# Effect of Heat Source and Imperfect Contact on Simultaneous Estimation of Thermal Properties of High-Conductivity Materials

Authors: Giampaolo D’Alessandro; Filippo de Monte; Donald E. Amos
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the current paper a novel methodology accounting for both the heater heat capacity and the imperfect thermal contact between a thin heater and a specimen is proposed. In particular, the volumetric heat capacity of the heater is considered by modelling it as a lumped capacitance body, while the imperfect thermal contact is considered by means of a contact resistance. Thus, the experimental apparatus consisting of three layers (specimen-heater-specimen) is reduced to a single finite layer (sample) subject to a “nonclassical” boundary condition at the heated surface, known as sixth kind. Once the temperature solution is derived analytically using the Laplace transform method, the scaled sensitivity coefficients are computed analytically at the interface between the heater and the sample (heater side and sample side) and at the sample backside. By applying the proposed methodology to a lab-controlled experiment available in the specialized literature, a reduction of the thermal properties values of about 1.4% is observed for a high-conductivity material (Armco iron).


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## Body

## 1. Introduction

Sensitivity coefficients are used in parameter estimation [1], optimal experimental design [2], and uncertainty analysis [3]. In parameter estimation, a sensitivity analysis of the temperature to the unknown parameters, such as thermal conductivity, volumetric heat capacity, or thermal diffusivity, provides valuable insight. For example, in an experimental apparatus for which sensitivity coefficients had not been studied adequately it is possible that additional materials used in the experimental set-up (for instance, the thin heater giving up heat to the sample) may affect the temperature more than the material under investigation (sample); also, the temperature sensor location might not be the optimal one. Even the uncertainty in the profile of the applied heat flux may represent a mode of noise in the experiment [4]; in such a case an analysis of the temperature sensitivity to the applied heat flux is needed. All these features can compromise the quality of the results [2].Also, the sensitivity coefficients not only make possible a preliminary evaluation of the goodness of the experimental results (at least from a qualitative point of view), but are also directly involved in the estimation of the parameters by means of the sensitivity matrix when minimizing the ordinary least square norm [5–7] whose convergence difficulties may be predicted by establishing whether linear dependence between sensitivity coefficients exists [4].As known, the parameter estimation technique requires measured temperature values obtained by transient [8] measurements. One of the principles of parameter estimation is that the measured temperature values are sensitive to the parameters of interest: the more sensitive the temperature (or large the sensitivity coefficient) is, the more valuable the temperature measurements are [2, 8], though the accuracy of the estimates may adversely be affected by a large magnitude difference between the same coefficients [4]. Furthermore, the sensitivity coefficients have to be uncorrelated; otherwise it is impossible to estimate two or more parameters simultaneously [2, 8].The focus of the current paper is to develop a new procedure accounting for both the heater heat capacity and the imperfect thermal contact between the thin heater and the specimen. For this reason, the dependence of the thermal properties on temperature is not considered here. Research work regarding the estimation of temperature-dependent properties is available in literature as Refs. [7, 9]. Then, if the thin heater can be modelled as a lumped capacitance body, the addressed transient heat conduction problem concerns a one dimensional (1-D) finite single-layer rectangular body (sample) subject to a boundary condition of the sixth kind at the heated boundary and insulated at the backside.The boundary condition of the 6th kind is somewhat neglected in the specialized heat conduction literature with the exception of Refs. [10, p. 22], [11]. Note that the word sixth kind is not used in Ref. [10], while it is used in [11, p. 160].The governing equations are conveniently solved by using the Laplace Transform method. In particular, the solution calculated in the Laplace domain is inverted by means of the residues theorem which requires the computation of the poles and of the corresponding residues. Once the thermal field is obtained, the so-called “scaled” sensitivity coefficients are computed analytically by performing the partial derivative of the temperature with respect to the parameters of interest. The coefficients are presented in a graphical form for three different locations: (1) at the interface between the heater and the sample (heater side), (2) at the interface between the heater and the sample (sample side), and (3) at the sample back side. A discussion about the better locations of the temperature sensors is also given.The results of the proposed procedure are then applied to a lab-controlled experiment available in the parameter estimation literature [8, Chap. 7]. By comparing measured and calculated temperatures through the ordinary least squares norm and, then, minimizing this by the well-established Gauss method, a reduction of about 1.4% was observed for the thermal properties values of the sample considered (Armco iron). This slight reduction is however in accordance with the very thin heater and the very low contact resistance of the experimental apparatus considered.


## 2. Mathematical Formulation

A schematic of the experimental apparatus for the measurement of the thermal properties of a solid specimen is depicted in Figure1(a). This figure shows a thin layer heater (metallic or kapton/silicon) located between two high-conductivity samples of the same material and thickness which releases heat at the surfaces of both samples. This three-layer configuration (specimen-heater-specimen) reduces to the simplified configuration depicted in Figure 1(b) by virtue of a thermal symmetry. In particular, the finite body is in imperfect thermal contact with a thin layer (representing the heater as a lumped body) at the boundaryx=0 through a surface contact resistance Rc. In particular, this resistance increases when using metallic resistive heaters (made of copper or aluminium) as they are rigid, while it decreases if kapton or silicon resistive heaters are utilized as they are flexible. Then, the finite body and the thin layer are at the same uniform initial temperature Tin. Also, att=0 a step change in the surface heat flux is applied to the thin layer, whose thickness and volumetric heat capacity are denoted by Lf and Cf, respectively. In addition, the finite body is insulated at the backsidex=L, and its properties k and C are considered temperature-independent, as is Cf.Figure 1
Schematic of the experimental apparatus for thermal properties measurements (a) and simplified schematic for the addressed problem (b).
(a) (b)The mathematical formulation of this transient, linear, 1-D heat conduction problem is defined as(1a)∂2T∂x2=1α∂T∂t0<x<L;t>0(1b)-k∂T∂xx=0+LfCfdTfdt=qf,0′′t>0(1c)-k∂T∂xx=0=1RcTft-T0,tt>0(1d)-k∂T∂xx=L=0t>0(1e)Tx,0=00<x<L(1f)Tf0=0where Eqs. (1b) and (1c) represent the boundary condition of the sixth kind atx=0. In particular, the former equation is obtained by applying the first law of thermodynamics to the thin layer assumed as a lumped-capacitance body, where qf,0′′=gf,0Lf (gf,0 denotes the volumetric heat generation within the heater), while the latter accounts for the imperfect contact between the thin layer and the finite body. Also, the perfect insulated condition presented in Eq. (1d) is consistent only with high-thermal conductivity samples.Then, according to the numbering system devised in [12, Chap. 2], [13], the above problem may be denoted by X62B10T11. In particular, this number denotes a transient heat conduction problem concerning a 1D rectangular finite body (by the X), subject to a boundary condition of the sixth kind at the surfacex=0 (by the 6 in X62) with a constant applied heat flux (by the B1), and having an insulated boundary atx=L (type 2 boundary condition by the 2 in X62, and zero heat flux by the 0 in B10); also, T11 denotes a nonzero uniform initial temperature for both layers.By using the following dimensionless variables,(2)T~=T-Tinqf,0′′L/k,x~=xL,t~=αtL2,T~f=Tf-Tinqf,0′′L/k,P=CfLfCL,R~c=RcL/kthe governing equations given by Eqs. (1a)–(1f), in dimensionless form, become (3a)∂2T~∂x~2=∂T~∂t~0<x~<1;t~>0(3b)-∂T~∂x~x~=0+P∂T~f∂t~=1t~>0(3c)-∂T~∂x~x~=0=1R~cT~ft~-T~0,t~t~>0(3d)-∂T~∂x~x~=1=0t~>0(3e)T~x~,0=00<x~<1(3f)T~f0=0Moreover, as the initial conditions are now homogeneous, Eqs. (3e) and (3f), the problem notation becomes X62B10T00.


