The thermal conductivity of carbon | My Assignment Tutor

JOURNAL OF MATERIALS SCIENCE 14 (1979) 1326-1338The thermal conductivity of carbonfibre-reinforced compositesM. W. PILLING*, B. YATES, M. A. BLACKDepartment of Pureand Applied Physics, University of Salford, Salford, UKP. TATTE RSALLFothergill and Harvey Ltd, Littleborough, Lancashire,UKMeasurements of the thermal conductivity between approximately 80 and 270 K of aseries of unidirectional and bidirectional specimensof epoxy resin DX210/BF3400reinforced with Morganite high modulus (HMS) and high strength (HTS) carbon fibres arereported for in-plane and out-of-plane directions. The main features of the resultsconform with expectations based upon known structural properties of the fibres andpredictions basedupon current theoretical models. Employing the results for thecomposites in associationwith results for the pure resin, the account concludes with anassessmentof some of the heattransmission characteristics of the fibres.1. IntroductionA recognition of the importance of the highstiffness and strength weight ratios of carbon fibrereinforced plastics (CFRP)has led to extensiveinvestigations of their mechanical properties andassociated effects of environmental degradation(e.g. [1–4] ). More recently, an appreciation of theinfluence of temperature change has led to investigations of the thermal expansion characteristicsof various CFRP laminae and laminates [5, 6].One outcome of these investigations has been thatfeatures of the behaviour of CFRP structuresoperating under realistic working conditions cannow be assessed with a reasonable degree ofconfidence.The dissipation of heat by these materials is afurther subject of some importance, but theshortage of experimental thermal conductivitydata has so far prevented a comprehensive appraisalof their heat transmission characteristics. Theinvestigation to be described was undertakenagainst this background, the object being toprovide characterization data for immediatetechnological application, which might also beemployed for assessing the standing of currenttheoretical models of heat conduction in compositesolids.*Presentaddress: ChlorideTechnical Ltd, Manchester,UK.13262. The specimens2.1. Constituent materialsFor the purpose of the present investigationattention was concentrated upon one resin systemand two types of carbon fibre. The resin wasDX210, the hardener BF3400, both of which wereproduced by Shell Chemicals Ltd. The type I(HMS) and type II (HTS) carbon fibres employedwere both manufactured by Morganite ModmorLtd from polyacrilonitrile precursor.2.2. Pure resinSpecimens of the pure resin were required in theform of circular discs, 35ram diameter. Thesewere cut from a cylinder which was cast in arubber mould having this internal diameter. Theprocedure adopted was to add the appropriateamount of hardener to a quantity of 80% resin in asolution of methylethylketone and to place theresulting mixture in a vacuum oven at 100~together with the mould, the inside of which hadfirst been sprayed with a silicone release agent.The oven was pumped for approximately 10rain,until the resin had ceased to bubble. After ventingthe oven the resin was poured into the mouldwhere it was pumped again until bubbling ceased,following which the oven was again vented to the9 1979 Chapman andHall Ltd. Printed in Great Britain.atmosphere and the curing of the resin was allowedto proceed for 1 h, during which it hardened. Theresin cylinder was then removed from the mouldand cooled. Upon being found to be void-free andglass-clear its temperature was finally raised to150 ~C, at which it was held for 2h in order tocomplete the curing cycle.2.3. C o m p o s i t e barsThe composite bars were assembled from sheets offibre pre-impregnated with resin (pre-preg). Havingdecided upon the fibre volume fraction required ina bar, the required volume of 40% resin in solutionwas calculated. This figure was then increased by25% in order to allow for losses and to ensure agood resin flow during the final moulding process.This was poured evenly over the fibre tows, whichwere laid on a non-stick surface of siliconizedpaper. After manipulation, suitably wetted towswere selected and laid parallel to and equidistantfrom one another on another sheet of siliconizedpaper. A further sheet of siliconized paper was laidon top and the sandwich was transferred to a hotplate, where it was rolled to remove trapped airand to produce a more nearly uniform distributionof fibres. The top sheet of siliconized paper wasremoved and the pre-preg was heated at 80 ~ C for2min in order to achieve a partial cure and todrive off most of the solvent. It was then removedfrom the oven, cooled, the top sheet of siliconizedpaper was replaced and the pre-preg sheet wasreturned to the hot plate, where it was rolled again.This process was repeated until the pre-preg wasfirm, tacky and contained a uniform dispersion offibres, following which it was stored in a freezingcabinet until required.For thermal conductivity measurements in adirection parallel to the fibres bar specimens wererequired, while for measurements perpendicular tothe fibres plates were required. The leaky