DEVELOPMENT OF THIN HEATER APPARATUS FOR
THERMAL TRANSMISSION PROPERTIES ABOVE 1500 K
E.G. Wolff, J.E. Sharp, B.C. Nielsen and D.A. Schneider
Precision Measurements and Instruments Corporation
R.B. Guenther
Department of Mathematics, Oregon State University
Corvallis, OR 97331
ABSTRACT
The thin heater Apparatus (ASTM C1114, ASTM C1044) has numerous
advantages over the guarded hot plate (ASTM C177, ISO 8302) method for
measurement of steady state thermal transmission properties. The principle one is
the potential extension to very high temperatures (>1500 K) in air. The standard
design, however, gives k values which are too high. This work uses a general
three dimensional solution based on eigenvalues to develop a numerical scheme
to correct for edge losses. A method to evaluate the edge heat transfer coefficient
is also outlined and applied to measurements of a shuttle tile material. Initial tests
were conducted with Nichrome wire thin heaters using silica or alumina plates
to distribute the input power from the thin heater. The effect of rhodium plating
of interface plates for temperature uniformity was studied and an outline for
system materials substitution was made to enable measurements up to >2000 K in
air.
INTRODUCTION
The Guarded Hot Plate Method
The guarded hot plate method using one-dimensional heat flow to determine the
through-thickness absolute thermal conductivity of plate samples is well known.
[e.g., Salmon (2005), Stavcy (2006)]. . Measurement uncertainties on insulation
materials such as a rockwool fibre board, based on intercomparisons of different
laboratories, are on the order or ± 15% over the range 373 to 773K. No one
factor was held responsible for these differences. We note, however, that this
method requires as separate heater, careful temperature control, careful gap
dimensions and is complex and expensive to construct and operate. Difficulties
arise from design and use of the hot plate, guard heaters and gap configurations.
It requires great effort to obtain accurate data, even at room temperature, where
standard reference materials are readily available.
Moreover, this technique has rarely been used above about 800K. The apparatus
of Ferro (1968) for use up to 1300K requires complex thermocouple grooves and
thermal equalization techniques. The apparatus of Filla (1997) is reported to
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work to 1400K but special modifications are needed to achieve data at higher
temperatures, including smaller specimens and higher thermal conductivity
values.
The thin heater approach
The thin-heater method is an absolute method and is simpler and more amenable
to high temperature use for determining the thermal conductivity of plate type
samples. Figure 1 shows the basic configuration. A thin flat plate heater is
sandwiched between two identical sample plates. Problems associated with a
guard heater and heater-guard gap and thermocouple placement do not arise. A
thin heater has considerably less mass than a combined central heater and a guard
heater, thus minimizing drift errors and shortening times to reach steady state.
Heat flow may be assumed to be one dimensional because of the geometry – edge
area is minimal so that no heater guard is required. The lateral or inplane
conductivity of a thin heater is also relatively low, promoting heat flow to the
samples and not the edges. Thin heaters for the purpose of measuring thermal
conductivity have been made from stainless steel foil [Hager (1960), Hager
(1985), Sirdeshpande (1993)], Nichrome mesh [McElroy (1985)], gold films
[Zeng (1996)] and mica insulated electric resistance heaters [Dowding (1995)].
Mathematical analysis is difficult because of the square configuration and threedimensional nature of the thermal gradients. Somers (1951) analyzed the error due
to edge losses in a guarded hot plate system, introduced the use of transcendental
equations and expressed the results as a formulation of k/kexp. However, special
restrictions were made on the surface conditions. Woodside (1957) provided
similar insight with extension to edge loss without a guard ring. Pratt (1962)
questioned the accuracy of the assumptions made and further developed a
correction for flat plate edge losses without a guard ring. However, Pratt
considered heat loss only from two sides and perfect insulation on the other two,
and it is not clear how kexp was calculated.
