Applicability of the Transient Plane Source Method To
`Measure the Thermal Conductivity of Low-Density
`Polyethylene Foams
`
`O. ALMANZA,1 M. A. RODRI´GUEZ-PE´REZ,2 J. A. DE SAJA2
`
`1Departamento de Fı´sica, Universidad Nacional de Colombia, Bogota´, Colombia
`
`2Departamento de Fı´sica de la Materia Condensada, Facultad de Ciencias, Universidad de Valladolid,
`47011 Valladolid, Spain
`
`Received 2 July 2003; revised 25 September 2003; accepted 19 November 2003
`
`ABSTRACT: The thermal conductivity at constant pressure of a collection of crosslinked,
`closed-cell polyethylene foams were measured at room temperature with the transient
`plane source (TPS) method. The experimental results were compared with those deter-
`mined by a standard steady-state technique. The results showed that the values
`measured by the TPS method follow the same trends as those measured by a heat-flow
`meter. Therefore, with the TPS technique it is possible to observe the influence of
`structural characteristics such as cell size, black carbon content in foams, density, and
`so forth on thermal conductivity. However, the values obtained by the transient method
`were approximately 20% higher than those given by the standard method. Possible
`reasons for these variations are discussed. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B:
`Polym Phys 42: 1226–1234, 2004
`Keywords:
`thermal properties; polyethylene; polyolefins; foams; transient plane
`source method
`
`INTRODUCTION
`
`Thermal conductivity is an important property in
`the application and use of cellular plastics. This
`property changes extensively depending on the
`density, cellular structure, and morphology of the
`polymeric matrix and so forth.1– 4 Moreover, ther-
`mal conductivity also exhibits a strong depen-
`dence on temperature and pressure.
`To analyze the thermal properties of cellular
`polymers, it is necessary to evaluate the three
`mechanisms whereby, under the influence of a
`temperature gradient, energy can be transported
`
`to: M. A. Rodriguez-Perez
`
`Correspondence
`marrod@fmc.uva.es)
`Journal of Polymer Science: Part B: Polymer Physics, Vol. 42, 1226–1234 (2004)
`© 2004 Wiley Periodicals, Inc.
`
`(E-mail:
`
`1226
`
`from one region of space to another. These mech-
`anisms are radiation, conduction through the
`solid and gaseous phases, and convection. These
`processes of heat transfer are often very impor-
`tant in a wide variety of scientific and industrial
`applications. Because of this reason, a number of
`different experimental techniques have been de-
`veloped5,6 to measure the thermal conductivity
`for different experimental conditions and for dif-
`ferent materials. The fact that in most practical
`situations all three heat-transfer mechanisms are
`present greatly complicates the process of mea-
`surement of this property.
`The overall range of thermal conductivity for
`cellular plastics is one order of magnitude above
`approximately 0.015 W/m K [the lowest value for
`a low-density (⬍50 kg/m3) phenolic urethane or
`polyisocyuretane closed-cell
`insulation product
`
`BOREALIS EXHIBIT 1045
`
`Page 1 of 9
`
`

`
`TRANSIENT PLANE SOURCE METHOD
`
`1227
`
`The steady-state techniques have found wide
`applications. In fact, there are a number of stan-
`dard methods based on this procedure (ASTM
`C177 or ISO DIS 8302, ASTM C518 or ISO 8301,
`and ASTM F433).
`Whether the sample is a solid, the heat gener-
`ated in the upper plate is not all conducted to the
`lower plate. Thus, it is necessary in all cases to
`account for heat losses. Moreover, it is not always
`true that the heat flow is normal to the heat
`surfaces and there is a small gap between the two
`heater surfaces and those of the sample. This gap
`contributes to the reduction of the heat trans-
`ferred (interfacial heat-transfer resistance).
`There are a wide number of commercial equip-
`ments designed to work under steady-state con-
`ditions, for instance, the guarded hot plate, the
`unguarded hot plate, different arrangements for
`determining linear heat flow, or radial heat flow.5
`Although the heat-flow meters are relatively fast
`in operation, there is still a need to reduce such
`times, particularly for quality-control applica-
`tions.
