J Therm Anal Calorim DOI 10.1007/s10973-014-3837-9

Development of the hot wire technique for finite geometry samples Estimation of thermal properties of soil Mohamed B. H. Sassi • M’hamed Boutaous

Received: 25 November 2013 / Accepted: 22 April 2014  Akade´miai Kiado´, Budapest, Hungary 2014

Abstract The hot wire technique is widely used to determine the thermal properties of materials. Commonly, this technique is developed for considered infinite radius of cylindrical mediums. Here, we propose an analytical solution of the heat conduction problem in an insulated finite sample. The derived temperature solution is found mathematically to be non-regular convergent series, and lead to avoid the assumption of infinite geometries, difficult to realize in practice. The first part of this paper concerns the study of the calculability of this series. The second one deals with the thermal properties estimation. The sensitivity study shows that the estimation procedure is feasible either using the concept of the time of the maximum rising temperature or, when exploring large time measurements, using the least squares minimisation that has an equivalent pure graphical procedure. Keywords Heat conduction  Parameters estimation  Granular mediums  Sensitivity study  Hot wire technique  Thermophysical properties

Introduction Analytical solutions are very useful for parameter estimation in many problems. The accuracy and the domain of validity of such solution must be specified. This study deals M. B. H. Sassi Departement de Physique et Instrumentation, Laboratoire MMA, INSAT, Universite´ de Carthage, Carthage, Tunisia M. Boutaous (&) Universite´ de Lyon, CNRS, CETHIL, UMR 5008, INSA-Lyon, 69621 Villeurbanne, France e-mail: [email protected]

with the development of an accurate analytical solution for the classical heat conduction transient problem, in order to extend the hot wire technique to identify thermal parameters even in case of finite samples. The parameters to be estimated are the thermal conductivity, the diffusivity, and eventually the specific heat, using finite cylindrical geometry, avoiding the constraining condition for the hot wire technique, which consists in considering an infinite radius. The techniques developed in literature for such estimation are principally classified into two groups: the steady state techniques and the transient ones. Generally, the first kind permits to determine only one thermal parameter [1, 2]. The typical heating source signals investigated in practice are generally in the form Dirac, Heaviside, and the sinusoidal one known as 3-Omega method [3, 4]. The Flash method, initially proposed by Parker to estimate the diffusivity, is found to be not appropriate for granular mediums. One of the main reasons is that the used radiation heating flash will spread through the porosities and so that the heat transfer within the domain is so complicated and not a pure conduction phenomena. Note that more detailed discussion on this method can be found in references [5–7]. On the contrary, the hot wire technique is efficient and commonly used for this type of materials because the heat source is generally ensured by an electric resistance [8, 9]. First, it has been developed to determine the apparent heat conductivity. The studied medium is considered infinite radial domain, and the mathematical model is asymptotic and available only when using temperature profile for large time. In these assumptions, the conductivity is proportional to the derivative of the temperature with respect to the logarithm of time ddTln t. Eventually, an instantaneous analytical solution of this model and the simultaneous parameter estimation are recently developed in Ref. [9]. In

123

M. B. H. Sassi, M. Boutaous

the present work, the medium will be considered as an insulated finite radial domain. Compared to numerical resolution, the analytic solution that will be derived herein has the following features: • •

•

It allows graphical parameter estimation using the practical measured temperature curves; From the mathematical point of view, the linearity between the sensitivity coefficients is easily analysed since they are expressed explicitly as functions of time; and The temperature expression includes the geometrical and the theoretical boundaries conditions. So that, the sensitivity of the estimated parameters to the relative sensor position errors is easy to determine.

The analytic formulation and resolution will be developed only for the Dirac and the Heaviside types of heat source generation. In the last section, practical measurements will be investigated to estimate thermal properties of a dry soil in order to demonstrate the consistency of the proposed model and the inverse problem procedure.