mouldtechnique was employed in the production of bothsets of specimens. The temperature of theassembly was raised to 170 ~ C under high pressure,following which the composite block was removedfrom the mould, which was then sprayed withrelease fluid for a second time before the blockwas replaced and cured under closing pressure for1 h at 170~ C. The pressure was controlled bystops which defined the final lateral specimendimensions. The block was then removed from themould and cured for a further hour at 170 ~ C.Great care was taken in accurately aligning theconstituent laminae of the different compositeblocks before these were assembled in the moulds.Precautions were also taken to standardize thefibre types, as far as this was possible, and all thettMS fibre employed in the investigation camefrom the same batch. In the case of the HTS fibrepractical problems necessitated the fibre requiredfor the -+45~ bar being taken from a differentbatch from that employed in the unidirectionaland 0~ – 9 0 ~ bar.2.4. S p e c i m e n preparationAll dimensions of the bar specimens were intentionally oversize. Approximately 7mm was cutfrom the end of each bar with a slicing wheel andthe remaining faces were reduced appropriatelywith the aid of a surface grinder until the dimensions had been reduced to 100minx 6 m m x6mm, by which time the fibre lay-up was properlybalanced in the cases of the cross-plies. Thesedimensional reductions were undertaken veryslowly and the bar temperature was monitored inorder to ensure that there was no overheating,which might have affected the physical characteristics of the bars. Disc-shaped specimens, approximately 40ram diameter, were produced from themoulded plates with the aid of a disc cutter. Thediameters were reduced to 35 mm and the thicknessof each disc was reduced to the values required bygrinding and polishing. A similar procedure wasadopted when adjusting the thicknesses of thespecimens of pure resin.Final determinations of fibre volume fractionwere effected by density determinations, employedin association with acid digestion. In the case ofthe bars, tests for voids were applied by the microscopic examination of polished surfaces of specimens, whereas in the case of the plates ultrasonicc-scan techniques were employed. Following thesetests, attempts were made to use void-free regionsfor specimen preparation, as far as this was possible.A list of the specimens prepared and investigated issummarized in Table h3. Experimental details3.1. The apparatusThe main features of the original apparatushavebeen described in detail by German [7]. Modifications were necessary in order to adapt thesystem so as to be able to accommodate andinvestigate the bar specimens, which had fairlyhigh conductivities, and the plate specimens, the1327TABLE I The specimens SpecimendesignationFibretype Anglebetweenfibre directions(deg)Direction of thermal Fibre Voidconductivity measurements volume content(%) (%)1 No fibres2 HMS 03 HMS 04 HMS 905 HMS 906 HTS 07 HTS 08 HTS 09 HTS 010 HTS 9011 HTS 900.0 Visuallyassessedtobe negligibleParallel to fibres 60.7 0.1Perpendicular to fibres 57.9 0.3Parallel to one set of fibres 55.7 0.8Bisectinganglebetween fibres 56.1 0.0Parallel to fibres 58.4 2.3Perpendicular to fibres 45.9 0.0Perpendicular to fibres 59.1 0.3Perpendicular to fibres 71.9 0.0Parallel to one set of fibres 62.1 1.2Bisectinganglebetween fibres 59.3 2.8thermal conductivities of which were much lower.For the sake of completeness both of the alternative modifications employed to accommodate thespecimens will be described briefly.& 1.1. Searlo’s bar systemThe modification employed for investigating thehigher conductivity bar-shaped specimens took theform of a Searle’s bar system. The cryostatassembly, which was suspended from the top capr20ramFigure 1 Cryostat assembly for the Searle’sbar system. S,specimen; Y, platinum resistance thermometer yokes; H,specimen heating coil; O, specimen mounting block; U,platinum resistance temperature sensor; T, top cap ofradiation shield R; A, thermal anchoringpost; C, specimenchamber;B, outer can.1328of a Dewar vessel by three stainless steel tubes, isillustrated in Fig. 1. It consisted of a copperspecimen chambeL C, surrounded by an outerbrass can, B. The specimen, S, was positionedaxially in a holder which was located at the centreof the base of the specimen chamber. It wassurmounted by a copper heater block, H, andsurrounded by a stainless steel radiation shield, R,which terminated in a copper top cap, T, carryinga thermal anchoring post, A, for the electricalleads. Two copper specimen mounting blocks werefabricated. One of these had a circular hole andwas used to hold an electrolytic iron reference bar;the other had a square hole and was used to holdthe CFRP bar specimens. Good thermal contactbetween the various demountable components wasachieved with the aid of Apiezon vacuum grease, Acopper block U was hard soldered to the base ofthe specimen chamberin order to provide a housingfor a platinum resistance thermometer which wasused as the sensing element of the specimenchamber temperature controller.A system of pumping tubes allowed thespecimen chamber and outer