THEORY
List of symbols
A = area in xy plane of heater and samples = ab
a = side length of sample (along x-axis, Fig.2)
b = side length of sample (along y-axis, Fig.2)
c = height of sample (along z-axis , Fig 2)
h = heat transfer coefficient (m-1)
i, int, s = subscripts denoting insulation, interface and sample, resp.
kt = true thermal conductivity of samples in z-direction (W/m-K)
kexp = experimental or apparent sample thermal conductivity (W/m-K)
∇2 = Laplace operator
q = heat flux (Watts/m2)
2
Qa = heat flow (watts) between sample faces
QT = total power output of thin heater
θ1 = measured lower sample temperature before use of thin heater
θ’1 = measured lower sample temperature after use of thin heater
θ2 = higher sample temperature (kept constant)
θa = average surface temperature on outside of insulation (see Fig.1)
u,w = temperature variables in the “n” direction , where n = x,y
Δx = thickness of surrounding insulation
Derivation of Edge Heat Loss Correction
From Fourier’s first law (q = - kexp ∇ θ ) the heat flow from this heater then
creates or changes a transverse temperature gradient in the sample.
kexp = (c/ab) (QT/2) [θ’1 - θ1] –1
(1)
Figure 1 shows that the surface area of the heater equals the lateral sample area
(A).( The C1114 standard calls the heater area the “metered” area but allows for
“some defined portion of that area”). The factor of 2 is needed since half the thin
heater power heats each sample. For a thin resistance heater heating two samples,
QT = E(Volts)* I (Amps).
(2)
Without a guard heater, there will be thermal gradients near the sample edges,
both through thickness (z) and laterally in the plane (x,y) of the sample heater.
These will change after the thin heater is turned on, so that simple use of Eq. 1
will produce an error. The error in thermal conductivity measurement decreases
as the edge insulation increases, as the specimen thickness decreases and distance
from the center decreases [McElroy (1985)]. While edge losses may be minimal
at the samples’ center, they always lead to a value of thermal conductivity which
is too high. Furthermore, these losses would increase as the overall temperatures
increases, beyond the 0.5% edge heat losses recommended in C1114. The heat
flow situation may then be described as:
(QT - Q (edge losses))/2 = Qs /2 = kexp f(a,b,c, h, θ’1, θ1) = kt (ab/c) (θ2 – θ1)
(3)
Hence kexp needs to be reduced by a function that increases with heat transfer
from the sides (h) and therefore with QT itself. A general solution which
considers interactions with the surroundings on all four sides is given as follows.
The mathematical formulation is based on eigenfunction expansions. Consider
Figure 2 for the coordinate system at the sample edges:
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Figure 1 Schematic of Thin Heater Thermal Conductivity Apparatus
Figure 2 Coordinate system for edge heat loss corrections
The time independent problem to be solved is:
∇2θ = 0
in the space {0<x<a, 0<y<b, 0<z<c},
(4)
The initial conditions are ;
θ1 = θ (x,y,0)
(0<x<a, 0<y<b)
(5)
θ2 = θ (x,y,c).
(0<x<a, 0<y<b)
(6)
∂θ/∂n
+ h (θ – θa) = 0
on all four lateral sides.
(7)
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Rigorous solution of equation (4) combined with equation (3), and assuming c <
a,b, leads to a summation form of f(a,b,c,θ2, θ1,h). Thus
∞
k / kexp = [(ab/c) (θ2 – θ1)] –1
∑
Bpq λpq I p J q
(8)
p,q =1
where
Bpq =
[θ2 Ip Jq - Apq mpnq cosh (λpq c) ] [mpnq sinh (λpq c) ] -1
(h/ μp) ( μ2p + h2) –0.5
Ip =
Jq = (h/νq) ( ν2q + h2)-0.5
{
}
(μ2p + h2) [(2μph)2 + (μ2p – h2)2]-0.5 + 1
{
(ν2q + h2 ) [(2μqh)2 + (ν2q – h2)2] -0.5 + 1
}
μ = roots of tan μa = 2μh / (μ2 – h2)
ν = roots of
λpq2
tan νb = 2νh / (ν2 – h2)
= μ2p + ν2q
a
mp = ∫o
μp2 (x) dx
= a/2 + (4 μp ) -1 [ sin (2 μpa – 2 αp) + sin 2 αp]
b
nq = ∫o νq2 (y) dy = b/2 + (4νq) –1 [sin (2 νqb - 2βq) + sin 2βq ]
cos αp = μp / ( μp2 + h2)-0.5
cos βq
= νq / (νq2 + h2) –0.5
Apq = ( Ip Jq θ1) / (mp nq )
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Figure 3 Variation of predicted (Eq. 8) k/kexp with h and (θ2 - θ1). Corrections for
θ2 = 1073K, a = b = 0.1524 m and c = 0.0254 m.
