Off-center hot spots: Double thermocouple determination of

American Mineralogist, Volume 83, pages 228-235, 1998
Off-centerhot spots:Doublethermocouple
determinationof the thermalgradientin
1.27cm (ll2 in.) CaF,piston-cylinder
furnaceassembly
JrNNrrnn M. Prcxnnnvc, Bn-LxooN E. ScrrwAB, ANDA. DaNa JoHNsroN
Departmentof Geological Sciences,University of Oregon, Eugene,Oregon 97403-12'72,U.S.A.
Ansrnacr
Double-thermocoupleexperiments were carried out at l0 kbar in a 1.27 cm (ll2 in.)
piston-cylinder apparatusto map the thermal gradient and locate the peak temperatureof
a CaFr-graphitefurnace assembly over the temperatureinterval 900-1500 oC. An unexpected but reproducible result was that the thermal peak is displaced upward significantly
(toward the steel base plug) from the vertical center of the graphite heater tubes.
The hot spot of our assembly,defined as the region no more than 10 oC cooler than the
peak temperature,is displaced upward from the center of the furnace and moves slightly
toward the center with increasingtemperature.At a maximum temperatureof 914 oC, the
centerof the hot spot was 2.6 mm abovethe mid-point of the furnace,while at 1510'C,
the highest temperatureinvestigated, the center of the hot spot was 2.2 mm above the
center of the furnace. The hot spot varies in width from3.2 fiun at lower temperafuresto
2.2 mm at 1510 'C. We attribute the upward displacementof the hot spot to the greater
thermal conductivity of the lower tungsten-carbidepiston (ft : 100WmK at 2fi) "C)
relative to the upper stainless steel base plug (/c : 25 WmK at 600 "C). The piston
apparentlyconductsheat more efficiently downward than the steel plug does upward, thus
displacing the hot spot upward from the furnace center.
InrnonucrroN
Steep thermal gradients are intrinsic to the design of
the piston-cylinder apparatus.In most such devices,iamples are enclosedin an approximately 3_-4cmlong, 1.27
cm (l/2 in.) or 1.91 cm 8/4 in.) diameter furnace irr.-bly that is then placed inside the axial hole of the pressur"
vessel.This vessel(or bomb plate as it is known in m-y
labs) is typically about 5 cm thick and is cooled along
the top and bottom surfaceswith flowing water. In many
configurations,a sheet of mylar lies between the cooling
water and the top of the vessel. Heat is generatedwithin
the furnace assembly by running an electrical current
through a thin-walled graphite tube, which heats resistively. As the graphite furnace tube is the only assembly
part that generatessignificant heat, the thermal gradient
along the vertical axis of the full assembly is greatest
within the portion spannedby the graphite furnace. Previous studiesin talc assemblies(Boyd and England 1960,
1963; Cohen et al. 1967) suggestedthat the gradient is
symmetrical with respect to the furnace center. Recent
studies of NaCl-Pyrex assemblies (Hudon et al. 1994;
Dunn 1993) also infer a hot spot centeredwithin the furnace.However, with the exception of Hudon et al. (1994),
these studies do not address the effect of compression
during experimentson the position of the thermocouples.
In particular, it is unclear whether reported thermocouple
positions are measuredpre- or post-experiment,leaving
the data ambiguousto interpret. Nevertheless,as a result
of thesefindings, most piston-cylinder laboratoriescenter
0003-004x/98/0304-0228$0s.00
the sample capsule within the furnace tube of their particular assembly.We show that assuming that the peak
temperatureis centered vertically in the furnace is not
always valid and can have undesirable consequencesif
large sample capsulesare used.
During an experiment,the crimped and folded capsules
are positioned very close to the thermocouple with their
narrow dimensions oriented vertically. These capsules
probably do not experiencelarge thermal gradients simply becausetheir smaller dimensions are aligned parallel
to the long axis of the furnace. In addition, at the lower
temperatureswhere gold capsulesare often used, thermal
gradients are less steep and the hot spot region is larger
(Boyd and England 1963, Fig. 1; Hudon et al. 1994).