can to be evacuatedand further tubes provided entry for the variouselectrical leads. The specimen chamber was sealedto the bottom plate by “Loctite Stud Lock”,which provided convenient access when changingspecimens. With liquid nitrogen in the Dewarvessel, mean specimen temperatures betweenapproximately 80 and 270 K could be achieved bypassing current through a series of five heatingcoils which were wound on the outside of thespecimen chamber, C, while the enclosures withinB were under high vacuum. Temperatures could beheld steady within a few millidegrees with the aidof the sensor located in U, which formed one armof an a.c. Wheatstone bridge. The out-of-balancesignal from the bridge ultimately activated thecontroller of current supplied to the specimenchamber heating coils. In order to make a measurement, current was passed through the specimenheating coil, H, and the temperature gradientestablished along the specimen was measured withthe aid of platinum resistance thermometerslocated within the copper yokes, Y. The yokeswere bolted to the specimen with which theymade contact through blunted knife edges. It wasnecessary to ensure that the temperature gradientalong the radiation shield, R, was identical to thatalong the specimen. This was achieved at its lowerend by thermally anchoring the base of the shieldto the specimen holder, while at the upper end,the out-of-balance signal from a differentialthermocouple between the heater block, H, andthe top cap, T, of the radiation shield activated thesupply of current to a heating coil wound on thetop section of the radiation shield.3. 1.2. Lees” disc systemFor the investigation of the lower conductivitydisc-shaped specimens, a Lees’ disc arrangementwas adopted. This is illustrated in Fig. 2, in whichthe outer brass can, B, the specimen chamber, C,8 ~C ~S ~UFigure 2 Cryostat assembly for the Lees’ disc system. S,specimen; H, specimen heating block;. U, platinumresistance temperature sensor; R, radiation shield; N,nylon ring; A, thermal anchoring post; L, specimensupport block;C, specimen chamber; B, outer can.and the temperature sensor at U are the same asbefore. The specimen disc, S, was sandwichedbetween two copper blocks, which contained theplatinum resistance thermometers. The upperblock terminated in a pillar, H, on which waswound a heater. The lower block, L, fitted into thebase of the specimen chamber and good thermalcontact between the various surfaces was achievedwith the aid of low vapour pressure greases,“Nonaq” grease being used between the copperblocks and the specimen because of its relativelylow viscosity at reduced temperatures. Theradiation shield, R, consisted of three sections, theupper and lower members of which were constructed from copper. A nylon insert, N, corresponded in length and position with the specimen,and temperature equality between the uppersection of the shield and the upper sandwich blockwas achieved with the aid of a differential thermocouple which activated a heater wound on theupper section of the shield. The electrical leads tothe specimen heater, H, were thermally anchoredto the pin, A, and temperature equality betweenthe lower secti6n of the shield and the blocksupporting the specimen was achieved by thermallyconnecting these through the base of the specimenchamber.3.2. Performance of the apparatusBy way of assessing the accuracy with which bothforms of the apparatus would reproduce established results for well-defined substances, theinvestigation was preceded by a preparatoryproving programme.In the case of the Searle’s bar system the testsubstance employed was a rod of electrolytic iron,SRM 734, obtained from the National Bureau ofStandards. A comparison of the present primarydata with the smoothed NBS data [8], illustratedin Fig. 3, reveals agreement within the combinedexperimental uncertainties at all temperatures.In the case of the Lees’ disc system, vitreoussilica was adopted as the test substance. Thethermal conductivity of this material has beeninvestigated by a number of workers, employingsamples drawn from a variety of sources. The mostmeaningful comparison involving the presentresults can probably best be made with themeasurements reported by Ratcliffe [9] at theNational Physical Laboratory, whose specimen ofVitreosil came from the same supplier as thepresent specimens, Thermal Syndicate Ltd. From!329Figure 3 The thermal conductivity K of electrolytic iron, (SRM 734 from the National Bureauof Standards) , in which the vertical lines indicatethe estimated uncertainty limits: ~ presentprimary data; I—{ smoothed NBS data [8].120TE 1008Coo o o100 150 200 250T (K)1,2TT~ e1″11’Coo oo 0 0 0 01″/5 260 225 250 275T (K)Figure 4 The thermal conductivity K of vitreoussilica, in which the vertical lines indicate theestimated uncertainty limits: r ~ presentprimary data employing Rosemount and Mincothermometers respectively;I—{smoothed NPLdata [91.the comparison illustrated in Fig. 4 it may be seenthat the two sets of results agree within thecombined experimental uncertainties at alltemperatures.Platinum resistance thermometers from twosources were employed during the course of thework, all of which were calibrated