`Transitory methods are based on the analysis
`of the transient term solution of eq 1, which re-
`lates change in temperature with time. Hot wire,
`transient hot strip, and transient plane source
`are techniques based on measuring the sample
`behavior in the transient regime of heat flow.
`These methods have several advantages; for
`example, it is possible to obtain values of thermal
`conductivity, thermal diffusivity, and specific
`heat simultaneously. The range of measuring of
`these properties is wide (ca. 0.02–150 W/m K).
`Other benefits are that these methods can be used
`to measure properties of inhomogeneous and/or
`anisotropic materials, can offer the ability to mea-
`sure in small samples, and measurements are in
`general fast.
`The transient methods are not yet standard.
`However, several efforts are being developed to
`standardize these transient methods.7 In the last
`few years, different kinds of materials (butadiene
`rubber compounds, pineapple leaf fiber-reinforced
`composites, and highly porous building materials)
`have been characterized by the transient plane
`source (TPS) technique.4,8 –12
`In this work, an experimental study on the
`thermal conductivity of a collection of low-density
`polyethylene (LDPE)
`foams, produced from a
`high-pressure nitrogen gas solution process, is
`presented. Measurements were performed both
`with a steady-state technique and with TPS. The
`main goal of this investigation was to check and
`
`Figure 1. Diagram of a heat-flow meter.
`
`containing a low thermal conductivity blowing
`agent (CFC, HCFC, etc.) in the cells]. Practically,
`there is no ideal method of measurement for this
`range of thermal conductivity, ␭. It is necessary to
`choose a procedure depending on factors such as
`the expected conductivity, the shape of the mate-
`rial, its density, and its availability in a suitable
`size to be considered representative of the bulk or
`for the application under analysis.
`To measure thermal conductivity or a related
`property by a steady-state or a transient method,
`the experimental arrangement must simulate a
`solution to the basic heat-conduction equation for
`a homogeneous isotropic solid:
`
`(1)
`
`⫽ ⵜ2T ⫹
`
`A共x,y,z,t兲
`␭
`
`⭸T
`⭸t
`
`1 k
`
`where T is the temperature, t is the time, k is the
`thermal diffusivity (k ⫽ ␭/␳cP), cP is the specific
`heat capacity at constant pressure, ␳ is the den-
`sity of the sample, and A(x,y,x,t) is the heat gen-
`eration per unit volume per unit time in the me-
`dium.
`For longitudinal unidirectional heat flow, no
`radial losses, and without power supplied into the
`solid, analysis of the steady-state term leads to a
`lineal dependence between temperature and di-
`mension. Then the heat per unit time and unit
`area through a sample can be determined by Fou-
`rier’s law:
`
`⌬T
`Qa ⫽ G␭
`d
`
`(2)
`
`where Qa is the heat flow generated by the appli-
`cation of a temperature difference (⌬T) between
`the two sides of the sample material, separated
`over a distance d, and G is a constant, evaluated
`by calibration, for a given apparatus (see Exper-
`imental). Figure 1 shows a diagram of a typical
`heat-flow meter.
`
`Page 2 of 9
`
`

`
`1228
`
`ALMANZA, RODRI´IGUEZ-PE´ REZ, AND de SAJA
`
`Figure 2.
`(a) Diagram of a TPS sensor and (b) exper-
`imental arrangement of the sensor and samples.
`
`analyze the ability of the TPS technique as a
`scientific instrument for the analysis of the ther-
`mal properties of low-density foams, while trying
`to point out the advantages and disadvantages of
`this technique as compared with a standard
`method (heat-flow meter) widely known and used
`for both scientific and industrial investigations
`and for quality assurance.