However, when graphite liners are used to isolate samples
from outer platinum capsules, significantly larger capsules must be used to accommodateboth the liner and
sample volume. GraphiteJined platinum sample capsules
used in this study, initially 4 to 4.5 mm in length, compress to 2.8 to 3.2 mm during experiments.If not placed
correctly within the hot spot, such large capsulescan experience large thermal gradients,especially when carried
out at the high temperaturesnecessaryin peridotite melting experiments(>1200'C).
In Figure I we show spinel compositions from experiment FER-A-6 carried out at 1280 "C for 96 h. The disconcerting variability of these data motivated this study.
With increasing temperatureand concomitant increasein
panial melting, the Mg/(Mg + Fe,) of spinel decreases
228
PICKERING ET AL.: TIIERMAL GRADIENT IN A PISTON-CYLINDER APPARATUS
229
0.65
0.55
AAA
H Stuint"rs steel
0.45
%Pyr"*
E tutgo
ffi Samplecapsule
ffi curt
+
6 0.35
ts
AV
U
0.25
q"\
tg
0.15
0.05+0.7
I
0.75
0.8
MglMg+Fe1)
0.85
Graphite
o
C)
o
.o
ts
o
Cm
012
1.27
Frcunn 2. Crosssectionof pre- and post-experiment
(ll2
the
illustration
in.)
fumace
assemblies.
Drawn
to
scale,
cm
Frcunn1. A plot of atomicCrl(Cr+ Al) vs.Mg/(Mg + Fe,)
we see
of spinelfrom l0 kbar meltingexperiments
on KilbourneHole at right is typicalof the average307ooverallshortening
out
at
10
kbar.
Note
the
compaction
in
our
experiments
carried
:
xenolith[mode Olivine(507o),Opx (30Vo),Cpx(17Vo),Spinel
furnaceextrusion,
(37o)1.Filled trianglesare meanspinelcompositions
from the of the lower graphitepartsto accommmodate
is
discussed
in
the
text.
which
meltingexperiments
of Bakerand Stolper(1994)from l27Oto
1390'C. Opencirclesaremultiplespinelanalyses
from FER-Aproduct.Designedspecificallyto piece
6, our 1280'C experimental
had an axial hole for the thermocouple,and a dumreplicatetheir 1280"C/10kbar experiment(indicatedby the inmy
sample,
contained in a graphite-lined platinum capvertedtriangle),our 1280'C experiment
utilizedidenticalstartset
into a cavity drilled into the solid lower
sule,
was
ing materialto thatusedby BakerandStolper(1994).Thearrow
piece
of
MgO
and surroundedby MgO powder. A thin
indicatesthetrendof changingspinelcompositions
with increas(0.5 mm) wafer of MgO separatedthe end of the thering temperatureand extentof partial melting.
mocouple from the top of the sample capsule.In all but
one experiment, the length of the upper MgO piece
and the Crl(Cr + Al) increases,as indicated by the arrow equaledthe combined length of the lower MgO piece plus
in Figure 1. Thus the trend spannedby the spinel analyses the thin wafer. In one experiment conducted without a
from our single 1280'C experiment product (open cir- sample capsule,the upper MgO piece with the axial hole
cles) suggeststhat a large thermal gradient was present for the thermocouplewas cut a little longer than half the
across the sample capsule and possibly that the sample length of the furnace, to allow measurementof the thertemperaturewas <1280'C at all points along the length mal gradient below the furnace center.
of the capsule. Dissecting experiment FER-A-6 after
Our assembly,modified after that used by Boettcher et
quench conflrmed that the sample capsule was centered al. (1981), incorporates a graphite ring and plug at the
in the furnace during the experiment, and that the ther- baseof the assembly,which provides a small empty space
mocouple had indented the top of the capsule.Hence, we into which the furnace slides during compression (Fig.
infer that the spatial variation in spinel composition with- 2). To ensurethat the furnace seatsproperly in this space,
in the experimentalproduct depicted in Figure 1 resulted furnace tubes are cut I mm longer than the combined
from a significant thermal gradient acrossthe 3 mm ver- inner MgO parts. This configuration results in unsheared,
tical dimension of the sample capsule. This deduction uniformly compressedfurnaces,which contribute to powleads to the conclusion that sampleFER-A-6 was not cen- er stability during experiments.Before inserting the astered on the hot spot, and therefore the hot spot is not sembliesinto the pressurevessel,they are wrapped in 0.7
centeredwithin the furnace in our assembly.To determine mm thick lead foil, and the inner wall of the carbide core
where the hot spot is in our CaFr-graphiteassembly,we is coated with a thin layer of MoS, lubricant.
carried out the double-thermocouple experiments disExpnnruoNTAL PROCEDURE
cussedin the remainder of this paper.
