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A Conjugate Thermo-Electric Model for a Composite Medium

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A Conjugate Thermo-Electric Model for a Composite Medium
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  A Conjugate Thermo-Electric Model for a CompositeMedium Oscar Cha´ vez 1 , Francisco A. Godı´ nez 2 , Alberto Beltra´ n 1 , Armando Garcı´ a 3 , Roberto Zenit 1 * 1 Instituto de Investigaciones en Materiales, Universidad Nacional Auto´noma de Me´xico, Ciudad Universitaria, D.F., Me´xico,  2 Instituto de Ingenierı´a, Universidad NacionalAuto´noma de Me´xico, Ciudad Universitaria, D.F., Me´xico,  3 Tecnologı´a Aplicada en Exploracio´n y Produccio´n Petrolera, Inc., Polanco, D.F., Me´xico Abstract Electrical transmission signals have been used for decades to characterize the internal structure of composite materials. Wetheoretically analyze the transmission of an electrical signal through a composite material which consists of two phases withdifferent chemical compositions. We assume that the temperature of the biphasic system increases as a result of Jouleheating and its electrical resistivity varies linearly with temperature; this last consideration leads to simultaneously study theelectrical and thermal effects. We propose a nonlinear conjugate thermo-electric model, which is solved numerically toobtain the current density and temperature profiles for each phase. We study the effect of frequency, resistivities andthermal conductivities on the current density and temperature. We validate the prediction of the model with comparisonswith experimental data obtained from rock characterization tests. Citation:  Cha´vez O, Godı´nez FA, Beltra´n A, Garcı´a A, Zenit R (2014) A Conjugate Thermo-Electric Model for a Composite Medium. PLoS ONE 9(5): e97895. doi:10.1371/journal.pone.0097895 Editor:  Aristides Docoslis, Queen’s University at Kingston, Canada Received  January 7, 2014;  Accepted  April 24, 2014;  Published  May 27, 2014 Copyright:    2014 Cha´vez, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the srcinal author and source are credited. Funding:  This research was funded by CONACyT-Mexico through its SENER-HIDROCARBUROS program (grant number: 143927). The funders had no role in studydesign, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests:  Armando Garcı´a is an industrial collaborator. The fact that he works in a private company does not alter the authors’ adherence to PLOSONE policies on sharing data and materials.* E-mail: zenit@unam.mx Introduction  A composite medium can be defined as that made of at least twophases of different chemical compositions [1]. The study of composite media are of great interest to various areas such asphysics, chemistry and materials science, among others [1–3].The response of composite media when transmitting-absorbing waves of different intensities and frequencies has been analyzed bymany previous studies. In [4], a fiber reinforced epoxy matrixcomposite is studied as electromagnetic wave absorbing material ina wide frequency range. Carbon fiber reinforced concrete [5] andmetallic wire structures [6] have been characterized in terms of their capacity to absorb electromagnetic fields. It is interesting tonote that some materials have the ability to shield electromagneticwaves, examples of these are styrene-butadiene rubber composites[7], wood-cement boards [8] and nanocomposites [9].The prediction of the effective properties of a compositemedium is commonly based on the knowledge of both the volumefraction distribution and the value of the property of each of itsphases [1]. A series of mathematical models have been developedto define global