Thermal characterization of an insulating material through a three- - - PowerPoint PPT Presentation

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Thermal characterization of an insulating material through a three- layers transient method (experimental)

METTI 5 Mesures en Thermique et Techniques Inverses Roscoff, June, 13-18 2011 Tutorial T9 – 1h30

Vincent FELIX Yves JANNOT Alain DEGIOVANNI

Laboratoire d’Energétique et de Mécanique Théorique et Appliquée

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Plan of the presentation

I. Introduction II. The classical approach

1. Example : the flash method 2. Example : the hot plate method

III. The three-layers device method

1. Proposed approach 2. Principle 3. Modeling 4. Sensitivity analysis 5. Numerical simulations

IV. Experimental part

1. The device 2. Materials 3. Measurements and data processing

V. Conclusion

2

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I. Introduction

3

Classical approach in thermal characterization T(x,y,z,t) Φ

λ,ρc,a… h,Rc,mcs…

• Model under the form of a

heat flow / temperature relationship

( ) ( )

... , , , , , , , ,

S

mc Rc h c f t z y x T ρ λ Φ =

• The estimation is performed on the whole unknown parameters with two risks:

The accuracy is decreasing with the number of parameters (correlations, local minimums) Some parameters are not easily quantifiable: hypothesis may introduce bias in the model

• What if the material is very insulating, has a low density or a high porosity ?

A qualitative answer through 2 examples : the flash method and the hot plate method

• How does the proposed three-layers device method avoid these drawbacks ?
• The technical specificity
• The original modeling approach : a temperature / temperature relationship

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II. Classical approach

4

• 1. Example: the flash method

T(t) Φ

a

h1 h2

• Model under the form:

( ) ( )

2 1,

, , h h a f t T Φ =

• In case of an insulating material,

Theated surface >> T and thus h1 ≠ h2 Theated surface has to be very high to have a good signal T The more insulating the material is, the more the hypothesis is contestable. If h1 and h2 are estimated separately, h1, Φ and a get correlated between them.

• In case of a low density material

The measurement of the surface temperature is difficult The heat flow may not be absorbed only at the surface but inside the sample (volume absorbtion)

cst h h h = = =

2 1

• Assuming the hypothesis that:
• The following parameters are estimated:

Φ Φ Φ Φ, a and h.

r x

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II. Classical approach

5

• 2. Example: the hot plate method
• Model under the form:

( ) ( )

s

mc Rc E f t T , , , Φ =

• In case of a low density material

The thermal capacity of the heating element is no more negligible compared to the capacity of the sample The effusivity may be correlated with the thermal capacity of the heating element

• In case of an insulating material

The transfer into the heating element (parallel to the contact surface) is not negligible To limit bounds effects, large sample are needed

• Assuming the following hypothesis:
• Uniform contact resistance
• Homogeneous heating element
• Thin heating element (capacity)

T(t) Φ

E

Rc mcs

r x

• The estimated parameters are: E, Rc and

mcS.

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III. The three-layers method

6

1. Proposed approach: temperature / temperature relationship

• Time dependent model under the form of a convolution product:

( ) ( ) ) ( , , , , ,

1 3 2 2

t T h h a c t F t T ⊗ = ρ λ

As a consequence: the model is independent of all the parameters before the input T1: Φ,h1,mcS,Rc… h1,Rc,mcs…

T2(x,y,z,t) Φ

λ,ρc,… h2,h3,…

T1(x,y,z,t)

Instead of:

( ) ( )

S

c m c R h h h a c t F t T , , , , , , , , ,

3 2 1 2

ρ λ φ =

Input / Output representation where the system (sample) is characterized by its transfer function H(p):

( ) ( ) [ ] ( ) [ ] t T L t T L p H

1 2

=

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7

• 2. Principle
• Flash method = surface temperature measurement impossible for low density materials

Technical solution: flash method on a three-layers system

• The problem of the correlation between h1, Φ and λ remains if a standard model is considered
• Trying to describe the heat flow is difficult: is it a perfect pulse ? A rectangular function ?