## 3. Temperature Solution

The temperature solution may be derived by applying the Laplace Transform method to the governing equations, Eqs. (3a)–(3f), as shown below. In fact, taking into account the initial conditions Eqs. (3e) and (3f), Eqs. (3a)–(3f) yield (4a)∂2ϑ∂x~2=sϑ0<x~<1(4b)-∂ϑ∂x~x~=0+Psϑf=1s(4c)ϑf=ϑ0,s-R~c∂ϑ∂x~x~=0(4d)∂ϑ∂x~x~=1=0where ϑx~,s=LT~x~,t~.The solution of the problem defined by Eqs. (4a)–(4d) results in(5)ϑx~,q=B1cosh⁡qx~+B2sinh⁡qx~=A1cosh⁡qx~+φwhere q=s and the phase φ can be determined by applying the boundary condition at x~=1, i.e., Eq. (4d). It follows that φ=-q.Also, it is possible to determine the constantA1 by using the boundary condition of the sixth kind defined by Eqs. (4b) and (4c) in the Laplace domain. Then, bearing in mind that the hyperbolic cosine is an even function, the sought solution ϑx~,q is(6)ϑx~,q=cosh⁡q1-x~q2Pq2cosh⁡q+qPR~cq2+1sinh⁡q︸Dq=cosh⁡q1-x~DqNow a Taylor series expansion of the hyperbolic functions allows us to gain insight on the poles of the solution defined by Eq. (6). Hence, the function Dq appearing at the denominator of the above equation becomes(7)Dq=q2Pq21+q22+Oq4+qPR~cq2+1q+q33!+Oq5=q4P+1+q2P2+PR~c+16+Oq4Therefore, this function admits a quadruple pole atq=0 (which implies a double pole ats=0). Also, it has simple zeros along the negative real axis that may conveniently be assumed as q=iβ (or s=-β2). Then, the value of β can be computed by solving numerically the eigencondition Dq=iβ=0; that is, (8)Pβcos⁡β+1-PR~cβ2sin⁡β=0where the relations cosh⁡iβ=cos⁡β and sinh⁡iβ=isin⁡β have been used. Eq. (8) admits an infinite number of simple poles at q=iβm, βm>0,m=1,2,… (or s=-βm2), where βm is them-th eigenvalue whose computation is discussed in Section 3.1.Since the functionϑx~,q in Eq. (8) is analytic except at polesq=0 (s=0) and at poles q=iβm (s=-βm2), βm>0,m=1,2…, the inverse of such a function may be obtained using the residue theorem [14, pp. 384-385](9)T~x~,t~=L-1ϑx~,s=12πi∫γ-i∞γ+i∞ϑx~,sestds=∑encompassingthelefthalfplaneResiduesintheclosedcontour=Rs=0+∑m=1∞Rs=-βm2where the calculation of the residues, Rs=0 and Rs=-βm2, is given in the Appendix. In particular, they result in(10a)Rs=0=t~P+1+x~22P+1-x~P+1+1-3PR~c3P+12(10b)Rs=-βm2=2PR~cβm2-1cos⁡βm1-x~e-βm2t~PR~c2βm4+P+PR~c-2R~cPβm2+P+1βm2cos⁡βmSubstituting Eqs. (10a) and (10b) in Eq. (9) yields the temperature solution in the time domain.(11a)T~x~,t~=t~P+1+x~22P+1-x~P+1+1-3PR~c3P+12+2∑m=1∞PR~cβm2-1cos⁡βm1-x~N~mβm2cos⁡βme-βm2t~0≤x~≤1;t~≥0where N~m is the dimensionless norm given by(11b)N~m=PR~c2βm4+P+PR~c-2R~cPβm2+P+1Once T~x~,t~ is known, the thin layer temperature T~ft~ may be obtained by using the interface condition defined by the latter of Eq. (3b). After some algebra, it is given by(11c)T~ft~=t~P+1+R~c+1/3P+12-2∑m=1∞e-βm2t~N~mβm2t~≥0


### 3.1. Computation of the Eigenvalues

The eigencondition defined by Eq. (8) may be rewritten as (12)tan⁡βm=βm/R~cβm2-1/PR~cIt is worth noting that Eq. (12) is similar to the corresponding equation of the X33 case treated in [15], where “X33” denotes a 1D rectangular finite body with boundary conditions of the third kind on both sides (assuming Bi1 atx=0 and Bi2 atx=L). In fact, it is sufficient to replace the sum of the Biot numbers Bi1+Bi2 with the term 1/R~c, and their product Bi1Bi2 with the term 1/PR~c. Therefore, its roots may be computed by using the same explicit approximate relations based on the third-order modified Newton method [15]. In particular, the approximate eigenvalue ζm is obtained by using an iteration process as shown below:(13)βm≈ζm=zm+εmzmwhere βm is the exact eigenvalue, zm is the initial guess value used in the first iteration, and εm is the deviation which yields the update value for ζm.These relations provide an approximate value of the exact eigenvalue with high accuracy (to at least 7th decimal place whenm=1, and a higher decimal place form>1, after one iteration) for P∈0,∞ and R~c∈0,∞. To obtain eigenvalues with an accuracy of 10-15 two more iterations may be required.In particular, a computer code called “fdeigX62” for calculating these eigenvalues is available in Ref. [16, App. B] for ease of use. Some of them computed for P=R~c=1 are shown in Table 1, where the exact eigenvalues βm listed in the last column are obtained by using the internal Matlab function “fsolve” setting the tolerance parameter equal to 10-15. Note that ζm after two iterations is exactly the same as βm, but βm is obtained after three iterations starting from the same initial guess zm.Table 1
Comparison of the first ten calculatedζm and exact βm eigenvalues for the X62 case.
m z m ζm ζ m β m (initial guess) (after 1 iteration) (after 2 iterations) (exact) 1 1.195476244024383 1.207792438624699 1.207792655153241 1.207792655153241 2 3.486925937947245 3.448242071207367 3.448237983351985 3.448237983351985 3 6.468623593104900 6.440956097445878 6.440954447908016 6.440954447908016 4 9.550944741970163 9.530477847514561 9.530477156207574 9.530477156207574 5 12.662238896069196 12.645779030853410 12.645778666218552 12.645778666218552 6 15.785462794465527 15.771538985089661 15.771538762863299 15.771538762863299 7 18.914707557510582 18.902557753172037 18.902557604969395 18.902557604969395 8 22.047411709862786 22.036590088566303 22.036589983609232 22.036589983609232 9 25.182288796593070 25.172508998808841 25.172508921226136 25.172508921226136 10 28.318621989478178 28.309686934070488 28.309686874845468 28.309686874845468


