Research Article Numerical Investigation of Characteristic of Anisotropic Thermal Conductivity of Natural Fiber Bundle with Numbered Lumens

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1 Mathematical Problem in Engineering, Article ID , 8 page Reearch Article Numerical Invetigation of Characteritic of Aniotropic Thermal Conductivity of Natural Fiber Bundle with Numbered Lumen Guan-Yu Zheng Department of Building Engineering, College of Civil Engineering, Tongji Univerity, Shanghai , China Correpondence hould be addreed to Guan-Yu Zheng; zheng guanyu@163.com Received 6 July 2014; Accepted 15 July 2014; Publihed 6 Augut 2014 Academic Editor: Song Cen Copyright 2014 Guan-Yu Zheng. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Natural fiber bundlelikehemp fiber bundleuuallyinclude manymall lumen embedded in olid region; thu, it can preent lower thermal conduction than that of conventional fiber. In the paper, characteritic of aniotropic tranvere thermal conductivity of unidirectional natural hemp fiber bundle wa numerically tudied to determine the dependence of overall thermal property of the fiber bundle on that of the olid region phae. In order to efficiently predict it thermal property, the fiber bundle wa embedded into an imaginary matrix to form a unit compoite cell coniting of the matrix and the fiber bundle. Equally, another unit compoite cell including an equivalent olid fiber wa etablihed to preent the homogenization of the fiber bundle. Next, finite element thermal analyi implemented by ABAQUS wa conducted in the two etablihed compoite cell by applying proper thermal boundary condition along the boundary of unit cell, and influence of the olid region phae and the equivalent olid fiber on the compoite were invetigated, repectively. Subequently, an optional relationhip of thermal conductivitie of the natural fiber bundle and the olid region wa obtained by curve fitting technique. Finally, numerical reult from the obtained fitted curve were compared with the analytic Haelman-Johnon reult and other to verify the preent numerical model. 1. Introduction Natural fiber like kenaf fiber [1], hemp fiber [2], ial fiber [3], date palm fiber [4], wood fiber [5], and bamboo fiber [6] have unique advantage of low denity, high pecific propertie, biodegradable nature, and low cot; thu, compoite filled with natural fiber, uch a natural fiber reinforced polymer/cement compoite, are uually viewed a green and environmentally friendly compoite and have attracted much attention of reearcher for potential engineering application. A one of inherent material propertie of natural fiber, thermal property of natural fiber i of great importance in natural fiber reinforced compoite, due to inherent hollow microtructure of natural fiber. Recent reearche have hown that natural fiber coniting of celluloe or lumen can preent extremely lower thermal conduction than conventional fiber like gla fiber and carbon fiber [7]; thu, natural fiber reinforced compoite can be conidered to be thermal inulator in uch engineering a building and furniture. In addition, it i viable to achieve the aim of lightweight and proenvironment compoite material by conidering nature of hollow microtructure and biodegradability of natural fiber. In the pat few year, many reearcher invetigated thermal propertie of natural fiber and compoite filled with them. For example, El-Shekeil et al. experimentally invetigatedtheinfluenceoffibercontentonthemechanical and thermal propertie of kenaf fiber reinforced thermoplatic polyurethane compoite [8]. Liu et al. evaluated the tranvere thermal conductivity of Manila hemp fiber in olid region by the finite element method and analytical Haelman-Johnon model [2].Alo,they tudied the effect ofthemicrotructureofnaturalfiberonthetranverethermal conductivity of unidirectional compoite with abaca and bamboo fiber, by experiment and finite element imulation [9]. Behzad and Sain predicted the thermal conductivity for hemp fiber reinforced compoite by experimental meaurement [10], and ubequently they developed a finite element