by CryogenicCalibrations Ltd. In the earlier part of the investigation, thermometers supplied by the RosemountEngineering Co Ltd were employed in the investigation of all the bar specimens, including theelectrolytic iron reference specimen. With theexception of specimens 3, for which RosemountThermometers were used, all the disc specimenswere investigated using thermometers produced byMinco Products Inc and supplied by CryogenicCalibrations Ltd. For this reason the provingprogramme undertaken with vitreous silicaincluded the use of thermometers drawn fromboth sources, the results of measurementsemploying which are illustrated in Fig. 4.13304. Results4.1. Preliminary considerationsFor convenience of analysis the thermal conductivities of the specimens are summarized insmoothed form in Table II. These values were readfrom smooth lines drawn by eye through plots ofthe primary experimental data.In the case of the results taken with the Searle’sbar system, after confirming that the apparatuswas producing reproducible results from oneoccasion to another, the accuracy to which theeffective separation of the thermometers could bemeasured was examined by altering the separationof the thermometer yokes and repeating measurements during a completely separate run. Anadditional precaution was applied by testing forany possible influence of the method of specimenpreparation Upon the uniformity of temperaturethrough a plane at right angles to the direction ofheat flow. This was done by releasing the boltssecuring the thermometer yokes to a specimen,TABLE II Smoothed values of the thermal conductivity K of the specimens described in Table I T(K)80K (Wm-1 K-I) for the specimens numbered below6(1.55)7891234(3.30)5(3.64) 10 11(0.94) (1.02)90 9.4 4.30 4.61 1.90 1.20 1.24100 12.0 5,30 5.64 2.27 1.45 1.46110 14.6 6.35 6.69 2.65 1.70 1.70120 17.2 7,40 7.75 3.05 1,95 1.94130 19.7 8.50 8.80 3.43 2.20 2,18140 22.2 9.66 9.89 3.84 2.45 2,45150 24.6 10.9 11.0 4.25 2.70 2,71160 27.1 12.0 12.0 4.66 2.95 2.99170 29.5 13.1 13.1 5.08 3.20 3.28175 0.194 30.7 1.194 13.6 13.6 5.29 0.457 0.506 0.746 3.33 3.42180 0.196 32.0 1.219 14.1 14.1 5.50 0.463 0.523 0.764 3.46 3.56190 0.198 34.4 1.289 15.1 15.2 5.93 0.473 0.553 0.793 3.71 3.87200 0.201 36.6 1.331 16.1 16.2 6.35 0.485 0.575 0.829 3.96 4.17210 0.204 38.9 1.378 17.1 17.1 6.77 0.499 0.598 0.874 4.21 4.48220 0.207 41.1 1.413 18.0 18.0 7.20 0.514 0.623 0.926 4.46 4.79230 0.210 43.2 1.423 18.9 18.9 7.62 0.530 0.649 0.968 4.71 5.10240 0.214 45.2 1.403 19.8 19.7 8.05 0.551 0.676 1.010 4.46 5.42250 0.218 47.1 1.410 20.7 20.6 8.48 0.574 0.701 1.048 5.21 5.75260 0.222 48.9 1.447 21.7 21.3 8.90 0.593 0.726 1.089 5.47 6.06270 0.226 50.6 1.463 22.6 22.0 9.33 0.607 0.747 1.133 5.73 6.39~” 4[‘E2O []u ~SBQoo u~Dt~o ~ / p u0ooo ~ oo~ o~oOf~100 150 200 250T [K)Figure 5 The thermal conductivity K of specimen 2: o run1 (thermometer separation 1); [] run 2 (thermometerseparation 1); ~ run 3 (thermometer separation 2); o run4 (thermometer yokes clamped to the other pair ofopposite faces, at separation 3).turning the specimen through 90 ~, tightening thebolts and taking the specimen through an additional sweep of the temperature range, duringwhich a further sequence of measurements wastaken under conditions of temperature equilibrium.The results from these four runs are plotted inFig. 5, from which it may be seen that they agreewith one another within the limits of experimentaluncertainty. A detailed consideration of theexperimental uncertainties involved in measurements made with the Searle’s bar system indicatesthat results taken with the electrolytic ironreference sample should be accurate to within-+3% at 80 K, rising to + 4% at 270 K. In the caseof the composite bars containing HMS fibre, thefigures are similar, while for the composite barscontaining HTS fibre, the corresponding figuresrange between -+5% and + 7% at 80 K, to between-+7% and -+10% at 270K.In the case of the disc specimens it was necessaryto correct for the thermal resistance of the layersof grease separating the blocks carrying the thermometers from the specimens. From a considerationof the thermal resistance of the sandwich consistingof a disc specimen flanked by layers of grease, it isnot difficult to show that the thermal conductivity,K, of the specimen of thickness ts is related to thethermal conductivity, Kg, of the grease layers,having an overall thickness tg, by the equationts= gr.]in which Km is the measured thermal conductivityof the sandwich. Measurements of values of K ~for discs of different thickness t s clearly providedata for a graph of t s against t s / K m , the slope ofwhich should give the required thermal conductivity, K, of the specimens directly. Preliminarymeasurements indicated that the influence of smallvariations in tg from one sample to another weresufficiently small to be discounted. This strength-1331ened the basis upon which corrections wereapplied for the grease layers in subsequent measurements upon the discs, groups of between three andfive of which were employed, having thicknessesranging between approximately 1 and 5mm. Adetailed consideration of the experimental uncertainties in the case of