A thermocouple is used to measure the temperature of air...

A thermocouple is used to measure the temperature of air inside a long pipe with black walls. The pipe walls are at $100^{\circ} \mathrm{C}$, and the air is at $20^{\circ} \mathrm{C}$. Treating the thermocouple as black, what temperature will it indicate if it exchanges heat primarily by radiation and by convection in the air? (The convection may be modelled by Newton's law of cooling, with a coefficient $h=14 \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}$.) What will be the temperature reading if the thermocouple is surrounded by a tightly fitting shiny envelope of emissivity 0.05 ?

Key Concepts

Emissivity

Emissivity is a measure of a material's ability to emit or absorb thermal radiation relative to that of an ideal blackbody. It takes values between 0 and 1, where a high emissivity indicates strong radiation emission and absorption, and a low emissivity indicates that the material reflects most radiation. This property critically influences the net radiative heat exchange of a surface, and thereby affects temperature measurements in radiative environments.

Thermocouple Measurement Principles

Thermocouples operate on the principle of creating a voltage difference at the junction of two dissimilar metals, which correlates to the temperature difference between that junction and a reference. When used in environments where energy exchange occurs via multiple modes such as radiation and convection, the actual temperature sensed by the thermocouple can be influenced by its radiative absorption properties and any surrounding coatings or enclosures, which modify its effective surface properties.

Radiative Heat Transfer

Radiative heat transfer involves the exchange of energy between surfaces by emission and absorption of electromagnetic radiation, which is governed by the Stefan-Boltzmann law. It is significant in situations where bodies at different temperatures exchange energy even across a vacuum or an intervening medium, and the effectiveness of this exchange greatly depends on the temperature and the surface properties, such as emissivity, of the bodies involved.

Convective Heat Transfer

Convective heat transfer describes the energy exchange between a surface and a fluid (such as air) in motion around it, and is commonly modeled by Newton’s law of cooling. This principle states that the rate of heat transfer is proportional to the temperature difference between the surface and the ambient fluid, with the proportionality constant being the heat transfer coefficient.

Thermal Energy Balance

A thermal energy balance is the analysis of the equilibrium condition where the net energy gained and lost by a system is zero. In scenarios involving combined effects of radiation and convection, the sensor or device reaches an equilibrium temperature at which the energy absorbed equals the energy lost through both radiative and convective mechanisms.

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