As explained in the references [10, 11], the Green’s function can be obtained by setting the homogeneous model where the generation and the non-homogeneous boundary condition functions are lumped, except that the initial condition function can be considered arbitrary to derive first the ¨ zisik [12], function Gðr; t=r 0 ; sÞjs¼0 . Second, as shown by O 0 the Green’s function Gðr; t=r ; sÞ for the transient heat conduction is obtainable from Gðr; t=r 0 ; sÞjs¼0 by replacing t by (t-s) in the latter. The use of the variables separation method for the homogeneous model gives ! 1 0 X 2 0 ab2m ðtsÞ J0 ðbm r ÞJ0 ðbm r Þ Gðr; t=r ; sÞ ¼ 2 1 þ e ; b J02 ðbm bÞ m¼1 ð5Þ where J0 is the zero-order Bessel function and bm are the roots of J0 ðbm bÞ ¼ 0. Note that the first Eigen-value is zero b0 ¼ 0; and it has been included in the Green’s function by means of the unity term. The rest of the Eigen-values bm can be deduced from the roots vm of the first-order Bessel function J1 as follows: J1 ðbm bÞ ¼ J1 ðvm Þ ¼ 0:

The analytical solution and its numerical calculation Consider a finite radial cylindrical medium 0  r  b, initially at a homogeneous temperature. The outer surface r ¼ b is thermally insulated. The contact between the thin axial electric resistance and the medium is supposed to be perfect. In the assumption of negligible heat inertia of the electric resistance and constant thermo physical properties, the mathematical model describing the one-dimensional heat conduction through the full cylinder is o2 T 1 oT 1 1 oT þ gðr; tÞ ¼ þ or 2 r or k a ot oT ¼ 0 at r ¼ b or

in 0  r  b

T ðr; t ¼ 0Þ ¼ T0 ;

ð1Þ ð2Þ ð3Þ

where k denotes the thermal conductivity and a the thermal diffusivity of the equivalent homogeneous medium. The initial temperature value T0 can be taken as the origin of the temperatures. By setting h ¼ T  T0 , it will be set equal to zero for the rest of this analysis. The term g represents the distributed heat source generated by the electric resistance. Its dimension is (w/m3). The solution of this non-homogeneous problem in terms of the Green’s function is T ðr; tÞ ¼

a k

Z t Zb

r 0 Gðr; t=r 0 ; sÞgðr 0 ; sÞdr 0 ds;

s¼0 r¼0

where Gðr; t=r 0 ; sÞ is the Green’s function.

123

ð6Þ

The twelve first roots of this function can be found in Ref. [13]. To determine the rest of Eigen-values, we can use the asymptotic behavior of the Bessel’s functions. Eventually, for a large value of a real z  15:9; we have rffiffiffiffiffiffi rffiffiffiffiffiffi   2 p p 2 p p J m ðzÞ ffi cos z   m sin z þ  m ; ¼ pz 4 2 pz 4 2 ð7Þ where v is the order of the Bessel function. So, for m [ 20, the roots values vm of the Bessel’s function J1 are given by p ð8Þ vm ¼ mp þ : 4 Two cases of the heat flux delivered by the electric resistance will be investigated, the Dirac and the Heaviside kinds. To determine the temperature solution for a Dirac heat flux, we set gðr; tÞ ¼

Q dðtÞdðr  RÞ; 2p r

ð9Þ

where R is the electric resistance radius. Hence,

! 1 X Q ab2m t J0 ðbm r ÞJ0 ðbm RÞ Td ðr; tÞ ¼ 2 1þ e ; pb qc J02 ðbm bÞ m¼1 ð10Þ

ð4Þ

in which qc denotes the volume heat capacity. In the same manner, the solution for a Heaviside heat flux generation is derived by setting the term source:

Hot wire technique for finite geometry samples

Q dðr  RÞY ðtÞ: 2pr Q t pb2 qc 1 h i J ðb r ÞJ ðb RÞ 2 Q X 0 m 0 m þ 2 : 1  eabm t 2 2 pb k m¼1 bm J0 ðbm bÞ

Th ðr; tÞ ¼

ð12Þ

Introducing dimensionless parameters, the Fourier number for the time is at : b2

ð13Þ

For space variables,

1 0.8 0.6 0.4 0.2 0 0

r R r ¼ ; R ¼ ; b b 

0.1

ð14Þ

0.2

0.3

0.4

0.5

0.6

Fourier number F Fig. 1 Typical temperature plots for a Heaviside flux generation

and for the temperatures, we set Td  T 0 ¼

r * = 0.1 r * = 0.2 r * = 0.3 r * = 0.6

1.2

So that we obtain

F ðt Þ ¼

1.4

ð11Þ Dimensionless temperature

gðr; tÞ ¼

Q T pb2 qc d

ð15Þ 2

evm F ðtÞ 1 Mm ¼ pffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi :   r  R r R

