Estimation of a Thermal Conductivity in a Stationary Heat Transfer Problem with a Solid-Solid Interface

Thermal conductivity estimation problems under different conditions have been extensively studied in engineering for different applications. In some works, this parameter was determined from experimental, numerical and analytical techniques [1-10].

On the other hand, heat transfer problems in multilayer or solid-solid interface materials have been studied in recent years due to the multiple and different applications in science and engineering. These problems have direct applications in several industries, including metallurgic, technologic, electronic, the automotive, aerospace and aviation [11-16]. The advancement of technology requires materials with particular thermal, electrical, magnetic, acoustic and optical properties due to which the interface properties of different material combinations have been studied, for instance, Cu-Al [17, 18], Si-Ge [19], Al-Si [20], Pb-Sn [21, 22], Sn-Pb [23], Ti-Al [24].

Thermal conductivity is one of the most studied physical properties in heat transfer processes in composite materials, as it can be seen in ref. [25-28], to mention some of the works found in the literature.

Under certain conditions, a process of heat transfer throughout a multilayer material can be seen as a process that develops in one direction through the different layers. In that case, it can be modeled as a 1D process with a solid-solid interface, see for instance [29, 30].

In this work, we consider a stationary heat transfer process in a bar made up of two consecutive sections of different, isotropic and homogeneous materials, totally isolated on its lateral surface. The length of the bar is assumed to be much greater that the diameter so that a 1D heat transfer process is considered. The section on the left side of the bar, which is assumed to be unreachable, is composed by an unknown material and its left end is assumed to have a constant temperature, represented by a Dirichlet condition. The thermal resistance at the interface is neglected and the right end of the bar is let free, in contact with an external fluid, giving rise to the phenomenon of convection, so a Robin type condition that models the heat dissipation is considered.

We propose to determine the thermal conductivity of unknown (left) material based on a noisy measurement of the heat flux at the available (right) body end. The approach presented here is quite simple and can be easily applied as a complement or alternative way for the determination of thermal conductivity in new materials.

Necessary and sufficient conditions for its application are derived together with an expression for a bound of the estimation error. In order to study the local influence of the heat flux measurement in the estimation, an elasticity analysis is performed and its properties are discussed. Some examples considering different situations are included to illustrate the estimation procedure proposed here. Absolute and relative errors are calculated from the results of the numerical experiments where noisy values of the heat flux are assumed for the estimation. In addition, we show that the errors obtained agree with the analysis of the elasticity function.

The main objective of this work is developed in this section. The thermal conductivity ${{\kappa }_{A}}$ of the material A is estimated using a noisy heat flux data on x = L subjected to Eqns. (1)-(6). Necessary and sufficient conditions for the estimation of the thermal conductivity are derived and a bound for the estimation error is provided.

3.1 Estimation

By definition, the thermal flux $q$ at $x=L$ is given by the following expression [29]:

$q=-{{\kappa }_{B}}u'(x),\text{   }x=L.$    (10)

From Eq. (7), the thermal flux becomes 

$q=\frac{{{\kappa }_{B}}{{\kappa }_{A}}h(F-{{T}_{a}})}{{{\kappa }_{A}}{{\kappa }_{B}}+{{\kappa }_{A}}hL+({{\kappa }_{B}}-{{\kappa }_{A}})hl}$         (11)

Then, the parameter ${{\kappa }_{A}}$ is obtained by the following expression, 

${{\kappa }_{A}}=\frac{qhl{{\kappa }_{B}}}{h{{\kappa }_{B}}(F-{{T}_{a}})-qh(L-l)-q{{\kappa }_{B}}}$          (12)

Therefore, the estimate ${{\hat{\kappa }}_{A}}$ of the thermal conductivity as a function of the heat flux measurement $\hat{q}$ at x=L is expressed as:

${{\hat{\kappa }}_{A}}=\frac{\hat{q}hl{{\kappa }_{B}}}{h{{\kappa }_{B}}(F-{{T}_{a}})-\hat{q}h(L-l)-\hat{q}{{\kappa }_{B}}}$      (13)

where it is assumed that:

$\left| q-\hat{q} \right|\le \varepsilon ,$     (14)

and $\varepsilon >0$ represents the noise level in the data.

