The author and his then 6-year-old son have conducted the...

The author and his then 6-year-old son have conducted the following experiment to determine the thermal conductivity of a hot dog. They first boiled water in a large pan and measured the temperature of the boiling water to be $94^{\circ} \mathrm{C}$, which is not surprising, since they live at an elevation of about 1650 m in Reno, Nevada. They then took a hot dog that is 12.5 cm long and 2.2 cm in diameter and inserted a thermocouple into the midpoint of the hot dog and another thermocouple just under the skin. They waited until both thermocouples read $20^{\circ} \mathrm{C}$, which is the ambient temperature. They then dropped the hot dog into boiling water and observed the changes in both temperatures. Exactly 2 min after the hot dog was dropped into the boiling water, they recorded the center and the surface temperatures to be $59^{\circ} \mathrm{C}$ and $88^{\circ} \mathrm{C}$, respectively. The density of the hot dog can be taken to be $980 \mathrm{~kg} / \mathrm{m}^3$, which is slightly less than the density of water, since the hot dog was observed to be floating in water while being almost completely immersed. The specific heat of a hot dog can be taken to be $3900 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, which is slightly less than that of water, since a hot dog is mostly water. Using transient temperature charts, determine (a) the thermal diffusivity of the hot dog; (b) the thermal conductivity of the hot dog; and (c) the convection heat transfer coefficient.

Key Concepts

Convection Heat Transfer Coefficient

The convection heat transfer coefficient is a parameter that quantifies the convective heat transfer between a surface and a surrounding fluid. It characterizes how effectively heat is carried away from or supplied to a surface through fluid motion, and it plays a key role in setting the boundary conditions for transient heat conduction problems.

Transient Heat Conduction

Transient heat conduction refers to the study of how temperature within a material changes over time when subjected to a thermal disturbance. Unlike steady-state conduction, transient analysis involves time-dependent factors and requires understanding the interplay between internal conduction, material properties, and boundary conditions, such as convective heat transfer.

Density

Density, defined as mass per unit volume, is an important material property in heat transfer analysis when coupled with specific heat capacity. Together, they form part of the denominator in the expression for thermal diffusivity, influencing the rate at which a material can absorb or release heat during transient thermal processes.

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius. It plays a vital role in transient heat transfer by determining how much energy is stored or required to change the temperature of a material, thus influencing the speed at which a material heats up or cools down.

Thermal Conductivity

Thermal conductivity is the measure of a material's ability to conduct heat. It represents the rate at which heat is transferred through a material per unit area per unit temperature gradient. This property is central to understanding heat conduction processes and is essential for modeling steady?state and transient temperature distributions in various materials.

Thermal Diffusivity

Thermal diffusivity is a material property defined as the ratio of thermal conductivity to the product of density and specific heat capacity. It quantifies how quickly heat diffuses through a material. In transient heat conduction problems, a higher thermal diffusivity means the material will respond more rapidly to changes in temperature, making it a crucial parameter for analyzing time?dependent temperature profiles in objects.

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