Temperatureswere measured using W.-Re/!Vru-Re
Tnn assnnrnr.y
thermocouple (tc) wires calibrated by Englehard IndusExperimentswere conductedusing a cell with an outer tries. Calibrations are traceable to National Bureau of
CaF, sleeve, a straight-walled graphite furnace, and two Standardstests nos. 243484A and 2330648, with estiinner pieces of crushableMgO (Fig. 2). The upper MgO mated uncertaintiesnot exceeding O.4Voof the tempera-
230
PICKERING ET AL.: THERMAL GRADIENT IN A PISTON-CYLINDER APPARATUS
Frcunr 3. The results of six double-thermocoupleexperiments are plotted as temperature('C) vs distance(mm) from the
center of the post-experimenteffective furnace.Each thick curve
representsa second-orderpolynomial fit to all six experiments
at a given mV step for the centered control tc. Each individual
experiment is represented by the control tc in the center and one
point offset above or below 0 mm on each curve. Errors in temperature measurementsare less than the vertical dimension of the
symbols. The +0.5 mrn distance error bars representa combination of the authors' ability to reproduce independently a given
measurementof the post-experimentassembly (+0.2 mm) and
the allowed +0.3 mm variance on the control thermocouplebeing out-of-center(seetext). The bars below each curve represent
the width of the l0 "C hot spot for that curve. The tick mark in
the center of each bar represents the position of the peak temperature. The thin curves are calculated model gradients, discussedin the text.
I 550
I 500
1450
1400
1350
l 300
1250
(J
p) r200
(.)
(.)
I 150
t1
l 100
I 050
1000
950
900
850
800
3210-t
mm from centerof
effectivefurnace
ture in the range investigated.All tc voltages were measured without cold junction compensation, and no
pressurecorrections were applied to the emf of the thermocouples. To compensatefor the cold junction of the
thermocouplesbeing at room temperaturerather than 0
oC, a value of 0.2 mV was added to every ernf measurement taken. Second-orderpolynomial regressionsof temperature vs. the calibrated mV values provided by Engelhard allowed quantitative calculations of the
temperaturescorrespondingto the emf readings.
Four-bore mullite insulators housed the thermocouple
wires. Two thermocoupleswere welded into a single tube
by grinding away one side of the four-bore tubing to expose two of the holes a few millimeters from the end,
then inserting two sets of wires and spot welding under
argon gas, one pair at each exposure.This procedureresulted in small shiny beadsthat did not protrude outside
the outer diameter of the mullite tubing. During all experiments,a Eurotherm 808 digital temperaturecontroller
maintained the temperature of the thermocouple located
in the center of the furnace. Temperatures at incremental
distancesabove or below the central control thermocouple were measuredpassively with the secondthermocouple, using a digital multimeter.
Experimentswere maintainedat 10 kbar +83 bars.Reported pressuresare nominal Heise gaugepressures,with
no friction correction. Experiments were pressurized at
room temperatureto approximately 9.5 kbar, heatedto an
emf of 21.0 mV (1164 oC,after applying the cold junction
compensation),and allowed to equilibrate thermally. In
every case, thermal expansionresulted in the final pressure adjustmentto 10 kbar being a pressurerelease(i.e.,
hot piston-out). The emf of the control thermocouplewas
then increasedin 1.0 mV incrementsat a ramping rate of
1.5 mV/min, to a maximum of 26.0 mY (1469 "C). Temperature was then ramped downward in 2.0 mV increments to 16.0 mV (893 'C), and the emf of the passive
tc was recorded 90 s after reaching each target temperature. This procedure was repeated once more, ramping
PICKERING ET AL.: THERMAL GRADIENT IN A PISTON-CYLINDER APPARATUS
231
ner MgO parts. We consider this length the effective furnace, and believe that it is along this length that heat is
generated,rather than the length of the pre-experiment
furnace. Henceforth, we refer to the post-experiment
length of the inner MgO parts as the effective furnace
and the full length ofthe graphite tube as the full furnace.