### 3.2. Computational Solution

The exact analytical solutions given by Eqs. (11a) and (11c) exhibit an infinite series, while the computational solution requires a finite number of terms, saymmax terms. Therefore, a convergence criterion for the series solutions is needed and it is defined as follows:(14)
ε
=
T
~
c
x
~
,
t
~
,
A
-
T
~
x
~
,
t
~
T
~
f
t
~
=
2
∑
m
=
m
m
a
x
+
1
∞
P
R
~
c
β
m
2
-
1
cos
⁡
β
m
1
-
x
~
/
N
~
m
β
m
2
cos
⁡
β
m
e
-
β
m
2
t
~
︸
S
~
T
t
~
/
P
+
1
+
R
~
c
+
1
/
3
/
P
+
1
2
-
2
∑
m
=
1
∞
e
-
β
m
2
t
~
/
N
~
m
β
m
2
≤
1
0
-
Awhere T~x~,t~ denotes the exact solution (with an infinite number of terms), T~cx~,t~,A represents the computational one with a finite number of termsmmax for a truncation error ε of 10-A (A=2,3,…15), and S~T is the “tail” of the series solutions that may be calculated by using the Euler-McLaurin formula [17]. Then, by solving numerically Eq. (14) formmax and by fitting properly the obtained numerical data, the maximum number of terms may be taken in a conservative way as(15)mmax=1+ceil1πAln⁡10+HP-1-1ln⁡Pt~where H· denotes the Heaviside step function, while “ceil(z)” is a Matlab function that rounds the number z to the nearest integer greater than or equal toz.It is worth noting thatmmax is independent of both the thermal contact resistance R~c and the location x~. WhenP>1, it is independent of P too.According to Eq. (15), a large number of terms may be required for early times. Therefore, for times less than the so-called deviation time t~d defined as [13, 18](16)t~d=110A2-x~2,the temperature solution can be replaced by a semi-infinite transient solution available in Ref. [19] with an error less than 10-A (A=2,3,…15), where an intrinsic verification of the solution was proven. A computer code in Matlab ambient for calculating the solution of the current X62B10T00 problem is provided in [16, App. E] for ease of use. A plot of dimensionless temperature and heat flux as a function of time at the heated surface of the sample for different P values and R~c=1 is given in Figure 2. It is evident that the heat flux at x~=0 is time-dependent and this dependence is due to both the thermal inertia of the heat source and the contact resistance.Figure 2
Dimensionless temperature (a) and heat flux (b) as a function of the dimensionless time atx~=0, for different P values and R~c=1.
(a) (b)


## 3.1. Computation of the Eigenvalues

The eigencondition defined by Eq. (8) may be rewritten as (12)tan⁡βm=βm/R~cβm2-1/PR~cIt is worth noting that Eq. (12) is similar to the corresponding equation of the X33 case treated in [15], where “X33” denotes a 1D rectangular finite body with boundary conditions of the third kind on both sides (assuming Bi1 atx=0 and Bi2 atx=L). In fact, it is sufficient to replace the sum of the Biot numbers Bi1+Bi2 with the term 1/R~c, and their product Bi1Bi2 with the term 1/PR~c. Therefore, its roots may be computed by using the same explicit approximate relations based on the third-order modified Newton method [15]. In particular, the approximate eigenvalue ζm is obtained by using an iteration process as shown below:(13)βm≈ζm=zm+εmzmwhere βm is the exact eigenvalue, zm is the initial guess value used in the first iteration, and εm is the deviation which yields the update value for ζm.These relations provide an approximate value of the exact eigenvalue with high accuracy (to at least 7th decimal place whenm=1, and a higher decimal place form>1, after one iteration) for P∈0,∞ and R~c∈0,∞. To obtain eigenvalues with an accuracy of 10-15 two more iterations may be required.In particular, a computer code called “fdeigX62” for calculating these eigenvalues is available in Ref. [16, App. B] for ease of use. Some of them computed for P=R~c=1 are shown in Table 1, where the exact eigenvalues βm listed in the last column are obtained by using the internal Matlab function “fsolve” setting the tolerance parameter equal to 10-15. Note that ζm after two iterations is exactly the same as βm, but βm is obtained after three iterations starting from the same initial guess zm.Table 1
Comparison of the first ten calculatedζm and exact βm eigenvalues for the X62 case.
m z m ζm ζ m β m (initial guess) (after 1 iteration) (after 2 iterations) (exact) 1 1.195476244024383 1.207792438624699 1.207792655153241 1.207792655153241 2 3.486925937947245 3.448242071207367 3.448237983351985 3.448237983351985 3 6.468623593104900 6.440956097445878 6.440954447908016 6.440954447908016 4 9.550944741970163 9.530477847514561 9.530477156207574 9.530477156207574 5 12.662238896069196 12.645779030853410 12.645778666218552 12.645778666218552 6 15.785462794465527 15.771538985089661 15.771538762863299 15.771538762863299 7 18.914707557510582 18.902557753172037 18.902557604969395 18.902557604969395 8 22.047411709862786 22.036590088566303 22.036589983609232 22.036589983609232 9 25.182288796593070 25.172508998808841 25.172508921226136 25.172508921226136 10 28.318621989478178 28.309686934070488 28.309686874845468 28.309686874845468