properties of a composite medium. In [1,10] theeffective electrical conductivity is obtained from the individualelectrical properties, also, heat transfer studies at macroscopic andmicroscopic levels have been conducted to determine the effectivethermal conductivity [11,12]. To our knowledge the interactionbetween the heat diffusion and transmission of electric currentthrough composite media has not been studied fully to date.The present theoretical model considers two non-deformablephases which are modeled as continua, and it is based on aprevious model developed by Cha´vez and Me´ndez [13] whoanalyzed the conjugate heat and electromagnetic transfer mech-anism in a bimetallic conductor. Our system consists of a cylindralexternal phase confining a second cylindrical internal phase,henceforth external and internal phases, as depicted schematicallyin Figure 1.Maxwell’s equations are coupled with the heat conductionequation considering a heat source term to account for Joule’seffect. This coupling is performed for each phase; also, both phasesare coupled with each other through the boundary conditions atthe common interface.Particularly, the proposed model is intended to explain the roleof some effects that occur during the electrical conductionprocesses in composite media, which to our knowledge have notbeen fully addressed:1. The so-called skin effect observed in power transmission lines[14].2. The Joule heating effect [15].3. The effect of frequency on the current density and temperaturedistribution.4. The effect of volumetric fraction (porosity) on the bulk electrical resistivity. Modelling The physical model under study is a composite medium like thatshown in Figure 1. We consider an external phase with radius  b and an internal one with radius  a . A sudden alternating electriccurrent through this biphasic conductor is established. Thus, a riseof temperature results from the flow of electric current, caused by Joule’s effect. We assumed that the resistivity in both phases varies PLOS ONE | www.plosone.org 1 May 2014 | Volume 9 | Issue 5 | e97895  linearly with temperature. For large values of the frequencyassociated to the alternating current, a redistribution of the currentdensity is inevitable and the skin effect yields a tendency of theelectric current to flow over the outer surfaces of both phases. Inreal conditions, the current density distribution depends on the values of electrical resistivities of each phase and the heatgenerated by Joule’s effect which is transferred by heat conduction.Finally, we model the heat transferred to the environment by aconvection process. Electromagnetic model Using Maxwell’s equations, we can readily derive a waveequation to analyze the electromagnetic propagation. Therefore,the current density is governed by the following equation [16]: + 2 l ~ J J  ~ m  L ~ J J  L t z c L 2 l ~ J J  L t 2  ! ,  ð 1 Þ where,  l  is the electric resistivity,  ~ J J   is the current density,  m  is themagnetic permeability,  c  is the electric permitivity and  t  is thephysical time.We consider only variations of the current density in the radialdirection and the alternating current behaves like a sinusoidalwave. Therefore, the current density can be written as ~ J J  ~ J  s ( r ) e i  v t . On the other hand, the electrical resistivity has alinear variation with temperature [16] which can be written as l ~ l ?  1 z w ( T  { T  ? ) ½  , and introducing it into Eq. (1) we obtainthat, d  2 J  s dr 2  z 2 w 1 z w ( T  { T  ? ) L T  L r z 1 r   dJ  s dr z w 1 z w ( T  { T  ? ) L 2 T  L r 2  z 1 r L T  L r " # J  s ~ mvl  i  z cvl ? wv 2 L 2 T  L t 2  z 2 i  wv L T  L t { (1 z w ( T  { T  ? ))  !" #( ) J  s , ð 2 Þ where  v  is the frequency of the electrical signal,  r  is the radialcoordinate,  T   is the temperature,  T  ?  