Preface
`
`In many demanding applications such as automotive or aerospace, common engi-
`neering materials reaching their limits and new developments are required to fulfill
`the increasing demands on performance, characteristics, and functions. The proper-
`ties of materials can be increased, for example, by combining different materials to
`achieve better properties than a single constituent or by shaping the material or
`constituents into a specific structure. Many of these new materials reveal a much
`more complex behavior than traditional engineering materials due to their advanced
`structure or composition. The purpose of this book is to cover one of the important
`physical characteristics, that is thermal properties, in detail from different points of
`view. This book aims to provide readers not only with a good understanding of the
`fundamentals but also with an awareness of recent advances in properties determi-
`nation and applications of multiphase materials. The book contains 14 chapters
`written by experts in the relevant fields from academia and from major national
`laboratories/research institutes.
`The first part of the book covers materials where two or more solid phases form
`the composite. The second part is related to porous and cellular materials where two
`or more solid phases form certain shapes of cells with an empty or air-filled space.
`Typical representatives of this group are foamed polymers or metals, which have a
`significant potential in multifunctional applications. The last part of the book covers
`problem where fluids in a solid structure fulfill technical functions – such as in the
`case of combustion – or significantly determining the overall characteristics of the
`material.
`The editors wish to thank all the chapter authors for their participation and
`cooperation, which made this text possible.
`Finally, we would like to thank the team at Springer, especially Dr. Christoph
`Baumann, for their excellent cooperation during the whole phase of the project.
`
`January 2011
`
`Andreas O¨ chsner
`Graeme E. Murch
`
`v
`
`Page 1 of 34
`
`BOREALIS EXHIBIT 1092
`
`

`
`Contents
`
`Part I Composite Materials (two or more solid phases)
`
`Continuum Modeling of Diffusive Transport in Inhomogeneous Solids . . . 3
`Helmut J. Bo¨hm, Heinz E. Pettermann, and Sergio Nogales
`
`Thermal Residual Stresses in Aluminium Matrix Composites
`F. Teixeira-Dias and L.F. Menezes
`
`. . . . . . . . . . . . 33
`
`Heat Conduction in Two-Phase Composite Materials with
`Three-Dimensional Microstructures and Interfacial
`Thermal Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
`Carlos Frederico Matt and Manuel Ernani Cruz
`
`Part II Porous and Cellular Materials (one or more
`solid phases and free volume)
`
`Heat Transfer in Graphitic Foams
`Anthony G. Straatman
`
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
`
`Heat Transfer in Polyolefin Foams
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
`Marcelo Antunes, Jose´ Ignacio Velasco, Eusebio Solo´rzano,
`and Miguel A´ ngel Rodrı´guez‐Pe´rez
`
`Heat Transfer in Polymer Composites Filled with Inorganic
`Hollow Micro-Spheres
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
`J.Z. Liang
`
`Radiative Transfer in Two-Phase Dispersed Materials
`. . . . . . . . . . . . . . . . . . . 187
`Jaona Randrianalisoa, Re´mi Coquard, and Dominique Baillis
`
`vii
`
`Page 2 of 34
`
`

`
`viii
`
`Contents
`
`Predictions of Effective Thermal Conductivity of Complex Materials . . . . 235
`Ramvir Singh
`
`Lattice Monte Carlo Analysis of Thermal Diffusion
`in Multi-Phase Materials
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
`T. Fiedler, I.V. Belova, A. O¨ chsner, and G.E. Murch
`
`Optimization of a Unit Periodic Cell in Lattice Block Materials Aimed
`at Thermo-Mechanical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
`Pablo A. Mun˜oz-Rojas, Thiago A. Carniel, Emilio C.N. Silva,
`and Andreas O¨ chsner
`
`Computational Model of Porous Media Using True 3-D Images . . . . . . . . . 347
`Khairul Alam, Mihnea S. Anghelescu, and Adrian Bradu
`
`Part III Consideration of a Fluid (solid phase [structure] and fluid
`[in liquid and/or gaseous aggregate state])
`
`Thermal Instabilities in a Fluid Saturated Porous Medium . . . . . . . . . . . . . . 381
`A. Barletta
`
`New Bio-Inspired Multiphase Thermal Functional Fluid . . . . . . . . . . . . . . . . . 415
`Jose´ L. Lage
`
`Simulation of Turbulent Combustion in Porous Materials
`with One- and Two-Energy Equation Models
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
`Marcelo J.S. de Lemos
`
`Page 3 of 34
`
`