`
`THEORY OF TPS
`
`Measurements of both thermal conductivity and
`thermal diffusivity are possible by means of the
`hot-disk method.8 In this method, the TPS ele-
`ment behaves both as a temperature sensor and
`as a heat source. The TPS element consists of an
`electrical conducting pattern of thin nickel foil (10
`␮m thick) in the form of a double spiral, which
`resembles a hot disk, embedded in an insulating
`layer made of Kapton (70 ␮m thick) [Fig. 2(a)].
`Two samples of the same material are located in
`contact with the two faces of the sensor [Fig. 2(b)].
`A constant electric power is supplied to the
`hot-disk sensor, and the increase in temperature
`⌬T(t) is directly related to the variation in the
`sensor resistance [⌬R(t)]. The time-dependent re-
`sistance [R(t)]is given by8
`
`R共t兲 ⫽ R01 ⫹ ␣⌬T␶)
`
`(3)
`
`where R0 is the hot-disk resistance in the begin-
`ning of the recording (initial resistance), ␣ is the
`temperature coefficient of resistance of the nickel
`foil, and ⌬T(␶) is the temperature increase of the
`sensor expressed in terms of an only variable ␶,
`defined as
`
`␶ ⫽ 共t/␪兲1/2
`
`␪⫽ a2/k
`
`(4)
`
`where t(s) is the measurement time from the start
`of the transient heating; ␪ is the characteristic
`time, which depends both on parameters of the
`sensor and the sample; a (millimeters) is the hot-
`disk radius; and k (millimeter squared per sec-
`ond) is the thermal diffusivity of the sample.
`Assuming an infinite sample and the conduc-
`tive pattern to be in the Y–Z plane of a coordinate
`system, the temperature rise at a point (y,z) at
`time t due to an output of power per unit area Q
`is given by an expression obtained from the heat-
`conduction equation solution8,13
`
`⌬T共y,z,t兲 ⫽ 共8␲3/2␳cP兲⫺1冕
`⫻冕
`
`t
`
`0
`
`dt关k共t ⫺ t⬘兲兴⫺3/2
`
`dy⬘dz⬘Q共y⬘,z⬘,t⬘兲exp兵 ⫺ 关共y ⫺ y⬘兲2 ⫹ 共z ⫺ z⬘兲2兴
`
`S
`
`⫻ 关4k共t ⫺ t⬘兲兴⫺1其
`
`(5)
`
`where S is the total area of the conducting pattern
`that is exposed to a certain temperature increase.
`Previous expression can be simplified by taking
`k共t ⫺ t⬘兲 ⫽ ␴2a2:
`
`␶
`
`⌬T共y,z,t兲 ⫽ 共4␲3/2a␭兲⫺1冕
`⫻ exp再 ⫺
`
`␴2冕
`
`d␴
`
`dy⬘dz⬘Q共y⬘,z⬘,t⬘兲
`
`0
`A
`关共y ⫺ y⬘兲⬘2 ⫹ 共z ⫺ z⬘兲2兴
`4␴2a2
`
`冎
`
`(6)
`
`In the case of disk geometry, consisting of n con-
`centric ring sources, ⌬T can be related to ⌬T(␶)
`through the equation
`
`Page 3 of 9
`
`

`
`⌬T共␶兲 ⫽ P0共␲3/2a␭兲⫺1D共␶兲
`
`(7)
`
`where P0 is the total output power and D(␶) is a
`geometric function obtained from eq 5 given by
`the next equation
`
`␶
`
`d␴
`␴2
`
`
`
`k 䡠 exp冋共 ⫺ l2 ⫹ k2兲册L0冉 lk2␴2n2冊冎冊
`
`n
`
`n
`
`l⫽l
`
`k⫽1
`
`(8)
`
`TRANSIENT PLANE SOURCE METHOD
`
`1229
`
`a calibration is needed (determination of the con-
`stant G). A fibrous board insulation material
`(NIST SRM 1450c) was used as the standard ma-
`terial. The calibration constant is an apparatus
`constant. The major advantage of the in situ cal-
`ibration is that the specimen may be considered
`somewhat self-guarding, and the heat losses can
`be considered to be eliminated for each particular
`condition at which calibration is undertaken.