2 2 Mathematical Problem in Engineering (a 50 μm Lumen Solid region (b Figure 1: (a Cro-ection morphology of the hemp fiber bundle [2] and (b chematic verion of the hemp fiber bundle. imulation procedure to predict the temperature profile and the curing behavior of the hemp fiber/thermoet compoite during the molding proce [11]. Mangal experimentally meaured the thermal propertie of pineapple leaf fiber reinforced compoite [12]. Takagi and cooperator analyzed thermalconductivityofpla-bamboofibercompoiteuing ahot-wiremethod[13]. All thee work mentioned above have been beneficial in undertanding of thermal tranfer mechanim in the natural fiber and deign of natural fiber filled compoite with deirable thermal propertie. A important filler of green compoite, it i neceary to etablih comprehenive undertanding of thermal propertie of natural fiber or fiber bundle. In thi paper, the emphai i put on the tudy of thermal propertie of the natural hemp fiberbundle.fromtheviewpointofcompoite,thenatural hemp fiber bundle can be viewed a compoite material with outtanding thermal propertie, becaue, in the natural fiber bundle of interet, there are a large number of lumen filled with air in tranvere direction of it (ee Figure 1(a for the cro-ection morphology of the hemp fiber bundle [2]. The thermal propertie of the natural fiber bundle vary coniderably depending on lumen volume and ize and alo the thermal property of olid region phae, which encloe the lumen in the natural fiber bundle. Figure 1(b how a chematic illutration of the natural fiber bundle coniting of lumen and olid region. It i oberved in Figure 1 that the large-cale fiber bundle i filled with many mall-cale lumen in the olid region; thu, thermal propertie of the lumen and the olid region are important parameter of the natural fiber bundle. In practice, the lumen filler i filled with air; thu, it thermal conductivity i uually pecified with very mall value, for example, W/(mK, which i normally the thermal conductivity of air meaured at the tandard atmophere. Therefore, the thermal conductivity of the fiber olid region more ignificantly affect the whole thermal performance of the fiber bundle than that of lumen. Here, the main purpoe of thi tudy i to invetigate the effect of material thermal property of the olid region phae on the equivalent aniotropic thermal property of the natural fiber bundle by finite element imulation [14, 15]of compoite microtructure [16] and then etablih an optional interrelationhip between them by curve fitting technique [17]togive a rapid and highly accurate prediction of material thermal propertie for both of them. 2. Finite Element Model for Aniotropic Natural Fiber Bundle Reinforced Compoite In thi paper, the natural fiber bundle i aumed to be embedded into a polymer matrix with contant thermal conductivity to form a quare repreentative volume element (RVE (or unit cell (ee Figure 2, a wa done by many reearcher in the analyi of heterogeneou material [18 20]. Then, the finite element model of natural fiber bundle reinforced compoite [11, 15, 16] will be etablihed to invetigate the influence of olid region phae on the fiber bundle. The etablihed unit cell conit of three different region, that i, matrix, olid region, and lumen. Each region ha iotropic thermal conductivity. Theoretically, the ditribution of lumen in the natural fiber bundle will caue aniotropy of the compoite under conideration. Therefore, in thi tudy, the aniotropic thermal conductivitie of the compoite will be invetigated. In teady-tate heat conduction problem, the temperature field within the aniotropic repreentative volume element atifie the quaiharmonic differential equation: x (k T x x + y (k T y =0, (1 y where the thermal conductivitie k x and k y are piecewie contant. Becaue the matrix, the olid region, and the lumen are aumed to be locally iotropic and homogeneou, k x = k y =k m in the matrix, k x =k y =k in the olid region and equal to k l in the lumen, repectively.

