The temperature of a gas stream is to be measured by a...

The temperature of a gas stream is to be measured by a thermocouple whose junction can be approximated as a $1.2-\mathrm{mm}$-diameter sphere. The properties of the junction are $k=35 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}, \rho=8500 \mathrm{~kg} / \mathrm{m}^3$, and $c_p=320 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, and the heat transfer coefficient between the junction and the gas is $h=90 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Determine how long it will take for the thermocouple to read 99 percent of the initial temperature difference.

Key Concepts

Transient Heat Conduction

This concept deals with the study of temperature changes within an object as it exchanges heat with its surroundings over time. It involves solving the time?dependent heat transfer equations to determine how temperatures evolve after a thermal disturbance or change in boundary conditions.

Lumped Capacitance Model

When the internal resistance to heat conduction within an object is small compared to the resistance to heat transfer across its boundary, the temperature inside the object can be assumed uniform. This simplifies the analysis to a first?order ordinary differential equation, treating the object as having a spatially uniform, time-varying temperature.

Biot Number

The Biot number is a dimensionless parameter that compares the internal conductive resistance to the external convective resistance. A small Biot number indicates that temperature gradients inside the object are negligible, which justifies the use of the lumped capacitance model for transient heat transfer analysis.

Thermal Time Constant

The thermal time constant quantifies the response speed of an object to changes in its thermal environment. It is determined by the object's mass, specific heat, geometry, and the convective heat transfer conditions, representing the time it takes to reach a significant fraction of the final equilibrium temperature.

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