𝑉k𝜓k||𝑥+𝑦k𝑑𝑉.(6) The intrinsic average is expressed in the form 𝜓k=1𝑉k𝑉k𝜓k||𝑥+𝑦k𝑑𝑉.(7) These averages will be used in the theoretical development of the two-phase flow transport equations and are related by 𝜓ks=𝛼k𝜓k.(8) With 𝜓k=1, the result leads to 1s=𝛼k.(9) 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] 𝜓ks=𝜓ks+1𝑉𝐴k𝜓k||𝑥+𝑦k𝑛k𝑑𝐴,(10) where 𝜓k is a quantity associated with the k phase, 𝑛k 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]: 𝜕𝜓k𝜕𝑡s=𝜕𝜓ks𝜕𝑡1𝑉𝐴k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴,(11) where 𝑊k is the velocity of the k-f interface in 𝑉,𝐴k=𝐴kw+𝐴lg.(12) If 𝜓k=1, the previous theorems lead to 𝛼k=1𝑉𝐴k𝑛k𝑑𝐴,(13)𝜕𝛼k𝜕𝑡=1𝑉𝐴k𝑊k𝑛k𝑑𝐴.(14) 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+𝜓k) [5], 𝜓ks=𝜓ks+1𝑉𝐴k𝜓k||𝑥+𝑦k𝑛k𝑑𝐴+1𝑉𝐴k𝜓k||𝑥+𝑦𝑛k𝑑𝐴.(15) In the homogeneous regions of the system, the following length-scale restriction given by (1) is usually satisfied, and the following simplification is considered: 𝜓k||𝑥+𝑦k𝜓k,for1.(16) Then, the second term on the right side of (15) can be written as 1𝑉𝐴k𝜓k||𝑥+𝑦k𝑛k𝑑𝐴=1𝑉𝐴k𝑛k𝑑𝐴𝜓k=𝛼k𝜓k,for1.(17) 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:𝜓ks=𝛼k𝜓k+1𝑉𝐴k𝜓k||𝑥+𝑦𝑛k𝑑𝐴,for1.(18) The similar form, the theorem given by (11) can be rewritten as 𝜕𝜓k𝜕𝑡s=𝛼k𝜕𝜓ks𝜕𝑡1𝑉𝐴k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴,for1,(19) 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. Nonlocal Averaged Volume

The spatial decomposition given by 𝜓k=𝜓k+𝜓k 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 𝜓𝑘 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+𝑦k). 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+𝑦k𝑛k is evaluated in the k phase indicated by position vector 𝑦k shown in Figure 1. Then, 1𝑉𝐴k𝜓k||𝑥+𝑦k𝑛k𝑑𝐴,(20) 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+𝑦k can be known applying a Taylor series expansion about the centroid of the averaging volume [33]: 𝜓k||𝑥+𝑦k=𝜓k+𝑦k𝜓k+12𝑦k𝑦k𝜓k+.(21) The second, third, and following terms on the left side correspond to nonlocal effects. Then, this equation can be approximate by 𝜓k||𝑥+𝑦k=𝜓k||𝑥+𝜓kNL,(22) where 𝜓kNL=𝑦k𝜓k+(1/2)𝑦k𝑦k𝜓k+. Inclusive can be treated as source term formed by 𝜓k|𝑥+𝑦k𝜓k. 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 𝜓kNL=𝜓k||𝑥+𝑦k𝜓k,withoutlength-scalerestrictions,0,for1.(23) 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+𝑦k𝜓k is important and the length-scale constraints given by (1) are not valid.