`
`Heat Transfer in Polyolefin Foams
`
`Marcelo Antunes, Jose´ Ignacio Velasco, Eusebio Solo´rzano,
`and Miguel A´ ngel Rodrı´guez‐Pe´rez
`
`Abstract This chapter is dedicated to the study of heat transfer in polyolefin-based
`foams, particularly thermal conductivity, as a function of their structure and
`chemical composition. A small review of the main experimental techniques used
`to measure the thermal conductivity of these materials is also given, focusing on the
`transient plane source method (TPS), as well as different theoretical models com-
`monly used for estimating its value. Alongside cellular structure (cell size, anisot-
`ropy, etc) and composition considerations, particular importance is given to the
`analysis of the presence of micrometric and nanometric-sized fillers in the resulting
`cellular composite thermal properties. This is a novel research field of particular
`interest, thought to extend the application range of these lightweight materials by
`tailoring their conductivity.
`
`1 Introduction
`
`It is well known that heat transfer is one the most important fields of research for
`cellular polymers due to the wide number of applications and uses of these
`materials as thermal insulators. Heat transfer in these materials strongly depends
`on relative density, cellular characteristics such as cell size, cell density, cell
`anisotropy, etc, and presence of additional phases and/or fillers (concentration,
`orientation and dispersion of these additional phases) [1–3].
`
`M. Antunes (*) and J.I. Velasco
`Centre Catala` del Pla`stic, Departament de Cie`ncia dels Materials i Enginyeria Metal·lu´rgica,
`Universitat Polite`cnica de Catalunya, C/Colom 114, 08222 Terrassa, Barcelona, Spain
`e-mail: marcelo.antunes@upc.edu; jose.ignacio.velasco@upc.edu
`
`E. Solo´rzano and M.A. Rodrı´guez-Pe´rez
`Cellular Materials Laboratory (CellMat), Condensed Matter Physics Department, University of
`Valladolid, 47011 Valladolid, Spain
`e-mail: esolo@fmc.uva.es; marrod@fmc.uva.es
`
`A. O¨ chsner and G.E. Murch (eds.), Heat Transfer in Multi-Phase Materials,
`Adv Struct Mater 2, DOI 10.1007/8611_2010_44,
`# Springer-Verlag Berlin Heidelberg 2010, Published online: 26 January 2011
`
`131
`
`Page 4 of 34
`
`

`
`132
`
`M. Antunes et al.
`
`This chapter is focused on the study of heat transfer in polyolefin-based foams,
`although most of the concepts and trends presented are applicable to most of the
`polymeric cellular materials available in the market. It displays the main tendencies
`of heat transfer, focusing on thermal conductivity, as a function of the structure and
`chemical composition for different types of polyethylene and polypropylene foams
`with densities ranging from 20 to 600 kg/m3. It also shows some strategies to
`modify the thermal conductivity in terms of structure, compounding and production
`techniques. A small review of the main experimental techniques to measure the
`thermal conductivity of these materials is also given and different theoretical
`models commonly used for determining the thermal conductivity of polymer
`foams have been applied.
`Part of the chapter is dedicated to the analysis of the presence of third phases
`(micrometric and nanometric-sized fillers) in the resulting cellular composite ther-
`mal properties. This is a novel research field of particular interest, thought to extend
`the application range of lightweight materials by tailoring their conductivity, and
`actually scarce information about the thermal behaviour of thermoplastic foams
`with conductive fillers has been published [3–5].
`
`1.1 The Concept of Cellular Solid
`
`A cellular solid is a two-phase material in which a gas has been dispersed in a solid
`continuous matrix. If the matrix is polymeric in nature, the material is known as
`cellular polymer or polymer foam.
`Among the most important parameters that modify the physical and transport
`properties of these materials are the nature and morphology of the base material,
`type of gas entrapped inside the cells, density, and the cellular architecture and
`topology, such as cell connectivity (closed, open or partially interconnected cells),
`cell size (f) and distribution of cell sizes, cell wall thickness (d) and respective
`distribution, fraction of solid in the cell struts (fs) and cell geometry and shape [6].
`Some of these basic parameters used to characterize the cellular structure are
`related by the following expression:
`ð
`f 1 fs
`
`Þ r
`r
`
`s
`
`¼ Cd;
`
`(1)
`
`where r/rs is the so-called relative density of the cellular material (r: density of
`the foam and rs: density of the respective unfoamed solid matrix) and C is a
`constant that depends on the cell’s shape and geometry. For instance, this constant
`has a value of 3.46 for pentagonal dodecahedron [1] and 3.35 for tetrakaidecahedral
`cells [7].
`The concept of foam as a two-phase material is important to understand their
`behaviour as that resulting from the combination of the properties of both phases
`and their relative content. Due to this reason, the relative density, and analogously
`
`Page 5 of 34
`
`