3 Mathematical Problem in Engineering 3 Inulator T 0 Inulator T 1 T 0 y x Inulator y x Inulator (a (b T 1 Figure 2: Schematic illutration of a quare cell embeded with the natural hemp fiber bundle. (a Boundary condition for k e x.(bboundary condition for k e y. Beide, in the heat conduction ytem, the heat flux component q x and q y are, repectively, defined by temperature gradient: q x = k x T x, q y = k y T y. (2 In the preent compoite computational model, the proper thermal boundary condition hould be applied along the boundary of the cell hown in Figure 2 to contruct a complete compoite heat tranfer ytem being olved by finite element technique [15, 21, 22], which ha been uccefully employed by many reearcher for the analyi of effective thermal propertie of unidirectional fiber compoite [23 25]. According to the work of Ilam and Pramila [24], the precribed temperature boundary condition T 1, T 0 on the vertical or horizontal boundarie of the cell can produce the mot accurate reult up to a relatively high fiber volume fraction, and the remaining boundarie of the cell are aumed to be inulating, a hown in Figure 2. It i aumed that T 1 >T 0 ; thu, the average heat flux component Q x and Q y loaded on the data collection face, for example, the left ide face for the cae (a and the bottom ide face for the cae (b, are poitive. Aareult,baedonFourier lawinheattranfer(ee(2, the effective thermal conductivitie of the compoite can be given a [24, 25] k e x = Q x L (T 1 T 0, ke y = Q y L (T 1 T 0, (3 where L i the ide length of the quare cell, Q x = 1 L L q x (0, y dy > 0 (4 0 for the cae in Figure 2(a,and Q y = 1 L L q y (x, 0 dx>0 (5 0 for the cae in Figure 2(b. The integral in (4and(5canbe evaluated by trapezoidal numerical integration. In the practical computation, the ide length of the quare cell i et to be 1, which i a normalized length. If the volume fraction of the fiber bundle to the cell i aumed to be a moderate value of 50%, the normalized radiu of the fiber bundle i Furthermore, if the volume fraction of lumen to the fiber bundle keep contant, that i, 30.87%, which i the experimental reult [2], the normalized radiu ofthelumencanbeevaluatedby /N, wheren i the number of lumen in the fiber bundle. For example, if thenumberoflumeni106,whichicloetotheactual dipere of lumen in the fiber bundle (ee Figure 1, then the normalized radiu of each lumen i Beide, the pecified temperature boundry condition along the two vertical edge of the unit cell are et to be 20 and 0, repectively. Moreover, the thermal conductivitie of matrix, olid region, and lumen are, repectively, normalized with the reference value W/(mK, which i the thermal conductivity of lumen, in the finite element procedure below. In the paper, the ymbol k m, k,andk l, repectively, indicate the thermal condctivitie of matrix, olid region, and lumen. 3. Numerical Reult and Dicuion 3.1. Convergence Invetigation. Generally, the finite element (FE olution will be more accurate a the model i ubdivided into maller element. The only ure way to know if we have ufficiently dene meh i to make everal model with different grid of element and check the convergence of the olution. In order to invetigate the convergence of the FE

4 4 Mathematical Problem in Engineering Approximated element ize Table 1: Summary of the numerical tet for the olution meh ize independence. Number of element Number of node Average heat flux component Q x Deviation (% Size Size Size Size Reference (a (b Figure 3: Finite element model of the compoite with natural fiber bundle including 106 lumen. (a Computational domain. (b Computational meh. olution, the compoite model in Figure 2(a i tudied by ABAQUS and the FE ize i changed from very coare to very fine. The element type employed in ABAQUS i DC2D8. In each of FE ize level, the average horizontal heat flux component Q x at the left wall of the quare unit cell i calculated. Table 1 give a ummary of the output of thee ize level indicating the number of element and node ued in the computational domain correponding to each element ize. In thi table, Size 1 tand for very coare element and Size 4 mean very fine element. The table alo indicate the deviation between the average heat flux component at theleftwallofthequareunitcellcalculateduingvariou element ize and that calculated uing the finet element ize of 4. The ummarized reult in Table 1 indicate that the maximum deviation between the olution uing the finet element of 4 that correpond to element and the coaret element of 1 that correpond to 2124 element i %. Thi reflect clearly that the numerical olution obtained via thi FE imulation i meh ize independence. Additionally, looking for high accuracy, the author decided to ue a fine element ize of 4 in the following computation Aniotropy Invetigation. It i known that the ditribution mode of lumen may caue aniotropy of both fiber bundle and compoite. To invetigate thi effect, let u conider the compoite model involving polymer matrix, olid region, and lumen, a diplayed in Figure 3(a, inwhich106lumen are regularly ditributed in the fiber bundle to approximate the real ditribution of lumen in the practical natural fiber Table 2: Aniotropic thermal conductivitie of the compoite for variou thermal conductivitie of the olid region phae. k /k l k e x /k l k e y /k l bundle (ee Figure 