1. Accurate Simultaneous Measurements of Thermal Conductivity and Volumetric Specific Heat Of Rubber, Elastomers, and Other Materials
Akhan Tleoubaev, Andrzej Brzezinski (LaserComp, Inc., USA),
and Luiz Claudio Braga (Elittec, Brazil)
Presented at the 12th Brazilian Rubber Technology Congress
May 2008, São Paulo, Brazil

2. The Most Widely Used and Reliable Method for Thermal Conductivity Measurements of Insulation Materials
ASTM C 518 “Standard Test Method for Steady-State Thermal Transmission Properties by Means of the Heat Flow Meter Apparatus”
ISO 8301 “Thermal insulation – Determination of steady-state thermal resistance and related properties – Heat flow meter apparatus”
EN “Thermal performance of building products and components …– Part 3: Measurements by heat flow meter method”

3. Heat Flow Meter (HFM) Method: ASTM C 518, ISO 8301, EN 1946-3.
LaserComp’s FOX Instruments
Two isothermal plates - hot and cold at temperatures TH and TC
Heat Flow Meters: QH , QC
Flat sample of thickness DX
Result -thermal conductivity  after reaching final thermal equilibrium

4. Heat Flow Meters Signals - (in microvolts) versus time (in seconds) Thermal conductivity = 33.3 mW/(m*K)

5. Calibrations: Scal = cal ΔT / (xcal Qcal)
Regular simple formulas are valid only when thermal contact resistance 2R is negligible (i.e. when 2R<< x/)
Calibrations: Scal = cal ΔT / (xcal Qcal)
Tests: test = Scal xtest Qtest / ΔT
test = cal (xtest /xcal ) (Qtest )/Qcal)

6. ASTM Active Standards:
C “Standard Test Method for Steady-State Thermal Transmission Properties by Means of the Heat Flow Meter Apparatus”
E “Standard Test Method for Evaluating the Resistance to Thermal Transmission of Materials by the Guarded Heat Flow Meter Technique”

7. Thermal Resistances
Sample’s resistance - thickness x divided by thermal conductivity : x/
Thermal contact resistance (neglected in the Standards) - temperature drop T divided by heat flux q: R = T/q
Total thermal resistance - sample’s resistance x/ plus thermal contact resistance of two surfaces 2R : x/+2R

8. FOX50 Heat Flow Meter Instrument for medium conductivity materials – rubber, plastics, ceramics, rocks, glass, composites, polymers, etc.

9. FOX50 Heat Flow Meter Instrument LaserComp, Inc.

10. System of Two Equations
To find the two unknowns ( and 2R) we have system of two equations:
1) For thin sample:
q1 = T / (x1/ + 2R) = Scal Q1
2) For thick sample:
q2 = T / (x2/ + 2R) = Scal Q2

11. Solution of the System of Two Equations for Calibrations:
Calibration Factors:
Scal = T cal (Q1–Q2)/[Q1 Q2 (x2 – x1)]
Thermal Contact Resistance:
2Rcal = (x2 Q2 – x1 Q1) /[cal (Q1 – Q2)]

12. For high temperature versions of the FOX50 HFM Instrument:
THERMAL CONDUCTIVITY OF CALIBRATION MATERIALS, W/(m K)
T, 0C
Perspex
Vespel
Pyrex 7740
Pyro-ceram 9606
0.1860
0.365
1.063
4.15 20
0.1885
0.371
1.086
4.04 40
0.1909
0.377
1.115
3.94 60
0.1933
0.386
1.145
3.85 80 -
0.389
1.175
3.78
100
0.396
1.203
3.71
For high temperature versions of the FOX50 HFM Instrument:
150
0.411
1.270
3.58
200
0.426
1.330
3.49
250
0.441
1.391
3.42
300
0.457
1.452
3.34

13. Solution of the System of Two Equations for Tests:
Thermal Conductivity:
test = Scal Q1 Q2 (x2 – x1) / [T(Q1 –Q2)]
test= cal [(Q1cal – Q2cal )/(Q1test –Q2test )]
[(Q1test Q2test )/( Q1cal Q2cal )]
[(x2test - x1test )/(x2cal - x1cal )]
Thermal Contact Resistance:
2Rtest =(x2Q2–x1Q1)T/[Q1 Q2 Scal (x2– x1)]

