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Experimental Determination of the Pressure Dependence of The VOL. 81, NO. 17 JOURNAL OF GEOPHYSICAL RESEARCH JUNE 10, 1976 ExperimentalDetermination of the PressureDependence of the Thermal Diffusivity of Teflon, Sodium Chloride, Quartz, and Silica SUSAN WERNER KIEFFER Departmentof Geology,University of California,Los Angeles,California 90024 IVAN C. GETTING AND GEORGE C. KENNEDY Instituteof Geophysics,University of California,Los Angeles,California 90024 The thermaldiffusivity of Teflon,sodium chloride, quartz, and silicaglass was measured at 40øCto pressuresof 35, 18,30, and 36 kbar, respectively.A transientline sourcemethod was modified for usein a piston-cylinderhigh-pressure cell. Pressure gradients were determined by experimentswith bismuthfoils. The pressuredependence of thethermal diffusivity at 40øCfor thesubstances studied may be represented as follows(K in squarecentimeters per second,P in kilobars):for the low-pressurephases of Teflon, Teflon I-II, P < 5.5 kbar, K = 0.0012+ 3.6 X 10-sP;for the high-pressurephase, Teflon III, 5.5 kbar < P < 35 kbar, • = 0.0012 + 8.0 X 10-5 P; for polycrystallinehalite, P < 18 kbar, • = 0.0031 + 9.5 X 10-4 P; for quartz,perpendicular to the c axis,P < 30 kbar, • -- 0.031+ 5.3 X 10-4 P; for silicaglass, P < 36 kbar, • = 0.0068- 6.7 X 10-6 P. The diffusivityof silicaglass decreases with pressure,in contrastto the diffusivityof its crystallinecounterpart, quartz, which increaseswith pressure.In addition to the diffusivitythe thermalconductivity of Teflonwas determined by measuringthe powerapplied to the heaterwire. The thermalconductivity of a TeflonI-II mixtureis approximately constant at 0.0075-0.0078 cal/cms øK to 5.5 kbar.Above 5.5 kbarthe conductivityof TeflonIII isgiven by K: 0.0062+ 4.0 X 10-sP.The specificheat of Teflondecreases with pressureand decreases discontinuously by 15%across the Teflon lI-lII phasechange, in goodagreement with the decreasepredicted from thermalexpansion and compressibilitydata. INTRODUCTION determined,either allowing the specificheat to be determined There are only a few reported data in the literature on the through c = K/pK or providing a consistencycheck if the variation of the thermal conductivityor diffusivityof minerals specificheat is known; (2) the early part of the temperature- with pressure[Bridgman, 1924; Fujisawa et al., 1968;Schloessin time history may be used, thus minimizing violations of and Dt;orak, 1972].We report herenew thermaldiffusivity (K) boundaryconditions which are in generaldifficult to maintain data for Teflon, sodiumchloride, quartz, and silicaglass and in the pressureapparatus; (3) thermal propertiesare measured thermalconductivity (K) and specificheat (c) data for Teflon. in a singleplane, perpendicularto the line source;(4) sample Thesesubstances were chosenfor measurementsfor a variety designis amenableto the inherent pressurevessel geometry; of reasons:(1) Teflon, as a material that is commonly used (5) sample size maximized so that experimental errors due to in pressure cells, as a material for which values had been the finite size of the heater wire and the thermocoupleare published[,4ndersson and Bg'ckstr•m,1972], and as a material minimized; (6) sample preparation, while it is not trivial, is with phase transitions in the range of pressuresattainable relatively simple.The main disadvantageof the method is that [Pistorius,1964]; (2) sodium chloride (polycrystalline),as a data reduction is complex. crystalline substance whose atomic structure allows theo- An inherent difficulty with the application of most methods reticalcalculations of the thermal properties[e.g., Mooney and of simultaneouslymeasuring the thermal diffusivityand con- Steg, 1969]; (3) quartz, as a framework silicaterepresentative ductivity to high pressureis in providing an accuratedescrip- of minerals in the earth's crust; and (4) silica glass, as an tion o[ the boundaryconditions on the sample.The original amorphousmaterial for comparisonwith its crystallinecoun- line source method of Jaeger and Sass [1964] applies to a terpart, quartz. sample which is perfectly insulated or has only small heat losses.It is not possibleto insulatea mineral sample(which is THEORY: LINE SOURCE METHOD itself a good insulator) in a high-pressurevessel. We per- We modifiedthe transientline sourcemethod first proposed formed severalexperiments which showedthat the assump- by daeger[1959]and later developedby daegerand Sass[1964] tions of perfectinsulation or small heat lossesfrom the sample for application to a piston-cylinderhigh-pressure cell. In this are not even approximated at high pressure. We therefore method a thin linear heater serves as the heat source; the decidedto make the cylindricalsurface of the sampleas nearly temperature rise (t;) above ambient is recordedas a function