A simple device for determining thermal effusivity of thin plates

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https://hal-mines-albi.archives-ouvertes.fr/hal-01651381

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A simple device for determining thermal effusivity of thin plates

Elian Coment, Jean-Christophe Batsale, Bruno Ladevie, Jean-Luc Battaglia

To cite this version:

Elian Coment, Jean-Christophe Batsale, Bruno Ladevie, Jean-Luc Battaglia. A simple device for determining thermal effusivity of thin plates. High Temperatures-High Pressures, Old City Publishing, 2002, 34 (6), pp.627 - 637. �hal-01651381�

Abstract.We measured thermal effusivity of thin plates in the longitudinal direction. This parameter is strongly correlated with lateral convection, so its determination is very difficult. Thermal quadrupole theory is presented which allows the theoretical expression of the problem to be determined. By applying a heat flux without a low-frequency component, only some characteristic times of the system are excited. A real-time method of estimation is developed by using a non-integer derivative signal.The results obtained for a copper plate are in good agreement with the literature.

A simple device for determining thermal effusivity of thin plates

Elian Coment, Jean-Christophe Batsale}, Bruno Ladevie, Jean-Luc Battaglia}

Centre Eènerge¨tique Environnement, LGPSD, UMR 2392 CNRS, Eècole des Mines d'Albi, Campus Jarlard, F-81013 Albi Cedex 09, France

}Laboratoire Eènerge¨tique et Phe¨nome©nes de Transfert, UMR CNRS-Bx1-ENSAM8508, Eècole Nationale Supe¨rieure d'Arts et Me¨tiers, Esplanade des Arts et Me¨tiers, F-33405 Talence Cedex, France;

email:coment@enstimac.fr

Presentedat the 16th European Conference on Thermophysical Properties, Imperial College, London, England, 1 ^ 4 September 2002

Nomenclature

Symbol Quantity Units

a thermal diffusivity m² s ¹

Cp specific heat J kg ¹ K ¹

Dⁿ non-integer derivative operator

e thickness m

f frequency Hz

h convection heat transfer coefficient W m ² K ¹

H transfer function

L Laplace transform

p Laplace variable

R contact resistance m² W ¹ K ¹

t time s

T temperature K

x, y coordinates

Z thermal impedance

a, b parameters

D sampling

e residues

x error modelling

z measurement noise

l thermal conductivity W m ¹ K ¹

f heat flux density W m ²

F Laplace transform of heat fluxdensity

r density kg m ³

y Laplace transform of temperature

o rate of pulsation rad s ¹

Subscripts

M plate medium

s sensor

f filtered

Superscripts

^ estimated magnitudes

* magnitudes with noise

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shows that we have to derive the physical parameters of our systemöboth the temperature and the heat fluxöin a non-integer form:

…a1p^0:5‡a2p‡a3p^1:5†y…p† ˆ …b1‡b2p^0:5†F…p†

, (4) where

a1 ˆ1, a2 ˆ(rCp)_ses=(lrCp)¹⁼²

,

a3 ˆR(rCp)_ses

,

b1 ˆ(lrCp) ¹⁼²

,

b2 ˆR. The

fractional differential equation is of the form

a1

D

^0:5T…t† ‡a2

D

¹T…t† ‡a3

D

^1:5T…t† ˆb1f…t† ‡b2

D

^0:5f…t†

. (5) The general non-integer derivative of a function

g…t†

is written with the use of the discrete definition of Gru«nwald (see Miller and Ross 1993):

Dⁿg…t† ˆ 1 Dtⁿ

X^K

kˆ0

… 1†^k n

k g…t kDt†, n 2 <, tˆKDt

, (6) where

kn ˆn…n 1† …n k‡1†

k!

is Newton's binomial and

Dt

assigns the sampling.

The goal of the identification is to estimate, from observed data (temperature and heat flux) of the system to be studied (figure 1), the coefficients of the fractional differential equation (5). Here, this model is characterised by the vector

##ˆ ‰a2 a3 b₁ b₂Š^T

.

Applying a linear transformation [low-pass filter

F(p)

in our case] to equation (5) and isolating D

^0:5Tf(t), we obtain a linear continuous regression with null initial

conditions, of the form:

D

^0:5Tf…t† ˆ a2

D

¹Tf…t† a3

D

^1:5Tf…t† ‡b₁

D

⁰f_f…t† ‡b₂

D

^0:5f_f…t†

ˆ ‰

D

¹Tf…t†

D

^1:5Tf…t†

D

⁰f_f…t†

D

^0:5f_f…t†Š##

ˆuf##

,

(7)

where

Tf(t)

and

f_f(t)

are, respectively, the filtered temperature and the heat flux. Even though equations (5) and (7) are equivalent, only the form of equation (7) allows us to compute the various derivatives from observed data.

4.2

Fractional state-variable-filter design

In order to estimate parameters, the fractional derivatives of the filtered data

Tf(t)

and

f_f(t), included in the vector uf

, require to be computed. Indeed, computation of fractional derivatives with the use of approximation (6) generates significant errors. The method consists of using a fractional state-variable-filter (SVF), which is composed of the fractional derivatives of the filtered data:

xˆ ‰D^{Nf 1†n}zf…t†,

D

^{Nf 2†n}zf…t†, . . . ,Dⁿzf…t†, zf…t†Š^T

, (8)

where

zf(t)

denotes either

Tf(t)

or

f_f(t);n ˆ0:5, and Nf 2

.

The state-variable-filter used here results from an extension of the Poisson filter to fractional systems:

F…p† ˆ p

of

n

‡1 Nf

ˆ o^nN_f ^f

p^nNf ‡

C

^N₁^foⁿ_fp^{n…Nf 1†}‡. . .‡

C

^N_N^f_{f 1}o^n…N_f ^{f 1†}pⁿ ‡o^nN_f ^f

,

(9)

where

of

is the characteristic pulsation frequency of the system, and the filter order is

defined by

n Nf

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