`A hot-disk TPS, thermal-constant analyzer,
`was used for the measurement of thermal conduc-
`tivity of LDPE foams. Experiments were carried
`out at room temperature (24 ⫾ 2 °C). The mea-
`surements were performed in two square samples
`of 36 mm side and 11 mm thick. The samples were
`much smaller than those used for the heat-flow
`meter. The thickness of the samples should pref-
`erably not be less than the diameter of the hot-
`disk sensor, and this must always be much larger
`than the porosity or the cellular structure of the
`sample if the material is not dense or homoge-
`neous (ca. an order of magnitude higher than the
`cell size). A disk-shaped TPS sensor with a diam-
`eter of 12.806 mm provided with 16 rings was
`used in all measurements. The TPS element was
`made of 10-␮m-thick nickel foil with electric in-
`sulation on each side of 30-␮m Kapton.
`The calculations of the thermal properties were
`performed according to the equations outlined in
`theory of TPS. The first 50 points of each record-
`ing (total number of points in each measurement
`was 200) were not used in each calculation. This
`procedure reduced the effect of the contact resis-
`tance between the sensor and sample.
`
`MATERIALS
`
`The product code, base polymer, measured den-
`sity (␳f), mean cell diameter (␾), thickness L, and
`apparent color of the industrial materials under
`study are summarized in Table 1.
`Foams produced from 100% LDPE and blends
`of LDPE and high-density polyethylene (HDPE)
`were examined. The density ranged between 15
`and 83 kg/m3 and the cell diameter was between
`313 and 1000 ␮m. The thickness of the foams was
`approximately 11 mm. Moreover, foams of differ-
`ent colors were characterized; in particular, black
`foams with different black carbon contents were
`used in this analysis. The solid sheets that were
`used to manufacture some of the foams were also
`considered in this investigation.
`
`D共␶兲 ⫽ 关n共n ⫹ 1兲兴⫺2冕
`⫻冉冘
`
`l再冘
`
`0
`
`2␴2n2
`
`in which L0 is the modified Bessel function.8,9,13
`Thermal conductivity can be obtained by fitting
`the experimental data to the straight line given
`by eq 7, and thermal diffusivity is calculated from
`eq 4 considering the ␶ value determined in the
`previous fit. Finally, the heat capacity was de-
`rived from previous values with the relation k
`⫽ ␭/␳cP, where ␳ is the sample density. In this
`article the diffusivity and cP values are not dis-
`cussed.
`
`EXPERIMENTAL
`
`A rapid K heat-flow meter from Holometrix was
`used for the thermal-conductivity measurements
`under steady heat-flow conditions. The experi-
`ments were performed in accordance with the
`ASTM C518 and ISODIS 8301 methods. The mea-
`surements were performed in square samples of
`30 cm side and 11 mm thick. A dispersion less
`than 1% in two consecutive readings was taken as
`the criterion to ensure that the measurements
`were made under steady-state conditions. The
`time lapse between readings was 20 min. Mea-
`surements were performed at 24 °C with a tem-
`perature difference between plates of 30 K. The
`time for conducting an experiment was approxi-
`mately 7 h. Therefore, if three data are needed to
`characterize one material, the time per material
`is approximately 21 h.