To get an accurateprofile of the thermal gradient in an
assemblyduring an experiment, we dissectedall charges
post-experimentand carefully measuredthe length of the
effective furnace, the position of the control thermocouple, and the distance separatingthe two thermocouples
during the experiment with a digital micrometer. Hence
the distance of the passivethermocouple(in millimeters)
DrscussroN oF THE EFFECTSoF coMpREssroN
from the center reported in Figure 3 representsthe posiAfter a 10 kbar experiment,we typically observe 307o tion of the passivethermocouple,relative to the center of
shortening of our CaFr-graphite assembly. We believe the compressedeffective furnace, during the actual conthat the shortening is due primarily to our inability to ditions of the experiment.Neglecting the effects of commanufactureCaF, parts with densitiesgreater than about pression on double-thermocoupledeterminationsof ther85Vo of the density of single-crystal fluorite using our mal gradients results in broader curves and ambiguous
bench pressand die. As we pressurizeour assemblies,the peak temperaturepositions with respect to the center of
CaF, further compactsas the furnace tube slides into the the furnace. Such inaccuracieslead to underestimatesof
space provided by the graphite ring and plug at the as- the thermal slope of the gradient and hence overestimates
sembly bottom. When starting an experiment, we pump of the length of the hot spot and possibly lead to an erthe master ram to within 0.5 kbar of the target pressure, roneous placement of the sample in the assembly with
then begin ramping up to the target temperature.During respectto the thermal peak.
temperatureramping, the master ram pressure typically
Rnsur,rs
falls due to compaction of the assembly,requiring continuous pumping to maintain the samplepressurenear the
In contrastto previous studies(e.g., Boyd and England
desired value. When the target temperatureis reached,or 1960, 19631Cohen et at.1961; Dunn 1993),the thermal
soon after, thermal expansion of the assemblyresults in peak in the assemblydescribedhere is not located in the
master ram pressureincreasing, requiring a presslue re- center of the effective furnace, but is displaced upward
lease to maintain the target pressureuntil thermal equi- toward the steelbaseplug by approximately2.5 mm. Figlibrium is achieved. Thus it appearsthat the shortening ure 3 shows the vertical thermal gradientsmeasuredover
of the assembly occurs almost entirely during the tem- a 7 mm span from just below the center of the effective
perature ramping to the desired set point and that the furnace upward toward the base plug. The thermal graeffective geometry during the experiment is that of the dient was measuredat 10 kbar over the temperatureincompressedassembly.
terval 900-1500 "C.
We carefully monitored the effects that compressionof
With increasing temperature,the location of the therthe assembly during the experiment had on the results, mal peak moves toward the center of the effective furnace
which were twofold. First, compaction of the thermocou- (Fig. 3). In the lowest temperatureexperiment reported,
ple welds into the mullite tubing resulted in the welds a peak temperatureof 914 oC was located 2.6 mm above
'C
being approximately I mm closer together after each ex- the center of the effective furnace, whereas at 1510
periment, regardlessof how far separatedthey were be- the peak had moved to within 2.2 mm of the center.For
fore compression.Second,differential compressionof the each experiment we calculated the length of the zone
inner MgO parts may result in the control thermocouple within which the temperaturewas no cooler than 10 "C
moving to an out-of-center position during the experi- below the peak temperature, which we consider as an
ment. To compareresults from one experimentto anothe! acceptablehot spot. This hot spot zone was 3.2 mm in
the control thermocouple must always be located in the length at lower temperatures,decreasingto -2.7 mm for
same place within the assembly. If the control thermo- temperatures
in the range 1000-1300'C. Above 1300'C,
couple was >0.3 mm away from center, the experiment the hot spot zone shortenedto 2.4 mm and was only 2.16
was discarded and repeated.Furthermore, furnace extru- mm long at 1510 'C. This shortening of the hot spot resion into the empty spaceprovided by the basal graphite sults from the steepeningof the thermal gradient with
ring and plug during initial compressionallows the inner increasing temperature, as has been observed in other
MgO parts to compressmore than the furnace.The short- studies(e.g.,Boyd and England 1963;Hudon et aL.1994).