Note that the estimation of ${{\hat{\kappa }}_{A}}$ depends on the heat flux measurement and the parameters of the problem.

3.2 Necessary and sufficient conditions

There exist necessary and sufficient conditions for the estimation of the thermal conductivity.

The thermal conductivity values $\kappa_{A}, \kappa_{B}$ and $\hat{\kappa}_{A}$ are assumed to be bounded, that is, there exist positive constants $\kappa_{m}$ and $\kappa_{M}$ that satisfy

$0<{{\kappa }_{m}}<{{\kappa }_{A}},{{\kappa }_{B}},{{\hat{\kappa }}_{A}}<{{\kappa }_{M}}$     (15)

By Eq. (13) it follows that,

$0<{{\kappa }_{m}}<\frac{\hat{q}hl{{\kappa }_{B}}}{h{{\kappa }_{B}}\left( F-{{T}_{a}} \right)-\hat{q}h\left( L-l \right)-\hat{q}{{\kappa }_{B}}}<{{\kappa }_{M}}.$        (16)

Algebraic computations yield:

${{q}_{m}}<q,\hat{q}<{{q}_{M}}<{{\bar{q}}_{M}},$    (17)

where,

${{q}_{m}}=\frac{1}{l}\left( \frac{\left( F-{{T}_{a}} \right)}{\frac{1}{{{\kappa }_{m}}}+\frac{1+\frac{h\left( L-l \right)}{{{\kappa }_{B}}}}{hl}} \right),$        (18)

${{q}_{M}}=\frac{1}{l}\left( \frac{\left( F-{{T}_{a}} \right)}{\frac{1}{{{\kappa }_{M}}}+\frac{1+\frac{h\left( L-l \right)}{{{\kappa }_{B}}}}{hl}} \right)$        (19)

and

${{\bar{q}}_{M}}=\frac{\left( F-{{T}_{a}} \right)h}{1+\frac{h\left( L-l \right)}{{{\kappa }_{B}}}}.$       (20)

Therefore, there exists the estimate value $~{{\hat{\kappa }}_{A}}$ and Eq. (15) is satisfied if and only if Eq. (17) holds. In other words, Eq. (17) is a necessary and sufficient condition for the existence of $~{{\hat{\kappa }}_{A}}.$

3.3 Estimation error

Note that from Eqns. (12)-(13) it follows:

$\left| \frac{1}{{{\kappa }_{A}}}-\frac{1}{{{{\hat{\kappa }}}_{A}}} \right|=\frac{F-{{T}_{a}}}{lq\hat{q}}\left| q-\hat{q} \right|$          (21)

Eq. (17) leads to,

$\left| \frac{1}{{{\kappa }_{A}}}-\frac{1}{{{{\hat{\kappa }}}_{A}}} \right|<\left( \frac{F-{{T}_{a}}}{l} \right)\frac{|q-\hat{q}|}{q_{m}^{2}},$         (22)

which along with Eq. (18) it results:

$\left| \frac{1}{{{\kappa }_{A}}}-\frac{1}{{{{\hat{\kappa }}}_{A}}} \right|<\frac{|q-\hat{q}|l}{\kappa _{m}^{2}(F-{{T}_{a}})}{{\left[ 1+\frac{{{\kappa }_{m}}}{hl}\left( 1+\frac{h(L-l)}{{{\kappa }_{B}}} \right) \right]}^{2}}$          (23)

