Specimen Size Effects in the Determination of Nuclear Grade Graphite Thermal Diffusivity

1. Specimen Size Effects in the Determination of Nuclear Grade Graphite Thermal Diffusivity ASTM D02F000 Symposium on Graphite Testing for Nuclear Applications: the Significance of Test Specimen Volume and Geometry and the Statistical Significance of Test Specimen Population September 19-20, 2013 Seattle Hilton; Seattle, WA Dave Swank Will Windes

2. Outline: • Description of measurement technique • Sources of Uncertainty • Limitations of heat loss correction models • Limitations of finite laser pulse corrections • Example of estimating measurement uncertainty • Summarize and conclude

3. Why do we need to measure thermal diffusivity? • Thermal conductivity • Conduction through the graphite is how we get the energy out of the fuel • Diffusivity of graphite is significantly reduced by irradiation • Engineers need to understand this relationship for design • Passive safety of system – get the heat out • Measurement is performed to ASTM E 1461 • Generic standard covering the measurement of diffusivity by the laser flash technique for all materials. • Graphite and irradiation experiments of graphite have some special considerations - • specimen geometry and homogeneity

4. Laser Flash Apparatus (LFA) Operation • Small, thin, disk-shaped specimen held in a controlled atmosphere furnace. • Nd-YAG pulsed laser is used to subject one surface of the specimen to a high-intensity, short-duration energy pulse. • Energy is absorbed on the front surface of the specimen’ • Resulting rise in rear-face temperature is recorded with a sensitive IR detector. Radiation to detector Specimen Laser

5. Radiation to detector L Specimen Laser Thermal Diffusivity • Thermal Diffusivity for a Laser Flash Apparatus (LFA) solved analytically for adiabatic conditions by Parker et. al., 1961 Laser Pulse Detector Signal t1/2 • One-dimensional heat flow • No heat loss • Homogenous specimen • Uniform absorption of the laser energy • Short pulse length of the laser compared to the heat transport times

6. Where are the Sources of Uncertainty • Length measurement - L • ASTM E 1461-07 : L ± 0.2% • Realistically we can machine and measure specimens down to ± ~20 µm • Heat transport time - t1/2 • Non uniform heating • Multi directional conduction • Heat Loss: Radiation, Conduction, Convection • Finite laser pulse width • Heterogeneity - # of grains, cracks/pores size and density

7. Adiabatic model Detector signal Effects of Heat Loss: Adiabatic Conditions? (AXF-5Q graphite, 12.7mm diameter at 800°C) L= 1.6 mm L = 3.2 mm L= 6.4 mm

8. Radiation to detector Radiation heat loss L Specimen Laser Sources of Specimen Heat Loss • Convection – negligible if purge gas flow rates are kept low • Conduction – negligible if specimen holder is properly designed • Radiation – • Top and bottom surface – early in the develop of LFA it was determined this can have a significant effect (1963 Cowan). • Circumferential – specimen holder can be designed to minimize exposure to other surfaces

9. Radiation Heat Loss Correction Models • Cowan, 1963 • Assumes a finite square wave impulse of energy • Linearizes the radiation heat loss based on data at 5t1/2 and 10t1/2 • Assumes one dimensional conduction heat transfer in the specimen • Therefore radiation loss from the circumference is not considered • Only radiation from the top and bottom surfaces is considered.

10. Radiation to detector Radiation heat loss L Specimen Laser Radiation Heat Loss Correction Models (cont.) • Cape-Lehman, 1963 • Assumes Two dimensional conduction • Therefore considers radiation exchange at the circumference of the specimen • Maintains higher order terms and therefore is a nonlinear solution which is more accurate at higher temperatures

11. Model Comparison for AXF-5Q 12.7 mm diameter x 12.7 mm thick • Cowan method chosen here because: • Adequate for current specimen fixturing designs • Relative simplicity • Universal availability • Proven results

12. Adiabatic model Detector signal Adiabatic 9.6 mm 900°C Adiabatic 0.25” (6.4 mm) 800°C Cowen 9.6 mm 900°C Cowan 0.25” (6.4 mm) 800°C Application of the Cowan Heat Loss model (AXF-5Q graphite)