ening of the inner MgO parts draws the previously menA displacementof the hot spot approximately l-2 mm
tioned small graphite plug into the furnace (Fig. 2). Thus upward toward the base plug also has been observed in
the length over which the furnace is actually thin-walled NaCl and NaCl-Pyrex assemblieshaving configurations
is equivalent to the combined length of the shortenedin- almost identical to that shown in Figure 2 (T.C. McCartemperatureback up to 26.0 mV, then down to 16.0 mY
and recording the emf of the passive tc at each temperature step. A time interval of 90 s was sufficient for the
emf outputs of both thermocouplesto stabilize. This was
testedby taking a measurementafter 90 s, then monitoring the values over a 30 min period, over which the emf
of the passivetc fluctuated no more than +0.03 mV. The
emf measured at each temperature step during upward
rampings varied no more than +0.10 mV from the emf
measured at the same step during downward rampings.
The emf measurementsreported in Figure 3 were all recorded on the first downward ramp of each experiment.
232
PICKERING ET AL.: THERMAL GRADIENT IN A PISTON-CYLINDER APPARATUS
thy, personal communication, 1997). Although all three
assemblies started with a pre-experiment full furnace
length of 2l .5 mm, the NaCl-Pyrex and NaCl assemblies
shortenedconsiderably less during experiments because
NaCl powder can be pressedto nearly the density of single-crystal halite using a bench press (post-experiment
full furnace lengths averaged22.5 mm in NaCl-Pyrex and
24.8 mm in NaCl, compared with 20 mm in our experiments in CaF,). It thus appearsthat off-center hot spots
may occur even in assemblieswith longer furnaces.However, effective furnace lengths were not measuredpostexperiment in the NaCl and NaCl-Pyrex assemblies,precluding a rigorous comparison of the results of the two
studies.
solve the case where conductivity in all three layers is
the same.This gives a curve (Fig. ab) with a peak temperature located 1.4 mm below the center of the middle
layer (the furnace), reflecting the fact that the layer representing the portion of the piston within the vessel is
thicker than that representingthe steel base plug. As the
conductivity ratio (Ak1 between the combined baseplug
and furnace layers and the piston layer decreases,the
peak temperature is displaced away from the piston.
which is the material with higher conductivity. At a conductivity ratio klk" : 0.6, the peak temperatureis centered in the furnace layer. As Ak" continuesto decrease,
the peak temperature continues to migrate toward the
steel plug, as observed in our assembly.The position of
the peak temperaturerelative to the center of the furnace
Tnnnnr,q.L MoDEL
layer is plotted in Figure 4c as a function of the conducIn collaboration with Michael Manga at the University tivity ratio Ak" for an assemblywith the samedimensions
of Oregon, we simplified the assemblyinto a one-dimenas ours.
sional (parallel to the axis of symmetry) model in an atTo model our specific assembly,we need estimatesof
tempt to understandthe cause of the upward shift of the
the actual thermal conductivities of the three model laythermal peak of our assemblyrelative to the midpoint of
ers. For the thermal conductivity of the base plug and
the effective furnace. The symmetry of the assemblyand
WC-piston layers, we use values of 25 WmK (AISI
pressurevessel about the vertical axis implies that heat
stainlesssteel304 at 600'C, Bejan 1983)and 100 WmK
flow in the radial direction is symmetric. Thus we assume
(at2OO"C, Carboloy, product information, 1997), respecthat heat flow in the radial direction does not contribute
tively. We assumethesevalues are valid to elevatedtemto displacementsof the thermal peak up or down along
peratures as both materials are metals without a strong
the vertical axis of the assembly.
