A Steady-state Method for the Estimation of the Thermal Conductivity of a Wire

The measurement of thermal properties of wires (or fibers) is essential when there are used in temperature sensors (as thermocouples for example), superconducting materials [1] or microelectronic devices. 

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Figure 1. Schema of the method used for thermal conductivity measurement of an electric conducting wire

Most of the methods used for the estimation of the thermal conductivity of electric conducting wires are based on the typical experimental device represented in Figure 1: The two extremities of a wire of length L are kept at a constant temperature T_a  and they are connected to an electrical generator. An electric current l gets through the wire, producing by Joule effect.

The mean temperature rise in the wire is deduced from the variation of its electrical resistance R measured by a four probe method. It is assumed that the wire electrical resistance varies linearly with its mean temperature as follows:

$R={{R}_{0}}[1+\alpha (\overline{T}-{{T}_{a}})]$ (1)

where $\overline{T}$ is the wire mean temperature defined as:

$\bar{T}\text{=}\frac{1}{L}\int _{0}^{L}T(x)dx$ (2)

α is the temperature coefficient of the electrical resistance and R₀ is the electrical resistance of the wire at the temperature T_a. The heat produced inside the wire is transferred to its environment by two ways:

• Heat conduction from the wire to the extremities maintained at ambient temperature T_a.
• Combined convection-radiation heat transfer with a global coefficient  h to the surrounding environment. Thus the mean temperature rise $\overline{θ}$ depends on the following parameters:
• Time t (or pulsation ω in case of a periodic heat flux)
• Thermal conductivity λ (W m^-1 K^-1) and volumetric heat capacity ρc (J m^-3 K^-1) of the wire
• External (convection+radiation) heat transfer coefficient h (W m^-2 K^-1)
• Dimensions of the wire: diameter D and length L

The methods based on the previously described device may be classified in two types:

• Steady-state [2]: in this case the electrical current intensity in the wire is constant: I=I₀ and the steady state value $\overline{θ}$ of the mean temperature rise only depends on the wire thermal conductivity λ and on the external heat transfer coefficient h, Thus, λ can be deduced from a the steady-state measurement only if h is known. Most often the experiment is carried out under vacuum and h is considered to be null.
• Periodic [3] to [9]: in this case I=I₀  cos(ωt) and the measurement is repeated for several values of the electrical current pulsation w. The third harmonic U_3ω of electric potential difference U_L is measured when the steady-state periodic regime is reached. A theoretical relation U_3ω=f(ω,λ,ρc,h) may be established from heat transfer modeling in the wire. The thermal conductivity λ may then be estimated from measurements corresponding to different values of ω if the sensitivities of U_3ω to λ, ρc and h are not correlated.

The proposed method intends to conserve the simplicity of the stationary method but without being forced to neglect the external heat transfer. This is made possible by realizing several steady-state measurements for different lengths of the same wire. The idea has already been used by Volkein and Kessler [10] for thin films thermal conductivity measurements but the applicability was not fully demonstrated. This paper will to demonstrate that thermal conductivity of wires may be estimated with a good precision using the proposed method. 

The remainder of this paper is organized as follows: Section 2 describes the experimental device and the measurement method. Section 3 presents the modeling of the wire mean temperature rise θ by heat transfer analysis, both with a sensitivity study of this temperature rise θ to the length L of the wire, to its thermal conductivity λ and to the heat transfer coefficient h. Section 4 presents the experimental results obtained with Copper and Chromel wires, confirming the feasibility of thermal conductivity estimation by the proposed method.

The following hypotheses have been assessed:

• The temperatures of the brass blocks are constant and equal to the surrounding temperature T_a.
• The radial thermal gradient inside the wire is negligible. Considering the following limit values of the parameters: h<10 W m^-2 K^-1, D<1 mm and λ>1 W m^-1 K^-1, the Biot number ${{B}_{i}}=\frac{hD}{\lambda}$ is lower than 0.01. This value justifies the hypothesis that the temperature radial gradient is negligible inside the wire.

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Figure 3. Schema of the thermal balance on an infinitesimal part dx of the wire

For modeling the temperature T(x) of the wire, where x is the distance from the middle of the wire, a thermal balance has been realized on the wire part between x and x+dx  as represented on figure 3, it may be written:

${{\phi }_{con{{d}_{x}}}}+{{\phi }_{gen}}={{\phi }_{con{{d}_{x}}}}+{{\phi }_{conv}}$ (5)

${{\phi }_{con{{d}_{x}}}}\text{=-}\lambda \text{S(}\frac{dT}{dx}\text{)}$ (6)

${{\phi }_{gen}}={{\phi }_{0}}Sdx$ (7)