1. Figure 3(b preent the computational meh of element Size 4. Reult in Table 2 diplay the change of aniotropic thermal conductivitie of the compoite for variou thermal conductivitie of the olid region phae in the natural fiber bundle. It i obviou that the thermal conductivitie of the compoite along two direction are extremely imilar, o it i concluded that the approximated practical ditribution of lumen in the fiber bundle caue thecompoiteandfiberbundleiotropy.theimilarconcluion wa drawn by Liu et al., who predicted that the aniotropy of the compoite became maller with the number of lumen increaing [2]. Therefore, it i reaonable to aume theiotropyofthecompoiteandthefiberbundleinthe following analyi Effect of the Thermal Conductivity of the Solid Region in the Natural Fiber Bundle. To etimate the effect of the olid region on the compoite, it i aumed that the normalized

5 Mathematical Problem in Engineering 5 k e /k l FEM Quadratic polynomial Cubic polynomial k /k l Figure 4: Variation of the effective thermal conductivity of the compoite againt the olid region. thermal conductivity of the olid region change in the interval [1, 10]. By finite element computation, the ditribution of horizontal heat flux component and the correponding averagevalueofitcanfirtlybeobtainedonthedata-collection urface for each pecific value of the normalized thermal conductivity of the olid region. Then, the normalized effective thermal property of the compoite can be evaluated by (3. The variation of the effective thermal conductivity of the compoite i given in Figure 4,fromwhichitiobervedthat the imulated effective thermal conductivity of compoite increae with the increaing thermal conductivity of olid region, a we expect. Simultaneouly, it i found that the variation hown in Figure 4 how light nonlinearity, intead of linearity. Thu, to decribe the nonlinear variation hown in Figure 4, the following quadratic and cubic polynomial expreion from curve fitting technology are, repectively, employed. (i Quadratic polynomial fitting: k e = ( k ( k (6 k l (ii Cubic polynomial fitting: k e = ( k ( k 2 k l ( k k l (7 fiber with the ame ize. Alo, the ame thermal boundary conditionathoeuedinthecompoitemodelaboveare applied along the outer boundarie of the cell. To conduct the finite element analyi, a total of quadratic quadrilateral element of type DC2D8 with node are generated by ABAQUS to dicretize the computational domain (ee Figure 5(b. It i aumed that the normalized thermal conductivity of the olid fiber change in the interval[1, 10]; thu, the effective thermalconductivityofthecompoitecanbeevaluatedby finite element imulation for any pecific value of thermal conductivity of the equivalent olid fiber. The variation of the effective thermal conductivity of the compoite againt the equivalent olid fiber i diplayed in Figure 6, which clearly howthattheimulatedeffectivethermalconductivityof the compoite nonlinearly increae with the increaing value of the thermal conductivity of the equivalent olid fiber. To accurately capture the nonlinear variation hown in Figure 6, the following quadratic and cubic polynomial curve are, repectively, employed by mean of curve fitting technology. (i Quadratic polynomial fitting: k e = ( k ( k (8 k l (ii Cubic polynomial fitting: k e = ( k ( k 2 k l ( k k l Optional Interrelationhip between the Thermal Conductivity of the Solid Region and the Fiber Bundle. Finally, the equivalence of the two compoite model, repectively, hown in Figure 3(a and 5(a require that the two compoite model hould have ame effective thermal conductivitie. Therefore, combining (6 (9, we have an optional interrelationhip between the thermal conductivity of the olid region and the fiber bundle; that i, ( k ( k = ( k ( k or ( k ( k ( k k l (9 ( Effect of the Thermal Conductivity of the Homogenized Fiber Bundle. In thi ection, the compoite model hown in Figure 5(a i taken into conideration to invetigate the effect of the homogenized fiber bundle on the compoite. In the model, the homogenized fiber bundle i repreented by a olid = ( k ( k ( k k l , (11

6 6 Mathematical Problem in Engineering (a (b Figure 5: Unit quare cell embeded with a olid fiber to repreent the homogenized fiber bundle. (a Computational domain. (b Computational meh. k e /k l k /k l FEM Quadratic polynomial Cubic polynomial Figure 6: Variation of the effective thermal conductivity of the compoite againt the homogenized fiber bundle. k /k l k /k l Haelman-Johnon model Reult of cubic fitting curve Reult of quadratic fitting curve Figure 7: Approximated relation of thermal conductivitie of the natural fiber bundle againt the fiber olid region. from which the variation of k in term of k i plotted in Figure 7. To verify the obtained relation of thermal conductivity betweentheolidregionandfiberbundle,thetheoretical Haelman-Johnon model derived from the interface interaction between the circular matrix and circular incluion embedded in the matrix [2] i taken a reference for the purpoe of comparion. Here, an analytical expreion from the Haelman-Johnon modelcanbeemployedtoinvetigate the thermal conductivity of the olid region k with repect to that of the fiber bundle k ;thati, k = ( k 1 (1 + V l 2(1 V l + (1 + V l 2 2 4(1 V l 2 (k 1 + k, (12 where V l repreent the volume fraction of lumen to the fiber bundle.