is the environmenttemperature,  J  s  is the current density function depending onlyon radial coordinate,  w  is the temperature coefficient for resistivityand  i  ~  ffiffiffiffiffiffiffiffi { 1 p   .In most practical cases, the term  cv 2 m  is smaller than  i  vm = l and can be neglected as a first approximation. In addition, we alsointroduce the well-known conductor skin depth parameter,  d ,defined by  d ~ (2 l = mv ) 1 = 2 [17]. Thus, Eq. (2) can be rewritten forthe internal phase as: d  2 J  s , I  dr 2  z 2 w I  1 z w I  ( T  I  { T  ? ) L T  I  L r  z 1 r   dJ  s , I  dr z w I  1 z w I  ( T  I  { T  ? ) L 2 T  I  L r 2  z 1 r L T  I  L r " # J  s , I  ~ 2 i  d 2 I  (1 z w I  ( T  I  { T  ? )) J  s , I  , ð 3 Þ and for the external phase, d  2 J  s , E  dr 2  z 2 w E  1 z w E  ( T  E  { T  ? ) L T  E  L r  z 1 r   dJ  s , E  dr z w E  1 z w E  ( T  E  { T  ? ) L 2 T  E  L r 2  z 1 r L T  E  L r " # J  s , E  ~ 2 i  d 2 E  (1 z w E  ( T  E  { T  ? )) J  s , E  , ð 4 Þ where the subscript ‘‘ s ’’ is used to denote the spatial dependenceand the subscripts ‘‘ E  ’’ and ‘‘ I  ’’ are used to denote external andinternal phases, respectively. The ‘‘ ? ’’ subscript refers to externalenvironmental conditions.The above equations system must be solved considering thefollowing boundary conditions:at  r ~ 0  : dJ  s , I  dr  ~ 0,  ð 5 Þ at  r ~ a  :  l ? , I  ½ 1 z w I  ( T  I  { T  ? )  J  s , I  ~ l ? , E  ½ 1 z w E  ( T  E  { T  ? )  J  s , E  , ð 6 Þ l ? , I  m I  (1 z w I  ( T  I  { T  ? )) dJ  s , I  dr  z w I  J  s , I  L T  I  L r   ~ l ? , E  m E  (1 z w E  ( T  E  { T  ? )) dJ  s , E  dr  z w E  J  s , E  L T  E  L r   , ð 7 Þ Figure 1. Schematic representation of a composite medium. doi:10.1371/journal.pone.0097895.g001A Model for the Resistivity of Porous MediaPLOS ONE | www.plosone.org 2 May 2014 | Volume 9 | Issue 5 | e97895  atthe surface  r ~ b  :  J  s , E  ~ J  b :  ð 8 Þ The boundary condition at the centre (Eq. (5)) is the symmetrycondition, while the continuity of the electric field at the interfaceis expressed by Eq. (6). Eq. (7) refers to the continuity of themagnetic field, while Eq. (8) expresses a characteristic currentdensity.  J  b  is the current density at the outer surface of the externalphase and should be determined with the following restriction: I  ~ 2 p ð  a 0 J  s , I  rdr z ð  ba J  s , E  rdr    ð 9 Þ Thermal model The general heat diffusion equation can be expressed as [18]: + : ½ k  + T   z _ qq  gen ~ ( r c ) L T  L t  :  ð 10 Þ With this equation we can determine the gradients of temperature in both phases by taking into account their thermalproperties and the amount of heat generated on each of them.Equation (10) is simplified by considering only temperature variations in the radial direction and exclusively heat generationby Joule’s effect. Therefore, for the internal phase we have: k  I  r LL r r L T  I  L r   z l ? , I  ½ 1 z w I  ( T  I  { T  ? ) j J  s , I  j 2 ~ ( r c ) I  L T  I  L t  ,  ð 11 Þ and for the external phase: k  E  r LL r r L T  E  L r   z l ? , E  ½ 1 z w E  ( T  E  { T  ? ) j J  s , E  j 2 ~ ( r c ) E  L T  E  L t  ,  ð 12 Þ subjected to the following boundary conditions:at  r ~ 0  : L T  I  L r  ~ 0,  ð 13 Þ at  r ~ a  :  T  I  ~ T  E  ,  ð 14 Þ { k  I  L T  I  L r  ~{ k  E  L T  E  L r  :  ð 15 Þ For the outer surface of the external phase, we have r ~ b  :  { k  E  L T  E  L r  ~ h ( T  E  { T  ? ) :  ð 16 Þ  Also an initial condition is necessary. Here we have consideredthat the composite media is initially at ambient temperature: t ~ 0  :  T  I  ~ T  E  ~ T  ? :  ð 17 Þ In the above equations,  k   is the thermal