`The performance of any heat-flow-meter appa-
`ratus is unique. It depends on factors such as the
`type and form of heat-flow transducer, its thermal
`resistance in relation to the specimen resistance,
`possible contact resistances in the system, heat
`losses, position of temperature sensors, and the
`overall calibration factor. To minimize this effect,
`
`Page 4 of 9
`
`

`
`1230
`
`ALMANZA, RODRI´IGUEZ-PE´ REZ, AND de SAJA
`
`Table 1. Main Characteristics of the Foams under Study
`
`Foams
`
`LD15W
`LD24W
`LD29W
`LD33 (1)W
`LD50CNB
`LD60 G
`LD70B
`LD24(FC)
`LD24(LC)
`LD33 (2)W
`LD33B
`LD sheet W
`LD sheet B
`HL79(1)W
`HL79(2)W
`HL sheet W
`
`Chemical
`Composition
`
`100% LDPE
`
`50% LDPE
`⫹ 50% HDPE
`
`Apparent
`color
`
`␳f
`(kg/m3)
`
`␾ (␮m)
`
`L (cm)
`
`␪carbon
`(%)
`
`White
`White
`White
`White
`Black
`Green
`Black
`Black
`Black
`White
`Black
`White
`Black
`White
`White
`White
`
`16.7
`24.6
`30.7
`32.0
`52.3
`58.5
`69.5
`23.6
`23.7
`31.4
`32.5
`910
`910
`81.0
`83.0
`926
`
`313
`312
`528
`424
`910
`773
`528
`315
`956
`377
`337
`—
`—
`1006
`1076
`—
`
`1.12
`1.02
`1.11
`1.10
`1.04
`1.02
`1.10
`1.02
`1.02
`1.10
`1.10
`0.93
`1.03
`0.99
`1.70
`1.31
`
`0
`
`12
`0
`2
`
`2
`0
`2
`0
`2
`0
`
`␳f is the foam density, ␾is the cell size, L(cm) is the thickness, and ␪carbon is the additional black carbon content in the polymeric
`matrix.
`
`crosslinked,
`foamed samples were
`These
`closed-cell polyolefin foams manufactured by a
`high-pressure nitrogen gas-solution process and
`were provided by Z-Foams Plc. (Croydon, United
`Kingdom). In this process, a polyolefin is com-
`pounded with a peroxide curing agent and ex-
`truded as a thick sheet, which is passed through a
`hot oven to effect crosslinking (gel content was ca.
`40%). Slabs are cut from the extruded sheet and
`placed in an autoclave where they are subjected
`to high pressure (several hundred bars) nitrogen
`gas at temperatures above the polymer softening
`point. Under these conditions, the nitrogen dis-
`solves in the polymer. At the end of the solution
`stage and after cooling, the pressure is reduced to
`zero gauge. The slabs, now containing nitrogen
`gas for expansion, are then placed under low pres-
`sure in a second autoclave and again heated
`above the polymer melting point. Release of the
`pressure then results in full expansion. By alter-
`ing the saturation gas pressure, the amount of
`gas dissolved in the polymer and thus, the final
`foam density, is varied. The cell size can also be
`controlled by changing some industrial process
`parameters.
`
`RESULTS
`
`The thermal conductivity for all the foams under
`study as a function of the foam density, obtained
`by both methods, is shown in Table 2 and Figure 3.
`
`Table 2 also includes the experimental power,
`measuring times, and time between experiments
`used for each material. The whole time for an
`experiment in which five recordings were mea-
`sured was approximately 2.5 h.
`The first important result is that the trends
`observed by both methods as a function of foam
`density are very similar (Fig. 3). Therefore, if the
`interest is focused on the analysis of the relation-
`ships between structure and thermal properties,
`it seems clear that very similar conclusions would
`result from the data obtained by both techniques.
`The qualitative influence of structural parame-
`ters, such as cell diameter, density, black carbon
`content, kind of base polymer, and so forth, that it
`is possible to deduce from the data collected by
`the TPS method were similar to those we ana-
`lyzed in previous investigations1–3 in which the
`measurements were performed under steady-
`state conditions.
`In that research, the effect of several struc-
`tural characteristics that influences the ther-
`mal conductivity was studied. It was estab-
`lished that cell size and black carbon content
`were the two most important factors that affect
`the thermal conductivity through their effect on
`the radiative heat-flow contribution. In partic-
`ular, it was proven that increasing the cell size
`results in higher conductivities and adding a
`low proportion of black carbon content reduces
`the conductivity.
`
`Page 5 of 9
`
`

`
`Table 2.