and Th  T0 ¼

Q  T : pk h

ð16Þ

Then, these non-dimensional temperatures are as follows: Td ðr; tÞ ¼ 1 þ

1 X

2

evm F ðtÞ

m¼1

J0 ðvm r  ÞJ0 ðvm R Þ J02 ðvm Þ

ð17Þ

and Th ðr; tÞ

¼ F ðtÞ þ

1 h X

1e

m¼1

v2m F ðtÞ

i J ðv r  ÞJ ðv R Þ 0 m 0 m : v2m J02 ðvm Þ ð18Þ

It is obvious that accuracy of these solutions will depend strongly on the truncation of the infinite summation. In practice, these series are calculated numerically using FORTRAN 77 subroutines, and the calculation of the Bessel functions and their roots is a double precision one. Let m0 be the last kept term of this sum. For large value of vm , using the asymptotic value of the zero-order Bessel’s function, rffiffiffiffiffi  2 p sin z þ : J 0 ðzÞ ffi ð19Þ pz 4

In such a case, the rest of the series sum does not converge: 1  X p  p Mm sin vm r  þ sin vm R þ 4 4 m¼m0 1    X 1 p p ffi pffiffiffiffiffiffiffiffiffi sin vm r  þ sin vm R þ : ð22Þ 4 4 r  R m¼m0 It is important to note that the modulus diverges when extending the product terms ðr  :R Þ to zero, consequently the calculus of the summation will do too. It is an expected result, because in this case, the model corresponds to the hot wire technique dealing with an infinitely large medium [9]. Further more, the modulus Mm decreases exponentially when increasing m. Eventually, it can be easily demonstrated that 3 2 Mmþ1 ¼ ep F ðtÞð2mþ2Þ : Mm

2



v2m F ðtÞ



2

p

J0 ðvm r ÞJ0 ðvm R Þ e ffi pffiffiffiffiffiffiffiffiffi sin vm r  þ 4 J02 ðvm Þ r  R   p sin vm R þ : 4

evm F ðtÞ

So, for Fourier numbers closer to zero, we have

ð23Þ

So that, for a given lower limit of the Fourier number Fl and to guarantee an appropriate accuracy of the calculated temperature of order e = 10-a, we should choose the maximum of the truncated sum Mm0 that is obtained for r* = R* respecting

The corresponding terms of the series are 

ð21Þ

Mm0 ¼

evm0 Fl ¼ e; R

ð24Þ

which is equivalent to m20 ffi 

lnðeR Þ : p2 Fl

ð25Þ

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M. B. H. Sassi, M. Boutaous

Thermal properties estimation procedures Considering the problem of heat transfer in a cylindrical medium, we assumed to have homogeneous thermal properties to be identified using the hot wire technique. Let I stands for a number of interior locations where the temperature is recorded and N denotes the data number of measurements of each sensor. The least square method consists of minimising the error between the measured and computed temperatures given as follows: W¼

I X N X

ðTci  Tmi Þ :

oT : og

6

4

2

0

ð27Þ

Feasibility of the three parameters simultaneous estimation At first, the temperature Th can be seen as a function of the three independent parameters Th ðk; qc; aÞ since it can be written as Q Th ðr; tÞ ¼ 2 t pb qc 1 h i J ðb r ÞJ ðb RÞ 2 Q X 0 m 0 m þ 2 : 1  eabm t 2 2 pb k m¼1 bm J0 ðbm bÞ

0.2

0.3

0.4

Fourier number F

estimated. Furthermore, for short time, we have Sk 0 and the others are linearly dependant: Sa ðtÞ ¼ Sqc :

1 X J0 ðvm r  ÞJ0 ðvm R Þ : J02 ðvm Þ m¼1

ð32Þ

It follows that the simultaneous estimation is not feasible. Eventually, only two parameters will be simultaneously estimated, and the third one will be deduced from the expression defining the diffusivity a ¼ k=qc. When substituting this equation into the expression of the temperature Th gives three required possibilities as described follows. Required Th ðk; aÞ function The corresponding temperature expression is Th ðr; tÞ ¼

ð28Þ

Let Sk, Sqc ; and S be the sensitivity coefficients of the temperature Th, respectively, to the parameters k, qc, and a: 1 h i J ðv r  ÞJ ðv R Þ QX 2 0 m 0 m Sk ð t Þ ¼  1  evm FðtÞ ð29Þ pk m¼1 v2m J02 ðvm Þ Q F ðt Þ pk 1 X Q J0 ðvm r  ÞJ0 ðvm R Þ 2 Sa ð t Þ ¼ F ðtÞ: evm F ðtÞ : pk J02 ðvm Þ m¼1