of isothermalas possibleby placing the samplein contact with time (t) at a known distance from the heater. The thermal good thermal conductors.The following theory, then, is a diffusivityand conductivitymay be determinedfrom this tem- modification of the Jaeger and Sass [1964] theory to this perature-time(t;-t) curve and a knowledgeof the power output boundary condition. of the heater. Many advantagesof the method as used for An infinitely long circular cylinder 0 < t, < a (or equiva- work at 1-atm pressurewere listedby daegerand Sass[1964]. lently, one which is perfectly insulatedat the endsso that the In addition,the methodhas the followingadvantages for high- axial flux is zero, dv/dz = 0) is assumedto have a thermal pressurework: (1) the diffusivity and conductivityare both conductivityK, densityp, specificheat c, and thermal diffusiv- ity • = K/pc. K, •, p, and c are assumedto be independentof Copyright¸ 1976by the AmericanGeophysical Union. temperature. The initial differential temperature between the 3018 KIEFFERET AL.' EXPERIMENTALDETERMINATION OF THERMAL DIFFUSIVITY 3019 cylinder and its surroundings(v) is assumedto be zero (v = +z vo(r) = 0) at time zero (t = 0). The outer surface(r = a) is Ht.'• T.C. assumedto be an isothermalsurface (v = v(a) = 0 for all t > 0). For all time t > 0, there is a line sourceof infinitelysmall diameter parallel to the axis through the point whose polar coordinates are (r', 0). This sourceemits heat at the constant •r rate Q per unit length per unit time. The basicequation of heatconduction in cylindricalcoordi- tainless nates is --Pyro. 7 - = o + --Brass The ditTerentialtemperature v (henceforthreferred to simply Teflon as the temperature) at the point (r, 0) at time t (t • 0) is obtained by applying, successively,a Laplace transform, the dv =0 --Lead addition theorem, the inversiontheorem, and the theory of dz residues,or by direct integrationof the Green's function [Car- --Silver slaw and Jaeger, 1959, p. 385]. For a line source and the thermocoupleat equal distances,r' = r, the solution is ChromeI-Alumel v(t)=0-- hermocouple •Kv(r)_o 21 {--ln (•)+ In -- Z• qnCOS H0Z• •xp[-- K.n,,• 2t/•2] Jn2[a .... (r/a)] •Heater n=o m=o (•n,m•/a•)Jnt• (•n,m) ' (2) d._•v=0 where O is the heat emissionper unit length,K is the con- dz ductivity, • is the diffusivity,r is the radiusto the heaterand foil thermocouple(assumed equal), a is the cylinderradius, •, = 1 + Moly-coat for all n = 0, •, = 2 for n 2 1, andan,m are the positiveroots of J,'(a) = 0. This solution consistsof two terms: the first, a steady state term; the second, a transient term which is a flon complicatedfunction of the samplegeometry. EXPERIMENTALPROCEDURE AND DATA REDUCTION The experimentswere conducted in an end-loadedpiston- cylinder apparatusto a maximum pressureof 35 kbar. The initial temperature(To) wasroom temperature (about 27øC). 2.54 cm (1 in) The high-pressurecell whichcontained the samplewas placed Fig. 1. High-pressurecell for measurementof K, K, and pc by the in a largepressure vessel (3.18 cm (1{ inches)in bore diameter line source method. by 15.24 cm (6 inches)) in length. The outside of the vessel was cooled by continuouslycirculating water. The high-pressurecell is shown in detail in Figure 1. The immediatelyin contact with the samplebecause of its high sample(5.08 cm (2 inches)in length, 1.27cm (• inch) in diam- thermal conductivity.To assurereasonable stress distribution, eter) contained two diametrically opposing slots, passing however, a weaker bushing of lead (a slightly poorer con- through the points (r, 0) and (r, •); r was generallychosen ductor) was placed outside the sample over a substantial to be 0.7 of the sample radius. The heater wire and thermo- length of the cell. An insulator,Teflon, was placedat the ends couple (Chromel-Alumel) were placedi n opposedslots. For of the sampleto givea reasonableapproximation to the Teflon samples the wires were packed into the slots with boundary condition that the axial heat flux be zero (dv/dz = Teflon; glass was melted into the sIOtsof quartz and glass 0). samples;and sodiumchloride was packedinto the slotsof Ideally, the sampleenvironment should be hydrostatic.This the sodium chloride sample. The four wires from the heater conditionis impossibleto attain in a solid mediumpressure and thermocouplewere brought out from the high-pressure apparatus becauseof the finite shear strengthsof the various environment through a four-hole ceramic tube at the top of cell components.Weak materials(lead and Teflon) were used the assembly. wherever possiblein order to minimize shear stresses. The cell was designedto achievetwo conditions:(1) the Stress distribution in the cell was studied in a separate desiredthermal boundary conditionson the sidesand endsof experimentof geometry similar to the
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