`
`Foams
`
`LD15W
`LD24W
`LD29W
`LD33 (1)W
`LD50CNB
`LD60G
`LD70B
`LD24(FC)
`LD24(LC)
`LD33 (2)W
`LD33B
`LD Sheet W
`LD Sheet B
`HL79(1)W
`HL79(2)W
`HL Sheet W
`
`TRANSIENT PLANE SOURCE METHOD
`
`1231
`
`␭RK (W/m K)
`Rapid K
`
`␭HD (W/m K)
`Hot Disk
`
`⌬␭ (%)
`
`tm (s)
`
`0.0374
`0.0372
`0.0441
`0.0407
`0.0413
`0.0475
`0.0456
`0.0364
`0.0413
`0.0401
`0.0364
`0.2140
`0.2140
`0.0565
`0.0592
`0.2246
`
`0.0439
`0.0468
`0.0516
`0.0503
`0.0530
`0.0599
`0.0581
`0.0460
`0.0497
`0.0519
`0.0484
`0.3729
`0.3773
`0.0705
`0.0767
`0.4727
`
`14.8
`20.5
`14.6
`19.1
`22.0
`20.7
`21.5
`20.8
`16.9
`22.6
`24.7
`42.6
`43.3
`19.9
`22.8
`52.5
`
`30
`30
`30
`40
`40
`40
`50
`30
`30
`40
`40
`110
`110
`40
`40
`90
`
`␴
`
`0.0002
`0.0002
`0.0003
`0.0001
`0.0005
`0.0002
`0.0004
`0.0003
`0.0004
`0.0003
`0.0002
`0.0015
`0.0033
`0.0005
`0.0003
`0.0048
`
`␭ is the thermal conductivity for all the foams under study. ⌬␭ (%) is the difference between the values of ␭ obtained with hot
`disk and rapid K equipments, ⌬␭ ⫽ 100 䡠 (1 ⫺ ␭RK/␭HD). tm (s) is the measuring time for each sample. Five measurements were
`obtained, and the time between experiments was 30 min. The deviations of these five measurements are also given in the table (␴).
`The applied power was 0.02 W.
`
`Similar trends can be inferred from the exper-
`imental data measured by the TPS method. The
`thermal conductivity of the low-density polyeth-
`ylene foams (LD) measured by the TPS method as
`a function of the foam density is displayed in
`Figure 4. The black foams have a slightly lower
`conductivity than the white foams of similar den-
`sity. Moreover, for this kind of material, reducing
`
`the density involves an improvement of the ther-
`mal-insulation capability.
`Figure 5 depicts the thermal conductivity of
`the foams measured by the TPS method as a
`function of the cell diameter. A linear increase of
`this thermal property was observed for the LD
`white foams. It was also possible to observe that
`black foams present lower values of the conduc-
`tivity. As explained previously, these trends were
`due to the thermal-radiation heat flow. The out-
`
`Figure 3. Experimental thermal conductivity, ob-
`tained by both the hot-disk and rapid K methods, as a
`function of the density for all foams under study.
`
`Figure 4. Thermal conductivity of the white and
`black LD foams measured by the TPS method as a
`function of foam density; linear fits for the white and
`black foams are included.
`
`Page 6 of 9
`
`

`
`1232
`
`ALMANZA, RODRI´IGUEZ-PE´ REZ, AND de SAJA
`
`Figure 5. Thermal conductivity measured by the
`TPS method as a function of the cell diameter; the
`linear fit corresponds to the LD white foams.
`
`come of reducing the cell size is to increase the
`number of cell walls the radiation has to pass
`across. As the heat flow by radiation is absorbed
`or scattered in the cell walls, reducing the cell
`diameter results in a lower conductivity. More-
`over, adding black carbon to the cell walls in-
`creases dramatically the extinction coefficient of
`the polymer; therefore, the radiation is more ab-
`sorbed in black foams decreasing the heat flow.