conductivity,  r  is thedensity,  c  is the specific heat,  h  is the convective heat transfercoefficient. Dimensional Analysis In order to reduce the number of physical parameters, we canperform a dimensional analysis. We first identify the characteristicconvective time scale  t c ~ ( r c ) I  b = h . On the other hand, thesuitable spatial scale is chosen as the radius of the external phase, r ~ b . Furthermore, the characteristic temperature drop  D T  c  canbe obtained through an energy balance between the heatgeneration term and the transient term, i.e.: D T  c ~ l ? , I  J  2 a a 2 k  E  Bi   ,  ð 18 Þ where  J  a  is the current density at the surface of the internal phaseand  Bi   is the Biot number which measures the environmentalconditions and is defined as Bi  ~ hbk  E  ,  ð 19 Þ With the above set of characteristic geometrical and physicalscales, the electromagnetic and thermal models can be simplifiedby introducing the following dimensionless variables and param-eters: t ~ th ( r c ) I  b ,  h I  , E  ~ T  I  , E  { T  ? D T  c ,  j ~ rb Q I  , E  ~ J  s ,( I  , E  ) J  a ,  k ~ w I  , E  D T  c ,  W ~ ab   2 G  , e I  ~ d I  a  ,  e E  ~ d E  b { a ,  where  d I  , E  ~  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l ? ,( I  , E  ) m I  , E  v s   , where  W  is the volumetric fraction defined as the ratio of the volume of the internal phase and the total volume of the compositemedia, while  G   is defined as the actual lenght of the inner pathdivided by the straight-line distance between the ends of the innerpath, for porous media applications this parameter is known astortuosity and is always larger or equal than one. Dimensionless electromagnetic model The system (3)–(8) can be rewritten in dimensionless form byusing the above dimensionless parameters and variables. A Model for the Resistivity of Porous MediaPLOS ONE | www.plosone.org 3 May 2014 | Volume 9 | Issue 5 | e97895  The internal phase dimensionless model is: d  2 Q I  d  j 2  z  2  k I  (1 z k I  h I  ) L h I  L j z 1 j   d  Q I  d  j z k I  1 z k I  h I  L 2 h I  L j 2  z 1 j L h I  L j  ! Q I  ~ 2 i  (1 z k I  h I  ) e 2 I  W = G  Q I  , ð 20 Þ and external phase dimensionless model is: d  2 Q E  d  j 2  z  2  k E  (1 z k E  h E  ) L h E  L j  z 1 j   d  Q E  d  j  z k E  1 z k E  h E  L 2 h E  L j 2  z 1 j L h E  L j  ! Q E  ~ 2 i  (1 z k E  h E  ) e 2 E  (1 {  ffiffiffiffiffiffiffiffiffiffi W = G  p   ) 2  Q E  , ð 21 Þ The corresponding dimensionless boundary conditions are: j ~ 0  : d  Q I  d  j ~ 0,  ð 22 Þ j ~  ffiffiffiffi W G  r   :  Q I  ~ l E  ½ 1 z k E  h E   l I  ½ 1 z k I  h I    Q E  ,  ð 23 Þ m E  l I  m I  l E  (1 z k I  h I  )(1 z k E  h E  ) d  Q I  d  j z k I  1 z k I  h I  L h I  L j Q I    ~ d  Q E  d  j  z k E  1 z k E  h E  L h E  L j Q E    , ð 24 Þ j ~ 1  :  Q E  ~ l I  l E  G  :  ð 25 Þ Dimensionless thermal model In the same manner, we can use the dimensionless variables andparameters in order to obtain the following dimensionless thermalmodel:Therefore, the internal phase equation can be written as: 1 j LL j j L h I  L j   z k  E  k  I  (1 z k I  h I  ) W = G  Bi  j Q I  j 2 ~ k  E  k  I  Bi  L h I  L t  ,  ð 26 Þ and external phase as: 1 j LL j j L h E  L j   z l E  l I  (1 z k E  h E  ) W = G  Bi  j Q E  j 2 ~ ( r c ) E  ( r c ) I  Bi  L h E  L t  :  ð 27 Þ The associated boundary and initial conditions are given by: j ~ 0  : d  h I  d  j ~ 0,  ð 28 Þ j ~  ffiffiffiffi W G  r   : L h I  L j ~ k  E  k  I  L h E  L j  ,  ð 29 Þ h I  ~ h E  ,  ð 30 Þ j ~ 1  : L h E  L j  ~{ Bi  h E  ,  ð 31 Þ t ~ 0  :  h I  ~ h E  ~ 0 :  ð 32 Þ The system of equations to be solved is formed by Eqs. (20) and(21) for the current density distribution coupled through