0.1

Fig. 2 Typical temperature plots for a Dirac flux generation

Sensitivity study

Qa t pb2 k   1  QX av2m t J0 ðvm r  ÞJ0 ðvm R Þ þ 1  exp  2 ; pk m¼1 b v2m J02 ðvm Þ ð33Þ

which leads to " # 1 h i   X Q v2m F ðtÞ J0 ðvm r ÞJ0 ðvm R Þ F ðt Þ þ 1e Sk ð t Þ ¼  pk v2m J02 ðvm Þ m¼1 ð34Þ

ð30Þ and ð31Þ

It is easy to see that for large time, the sensitivity coefficient S decreases when increasing time, and it vanishes to zero. Thus, for this range of time, the temperature is insensitive to the diffusivity and consequently it cannot be

123

8

ð26Þ

It is well known that the simultaneous parameters estimation is not successful if the temperature derivatives with respect to the different parameters are linearly dependent [14–16]. In this section, we will deal with the sensitivity coefficient of the temperature to any parameter g defined as follows:

Sqc ðtÞ ¼ 

r * = 0.2 r * = 0.4 r * = 0.6 r * = 0.8

0 2

i¼1 j¼1

Sg ð t Þ ¼ g

10

Dimensionless temperature

The general shapes of the temperature curves are plotted in Figs. 1 and 2. We have fixed for this example R* = 10-3, Fl = 10-3, and e = 10-4; this leads to m0 = 41.

" # 1   X Q v2m F ðtÞ J0 ðvm r ÞJ0 ðvm R Þ F ðt Þ 1 þ Sa ð t Þ ¼ e : pk J02 ðvm Þ m¼1 ð35Þ As shown in Fig. 3, for large Fourier’s numbers, these two sensitivity coefficients are linearly dependant:

Hot wire technique for finite geometry samples 0.6

0.6 R * = 10–3, r * = 0.5

R * = 10–3, r * = 0.5

0.5

S*α S*λ

Sensitivity coefficients

Sensitivity coefficients

0.5

0.4

0.3

0.2

S*ρc S*λ

0.4 0.3 0.2 0.1

0.1

0 0 0

0.1

0.2

0.3

0.4

0.5

–0.1 0

Fourier number F

0.1

0.2

0.3

0.4

0.5

Fourier number F Fig. 3 Sensitivity coefficients for required Th ðk; aÞ function Fig. 4 Sensitivity coefficients for required Th ðk; qcÞ function

Sa ðtÞ ¼ Sk ðtÞ þ

1 QX J0 ðvm r  ÞJ0 ðvm R Þ : pk m¼1 v2m J02 ðvm Þ

ð36Þ

In the case of reduced Fourier’s numbers, they are linearly dependent too and we have " # 1 X J0 ðvm r  ÞJ0 ðvm R Þ Sa ð t Þ ¼  1 þ ð37Þ  Sk ðtÞ: J02 ðvm Þ m¼1 Figure 3 shows the curves of the dimensionless sensitivity coefficients Sg ¼ pk Q Sg . Note that the sign (-) of the one relative to the conductivity is omitted. Obviously, this constraints show that the simultaneous estimation of the conductivity and the diffusivity is delicate. Required Th ðk; qcÞ function

The estimation is efficient for large time because these coefficients behave as 1 QX J0 ðvm r  ÞJ0 ðvm R Þ : pk m¼1 v2m J02 ðvm Þ

ð42Þ

That is evidently a constant, and the second increase linearly with respect to time as

ð38Þ

Consequently, the estimation of the parameters is efficient when using temperature measurements corresponding to a large range of time.