Mathematical solution: to measure T1(t) and to use it as known data in a convolution model Brass discs: thin, high thermal conductivity so their temperature T1 and T2 are uniform

( ) ( ) [ ]

) ( , , , ,

1 3 2 1 2

t T h h c p H L t T ⊗ =

−

ρ λ

T2(t) Φ(t)

λ,ρc

h2 h1

Brass,λb,ρcb

T1(t)

r x

III. The three-layers method

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8

• 3. Modeling
• 3D heat transfers with cylindrical symmetry: T=T(r,x,t)
• Contact resistances between the brass discs and the sample are neglected (insulating sample)
• Isotropic and macroscopically homogeneous sample
• Optically thick sample, no radiative transfers at the considered temperatures (ambient)
• Uniform brass discs temperature
• The convective and radiative exchange with the ambient medium are considered in a constant h
• The system is at thermal equilibrium at initial time

Hypothesis

III. The three-layers method

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III. The three-layers method

9

• 3. Modeling

Energy conservation equation for a single layer system

2 2 2 2

) , , ( ) , , ( 1 ) , , ( ) , , ( 1 x t x r T r t x r T r r t x r T t t x r T a ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂

( ) ( )

t T t r T h x t x r T

x

ϕ λ + − − = ∂ ∂ −

∞ =

) , , ( ) , , (

1

( )

∞ =

− = ∂ ∂ − T t e r T h x t x r T

e x

) , , ( ) , , (

2

λ ) , , ( = ∂ ∂ −

= r

r t x r T λ

( )

∞ =

− = ∂ ∂ − T t x R T h r t x r T

R r

) , , ( ) , , (

3

λ

∞

= T x r T ) , , (

2 2 2 2

) , , ( ) , , ( 1 ) , , ( ) , , ( x p x r r p x r r r p x r p x r a p ∂ ∂ + ∂ ∂ + ∂ ∂ = θ θ θ θ ) ( ) , , ( ) , , (

1

p p r h x p x r

x

Φ + − = ∂ ∂ −

=

θ θ λ ) , , ( ) , , (

2

p e r h x p x r

e x

θ θ λ = ∂ ∂ −

=

) , , ( = ∂ ∂

= r

r p x r θ ) , , ( ) , , (

3

p x R h r p x r

R r

θ θ λ = ∂ ∂ −

=

Laplace transform [ ]

∞

− = T t x T L p x ) , ( ) , ( θ

Φ(t)

λ,ρc

h2 h1 h3

r x R e T(r,x,t)

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10

• 3. Modeling

Solution of the single layer system

Integral transform + separataion of the variables:

) , ( ) , ( ) , , ( p r R p x X p x r = θ        = − ∂ ∂ − =         ∂ ∂ + ∂ ∂ ) , ( ) , ( 1 ) , ( 1 ) , ( ) , ( 1

2 2 2 2 2 2

γ α x p x X p x X r p r R r r p r R p r R

( ) ( ) ( ) [ ]

∑

∞ = =

+ =

n n n n n n n

ch sh H r J F p r

1 2

) , , ( β β β α θ

( )

∑

∞ = =

=

n n n n n

r J F p e r

1

) , , ( β α θ

Solutions for x=0 and x=e as a function of r: A system of two differential equations to solve: Solutions for x=0 and x=e averaged on r: ( )

∑

∞ = =

=

n n n n n n moy

J F p

1

2 ) , ( ω β ω θ

( ) ( ) [ ] ( )

∑

∞ = =

+ =

n n n n n n n n moy

J sh H ch F p e

1 2

2 ) , ( ω β β β ω θ

III. The three-layers method

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• 3. Modeling

Including the brass discs

Brass layers heat balance ( )