## 3.2. Computational Solution

The exact analytical solutions given by Eqs. (11a) and (11c) exhibit an infinite series, while the computational solution requires a finite number of terms, saymmax terms. Therefore, a convergence criterion for the series solutions is needed and it is defined as follows:(14)
ε
=
T
~
c
x
~
,
t
~
,
A
-
T
~
x
~
,
t
~
T
~
f
t
~
=
2
∑
m
=
m
m
a
x
+
1
∞
P
R
~
c
β
m
2
-
1
cos
⁡
β
m
1
-
x
~
/
N
~
m
β
m
2
cos
⁡
β
m
e
-
β
m
2
t
~
︸
S
~
T
t
~
/
P
+
1
+
R
~
c
+
1
/
3
/
P
+
1
2
-
2
∑
m
=
1
∞
e
-
β
m
2
t
~
/
N
~
m
β
m
2
≤
1
0
-
Awhere T~x~,t~ denotes the exact solution (with an infinite number of terms), T~cx~,t~,A represents the computational one with a finite number of termsmmax for a truncation error ε of 10-A (A=2,3,…15), and S~T is the “tail” of the series solutions that may be calculated by using the Euler-McLaurin formula [17]. Then, by solving numerically Eq. (14) formmax and by fitting properly the obtained numerical data, the maximum number of terms may be taken in a conservative way as(15)mmax=1+ceil1πAln⁡10+HP-1-1ln⁡Pt~where H· denotes the Heaviside step function, while “ceil(z)” is a Matlab function that rounds the number z to the nearest integer greater than or equal toz.It is worth noting thatmmax is independent of both the thermal contact resistance R~c and the location x~. WhenP>1, it is independent of P too.According to Eq. (15), a large number of terms may be required for early times. Therefore, for times less than the so-called deviation time t~d defined as [13, 18](16)t~d=110A2-x~2,the temperature solution can be replaced by a semi-infinite transient solution available in Ref. [19] with an error less than 10-A (A=2,3,…15), where an intrinsic verification of the solution was proven. A computer code in Matlab ambient for calculating the solution of the current X62B10T00 problem is provided in [16, App. E] for ease of use. A plot of dimensionless temperature and heat flux as a function of time at the heated surface of the sample for different P values and R~c=1 is given in Figure 2. It is evident that the heat flux at x~=0 is time-dependent and this dependence is due to both the thermal inertia of the heat source and the contact resistance.Figure 2
Dimensionless temperature (a) and heat flux (b) as a function of the dimensionless time atx~=0, for different P values and R~c=1.
(a) (b)


## 4. Sensitivity Coefficients

The scaled sensitivity coefficients, that is, the partial derivatives of the temperature with respect to the model parameterγ of interest (e.g., qf,0′′,k,C,Cf,Rc) multiplied by the parameter itself, are defined as follows:(17)Xγ=γ∂T∂γ,Xγ,f=γ∂Tf∂γTherefore, as suggested by the above equation the scaled sensitivity coefficients have units of °C and their calculation requires the use of the chain rule [20]. In dimensionless form they become (18)X~γ=Xγqf,0′′L/k,X~γ,f=Xγ,fqf,0′′L/k


### 4.1. Heat Flux as a Parameter

If the parameter of interest is the applied heat flux at the boundaryx~=0, the corresponding sensitivity coefficient is (19)Xqf,0′′=qf,0′′∂T∂qf,0′′=qf,0′′∂∂qf,0′′T~qf,0′′Lk=qf,0′′qf,0′′Lk∂T~∂qf,0′′︸=0+T~∂∂qf,0′′qf,0′′Lk=qf,0′′LkT~which in dimensionless form results in X~qf,0′′=T~, where T~ is defined by Eq. (11a). In the same way it is found that X~qf,0′′,f=T~f where now T~f is defined by Eq. (11c). Therefore, the curves shown in Figure 2 also represent the sensitivity coefficient for the finite body to the applied heat flux.


### 4.2. Thermal Conductivity and Volumetric Heat Capacities as Parameters

AsT~ and T~f temperatures depend on the thermal conductivity of the specimen k and volumetric heat capacity of both specimen and heater,C and Cf, respectively, six sensitivity coefficients should be studied.When the considered parameter is the thermal conductivity, the corresponding sensitivity coefficient is given as(20)Xk=k∂T∂k=k∂∂kT~qf,0′′Lk=kqf,0′′Lk∂T~∂k+T~∂∂kqf,0′′Lk=kqf,0′′Lk∂T~∂k-qf,0′′Lk2T~where T~=T~t~,P,R~c,βmP,R~c, with t~αk,C,PC,Cf,R~cRc,k. Thus, by using the chain rule,(21)∂T~∂k=∂T~∂t~∂t~∂α∂α∂k+∂T~∂R~c∂R~c∂k+∑m=1∞∂T~∂βm∂βm∂R~c∂R~c∂kSubstituting Eq. (21) in Eq. (20) gives the scaled sensitivity coefficient with respect to the thermal conductivity. In dimensionless form it becomes (22)X~k=k∂T~∂k-T~=t~∂T~∂t~+R~c∂T~∂R~c+∑m=1∞∂T~∂βm∂βm∂R~c-T~In the same way, the sensitivity coefficient with respect to the volumetric heat capacityC may be obtained.(23)X~C=C∂T~∂C=-t~∂T~∂t~-P∂T~∂P-P∑m=1∞∂T~∂βm∂βm∂PSimilarly the sensitivity coefficient with respect to the volumetric heat capacity of the heaterCf, in dimensionless form, results in (24)X~Cf=Cf∂T~∂Cf=P∂T~∂P+P∑m=1∞∂T~∂βm∂βm∂PAs regards the thin layer sensitivities, they are defined as(25a)X~k,f=k∂T~f∂k-T~f=t~∂T~f∂t~+R~c∂T~f∂R~c+∑m=1∞∂T~f∂βm∂βm∂R~c-T~f(25b)X~C,f=C∂T~f∂C=-t~∂T~f∂t~-P∂T~f∂P-P∑m=1∞∂T~f∂βm∂βm∂P(25c)X~Cf,f=Cf∂T~f∂Cf=P∂T~f∂P+P∑m=1∞∂T~f∂βm∂βm∂P


### 4.3. Contact Resistance as a Parameter

If the parameter of interest is the thermal contact resistanceRc, the corresponding sensitivity coefficients are (26)X~Rc=RC∂T~∂RC=R~c∂T~∂R~c+R~c∑m=1∞∂T~∂βm∂βm∂R~c(27)X~Rc,f=Rc∂T~f∂Rc=R~c∂T~f∂R~c+R~c∑m=1∞∂T~f∂βm∂βm∂R~cIt is interesting to note that all scaled sensitivity coefficients discussed above are linearly dependent; in fact, they sum to zero for all values of time and position; i.e.,(28)X~qf,0′′+X~qf,0′′,f+X~k+X~k,f+X~C+X~C,f+X~Cf+X~Cf,f-X~Rc+X~Rc,f=0The same condition is valid for the thin layer (heater) and the finite body (sample) considered individually:(29a)X~k,f+X~C,f+X~Cf,f-X~Rc,f=-T~f(29b)X~k+X~C+X~Cf-X~Rc=-T~


### 4.4. Computation of the Scaled Sensitivity Coefficients

The sensitivity coefficients listed above may be computed analytically. In fact, the temperature derivatives with respect tot~,P, and R~c can be taken directly using Eqs. (11a) and (11c). As regards the derivatives ∂βm/∂P and ∂βm/∂R~c, from the eigencondition defined by Eq. (12) one can obtain (30)P=tan⁡βmβm2R~ctan⁡βm-βm,R~c=βm+tan⁡βm/Pβm2tan⁡βmThen, by differentiating Eq. (30) with respect to the eigenvalue βm and by taking the corresponding reciprocals, it is found that (31)∂βm∂P=βm2R~ctan⁡βm-βm2tan⁡βm1-βmtan⁡βm1+2R~c-βm,∂βm∂R~c=-βm2tan⁡βm1+tan⁡βm2/Pβm+βmIn addition, for early times (less than the deviation timet~d) the temperature field within the specimen and the thin layer temperature may be evaluated accurately by using the semi-infinite transient solution defined in [19].