7 Mathematical Problem in Engineering 7 According to the experiment data in [2], the practical volume content of the lumen to the fiber bundle i about 30.87%; thu, the ubtitution of V l = 30.87%into(12 yield k = ( k ( k k (13 from which one can get the variational curve of k in term of k,ahowninfigure 7 forthepurpoeofcomparion. Specially, if k i taken to be W/(mK [2, 10], the thermal conductivity of the olid region k i calculated by (13 a W/(mK. In Figure 7, it i oberved that there i good agreement between the numerical reult from either quadratic or cubic curve and the theoretical reult of Haelman-Johnon model for the cae of moderate change of k. For example, for the cae of k = W/(mK, the thermal conductivity of the olid region k i calculated a W/(mK for the quadratic approximation and W/(mK for the cubic approximation, which ha relative derivation of 0.21% and 0.54% of the theoretical olution W/(mK, repectively. Therefore, both quadratic and cubic relation of thermal conductivity of the olid region and the fiber bundle can be ued to evaluate thermal propertie of the natural fiber bundle or the olid region in the bundle. Alo, in contrat to the analytical expreion (13, it can be een from (10and(11 that either k or k i given, and one can eaily determine another. Thi mean that we can do invere prediction conveniently by a pecified material thermal conductivity of the olid region. Thi i an advantage of the optional relation preented in the paper over the analytical olution. Beide, it i obviou in Figure 7 that the exitence of lumen ignificantly weaken the capacity of heat tranmiion in the fiber bundle. A a reult, the thermal conductivity of the fiber bundle k i greatly le than that of k. 4. Concluion In thi paper, 2D computational compoite model of the natural fiber bundle including number of lumen i developed. Due to the geometrical limitation of the fiber bundle, it i not convenient to directly apply thermal boundary condition to it to perform finite element analyi of compoite. To treat thi,thefiberbundleiaumedtobeembeddedintoa matrix with known thermal conductivity to contruct unit compoite cell, which i numerically analyzed by applying proper thermal boundary condition along the cell boundary. By mean of the developed finite element computational compoite model, the effect of the olid region in the bundle on the overall thermal property of the compoite i tudied. Simultaneouly, a homogenized compoite model i contructed, in which the fiber bundle i replaced by an equivalent olid fiber to invetigate the influence of the homogenized fiber bundle. By comparing the two compoite model developed in thi tudy, an optional interrelationhip between thermal conductivitie of the olid region and the homogenized fiber bundle wa obtained by curve fitting technique. Finally, the preent computational compoite model i verified and numerical experiment how that either quadratic or cubic prediction can produce almot imilar reult for the olid region in the fiber bundle, in contrat to the theoretical Haelman-Johnon model and other numerical reult. Moreover, the direct or invere prediction can be eaily performed to evaluate the thermal conductivity ofthefiberbundleortheolidregioninthepractice,ifoneof them i given. More importantly, the preent computational method can be eaily extended for the prediction of thermal property of other natural fiber bundle with variou lumen pattern. Conflict of Interet The author declare no conflict of interet regarding the publication of thi paper. Reference [1]H.Akil,M.Omar,A.Mazuki,S.Safiee,Z.A.M.Ihak,and A. Abu Bakar, Kenaf fiber reinforced compoite: a review, Material and Deign, vol. 32, no. 8-9, pp , [2] K. Liu, H. Takagi, and Z. 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