1 h

2 i J0 ðvm r ÞJ0 ðvm R Þ QX 1  1 þ v2m F ðtÞ evm F ðtÞ ; pk m¼1 v2m J02 ðvm Þ

ð39Þ and

They are represented in Fig. 4 with omitting their sign (-). For reduced Fourier numbers, these two coefficients are linearly dependent and they respect " # 1 X J0 ðvm r  ÞJ0 ðvm R Þ Sqc ðtÞ J02 ðvm Þ m¼1 " # 1 X J0 ðvm r  ÞJ0 ðvm R Þ ¼ Sk ðtÞ 1 þ : ð41Þ J02 ðvm Þ m¼1

Q t pb2 qc    1 QX kv2m t J0 ðvm r  ÞJ0 ðvm R Þ þ 1  exp  2 : pk m¼1 b qc v2m J02 ðvm Þ

So, the sensitivity coefficients are Sk ðtÞ ¼ 

ð40Þ

Sk ð t Þ ¼ 

In this case the temperature is expressed as Th ðr; tÞ ¼

" # 1   X Q v2m F ðtÞ J0 ðvm r ÞJ0 ðvm R Þ e : Sqc ðtÞ ¼  F ðtÞ 1 þ pk J02 ðvm Þ m¼1

Sqc ðtÞ ¼ 

Q F ðtÞ: pk

ð43Þ

Required Th ðqc; aÞ function Following the same procedure, the temperature is expressed as

123

M. B. H. Sassi, M. Boutaous

Q Q Th ðr; tÞ ¼ 2 t þ pb qc paqc   1  X av2m t J0 ðvm r  ÞJ0 ðvm R Þ 1  exp  2 : b v2m J02 ðvm Þ m¼1

W¼ ð44Þ

and



J0 ðvm r ÞJ0 ðvm R Þ : v2m J02 ðvm Þ

ða1 ; a2 Þ ¼ 1 0 N N N N N N N P P P P P P P 2 Thmi ti  ti Thmi ti C BN Thmi ti  Thmi ti C B i¼1 i¼1 i¼1 i¼1 i¼1 i¼1 ; i¼1 C: B    N 2 2 N N N A @ P P P P N ti2  ti N ti2  ti i¼1

ð46Þ

However, when increasing time, these coefficients behave as " # 1 X Q J0 ðvm r  ÞJ0 ðvm R Þ F ðt Þ þ Sqc ðtÞ ¼  ð48Þ pk v2m J02 ðvm Þ m¼1 and 1 QX J0 ðvm r  ÞJ0 ðvm R Þ : pk m¼1 v2m J02 ðvm Þ

ð49Þ

It follows that the estimation is successful only when using temperatures measurements for large range of time. Graphical procedures for the estimation

i¼1

i¼1

ð52Þ

qc ¼

Q : pb2 a1

ð53Þ

It is of interest to note that the specific heat given by the above equation does not depend explicitly of the sensor position. In similar manner, the conductivity is found as k¼

1 Q X J0 ðvm r  ÞJ0 ðvm R Þ ; pa2 m¼1 v2m J02 ðvm Þ

ð54Þ

from which it follows that the diffusivity is obtained independently of the heat rate Q: a¼

1 a1 b2 X J0 ðvm r  ÞJ0 ðvm R Þ : a2 m¼1 v2m J02 ðvm Þ

Let us note this full range series R ¼

ð55Þ P1

m¼1

J0 ðvm r ÞJ0 ðvm R Þ , v2m J02 ðvm Þ

and its calculus truncation term is bounded as follows:    X 1 1 J0 ðvm r  ÞJ0 ðvm R Þ 1 X 1  ffiffiffiffiffiffiffiffiffi p    2 J 2 ðv Þ     m v v r R m0 2m m 0 m 0 1  p  p 1 X 1

sin vm r  þ : sin vm R þ   pffiffiffiffiffiffiffiffiffi   4 4 v r R m0 2m ð56Þ Substituting the vm values for large numbers m, one can easily obtain

As seen in the preceding sections, the estimation using least square method is feasible only when using large time measurements for both cases Th ðqc; aÞ and Th ðk; qcÞ. It is important to note that for this range of large Fourier number, the temperature Th behaves in a linear manner with time as follows: 1 Q QX J0 ðvm r  ÞJ0 ðvm R Þ Th ðr; tÞ ¼ 2 t þ ¼ a1 t þ a2 ; pb qc pk m¼1 v2m J02 ðvm Þ ð50Þ so that the practical temperature curve Thm ðti Þ can be used to determine the values of the constants a1 and a2 that minimise the error function:

123

i¼1

Thus, the thermal parameters are given as

Similarly, to the previous case, the estimation is not efficient in the range of short times because the sensitivity coefficients are linearly dependent since they respect " # 1 X J0 ðvm r  ÞJ0 ðvm R Þ Sa ðtÞ ¼ Sqc ðtÞ : ð47Þ J02 ðvm Þ m¼1