### 4.5. Sensitivity Analysis Results

Three different locations are considered in the analysis: the interface between the heater and the sample, i.e.,x~=0 (both heater side and sample side) and the sample backside (x~=1). Note that two sensors are used at the interface x~=0, one located on the surface of the heater and the other on the surface of the sample. Then at this interface a silicon grease is used to reduce the contact resistance. Note also that the sensor location on the surface of the heater (interface, heater side) is realistic only when using kapton or silicon resistive heaters rather than metallic ones.Figures3–5 show the sample temperature sensitivities as a function of time and for different values of the heat capacity ratioP; also, the thin heater sensitivities X~C,f and X~Cf,f are shown by Figure 6. For the first group of figures a curve for P=0 is plotted as a reference case; in fact, in such a case, the thin heater is neglected and the analyzed problem reduces to the X22B10T0 case whose solution is discussed in [21].Figure 3
Sensitivity coefficient to the thermal conductivity as a function of time, for differentP values and R~c=1, at x~=0 (a) and at x~=1 (b).
(a) (b)Figure 4
Sensitivity coefficients to the sample heat capacity (a) and to the heater heat capacity (b) as a function of time, for differentP values and R~c=1, at x~=0.
(a) (b)Figure 5
Sensitivity coefficient to the thermal contact resistance as a function of time, for differentP values and R~c=1, at x~=0 (a) and at x~=1 (b).
(a) (b)Figure 6
Heater sensitivity to the sample heat capacity (a) and to its heat capacity (b) as a function of time, for differentP values and R~c=1.
(a) (b)Figure3 shows the sensitivity coefficient X~k. In particular, for x~=0, it decreases in absolute value when the P parameter increases, while at the backside (x~=1) this behaviour cannot be observed forP=0.1. Also, at x~=1 the sensitivity to the thermal conductivity k is less affected by the P parameter than the sensitivity at x~=0, when large times are considered.The sensitivity coefficientsX~C and X~Cf for x~=0 are depicted in Figure 4. The corresponding graphs for x~=1 are not reported here as they are similar to ones given in Figure 4. A comparison among Figures 4(a) and 4(b) suggests that, at x~=0, the sample temperature is more sensitive to its heat capacity C whenP<0.5, while for P≥0.5 it becomes more sensitive to that of the thin heaterCf. Also, the sample temperature sensitivity to the contact resistance Rc is shown in Figure 5. As indicated by the figure, the highest sensitivity occurs at the interface between the heater and the sample (x~=0).The coefficientsX~C,f and X~Cf,f related to the thin heater are shown by Figure 6. It is worth noting that, at large times and forP<0.5, the heater temperature is more sensitive to the volumetric heat capacity of the sample C than to its volumetric heat capacity Cf; on the contrary, for P≥0.5 it becomes more sensitive to Cf.Figures7 and 8 show the sensitivities X~k (at x~=0) and X~k,f and the other heater coefficients X~Cf,f and X~Rc,f, respectively, as a function of time with the dimensionless contact resistance R~c as a parameter. In these figures a curve for R~c=0 is plotted as a reference case; in fact, in such a case, the contact between the heater and the sample is perfect and the problem reduces to the X42B10T00 case, whose solution is provided in [10] and discussed in [22]. Also, a sensitivity analysis for this heat conduction problem is carried out in [23]. Note that the heater sensitivities are more affected by the thermal contact resistance R~c, than the sample sensitivity coefficients.Figure 7
Sensitivity coefficients to the thermal conductivity as a function of time, for differentR~c values andP=0.1, for the sample at x~=0 (a) and for the heater (b).
(a) (b)Figure 8
Sensitivity coefficients for the heater as a function of time, for differentR~c values andP=0.1, with respect to its heat capacity (a) and to the contact resistance (b).
(a) (b)A comparison of the sensitivity coefficients related to the sample and to the heater atx~=0 and for P=R~c=0.1 is given by Figure 9. In particular, Figure 9(a) compares the coefficients X~k, X~C, X~k,f and X~C,f, while Figure 9(b) compares the coefficients X~Cf, X~Rc, X~Cf,f, and X~Rc,f. The former shows that, for times less than the deviation time defined by Eq. (16), the coefficients X~k and X~C are exactly equal for x~=0, and the same equality is also true for X~k,f and X~C,f. Moreover, the sensitivities to the thermal conductivityk, i.e., X~k and X~k,f, respond in the same way; it also occurs for the coefficients X~C and X~C,f. This suggests that for small values of P and R~c, it is indifferent to place the temperature sensor on the heater side or on the sample side. In addition, for large values of these parameters, the sensitivities to k and C at the sample side are greater than the corresponding heater coefficients (i.e., X~k,f and X~C,f). On the contrary, Figure 9(b) shows that the greatest sensitivities to the heater heat capacity Cf and to the thermal contact resistance Rc occur at the heater side. This consideration is always valid for the sensitivity to Cf regardless of the value of P and R~c; as regards the sensitivity to the contact resistance it is valid only for P≤0.5.Figure 9
Comparison of the sensitivity coefficients of the heater and of the sample atx~=0, with respect to the thermal conductivity and to the sample heat capacity (a), and with respect to the heater heat capacity and to the contact resistance (b) for P=R~c=0.1.
(a) (b)Figure10 compares the sample sensitivities with respect tok,C, Cf, and Rc obtained for P=R~c=0.1, at the sample backside (x~=1). As shown by the figure the coefficients X~k, X~C, X~Cf, and X~Rc respond initially in the same way, and then they split up and in particular the sensitivity to k approaches to a constant value, while the sensitivity to C increases linearly with the time. Therefore, at the sample backside the coefficients X~k and X~C are uncorrelated, and the thermal properties k and C can be estimated simultaneously. Furthermore, as no large magnitude difference exists between them the estimates for k and C may be accurate.Figure 10
Comparison of the sensitivity coefficients related to the sample, atx~=1 and for P=R~c=0.1.Also, by comparing Figures9(a) and 10, it is possible to state that the greatest sensitivity to the sample heat capacity C always occurs at the backside x~=1 (except at early times). As far as the sensitivity to k is concerned, for small values ofP (around 0.1) it is greater at the sample side (x~=0), while, for large P values and at large times, it becomes greater at the sample backside. Also, the analysis shows that for estimating the thermal conductivityk, in the event that the heat capacity ratio (P) is unknown, the temperature sensor should be placed at x~=1, where the influence of this parameter is lower (see Figure 3).Moreover, at the sample backside the sensitivities to the heater heat capacityCf and to the contact resistance Rc are lower than the corresponding coefficients obtained for x~=0. In particular, for estimating Cf the temperature sensor should be placed at the heater side when a thin enough heater (P≤0.5) is used and large times are considered. In fact, at large times the influence of Rc is the same either at the heater side or at the sample side, but at the heater side the sensitivity to Cf is higher. Also, it is even the best location for the temperature sensor when estimating the contact resistance, in the event that a heater with known properties is used.