the Lees’ disc systemyielded figures ranging between +-3% and +-10%over the temperature range of the measurements,the low temperature limit of which was limited bythe viscosity of the grease.4.2. Comparison with related w o r kThe value of a comparison of the results of aninvestigation such as the present one with those ofother workers is limited by the number of variablesinvolved, although it does help to create a contextwithin which a perspective may be gained in thelong tenn. Perhaps the most relevant investigationfor the present purpose is that of Knibbs et aLFigure 6 The thermal conductivity Krof epoxy resins DX 210/BF3400 andMY750/DDM, in which the estimateduncertainties are indicated by thevertical lines: ~ present results forspecimens 1 (DX210/BF3400), ~ theresults of Knibbs et aL [10] forMY750/DDM.03E030TTE0″250.20[10] with whose results the present thermalconductivities are compared in Figs. 6 to 8.The temperature dependence of the thermalconductivity of the resin DX210/BF3400,displayed in Fig. 6, contains no unusual featuresand an extrapolation of the results to the assumedtemperature of the isolated result for epoxy resinMY750/DDM[10] indicates that these havesimilar magnitudes. The results for the presentunidirectional specimen reinforced with HMS fibre,displayed in Fig. 7, extrapolate to a value similarto that of the corresponding result of Knibbs et al.[10], the fibre volume fraction of whose specimenwas comparable, though the origin of whose fibresis not known. Combining this uncertainty with theknown difference of identity of the resins, thecomparison reveals nothing surprising. Similarremarks apply to a comparison of the corresponding results for unidirectional compositescontaining HTS fibre, displayed in Fig. 8.26o 2~ 3~T (K)6040TE201332cooo0% d~%o c~~oo oooO oo o~ooco~0 ~ ~ o~0~0 ~O~Q~O~~176o cD 0r OZX O~ 0 ~ ~0 ~16o go z6o 2~oT (K)300Figure 7 The thermal conductivities K ofHMS carbon fibre-reinforced plastics: opresent results for specimen 2; [] presentresults for specimen 4; Apresent results forspecimen5; o Knibbset aL [10].YTE“~ 5000o ~00O0 0oo ~00000 o0 0o I~o is’o 2~o AoT (K)AAFigure 8 The thermal conductivities K of HTScarbon fibre-reinforced plastics: o presentresults for specimen 6; D present results forspecimen 10; z~present results for specimen 11;r KnibbsetaL [10].3005. Discussion5.1. Qualitative observationsA number of qualitative observations are possible:(i) the thermal conductivity of the pure resin isthe lowest of all the specimens examined;(fi)the thermal conductivity in the fibredirection of the HMS carbon fibre unidirectionallyreinforced plastic specimen 2 exceeds that of eachof the corresponding 90~ cross-plies, i.e. specimens4 and 5, which in turn exceed that in the transversedirection of the corresponding unidirectionallyreinforced specimen 3;(iii) the thermal conductivities of the 0o-90 ~and +45 ~ HMS carbon fibre-reinforced plasticspecimens 4 and 5 are identical, within the limitsof experimental uncertainty;(iv)the thermal conductivity in the fibredirection of the HTS carbon fibre unidirectionallyreinforced plastic specimen 6 exceeds that of eachof the corresponding 90~ cross-plies, i.e. specimens10 and 11, which in turn exceed that in thetransverse direction of the corresponding unidirectionally reinforced specimen 8;(v) the thermal conductivities of the 00-90 ~and +45 ~ HTS carbon fibre-reinforced plasticspecimens 10 and 11 are identical, within thelimits of experimental uncertainty;(vi)the thermal conductivity in the fibredirection of the HMS carbon fibre unidirectionallyreinforced plastic specimen 2 exceeds that of thecorresponding HTS fibre-reinforced specimen 6;(vii) the thermal conductivities of the 90~ crossplied laminates containing HMS fibre, i.e.specimens 4 and 5, exceed that of the corresponding HTS fibre-reinforced specimens 10 and11;(viii) the thermal conductivities in the transverse direction of the HTS carbon fibre unidirectionally reinforced specimens 7, 8 and 9 increasewith fibre volume fraction.None of these observations conflict withexpectation and the comparative observationsinvolving the HMS and HTS fibres accord with theidea of the c-crystal!ographic axes of the graphitecrystallites in the HMS fibres being more nearly atright angles to the fibre axis than those in the HTSfibres.5.2. Analysis of data5.2. 1. The Searle’s bar results5.2.1.1. The longitudinal thermal conductivities o fthe carbon fibres. The thermal conductivity of aunidirectionally reinforced composite in a directionparallel to the fibres, KI’1, may be expressed interms of the corresponding conductivities of thefibres, K~, and the resin matrix, Kr, by theequation,K]I = K~vf + Kr (1 — vf), (2)in which vf is the fibre volume fraction. Applyingthe results for specimens 1,2 and 6 to this equationhas permitted the calculation of the longitudinalconductivities, K~, of the fibres. These are illustrated in Fig. 9 together with the results of Volgaet al. [11]. The orientation of the c-crystallographic axis of the graphite crystallites to thefibre axis, which has been indirectly shown to berelated to their longitudinal thermal conductivity,may be expected to be directly related