laser flash method to determine thermal conductivity flash method to determine thermal... ·...

[email protected]://phox.at

1

Laser Flash Method to Determine Thermal Coductivity

A today's philosophy to use Flash Experiments top determine Thermal Conductivity

Motivation & Requirements Theoretical RemarksMeasuring Techniques

Dr.-Ing. Wolfgang Hohenauer


2

Content

• Motivation and Requirements• Basics

– Uncertainty in Measurement Results– Theoretical Remarks

• Laser Flash Technique– Prinziples of the Method– Uncertainty– Limitations of Flash Methods– Experimental results

• Dynamic Scanning Calorimetry• Push Rod Dilatometry• Combined Methods to Determine Thermal Conductivity

• Summary


3

Motivation and Requirements

• It is an integral part of the description of a material.

• To understand the behaviour of a material.• To influence processing steps systematically.• To optimise processes.• Quality management.• Materials selection in technical Design.• Input data for mathematical simulations:

Temperature distribution, deformation of specimen, stress analysis, life time estimation, kinetics of processes, etc.

• Failure analysis.• Lots of other purposes …

Measuring Techniques have to be…


4

Measurement Results & Uncertainty

GUM*) - ENV 13005 (1999) Guide to the expression of uncertainty in measurement:

The result of any measurement is an approximation of the real value and represents an estimate value only…… The best available estimate of the expected quantity is the mean value (of a number of measurement results).… The best estimate of the uncertainty is the positive square root of the experimental variance of the observations – termed: experimental standard deviation

*) ENV 13005; Guide to the expression of uncertainty in measurement (GUM)


5

Measurement Results & UncertaintyENV 13005; Guide to the expression of uncertainty in measurement (GUM)

… The best representation of an estimated value is the mean value (of a number of individual measurement results).

… The uncertainty of an individual measurement result:

… The uncertainty of the estimated value(=mean value):

… The standard uncertainty of a function of input values: “Gaussian error propagation”

… To obtain the expanded uncertainty onemultiplies by an coverage factor U(y) = k . uc(y)

∑=

=n

kkq

nq

1.1

∑=

−−

=n

kkk qq

nqs

1)²(.

11)(

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=N

ii

ic xu

xfyu

1

22

2 )(.)(

)(:)(.1)( ik xuqsn

qu ==

Detailed information available as download: http://phox.at


6

Direct Methods to Measure Thermal Conductivity

• Principle:They realise thermal conductivity situations which allow simple mathematical description.

• Methods:– Comparative method– Guarded Heat Flow method– Hot wire method– Hot strip method

• Problem:– These methods often suffer from

significantly high uncertainties

S

SeffA

S

effA

S TlTP

TTPT

ΔΔ

⋅=∇

= )(||

|)(|)( ./

./

r

λ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΔΔ

+ΔΔ

⋅ΔΔ

⋅⋅=lR

lR

uR

uR

S

SRS l

TlT

TlTT

;

;

;

;)(21)( λλ

upper heater

lower heater

lowerreference

upperreference

specimen

TC 1

TC 3

TC 5

TC 2

TC 4

TC 6

ΔT

≅80

°C

upper heater

lower heater

lowerreference

upperreference

specimen

TC 1

TC 3

TC 5

TC 2

TC 4

TC 6

ΔT

≅80

°C


7

Direct Methods to Measure Thermal Conductivity (3)

Real Conditions in Temperature Measurementcause minimum uncertainties of > 20% of λ


8

Balancing Thermal Flow Effects


9

Thermal Conductivity Equation

0.),(.),(),(),(),( =∇−∂

∂⋅⋅⋅ ∫∫∫

FVp fdtxTTx

ttxTTxcTxdV

rrrr

rr λρ

),()(),()()(

)(),( txTTatxTTcT

Tt

txT

p

rrr

Δ⋅=Δ⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

⋅=

∂∂

ρλ

{ 0

/ ),()(∑ ∑

=⋅+ −++∫∫

i jji SS

txPfddt

tdQA

F

rrr

Conservative systemLinear descriptionMaterial:

HomogeneousIsotropic



10

Principles of a Flash Experiment

Specific experimental setup

1. One-dimensional experiment2. Homogeneous material3. Samples prepared in principle

axis of the material4. Adiabatic boundaries5. Infinite duration of heat impact

This leads to

( ))()(41ln)(

21

2

2 TtThTa ⋅−≅

π


11

Physics / Mathematics of a Flash Experiment

),()(),()()(

)(),( txTTatxTTcT

Tt

txT

p

rrr

Δ⋅=Δ⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

⋅=

∂∂

ρλ

( ))()(41ln)(

21

2

2 TtThTa ⋅−≅

π

2

2 ),().(),(x

txTTat

txT∂

∂=

∂∂

⎥⎦

⎤⎢⎣

⎡⋅⋅+⋅⋅⋅= ⋅− x

acBx

acAetxT tc cossin),(

2

22

0cos),( h

tan

nn e

hxnBtxT

⋅⋅⋅−∞

=

⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅

⋅= ∑ππ

( ) ⎥⎦

⎤⎢⎣

⎡⋅−⋅+⋅Δ==Δ ∑

∞

=∞

⋅⋅⋅⋅−

1

2

22

121),(n

n h

tan

eTthxTπ

D1 Problem

General solution

Adiabatic boundaries

Infinitesimal impact duration

Axial thermal flux

No heat flux at surfaces; coplanar specimen

Heat pulse

O(1)



12

Experimental Data of a Flash Experiment

( ))()(41ln)(

21

2

2 TtThTa ⋅−≅

πΔ

T max

= Δ

T ∞

ΔT(

t 1/2)

t1/2

theoretical adiabatic curve

experimental curve

heat pulse

Modelled curve:based on theoretical models as- Parker, Clark, Taylor, Cowan- Cape & Lehmann- Radiation model- Finite heat pulse length

h


13

The Laser Flash Method - Optimisations

• There are non-conservative systems• Local inhmogenities occur• Sometimes no easy interpretation of

transient temperature curve possible

• Adiabatic boundaries do not exist• Infinitesimal initial heat impacts

cannot be realised• First order solution does not describe

reality – BUT it helps to understandthe fundamentals!

• Up to date LF software calculates amodel of the experimental data

• and includes non adiabatic effectsand finite pulse durations as well


14

The Laser Flash Method - Requirements

• There are non-conservative systems• Local inhmogenities occur• Sometimes no easy interpretation of

transient temperature curve possible

• Adiabatic boundaries do not exist• Infinitesimal initial heat impacts

cannot be realised• First order solution does not describe

reality – BUT it helps to understandthe fundamentals!

• Up to date LF software calculates amodel of the experimental data

• Models include non adiabatic effectsand finite pulse durations as well

Take it or leave it

• This causes limitations of the method- e.g. high conductivity thin layers

• Specific problems often do not allowautomatically controlled procedures

• Competent interpretation of thetemperature response is needed

• Experimentalist have to understand physics and the materials they measure

• Numeric models must be compatible with the measured material (glass !)


15

To Interpret a Flash Measurement

Solid sample measurement

Liquid sample measurement

Curie transition

Phase transition

Melting area


16

Diffusivity & Uncertainty – Thermal Insulators

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=N

ii

ic xu

xfyu

1

22

2 )(.)(

Data acquisitionTime step: 2 µs

a > 5 x 10-9 m²/s


17

Diffusivity & Uncertainty – High Diffusivity

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=N

ii

ic xu

xfyu

1

22

2 )(.)(

Data acquisitionTime step: 12 µs


18

Minimum Possible Sample Thickness

• Modern LFA performance:

– Assume perfect mathematical modelling of T(t)

– Infrared detector: Possible data acquisition rate ~1 MHz

– LFA :Possible data acquisition rate is ~500 kHz

– Time distance between the points is ~2 μs

– Approximately 250 points are required between the release of the laser flash and the half time t1/2.

