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Abstract

In this article, a three-dimensional (3D) finite elements method has been developed for predicting the effective thermal conductivity (ETC) of a conductive hollow tube polymer composite. 3D Representative Volume Element (3D-RVE) was used to represent the composite material. Governing heat transfer equations in both transverse and longitudinal directions for predicting the effective thermal conductivities of composites are used. ETC was numerically calculated using COMSOL^TM software. The guarded hot plate method was used to measure the composites conductivities consisting of epoxy resin matrix filled with metallic hollow tube. A comparison between the numerically calculated thermal conductivities, measured and analytical ones for various samples was made. A satisfactory agreement between numerical and experiment takes place.

Keywords

Composite materials hollow tube thermal properties finite elements method

Introduction

Various approaches for characterizing the effective thermal, electrical, and elastic properties of composite media have been introduced since Maxwell’s seminal work on spherical particle suspensions more than a century ago.¹ Apart from the large number of results, which are based on either highly approximated micro-models or ad hoc mixing rules, several approaches based on reasonable physical models have also been established and refined to include the effects of various particle shapes and of an imperfect thermal contact between the particle surface and the continuous medium. Such studies include the exact mathematical analysis of dilute systems, in which bounds are established on the effective properties, and asymptotic behavior is determined (Beasley and Torquato² and Thovert and Acrivos³); exact mathematical analysis of various regular arrays of particles (McKenzie et al.,⁴ Gu and Tao,⁵ Gu and Liu,⁶ Torquato and Rintoul,⁷ and Cheng and Torquato⁸); numerical simulation of Brownian diffusion of pulsed sources through particle arrays (Torquato and Kim,⁹ and Kim and Torquato¹⁰); the self-consistent field concept for analyzing non-dilute, non-regularly spaced particles (Hashin,¹¹ Benveniste and Miloh,¹² and Benveniste¹³); and the technique of successive embedding of effective media to treat multicoated cylinders or spheres (Schulgasser,¹⁴ and Milgrom and Shtrikman¹⁵).

Numerous numerical studies of thermal conductivity of filled polymer were also conducted in the past. Thus, Deissler’s works¹⁶ were extended by Wakao and Kato¹⁷ for a cubic or orthorhombic array of uniform spheres in contact. Shonnard and Whitaker¹⁸ have investigated the influence of contacts on two-dimensional (2D) models. They have developed a global equation with an integral method for heat transfer in the medium. Auriault and Ene¹⁹ have investigated the influence of the interfacial thermal barrier on the effective conductivity and on the structure of the macroscopic heat transfer equations. Using the finite elements method (FEM), Veyret et al.²⁰ studied the heat conductive transfer in the periodic distribution of the filler in the composite materials. In their study, calculation was carried out on 2D and 3D geometric spaces. The same method was used by Ramani and Vaidyanathan²¹, who incorporated the effect of microstructural characteristics such as filler aspect ratio, interfacial thermal resistance, volume fraction, and filler dispersion to determine the effective thermal conductivity (ETC) of a composite with spherical and parallelepiped fillers. The thermal conductivity increased from 0.32 W m⁻¹ K⁻¹ for pure PA6 to 2.09 W m⁻¹ K⁻¹ by adding spherical copper powder filler with a 50% volume fraction. Terada et al.²² have generated an FEM by identifying each pixel with a finite element and accompanying appropriate image processing. Bakker²³ has calculated the thermal conductivity coefficient of porous materials by using FEM and has looked for the relationship between 2D and 3D values. A numerical approach to calculate the ETC of granular-reinforced composite was proposed by Cruz.²⁴ Many other contributing works were attributed to Yin et al.,²⁵ Kumlutas et al.,²⁶ Jiang et al.,²⁷ Wei S et al.,²⁸ Aadmi et al.,²⁹ and Bohayra M et al.³⁰ Recently, ANSYS software was used by Liang³¹ to perform the numerical simulation of the heat-transfer process in hollow–glass–bead (HGB)-filled polymer composites. The effects of the content and size of the HGB on the ETC was identified. The ETC of the polypropylene (PP)/HGB composites was estimated at temperatures varying from 25 to 30°C. Lattice Monte Carlo (LMC) and finite element analyses were used on the ETC of sintered metallic hollow spheres structures by Fiedler et al..³² In their work, the LMC calculation strategy is enhanced in order to incorporate temperature dependence of thermal conductivity and specific heat in transient thermal analyses.³³ Leclerc³⁴ describes an efficient numerical model to better understand the influence of the microstructure on the thermal conductivity of heterogeneous media. This is the extension of an approach recently proposed for simulating and evaluating effective thermal conductivities of alumina/Al composites. A C++ code called MultiCAMG, taking into account all steps of the proposed approach, has been implemented in order to satisfy requirements of efficiency, optimization, and code unification. Ji³⁵ developed a new random model to simulate the microstructures and the effective thermal conductivities of the insulation materials based on the assumption of either two phases or multiphase. To support the validity of the proposed approach, comparisons with experiments were made. Karkri et al.³⁶ conducted a 3D numerical finite element study on the ETC of composites filled with spherical particles, where the effect of a thermal contact resistance and of the relative resistivity of the filler and geometrical parameter was investigated. A deeper understanding of thermal transport in composite materials requires simultaneous modeling and experimental approaches by considering the influence of various parameters (interaction between the components, filler distribution, and geometry).