13. Empirically evaluate the Cowan heat loss correction (AXF-5Q graphite) • 12.7 mm diameter • Apparent lower diffusivity for thicker samples. • Deviation >300°V

14. Stefan-Boltzmann Law Eb = σT4 Temp (°C) 9.6 (mm) Empirical test (cont.) • AXF-5Q • 12.7 mm diameter specimens • With Cowan radiation heat loss Radiation heat transfer becomes significant at 400°C and above

15. 9.6 (mm) Temp (°C) PCEA Graphite (12.7mm dia.)

16. 9.6 (mm) Temp (°C) Gilso Graphite (12.7mm dia.)

17. 6.4 (mm) Temp (°C) 12.7 (mm) Temp (°C) NBG-18 Graphite 25.4 mm diameter 12.7 mm diameter

18. Summary of Thickness Limitations(Due to radiation heat loss up to 1000°C)

19. Specimen Minimum Thickness? Specimen Thickness NBG-18 (12.7 mm dia.)

20. Material Effects on Measurement Uncertainty (cont.) • Samples above ~400°C but 1 mm thick do not exhibit the error • Similar results seen for PCEA, AXF-5Q, and Gilso graphite 20% • Sources of error come from breakdowns in assumptions? • Heat loss • Heterogeneity • # of grains • Cracks/pores • Non uniform heating • Multi directional cond. • Finite laser pulse width NBG-18 (12.7 mm dia.) (1.7 mm max, 0.6 mm avg. grain size)

21. Laser Pulse Width Effects on Half Rise Time • Laser pulse, fit and smoothed detector data for 1mm specimen at 200°C • Graphite thermal conductivity at RT is similar to Cu. “Fast Material” • τ is 15-20% of t1/2 • Over prediction of t1/2 would result in erroneously low calculation of the diffusivity.

22. Material Effects on Measurement Uncertainty (cont.) • Finite Laser pulse corrections: • Cape-Lehman 1963 • Square pulse • Azumi-Takahashi 1981 • Delta function 6% • Finite pulse corrections have a limit • Establish a more generic limit for τ/t1/2 Solid = Azumi laser pulse corrected , Hollow = uncorrected NBG-18 12.7mm dia. With Cowan heat loss correction applied

23. Limit of Laser Pulse Correction to Half Rise Time • For T > 400°C and L>4 mm defines a limit of: • τ/t1/2 < 0.025 • For τ= 0.5 mSec • t1/2= 20 mSec

24. Propagation of Error/Uncertainty Estimate (after Kline and McClintock 1953) Where: α = Thermal diffusivity ω = Uncertainty L = Specimen thickness t1/2 = Half rise time • Rules: • D/L > 2 • τ/t1/2 < 0.025 *Based on the standard deviation of t1/2 (length normalized). **Based on ½ of the manufactures specified laser pulse width of 0.5 msec.

25. Summary and Conclusions • ASTM E 1461-11 guide lines: • L = 1 to 6 mm • L ± 0.2% • t1/2 = 10 to 1000 ms • Heat Loss Correction Limit: (upper limit on thickness) • The extent to which any of the heat loss models tested can correct for radiation heat loss is limited. • Specimen dimensions with a D/L > 2 will result in acceptable heat loss corrections when using the Cowan model. • Finite Laser Pulse Correction: (lower limit on heat diffusion time) • As with the heat loss models, the accuracy of the laser pulse width correction is limited. • The Azumi pulse width correction to the t1/2 timing start position is acceptable for τ/t1/2 > 0.025. (t1/2 > 40τ)

26. Summary and Conclusions (cont.) • Comment on representing the bulk material: • The thermal diffusivity remained unchanged for specimens of PCEA and NBG-18 down to 1 mm thick when the condition of τ/t1/2 > 0.025 was met (T>400°C). This indicates that the homogeneity of these relatively large grained graphite's is sufficient down to 1mm thick for LFA determination of diffusivity.

27. Thank you For you Attention… Dave Swank w.swank@inl.gov (208) 526-1698