Estimation of the edge heat transfer coefficients
An initial correction for the measurement of k can be made by reducing the QT
value in Eq.1 by the ratio of thin heater edge area to surface area (here 0.943).
This accounts for heat loss which goes directly to the side insulation and not in
the z direction. The second correction is made for a finite heat transfer coefficient
at the edges with Eq. 8, and as shown in Figure 3. We note that for a given value
of h, the correction increases as the gradient θ2 – θ1 decreases. This implies that
if the edge heat losses are constant, they will constitute a greater proportion of the
total heat input as the gradient decreases. A suitable choice of the edge heat
transfer coefficient h requires a measurement, since changes in system
dimensions and heat input can be expected to change the magnitude of the edge
losses. Let u and w be the temperature variations in the sample and the adjacent
insulation., resp. Since the heat flow q is continuous across their interface, we
can say
q = ks ∂u/∂n = ki ∂w/∂n = ki [ θa – θint] / ∆x
(9)
Thus ∂u / ∂n - ki [ θa – θint ] / ks ∆x) = 0
(10)
which is equivalent to Eq.7 with h = ki / ( ks Δx). A useful expression for h can
be derived from Eq. 9 by noting
ks (θ1 – θint) = - ki ( θa - θint)
(11)
Substituting h Δx ks for ki leads to
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h = - [( θ1 – θint) / (θa – θint) ] / Δx
(12)
This is the heat transfer that applies before the thin heater is turned on and
accounts for the x,y -direction temperature gradients (Eq. 7). θ1 is the average
temperature at the cooler sample surface. The 2QT power from the thin heater is
a small fraction of the total heat flow which comes from the end heater but it
causes a small change in the edge losses. To correct for the kexp of Eq. 1, we need
only use the change in h caused by the additional QT from the thin heater.
EXPERIMENTAL
Apparatus
The objectives included operation at maximum possible temperatures in air. The
test system (Figure 1) was designed to allow later substitution with higher melting
materials. The end heaters consisted of Kanthal coils. The thin heater was a
Nichrome wire grid encased in 2.36 mm alumina insulators supported between 1
mm thick silica plates. K-type thermocouples (Ni-Cr vs. Ni-Al), 0.1 mm diameter,
were placed between the silica plates and the samples surfaces, giving the initial
test system a capability to about 1500K. With a sinusoidal wire thin heater, two
thermocouples placed a distance of the wire spacing will always yield the
average temperature, irrespective of exact position. If a thermocouple is placed at
an arbitrary point x on the x-axis (perpendicular to the wire direction), it will
measure
T = Tavg + (Tm /2) cos (2πx/λ)
(13)
Where Tm is the total temperature amplitude and λ/2 is the wire spacing. If the
second thermocouple is placed at (x + λ/2), it is readily shown that the average of
the two temperatures is the average temperature along the x-axis.
Mesh heaters
Nichrome mesh heaters were studied but were found to exhibit variable
resistance with time, making steady state conditions difficult to achieve. Special
care is also needed to obtain a uniform temperature distribution. The ASTM
C1114 recommends a constant pressure of 2.5 kPa (0.36 psi) and this was
maintained in all tests by use of dead weight loading. Use of a mesh heater allows
use of higher pressures than the wire/insulators/thin plates heater combination.
Use of High Temperature Oven
The use of temperature controlled end heaters were compared to use of a
surrounding uniform temperature oven. An oven would eliminate the initial zdirection gradient in the samples, so that when the thin heater is turned on, the
ensuing samples’ gradient would reflect only the input power – one-dimensional
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heat flow only would ensue. However, it was found that the additional power
provided by the thin heater also raised the end plate temperatures and the position
of the furnace temperature controls became critical.
RESULTS
Measurements on Shuttle Tiles
Figure 4 shows thermal conductivity data for a shuttle tile material (Lockheed
Martin). The HTP-16-22 has a density 16 lbs/ft3 (0.256 g/cc) and has improved
thermal stability over the first generation LI-900 shuttle tiles. The uncorrected
data show the effect of edge losses. Table I includes additional data points near
1000K and outlines typical values needed to obtain the heat transfer coefficient
and the resultant corrections for kexp. The tests at 6.34 and 17.47 watts were made
with alumina rather than silica plates, and this changes the heat transfer behavior,
as discussed below. Figure 5 illustrates these data which suggest that a proper
extrapolation to zero thin heater power input should give the correct value of k.
