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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 SwankWill Windes

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Temperature (°C)

Ther

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AXM-5Q-06

AXM-5Q-07

AXM-5Q-08

AXM-5Q-08

AXM-5Q-08

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

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

PCk

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

Laser Flash Apparatus (LFA) Operation

Radiation to detector

Laser

Specimen

• 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.

Thermal Diffusivity

• 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

Thermal Diffusivity for a Laser Flash Apparatus (LFA) solved analytically for adiabatic conditions by Parker et. al., 1961

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L

Radiation to detector

Laser

SpecimenL

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Laser Pulse

t1/2

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

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LWhere 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

L= 1.6 mm

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L = 3.2 mm

L= 6.4 mm

Effects of Heat Loss: Adiabatic Conditions? (AXF-5Q graphite, 12.7mm diameter at 800°C)

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Adiabatic model

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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

Radiation to detector

Radiation heat loss

Laser

SpecimenL

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

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• Therefore radiation loss from the circumference is not considered

• Only radiation from the top and bottom surfaces is considered.

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

Radiation to detector

Radiation heat loss

Laser

SpecimenL

Radiation Heat Loss Correction Models (cont.)

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

Application of the Cowan Heat Loss model (AXF-5Q graphite)

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Cowan 0.25” (6.4 mm) 800°C

Adiabatic 0.25” (6.4 mm) 800°C

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Cowen 9.6 mm 900°C

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Adiabatic model

Empirically evaluate the Cowan heat loss correction (AXF-5Q graphite)

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Temperature (°C)

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• 12.7 mm diameter

• Apparent lower diffusivity for thicker samples.

• Deviation >300°V

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Em

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Stefan-Boltzmann Law Eb = σT4

• AXF-5Q

• 12.7 mm diameter specimens

• With Cowan radiation heat loss

Radiation heat transfer becomes significant at 400°C and above

Empirical test (cont.)

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9.6 (mm) Temp (°C)

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PCEA Graphite (12.7mm dia.)

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Gilso Graphite (12.7mm dia.)

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6.4 (mm)

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NBG-18 Graphite

12.7 mm diameter

25.4 mm diameter

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

Graphite Type Average grain size

Maximum thickness

(mm)

Diameter (mm)

Minimum diameter to

thickness ratio

AXF-5Q 5 µm 9.6 12.7 1.3

Gilso Carbon 134 µm 9.6 12.7 1.3

PCEA 750 µm 9.6 12.7 1.3

NBG-18600 µm

1.7 mm max6.4 12.7 2

NBG-18600 µm

1.7 mm max12.7 25.4 2

Specimen Minimum Thickness?

NBG-18 (12.7 mm dia.)

Specimen Thickness

Material Effects on Measurement Uncertainty (cont.)

20%

• Samples above ~400°C but 1 mm thick do not exhibit the error

• Similar results seen for PCEA, AXF-5Q, and Gilso graphite

NBG-18 (12.7 mm dia.)(1.7 mm max, 0.6 mm avg. grain size)

• 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

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”

• Over prediction of t1/2 would result in erroneously low calculation of the diffusivity.

•τ is 15-20% of t1/2

6%

Material Effects on Measurement Uncertainty (cont.)

Solid = Azumi laser pulse corrected , Hollow = uncorrected

• Finite Laser pulse corrections:

— Cape-Lehman 1963Square pulse

— Azumi-Takahashi 1981Delta function

NBG-18 12.7mm dia. With Cowan heat loss correction applied

• Finite pulse corrections have a limit

• Establish a more

generic limit for τ/t1/2

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

Propagation of Error/Uncertainty Estimate(after Kline and McClintock 1953)

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Where:α = Thermal diffusivityω = UncertaintyL = Specimen thicknesst1/2 = Half rise time

*Based on the standard deviation of t1/2 (length normalized).**Based on ½ of the manufactures specified laser pulse width of 0.5 msec.

Rules:• D/L > 2

• τ/t1/2 < 0.025

Case L (mm) Dia. (mm) D/L Temp.

(°C) ωL

(mm) t1/2

(mSec) τ/t1/2 ωt1/2

(mSec) ωα/α

A 6 12.7 2.1 800 0.02 250 0.002 4* ~2% B 1 12.7 12.7 25 0.02 2 0.25 0.25** ~13% C 12.7 12.7 1 800 0.02 1350 0.0004 158* ~12%

Summary and ConclusionsASTM 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)

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.

Thank you For you Attention…

Dave Swankw.swank@inl.gov (208) 526-1698