) ( ) ( 2 ,

1 3 1 1 1 1

p p R h e h p c e x p x

isomoy

Φ +             + + − = ∂ ∂ − θ ρ θ λ

( )

) ( 2 ,

2 3 2 2 2 2

p R h e h p c e x p x

isomoy

θ ρ θ λ             + + = ∂ ∂ −

Considering the average value of the temperature on the variable r, the brass layers heat balance is similar to the limit conditions at x=0 and x=e of the single layer system (the sample) The brass discs are taken into account in the single layer solution by introducing the modified coefficients:

      + + = R h e h p c e h

3 1 1 1 1 * 1

2 ρ       + + = R h e h p c e h

3 2 2 2 2 * 2

2 ρ ) ( ) , , ( ) , , (

1

p p r h x p x r

x

Φ + − = ∂ ∂ −

=

θ θ λ ) , , ( ) , , (

2

p e r h x p x r

e x

θ θ λ = ∂ ∂ −

=

Single layer limit conditions

III. The three-layers method

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• 3. Modeling

Solution of the three-layers system

With ωn which are the N first solutions of the transcendent equation: ( ) ( ) [ ] ( )[

]

( ) ( ) [ ]

[ ]

∑

∞ = =

+ + +         + + Φ =

n n n n n n n n n n n moy

H H ch H H sh H sh H ch e p p

1 1 2 2 1 2 2 3 2 2 2

1 ) ( 4 ) , ( β β β β ω ω β β β λ θ

( )[

]

( ) ( ) [ ]

[ ]

∑

∞ = =

+ + +         + Φ =

n n n n n n n n n moy

H H ch H H sh H e p p e

1 1 2 2 1 2 2 3 2 2 1

) ( 4 ) , ( β β β β ω ω β λ θ ) ( ) (

3 1

ω ω ω J H J =

2 2

      + = R e a pe

n n

ω β

And:

λ e h H

* 1 1 =

λ e h H

* 2 2 =

λ R h H

3 3 =

The front and rear brass discs temperature:

III. The three-layers method

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• 3. Modeling

The convolution model

[ ]

) ( ) , , , ( ) (

1 1 2

t T h c p H L t T ⊗ =

−

ρ λ

• h1 and the properties of the front brass disc have no effect on T2
• No hypothesis about the heat flow (space and time)
• 3 parameters to estimate

Considering the transfer function:

) ( ) , , , , ( ) (

1 3 2 2

p h h c p H p θ ρ λ θ ⋅ =

The time dependent expression of the rear brass disc temperature: T1 is assumed to be a known parameter of the model and is measured: ( )

t T t T

exp 1 1 )

( =

[ ] ( )

t T h c p H L t T

exp 1 1 2

) , , , ( ) ( ⊗ =

−

ρ λ

The remaining unknown parameters are λ, ρc and h which are estimated minimizing: ( ) ( )

[ ]

∑

−

2 exp 2 2

t T t T ) , ( ) , ( ) ( p p e p H

moy moy

θ θ =

The rear brass disc temperature: The direct model explains T2(t) as:

III. The three-layers method

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4. Sensitivity analysis

III. The three-layers method

20 40 60 80 100 120 140 160

• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 Time (s) Reduced sensitivity 20 40 60 80 100 120 140 160

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 Time (s) Reduced sensitivity 20 40 60 80 100 120 140 160

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 Time (s) Reduced sensitivity 20 40 60 80 100 120 140 160

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 Time (s) Reduced sensitivity

λ λ ∂ ∂

2

T c T c ρ ρ ∂ ∂

2

h T h ∂ ∂

2

Super-insulating material: e = 5mm

λ = 0.02 W.m-1.K-1 , ρc = 5 000 J.m-3.K-1

Super-insulating material: e = 10mm

λ = 0.02 W.m-1.K-1 , ρc = 5 000 J.m-3.K-1

Polystyrene: e = 10mm

λ = 0.035 W.m-1.K-1 , ρc = 45 000 J.m-3.K-1

Cellular concrete: e = 10mm

λ = 0.15 W.m-1.K-1 , ρc = 300 000 J.m-3.K-1

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4. Numerical simulations

III. The three-layers method

Thermograms T1exp and T2exp computed with COMSOL considering the following nominal data

