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Title
Abstract
Figures
Introduction
Elasticity Analysis
Conclusion
References
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Physics
Thermodynamics

Determination of Thermal Conductivities in Multilayer Materials

Profile image of Domingo A . TarziaDomingo A . Tarzia

WSEAS TRANSACTIONS ON HEAT AND MASS TRANSFER

https://doi.org/10.37394/232012.2022.17.20
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Abstract

The objective of this work is the determination of the materials that make up a three-layer body, based on the simultaneous estimation of the thermal conductivity of the material of each layer. The body is exposed to a one-dimensional stationary, non-invasive, heat transfer process. It is assumed that the union of each pair of consecutive materials does not present thermal resistance. The parameters to be determined are estimated using three temperature measurements, one at each interface and another at the right edge of the body. The estimation is calculated analytically and a bound is given for the estimation error. In addition, an elasticity analysis is carried out to analyze the local dependence of each estimated parameter with respect to the data. A numerical example is included to illustrate and discuss the method proposed here.

Figures (8)
where u [°C] represents the stationary temperature, h [W/(m?°C)] the coefficient of heat transfer by con- vection, Ty, [°C] the temperature outside the body and The problem described above can be modeled with the following system:
where u [°C] represents the stationary temperature, h [W/(m?°C)] the coefficient of heat transfer by con- vection, Ty, [°C] the temperature outside the body and The problem described above can be modeled with the following system:
Proof. The theorem can be easily proved following the idea developed in [17]. Observe that Co is strictly positive, which assures the existence and uniqueness of the solution.
Proof. The theorem can be easily proved following the idea developed in [17]. Observe that Co is strictly positive, which assures the existence and uniqueness of the solution.
From (10) and (23), the estimation errors are obtained as follows Proof: By using the noisy temperature data Ty, T5 and Ts in (10) it results
From (10) and (23), the estimation errors are obtained as follows Proof: By using the noisy temperature data Ty, T5 and Ts in (10) it results
Table 1 shows the relative estimation errors for some values 77, 75 and T3 close to 7), T> and 73. It can be seen that good relative estimation errors are obtained for the conductivity values. Furthermore, it is observed that the estimate worsens as the error in data increases. In this range of temperature values, a maximum error of 6% is obtained for the estimate of kA, of 2% for the estimate of « and of 2.3% for that of Ko. Firstly, the estimation errors for different values of Ty, T5 and T3 close to 7), T> and T3, are analyzed. Let us define
Table 1 shows the relative estimation errors for some values 77, 75 and T3 close to 7), T> and 73. It can be seen that good relative estimation errors are obtained for the conductivity values. Furthermore, it is observed that the estimate worsens as the error in data increases. In this range of temperature values, a maximum error of 6% is obtained for the estimate of kA, of 2% for the estimate of « and of 2.3% for that of Ko. Firstly, the estimation errors for different values of Ty, T5 and T3 close to 7), T> and T3, are analyzed. Let us define
Figure 1: Gradients of the estimation errors at (87.71, 64.01, 52.65) (Example 1). It is interesting to determine the directions of maximum increase in estimation errors. The theory of calculus in several variables proves that the direc- tion of maximum growth of a function at a point is the direction of the gradient at that point. In this case, they are the directions of VErr,,,(I*),VErr,.,(I*) and VErr,,.(I*), respectively, shown in Figure 1.
Figure 1: Gradients of the estimation errors at (87.71, 64.01, 52.65) (Example 1). It is interesting to determine the directions of maximum increase in estimation errors. The theory of calculus in several variables proves that the direc- tion of maximum growth of a function at a point is the direction of the gradient at that point. In this case, they are the directions of VErr,,,(I*),VErr,.,(I*) and VErr,,.(I*), respectively, shown in Figure 1.
Figure 3: Elasticity of the conductivities with respect to T> for a Nickel-Lead-Iron material (Example 1).
Figure 3: Elasticity of the conductivities with respect to T> for a Nickel-Lead-Iron material (Example 1).
Figure 2: Elasticity of the conductivities with respect to T; for a Nickel-Lead-Iron material (Example 1). of T; leads to an error of about 7% in the estimate of kK, and 3.7 % in the estimate of «, which means that «A is more sensitive to the error in Tf than the &g. As it was mentioned before Kc does not depend on TY.
Figure 2: Elasticity of the conductivities with respect to T; for a Nickel-Lead-Iron material (Example 1). of T; leads to an error of about 7% in the estimate of kK, and 3.7 % in the estimate of «, which means that «A is more sensitive to the error in Tf than the &g. As it was mentioned before Kc does not depend on TY.
Figure 4: Elasticity of the conductivities with respect to T3 for a Nickel-Lead-Iron material (Example 1). Figure 4 indicates that the estimate of Kc is also more sensitive to the measurement of 73 than the other estimates since BS 2 (I*) = 6.5 (about 6.5% of esti- mate error when there i isa 1% in the measurement er- ror), while ES (T*) = EA 13 (I*) = 2 (about 2% of error in the estimates when there | is a 1% in the mea- surement error).
Figure 4: Elasticity of the conductivities with respect to T3 for a Nickel-Lead-Iron material (Example 1). Figure 4 indicates that the estimate of Kc is also more sensitive to the measurement of 73 than the other estimates since BS 2 (I*) = 6.5 (about 6.5% of esti- mate error when there i isa 1% in the measurement er- ror), while ES (T*) = EA 13 (I*) = 2 (about 2% of error in the estimates when there | is a 1% in the mea- surement error).

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