`
`Heat Transfer in Polyolefin Foams
`
`133
`
`the expansion ratio, ER (ER = rs/r), is a crucial parameter when studying these
`materials, being directly related to the relative volume fraction of both phases:
`Vgas ¼ (rs r)/(rs rgas)  1 r/rs and Vsol ¼ (r rgas)/(rs rgas)  r/
`r
`s. It is important to mention that these approximations are only valid for relative
`densities over 5%.
`
`1.2 Polyolefin Foams: Production Techniques
`
`In order to fully understand how the thermal properties vary with the structure, it
`is also necessary to explain the production methods of these materials. A small
`introduction on the most common foaming processes and typical resulting cellular
`structures is presented here. Polyolefins such as polypropylene (PP) or polyethyl-
`ene (PE) are commercially foamed using one of the basic foaming processes
`described in this section [8]. All results and microstructures shown here for the PP
`foams come from lab-produced materials, whereas PE foams results were mainly
`obtained from commercially available materials. As we will see, cellular structure
`variations induced by the different processing techniques or by process parameter
`modifications may slightly modify the thermal conduction behaviour of the whole
`system.
`
`– Foaming by direct extrusion, the foam is directly obtained by a sudden decom-
`pression at the exit of an extrusion die, normally using a physical blowing agent
`(PBA) such as CO2 or n-butane [9, 10] as seen in Fig. 1a. The physically foamed
`extruded foams tend to show a rather anisotropic cellular structure with cells
`elongated in the flow direction due to the stresses applied during the extrusion
`process.
`– Foaming by injection moulding,
`the polymer expansion is adjusted by
`controlling chemical blowing agent (CBA) thermal decomposition or PBA
`expansion inside a closed injection mould. A variation of the conventional
`1
`injection moulding is the microcellular injection moulding or Mucell
`tech-
`nique. In this method, supercritical N2 or CO2 is introduced in the plasticizing
`injection unit and mixed with the melted polymer before injecting in the mould
`[11] (see Fig. 1b). Small cell sizes are typically obtained, although this foaming
`technique is limited to rather high density materials (>300 kg/m3).
`– Foaming by compression-moulding, the material is foamed by simultaneously
`applying heat and pressure in order to decompose the CBA, nucleate the cells
`and subsequently expand the material by sudden decompression, as depicted in
`Fig. 1c. This process commonly uses exothermic CBAs such as azodicarbona-
`mide (ADC) [12]. Generally speaking, foams produced using this technique
`present small cell sizes and certain cell-size gradients, with smaller cells close to
`the mould’s surface.
`– Gas dissolution foaming, the material is foamed inside an autoclave by a high-
`pressure gas dissolution process. This process, commercially developed by
`
`Page 6 of 34
`
`

`
`134
`
`a
`
`b
`
`Mould
`
`c
`
`M. Antunes et al.
`
`Physical blowing
`agent (PBA)
`
`Foamed
`polymer
`
`Plastic pellets
`
`Melted plastic
`
`Physical blowing
`agent (PBA)
`
`Rotating screw
`
`Melted plastic
`
`Pressure
`
`Foam
`Temperature and
`time
`
`Foamed component
`
`Pressure
`release
`
`VD
`
`WD
`
`Circular mould
`
`d
`
`CO2
`
`Compression-
`moulded disc
`
`Temperature and
`time
`
`ΔP/Δt
`
`High-pressure
`gas chamber
`
`CO2
`decompression
`
`VD
`
`Foam
`
`Fig. 1 Schematics showing (a) direct extrusion, (b) injection moulding, (c) compression-mould-
`ing and (d) CO2 dissolution pressure-quench foaming processes
`
`Zotefoams [13], uses N2 or CO2 as physical blowing agent dissolving the gas in
`the polymer in a semi-solid state and afterwards allowing the material to expand
`by heating at a temperature above the softening temperature of the polymer–gas
`mixture. Although not available commercially, a second strategy, known as the
`pressure-quench method (see Fig. 1d), considers a one-step gas dissolution
`process. In this method, the material is nucleated and foamed by carefully
`
`Page 7 of 34
`
`