14. Figure 2 – Total thermal resistance vs
Figure 2 – Total thermal resistance vs. thickness in millimeters for several calibration samples and solid silicon rubber (Cohrlastic  700) at 25C mean temperature. Extrapolations down to zero thickness give values of the thermal contact resistance 2R. Reciprocals of the slopes (divided by 1000) give accurate values of thermal conductivity  (e.g. 1/ =1117 mW/mK)

15. Figure 3 – Total thermal resistance vs
Figure 3 – Total thermal resistance vs. thickness in millimeters for samples of solid silicon rubber (COHRlastic  700 – 1, 2, 4, 5, and 7 layers) at different meant temperatures. Extrapolations down to zero thickness give values of the thermal resistance 2R. Reciprocals of the slopes (divided by 1000) give values of thermal conductivity  (e.g.1/ =316mW/mK).

16. Heat Flow Meters’ signals recorded vs
Heat Flow Meters’ signals recorded vs. time contain information about volumetric specific heat of the sample Cp [J/(m3K)]
After (in thermal equilibrium) plates’ temperatures change the heat flow meters measure amount of heat absorbed by the sample (and by the heat flow meters) – i.e. the regular HFM instrument can work like calorimeter (with some corrections for small edge heat losses)

17. Energy conservation equation:
Volumetric Specific Heat Measurements Using HFM Method Amount of heat H passed through the heat flow meters into the sample:
H=i [SUcal (QUi –QUequil )+ SLcal (QLi –QLequil )]
Energy conservation equation:
H = (Cpx1 + Cp’’2x ) T

18. Graphs of the H sums vs. time in seconds Expanded Polystyrene, 23
Graphs of the H sums vs. time in seconds Expanded Polystyrene, 23.9 mm thick, 51.5 kg/m3

19. System of the Energy Conservation Equations for Two Samples:
To find the two unknowns Cp and Cp’’2x we have two equations:
1) For heat absorbed by thin sample:
Cpx1 + Cp’’2x = H1 /T
2) For heat absorbed by thick sample:
Cpx2 + Cp’’2x = H2 /T

20. Solution of the System:
Volumetric specific heat: Cp = (H2-H1) / [(x2 -x1)T]
HFMs’ heat capacity: Cp’’2x =(H1x2 – H2 x1) / [(x2 -x1)T]
Or (with no sample): Cp’’2x =H/T
Single-sample volumetric specific heat:
Cp = (H /T - Cp’’2x ) / x

21. Several materials of known specific heat were tested using the new method:
Expanded Polystyrene (EPS)
(of ~2, ~3, and ~4 pounds per cubic foot density)
1450b SRM (NIST Standard Reference Material)
Vespel SP1
Pyrex 7740
Pyroceram 9606
Stainless Steel 304

22. Figure 4. Graph of the H sum divided by temperature step (20C) vs
Figure 4. Graph of the H sum divided by temperature step (20C) vs. thickness in millimeters for COHRlastic  Silicon Rubber (1, 3, 5 and 7 layers) for temperature change from 50C to 70C. Slope equals the volumetric specific heat C divided by Point at zero thickness is the measured heat capacity of the HFMs (per square area).

23. Figure 5. Volumetric specific heat C of COHRlastic  700 Silicon Rubber measured by FOX50 Heat Flow Meter instrument vs. temperature

24. Conclusions
Two-Thickness procedures of calibrations and tests used in the LaserComp’s FOX50 Heat Flow Meter instrument provide excellent accuracy of thermal conductivity  tests of materials like rubber, elastomers, polymers, glasses, ceramics, etc. compared to existing methods and procedures (e.g. ASTM E1530).
Possibilities of the traditional HFM method have been significantly extended by using recorded signals of the HFMs versus time, and accurate and reliable algorithm for the volumetric specific heat Cp measurements was developed and experimentally checked.
Two more thermal properties - thermal diffusivity a= /Cp and thermal effusivity =(Cp) then can be calculated.