## 4.1. Heat Flux as a Parameter

If the parameter of interest is the applied heat flux at the boundaryx~=0, the corresponding sensitivity coefficient is (19)Xqf,0′′=qf,0′′∂T∂qf,0′′=qf,0′′∂∂qf,0′′T~qf,0′′Lk=qf,0′′qf,0′′Lk∂T~∂qf,0′′︸=0+T~∂∂qf,0′′qf,0′′Lk=qf,0′′LkT~which in dimensionless form results in X~qf,0′′=T~, where T~ is defined by Eq. (11a). In the same way it is found that X~qf,0′′,f=T~f where now T~f is defined by Eq. (11c). Therefore, the curves shown in Figure 2 also represent the sensitivity coefficient for the finite body to the applied heat flux.


## 4.2. Thermal Conductivity and Volumetric Heat Capacities as Parameters

AsT~ and T~f temperatures depend on the thermal conductivity of the specimen k and volumetric heat capacity of both specimen and heater,C and Cf, respectively, six sensitivity coefficients should be studied.When the considered parameter is the thermal conductivity, the corresponding sensitivity coefficient is given as(20)Xk=k∂T∂k=k∂∂kT~qf,0′′Lk=kqf,0′′Lk∂T~∂k+T~∂∂kqf,0′′Lk=kqf,0′′Lk∂T~∂k-qf,0′′Lk2T~where T~=T~t~,P,R~c,βmP,R~c, with t~αk,C,PC,Cf,R~cRc,k. Thus, by using the chain rule,(21)∂T~∂k=∂T~∂t~∂t~∂α∂α∂k+∂T~∂R~c∂R~c∂k+∑m=1∞∂T~∂βm∂βm∂R~c∂R~c∂kSubstituting Eq. (21) in Eq. (20) gives the scaled sensitivity coefficient with respect to the thermal conductivity. In dimensionless form it becomes (22)X~k=k∂T~∂k-T~=t~∂T~∂t~+R~c∂T~∂R~c+∑m=1∞∂T~∂βm∂βm∂R~c-T~In the same way, the sensitivity coefficient with respect to the volumetric heat capacityC may be obtained.(23)X~C=C∂T~∂C=-t~∂T~∂t~-P∂T~∂P-P∑m=1∞∂T~∂βm∂βm∂PSimilarly the sensitivity coefficient with respect to the volumetric heat capacity of the heaterCf, in dimensionless form, results in (24)X~Cf=Cf∂T~∂Cf=P∂T~∂P+P∑m=1∞∂T~∂βm∂βm∂PAs regards the thin layer sensitivities, they are defined as(25a)X~k,f=k∂T~f∂k-T~f=t~∂T~f∂t~+R~c∂T~f∂R~c+∑m=1∞∂T~f∂βm∂βm∂R~c-T~f(25b)X~C,f=C∂T~f∂C=-t~∂T~f∂t~-P∂T~f∂P-P∑m=1∞∂T~f∂βm∂βm∂P(25c)X~Cf,f=Cf∂T~f∂Cf=P∂T~f∂P+P∑m=1∞∂T~f∂βm∂βm∂P


## 4.3. Contact Resistance as a Parameter

If the parameter of interest is the thermal contact resistanceRc, the corresponding sensitivity coefficients are (26)X~Rc=RC∂T~∂RC=R~c∂T~∂R~c+R~c∑m=1∞∂T~∂βm∂βm∂R~c(27)X~Rc,f=Rc∂T~f∂Rc=R~c∂T~f∂R~c+R~c∑m=1∞∂T~f∂βm∂βm∂R~cIt is interesting to note that all scaled sensitivity coefficients discussed above are linearly dependent; in fact, they sum to zero for all values of time and position; i.e.,(28)X~qf,0′′+X~qf,0′′,f+X~k+X~k,f+X~C+X~C,f+X~Cf+X~Cf,f-X~Rc+X~Rc,f=0The same condition is valid for the thin layer (heater) and the finite body (sample) considered individually:(29a)X~k,f+X~C,f+X~Cf,f-X~Rc,f=-T~f(29b)X~k+X~C+X~Cf-X~Rc=-T~


## 4.4. Computation of the Scaled Sensitivity Coefficients

The sensitivity coefficients listed above may be computed analytically. In fact, the temperature derivatives with respect tot~,P, and R~c can be taken directly using Eqs. (11a) and (11c). As regards the derivatives ∂βm/∂P and ∂βm/∂R~c, from the eigencondition defined by Eq. (12) one can obtain (30)P=tan⁡βmβm2R~ctan⁡βm-βm,R~c=βm+tan⁡βm/Pβm2tan⁡βmThen, by differentiating Eq. (30) with respect to the eigenvalue βm and by taking the corresponding reciprocals, it is found that (31)∂βm∂P=βm2R~ctan⁡βm-βm2tan⁡βm1-βmtan⁡βm1+2R~c-βm,∂βm∂R~c=-βm2tan⁡βm1+tan⁡βm2/Pβm+βmIn addition, for early times (less than the deviation timet~d) the temperature field within the specimen and the thin layer temperature may be evaluated accurately by using the semi-infinite transient solution defined in [19].


