A 2 cm-thick ceramic slab has a thin metal sheath for...

A $2 \mathrm{~cm}$-thick ceramic slab has a thin metal sheath for protection. The contact resistance between the ceramic and metal gives an interfacial conductance of $1600 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. The slab is initially at $300 \mathrm{~K}$ and is suddenly exposed to a hot air stream at 1500 $\mathrm{K}$ with a convective heat transfer coefficient of $800 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. How long will it take for the cen- ter of the slab to reach $1200 \mathrm{~K}$ ? The ceramic properties are $$ \begin{array}{l}\rho=2600 \mathrm{~kg} / \mathrm{m}^{3} \\ c=1150 \mathrm{~J} / \mathrm{kg} \mathrm{K} \\ k=3.0 \mathrm{~W} / \mathrm{m} \mathrm{K}\end{array} $$

A $2 \mathrm{~cm}$-thick ceramic slab has a thin metal sheath for protection. The contact resistance between the ceramic and metal gives an interfacial conductance of $1600 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. The slab is initially at $300 \mathrm{~K}$ and is suddenly exposed to a hot air stream at 1500 $\mathrm{K}$ with a convective heat transfer coefficient of
$800 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. How long will it take for the cen-
ter of the slab to reach $1200 \mathrm{~K}$ ? The ceramic
properties are
$$
\begin{array}{l}\rho=2600 \mathrm{~kg} / \mathrm{m}^{3} \\ c=1150 \mathrm{~J} / \mathrm{kg} \mathrm{K} \\ k=3.0 \mathrm{~W} / \mathrm{m} \mathrm{K}\end{array}
$$

Key Concepts

Finite Body Conduction Analysis

In solids with finite dimensions, heat conduction must be analyzed with appropriate geometric considerations. This involves solving the heat diffusion equation with boundary and initial conditions that accurately represent the physical setup. Such analyses yield characteristic time scales and temperature profiles that are essential for predicting interior temperature evolution.

Contact Thermal Resistance (Interfacial Conductance)

When two different materials are in thermal contact, imperfections at the interface can inhibit heat flow, giving rise to an additional resistance to heat transfer. Interfacial conductance, which is the inverse of contact resistance, quantifies the effectiveness of thermal contact between surfaces and is important in multi-material heat transfer analyses.

Transient Heat Conduction

This concept involves analyzing how temperature within a solid changes over time due to internal energy redistribution, governed by the heat diffusion equation. It is essential when predicting the temperature evolution in solids that are subjected to time-dependent boundary or initial conditions.

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 characterizes the rate at which a material conducts thermal energy relative to its ability to store thermal energy, and it is crucial for determining the speed of temperature changes within a body.

Convective Heat Transfer

Convective heat transfer refers to the transfer of heat between a solid surface and a moving fluid, governed by the convective heat transfer coefficient. This process defines one of the boundary conditions in heat conduction problems and plays a vital role in controlling the rate of heating or cooling at the surface of a material.

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