Experimental techniques for the study of heat conduction and ETC in composites materials are available.³⁷ Mirmira³⁸ has measured the thermal conductivity of aluminum (Al)-filled high-density polyethylene using a modified hot-wire technique. The ETC values obtained from the experimental study are compared with several thermal conductivity models. As seen from this study, Russell’s³⁹ and Cheng and Vachon’s⁴⁰ models predict fairly well thermal conductivity values up to 10% by volume of Al particles, whereas beyond 10% of particle content, all models underestimate the thermal conductivity of the composite. Weidenfeller et al.⁴¹ have measured thermal diffusivities, specific heat capacities, and densities of particle-filled polypropylene and thermal conductivities were derived. Thermal conductivity of the polypropylene is increased from 0.27 up to 2.5 W m⁻¹ K⁻¹ with 30 vol% of talc in the polypropylene matrix. In addition for a filler content of 44 vol.% of magnetite, the ETC increases from 0.27 to 0.93 W m⁻¹ K⁻¹. A simultaneous characterization of thermal conductivity and diffusivity (α) of high-density polyethylene composite was investigated by Trigui et al.⁴² The effects of the filler size and volume fraction were studied and compared with several cases.^{34

–46}

This article is a continuation of research published,^36,47 where a model was developed for describing thermal conductivity of composites with conductive particle reinforcement. The scope of this study is to numerically and experimentally calculate ETC of hollow metallic tube composite. One has to consider a contact resistance between the outer surface of the tube and the continuous medium in which it is embedded (epoxy resin) and the geometrical parameters of the tube. The ETC was calculated using the COMSOL^TM version 4.2a software. The obtained values were compared with experimental results and some existing theoretical and semi-empirical models.

Prediction methods of ETC

Mathematical modeling

A Representative Volume Element (REV) based on finite element software COMSOL^TM version 4.2a thermal analysis is developed to predict the ETC of hollow tube composites. About the geometry, we considered the parallelipipedic unit cells corresponding to the hollow tube composites when inclusions of conductivity (λ _f) of outer diameter d _ex = 2r _ex and of length L are embedded in a matrix of conductivity λ _m. The REV of such composite is presented in Figure 1. The heat transfer in the elementary cell is governed by the stationary heat transfer equations. At the interphase, the temperature potential jumps across the interface. The associated normal component of the heat flux is continuous and is proportional to the temperature potential jump. The boundary conditions at the edges of the elementary cell are of adiabatic type except at the upper and lower faces where temperature is prescribed with σ and τ, the filler and the matrix temperatures, respectively, and r _c is the thermal contact resistance. According to the symmetries, only one-fourteenth of the original parallelipipedic cell needs to be meshed (Figure 2). For comparison purpose, the same boundary conditions and methodology are preserved for all the basic cells. The mathematical equations representing the heat transfer model are given by the equations (1 to 8).

Figure 1.

Elementary cell model of composite: perpendicular model.

Figure 2.

Mesh of the composite elementary cell, F = 1. F: dimensionless wall thickness of the hollow conductive tube.

Matrix		Tube
$d i v (λ_{m} \vec{g r a d (} τ)) = 0$	(1)	$d i v (λ_{f} \vec{g r a d (} σ)) = 0$	(5)
$- λ_{m} \frac{\partial τ}{\partial n} = 0, l a t e r a l f a c e s$	(2)	$- λ_{f} \frac{\partial σ}{\partial n} = (τ - σ) / r_{c}, m a t r i x \cap t u b e$	(6)
$λ_{m} \frac{\partial τ}{\partial n} = (σ - τ) / r_{c}, m a t r i x \cap t u b e$ ,	(3)	$- λ_{f} \frac{\partial σ}{\partial n} = 0$ , inner wall of the tube	(7)
For perpendicular model $\begin{aligned} τ = τ_{1}, z = + a a n d τ = τ_{2}, z = - a \\ {τ = (τ_{1} + τ_{2}) / 2\|}_{z = 0} \end{aligned}$	(4)	For parallel model: $\begin{aligned} τ = τ_{1}, y = + b a n d τ = τ_{2}, y = - b \\ τ = {(τ_{1} + τ_{2}) / 2\|}_{y = 0} \end{aligned}$	(8)

Where n is the normal unit vector pointing from the tube to the matrix, a is the half larger elementary cell (m), b is the half height of elementary cell (m), r _c is the contact resistance, cube radius (m), τ ₁ is the upper cell face temperature (K), and τ ₂ is the lower cell face temperature (K).

To simplify the problem and to decrease the computing time, dimensionless parameters, and variables were used:

X = x / r_{e x}, Y = y / r_{e x} a n d Z = z / r_{e x} : t h e d i m e n s i o n l e s s s p a c e v a r i a b l e s .