## 4.5. Sensitivity Analysis Results

Three different locations are considered in the analysis: the interface between the heater and the sample, i.e.,x~=0 (both heater side and sample side) and the sample backside (x~=1). Note that two sensors are used at the interface x~=0, one located on the surface of the heater and the other on the surface of the sample. Then at this interface a silicon grease is used to reduce the contact resistance. Note also that the sensor location on the surface of the heater (interface, heater side) is realistic only when using kapton or silicon resistive heaters rather than metallic ones.Figures3–5 show the sample temperature sensitivities as a function of time and for different values of the heat capacity ratioP; also, the thin heater sensitivities X~C,f and X~Cf,f are shown by Figure 6. For the first group of figures a curve for P=0 is plotted as a reference case; in fact, in such a case, the thin heater is neglected and the analyzed problem reduces to the X22B10T0 case whose solution is discussed in [21].Figure 3
Sensitivity coefficient to the thermal conductivity as a function of time, for differentP values and R~c=1, at x~=0 (a) and at x~=1 (b).
(a) (b)Figure 4
Sensitivity coefficients to the sample heat capacity (a) and to the heater heat capacity (b) as a function of time, for differentP values and R~c=1, at x~=0.
(a) (b)Figure 5
Sensitivity coefficient to the thermal contact resistance as a function of time, for differentP values and R~c=1, at x~=0 (a) and at x~=1 (b).
(a) (b)Figure 6
Heater sensitivity to the sample heat capacity (a) and to its heat capacity (b) as a function of time, for differentP values and R~c=1.
(a) (b)Figure3 shows the sensitivity coefficient X~k. In particular, for x~=0, it decreases in absolute value when the P parameter increases, while at the backside (x~=1) this behaviour cannot be observed forP=0.1. Also, at x~=1 the sensitivity to the thermal conductivity k is less affected by the P parameter than the sensitivity at x~=0, when large times are considered.The sensitivity coefficientsX~C and X~Cf for x~=0 are depicted in Figure 4. The corresponding graphs for x~=1 are not reported here as they are similar to ones given in Figure 4. A comparison among Figures 4(a) and 4(b) suggests that, at x~=0, the sample temperature is more sensitive to its heat capacity C whenP<0.5, while for P≥0.5 it becomes more sensitive to that of the thin heaterCf. Also, the sample temperature sensitivity to the contact resistance Rc is shown in Figure 5. As indicated by the figure, the highest sensitivity occurs at the interface between the heater and the sample (x~=0).The coefficientsX~C,f and X~Cf,f related to the thin heater are shown by Figure 6. It is worth noting that, at large times and forP<0.5, the heater temperature is more sensitive to the volumetric heat capacity of the sample C than to its volumetric heat capacity Cf; on the contrary, for P≥0.5 it becomes more sensitive to Cf.Figures7 and 8 show the sensitivities X~k (at x~=0) and X~k,f and the other heater coefficients X~Cf,f and X~Rc,f, respectively, as a function of time with the dimensionless contact resistance R~c as a parameter. In these figures a curve for R~c=0 is plotted as a reference case; in fact, in such a case, the contact between the heater and the sample is perfect and the problem reduces to the X42B10T00 case, whose solution is provided in [10] and discussed in [22]. Also, a sensitivity analysis for this heat conduction problem is carried out in [23]. Note that the heater sensitivities are more affected by the thermal contact resistance R~c, than the sample sensitivity coefficients.Figure 7
Sensitivity coefficients to the thermal conductivity as a function of time, for differentR~c values andP=0.1, for the sample at x~=0 (a) and for the heater (b).
(a) (b)Figure 8
Sensitivity coefficients for the heater as a function of time, for differentR~c values andP=0.1, with respect to its heat capacity (a) and to the contact resistance (b).
(a) (b)A comparison of the sensitivity coefficients related to the sample and to the heater atx~=0 and for P=R~c=0.1 is given by Figure 9. In particular, Figure 9(a) compares the coefficients X~k, X~C, X~k,f and X~C,f, while Figure 9(b) compares the coefficients X~Cf, X~Rc, X~Cf,f, and X~Rc,f. The former shows that, for times less than the deviation time defined by Eq. (16), the coefficients X~k and X~C are exactly equal for x~=0, and the same equality is also true for X~k,f and X~C,f. Moreover, the sensitivities to the thermal conductivityk, i.e., X~k and X~k,f, respond in the same way; it also occurs for the coefficients X~C and X~C,f. This suggests that for small values of P and R~c, it is indifferent to place the temperature sensor on the heater side or on the sample side. In addition, for large values of these parameters, the sensitivities to k and C at the sample side are greater than the corresponding heater coefficients (i.e., X~k,f and X~C,f). On the contrary, Figure 9(b) shows that the greatest sensitivities to the heater heat capacity Cf and to the thermal contact resistance Rc occur at the heater side. This consideration is always valid for the sensitivity to Cf regardless of the value of P and R~c; as regards the sensitivity to the contact resistance it is valid only for P≤0.5.Figure 9
Comparison of the sensitivity coefficients of the heater and of the sample atx~=0, with respect to the thermal conductivity and to the sample heat capacity (a), and with respect to the heater heat capacity and to the contact resistance (b) for P=R~c=0.1.
(a) (b)Figure10 compares the sample sensitivities with respect tok,C, Cf, and Rc obtained for P=R~c=0.1, at the sample backside (x~=1). As shown by the figure the coefficients X~k, X~C, X~Cf, and X~Rc respond initially in the same way, and then they split up and in particular the sensitivity to k approaches to a constant value, while the sensitivity to C increases linearly with the time. Therefore, at the sample backside the coefficients X~k and X~C are uncorrelated, and the thermal properties k and C can be estimated simultaneously. Furthermore, as no large magnitude difference exists between them the estimates for k and C may be accurate.Figure 10
Comparison of the sensitivity coefficients related to the sample, atx~=1 and for P=R~c=0.1.Also, by comparing Figures9(a) and 10, it is possible to state that the greatest sensitivity to the sample heat capacity C always occurs at the backside x~=1 (except at early times). As far as the sensitivity to k is concerned, for small values ofP (around 0.1) it is greater at the sample side (x~=0), while, for large P values and at large times, it becomes greater at the sample backside. Also, the analysis shows that for estimating the thermal conductivityk, in the event that the heat capacity ratio (P) is unknown, the temperature sensor should be placed at x~=1, where the influence of this parameter is lower (see Figure 3).Moreover, at the sample backside the sensitivities to the heater heat capacityCf and to the contact resistance Rc are lower than the corresponding coefficients obtained for x~=0. In particular, for estimating Cf the temperature sensor should be placed at the heater side when a thin enough heater (P≤0.5) is used and large times are considered. In fact, at large times the influence of Rc is the same either at the heater side or at the sample side, but at the heater side the sensitivity to Cf is higher. Also, it is even the best location for the temperature sensor when estimating the contact resistance, in the event that a heater with known properties is used.


## 5. Experimental Results

The proposed methodology is now applied to a lab-controlled experiment based on the schematic of Figure1(a) whose measured results are available in Ref. [8, Chapter 7, p. 402]. In this experiment two adjacent, identical Armco iron disk-shaped flat specimens (thicknessL=2.54x10−2 m; diameter 7.62x10−2 m) were heated by a Kapton heater (2Lf =8.4x10−4 m; Cf =2000 kJ/m3°C) placed between them. Also, an imperfect contact between the heater and the sample is considered through a contact resistance Rc=1.34x10−4 m2°C/W. All the external surfaces of the samples were insulated. Therefore, the heat conduction may be considered one dimensional. Moreover four temperatures sensors were carefully attached to the heated surface of each specimen and four were attached at the insulated backsides. In particular, the sensors were located at angles of 0°, 90°, 180°, and 270°.Both specimens were at the same uniform initial temperature (about 27.5°C) before the start of the experiment. Though the heat flux of 30.3 kW/m2 was applied to each sample for a finite heating duration of 15.3 s, the measured data were collected for 27 s with a time step of 0.3 s.The average measured temperatures at both the heated (x=0) and insulated (x=L) surfaces are shown in Figure 11(a). It is evident that the heated surface temperature increases only during the heating period, while the insulated surface temperature continues to increase after heater is turned off until they approach the same equilibrium temperature. Figures 11(b) and 11(c) show the dimensional actual sensitivity coefficients with respect to k andC, respectively, for the X22B50T0 (heater completely neglected ⇒Cf=0 and Rc=0) and X62B50T00 cases, where, in the numbering system [13, Table 2], “B5” denotes a step change in time for a finite heating duration. It can be observed that the sensitivity coefficients are not very different at the same location as the heater used is very thin and the contact resistance is very low (in fact, it was reduced using a silicon grease).Figure 11
Average measured temperatures atx=0 andx=L (a), dimensional actual sensitivity coefficients for X22B50T0 and X62B50T00 cases with respect tok (b), and with respect toC (c).
(a) (b) (c)Sensitivity coefficients are involved in the parameter estimation procedure when minimizing the ordinary least squares norm(32)S=∑s=18∑n=190Ysn-Tsn2where Ysn is the measured temperature from thes-th sensor atn-th time and Tsn is the calculated temperature at the location of thes-th sensor atn-th time. Then, applying the Gauss method to minimize Eq. (32) yields the following recursive expression:(33)b_i+1=b_i+X_TiX_i-1X_TiY_-T_iwhere i is the iteration index, b_ denotes the estimated parameter vector (in the current case b_=kCT), Y_ is the measured temperature vector, T_ is the calculated temperature vector, and X_ is the sensitivity coefficient matrix. Also, in Eq. (33) X_i, X_Ti, and T_i are all computed using the b_i vector obtained at thei-th iteration.Moreover, the uncertainty related to the estimated parameters (k andC) is determined as discussed in the next subsection.