Sa ð t Þ ¼ 

ð51Þ

One can easily derive their known expression as developed in [17, 18]

ð45Þ



ðThmi  a1 ti  a2 Þ2

i¼1

The sensitivity coefficients are plotted in Fig. 5, and their mathematical expressions are, respectively, " # 1 h i   X Q v2m F ðtÞ J0 ðvm r ÞJ0 ðvm R Þ F ðt Þ þ Sqc ðtÞ ¼  1e pk v2m J02 ðvm Þ m¼1

1 h

2 i QX Sa ð t Þ ¼  1  1 þ v2m F ðtÞ evm F ðtÞ pk m¼1

N X

1 1 X 1 16 X 1 ¼ : 2 2 v p ð 4m þ 1Þ 2 m0 m m0

ð57Þ

It is easy to recognize that this series is convergent since the power of the denominator terms is greater than the P unity, so that the numerical calculability of does not imposes any difficulties. But, on other hand, this series depends on the sensor position, and its relative calculation error is related to the relative position error as follows: DR r  oR Dr  Dr  ¼   ¼ Hðr  ; R Þ   ;  R R or r r

ð58Þ

Hot wire technique for finite geometry samples 60

0.6 –3

R * = 10 , r * = 0.5

40

S*α S*ρc

0.4

Amplification function Θ

Sensitivity coefficients

0.5

0.3 0.2 0.1

20

0

–20

–40

0 –60 0

–0.1 0.0

0.1

0.2

0.3

0.4

0.5

Fourier number F

0.2

0.4

0.6

0.8

1

Relative position r * Fig. 6 The amplification relative error function

Fig. 5 Sensitivity coefficients for required Th ðqc; aÞ function

so that we also obtain Dk Da DR Dr  ¼ ¼ ¼ Hðr  ; R Þ   : k a R r

ð59Þ

The position function H can be seen as an amplification term of the relative position error. The graph of H is shown in Fig. 6 for a value of R* = 9.09.10-4. Clearly, we see that the plot presents a divergent point jHj ! 1 when r  ! 0:5495. Thus, only sensors having an attenuation H function ðjHðr  ; R Þj\1Þ are recommended to be used for the estimation of the conductivity and the diffusivity using the above equations. Let us now remember that the temperature field Td ðtÞ can be obtained by simple derivation of the curve Td ðtÞ: Td ðtÞ ¼

oTh ðtÞ : ot

ð60Þ

As a result, the second alternative for the estimation consists to use the maximum temperature rising of the temperature Td ðtÞ that corresponds to an inflexion point of Th ðtÞ. The Fourier number F0 corresponding to this point respects 1 X m¼1

oTd oF 2

ð63Þ In practice, the value Td(r,tmax) is small and obtained numerically as the derivative of Th with respect to time. So, the time increment Dt should be selected with certain care, and the temperature measurements must be so precise to guarantee reasonable accuracy of the calculated value of Td(r,tmax). For this reason, this equation will not be explored in the next section. In order to illustrate the above results, some experimental analyses are given in the next section, where only the concept of the maximum of Fourier number and the temperature curves asymptotes will be used to determine the thermal properties of a dry soil used as a building material.

¼ 0, which leads to

v2m evm F0

J0 ðvm r  ÞJ0 ðvm R Þ ¼ 0: J02 ðvm Þ

ð61Þ

Clearly, this Fourier number obtained from the above equation involves only the dimensionless sensor position r*and the radius of the electric resistance R*. So, let tmax be the practical time at which the maximum of Td is reached, then the diffusivity is obtained as a¼

It is also possible to use the maximum amplitude rising of the temperature Td(r,tmax) to determine the sensible heat (qc) as follows: ! 1   X Q v2m F ðr ;R Þ J0 ðvm r ÞJ0 ðvm R Þ 1þ qc ¼ 2 e : : pb Td ðr; tmax Þ J02 ðvm Þ m¼1

b 2 F0 ð r  ; R Þ : tmax

ð62Þ

Measurements and application for the estimation of soil properties Before starting, let us remember that the estimation of the conductivity and the sensible heat includes the value of the heat rate Q that is difficult to evaluate in general. Unless, if the electric resistance is made from a good heat conductor alloy, and it has a reduced radial dimension, then one can neglect the temperature gradient and the rate of energy storage within it. Furthermore, if the contact between this resistance and the medium is supposed perfect, so that, the