η = L / d_{e x} : t h e r e d u c e d r a t i o b e t w e e n l e n g t h a n d d i a m e t e r o f t h e t u b e .

\begin{aligned} S = (2 σ - τ_{1} - τ_{2}) / (τ_{1} - τ_{2}), T = (2 τ - τ_{1} - τ_{2}) / (τ_{1} - τ_{2}) : \\ t h e u n k n o w n i n n e r a n d o u t e r t e m p e r a t u r e s f i e l d . \end{aligned}

\begin{aligned} B_{x, z} = (2 a - 2 r_{e x}) / 2 r_{e x} a n d B_{y} = (2 b - L) / 2 r_{e x} : \\ t h e r e d u c e d r e s i s t a n c e o f t h e m a t r i x l a y e r b e t w e e n n e a r e s t t u b e . \end{aligned}

D = λ_{m} / λ_{f} : t h e c o n d u c t i v i t y r a t i o b e t w e e n t h e t w o p h a s e s .

C = r_{c} λ_{m} / r_{e x} : t h e r e d u c e d c o n t a c t r e s i s t a n c e l o c a t e d a t t h e t u b e i n t e r f a c e .

E_{⊥} = λ_{e f f ⊥} / λ_{m} : t h e p e r p e n d u c l a r E T C .

E_{/ /} = λ_{e f f / /} / λ_{m} : t h e p a r a l l e l E T C .

F = 1 - r_{i n} / r_{e x} : r e l a t i v e w a l l t h i c k n e s s o f t h e h o l l o w c o n d u c t i v e t u b e .

The ETC $E_{/ /, ⊥}$ is calculated versus the relative thermal contact resistance between hollow tube and matrix (C), the wall thickness F, the half distance between the tube divided by the tube radius B_x,y,z , the filler volume fraction ϕ, and the ratio of thermal conductivity between the two phases D. To obtain high accuracy for the ETC computation with each model, the refinement mesh around small geometrical features and on the upper face (Z = B_z + 1 for the perpendicular model and Y = B_y + η for the parallel one) was considered (Figure 2). In light of a previous work,³¹ the ETC for each model is calculated versus the heat flux crossing the elementary cell.

Perpendicular ETC (Figure 1):

The heat flux crossing the elementary cell is defined by:

\frac{λ_{e f f_{⊥}} s_{1} Δ τ}{a} = - λ_{m} \int \int_{S_{1}} \frac{d τ}{d z} d y d x

\frac{a \times b λ_{{e f f}_{⊥}} Δ τ}{a} = - λ_{m} \int_{0}^{b} \int_{0}^{a} \frac{d τ}{d z} d y d x

Introducing the lower cell face temperature and the dimensionless parameters, equation (10) can be re-written:

λ_{{e f f}_{⊥}} b (\frac{τ_{1} + τ_{2}}{2} - τ_{1}) = - \frac{λ_{m}}{2 r_{e x}} \int_{0}^{B_{y} + η} \int_{0}^{B_{x} + 1} \frac{d [(τ_{1} - τ_{2}) T + (τ_{1} + τ_{2})]}{d Z} r_{e x}^{2} d Y d X

b (\frac{τ_{2} - τ_{1}}{2}) \frac{λ_{e f f ⊥}}{λ_{m}} = - \frac{(τ_{1} - τ_{2}) r_{e x}}{2} \int_{0}^{B_{y} + η} \int_{0}^{B_{x} + 1} \frac{d T}{d Z} d Y d X

\frac{λ_{e f f ⊥}}{λ_{m}} = \frac{r_{e x}}{b} \int_{0}^{B_{y} + η} ({\int_{0}^{X = Y} \frac{d T}{d Z} d X|}_{Z = B_{z} + 1}) d Y

Thus we obtain the perpendicular ETC:

E_{⊥} = \frac{1}{(B_{y} + η)} \int_{0}^{B_{y} + η} ({\int_{0}^{X = Y} \frac{d T}{d Z} d X|}_{Z = B_{z} + 1}) d Y

The heat flux (Q) in the Z-direction over the upper face (S ₁) of the elementary cell is defined by:

Q_{⊥} = \int_{0}^{B_{y} + η} ({\int_{0}^{X = Y} \frac{d T}{d Z} d X|}_{Z = B_{z} + 1}) d Y

The perpendicular ETC and the filler volume fraction of the perpendicular model are given by:

E_{⊥} = Q_{⊥} / (η + B_{y}) a n d ϕ = π \times η [1 - (1 - F)^{2}] / 4 {(1 + B_{x, z})}^{2} (η + B_{y})

Parallel ETC.

In the same procedure, we calculate the parallel ETC:

E_{/ /} = Q_{/ /} {(η + B_{y}) / (1 + B_{x, z})}^{2} w i t h Q_{/ /} = \int_{0}^{B_{z} + 1} ({\int_{0}^{X = Z} \frac{d T}{d Y} d X|}_{Y = B_{y} + η}) d Z

Analytical models

Many analytical models were proposed in the past to predict thermal conductivity of composite materials. However, only few models take into account the fillers size and the thermal contact resistance between fillers and matrix. Hasselman and Johnson⁴⁸ have proposed a modification of the Hashin and Shtrikman model^49,50 to take into account these parameters and applied this model to the study of dispersions of Nickel particles into a sodium borosilicate matrix. According to this model, also used by Benveniste,¹² the ETC λ _eff of composites materials is defined by:

λ_{e f f} = λ_{m} \frac{1 + (d - 1) ϕ \overset{ˉ}{β}}{1 - ϕ \overset{ˉ}{β}}

with

\overset{ˉ}{β} = [\frac{λ_{f}}{λ_{m}} - (1 + \frac{λ_{f}}{r h_{c}})] / [\frac{λ_{f}}{λ_{m}} + (d - 1) (1 + \frac{λ_{f}}{r h_{c}})]

where ϕ is the filler volume fraction, λ _f is the filler thermal conductivity, λ _m is the matrix thermal conductivity, d is a parameter depending on the particle shape (d = 3 for spheres and d = 2 for cylinders), r is the radius of particles, and h _c is the contact conductance between the particle and the matrix and is defined by $h_{c} = 1 / r_{c}$ . Equations (18) and (19) can be re-written to introduce non-dimensional parameters E, C, and D previously defined and used for numerical simulations. Thus, we obtain:

E = \frac{1 + (d - 1) ϕ \overset{ˉ}{β}}{1 - ϕ \overset{ˉ}{β}}

with

\overset{ˉ}{β} = [\frac{1}{D} - (1 + \frac{C}{D})] / [\frac{1}{D} + (d - 1) (1 + \frac{C}{D})]

The model of Lewis & Nielsen is a model originally used for the prediction of mechanical properties of composites.⁵¹ This model was also applied to the prediction of the ETC of composites⁵¹ by taking into account different parameters for the prediction of ETC such as the shape, distribution and orientation of fillers into the matrix and also the thermal conductivity values of filler and matrix. The main interest for the use of such a model is that predictions also consider the value of the maximum packing fraction of fillers $ϕ_{max}$ . This model is defined by¹⁷:

λ_{e f f} = λ_{m} \frac{1 + A_{L N} B_{L N} ϕ}{1 - B_{L N} ψ ϕ}

with

\{\begin{matrix} B_{L N} = (\frac{λ_{f}}{λ_{m}} - 1) / \frac{λ_{f}}{λ_{m}} + A_{L N} \\ ψ = 1 + \frac{(1 - ϕ_{max}) ϕ}{ϕ_{max}^{2}} \end{matrix}

where $ϕ_{max}$ is the maximum packing fraction of fillers. The shape, distribution, and orientation allow defining the value of the parameter A _LN; in the case of fillers with spherical shape, $A_{L N} = 1.5$ , for parallel and perpendicular filler to heat flux $A_{L N} = 2 η$ and $A_{L N} = 0.5$ , respectively.⁵⁰ Others symbols in equations (22) and (23) have the same significance as defined previously. Equations (22) and (23) can be rewritten to introduce nondimensional parameters E and D previously defined and used for numerical simulations; in the case of the Lewis & Nielsen model, parameter C is not taken into account in the predictions. Thus, we obtain:

E = \frac{1 + A_{L N} B_{L N} ϕ}{1 - B_{L N} ψ ϕ}

with:

\{\begin{matrix} B_{L N} = (\frac{1}{D} - 1) / (\frac{1}{D} + A_{L N}) \\ ψ = 1 + (1 - φ_{max}) ϕ / φ_{max}^{2} \end{matrix}

Unlike the Maxwell formula, the model proposed by Bruggeman takes care of the shape factor of the second-phase inhomogeneities.⁵² Moreover, it has been argued that this model can be used to estimate the ETC with both low- and high-volume fraction of particles.

1 - ϕ = [\frac{λ_{f} - λ_{e f f}}{λ_{f} - λ_{m}}] {[\frac{λ_{m}}{λ_{e f f}}]}^{\frac{1}{d}}

d = 2 for cylinders oriented perpendicular to heat flow and $d = (1 + β) / β$ for oriented ellipsoids inclusion, where: $β = R \times [(1 / x) / ((1 / y) + (1 / z))]$ and x is the ellipsoidal axis parallel to heat flow, y and z are the other two ellipsoid axes, and R is a surface roughness factor (approximately 1).

For spherical inclusion, d = 3, and the ETC is defined by the following equation:

E = \frac{λ_{e f f}}{λ_{m}} = {[u - \frac{(1 - ϕ) (\frac{λ_{f}}{λ_{m}} - 1)}{3 u}]}^{3}, u = {[\frac{λ_{e f f}}{2 λ_{m}} + \sqrt{\frac{λ_{e f f}^{2}}{4 λ_{m}^{2}} + \frac{{(1 - ϕ)}^{3} {(\frac{λ_{f}}{λ_{m}} - 1)}^{3}}{27}}]}^{1 / 3},

The Hatta and Taya model⁵³ describes an ETC $λ_{e f f}$ ; this model takes into account the shape and interactions between particles. In the general case, Hatta and Taya have proposed the following equations (28) and (29).

\frac{λ_{e f f}}{λ_{m}} = [1 + φ_{f} \frac{(λ_{f} - λ_{m}) (2 S_{33} + S_{11}) + 3 λ_{m}}{J}]

where:

J = 3 (1 - φ_{f}) (λ_{f} - λ_{m}) S_{11} S_{33} + λ_{m} [3 (S_{11} + S_{33}) - φ_{f} (2 S_{11} + S_{33}) + 3 \frac{λ_{m}^{2}}{λ_{f} - λ_{m}}]

$S_{i i}$ is a factor related to the geometry of the inclusions.⁵³ For spherical inclusion, we have $S_{11} = S_{22} = S_{33} = 1 / 3$ , and for fiber, rod, or cylinder, we have $S_{11} = S_{22} = 1 / 2$ , $S_{33} = 0$ .