to thethermal history. Examination reveals two featureswhich are common to both sets of results. Thetemperature dependence of the longitudinalthermal conductivity is similar in both cases andthe absolute magnitude increases with graphitization temperature. Numerical differences13331334200TE~ =Y 100/ / 1 /#/ /// // z– bJ’” / A/ / c/ B,16o 1~o zbo 2~,o – –T (K)Figure 9 The longitudinal thermal conductivities K~ of carbon fibres expressed insmoothed form: A, present results for HMSfibre (approximate graphitization temperature2600~C); B, present results for HTS fibre(approximate graphitization temperature1500~C); a, b, c results of Volga et al. [11]for fibres heat-treated for 1h at temperaturesof 2800, 2600 and 1400~C, respectively.between the two sets of results probably arisefrom differences in the production of the fibres. The effect or fibre orientation. The basictransformation equation for the in-plane thermalconductivity of unidirectionaUy fibre-reinforcedcomposites may be written in the formK~ = K~I cos20-FK~2 sin20, (3)in which K~ is the thermal conductivity of thecomposite in a direction inclined at an angle 0 tothe principal fibre direction and K~2 is the value inthe transverse direction, i.e. when 0 = zr/2. Turningto the case of a bidirectional laminate consisting ofnl laminae orientated at an angle 0 to the directionof measurement and n2 exactly similar laminaeorientated at an angle ~, the contributions fromthe two laminae may be expressed in the terms ofEquation 3 and then added to give the thermalconductivityKo,e e = Po (K~I cos20 + K~2 sin20)+ pe (K~I cos2~ + K~2 sin2~b),(4)where Po =nl/(nx + n 2 ) and p~ =n2/(nl + n2)are called the packing fractions. All the bidirectional cases considered here were balanced, sothat Po = Pe = 1/2. In the cases of specimens 4and 5 Equation 4 reduces toKo:/2 = g+_Tr/4 = (K]Iwhich are particular cases of the more generalresultK coorl2 – o = ( K ~ , + K~2)/2, (6)20‘E15which follows directly from an addition ofappropriate terms having the form of Equation 3and illustrates the isotropy of the in-plane thermalconductivity of a balanced bidirectional laminate.The equality of the results for specimens 10 and11 is to be expected for the same reason. Figs. 7and 8 illustrate that the results for the bidirectionalcomposites containing HMS and HTS fibres bothconform with this expectation. Calculation of composite conductivities.Pursuing the self consistency of the results beyondthe establishment of equalities such as thoseconsidered above, Equation 5 permits the calculation of the in-plane thermal conductivities of thelaminates in terms of the corresponding values forthe appropriate laminae. The results of suchcalculations, after adjusting for differences of fibrevolume fraction and packing fraction, are illustratedoo oo300175 200 225 250 275T (K)Figure 10 The thermal conductivity K of specimen 4, inwhich the estimated uncertainties are indicated by thevertical lines: { measured; }—Icalctllated from the resultsfor specimens2 and 3.25x: 2015 ,752~o – –T (K)2:~52~,o27~ Figure 11 The thermal conductivity K of specimen 5, inwhich the estimated uncertainties axe indicated by thevertical lines: { measured; }—-tcalculated from the resultsfor specimens 2 and 3.the fibre for specimen 11 being drawn from adifferent batch from that employed for specimens6 and 10 and it seems likely that herein lies thecause of the outstanding difference.5.2.2. The Lees” disc results5.2.2.1. General. Various models have been putforward in an attempt to express the transversethermal conductivity of a unidirectional compositein terms of the thermal conductivities of thematrix and the included fibres. Some of these areexamined below, when the relative credibilities ofthe models will be assessed by the extent to whichthey provide a self consistent account of some ofthe experimental data.Z Y J2 17~ 26o ~’~s 2~oT (K)Figure 12 The thermal conductivity K of specimen 10, inwhich the estimated uncertainties are indicated by thevertical lines: { measured; I—-Icalculated from the resultsfor specimens 6 and 8. ‘E~4o{JJ 2 175 260 225 250T CK)Figure 13 The thermal conductivity K of specimen 11, inwhich the estimated uncertainties axe indicated by thevertical lines (see text): { measured; }—I calculatedfrom the results for specimens 6 and Figs. 10 to 13. In the first three of these figuresagreement between observation and calculationmay be seen to be within the limits of experimentaluncertainty. In the case of specimen 11 theagreement lies outside these limits. As explained inSection 2.3, practical considerations necessitated5.2.2.2. The series electrical resistance analogy.Drawing an analogy with an electrical circuit inwhich the components are connected in series,Thornburgh and Pears [12] derived the equation1 v~ (1 – v0 (7)K% = + – g – , ‘in which K~ is the thermal conductivity of thefibres in the transverse direction and the otherterms have their previous meanings. Re-arrangingthe terms in this equation and assuming variousvalues for the ratio KJK{, a comparison with theresults for specimens 7, 8 and 9 reveals an improving agreement with experiment as Kr/K~ increasesto infinity. However, even in this limiting case,agreement is poor and it is necessary to resort to amodified model in order to achieve improvedagreement. Ignoring the. influence of voids, “thevolume fractions of which are very low in thepresent cases, and making no allowance for directfibre-to-fibre contact, the form of their modifiedequation adopted for the comparison illustrated inFig. 