– The minimum theoretical half time of the sample is ~0,5 ms.

– Minimum possible sample thickness h can be calculated ….

atah ⋅≅⋅⋅≅ 06,04ln

(min)21

2

minπ

( ))()(41ln)(

21

2

2 TtThTa ⋅−≅

π

ΔT ma

x=

ΔT ∞

ΔT(

t 1/2)

t1/2

theoretical adiabatic curve

experimental curve

heat pulse

Modelled curve:based on theoretical models as- Parker, Clark, Taylor, Cowan- Cape & Lehmann- Radiation model- Finite heat pulse length

h


19


Material a(Tr) x 106 / m²/s hmin / mm

Diamond 1000 1,90Silver 174 0,79Copper 117 0,65POCO AXM 5Q 72 0,51Pure Iron 22 0,28Aluminia 10,5 0,19Stainless Steel; AISI 316 3,25 0,11Pyroceram 1,92 0,08Glass 0,7 0,05Filled Polymeres 0,5 0,04Polycarbonate 0,15 0,02Paper, PP, ÜTFE 0,1 0,02

65,0

67,5

70,0

72,5

75,0

0 200 400 600 800 1000 1200m / µm

Appa

rent

The

rmal

Diffu

sivity

: a /

mm

²/s

POCO AXM 5Q

Polynom. Approx. O(3)

T = 25°C<a> = 72,3 mm²/shmin = 0,52 mm

3,00

3,20

3,40

3,60

3,80

0 25 50 75 100 125 150 175 200 225T /°C

Appa

rent

Ther

mal D

iffus

ivity

: a / m

m²/s

Stainless Steel: AISI 316


h = 0,109 mm; hmin = 0,11 mma25°C; Touloukian = 3,25 mm²/s


20


( )

( )sWaermeatlaVDIsmma

smma

smmammdFilmPP

Lit

−=

±=

±=

=−

2.

22

21

142.0

005.0140.0

005.0142.0057.0

( )smma

mmdFilmLDPEStreched

21 002.0153.0

058.0±=

=−

( )

( )sWaermeatlaVDIsmmasmma

mmdFilmPVC

Lit −=

±=

=−

2.

2

125.0005.0126.0

072.0

Quelle: J. Blumm; NETZSCH Gerätebau GmbH


21

Magnesium Foam Material

0

5

10

15

20

-200 -100 0 100 200 300 400 500T / °C

λ / W

/m.K

0

10

20

30

l - LFA l - LFA-corr. l - Comp. a

a x 1

06 /m²/s

22,00

22,25

22,50

22,75

23,00

23,25

0,0 0,5 1,0 1,5 2,0Time /s

Appa

rent

Tem

pera

ture

/K

T_Sensor T_Steg T_Fuellung

SensorMetal StructureCeramic Filler

φ = 20 mmh = 10 mmρ ≅ 0,49 g/cm³

Experimental:

Simulation: Cubic: 10 x 10 x 10 x mm³ρ ≅ 0,6 g/cm³

λ LFA λ LFA; corr. λ Comp.


22

Diffusivity Results of a Steel Material

0,0

2,5

5,0

7,5

10,0

12,5

15,0

0 200 400 600 800 1000 1200 1400 1600T / °C

a x

106 /

m²/s

ecS 500 SF sample 1

S 500 SF sample 2

S 500 SF mean value

Melting AreaMelting area

Curie transition

Phase transition


23

Dynamic Scanning Calorimetry: DSC (1)

Measurement principle:• To heat up any material needs time – this

depends from:– mass– Temperature difference between the

sample and its environment (furnace with respect to the heating rate)

– Specific heat of the sample• One heats a furnace with a defined heating

rate. The difference between the temperature dependent temperature of an empty crucible and a crucible filled with some specific material characterises the thermal consumption of this material.

• To quantify this temperature difference with the behaviour with a reference material results the specific heat and transformation enthalpies of the unknown material.