Equivalently, by condition (14) we have,

$\left| \frac{1}{{{\kappa }_{A}}}-\frac{1}{{{{\hat{\kappa }}}_{A}}} \right|<{{K}_{1}}\varepsilon ,$      (24)

where, the constant value ${{K}_{1}}$ is given by,

${{K}_{1}}=\frac{l}{\kappa _{m}^{2}(F-{{T}_{a}})}{{\left[ 1+\frac{{{\kappa }_{m}}}{hl}\left( 1+\frac{h(L-l)}{{{\kappa }_{B}}} \right) \right]}^{2}}.$        (25)

Hence, $\left| \frac{1}{{{\kappa }_{A}}}-\frac{1}{{{{\hat{\kappa }}}_{A}}} \right|$  → 0 as ε → 0.

Furthermore, an analytical bound can be determined for the estimation error of the coefficient of thermal conductivity. From (24)-(25) the estimation error $\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|$ satisfies,

$\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|<{{\kappa }_{A}}{{\hat{\kappa }}_{A}}{{K}_{1}}\left| q-\hat{q} \right|$        (26)

or, by using the assumptions (14)-(15), the absolute estimation error satisfies,

$\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|<{{K}_{2}}\varepsilon ,$         (27)

where the constant ${{K}_{2}}$ is given by the following expression:

${{K}_{2}}=\kappa _{M}^{2}{{K}_{1}}=\frac{\kappa _{M}^{2}l}{\kappa _{m}^{2}(F-{{T}_{a}})}{{\left[ 1+\frac{{{\kappa }_{m}}}{hl}\left( 1+\frac{h(L-l)}{{{\kappa }_{B}}} \right) \right]}^{2}}.$         

Therefore, the estimation error $\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|$ is bounded by ${{K}_{2}}\varepsilon $, and thus, $\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|$  → 0 as ε → 0.

The following parameter values are imposed in the following numerical experiments $L=10 \mathrm{~m}, l=4 \mathrm{~m}, F=100{ }^{\circ} \mathrm{C}, T_{a}=25^{\circ} \mathrm{C}, h=10$ Wm^-2℃ ^-1.

For the error testing, for the bar different pairs of materials are considered where the thermal conductivity of the material occupying the left side must be estimated. The pairs of materials are chosen according to the different relationships between the coefficients of thermal conductivity, i.e., $\kappa_{A}<\kappa_{B}, \kappa_{A}>\kappa_{B}$ and $\kappa_{A} \approx \kappa_{B}$.

In order to simulate a noisy experimental measurement $\hat{q}$, the solution $u$ to the forward problem (1)-(6) is calculated by using expressions (7)-(8) and the heat flux $q$ is obtained by (10). Finally, a perturbation is added to $q$ so that the resulting value remains within the interval given by condition (17).

The estimate value $\hat{\kappa}_{A}$ is calculated by using the expression (13) for the data $\hat{q}$. Afterwards, the absolute and relative estimation errors are calculated. This procedure is repeated for ten data perturbated with different noise levels.

For each example, a table with the results and a figure showing the elasticity function are included.

Each line in the tables below contain the noisy heat flux $\hat{q}$ along with the data error $|q-\hat{q}|$, the estimate $\hat{\kappa}_{A}$ the absolute error $\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|$ and the relative error $\frac{\left| {{\kappa }_{A}}-{{{\hat{\kappa }}}_{A}} \right|}{{{\kappa }_{A}}}$.

5.1 Example 1 ( ${{\kappa }_{A}}<\ {{\kappa }_{B}}$ )

A Fe-Ag bar is considered to test the estimation procedure for the thermal conductivity of Fe, ${{\kappa }_{A}}={{\kappa }_{Fe}}=73$ W.m^-1.℃ ^-1. Since ${{\kappa }_{B}}={{\kappa }_{Ag}}=419$  W.m^-1.℃^-1, the analytical heat flux, calculated by using the expression (11), is q = 443.48 W.m^-2 .