`
`135
`
`AR
`
`01234567891
`
`0
`
`Heat Transfer in Polyolefin Foams
`
`Compression
`Moulding-PP
`
`Injection-PP
`
`G
`
`a s dis s olutio n - P
`
` PE
`Foams
`
`P
`
`Extrusion-PP
`
`10000
`
`1000
`
`f (µm)
`
`G as dissolutio n-P P
`
` PE
`Foams
`
`Extrusion-PP
`
`Compression
`Moulding-PP
`
`0.4
`0.3
`Relative density
`
`Injection-PP
`
`0.5
`
`0.6
`
`100
`
`0.0
`
`0.1
`
`0.2
`
`Fig. 2 Cell size, f, and anisotropy ratio, AR (AR = fVD/fWD, VD: vertical direction of foaming;
`WD: width direction) versus relative density for the analyzed foams
`
`controlling the sudden pressure drop and pressure drop rate applied during gas
`decompression [14]. This was the strategy used for preparing the PP lab-
`produced foams analyzed in this chapter.
`The main cellular characteristics (average cell size, f, and anisotropy ratio, AR)
`of the polypropylene and polyethylene based foams analyzed in this chapter are
`summarized in Fig. 2.
`Regarding the characteristic cellular anisotropies exhibited in Fig. 2, it is impor-
`tant to mention that lab scale-produced PP foams were particularly conditioned to this
`kind of anisotropies (with special mention to the ones prepared by the pressure
`quench method). Similar processes in other laboratories or industrial-scaled ones
`based on similar techniques may not exhibit such anisotropic cellular structures.
`Scanning electron micrographs showing typical cellular structures for the differ-
`ent processing techniques are shown in Fig. 3 for the lab-produced PP-based foams
`and in Fig. 4 for the PE-based commercial ones.
`
`1.3 Composite Polyolefin Foams: Production Techniques
`
`Polymeric cellular materials can also incorporate fillers, i.e., secondary solid
`phases, commonly inorganic, with the intention of extending their applicability
`window. We will focus our attention on the effects of the incorporation of these
`fillers in the thermal conduction behaviour of the foams.
`
`Page 8 of 34
`
`

`
`136
`
`a
`
`b
`
`c
`
`d
`
`VD
`
`WD
`
`e
`
`f
`
`VD WD
`
`M. Antunes et al.
`
`VD WD
`
`Fig. 3 SEM micrographs of the lab scale PP foams produced using different foaming processes:
`(a) direct extrusion, (b) injection-moulding, (c, d) compression-moulding, and (e, f) pressure-
`quench CO2 batch foaming
`
`a
`
`b
`
`Fig. 4 SEM micrographs showing typical cellular structures of commercial PE foams prepared by
`(a) compression-moulding and (b) N2 dissolution
`
`Polyolefin composites were initially prepared by melt-compounding the differ-
`ent fillers in a twin-screw extruder with a polypropylene-based matrix and a CBA
`(azodicarbonamide). The resulting composites were subsequently foamed by com-
`pression-moulding chemical foaming. Particularly, the influence of two different
`highly conductive fillers was experimentally evaluated:
`
`1. Incorporation of high amounts (50 and 70 wt.%) of a micrometric-sized filler,
`magnesium hydroxide (Mg(OH)2) (Fig. 5), commonly used as flame retardant
`[15, 16]. This kind of filler typically exhibits a particle size in the range of a few
`micrometers (<10 mm). The maximum theoretical thermal conductivity is
`
`1 K1 [17], although this value depends
`assumed to be approximately 130 W m
`highly on the crystalline orientation.
`
`Page 9 of 34
`
`