High-temperatureFurnace


24


Measurement procedure:1. Basis-line:

Characterises the behaviour of the empty DSC (zero-line)

2. Reference-line: Quantifies the response of the DSC when a reference material is measured:

1. Empty Reference crucible 2. Sapphire in sample crucible

3. Sample-line: measurement of a specimen:

1. Empty Reference crucible2. Sample in sample crucible

Calculation of the specific heat

High-temperatureFurnace

RP


25


Calculation of the specific heat:• Heat input of δQ causes an increase of temperature

in any substance:

• These equations can be transformed to express dT. With dT(Sa) = dT(Pr) one obtains:

• From this it follows the evaluating equation of a dynamic calorimeter:

)()()(

)()()(

)()(

)()(

)()(

)()(

TcmTDSCTDSC

TcmTDSCTDSC

Sp

S

BS

Rp

R

BR

⋅−

=⋅

−

)()()()( )()()()()()( TDSCTDSCdTTcmTQ BRRRP

RR −=⋅⋅=δ

)()()()( )()()()()()( TDSCTDSCdTTcmTQ BSSSP

SS −=⋅⋅=δ

( ) ( )Tcmm

TDSCTDSCTDSCTDSCTc R

pS

R

BR

BSS

p)(

)(

)(

)()(

)()()(

)()()()(

⋅⋅−−

=

RP


26


0

1

2

3

4

5

0 200 400 600 800 1000 1200 1400 1600T [ °C ]

c p [

J/g.

K ]

Alloy: Probe (1)Alloy: Probe (2)Alloy: Probe (3)Alloy: Mittelwert


27

Push Rod Dilatometry (1)

Position encoder Flushing Gas Sample Holder Furnace Sample Ventilator

Thermostat Touch Bed Furnace Tube (gas-proof)

DIL 402 C


28


Measurement principle:• The position encoder detects the temperature

dependent position P(T). This is the superposition of the unknown expansion of the specimen ΔL(T) and the intrinsic expansion of the internals of the dilatometer.

• The intrinsic expansion behaviour of dilatometer can be quantified by a correction function K(T).

• This correction function depends on:– Heating rate of the furnace,– Initial length of the sample L0– Material of the internals– Gas conditions in the dilatometer

• To determine the correction function K(T),one correlates the position curves P(T) of a reference-material with it’s recommended expansion data.

DIL 402 C


29


Measurement procedure:

• Examination of the intrinsic expansion behaviour of the dilatometer:– Reference-materials

• Calculation of the correction function K(T): • Measurements of samples:

– specimen

• calculations:– Thermal expansion of the material– CTE

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

Δ=

R

R

LitRR L

PL

LK;0;0

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

Δ+=+=

Δ

R

R

LitRSR

S

S

S

S

LP

LL

LPSK

LP

LL

;0;0;0;0;0

S

SSS L

TLT

TCTET;0

)(1:)()( ΔΔ

=≡α


30


0

5

10

15

20

0 200 400 600 800 1000 1200 1400 1600T [ °C ]

L/L 0

x 1

03

Alloy: Probe (1)

Alloy: Probe (2)

Alloy: Mittelwert

0

5

10

15

0 200 400 600 800 1000 1200 1400 1600T [ °C ]

CTE

[ K

-1 ]

x 10

6

Alloy: Probe (1)

Alloy: Probe (2)

Alloy: Mittelwert


31

Combined Methods to Determine the Thermal Conductivity

• Thermal conductivity equation correlates the relevant thermophysical material properties.

– Thermal density ρ(x,T)and thermal expansion

– Specific heat cp(x,T)and Enthalpy H; Entropy S

– Thermal conductivity λ(x,T)– Thermal diffusivity a(x,T)

• Thermal conductivity λ(x,T) can be calculated out of the properties ρ(x,T), cp(x,T), and a(x,T).