Applying these ideas the theorems can be expressed in nonlocal terms: 𝜓ks=𝛼k𝜓k+1𝑉𝐴k𝜓kNL𝑛k𝑑𝐴+1𝑉𝐴k𝜓k||𝑥+𝑦𝑛k𝑑𝐴,(24)𝜕𝜓k𝜕𝑡s=𝛼k𝜕𝜓k𝜕𝑡1𝑉𝐴k𝜓kNL𝑊k𝑛k𝑑𝐴1𝑉𝐴k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴.(25) 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 𝜑𝑘𝜓𝑘

The explicit representation of the average volume of the product of two local variables is given by 𝜑k𝜓ks=1𝑉𝑉k𝜑k𝜓k||𝑥+𝑦k𝑑𝑉.(26) Substituting the correspondent spatial deviations for the local variables 𝜑k and 𝜓k leads to 𝜑k𝜓ks=𝛼k𝜑k𝜓k+𝜓k𝜑ks+𝜑k𝜓ks+𝜑k𝜓ks.(27) This is rewritten as 𝜑k𝜓ks=𝜑k𝜓kNL+𝛼k𝜑k𝜓k+𝜑k𝜓ks,(28) 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 𝜑k𝜓kNL=𝜑k𝜓ks+𝜑k𝜓ks+𝜑k𝜓ks𝛼k𝜑k𝜓k.(29) It can be demonstrated that 𝜑k𝜓kNL=0,for1.(30)

3.3. Operators Applied to Two Local Variables 𝜑𝑘𝜓𝑘

The typical expressions in the transport phenomena in a two-phase flow involve an average differential operator with two local variables, 𝜑k𝜓ks=𝜑k𝜓ks+1𝑉𝐴k𝜑k𝜓k||𝑥+𝑦k𝑛k𝑑𝐴,𝜕𝜑k𝜓k𝜕𝑡s=𝜕𝜑𝑘𝜓𝑘s𝜕𝑡1𝑉𝐴k𝜑k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴.(31) With the previous ideas we obtain expanded form of the theorems for the product of two-local variables 𝜑k𝜓ks=𝜑k𝜓kNL+k𝜑k𝜓k+𝜑k𝜓ks+1𝑉𝐴k𝜑k𝜓k𝑛k𝑑𝐴NL𝑛𝑜𝑛𝑙𝑜𝑐𝑎𝑙+1𝑉𝐴k𝜑k𝜓k||𝑥+𝑦k𝑛k𝑑𝐴Ddispersion,(32)𝜕𝜑k𝜓k𝜕𝑡s=𝛼k𝜕𝜑k𝜓k𝜕𝑡+𝜕𝜑k𝜓kNL𝜕𝑡+𝜕𝜑k𝜓ks𝜕𝑡+1𝑉𝐴k𝜑k𝜓k𝑊k𝑛k𝑑𝐴NL𝑛𝑜𝑛𝑙𝑜𝑐𝑎𝑙+1𝑉𝐴k𝜑k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴Ddispersion.(33)

4. Nonlocal 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: 𝜕𝜌k𝜓k𝜕𝑡+𝜌k𝑈k𝜓k+𝐷k=𝜌k𝑓,(34) where 𝜓k is the quantity conserved, 𝐷k 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.

tab1
Table 1: Conservation terms.