Diameter (mm) Thickness (mm)

λ

(W.m-1.K-1)

ρc

(J.m-3.K-1) a (m2.s-1) Sample 35 5.6 0.02 5 000 4 x10-6 Metallic discs (copper) 35 0.4 397.5 3 440 000 1.15 x 10-4 The hypothesis of uniforms temperature in the metallic discs is valid (COMSOL)

Two estimation methods have been applied on these obtained thermograms:

The heat flow dependent model: Only T2exp has been considered as experimental data and

Φ,λ, ρc and h have been estimated using a least squares algorithm

The front temperature dependent model: T1exp has been considered as a time dependent known parameter and λ, ρc and h have been estimated using a least squares algorithm on T2exp

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4. Numerical simulations

III. The three-layers method

Results for different cases of exchange coefficient h1, h2 and h3 set in COMSOL

λ (W.m-1.K-1) ρc (J.m-3.K-1) h (W.m-1.K-1) Nominal value 0.0200 5 000 h1=h2=h3=10 Heat flow model 0.0165 3 929 11.3 Front temp. model 0.0199 5 237 11.2 λ (W.m-1.K-1) ρc (J.m-3.K-1) h (W.m-1.K-1) Nominal value 0.0200 5 000 h1=15 h2=h3=5 Heat flow model 0.0340 8 293 7.5 Front temp. model 0.0199 5 103 5.5

The heat flow dependent model is not precise and is very sensible to h1. The front temperature dependent model gives very good results and is not sensible to the conditions on the front surface (heat flow, heat losses) as announced before In addition, it has been verified that errors on thermal conductivity of the metallic discs have no consequences and an error of x% on thermal capacity of the disc induces the same error on the thermal conductivity of the sample

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• 1. The device

IV. Measurements

Computer Case

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• 1. The device

IV. Measurements

Computer Cold junction (thermocouple K) Signal amplifier and analog filter Direct voltage generator PC oscilloscope Support system Brass discs Sample Heating element

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2. Materials

IV. Measurements

Material D (mm) e (mm) ρ (kg.m-3) c (J.kg-1.K-1) ρc (J.m-3.K-1) Polyethylene foam 40 6.40 40 2 061 82 440 Depron (extruded polystyrene) 40 5.65 35 1 039 36 365 Spaceloft (silica aerogel) 40 2.75 145 1 095 158 775 PVC 40 2.95 1450 952 1 380 000

Proposed samples Metallic discs

Material D (mm) e (mm) ρ (kg.m-3) c (J.kg-1.K-1) ρc (J.m-3.K-1) Polished brass 40 0.38 8 400 377 3 166 800

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3. Measurements and data processing

IV. Measurements

Real time experiment will be carried out on one of our proposed materials

• PicoScope 6 software for the temperature signals acquisition
• Matlab for data processing

Pre-recorded data will be processed on the other materials

• Matlab for the data processing
• Our mind for discussion, questions and eventually answers

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3. Measurements and data processing

IV. Conclusions

Classical approach

• Limits in extreme cases : Very light and insulating materials

Advantages of the approach “temperature / temperature” relationship via a convolution model

• Low number of unknown parameters
• No hypothesis about the front surface exchange
• No information about the spatial and temporal forms of the heat flow needed

Advantages of the specific device “three-layers” using this approach

• Surface temperature measurement reliable
• Suited to small samples
• Estimation method is robust and the parameters are decorrelated

Limits of the three-layers method

• Not optimal for heavy materials
• Estimation of the heat capacity less precise

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Thank you for your participation !

End of the tutorial