Combined thermophysical measuring methods

),()(),()()(

)(),( txTTatxTTcT

Tt

txT

p

rrr

Δ⋅=Δ⋅⋅

=∂

∂ρ

λ

xxpxx TTcTaT rrrr |)(|)(|)(|)( ρλ ⋅⋅=

H Scp

aCTEλ

DSC DSCDSC (Kalorimetrie)

Laser Flash VerfahrenDilatometrie

λ(T) = a(T) . cp(T) . ρ(T)DSC DSCDSC DSC

DSC (Kalorimetrie)

Laser Flash VerfahrenDilatometrie Laser Flash VerfahrenDilatometrie

λ(T) = a(T) . cp(T) . ρ(T)


32

Diffusivity Results of a Steel Material

0,0

2,5

5,0

7,5

10,0

12,5

15,0

0 200 400 600 800 1000 1200 1400 1600T / °C

a x

106 /

m²/s

ecS 500 SF sample 1

S 500 SF sample 2

S 500 SF mean value

Melting AreaMelting area

Curie transition

Phase transition


33

Thermal Conductivity of a Steel Material

0,00

1,00

2,00

3,00

4,00

5,00

0 200 400 600 800 1000 1200 1400 1600T / °C

c p /

J/g

.K

S 500 SF DSC (deconvoluted)

S 500 SF DSC (not deconvoluted)

S 500 SF Specific Heat

0

25

50

75

100

0 200 400 600 800 1000 1200 1400 1600T / °C

π / W

/m.K

S 500 SF

S 500 SF


Melting Area

Curieu Transition

Phase Transition Phase Transition

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 1600T / °C

⎠L/

L 0 x

103

S 500 SF sample (1)S 500 SF sample (2)S 500 SF CTE . 10e6S 500 SF mean value

Melting Area

uc(λ) ≅ 5%

)()()()( TcTTaT p⋅⋅= ρλ

CTE

x 106

/ 1/K

Uncertainty Budget ESU*)

u(a)*) /% 1,00%u(cp)*) /% 1,50%u(ρ0) /% 1,00%u(CTE)*) /% 3,00%u(ΔT) /K 5*) Equipment Specific Uncertainty

uc( ) /% 3,22%


34

Summary (1)

• The thermal conductivity equation defines the structure of the transient spatial temperature field in a body.

• The thermal conductivity equation correlates the fundamental thermophysical properties.– Thermal density ρ(x,T); thermal expansion ΔL(x,T)/L0 or CTE(x,T)– Specific heat cp(x,T)– Thermal conductivity λ(x,T)– Thermal diffusivity a(x,T)

• The thermal conductivity equation - more or the less - points out how to measure thermophysical material properties.

),()(),()()(

)(),( txTTatxTTcT

Tt

txT

p

rrr

Δ⋅=Δ⋅⋅

=∂

∂ρ

λ


35

Summary (2)

• There exists a simple solution of the thermal conductivity equation for a special system:– Conservative …….– Homogeneous …..– Isotropic ……….....

• This solution can be specified to describe flash experiments by a simple formalism:– Adiabatic boundaries ………….…..– Infinitesimal initial heat impact …..

– First order series expansion ….…..( )

)()(41ln)(

21

2

2 TtThTa ⋅−=

π

Sample requirements

Sample and equipment requirements

Numericalrequirements


36

Summary (3)

• The most relevant thermophysical material properties are:– Thermal density and thermal expansion– Specific heat– Thermal diffusivity and thermal conductivity

• The most common measurement methods to determine these thermophysical properties are:

– Push rod dilatometry– Dynamic scanning calorimetry - DSC– (Laser) Flash methods - LFA– Calculation of the thermal conductivity out of these measurement results

• All these measurement methods enable uncertainties in measurement results near 1% of the analysed value (k=2; Conf.Int. 95% in accordance with DIN-V ENV13005). Thermal conductivities can be calculated with an level of uncertainty near 5%.

• Direct methods to measure the thermal conductivity are comparable expansive, time consuming and inaccurate.

laser flash method to determine thermal conductivity flash method to determine thermal... ·...

Documents