14 wasK~2 = BX~ (1 — Vd +[Vf + (1 –B)(1 –vf)] K~Kr[v~Kr + (1 –B)(1- v d K ~ l ” (8)In this equation, B is the volume fraction of theresin which is assumed to present a continuouspath for heat flow. A more sophisticated representation would involve taking account of a decreaseof B which must accompany an increase of vf, butfor the present purpose a constant value ofB = 0 . 4 was assumed. Within the limitationsimposed by the available experimental data and1 3 3 5 XI~1″0-~”-.T u~o.s . 7 . – . – ” –sb 6bvf (%)Figure 14 The transverse thermal conductivities Ke2 0fthe unidirectional composites reinforced with HTS fibre.Smoothed experimental results: o, 180K; % 225K, z~,270K. Results calculated on the basis of the series electrical resistance analogy; . . . . . .(K~ = 1.7 W m-~K-‘), – – – – – –Wm-1 K-l).( K f = 3.5Wm-‘ K-‘), – – ( K f =10.0 assumptions, the agreement achieved by assumingvalues for K~ of 1.7, 3.5 and 10.0Wm -1 K -1 at180,225 and 270 K, respectively,must be regardedas being reasonably favourable. The in-plane shear field analogy. The shearstress distribution in a filamentary composite hasbeen discussed in terms of a basic unit cell underthe assumptions of a regular periodic array ofparallel fibres, with both macroscopic homogeneity, local fibre isotropy and matrix isotropy,by Adams and Doner [13]. Springer and Tsai [14]showed that a series of transformations could beemployed to develop an analogous treatment oftransverse heat flow, in which displacement wasreplaced by temperature, the average shear stresswas replaced by the average heat transfer rate andthermal conductivity replaced shear modulus9Thetreatment of Adams and Doner proved to beinconvenient for immediate application and asimpler analysis of thermal conduction in compo-9 .sites was achieved by transforming the Halpin-Tsaiequations to the form: K~2 = 1 + ~r~v,(9)KrI — r/vf(K~IK~) — 1in which7? = (K~/Kr) + ~ ‘ and d/t was the aspect ratio of the fbres. Applyingthis treatment to the present results in a mannersimilar to that employed previously resulted in the1336comparison depicted in Fig. 15. The degree ofagreement is comparable with that achieved withthe earlier model, but this time the transversethermal conductivity of the fibre came out to be2.0,3.1 andh.7Wm -1 K-1 at 180,225 and 270K,respectively.5.2.29 The axial shear loading analogy. Adoptinga different approach to the problem through anessentially similar analogy, Hashin [15] derivedpredictive equations for upper and lower limits forK~2. Agreement between the experimental resultsand the upper limiting value involved assumingvalues for K~ which were much lower than thoseassumed in Sections 5.29 and and eventhen agreement was poor. Meanwhile the lowerlimiting value proved to be identical to thatemployed in Section The elastic moduli analogy. This model,developed by Nielsen [16], is essentially anextension of the previous models, in which accountis taken of the shape and packing of the fibres, aswell as an allowance being made for possiblevariations in fibre dispersion. In this case use ismade of a modifcation of the Halpin-Tsaiequations, which take the form: K ~Kr1 + CDvf1 –D~vf’(lO) where C = k E – 1, kE being the generalizedEinstein coefficient,D = (K~/Kr)– 1(K~/Kr) + C1’0 99 o~ux:~’O.5 ~ 9 o5b 6b 76vl (%)Figure 15 The transverse thermal conductivities K2e ofthe unidirectional composites reinforced with HTS fibre9Smoothed experimental results: o, 180K; o, 225K; a,270K. Results calculated on the basis of the in-plane shear field analogy: . . . . . .(K~= 2.0Win -1 K-l),– – – – – – (K_~= 39K-a).K-‘), – –(Kf = 5.7Wm -1 {1 – vm+ I T ) “,o = vm is the maximum packing fraction for fibres indifferent packing schemes and the other termshave the same definitions as before.Comparison with the in-plane shear fieldanalogy reveals that D, like r~, provides a measureof the relative conductivities of the two phases andC, like ~’, depends upon the shape and orientationof the fibres. Vrn is defined as the ratio of the truevolume of the dispersed phase to the volume itappears to occupy when at its maximum packingdensity and ff allows for the type of fibre packingin the system. For an ideal lay-up of uniaxial fibresthe appropriate value of C in the case of transverseheat flow is 0.5. However, Lewis and Nielsen [17]have shown that for composites containing fibreswhich are bunched, a value of C = 0.84 is morerealistic. Adopting this value of C and assumingthe typical value K[/Kr = 25 from the earliermodels, K~2/Kr has been evaluated at 270K as afunction of ve for values of vna corresponding tohexagonal (0.907), random (0.820) and square(0.785) packing, producing the results which areillustrated in Fig. 16. It is difficult to concludefrom this figure whether hexagonal packing isfavoured at all levels of fibre volume fractionexamined or whether the form of fibre dispersiontends to adjust itself towards the hexagonalmodification as the fibre volume fraction isincreased. More experimental evidence is requiredin order to be able to discriminate between thesepossible packing modes.2CE1cu ~/./C/ / . . A.”