Table 2 shows the performance for the estimation procedure for this particular case. It is observed that all relative errors are less than 4% for noise level less 1%.

The elasticity function is plotted in Figure 4. As noted before, the elasticity function is positive and strictly increasing in the interval $({{q}_{m}},{{\bar{q}}_{M}})$ where ${{\bar{q}}_{M}}=656.05$ W.m^-2 is the vertical asymptote (31)-(32). This function indicates that a measurement error of 1 % in the heat flux value $\hat{q}$ translates into an error of around 4 % in the estimation value, which was also noticed before from the results shown in Table 2.

Table 2. Example 1: Estimates for ${{\kappa }_{A}}=73$ W.m^-1.℃ ^-1, where the analytical heat flux is q = 443.48 W.m^-2

4.png

Figure 4. Elasticity function for Example 1

5.2 Example 2 ($\kappa_{A}>\kappa_{B}$)

A Al-Pb bar is now considered for the estimation of the thermal conductivity value ${{\kappa }_{A}}={{\kappa }_{Al}}=204$ W.m^-1.℃^-1. For this example ${{\kappa }_{B}}={{\kappa }_{Pb}}=35$ W.m^-1.℃^-1 hence, the relationship between the thermal conductivities is ${{\kappa }_{A}}>\ {{\kappa }_{B}}.$  For this example, the analytical heat flux is q = 257.69 W.m^-2.

Table 3. Example 2: Estimates for ${{\kappa }_{A}}=204$ W.m^-1.℃ ^-1, where the analytical heat flux is q = 257.69 W.m^-2

Ten simulated data are considered around the $q$ in the interval $\left(q_{m}, \bar{q}_{M}\right)$ and the results are shown in Table 3. The relative error for this example achieves 25% for an error of 2% in the data. The elasticity function for the Example 2 is plotted in the Figure 5 and it indicates that a measurement error of 1% in the heat flux value $\hat{q}$ produces an error of 15 % in the estimation value $\hat{\kappa}_{A}$. Note that for this example the vertical asymptote is located at $\bar{q}_{M}=276.31$  W.m^-2, not too far from the value of $q$ hence the estimation is not as good as for the previous example.

5.png

Figure 5. Elasticity function for Example 2

5.3 Example 3 ($\kappa_{A} \approx \kappa_{B}$)

Finally, the case $\kappa_{A} \approx \kappa_{B}$ is considered. The thermal conductivity estimation is analyzed for a Ag-Cu bar is where$\kappa_{A}=\kappa_{A g}=419 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot{ }^{\circ} \mathrm{C}^{-1}$ and $\kappa_{B}=\kappa_{\mathrm{cu}}=386 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot{ }^{\circ} \mathrm{C}^{-1}$.

The heat flux in this case is $q=599.56$ W.m^-2 and from the necessary and sufficient condition (17), the upper bound for the measured data is $\bar{q}_{M}=649.10$ W.m^-2. Ten simulated data are considered around the q, within the interval $\left(q_{m}, \bar{q}_{M}\right)$. The results are shown in Table 4. The relative error for the set of data considered here almost reaches 11%. The elasticity function for this example, shown in Figure 6, indicates that a measurement error of 1% in the heat flux value $\hat{q}$ produces an estimation error of around 13%. Again, as in the previous example, the value of $q$ is close to the vertical asymptote value $\bar{q}_{M}$ and therefore, the error in the data value is amplified in the estimation.

Table 4. Example 3: Estimates for $\kappa_{A}=419 $ W.m^-1.℃ ^-1, where the analytical heat flux is $q=599.56$ W.m^-2

6.png

Figure 6. Elasticity function for Example 3

The first and second authors acknowledge support from SOARD/AFOSR through grant FA9550-18-1-0523. The third author acknowledges support from European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH and by the Project PIP No. 0275 from CONICET-UA, Rosario, Argentina.

The authors appreciate the very useful comments made by the reviewer in order to improve this work.