${{\phi }_{con{{d}_{x}}}}\text{=-}\lambda \text{S(}\frac{dT}{dx}{{\text{)}}_{x+dx}}$ (8)

${{\phi }_{conv}}=hPdx[T(x)-{{T}_{a}}]$ (9)

and: P is the wire perimeter: P = πD (D is the wire diameter)

S is the wire cross section: $S\text{=}\frac{\pi {{D}^{2}}}{4}$

ϕ₀(x) is the heat flux per volume unit produced by Joule effect in the wire, given by:

${{\phi }_{0}}(x)=\frac{16{{\rho }_{0}}I_{0}^{2}}{{{\pi }^{2}}{{D}^{4}}}\{1+\alpha [T(x)-{{T}_{a}}]\}(W{{m}^{-3}})$ (10)

where ρ₀ is the electrical resistivity of the wire (Ω m).

Setting θ(x)=T(x)-T_a, this system of equations leads to:

$\frac{{{d}^{2}}\theta }{d{{x}^{2}}}-\frac{hP}{\lambda S}\theta +\frac{{{\phi }_{0}}}{\lambda }=0$ 0≤x≤L/2  (11)

The boundary conditions are:

$\frac{d\theta }{dx}=0$ (null flux for symmetry reason) at x=0 (12)

θ = 0 at  x=L/2 (13)

Eq. (11) can also be written:

$\frac{{{d}^{2}}\theta }{d{{x}^{2}}}-(\frac{4h}{\lambda D}-\frac{16{{\rho }_{0}}\alpha I_{0}^{2}}{\lambda {{\pi }^{2}}{{D}^{4}}})\theta +\frac{16{{\rho }_{0}}I_{0}^{2}}{\lambda {{\pi }^{2}}{{D}^{4}}}=0$ (14)

If: $\frac{4h}{\lambda D}-\frac{16{{\rho }_{0}}\alpha I_{0}^{2}}{\lambda {{\pi }^{2}}{{D}^{4}}}>0$, let us consider:

$K=\frac{16{{\rho }_{0}}I_{0}^{2}}{\lambda {{\pi }^{2}}{{D}^{4}}}$ (15)

And: $\beta _{1}^{2}=\frac{1}{\lambda }(\frac{4h}{D}-\frac{16\alpha {{\rho }_{0}}I_{0}^{2}}{{{\pi }^{2}}{{D}^{4}}})$ (16)

Taking into account the boundary conditions Eq. (12) and Eq. (13) the resolution of Eq. (14) leads to:

$\theta (x)=\frac{k}{\beta _{1}^{2}}[1-\frac{\cosh ({{\beta }_{1}}x)}{\cosh ({{\beta }_{1}}L/2)}$ (17)

The mean temperature rise can be calculated by:

$\bar{\theta}(x)=\frac{2}{L}\int_0^{L/2}\theta (x)dx$ (18)

And finally: 

$\overline{\theta }=\overline{T}-{{T}_{a}}=\frac{K}{\beta _{1}^{2}}[1-\frac{2\tanh ({{\beta }_{1}}L/2)}{{{\beta }_{1}}L}]$ (19)

If $\frac{4h}{\lambda D}-\frac{16{{\rho }_{0}}\alpha I_{0}^{2}}{\lambda {{\pi }^{2}}{{D}^{4}}}<0\\let us consider: \,\,\,\,\beta _{2}^{2}=\frac{1}{\lambda }(\frac{16{{\rho }_{0}}I_{0}^{2}}{{{\pi }^{2}}{{D}^{4}}}-\frac{4h}{\lambda D})$ (20)

Taking into account the boundary conditions Eq. (12) and Eq. (13) the resolution of Eq. (14) leads to:

$\theta (x)=\frac{k}{\beta _{2}^{2}}[\frac{\cos ({{\beta }_{2}}x)}{\cos ({{\beta }_{2}}L/2)}-1]$ (21)

Eq. (18) and Eq. (21) lead to:

$\bar{\theta }=\bar{T}-{{T}_{a}}=\frac{K}{\beta _{2}^{2}}[1-\frac{2\tan ({{\beta }_{2}}L/2)}{{{\beta }_{2}}L}-1]$ (22)

With the hypotheses that h=0 and that the heat flux ϕ₀ dissipated in the wire by Joule effect is constant and equal to:

${{\phi }_{0}}=\frac{4UI_{0}^{2}}{\pi {{D}^{2}}L}$ (23)

Eq. (14) becomes:

$\frac{{{d}^{2}}\theta }{d{{x}^{2}}}+\frac{{{\phi }_{0}}}{\lambda }=0$ (24)

Let us consider: ${{K}_{1}}=\frac{{{\phi }_{0}}}{\lambda }$  (25)

The resolution of Eq. (24) taking into account the boundary conditions Eq. (12) and Eq. (13) lead to:

$\theta (x)={{K}_{1}}(\frac{{{L}^{2}}}{4}-\frac{{{x}^{2}}}{2})$ (26)

And finally: $\bar{\theta }=\bar{T}-{{T}_{a}}=\frac{1}{3}\frac{U{{I}_{0}}L}{\lambda \pi {{D}^{2}}}$ (27)

Then the thermal conductivity λ of the wire can be deduced by:

$\lambda =\frac{1}{3}\frac{U{{I}_{0}}L}{\pi {{D}^{2}}\overline{\theta }}$ (28)

Several authors such as Moon [2] estimate the thermal conductivity l by using Eq. (28). It will be further shown than in some cases using Eq. (28) may lead to an estimation error up to several percents so that it is always better to use Eq. (19) or Eq. (22).

First, the temperature coefficients for the electrical resistance of each wire were measured. The experimental results are: α_Copper=3.88×10^-3 K^-1 and α_Chromel=3.26×10^-3 K^-1. The first value is very close to the value α_Copper=3.93×10^-3 K^-1 given by Fallou [13].

Eq. (15) shows that K is proportional to D⁴. Thus, the relative standard deviation of K would be four times the standard deviation of D if Eq. (15) was used to calculate K. Since the measurement of the electrical resistance of a thin wire is much more accurate than the measurement of its diameter, the resistance per length unit $\frac{R}{L}$ of each wire has been measured and then K has been calculated by:

$K\text{=}\frac{16(1+\alpha \theta )I_{0}^{2}}{\lambda \pi {{D}^{2}}}\frac{R}{L}$ (31)

Moreover, when using Eq. (31) instead of Eq. (15), it is no longer necessary to know the value of the electrical resistivity ρ₀ of the wire to calculate K.

Table 1 presents the measured values for each of the three wires.

The mean temperature $\overline{T}$ of a Copper wire with a length L=2.1 cm and a diameter D=0.025 mm has been measured using the experimental device described in Figure 1, under an air pressure of 3×10^-2 mbar. The measurement has been repeated three times.

Table 1. Experimental values of the linear resistance of the wires

The measured values of $\overline{T}$ and the values of the thermal conductivity λ calculated by Eq. (19) or Eq. (22) on one hand and by Eq. (28) on the other hand are presented in Table 2. For the estimation of using Eq. (19) or Eq. (22), two limit values h_min and h_max of the external heat transfer coefficient h have been considered, enabling the estimation of two extreme values λ_min and λ_max of λ.

Table 2. Experimental values of $\overline{T}-T_a$; estimated and reference value of the thermal conductivity λ of Copper

The low pressure value decreases the convective heat transfer and the external heat transfer is essentially done by radiation. The radiation heat transfer coefficient of a surface at temperature T is lower than 4σT³, where σ is the Stefan-Boltzmann constant (σ=5.67×10^-8 W m^-2 K^-4), corresponding to an approximate value of 6 W m^-2 K^-1 for a black body at a temperature of 25°C. Thus, the extreme values h_min=2 W m^-2 K^-1 and h_max=6 W m^-2 K^-1 have been considered.

Compared to a reference value of  386 W m^-1 K^-1 [14], the relative deviation is less than 2 % that is quite acceptable. It may be noted that using the approximate Eq. (28) would lead to a greater deviation around 4%.

For the Chromel wires the mean temperatures of wires with lengths between 15 mm and 85 mm have been measured under a constant pressure of 6.10^-2 mbar. An algorithm minimizing the sum of the quadratic errors between the experimental values and the values calculated with Eq. (19) enabled the simultaneous estimation of the thermal conductivity λ and of the external heat transfer coefficient h for each wire. The experimental points and the theoretical curves are presented in Figure 7. The values of λ  and h obtained using Eq. (19) are presented in Table 3.

Furthermore, the specific heat c of the Chromel has been measured with a Setaram mdsc3 differential scanning calorimeter and its density ρ has been estimated by weighing a sample and measuring its dimensions. The thermal diffusivity a has been estimated by the flash method [15] applied on a plane sample.  The thermal properties obtained at 20°C are:

ρc=3.83×10⁶ J m^-3 K^-1, a=4.54×10^-6 m² s^-1

leading to λ=17.4 W m^-1 K^-1.

The estimated values of the Chromel thermal conductivity obtained by our method are very close to the values obtained by coupling the results of the calorimetric and flash methods, with a deviation less than 4% that is quite acceptable.

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Figure 7. Experimental (□) and modeled ( — ) temperature for Chromel wires: a) D=0.127 mm and b) =0.254 mm

Table 3. Estimated values of  the wire thermal conductivitiy λ and of the external heat transfer coefficient h for two Chromel wires