theboundary conditions (23) and (24); and Eqs. (26) and (27) for thethermal behavior also coupled through their boundary conditions(29) and (30). It should be noticed that there is a coupling betweenthe electromagnetic and thermal model due to the dependence of resistivity with temperature which is expressed by  k   parameter,henceforth the coupling parameter. Solution Methodology The above dimensionless electromagnetic and heat conductionequations, together with their boundary and initial conditions,represented here by the system of Eqs. (20)–(32) was solved byusing a conventional iterative finite-differences method [19].The electromagnetic equations are given by the system of Eqs.(20)–(25). These equations are complex because the right-handsides of Eqs. (20) and (21) include, as a factor, the imaginarynumber  i  . Therefore, we separate for each region the electricalcurrent density  Q , in a real part,  Q R , and an imaginary part  Q I  ,through the relationship  Q ~ Q R z i  Q I  . The resulting equations arediscretized together with the boundary conditions (22)–(25)considering central differences. In this form, we can construct amatrix system which can be solved with a simple Gausselimination method.The corresponding equations for the thermal model given bythe set of Eqs. (26) and (27) together with the boundary and initialconditions (28)–(32), require a different treatment. In this case, theabove equations represent a non-stationary problem. Therefore,the numerical procedure is based on the well-known Crank-Nicholson finite difference scheme. In this manner, we obtain atridiagonal matrix which is solved by the tridiagonal matrixalgorithm (TDMA), also known as the Thomas algorithm.Finally, we introduce the following iterative scheme: firstly, auniform profile for the temperature is considered, then we solve forthe electrical current density. In this manner, we can obtain themodulus or absolute value of this function. Introducing the aboveresult into Eqs. (26) and (27), we obtain the first nonuniformtemperature profile. Again, we can obtain a new current densityand the foregoing procedure is repeated until a convergencecriterion is fulfilled. This criterion is based on the comparison of the temperature and current density profiles. Validation It is important to note that if the coupling parameter is equal tozero (  k ~ 0  ), the equations associated with the electromagneticbehavior, Eqs. (20) and (21), are no longer affected by the A Model for the Resistivity of Porous MediaPLOS ONE | www.plosone.org 4 May 2014 | Volume 9 | Issue 5 | e97895  temperature; thus, the system can be solved analytically. Thesolutions for the current density distributions,  Q l  ( j )  and  Q r ( j ) (liquid and rock phases, respectively) are: Q l  ( j ) ~ G   ffiffiffiffi W p   e l  m l  L J  0  g  j W U  z  ffiffiffiffi W p   e l  m l  V J  0 (  g  ),  ð 33 Þ and Q r ( j ) ~ G  l l  J  Y  0 ( {  f  j  ffiffiffi W p   ) z H J  0 (  f  j  ffiffiffi W p   ) l r ( U  z  ffiffiffiffi W p   e l  m l  V J  0 (  g  ))  ! ,  ð 34 Þ where  J  0 ,  J  1 ,  Y  0 ,  Y  1  denote the Bessel functions of first and secondkind and of zeroth and first order, respectively. The variables  f  , L , U  , V , J  and  H  are defined as:  f  ~ (1 { i  )  ffiffiffiffi W p  e r {  ffiffiffiffi W p   e r ,  g  ~ 1 { i  e l  ,  h ~ 1 { i  e r {  ffiffiffiffi W p   e r ,  ð 35 Þ L ~ J  1 (  f  ) Y  0 ( {  f  ) z J  0 (  f  ) Y  1 ( {  f  ),  ð 36 Þ U  ~ e r m r (  ffiffiffiffi W p   { 1)( J  0 (  f  ) J  1 (  g  ) Y  0 ( { h ) { J  0 ( h ) J  1 (  g  ) Y  0 ( {  f  )),  ð 37 Þ V ~ J  1 (  f  ) Y  0 ( { h ) z J  0 ( h ) Y  1 ( {  f  ),  ð 38 Þ J ~ (  ffiffiffiffi W p   { 1) e r m r J  0 (  f  ) J  1 (  g  ) z  