Determine the thermal conductivity of the carbon nanotube of...

Determine the thermal conductivity of the carbon nanotube of Example $3.4$ when the heating island temperature is measured to be $T_{h}=332.6 \mathrm{~K}$, without evaluating the thermal resistances of the supports. The conditions are the same as in the example.

Key Concepts

Thermal Conductivity

Thermal conductivity is a material property that quantifies the ability of a substance to conduct heat. It is defined as the amount of heat that passes in unit time through a unit cross-sectional area of a material per unit temperature gradient. In the context of nanoscale materials like carbon nanotubes, measuring thermal conductivity involves careful control and understanding of heat flow, as well as isolation of the contributions from the material itself from other components in the experimental setup.

Fourier's Law of Heat Conduction

Fourier’s law provides the fundamental relationship governing heat conduction, stating that the heat flux within a material is proportional to the negative gradient of temperature and the material's thermal conductivity. This law is essential when determining thermal conductivity because it allows for the calculation of heat transfer through a material given a known temperature difference, and it forms the basis for many experimental methods in thermal measurements.

Experimental Temperature Measurement Techniques

Experimental techniques for measuring thermal properties often involve creating a controlled temperature difference across a specimen. In this context, measuring the temperature of a ‘heating island’ is critical, as it provides a controlled parameter that can be used, along with known heat inputs, to infer the thermal properties of the material. Accurate temperature measurements and accounting for potential parasitic effects are fundamental to obtaining reliable results.

Thermal Resistance

Thermal resistance represents the opposition to heat flow through a material or a set of materials and is analogous to electrical resistance in circuits. In experimental setups that involve supports or contacts, the overall measured thermal resistance includes contributions from both the sample and the supports. However, in cases where the support resistances are neglected or decoupled from the measurement, the analysis focuses specifically on the intrinsic resistance of the sample, facilitating a more direct determination of its thermal conductivity.

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