dependenceof conductivity with temperature (Berman
As shown in Figure 4a, the model consists of three
1976). To estimate a bulk thermal conductivity for the
layers of different thickness. Layer one, 12 mm thick,
middle
layer, which consists of a graphite furnace tube
representsthe steel base plug, and layer two, l8 mm
by CaF, and filled with MgO, we used the
surrounded
thick, representsthe effective furnace. These layers have
given
method
in Kingery (1960) for calculating thermal
thermal conductivities ft and k', respectively. The third
layer, representingthe lower graphite parts and the por- conductivity of multiphase bodies. We assumethe contion of the tungsten-carbide(WC) piston within the pres- centric cylinders of materials in our assemblybehave as
sure vessel during a typical experiment with our assem- a series of slabs, with heat flow parallel to the plane of
bly, is 20 mm thick and has a different conductivity, k'1 the slabs. For the bulk thermal conductivity of this ge: EV,k,
The furnace layer (the middle layer) has constant heat ometry, Kingery (1960) gives the equation k'
(cross-sectional
V,
area)
where
is
the
volume
fraction
of
production, A. Heat production is zero in the base plug
(
and
is
the
thermal
conductivity
of
each
component,
that
layer and in the graphite plug + WC-piston layer. Using
the boundary conditions thatT :0 at the top and bottom component.
At temperaturesabove 727 "C, ft of CaF, approaches
of the model, and that heat flux and T are continuous
acrossthe boundariesbetween layers, one can solve the an approximately constant value of 2.3 WmK (Touloukian 1967). The thermal conductivity of MgO decreases
steady-stateheat flow equation:
with increasing 4 ranging from -54 WmK at 0'C to
-7.5 WmK at 1500 'C (Touloukian 1967). However, in
ox'
the temperaturerange of this study between9fi) and 1500
'C, ft of MgO varies only from -5.4 WmK to -7.5
for the temperaturedistribution in the axial (x) direction.
(Touloukian 1967). As MgO comprises only a
WmK
Despite its simplicity, the model offers a physical explanation for the observed displacementof our hot spot quarter of the cross-sectionalarea of the furnace layer,
toward the material with low thermal conductivity, the and the conductivity values for MgO are relatively low,
steel base plug. To illustrate this hypothesis, we show a this range in conductivity results in the bulk conductivity
few generic calculations in Figure 4b. We begin by in- estimate varying only + 0.5 WmK. Therefore, we have
vestigating the effect of varying the ratio between the chosen to do our calculations with an average value of
conductivities of the steel plug and the WC-piston. For 6.45 WmK for the conductivity of MgO. Including the
simplicity in these calculations,we consider the conduc- temperaturedependenceof k of MgO would likely cause
tivity of both the base plug and furnace layers to be the the temperaturedistribution curve to be slightly steeper
same (ft : k), and we vary the conductivity of the WC- and tighter than shown (e.g.,Li et al. 1996).We assume
piston layer (k') from this value to higher values.We first this effect would be roughly symmetrical about the peak,
rY: -r
(r)
PICKERING ET AL.: THERMAL GRADIENT IN A PISTON-CYLINDER APPARATUS
L53
a
AcrossInternalBoundaries:
T=0
'/ / \ '
Tiscontinuous
k dT/dx=constant
T=0
\
,/\
A=0lA=consentlA=0
conductivityI conductivity=k' I
=k
|
|
tl
I
baseplug !
fu.*,
o1
!l
H
o
conductivity=k'
.6
F
Wc-piston
t 0 15 20 25 30 35 40 45 50
Distance
fromtopof baseplug(mm)
(conductivilyk)
qF
:9
8ii
6H
hotspotmovestowards
low thennal
conductivity
baseplugaskft" decreases
a4)
ull
o
k=k'/k"
Distancefrom top of baseplug (mm)
Frcunn 4. (a) The geometry and boundary conditionsfor the
integration ofEquation 1. (b) Two curves generatedby integrating Equation 1. The symmetrical curve is for the simple case
where t : k'/k' : L The second curve illustrates the thermal
gradient when the k : k'/k" ratio is 0. 1, with the 2.8 mm upward
offset of the thermal peak indicated with the arrow labeled d.