`
`Heat Transfer in Polyolefin Foams
`
`137
`
`Fig. 5 SEM micrograph
`showing the typical
`hexagonal shape of Mg(OH)2
`particles
`
`0.5 μm
`
`a
`
`b
`
`c
`
`500 nm
`
`Fig. 6 (a, b) Transmission electron micrographs and (c) schematic showing the stacked-cup
`structure of the sub-micron vapour grown carbon nanofibres
`
`2. Incorporation of different amounts of carbon nanofibres (from 5 to 20 wt.%),
`1 K
`1) [18].
`a theoretically highly thermally conductive filler (>2,000 W m
`This fibrous-like nanometric filler was added with the objective of obtaining
`PP-based cellular materials with improved thermal conductivities [19, 20].
`The carbon nanofibres used here were sub-micron vapour grown carbon fibres
`(s-VGCF) with a stacked-cup structure produced using a floating catalyst tech-
`nique with a diameter of 20–80 nm, a fibre length higher than 30 mm and a
`graphitization degree of 70%. These nanofibres were kindly supplied by Grupo
`Antolı´n (Burgos, Spain). Figure 6 presents two different magnification transmis-
`sion electron micrographs, as well as a schematic displaying the stacked-cup
`structure of the carbon nanofibres.
`
`A graph indicating the main cellular structure characteristics, cell size and
`anisotropy, of these materials is shown in Fig. 7.
`It can be observed that the materials filled with Mg(OH)2 showed an expansion
`ratio between 2 and 3 and anisotropy ratios up to 3. The PP-CNF composite foams
`were intentionally produced with a fixed density (ER  3), although foams with
`
`lower densities could have been produced.
`
`Page 10 of 34
`
`

`
`138
`
`M. Antunes et al.
`
`AR
`
`0123
`
`50%-Mg(OH)2
`
`PP-CNF
`
`70%-Mg(OH)2
`
`70%-Mg(OH)2
`
`1000
`
`50%-Mg(OH)2
`
`PP-CNF
`
`f (μm)
`
`100
`
`0.2
`
`0.3
`
`0.5
`0.4
`Relative density
`
`0.6
`
`Fig. 7 Characteristic cell size, f, and anisotropy ratio, AR (AR = fVD/fWD, VD: vertical direction
`of foaming; WD: width direction) versus relative density for the composite foams analyzed in this
`chapter
`
`a
`
`b
`
`AR = 1.6
`c
`
`VD
`
`WD
`
`AR = 1.4
`d
`
`VD
`
`WD
`
`VD
`
`WD
`
`VD
`
`WD
`
`Fig. 8 SEM pictures of the 50 wt.% Mg(OH)2-PP foams: (a) 0.20 and (b) 0.23 relative density and
`70 wt.% Mg(OH)2-PP foams: (c) 0.47 and (d) 0.55 of relative density
`
`Figure 8 shows some examples of the cellular structure of Mg(OH)2 foams.
`Contrary to the foams filled with 70 wt.% Mg(OH)2 that show isometric-like cell
`structures with small cell sizes (180 mm), the foams with a 50 wt.% Mg(OH)2
`
`Page 11 of 34
`
`

`
`Heat Transfer in Polyolefin Foams
`
`139
`
`a
`
`b
`
`VD
`
`WD
`
`VD
`
`WD
`
`Fig. 9 Characteristic SEM pictures of (a) 5 and (b) 20 wt.% carbon nanofibre-reinforced polypro-
`pylene foams
`
`present higher cell sizes (from 700 to almost 1,000 mm) and increasingly higher cell
`anisotropies for lower relative densities.
`Some examples of the typical cellular structures of PP-CNF foams are shown
`in Fig. 9. The foamed nanocomposites were prepared with closed-cell structures
`
`and typical expansion ratios of 3. Isometric-like cellular structures (AR  1) with
`
`increasingly smaller cell sizes with gradually increasing the concentration of carbon
`nanofibres were obtained. For instance, the 5 wt.% CNF foam displayed an average
`cell size slightly above 500 mm, its value decreasing to 400 and around 250 mm
`respectively for the 10 and 20 wt.% CNF foams.
`
`2 Experimental Methods to Determine Thermal Conductivity
`
`The use of polymer foams is widespread in thousands of industrial applications and
`there is a continuous interest in regulating their thermal properties, in most cases
`with the objective of reducing the thermal conductivity. A wide variety of different
`experimental techniques to measure this property have been developed for different
`experimental conditions and materials [21, 22]. The process of measuring this
`property is complicated by the fact that in several practical situations most of the
`heat transfer mechanisms (conduction, convection and radiation) have to be con-
`sidered. Thus, for each material it is necessary to identify the ideal measuring
`procedure considering factors such as the expected conductivity, shape of the
`material, available sample size, density, etc.
`Generally speaking, in order to measure the thermal conductivity or a related
`property by a steady state or a transient method, the experimental arrangement must
`simulate a solution of the basic heat conduction equation:
`ð
`Þ
`¼ r2T þ A x; y; z; t
`l
`
`@T
`@t
`
`1 k
`
`;
`
`(2)
`
`where k and l are respectively the thermal diffusivity and conductivity, T is the
`temperature, t is the time, and A (x,y,z,t) is the heat generated per volume and time.
`
`Page 12 of 34
`
`