123

M. B. H. Sassi, M. Boutaous 32 r = 10 mm r = 20 mm r = 30 mm Asymptote Asymptote Asymptote

30

Temperature/°C

28 26 24 22 20 18 16 14 0

2000

4000

6000

8000

10000

12000

Time/s Fig. 7 Measured temperatures and their asymptotes

0.003

Temperature derivative/°C s–1

r = 10 mm r = 20 mm

0.0025

r = 30 mm

0.002 0.0015 0.001 0.0005 0 0

2000

4000

6000

8000

10000

12000

Time/s Fig. 8 The temperature derivatives with respect to time Table 1 Parameters associated to the estimation procedures ri

10 mm

20 mm

30 mm

F ðri ; RÞ 1 P J0 ðvm r  ÞJ0 ðvm R Þ

0.0083

0.0331

0.0766

0.4856

0.1638

0.0024

m¼1

v2m J02 ðvm Þ

jHðr  ; R Þj a1

0.98

2.53

126.4

7.0915 10-4

7.0613 10-4

7.0538 10-4

a2

5.0757

3.4033

2.5076

tmax

324

711

1341

heat generated by Joule effect inside the electric resistance can be considered entirely entering the medium, then the heat source term is Q¼

UI Ws=m, l

123

ð64Þ

where U stands for the practical electric voltage, I is the current intensity of the electric current through the resistance, and l denotes the resistance length. It is important to note that Q means the delivered heat during one second time (1 s). So, its practical numerical value corresponds to the heat rate delivered by meter of the resistance W/m. The electric resistances that we have used in our experiment were constructed from a Nickel–Chrome alloy. It is 500 mm long and 0.1 mm outer diameter. Three thermocouples K-type are inserted, respectively, at r1 = 10, r2 = 20, and r3 = 30 mm far from the axis of the external tube and midway along its length. The external tube to confine the soil is made from a thermoplastic insulator and having 55 mm inner radius. The used regulated d-c power supply is capable of varying the voltage from 0 to 30 V in steps of 0.01 V. Figure 7 shows the curves of the measured temperatures corresponding to U = 26.1 V and I = 0.33A and a constant time interval of acquisition Dt = 2 s. For the temperature calculation, we have fixed an accuracy e = 10-4 and a lower limit Fourier number Fl ¼ 104 , so that we have found the limit terms of the series sum truncation m0 = 128. The measured temperature curves and their asymptotes relatives to large time are presented in Fig. 7. The temperature derivatives with respect to time are plotted in Fig. 8. The maximum time of temperature rising and the asymptotes parameters are given in Table 1. When using the least square minimisation for large time, the three procedures explained in the previous section lead to the simultaneous estimated parameters using the first sensor k ¼ 0:5243 Wm1 K1 , a ¼ 2:0501 107 m2 s1 ; and q c ¼ 2:5573 106 J m3 K1 . For the second and third sensors, due to values of the sensitivity H higher than unity, only the specific heat is identified successfully. Its value is q c ¼ 2:564242 106  0:13 % J m3 K1 . The estimated diffusivities using the time of maximum temperature rising for the three sensors are closer and have the mean value a ¼ 2:02 107 m2 s1  1:77 %. Eventually, it is obvious to note that these different approaches to determine the thermal parameters lead relatively to the same values. It should be noted that these obtained values of the thermal parameters are in the same range compared to the obtained for the granular activated carbon in Ref. [9].

Conclusions The hot wire model to estimate thermal properties is developed for a finite insulated medium. The derived analytical solution of this model includes an infinite

Hot wire technique for finite geometry samples

summation of zero-order Bessel’s functions. The criteria of convergence and accuracy of this sum are studied. The study showed the existence of sensor positions where the sensitivity of the estimated conductivity and diffusivity is infinitely amplified. The sensitivity study demonstrates that only two thermal properties can be simultaneously estimated when exploring measurements corresponding to a large Fourier number. The third property is deduced from the definition of the thermal diffusivity as the ratio of the conductivity to the specific heat capacitance. This estimation is easy, and the procedure is successful when considering the unknown couple conductivity and the specific heat capacitance or the couple diffusivity and the specific heat. Finally, the estimation procedure permits to determine the diffusivity independently of the internal heat generation modulus and the sensible heat independently to the sensor position.

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