The volume averaged of the general balance equation can be expressed as 𝜕(𝜌k𝜓k)𝜕𝑡s+𝜌k𝑈k𝜓ks+𝐷ks=𝜌k𝑓ks.(35) 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: 𝜕𝜌k𝜓k𝜕𝑡s=𝛼k𝜕𝜌k𝜓k𝜕𝑡+𝜕𝜌k𝜓kNL𝜕𝑡+𝜕̃𝜌k𝜓ks𝜕𝑡1𝑉𝐴k𝜌k𝜓k𝑊k𝑛k𝑑𝐴NL1𝑉𝐴k̃𝜌k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴D.(36) 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 𝜌k𝑈k𝜓ks=𝛼k𝜌k𝑈k𝜓k+𝜌k𝑈k𝜓kNL+𝜌k𝑈k𝜓kD+1𝜐𝐴k𝜌k𝑈k𝜓k𝑛k𝑑𝐴NL+1𝜐𝐴k̃𝜌k𝑈k𝜓k|||𝑥+𝑦k𝑛k𝑑𝐴D,(37) where the dispersion term is given by 𝜌k𝑈k𝜓kD=𝜌k𝜓k𝑈ks+𝑈k̃𝜌k𝜓ks+𝜓k̃𝜌k𝑈ks+̃𝜌k𝑈k𝜓ks.(38) The nonlocal averaging theorem given by (24) with 𝜓k=𝐷k is used in order to obtain the diffusive term 𝐷ks=𝛼k𝐷k+1𝑉𝐴k𝐷kNL𝑛k𝑑𝐴+1𝑉𝐴k𝐷k||𝑥+𝑦𝑛k𝑑𝐴.(39) The terms 𝜌k𝑓ks are obtained with the application of (28) 𝜌k𝑓ks=𝜌k𝑓kNL+𝛼k𝜌k𝑓k+̃𝜌k̃𝑓ks.(40) In order to simplify the previous equations, the following representations are proposed: 𝜕𝜌k𝜓kNL𝜕𝑡=𝜂𝑎𝛼k𝜕𝜌k𝜓k𝜕𝑡,nonlocalaccumulation,(41)𝜕̃𝜌k𝜓ks||𝑥𝜕𝑡=𝛿𝑎𝛼k𝜕𝜌k𝜓k𝜕𝑡,dispersionforaccumulation,(42)𝜌k𝑈k𝜓kNL=𝜂𝑏𝛼k𝜌k𝑈k𝜓k,nonlocalconvection,(43)𝜌k𝑈k𝜓kD=𝛿𝑏𝛼k𝜌k𝑈k𝜓k,dispersionforconvection,(44)𝜌k𝑓kNL=𝜂𝑐𝛼k𝜌k𝑓k,nonlocalsource,(45)̃𝜌k̃𝑓ks=𝛿𝑐𝛼k𝜌k𝑓k,dispersionforsource,(46) 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 𝐷ks=𝛼k𝐷k+𝑀kNL+𝑀k.(47) In this equation the following definitions were used: 𝑀kNL=1𝑉𝐴k𝐷kNL𝑛k𝑑𝐴,interfacialNonlocaldiusion,(48)𝑀k=1𝑉𝐴k𝐷k||x+y𝑛k𝑑𝐴,interfacialDiusionduetodispersion.(49) Finally, substituting (41)–(49), the nonlocal volume-averaged of the general balance equation (without length-scale restriction) finally is obtained: 𝜆𝑎𝛼k𝜕𝜌k𝜓k𝜕𝑡+𝜆𝑏𝛼k𝜌kUk𝜓k+𝛼kDk=𝜆𝑐𝛼k𝜌k𝑓k𝑀kNL𝑀k𝑀ΓkD𝑀ΓkNL,(50) where 𝜆=𝜂+𝛿+1,(51)𝑀ΓkD=1𝑉𝐴k̃𝜌k𝑈k𝜓k|||𝑥+𝑦k𝑛k𝑑𝐴D1𝑉𝐴k̃𝜌k𝜓k||𝑥+𝑦k𝑊k𝑛k𝑑𝐴D,(52)𝑀ΓkNL=1𝑉𝐴k𝜌k𝑈k𝜓k𝑛k𝑑𝐴NL1𝑉𝐴k𝜌k𝜓k𝑊k𝑛k𝑑𝐴NL.(53) Recalling that a portion of 𝐴k is made of a liquid-gas interphase and a fluid-solid interphase 𝐴kw. Then, 𝑀ΓkD (and 𝑀ΓkNL) consider the transport phenomena related with interfacial mass transfer between fluid-fluid and fluid-solid interphase, that is, 𝑀Γk=𝑀ΓkfE+𝑀ΓwkE (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, 𝜌k𝑈k𝜓kNL0, 𝜌k𝑓kNL0, MkNL0, and MΓkNL0, the local averaging volume equation is recovered. Then, (46) simplifies to 𝛿𝑎+1𝛼k𝜕𝜌k𝜓k𝜕𝑡+𝛿𝑏+1𝛼k𝜌k𝑈k𝜓k+𝛼k𝐷k=𝛿𝑐+1𝛼k𝜌k𝑓k𝑀k𝑀ΓkD,1.(54) 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.

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