Experimental study

Sample preparation (epoxy resin/metallic hollow tube)

In our experimental setup, the matrix material is an epoxy resin of Vantico AG (Switzerland). The Araldite^® LY5052 is mixed to 42% weight of Aradur 5052. This resin with a low viscosity (0.8 Pa s at 23°C) is then placed under vacuum for about 50 min to remove the air bubbles before injection. The metallic hollow tubes (Goodfellow, UK) are placed in a mold (200 × 200 mm²) with an equidistant distribution of 1 mm between the tubes (Figure 3 (a) to (c)). To facilitate the release of the resin, a Teflon film polytetrafluoroethylene (PTFE) was stacked on the mold surface. Once the mold was closed and sealed, the resin is introduced from the injection gate at ambient temperature and under constant pressure. Besides, the sample is pressurized in the mold, and the injection of an excess of material will be introduced to compensate the shrinkage. After reticulation of the injected resin, the mold and PTFE film were removed. Three configurations of samples were prepared under the same manufacturing conditions (S1, S2, and S3). The first, resin/hollow copper tube (99.9% copper (Cu)) with internal and external diameter of 3.2 and 4.02 mm, respectively. The second resin/hollow brass tube (70% Cu, 30% zinc) with 2.74 and 3.56 mm of internal and external diameter, respectively. The third one is resin/hollow Al tube composite (99.5% Al) with 1.62 and 3.02 mm of internal and external diameter, respectively (Figure 3). Thermophysical properties of the tubes are presented in Table 1.

Figure 3.

Manufacturing of epoxy resin/metallic hollow tubes composites. (a and b) Positions of metallic tubes into a mold and (c) process of injection. (S1) epoxy resin + copper tubes, (S2) epoxy resin + brass tubes, and (S3) epoxy resin + aluminum tubes.

Table 1.

Sample parameter measurements.

Samples	Inner and outer diameters (mm)	Thickness e (mm)	C _p (kJ kg⁻¹ C⁻¹)	α _m (×10⁻⁷ m² s⁻¹)	λ _eff (W m⁻¹ K⁻¹)	ρ (kg m⁻³)
Sample (S1)	–	6.26	1.1	3.07	0.601 ± 0.06	1780 ± 74
Sample (S2)	–	5.91	0.9	2.94	0.466 ± 0.022	1758 ± 149
Sample (S3)	–	5.58	0.87	3.31	0.393 ± 0.028	1363 ± 84
Matrix epoxy	–	3.45	1.4	1.237	0.191 ± 0.004	1034 ± 28
Copper tube	$d_{i n} = 3.2$ , $d_{e x} = 4.02$		0.385	11.7	401	8933
Brass tube	$d_{i n} = 2.74$ , $d_{e x} = 3.56$		0.380	3.39	124	8530
Aluminum tube	$d_{i n} = 1.62$ , $d_{e x} = 3.02$		0.903	9.71	237	2702

Thermophysical properties measurements

For large size samples, testing using a noninvasive method for thermophysical properties determination is necessary. The proposed test bench for the parallelepiped-shape of composite provides temperature and heat flux measurements at the material borders (see Figure 4). The sample is located between two horizontal heat exchanger plates of Al connected to thermoregulated baths that allow for the fine regulation of the temperature of the injected oil H10 with a precision of approximately 0.1°C. Heat flux sensors and thermocouples (types T and K) are placed on each side of the composite sample to measure the heat flux $(ϕ_{1, 2})$ and temperature (T _1,2), respectively, on each face of the sample. The thickness of flux-meters is about 0.2 mm, and their sensitivity is about 202 μV W⁻¹ m⁻² for a sensor having an active surface area of 400 cm². The lateral sides of the samples are insulated by polyethylene-expanded (PE) foam, which creates an insulating ring around the sample and minimizes the heat transfer to the external environment. Thus, it reduces multidimensional heat transfer problem successfully to a one-dimensional problem. All sensors are connected to an acquisition device that is controlled using a LabVIEW program adapted to measure temperature fluctuations and heat flux exchanged during melting processes. Experimental data are recorded at regular and adjustable time steps (6 s).

Figure 4.

Experimental setup.

Results and discussion

Effect of filler volume fraction, thermal resistance, and D(F = 1)

In our previous study, the ETC for three elementary cells, such as simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC), were numerically and experimentally investigated.³⁶ The focus of this section is on two-phase composites material filled with solid tube (F = 1). In Figures 5 and 6, the effect of the ratio of thermal conductivities of filler to matrix material and the Kapitza resistance of the contact inclusion/matrix was investigated for perpendicular and parallel tube composites. Computation of about 150E values has showed that a decrease in the contact resistance r _c or of the inner resistance $λ_{m} / λ_{f}$ leads to an increase in the ETC. It is interesting to note that the perpendiuclar ETC $E_{⊥}$ is lower than the parallel one $E_{/ /}$ . The obtained parallel and perpendicular effective conductivities lie in the range of [4.72, 33.61] and [2.03, 3.17], respectively, covering, therefore, most of the eligible $E_{/ /, ⊥}$ range. In Figures 7 and 8, the effect of concentration ratio and thermal contact resistance r _c of different filler models are illustrated at a constant conductivity ratio of $λ_{m} / λ_{f} = 10^{- 3}$ . One can easily conclude from these figures that at low values of the thermal contact resistance (10⁻ ⁵ to 10⁻ ³), the volume fraction ϕ has a predominant effect on the heat transfer between the matrix and the fillers and increases leads to higher values of ETC. Figure 9 shows the dependence of the thermal conductivity on the reduced resistance of the matrix layer between nearest tubes (B) and thermal contact resistance at constant contrast D = 10⁻³. We could observe that the variations in the ETC become almost negligible, showing an asymptotic behavior. For thermal contact resistance values higher than C = 10⁻³, the contact resistance has a dominant influence on the ETC. This influence is more important when the volume fraction increases.