`
`140
`
`M. Antunes et al.
`
`For longitudinal unidirectional heat flow, no radial losses and disregarding the
`existence of a heat source into the solid, the analysis of the steady-state term leads
`to a linear dependence between temperature and dimension in the heat flow
`direction. The heat per area and time through a sample can be determined using
`Fourier’s law:
`
`Qa ¼ Gl
`
`DT
`d
`
`;
`
`(3)
`
`where Qa is the heat flow generated by the application of a temperature difference
`between the two sides of the material (DT), separated over a distance d (the material
`thickness), and G is a constant, determined by calibration for each given apparatus.
`Figure 10 shows a schematic diagram of the typical plate-like steady state equip-
`ment for determining the thermal conductivity.
`The steady state techniques are the most commonly used methods, and there are
`several standard methods (ISO and ASTM) based on this procedure [23, 24].
`Nonetheless, in some cases not all the heat generated in the upper plate is conducted
`to the lower one, thus being necessary to account for heat losses. Moreover, the heat
`flow is not always normal to the heat surfaces and there is a small gap between both
`heater surfaces and the surfaces of the sample. This gap contributes to the reduction
`of the effective transferred heat (interfacial heat transfer resistance). On the other
`hand, although this equipment is relatively fast in operation, there is still a need to
`reduce such times, particularly for quality control applications.
`Alternatively, transient methods based on the analysis of the transient term
`solution of (2) which relates change in temperature with time, are used. Transient
`hot wire, transient hot strip, transient plane source and laser flash methods are
`probably the most important techniques based on measuring the sample’s thermal
`behaviour under a transient heat flow regime. The laser flash method differs from
`the others since it is a non contact method and determines the thermal diffusivity of
`the sample instead of the thermal conductivity. Transient Plane Source (TPS) can
`be considered as the evolution of both transient hot wire and transient hot strip (by
`combining some aspects of the transient hot probe method, not mentioned before).
`
`Hot Plate
`
`Sample
`
`Cold Plate
`
`Heat flow
`meter
`
`Thermocouple 1
`
`Thermocouple 2
`
`Fig. 10 Schematic diagram of
`typical standard equipment
`conductivity based on Fourier’s law
`
`for determining the thermal
`
`Page 13 of 34
`
`