/ / ./ / 9 9 I *2~5 – – 50 7~5–vf (%)Figure 16 The reduced thermal conductivities Ke2/Kr ofspecimens 1, 7, 8 and 9: o experimental data. Resultscalculated from Equation 10: . . . . . . A (assuminghexagonal packing);– — — B (assumingrandom packing);– – C (assumingsquare packing). The long waves model. Employing amethod of analysis introduced earlier, Behrens[18] developed a model which involved calculations of the damping coefficients of thermalwaves having wavelengths which were longcompared with the inter-component spacing.Applying his treatment to the present situationfinally led to an equation for K~2 which wasidentical to the corresponding equation emergingfrom the in-plane shear field analogy.5.2.2. Z The transverse thermal conductivity ofcarbon fibres. Analysis of the experimentalcomposite results, obtained with the aid of theLees’ disc apparatus, provides the basis of assessments of the temperature dependence of thetransverse thermal conductivity of the carbonfibres. The approximations and assumptionsinherent in the models limit the reliance whichmay be placed upon any values derived with theiraid. Likewise structural imperfections in thespecimens and experimental uncertainties restrictthe applicability of any theoretical models.Subject to these constraints, the results displayedin Fig. 17 have been derived employing theequation K[ = Kz (1– v~) — K~2 (1 +v~)– vf) – (1 + v0″(11) This equation corresponds to the treatments basedupon analogies in elasticity, with which a reasonable degree of agreement with experiment hasbeen described. The small difference between theterms in the denominator prevents a realistic,~176 It4:X” 0″500.25175 200 225 250 275T (K)Figure 17 The transverse thermal conductivity K~ of HTSfibre calculated from Equation 11 and expressed inreduced form, in which the value of K~ at 270 K, i.e.(K~)~o, was taken as 6.0Win -1 K-1.1337assessment of K [ in the case of HMS fibre and thevalues displayed for HTS fibre were based uponresults for specimens 9, i.e. the specimenscontaining the highest fibre volume fraction, forwhich the uncertainty in the magnitude o f thedenominator of Equation 11 was least. The resultsare displayed in reduced form, in the derivation ofwhich a value of K [ = 6.0 W m -1 K-1 was assumedat 270K, in Fig. 17.6. ConclusionsReservations surrounding the application ofidealized theories to interpretations of experimental data for physically imperfect materialsinevitably qualify the value of deductions basedupon them. in spite of this limitation, in additionto augmenting the relatively few data available forthe thermal conductivity of carbon fibre-reinforcedplastics and their constituents, the present experimental results have served to confirm a number ofexpectations based upon analytical treatments ofthermal conduction in fibre.reinforced composites.A comprehensive appraisal of the standing oftheoretical models must await the production ofmore nearly structurally perfect materials andimprovements in experimental precision andaccuracy. Meanwhile it may be concluded fromthe self consistency of the interpretations examinedthat the degree of understanding afforded byanalogous treatments provides encouragement forthe further parallel development of experimentaland theoretical investigations in this field.AcknowledgementWe wish to express our gratitude to the ScienceResearch Council for a CAPS award received byone of us (M.W.P.).References1. R.B. PIPES, J. R. VINSON and TSU-WEI CHOU, J.Comp. Mater 10 (1976) 129.2, R.Y. KIM and J. M. WHITNEY, ibid 10(1976) 149.3. E. L. McKAGUE, J. D. REYNOLDS and J. E.HALKIAS, Trans. ASME Ser. H 98 (1976) 92.4. C. H. SHEN and G. S. SPRINGER, Jr. Comp. Mater.11 (1977) 2.5. K.F. ROGERS, L. N. PHILLIPS, D. M. KINGSTONLEE, B. YATES, M. ]. OVERY, J. P. SARGENT andB. A. McCALLA, J. Mater. ScL 12 (1977) 718.6. B. YATES, M. J. OVERY, J. P. SARGENT, B. A.McCALLA, D. M. KINGSTON-LEE, L. N. PHILLIPSand K. F. ROGERS, ibm 13 (1978)433.A. GERMAN, Ph.D. Thesis, University of Salford(1976).J. G. HUST and L. L. SPARKS, Data sheet for NBSSRM 734 (1971), Washington DC, USA.E. H. RATCLIFFE, Brit. J. AppL Phys. 10 (1959)22.R. H. KNIBBS, D. J. BAKER and G. RHODES,~ thermal and electrical properties of carbonfibre unidirectional reinforced epoxy composites”,26th Annual Technical Conference, ReinforcedPlastics/Composites Division, The Society of thePlastics Industry Inc. (1971).V. I. VOLGA, V. I. FROLOV and V. K. USOV,Inorg. Mater. 9 (1973) 643.J. D. THORNBURGH and C. D. PEARS, ASMEPaper 65-WA/HT-4 (1965).D. E. ADAMS and D. R. DONER, J. Comp. Mater. 1(1967) 4.G. S. SPRINGER and S.W. TSAI, ibid 1 (1967) 166.Z. HASHIN, NASA contractor report CR-1974(1972).L. E. NIELSEN, Ind. Eng. Chem. Fundam. 13(1974) 17.T. B. LEWIS and L. E. NIELSEN, Trans. Soc. Rheol.12 (1968) 421.E. BEHRENS, or. Comp. Mater. 2 (1968) 20 July and accepted 20 September 19791338


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