`For instance, the value of thermal conductivity
`obtained for the LD24(LC) foam (see Table 2), the
`material with a large cell size (956 ␮m), was
`clearly higher than the value of thermal conduc-
`tivity for the LD24(FC) sample. This last material
`had cells of smaller diameter (315 ␮m). A similar
`result was found when the influence of the black
`carbon content was analyzed. In Table 2 the val-
`ues of thermal conductivity obtained in a white
`LD33 foam and in a black LD33 foam are col-
`lected. This last sample presented a low content
`(2 wt %) of black carbon in their composition and
`had a thermal conductivity 6.7% smaller than the
`LD33 white foam.
`HL foams have higher thermal conductivities
`than LD ones. The main reason for this difference
`is the chemical composition of the polymer matrix
`of HL materials. These foams are produced from a
`blend of LDPE and HDPE. The HDPE phase has
`a higher thermal conductivity than the LDPE
`phase, which increases the thermal conductivity
`of the blend.
`As previously mentioned, the preceding trends
`can be deduced from both the data measured by
`TPS and the standard technique. From a scien-
`tific point of view, the effect of cell size and black
`
`carbon content on the conductivity are due to the
`contribution of the radiation term on the whole
`conductivity. For these materials, it can be stated
`that TPS technique detects the radiation heat
`flow.
`However, the thermal-conductivity values ob-
`tained with the TPS method are always higher
`than the values obtained with the rapid K equip-
`ment. The differences were for almost all the
`foams under study approximately 20% (Table 2).
`Also, these differences were clearly higher (ca.
`40–50%) for the solid sheets.
`To analyze the previous results, it is conve-
`nient to introduce several notions on the precision
`of these two techniques.
`Several interlaboratory studies were carried
`out in an attempt to determine the precision of
`the steady-state technique. Various types and
`sizes of heat-flow meter apparatus were involved
`in the American Society for Testing and Materi-
`als/National Mineral Wool Association round
`robin on fibrous blankets.14 The results indicated
`that the overall precision was ⫹2.5%, a similar
`order of the guarded hot plate. This was con-
`firmed by a study15 on random specimens of ex-
`panded polystyrene materials. Although differ-
`ences of 1–2% were found in the absolute values
`for five different specimens, all
`laboratories
`ranked the specimens in the same order illustrat-
`ing the value of the method for comparative stud-
`ies.
`As far as we know, there are no detailed stud-
`ies on the precision of equipments based on the
`TPS method. However, the precision on the line-
`source method, a procedure not equal but based
`on similar principles, was evaluated at 300 K on
`several materials.16 These included cellular ma-
`terials including a well-aged polystyrene board.
`The results of the investigations were not encour-
`aging. They indicated standard deviations of
`some 20% for the different foams, and for insula-
`tions the differences in results exceeded 10% from
`the accepted values obtained with steady-state
`methods.
`In our opinion, several reasons could explain
`the differences between the ␭values measured by
`both techniques.
`A first contribution has its origin on the tem-
`perature gap between the heat-flow meter sur-
`faces and those of the sample. Figure 6 illustrates
`this concept. The temperature difference contrib-
`uted to the reduction of the heat transferred by
`conduction and radiation.
`
`Page 7 of 9
`
`

`
`TRANSIENT PLANE SOURCE METHOD
`
`1233
`
`Figure 6. Diagram showing the temperature difference between the plates of the
`rapid K equipment and the temperature of the sample surface (contact resistance).
`
`It is possible to estimate a corrected thermal
`conductivity from the gap temperature and the
`measured conductivity. With the Fourier law, the
`following relationships can be deduced:
`
`␭corrected ⫽
`
`␭measured共TU ⫺ TI兲
`共TU ⫺ TI兲 ⫺ 共⌬T兲
`
`(9)
`
`⌬T ⫽ ⌬T1 ⫹ ⌬T2; ⌬T1 ⫽ TU ⫺ TUS;
`and ⌬T2 ⫽ TIS ⫺ TI
`
`where ␭measured is the thermal conductivity deter-
`mined by applying the procedure described in the
`Experimental section; ␭corrected is the value of the
`thermal conductivity corrected by the influence of
`the temperature difference previously mentioned;
`⌬T1 is the temperature difference between sam-
`ple and heater surface in the hot plate; ⌬T2 is the
`temperature difference between the sample and
`heater surface in the cold plate; TU and TI are the
`temperatures of the hot and cold plates, respec-
`tively; and TUS and TIS are the temperatures of
`the sample surface in the hot and cold plates,
`respectively.