Figure 5.

Perpendicular effective thermal conductivity. $ϕ = 50 %$ and $B = 0.2386$ , $F = 1$ . ϕ: filler volume fraction; B: dimensionless outer resistance; F: dimensionless wall thickness of the hollow conductive tube.

Figure 6.

Parallel effective thermal conductivity, $ϕ = 50 %$ and $B = 0.2386$ , $F = 1$ . ϕ: filler volume fraction; B: dimensionless outer resistance; F: dimensionless wall thickness of the hollow conductive tube.

Figure 7.

Perpendicular effective thermal conductivity, $D = 10^{- 3}$ , $F = 1$ . D: dimensionless inner resistance; F: dimensionless wall thickness of the hollow conductive tube.

Figure 8.

Parallel effective thermal conductivity, $D = 10^{- 3}$ , $F = 1$ . D: dimensionless inner resistance; F: dimensionless wall thickness of the hollow conductive tube.

Figure 9.

Perpendicular effective thermal conductivity, F = 1. F: dimensionless wall thickness of the hollow conductive tube.

Effect of the wall thickness of hollow tube, $F \in [0.05 - 1]$

In this section, we will present numerical results obtained by using the proposed mathematical model to examine the contribution of the various factors affecting heat transfer in hollow tube/polymer composites and their effect on the overall effective thermal properties. First, we consider the effect of tube wall thickness on the effective conductivity at constant thermal contact resistance C and inner resistance D. Figure 10(a) and (b) show the numerical results for D = 10⁻³ and C = 10⁻⁵ for perpendicular and parallel models, respectively. The wall thickness varies from 0.05 to 1. Figure 10(a) shows the evolution of the $E_{⊥}$ for different values of B and with a constant inner and thermal resistance D = 10⁻³ and C = 10⁻⁵. We can observe that the lowest and the highest limits of ETC $E_{⊥}$ are, respectively, 1.09 and 4.47 at B = 0.136. Note that as F > 0.2, it appears that the computed perpendicular thermal conductivities increase very slightly and tend to be at a constant value. It is clearly shown in Figure 10(b) that, for a given volume fraction (B = cst), the ETC increases with increasing wall thickness. The highest obtained ETCs lie in the range of [17.02, 54.7] at B = 0.136, covering, therefore, the $E_{/ /}$ range. The examination of these results shows that for a constant value of B, the parameter F has a predominant effect on the heat transfer between the matrix and the filler and its increase leads to higher values of the ETC. We can observe, for $B \in [0.136 - 0.573]$ , the wall thickness has a nonlinear evolution effect on the ETC, despite when $B \in [0.89 - 2.53]$ , the quasilinear evolution of parallel effective conductivity was observed.

Figure 10.

Effective thermal conductivities $E_{/ /}$ and $E_{⊥}$ versus wall thickness F $(B = 0.136)$ . (a and b) $D = 10^{- 3}$ , $C = 10^{- 5}$ ; (c and d) $D = 10^{- 3}$ ; (e and f) $C = 10^{- 5}$ . $E_{/ /}$ : dimensionless parallel effective conductivity; $E_{⊥}$ : dimensionless perpendicular effective conductivity; F: dimensionless wall thickness of the hollow conductive tube; B: dimensionless outer resistance; D: dimensionless inner resistance; C: dimensionless contact resistance.