`
`Heat Transfer in Polyolefin Foams
`
`141
`
`Therefore we can consider the TPS method as the most representative of these
`transient contact methods.
`In the TPS method a round and plane heat source is used (see Fig. 11, left). It acts
`as a transient plane source working simultaneously as a temperature sensor. This
`element consists of an electrical conducting pattern of thin nickel foil in the form of
`a double spiral inserted between two insulating plastic layers. The total thickness of
`this sensor is only a few tens of micrometers. The TPS element is placed between
`two samples with both sensor faces in contact with the two samples surfaces as
`depicted in Fig. 11, left. Two samples of similar characteristics are required for this
`purpose. During testing a constant electric power is supplied to the sensor and the
`temperature increase is recorded. To relate the change in temperature with time, the
`equation for the heat conduction assuming the conditions reported by Log et al. and
`Gustavsson et al. is applied [25, 26]. Although less known, this method also enables
`the possibility of detecting possible thermal anisotropies [27].
`Finally, the laser flash system is based in a laser beam pulse (typically <1 ms)
`focused on one of the faces of a relatively thin sample while the temperature
`increase in the opposite parallel face is recorded by a contactless method (IR
`pyrometer). From this temperature increase it is possible to determine the thermal
`diffusivity of the sample. The schematic description of the measurement procedure
`is shown in Fig. 11, right [28].
`Transitory methods present several advantages compared to steady state ones.
`For example, it is possible to simultaneously obtain values of the thermal conduc-
`tivity, thermal diffusivity and specific heat. The range of measurement comprises
`1 K
`1). These methods are also faster,
`five orders of magnitude (0.01–400 W m
`can be used to determine the influence of material inhomogeneities and/or aniso-
`tropic characteristics and offer the ability to measure in small samples compared to
`the thick samples conventionally needed for the steady state methods.
`On the other hand, it is important to remark that transient methods are not fully
`standardized. Nevertheless, in the last years some standard procedures have been
`approved for determining the thermal properties by the laser flash method [29, 30]
`and several efforts are being developed to standardize the TPS method [31]. It is
`
`Sample
`
`Sensor
`
`Sample
`
`Sample
`
`IR pyrometer
`
`Laser pulse
`
`t
`
`Fig. 11 Schematic diagrams for transient plane source (left) and laser flash (right) methods
`
`Page 14 of 34
`
`

`
`142
`
`M. Antunes et al.
`
`also interesting to comment that the thermal properties of non-conventional new
`materials have been studied in the last years by using transient methods [32–40]. In
`addition, several works have shown that these transient techniques give comparable
`results to the steady state ones [41, 42].
`
`3 Mechanisms and Models of Heat Transfer in Polymer Foams
`
`Generally speaking, the heat transfer in any cellular material is the result of a
`contribution of three different mechanisms, conduction, convection and radiation,
`and therefore the overall thermal conductivity can be depicted as the result of four
`additive terms:
`
`lfoam ¼ ls
`
`cnd
`
`þ lg
`
`cnd
`
`þ lrd þ lcnv;
`
`(4)
`
`cnd and lg
`where ls
`cndare respectively the thermal conductivities due to conduction
`through the solid and gas phases and lrd and lcnv the radiation and convection
`terms.
`
`3.1 Convection
`
`Convection due to gas movement inside the cells may be disregarded for cellular
`structures with cell sizes of less than 4–5 mm [1]. Considering that almost all
`polymer-based foams, independently of the final relative density, present cell sizes
`that are clearly below these values, heat transfer due to the movement of the gas
`molecules entrapped inside the cells (convection) can be considered minimal when
`compared to conduction and radiation.
`
`3.2 Conduction
`
`Several theoretical models have been proposed to estimate the conduction term of
`cellular polymers. Nonetheless, even the most recent ones, which take into account
`for instance arbitrary cell orientations or anisotropy geometrical parameters, tend to
`consider rather simple geometrical-shaped arrays representing the cellular structure
`(cubic or polyhedral-like) [1, 41, 43–45]. On the other hand, foams exhibit in many
`cases cell imperfections, gradients and inhomogeneities, far away from regular
`geometries, especially in the case of low density foams. Nevertheless, the following
`formula has been proven to give reasonable compliance for the prediction of the
`thermal conduction term [1]:
`
`Page 15 of 34
`
`

