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Fourier's Law of Heat Conduction is a fundamental principle that helps explain how heat transfers through materials. This law states that the rate at which heat flows through a material is directly proportional to the negative gradient of temperature and the area through which the heat flows. In mathematical terms, it is expressed as: \[ q = -k \frac{dT}{dx} \]where:

• \( q \) is the heat flux, or the rate of heat transfer per unit area (W/m²).
• \( k \) is the thermal conductivity of the material (W/m·K), a constant that measures a material's ability to conduct heat.
• \( \frac{dT}{dx} \) is the temperature gradient, the rate of temperature change with respect to distance.

In our context of cooling glass, Fourier's Law allows us to determine the maximum allowable rate of heat transfer to prevent the glass from cracking due to extreme thermal gradients. It helps set an upper limit on the heat flux based on the material's thermal conductivity and the maximum permissible temperature gradient.

Newton's Law of Cooling describes the rate at which an exposed body changes temperature through radiation, or when it is subject to airflow, via convective heat transfer. This law is formulated as:\[ q = h (T_s - T_{air}) \]where:

• \( q \) is the heat transfer rate per unit area (W/m²).
• \( h \) is the convective heat transfer coefficient (W/m²·K), characterizing the efficiency of heat transfer from the surface to the surrounding fluid.
• \( T_s \) is the surface temperature of the object (°C).
• \( T_{air} \) is the ambient temperature, or the temperature of the air (°C).

For the glass cooling process, Newton's Law was used to set up an equation where the heat flux equals the product of the convection coefficient and the temperature difference between the glass surface and air. Solving this equation gives the air temperature needed to ensure the glass doesn't exceed the critical temperature gradient, thereby preventing stress and cracks.

Thermal Conductivity is a material property that signifies its ability to conduct heat. It quantifies the ease with which heat flows through a material when subjected to a temperature difference. High thermal conductivity indicates that the material is good at conducting heat, whereas low conductivity means poor heat transfer.The formula associated with thermal conductivity is embedded in Fourier’s Law of Heat Conduction:\[ q = -k \frac{dT}{dx} \]Here, the thermal conductivity \( k \) acts as a proportional constant linking the heat flux \( q \) to the temperature gradient \( \frac{dT}{dx} \).In our glass example, the thermal conductivity of the glass is \( 1.4 \) W/(m·K). This value signifies the rate at which heat can pass through the glass. High thermal conductivity helps in distributing heat quickly through the material, which is essential for minimizing temperature gradients and avoiding thermal stress as the glass is cooled.