Material Selection for Maximum Temperature Drop

This question asks us to identify which material, given the same heat flow rate and thickness, will experience the largest temperature drop across it. This depends on the thermal properties of the materials listed, specifically their thermal conductivity.

Understanding Heat Transfer and Temperature Drop

Heat transfer through a material under steady-state conditions is governed by Fourier's Law of Heat Conduction. For a flat material with a constant cross-sectional area, the law can be expressed as:

$$ \frac{Q}{t} = -k \cdot A \cdot \frac{dT}{dx} $$

Where:

• $Q/t$ is the rate of heat flow (energy per unit time).
• $k$ is the thermal conductivity of the material (a measure of its ability to conduct heat).
• $A$ is the cross-sectional area through which heat flows.
• $dT/dx$ is the temperature gradient (change in temperature per unit thickness).

We can rearrange this formula to find the temperature drop ($\Delta T$) across a specific thickness ($\Delta x$):

$$ \Delta T = \frac{Q}{t} \cdot \frac{\Delta x}{k \cdot A} $$

In this problem, the heat flow rate ($Q/t$), the thickness ($\Delta x$), and the area ($A$) are the same for all materials. Therefore, the temperature drop ($\Delta T$) is inversely proportional to the thermal conductivity ($k$):

$$ \Delta T \propto \frac{1}{k} $$

This means that the material with the lowest thermal conductivity will have the highest temperature drop.

Comparing Thermal Conductivities of Materials

Let's examine the typical thermal conductivities ($k$) for the given materials. Lower values indicate better insulation and higher resistance to heat flow.

Determining the Material with Maximum Temperature Drop

From the table above, we can see that glass-wool has a significantly lower thermal conductivity compared to copper, steel, and even refractory brick. Since temperature drop is inversely proportional to thermal conductivity, the material with the lowest thermal conductivity (glass-wool) will experience the maximum temperature drop across the same thickness for a given heat flow.

Copper, being an excellent conductor, offers very little resistance to heat flow, resulting in a small temperature drop. Steel is less conductive than copper but still a relatively good conductor. Refractory bricks offer better insulation than metals but are less effective insulators than materials specifically designed for low thermal conductivity like glass-wool.

Therefore, glass-wool provides the most resistance to heat flow, leading to the largest temperature difference between its surfaces when subjected to the same heat flow and thickness as the other materials.