Critical Analysis of Dual-Probe Heat-Pulse Technique Applied to Measuring Thermal Diffusivity

Abstract

The paper presents an analysis of the experimental parameters involved in application of the dual-probe heat pulse technique, followed by a critical review of methods for processing thermal response data (e.g., maximum detection and nonlinear least square regression) and the consequent obtainable uncertainty. Glycerol was selected as testing liquid, and its thermal diffusivity was evaluated over the temperature range from − 20 °C to 60 °C. In addition, Monte Carlo simulation was used to assess the uncertainty propagation for maximum detection. It was concluded that maximum detection approach to process thermal response data gives the closest results to the reference data inasmuch nonlinear regression results are affected by major uncertainties due to partial correlation between the evaluated parameters. Besides, the interpolation of temperature data with a polynomial to find the maximum leads to a systematic difference between measured and reference data, as put into evidence by the Monte Carlo simulations; through its correction, this systematic error can be reduced to a negligible value, about 0.8 %.

Appendix 1: Analytical Evaluation of the Temperature Increase in the Dual-Probe Method

The solution of the conduction heat transfer equation of the “dual-probe” problem is obtained under the following assumptions:

• pure radial heat conduction propagation;

• homogeneous and isotropic medium;

• infinite line heat source, verified by the length to diameter ratio greater than 50 [30];

• square wave pulse with finite length.

Solution is obtained by overlapping two solutions, the first T₁(τ) corresponding to the response of a material when a heat source in the shape of step function is applied to it; the second is another identical solution but shifted in time of a τ₀ length and with negative value:

$$ T_{1} '\left( \tau \right) = - T_{1} \left( {\tau - \tau_{0} } \right) $$

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and

$$ \begin{array}{*{20}l} {T\left( \tau \right) = T_{1} \left( \tau \right)} \hfill & {\quad {\text{for}}\,\,\,\,\tau \le \tau_{0} } \hfill \\ {T\left( \tau \right) = T_{2} \left( \tau \right) = T_{1} \left( \tau \right) - T_{1} \left( {\tau - \tau_{0} } \right)} \hfill & {\quad {\text{for}}\,\,\,\,\tau > \tau_{0} } \hfill \\ \end{array} $$

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The analytical solution corresponding to the first input is obtained by the usual solution for the probe method, i.e.,

$$ T_{1} = \frac{{\dot{q}}}{4\pi \lambda }{\text{Ei}}\left( { - \frac{{r^{2} }}{4\alpha \tau }} \right) $$

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Being E_i the exponential integral:

$$ E_{i} \left( x \right) = - \int\limits_{ - x}^{\infty } {\frac{{e^{ - x} }}{t}} dt = - E_{1} \left( { - x} \right) $$

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The second solution is the same, shifted and with negative sign:

$$ T_{1} ' = - \frac{{\dot{q}}}{4\pi \lambda }{\text{Ei}}\left( { - \frac{{r^{2} }}{{4\alpha \left( {\tau - \tau_{0} } \right)}}} \right) $$

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Thus, the same Eq. 8 is the solution of the “dual-probe” problem for $0 \le t \le \Delta \tau$, while the following:

$$ T_{2} = \frac{{\dot{q}}}{4\pi \lambda }\left[ {{\text{Ei}}\left( { - \frac{{r^{2} }}{{4\alpha \left( {\tau - \tau_{0} } \right)}}} \right) - {\text{Ei}}\left( { - \frac{{r^{2} }}{4\alpha \tau }} \right)} \right] $$

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for $\tau_{0} \le \tau \le \infty$. This is the equation used as model for the nonlinear least square analysis (par. 3.2).

This equation presents a maximum, whose abscissa occurs after τ₀ (i.e., as far as the heat is supplied, the temperature can only increase). The value of the maximum abscissa is calculated by deriving and equating the derivative to zero, i.e., being:

$$ E_{i} \left( x \right) = - E_{1} \left( { - x} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{dE_{1} \left( x \right)}}{dx} = - E_{0} \left( x \right) $$

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and

$$ E_{0} \left( x \right) = \int\limits_{1}^{\infty } {\frac{{\exp \left( { - x\tau } \right)}}{{\tau^{0} }}} {\text{d}}\tau = \left. {\frac{{\exp \left( { - x\tau } \right)}}{ - x}} \right|_{1}^{\infty } = 0 + \frac{{\exp \left( { - x} \right)}}{x} $$

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Thus,

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$$ \exp \left( { - \;\frac{{r^{2} }}{{4\alpha }}\left[ {\frac{1}{{\tau _{{\max }} - \tau _{0} }} - \frac{1}{{\tau _{{\max }} }}} \right]} \right) = \frac{{\tau _{{\max }} - \tau _{0} }}{{\tau _{{\max }} }} \Rightarrow \alpha = \frac{{\frac{{r^{2} }}{4}\left[ {\frac{1}{{\tau _{{\max }} - \tau _{0} }} - \frac{1}{{\tau _{{\max }} }}} \right]}}{{\ln \frac{{\tau _{{\max }} }}{{\tau _{{\max }} - \tau _{0} }}}} $$

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where τ_max is the time when the maximum occurs.