`The temperature differences ⌬T1 and ⌬T2 were
`measured for the sample LD33(1). The results for
`these measurements are collected in Table 3. The
`temperature gap increased approximately 7% the
`value of the thermal conductivity.
`
`The temperature gap was much smaller in the
`hot-disk equipment. As mentioned in the Experi-
`mental, in this method this gap was reduced by
`cutting the first points in each transient record-
`ing.
`The temperature gap could also be one of the
`reasons for the large differences between the con-
`ductivity of the solid sheets measured by both
`methods. The solid sheets as compared with the
`foams were very stiff. As a consequence, a small
`deviation from a completely flat surface resulted
`in a considerable increase of the temperature gap
`existing in the rapid K measurements.
`A second possible cause of difference between
`the values obtained for the thermal conductivity
`with the heat-flow meter and the TPS method is
`the thermal state of the sample in which the
`equipment measured the values. Rapid K mea-
`sures a stationary heat flow, whereas TPS mea-
`sures the transient regimen. This dissimilarity
`could result in a systematic difference between
`the values of the thermal conductivity measured
`by both methods.
`Other possible sources of discrepancy are the
`slight differences in the average temperature of
`the samples; the size of the samples is much
`smaller in the transient method, which in case of
`inhomogeneous samples could also influence the
`difference between the methods and so forth. A
`deep analysis of these differences is outside the
`
`Table 3.
`
`Foams
`
`L
`(cm)
`
`TU
`(°C)
`
`TUS
`(°C)
`
`Gap between TU
`and TUS (K)
`
`LD33W 1.10
`
`39
`
`38.1
`
`0.9
`
`TI
`(°C)
`
`9
`
`TIS
`(°C)
`
`10.2
`
`Gap between
`TI and TIS (K)
`
`100*
`(1 ⫺ ␭corrected/␭measured)
`
`1.2
`
`7
`
`L(cm) is the sample thickness, TU is the temperature of the hot plate, TI is the temperature of the cool plate, TUS is the
`temperature of the face in contact with the hot plate, and TIS is the temperature of the face in contact with the cool plate.
`
`Page 8 of 9
`
`

`
`1234
`
`ALMANZA, RODRI´IGUEZ-PE´ REZ, AND de SAJA
`
`scope of this work. However, these differences
`should be analyzed systematically in the future.
`
`CONCLUSIONS
`
`The thermal conductivity of a collection of closed-
`cell polyethylene foams has been measured by
`both a steady-state method and a TPS technique.
`Both methods gave similar trends for this prop-
`erty, an important result, which validated the
`TPS method as a tool for analyzing the structure–
`property relationships for these two phase mate-
`rials. However, it has been also shown that there
`is a systematic difference between the numerical
`values given by the two methods. Several possible
`sources for these differences have been proposed.
`The TPS technique seems to be a powerful
`technique for comparative studies of the thermal
`properties of insulating materials because of sev-
`eral factors such as fast measurements, possibil-
`ity of using small samples, potential analysis for
`heterogeneity and anisotropy, and so forth. How-
`ever, considerable thought and effort must be ex-
`pended to analyze the absolute values given by
`the method.
`
`O. Almanza thanks La Secretaria de Estado de Educa-
`cio´n y Universidades of Spain for the concession of a
`postdoctoral grant. Financial assistance from the Junta
`de Castilla y Leon (VA026/03) is gratefully acknowl-
`edged.
`
`REFERENCES AND NOTES
`
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`1004.
`
`2. Almanza, O.; Arcos y Ra´bago, L. O.; Rodriguez-
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`mol Sci Phys 2001, 40, 603–613.
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`Page 9 of 9