ffiffiffiffi W p   e l  m l  J  0 (  g  ) J  1 (  f  ),  ð 39 Þ H ~  ffiffiffiffi W p   e l  m l  J  0 (  g  ) Y  1 ( {  f  ) { (  ffiffiffiffi W p   { 1) e r m r J  1 (  g  ) Y  0 (  f  ),  ð 40 Þ This analytical solution has great relevance since it serves as a validation test for our numerical results. As shown in Fig. 2, thenumerical simulations (open symbols) agree very well with theanalytical solutions (solid lines). Results and Discussion In the system of equations (26)–(32) we have two coupling parameters  k I   and  k E  . They, however, depend on each otherbecause they are affected by the same  D T  c , thus k E  ~ k I  w E  w I  ,  ð 41 Þ therefore it is necessary to know only one parameter. We choosethe level of coupling between thermal and electrical model as k ~ k I  .In the same manner the skin parameter for the internal phase, e I  , is related to that of the external one, because both phasestransmit a wave at the same frequency. Thus we have: e E  ~ e I  ( W = G  ) 12 1 { ( W = G  ) 12  ffiffiffiffiffiffiffiffiffiffi l E  m I  l I  m E  s   :  ð 42 Þ From now on,  e I   will be simply called  e , the depth of penetration of the electrical signal. Therefore, the main param-eters in this problem are:  l E  = l I  ,  k  E  = k  I  ,  ( r c ) E  = ( r c ) I  ,  m E  = m I  ,  e ,  Bi  , k  and  W .To study the transmission of the electrical signal in a compositemedium, we perform a parametric analysis based on the following parameters:  e , which is a function of the frequency, the ratio of resistivities,  l E  = l I  , and the ratio of thermal conductivities,  k  E  = k  I  .For simplicity, all these calculations assuming   k ~ 1 ,  W ~ 0 : 5 , ( r c ) E  = ( r c ) I  ~ 1 ,  m E  = m I  ~ 1  and  G  ~ 1 ; also, we consider  Bi  ~ 1 ,which assures that the heat is efficiently transferred by convectionfrom the external phase to the environment. To assure steady statesolutions we performed all calculations using   t ~ 20 . Effect of   e To analyze the effect of the skin parameter on the currentdensity and temperature, numerical results were obtained fromthree different values of   e  (0.1, 0.5 and 10) and are presented inFig. 3. The series of three temperature profiles shown in Fig. 3arepresents the steady-state solution for each corresponding value of  e . For lower values of the skin parameter (which correspond toelectric signals with high frequency), the temperature profilesbecome more uniform in both phases; on the contrary, whengrows a slightly parabolic temperature profile is exhibited in theinternal phase and even a steeper parabolic profile is observed forthe external one. This parabolic behavior comes from the Jouleeffect (source therms in equations (26–27)). Figure 3b shows thecurrent density distribution as a function of the dimensionlessradial coordinate for the same conditions shown in Figure 3a.Clearly, for small values of   e , the skin effect becomes noticeable inboth phases. In particular the current density in the internal phaseis higher in regions close to the interface (  e ? a = b  ). On the oppositeside, high values of   e  show nearly constant current distribution inboth phases, hence the electric power tends to be transmitted asdirect current. Figure 2. Dimensionless current density distribution  Q  as afunction of the radial coordinate  j  for both conducting phasesand three different values of the skin parameter  e .  Solid lines arecomputed from the analytical solution, the open symbols representnumerical simulation results.doi:10.1371/journal.pone.0097895.g002A Model for the Resistivity of Porous MediaPLOS ONE | www.plosone.org 5 May 2014 | Volume 9 | Issue 5 | e97895
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