`
`143
`
`(5)
`
` 1
`
`#
`
`4
`
`;
`
`1 A
`
`R
`
`þ 2 1 fs
`ð
`
`Þ
`
`"
`
`r
`
`ffiffiffiffiffiffiffi
`
`1 A
`
`R
`
`Vsol
`3
`
`fs
`
`Heat Transfer in Polyolefin Foams
`
`lfoam ¼ lgasVgas þ lsol
`
`where fs is the fraction of solid in the cell struts and AR is the geometrical anisotropy
`ratio, i.e., the quotient between the highest and smallest cell size in the direction of
`the heat flux (Fig. 12).
`In the particular case of isotropic-like cellular structures, (5) is reduced to the
`following equation:
`
`
`l ¼ lgasVgas þ 2
`
` fs
`3
`
`3
`
`
`lsolVsol
`
`(6)
`
`According to some other authors the thermal conductivity of a cellular solid can
`be modelled as:
`
`(7)
`
`l ¼ lgasVgas þ xlsolVsol;
`where x is the tortuosity, a parameter directly related to the foam’s inherent
`irregularity. The concept of tortuosity goes beyond the conventional geometrical
`tortuosity [46] and implicitly considers the effect of cellular structure (cell size, cell
`density and cell wall thickness, fs). This last equation is particularly interesting in
`the case of materials with an unknown fs parameter. On the other hand, in the
`particular case of anisotropic structures it is possible to consider the effect of
`anisotropic tortuosity in the different material directions, so under anisotropic
`conditions we could also talk about equivalency between (5) and (7).
`The influence of this mechanism can be modified by incorporating sec-
`ond phase constituents with different thermal conductivities, thus varying the
`expected conductivity, especially in the case of high density foams (higher
`contribution of the solid phase). Such is the case of the fillers considered in
`some of the next sections of this chapter. Particularly, the model used for these
`composite foams will be based on the experimental thermal conductivity of the
`solid composite (i.e., replacing l and V variables shown in previous equations by
`those of the corresponding composites).
`
`f2
`
`f2
`
`f1
`
`Heat flux
`
`AR =
`
`f1
`
`f2
`
`Fig. 12 Schematic showing
`cell anisotropy for a cubic cell
`geometry and the definition
`of AR
`
`Page 16 of 34
`
`

`
`144
`
`3.3 Radiation
`
`M. Antunes et al.
`
`To estimate this mechanism, the model proposed by Williams and Aldao will be
`adopted in this chapter. The reason is that this model has a high compliance with
`experimental results as has been shown in previous investigations [47–49]. One
`of its main advantages lies on the use of measured values for the cellular
`structure characteristics and polymer matrix properties instead of non-intuitive
`adjustable parameters. The model is based on a radiation term predicted as
`follows:
`
` ;
`
` 1
`
`(8)
`
`
` 
`4sT3L
`1 þ L
`
`1T
`
`N
`
`f
`
`lr ¼
`
`where s is the Stefan–Boltzmann constant, T is the temperature, L is the material
`thickness, f is the cell size and TN is the fraction of radiant energy sent forward by a
`solid membrane of thickness Ls. This energy fraction is given by:
`
`
`
`TN ¼ 1 rð Þ
`
`1 r
`ð
`
`
`Þ
`ð
`1 rt
`Þ
`ð
`1 þ rt
`
`(9)
`
`Þ þ 1 tðÞ t
`
`2
`
`;
`
`where r is the fraction of incident energy reflected by each gas–solid interface. This
`quantity is related to the refractive index of the solid matrix (o):
`
`
`r ¼ o 1
`o þ 1
`
`(10)
`
`2
`
`The coefficient t is the fraction of energy transmitted through the solid mem-
`brane of thickness Ls (cell wall thickness), which is given by:
`t ¼ expðaLsÞ;
`
`(11)
`
`where a is the absorption coefficient of the solid matrix.
`
`3.4 Evaluation of the Weight of the Different Terms
`
`This section considers the theoretical evaluation of the contribution of each heat
`transfer mechanism for PE foams as a function of relative density. The particular
`example uses real values, experimentally obtained for these materials. The follow-
`ing equation and the values presented in Table 1 were considered for the predictions
`of the overall thermal conductivity.
`
`Page 17 of 34
`
`

`
`Heat Transfer in Polyolefin Foams
`
`145
`
`Table 1 Properties of the
`foam considered for
`predicting the overall
`thermal conductivity
`
`1)
`
`r
`
`Value
`
`1
`
`1 K1)
`
`0.0263 (W m
`1 K0.30 (LDPE) (W m
`
`Closed cell
`150–500 mm
`ð
`Þ f 3:5347r
`1 fs
`0.2–0.4
`10 mm
`300 K
`1.51
`661 cm
`
`Property
`lgas
`lsol
`Cell type
`Cell size, f
`Cell wall, Ls (mm)
`fs
`Total material thickness
`Temperature
`o
`a
`
`Gas conduction
`Solid conduction
`Radiation
`
`100
`
`90
`
`80
`
`70
`
`60
`
`50
`
`40
`
`30
`
`20
`
`10
`
`0
`
`Contribution (%)
`
`0.0
`
`0.1
`
`0.2
`
`0.3
`
`0.6
`0.5
`0.4