The solid vertical line representsthe middle of the post-experiment effective furnace.(c) Displacementof the peak temperature
from the center of the effective furnace plotted as a function of
conductivity ratio, k : k'/k". Curve shown is a second degree
polynomial fit to the datz:f(x) : 2.588x'?- 7.558x + 3.570. (d)
Several calculated curves illustrating the effect of lengthening
the furnace. All five curves are calculatedusing the same conductivities as for the highest temperature (-1510 'C) model
curve shown in Figure 3 (l/k'/k": 25:13.3:100).The effective
fumace length was increasedin 2 mm increments from 18 mm
to a maximum length of 26 mm. (For clarity, only the 18 mm
and 26 mm furnace length calculations are labeled.) As the furnace lengthens,the bottom end of the furnace moves from 30
mm (from the top of the base plug) to a maximum of 38 mm,
as indicated by the arrow. For each calculation, the middle of
the effective furnace moves away from the base plug faster tlan
the calculated thermal peak, leaving the peak temperatue displaced above the effective furnace center by increasing amounts
with increasingeffective furnace lengths. In order of increasing
effective fumace length, calculated displacements upward from
center of the thermal peak in mm are 1.2 (18 mm effective furnace),1.4, 1.5, 1.7,and 1.8.
affecting the shape of the temperature distribution curve,
but not greatly affecting the position of the thermal peak.
The thermal conductivity of graphite has a strong dependenceon temperature,so a single averagevalue does
not suffice to cover the temperaturerange of this study.
However, including the temperaturedependenceof k of
graphite makes the model equation non-linear,thus greatly complicating the solution. For simplicity, we estimate
the conductivity of graphite at seven 100'C temperature
stepsfrom 900 to 1500 "C, roughly correspondingto the
peak temperaturesof the measured gradients shown in
Figure 3. Clark (1966) provides conductivity values in
the temperature range from 0 to 2500 'C for Acheson
graphite,which has a conductivity of 167 WmK at room
temperature.Similar to Acheson graphite, the high-density graphite comprising our furnaces has a conductivity
of 196 WmK at room temperature(Carbone,product information, 1997). To estimate the conductivity of our
graphite at higher temperatures, we applied the temperature dependenceof Acheson graphite and extrapolated
from the value of 196 WmK at room temperature to
higher temperatures.A second-orderpolynomial regression of the data projected in this fashion allowed us to
estimate the thermal conductivity of the graphite used in
our furnacesin 100'C stepsfrom 9fi) to 1500'C. We
use these estimatedvalues for k of graphite (in order of
increasingtemperaturein WmK: 91.0, 83.1, 75.8,69.3,
63.3, 58.1, and 53.5) in conjunctionwith the ft valuesof
2.3WlmK for CaF, and 6.45 WmK for MgO to calculate
a bulk conductivity for the furnace layer at the seven
temperature steps. The resultant estimates for the bulk
'C,
conductivity of the furnace layer (from 900 to 1500
zJ+
PICKERING ET AL.: THERMAL GRADIENT IN A PISTON-CYLINDER APPARATUS
in WmK: 20.5, 19.0, I1.6, 16.3, I5.2, 14.2,and 13.3) effect that differences in the conductivities of materials
were then used to calculate the seven model curves in the axial direction may have on the location of the
shown in Figure 3. Note however that this proceduredoes thermal peak. Primarily based on these measurements,
not account for the fact that temperature, and thus the and in part the model results, we recommend that other
thermal conductivity of graphite,vary along the length of labs measure the thermal gradient in their own assemthe furnace at any particular peak temperature.
blies, making allowances for any effects assembly comThe solutions specific to our assemblyparametersare pression might have.
plotted as thin lines on each gradient shown in Figure 3.