### 5.1. Uncertainty Quantification

Under standard statistical assumptions of additive, uncorrelated, normal errors with zero mean and constant variance the approximate covariance matrix of estimated parameters can be written as [8, p. 452, Eq. 8.5.17](34a)cov⁡b_=σk2σk,Cσk,CσC2=CkkCkCCkCCCC-1σ2=CCC-CkC-CkCCkkσ2Δwhere σ2 is the estimated variance of measurement errors, Δ=CkkCCC-CkC2 denotes the determinant of the X_TX_ matrix appearing in Eq. (33), and the elements Ckk, CCC, and CkC are defined as(34b)Ckk=∑s=18∑n=190∂Tsn∂k2;CCC=∑s=18∑n=190∂Tsn∂C2;CkC=CCk=∑s=18∑n=190∂Tsn∂k∂Tsn∂CAlso, ∂Tsn/∂k and ∂Tsn/∂C (with s=1,…,8) are the sensitivity coefficients of temperature measured by thes-th sensor with respect to the thermal properties k andC, respectively, at then-th time.Therefore, the standard deviations ofk and C can be obtained from Eq. (34a) and (34b) as shown below.(35a)σk≅σCCCΔ(35b)σC≅σCkkΔOnce the standard deviations fork and C are known, a 90% confidence interval of the estimates can be computed.The estimation procedure discussed above is applied to three different cases: (1) heater completely neglected (X22B50T0), (2) volumetric heater heat capacity and perfect thermal contact between the specimen and the heater (X42B50T00), and (3) volumetric heater heat capacity and imperfect contact between specimen and heater (X62B50T00). The results are shown in Table2 where a comparison between the estimates of k and C obtained using the exact temperature solution (X22B50T0) and the finite difference method [8] is provided too. As in the current experiment the standard deviations of the estimates are practically the same, the effect of heat source and contact resistance is of -1.3% for the thermal conductivity and of -1.5% for the volumetric heat capacity. However, the contact resistance between the heater and the specimen exhibits a lower influence. As already observed, this slight reduction is in accordance with the very thin heater and the very low contact resistance.Table 2
Estimation procedure results for three different cases.
Case k[W/(m °C)] Δ % C[kJ/(m3°C)] Δ % X22B50T0 74.10 ± 0.165  3736.95 ± 3.707 X22B50T0 [8] 75.01 1.2 3728.60 -0.2 X42B50T00 73.25 ± 0.163 -1.2 3692.35 ± 3.691 -1.2 X62B50T00 73.16 ± 0.163 -1.3 3680.98 ± 3.682 -1.5


## 5.1. Uncertainty Quantification

Under standard statistical assumptions of additive, uncorrelated, normal errors with zero mean and constant variance the approximate covariance matrix of estimated parameters can be written as [8, p. 452, Eq. 8.5.17](34a)cov⁡b_=σk2σk,Cσk,CσC2=CkkCkCCkCCCC-1σ2=CCC-CkC-CkCCkkσ2Δwhere σ2 is the estimated variance of measurement errors, Δ=CkkCCC-CkC2 denotes the determinant of the X_TX_ matrix appearing in Eq. (33), and the elements Ckk, CCC, and CkC are defined as(34b)Ckk=∑s=18∑n=190∂Tsn∂k2;CCC=∑s=18∑n=190∂Tsn∂C2;CkC=CCk=∑s=18∑n=190∂Tsn∂k∂Tsn∂CAlso, ∂Tsn/∂k and ∂Tsn/∂C (with s=1,…,8) are the sensitivity coefficients of temperature measured by thes-th sensor with respect to the thermal properties k andC, respectively, at then-th time.Therefore, the standard deviations ofk and C can be obtained from Eq. (34a) and (34b) as shown below.(35a)σk≅σCCCΔ(35b)σC≅σCkkΔOnce the standard deviations fork and C are known, a 90% confidence interval of the estimates can be computed.The estimation procedure discussed above is applied to three different cases: (1) heater completely neglected (X22B50T0), (2) volumetric heater heat capacity and perfect thermal contact between the specimen and the heater (X42B50T00), and (3) volumetric heater heat capacity and imperfect contact between specimen and heater (X62B50T00). The results are shown in Table2 where a comparison between the estimates of k and C obtained using the exact temperature solution (X22B50T0) and the finite difference method [8] is provided too. As in the current experiment the standard deviations of the estimates are practically the same, the effect of heat source and contact resistance is of -1.3% for the thermal conductivity and of -1.5% for the volumetric heat capacity. However, the contact resistance between the heater and the specimen exhibits a lower influence. As already observed, this slight reduction is in accordance with the very thin heater and the very low contact resistance.Table 2
Estimation procedure results for three different cases.
Case k[W/(m °C)] Δ % C[kJ/(m3°C)] Δ % X22B50T0 74.10 ± 0.165  3736.95 ± 3.707 X22B50T0 [8] 75.01 1.2 3728.60 -0.2 X42B50T00 73.25 ± 0.163 -1.2 3692.35 ± 3.691 -1.2 X62B50T00 73.16 ± 0.163 -1.3 3680.98 ± 3.682 -1.5


## 6. Conclusions

A novel mathematical modelling for simultaneous estimation of thermal properties of high-conductivity materials has been proposed. It allows the heater heat capacity and the surface contact resistance between sample and heater to be taken into account by using a nonconventional boundary condition, termed as a condition of the sixth kind.The results of the analysis have been obtained by means of the Laplace transform method and the related residues theorem. They have shown that the heater heat capacity can affect the sample sensitivity coefficients more than the contact thermal resistance at their interface; and these coefficients are enough large and uncorrelated at the sample backside. Thus, this side was found to be the best location for the temperature sensor when estimating simultaneously thermal conductivity and volumetric heat capacity.By comparing measured and calculated temperatures through the ordinary least squares norm and, then, minimizing it by the Gauss method, a reduction of about 1.4% was observed for the thermal properties values of Armco iron.



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*Source: 5945413-2019-06-23.xml*