Second, we examine the influence of the reduced contact resistance on the effective conductivity. The variation of the effective conductivity with the increase in C is shown in Figure 10 (c) and (d) for perpendicular and parallel models, respectively. However, it can be seen that the effective conductivity increases with the decrease in the interfacial contact resistance C. At low values of the thermal contact resistance $C \in [10^{- 15}, 10^{- 3}]$ , we observe the same effect on the heat transfer between the matrix and the filler, and its increase leads to higher values of both perpendicular and parallel ETCs. In addition, the variations of ETC become almost negligible, showing asymptotic behavior (for $E_{⊥}$ , it is observed for $F > 0.2$ , and for parallel model, it is observed when $F > 0.8$ ). This corresponds to the case of a perfect thermal contact between the tube and the matrix. For $C \in [10^{- 2}, 10^{- 1}]$ , the contact resistance has a dominant influence on the ETC. At given B = 0.136 and F = 0.6, we have $E_{/ /} \in [52.76 - 33.26]$ and $E_{⊥} \in [4.47 - 3.05]$ (e.g., −36% reduction and −32% from $C \in [10^{- 2}, 10^{- 1}]$ , respectively). It can be inferred that the interfacial resistance significantly affect the effective conductivity in parallel and perpendicular models and depend on the wall thickness of the tube. This is a very important result. Finally, using the same numerical results as before, the ETC of perpendicular and parallel models with varying conductivity ratio between the two phases is plotted in Figure 10 (e) and (f). Note that as $D = λ_{m} / λ_{f} \geq 10^{- 3}$ , it appears that the computed thermal conductivities increase very slightly and tend to be at a constant value. As seen from Figure 10(e), the wall thickness does not strongly affect the perpendicular ETC. Therefore, the use of higher conductive hollow filler $(λ_{f} \geq 10^{3} W m^{- 1} K^{- 1}, F > 0.2)$ is not interesting to enhance the perpendicular effective conductivity of the composite.³⁶ Figure 10(f) shows the evolution of parallel effective conductivity for different values of $D \in [10^{- 5}, 10^{- 1}]$ . A significant increase in the thermal conductivity with an increase in the wall thickness was observed. This increase was foreseeable because the tube has a significant higher thermal conductivity than the polymeric matrix. For higher conductive hollow tube, $D = λ_{m} / λ_{f} \geq 10^{- 4}$ , we can see that the conductivity is increasing by up to 157% by raising the wall thickness from 0.05 to 1 and remains negligible for F > 0.8.

Comparison between analytical models and numerical simulations

The effective conductivity of parallel reinforcement composite has been calculated using the proposed numerical model. A comparison has been made with the predictions of standard models like those of the Bruggeman,⁵² Hashin and Shtrikman,⁵⁰ Hatta and Taya,⁵³ and Lewis-Nielsen.⁵¹ The choice of these models was based on two reasons: the model of Hashin and Shtrikman is one of the bases of many recent thermal conductivity models, and the model of Hatta and Taya, among the models developed recently, provides good agreement between the calculated values and the numerical data. Figure 11 shows a comparison between theoretical models and numerical simulations. As seen in this figure, the ETC values $E_{⊥}$ are very close to the ones predicted by the analytical models at the lowest volume fractions ( $ϕ \leq 20 %$ ). Moreover, the estimated ETCs are systematically lower. This could be attributed to the fact that these models do not take into account the thermal resistance between filers and matrix. For higher values of the volume fraction, a continuously increasing gap between numerical and theoretical estimations is observed. It is seen that the maximum difference between numerical and predictions of Hashin and Shtrikman model is lower than 10%.

Figure 11.

Comparison between analytical models and the calculated effective thermal conductivity $E_{⊥}$ ( $C = D = 10^{- 3}$ , $F = 1$ ). $E_{⊥}$ : dimensionless perpendicular effective conductivity; C: dimensionless contact resistance; D: dimensionless inner resistance; F: dimensionless wall thickness of the hollow conductive tube.

ETC and heat capacity measurements

To experimentally determine thermal apparent conductivities of the hollow tube composites, a temperature difference was imposed between up (T ₁) and down (T ₂) sides of the samples until observing a zero heat flux (equilibrium state q ₁ = q₂). First, the temperature levels on both sides (T ₁ and T ₂) are fixed and maintained until thermal equilibrium. At time t = 0 h (Figure 12), the composite was heated by modifying the temperature on a single face only (T ₁ > T ₂) until obtaining a thermal steady state. A similar experiment was carried out for all samples; the material is subjected to a temperature difference. Several tests were carried out on the material to check the reproducibility of the measurement. The apparent thermal conductivities are calculated using equation (30). To measure heat capacity of the samples,⁵⁴ the experiment consists first in imposing on the sample a superficial temperature of $T_{i n i t} ≅ 15^{\circ} C$ on each one of its faces, until obtaining a thermal steady state corresponding to an isothermal material. The heat fluxes are then zero at the initial time t = 0. It is also confirmed that the thermal losses are negligible at the isolated side faces. At a particular moment ( $t_{i n i t} ≅ 0.1 h$ ), a sharp oil H10 temperature variation is imposed in the thermoregulated bath. The material will thus evolve from $T_{i n i t}$ to $T_{e n d} ≅ 20^{\circ} C$ . This induces a thermal evolution of the system (sensitive storage) until another state of equilibrium is obtained. Figure 13 presents an example of the variation of the heat storage capacity of the samples. Between these two permanent steady states, the material stores sensitive heat. The flux-meters make it possible to measure the heat fluxes exchanged at the edges of the sample ( $ϕ_{1}$ and $ϕ_{2}$ ). The total amount of energy per mass stored can then be obtained from the equation (31). The results were found to be satisfactory and provided values of apparent thermal conductivities and heat capacity that are listed in Table 1.

Figure 12.

Determination of thermal apparent conductivities of the composites (S1). S1: sample 1.

Figure 13.

Heat flux and temperatures evolution of S1 (15°C to 20°C). S1: sample 1.

λ_{e f f} = \frac{e . Σ φ_{}}{2. Δ T_{}}

Where e is the thickness of the sample; $Σ ϕ_{}$ is the sum of the measured heat fluxes.