Table I Typical results for θ2 = 1073K for two shuttle tile samples
QT (W)
2.96
6.11
θ1 - θ’1
8.6
18
θ1 (avg)
898
kexp (W/m-K) Eq.1
17.47
28.3
18.5
47.5
66.4
898
809
825
878
0.1884
0.1853
0.1876
0.201
θint (avg)
744
739
765
765
737
θa (avg)
574
573
555
559
490
Average θ
986
986
941
949
975
ho (m-1) Eq. 12
35.4
36.98
13.42
13.36
22.64
36.16
38.07
13.88
15.24
27.01
1.69
0.46
1.88
4.37
kexp(corrected x 0.943)- 0.1777
0.1747
0.1769
0.189
0.2198
k / kexp, Eq.8 or Fig 3 - 0.944
0.959
0.99
0.934
0.80
h’(m-1)
Δh (m-1)
0.75
6.34
0.2331
kt (predicted)
0.1677
0.1675
0.175
0.176
0.176
---------------------------------------------------------------------------------------------------
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Figure 4 Experimental data for a shuttle tile material showing that uncorrected
conductivity kexp exceeds referenced k values. (Pressure = 1 atm, air)
Figure 5
Variation of kexp with thin heater power input at about 970 K.
Initial correction is for thin heater side area (x 0.943) and second for edge
losses (Eq. 8 and Fig. 3).
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EXTENSION TO HIGHER TEMPERATURES
Extension of these techniques to the maximum temperatures possible requires
substitution with higher temperature materials which can operate in air. Type B
(Pt-30%Rh/Pt-6%Rh) thermocouples indicate an upper limit of about 2000K in
air. Insulation with MgO, zirconia or high alumina bricks should be adequate.
Molybdenum disilicide and silicon carbide heaters can also take the system to
above 2000K but their negative coefficients of resistance require complex
controls. Pt/Rh alloy wires could also be used for both the thin heater and the
controlled end heaters (for θ2).
The theory becomes more exact if the sample surfaces are at uniform
temperatures. In inert atmospheres this is aided by the use of copper plates above
and below the thin heaters, although this in-plane conductivity increases QT loss
from the side of the heater. We have used 1 mm silica and alumina plates and
note that the initially higher thermal conductivity of Al2O3 decreases to below
that of silica around 1400K. The use of a metallic coating on the alumina will
allow potential use to over 2000K. Rhodium has both higher melting point and
higher thermal conductivity than platinum and is more durable. At 1500K, a Rh
layer of 0.43 μm will equal the thermal conductance of the 1000 μm alumina
plate substrate.
Electron beam sputtering for 45 minutes from a bead of Rh in a water cooled Cu
holder in a vacuum of 6 x 10-7 TORR gave a uniform coating of 2.29 microns.
The effect on plate temperature uniformity was minimal below 1000K but
significantly reduced the standard deviation of temperatures from the several
thermocouples placed between the sample and the thin plate.
CONCLUSIONS
A thin heater system to measure the thermal conductivity of plate samples to >
1300K was constructed and tested. It meets all the requirements of ASTM C1114
but additionally demonstrates the corrections needed to account for edge losses.
The theory is self-consistent and allows parametric studies as well as
measurement corrections to be made based on sample dimensions, the edge heat
transfer coefficient changes and temperature gradients. Results on a shuttle tile
show good agreement with accepted values after corrections for thin heater side
loss power and sample edge losses are made. These corrections increase with
greater power applied to the thin heaters, but this effect may be reduced by
increasing the thermal conductivity of the thin heater plates. Initial tests with a
rhodium coating on alumina look promising. Directions for extending the
temperature range to >2000K through suitable materials substitution are given
Tests to calibrate such as system with the Pyroceram 9606 standard are underway.
10
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Filla B 1997 Rev. Sci. Instrum. 68 2822-2829
Hager N E, 1960 Rev.Sci. Instrum. 31 177-185
Hager N E, 1985 ASTM STP 879 180-190
Lockheed Martin (1997) HTP data sheets , LMSC, Sunnyvale, CA
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