Effect of a Window in a Door. A carpenter builds a solid...

Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions $2.10 \mathrm{~m} \times 0.91 \mathrm{~m} \times 6.0 \mathrm{~cm}$. Its thermal conductivity is $k=0.120 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} .$ The air films on the inner and outer surfaces of the door have the same combined thermal resistance as an additional $1.9 \mathrm{~cm}$ thickness of solid wood. The inside air temperature is $21.0^{\circ} \mathrm{C},$ and the outside air temperature is $-5.0^{\circ} \mathrm{C}$. (a) What is the rate of heat flow through the door? (b) By what factor

Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions $2.10 \mathrm{~m} \times 0.91 \mathrm{~m} \times 6.0 \mathrm{~cm}$. Its thermal conductivity is $k=0.120 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} .$ The air films on the inner and outer surfaces of the door have the same combined thermal resistance as an additional $1.9 \mathrm{~cm}$ thickness of solid wood. The inside air temperature is $21.0^{\circ} \mathrm{C},$ and the outside air temperature is $-5.0^{\circ} \mathrm{C}$.
(a) What is the rate of heat flow through the door? (b) By what factor

Key Concepts

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material that measures its ability to conduct heat. It establishes how quickly heat energy passes through a substance when there is a temperature difference across it. In heat transfer problems, higher thermal conductivity means faster heat transfer, while lower conductivity implies better insulation.

Fourier's Law of Heat Conduction

Fourier's Law is the fundamental principle for conduction heat transfer, stating that the heat flow rate through a material is proportional to the cross-sectional area, the thermal conductivity, and the temperature difference, and inversely proportional to the thickness of the material. It provides the basic formula used to calculate the rate at which heat moves through solids under a temperature gradient.

Thermal Resistance

Thermal resistance quantifies a material’s opposition to heat flow, analogous to electrical resistance in circuits. It depends on the material’s thickness and thermal conductivity and is crucial for determining the overall ease or difficulty of maintaining a temperature difference across an object. In layered or composite systems, individual thermal resistances add together to give the effective resistance.

Composite Thermal Systems

Composite thermal systems involve multiple layers or components through which heat is conducted. In such systems, the total thermal resistance is found by summing the resistances of each layer. This approach is essential when dealing with materials like a door that includes both solid material and additional resistive layers (like air films) that impact the overall heat transfer rate.

Steady-State Heat Transfer

Steady-state heat transfer assumes that the temperature distribution and the rate of heat flow remain constant over time. This condition allows for the use of simplified models, such as Fourier’s Law, where transient effects and energy storage within the materials are neglected, making the analysis of conduction through structures like doors and insulated walls more straightforward.

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