C_{p} = Q_{s e n s} / (T_{e n d} - T_{i n i t}) = [\frac{1}{ρ . e} \int_{t_{i n i t}}^{t_{e n d}} Δ ϕ . d t] / . (T_{e n d} - T_{i n i t})

$Δ ϕ$ represents the cumulated heat rate entering the sample; C _p is the apparent specific heat capacity composite (kJ/kg.°C).

Comparisons between simulations and experimental data

To illustrate the difference between the measured effective conductivities and the calculated values from FEM simulations, we have considered that the contact resistance is lower between the hollow tube and the epoxy matrix (C = 10⁻⁵). The choice of this value is inspired from Chapelle’s⁵⁵ study. His work is focused on the characterization of the interfacial thermal resistance between metallic wires and a polymer matrix and has found values between 3E−6 m² K⁻¹ m⁻¹ and 1.7E-5 m² K⁻¹ m⁻¹. Table 2 shows the experimental values $(E_{m, ⊥})$ at about 20°C that are compared with the calculated values $(E_{c, ⊥})$ from FEM simulations. The results show that the difference between $E_{m, ⊥}$ and $E_{c, ⊥}$ is lower than −3%. It is interesting to note that the parameters B _m and F play a fundamental role in the heat transfer between the matrix and metallic tube and thus greatly influences the value of ETC. For the sample (S1), the parameter B _m is low $B_{z, m} = 0.55$ $(B_{x, y, m} = 0.24)$ ; in this case, the maximum difference between $E_{m, ⊥}$ and $E_{c, ⊥}$ is about −1.58%. We can also observe the influence of this parameter on the ETCs of the samples (S2) and (S3). It may be seen that the difference between $E_{m, ⊥}$ and $E_{c, ⊥}$ decreases when B _m decreases ( $B_{z, m} (S 1) = 0.55$ and $B_{z, m} (S 3) = 0.84$ ), when the measured mass fraction $ϕ_{m}$ increases. Thus, it seems that the variation measurement model is lower at a weak mass fraction. This indicates that the control of the parameters B _m and F is significant to measure the ETC and to understand the heat transfer behavior of the hollow tube composites.

Table 2.

Comparison between calculated and measured parameters.

Samples	Diameters (mm)	$η = L / d_{e x}$	$ϕ_{m}$ (%)	D $\times 10^{- 4}$	F	$B_{x, y, m}$	$B_{z, m}$	$E_{m, ⊥}$	$E_{c, ⊥}$	$(E_{m, ⊥} - E_{c, ⊥}) /$ $E_{c, ⊥}$ (%)
Sample (S1)	$d_{i n} = 3.2$ $d_{e x} = 4.02$	49.75	55.81	4.76	0.2	0.24	0.55	3.10	3.15	−1.58
Sample (S2)	$d_{i n} = 2.74$ $d_{e x} = 3.56$	56.18	55.27	15.4	0.23	0.28	0.66	2.37	2.44	−2.86
Sample (S3)	$d_{i n} = 1.62$ $d_{e x} = 3.02$	66.22	33.08	8.06	0.46	0.33	0.84	2.01	2.06	−2.42

Conclusion

In this article, we have presented a mathematical model for calculating the ETC of hollow tube/resin composite. We have shown here, numerically, that the ETC will be calculated using a representative volume element based on the finite element software COMSOL^TM version 4.2a. It has also been shown that the interfacial resistance significantly affects the overall ETC. It was also found that the effective conductivity is especially sensitive to the parameter F and higher in the parallel model. This parallel thermal conductivity is increasing by up to 157% by raising the wall thickness from 0.05 to 1 (Figure 10(d); C = 10⁻⁵). Based on the results (Figure 10 (a), (c), and (e)), wall thickness does not strongly affect the perpendicular ETC. The rise in $E_{⊥}$ is less than 10% at D = 10⁻³ and is less than 0.5% for thermal conductivity ratio $D = λ_{m} / λ_{f} \geq 10^{- 4}$ . It can be concluded that in spite of the breakdown of conductive particle composite,^{36

–47} hollow particle composite can give a fairly good value of the ETC. This shows that the use of a high conductive hollow tube $(D = λ_{m} / λ_{f} \geq 10^{- 3})$ with wall thickness F > 0.2 instead of the solid one doses note significant to improve heat thermal transfer in the perpendicular hollow tube composite. The temperature fields in the composite are deformed, and heat transfer through the matrix is cut short by the tube. The tube acts as a bridge, transporting most of the heat fluxes through it shell. The temperature fields in the composite is deformed because heat is transferred more in tube than in the matrix. In heat transfer point of view, the center of the solid tube is not important; it can be filled with another material. In our project the center of the solid tube will be filled with a Phase change material (PCM).

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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A numerical and experimental study on the effective thermal conductivity of conductive hollow tube composite

Abstract

Keywords

Introduction

Prediction methods of ETC

Mathematical modeling

Analytical models

Experimental study

Sample preparation (epoxy resin/metallic hollow tube)

Thermophysical properties measurements

Results and discussion

Effect of filler volume fraction, thermal resistance, and D(F = 1)

Effect of the wall thickness of hollow tube, F ∈ 0.05 − 1

Comparison between analytical models and numerical simulations

ETC and heat capacity measurements

Comparisons between simulations and experimental data

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

Effect of the wall thickness of hollow tube, $F \in [0.05 - 1]$