A cooling unit for a new freezer has an inner surface area of 6.0 m^2, and is bounded by walls 1

step 1 We can write thermal conductivity as a QL over a temperature heated low, a temperature over time

A cooling unit for a new freezer has an inner surface area...

A cooling unit for a new freezer has an inner surface area of $6.0 \mathrm{m}^{2},$ and is bounded by walls 12 $\mathrm{cm}$ thick with a thermal conductivity of 0.050 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ . The inside must be kept at $-10^{\circ} \mathrm{C}$ in a room that is at $20^{\circ} \mathrm{C}$ . The motor for the cooling unit must run no more than 15$\%$ of the time. What is the minimum power requirement of the cooling motor?

Key Concepts

Fourier's Law of Heat Conduction

This fundamental principle of heat transfer states that the rate at which heat is conducted through a material is proportional to the temperature difference across it and the area through which the heat flows, and inversely proportional to the thickness of the material. It provides the mathematical basis for calculating the heat transfer in any system where conduction is the dominant mode of heat flow.

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material that measures its ability to conduct heat. Materials with high thermal conductivity allow heat to pass through them quickly, while those with low thermal conductivity act as insulators, slowing down the rate of heat transfer.

Thermal Resistance

Thermal resistance quantifies how effectively a material or a composite system resists heat flow. It is influenced by both the thickness of the material and its thermal conductivity, making it a crucial concept for designing systems that require precise thermal management, such as insulation in freezers or buildings.

Heat Transfer Rate Calculation

The calculation of the heat transfer rate involves determining how much heat energy is transferred per unit time through a material or system. This rate is essential for engineering applications where maintaining a specific temperature difference is critical, such as in cooling units or heat exchangers.

Duty Cycle

The duty cycle is the fraction of time a device, such as a cooling motor, is operating relative to a complete cycle. This concept is important for energy management because it allows engineers to calculate the average power consumption over time, ensuring that the system meets performance requirements while staying within energy efficiency guidelines.

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