A picture window has dimensions of 1.40 m ×2.50 m and is made of glass 5.20 mm thick. On a winte

step 1 First, we need to calculate the thermal resistance for glass before we can proceed further. The

A picture window has dimensions of 1.40 m ×2.50 m and is...

A picture window has dimensions of $1.40 \mathrm{m} \times 2.50 \mathrm{m}$ and is made of glass 5.20 $\mathrm{mm}$ thick. On a winter day, the outside temperature is $-20.0^{\circ} \mathrm{C}$ , while the inside temperature is a comfortable $19.5^{\circ} \mathrm{C}$ (a) At what rate is heat being lost through the window by conduction? (b) At what rate would heat be lost through the window if you covered it with a 0.750 -mm-thick layer of paper (thermal conductivity 0.0500)?

Key Concepts

Fourier’s Law of Heat Conduction

This fundamental principle states that the rate of heat transfer through a material is proportional to the negative gradient of the temperature and the area through which the heat flows. It is expressed mathematically as Q/t = -kA (?T/?x), where Q/t is the rate of heat transfer, k is the material's thermal conductivity, A is the cross-sectional area, and ?T/?x is the temperature gradient. This law underpins the calculation of heat conduction in any homogeneous or layered medium.

Thermal Conductivity

Thermal conductivity is a material property that quantifies its ability to conduct heat. Materials with high thermal conductivity allow heat to pass through them more easily, while those with low conductivity act as insulators. It plays a crucial role in determining the rate of heat loss by conduction through various materials and layers, as seen in calculations involving different substances like glass or paper.

Temperature Gradient

The temperature gradient represents the change in temperature per unit distance within a material. It is the driving force for conductive heat transfer, with a larger gradient leading to a higher rate of heat flow. Understanding the temperature difference across a material's thickness is essential for applying Fourier’s Law and computing the associated heat loss.

Composite Conduction and Thermal Resistance

In systems that involve multiple layers of different materials, the concept of thermal resistance becomes useful. Each layer offers a certain resistance to heat flow, analogous to electrical resistors in series. The overall rate of heat transfer can be determined by summing the thermal resistances of the individual layers, making it an essential concept when evaluating heat loss through composite walls or combined materials.

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