CoNcr,usroNs
At the lowest temperatureinvestigated, the model predicts a displacementof the 914 'C peak 1.6 mm upward
As the assembly and pressurevessel are symmetrical
from the center of the effective furnace, compared with in the radial direction, the variation in thermal parameters
the measured offset of 2.6 mm above center. With in- of materials in the axial direction has the greatesteffect
creasing temperature,the offset of the thermal peak in on the position of the thermal gradient along axis. Of
both the model and measuredgradientsdecreases.At the these,our results suggestthat the most significant factors
highest temperature investigated, the model predicts a are the thermal conductivity ratio between the steel plug
displacementof the 1510 'C peak 1.2 mm above the ef- and WC-piston and the length of the furnace. Shortening
fective furnace cente! comparedwith the measuredoffset the steel plug reduces the upward displacement of the
of 2.2 mm abovecenter.The approximationsand inherent thermal peak. The results of our simplified model do not
uncertaintiesin the model warrant caution in applying its corroborate the general assumption that longer furnaces
predictive power, but the relatively good agreementbe- may cause the peak temperatureto move closer to the
tween the measuredand predicted gradients suggeststhe effective furnace center. Further work on determining the
model capturesthe essenceof the problem.
thermal gradients in assemblieswith effective furnaces
It is important to note that all of the modeling results longer than 18 mm is required to resolve the effect of
presented above pertain to assemblieswith dimensions furnace length on the location of the thermal peak along
and materials like those used in our lab. We are aware the axis of the assembly.Regardlessof the effect on the
that several labs use furnace assembliesthat are signifi- position of the thermal peak, longer furnaces result in a
cantly longer than ours. Consequently,we have thus in- wider hot spot zone, making it easier to place a capsule
vestigated the effect of lengthening the furnace layer of entirely within the hot spot. By decreasingthe gradient
the overall assembly,leaving everything else the sameas in the center of the furnace, steppedor tapered furnaces
for the 1510 'C model curve (k:k':k" : 25:13.3:100). (e.g., Kushiro 1976) would likely dampen the effect of
Figure 4d shows the result of this exercise when the hot spot displacement on the thermal gradient experilength of the post-experiment effective furnace is in- encedby a sample centeredwithin the furnace. However,
creasedin 2 mm increments from 18 to 26 mm. As ex- stepped or tapered furnaces introduce additional chalpected,the position of the thermal peak moves away from lenges with their construction and may not be time or
the base plug with increasing furnace length. However, cost effective for many labs.
as the furnace lengthens,the center of the furnace moves
As a result of the observationsreported here, we have
away from the base plug faster than the thermal peak, relocated the sample-thermocoupleinterface in our exleaving the thermal peak displacedupward by increasing perimentsto 3.5 mm above the center of the furnace.This
amounts. Thus the calculations suggest longer furnaces relocation places the thermal peak within the sample voldo not lessen the hot spot displacement.As other labs ume, and also places the thermocouple and capsule botthat use longer furnacesthan ours have not reportedoffset tom at the edgesof the hot spot, ensuringthat every point
hot spots, this result may indicate rhat the simplifying within the sampleis within 10 "C of the temperaturemeaassumptionshave compromisedthe accuracyof the mod- sured by the thermocouple.To confirm that sample capel. Heat flow in the radial direction or the effect of tem- sules and thermocoupleswere positioned properly with
perature dependenceon thermal conductivities may have respect to the thermal gradient during experiments, we
a greater effect on the position of the thermal peak than carefully dissect all charges and measurethe lengths of
we have assumed. However, the fact that the model all assemblyparts post-compression.This procedure has
curves calculatedusing the parametersspecific to our as- eliminated data scattersuch as that shown in Figure 1 and
sembly are in relatively good agreementwith our mea- results in texturally and compositionally homogeneous
sured gradients(Fig. 3) leavesus no causeto dispute the experimentalproducts, even when relatively large graphsuggestionthat longer furnaces may also have offset hot ite-lined platinum capsulesare used.
spots. We cannot provide further empirical evidence eiAcxuowr,BncMENTs
ther supporting or casting doubt on this suggestion,as we
did not measurethe thermal gradient in a furnace with a
We gratefully acknowledgeMichael Manga for assistarcewith the therpost-experimentlength longer than 18 mm. We stressthat mal model. Dave Senkovich and Cliff Dax for technicalassistancein the
our principal aim with this contribution is to present the Machine Shopsat the University of Oregon,and TC. McCarthy for helpful
reviews of the manuscript and providing supporting data. We thank Mimeasuredgradients,documentingan offset hot spot in our chael Baker and John Ayers for caeful reviews of the manuscript This
assembly.We present the model merely to illustrate the work was supportedby NSF grant EAR-9506045.
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