A can of beer at 300 K is placed in a refrigerator that...

A can of beer at $300 \mathrm{~K}$ is placed in a refrigerator that maintains an air temperature [ $\bar{o}$ of $277 \mathrm{~K}$. The can is $8 \mathrm{~cm}$ in diameter and $12 \mathrm{~cm}$ high. The outside heat transfer coefficient is $5 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. After 6 hours the beer is removed from the refrigerator and poured into a glass. Estimate the beer temperature.

Key Concepts

Heat Transfer Coefficient

The heat transfer coefficient is a parameter that quantifies the efficiency of convective heat transfer between a surface and a fluid. It reflects how quickly heat is transferred due to convection and is influenced by factors such as fluid velocity, viscosity, and the nature of the flow. In transient cooling problems, the heat transfer coefficient directly affects the rate at which an object approaches the ambient temperature.

Lumped Capacitance Method

This method simplifies the analysis of transient heat transfer by assuming that the object being cooled maintains a uniform temperature throughout its volume at any point in time. It is valid when the internal resistance to conduction is much smaller than the convective resistance at the surface, which is often checked using the Biot number. This method allows for the use of an exponential decay model to predict the temperature change over time.

Newton's Law of Cooling

Newton's Law of Cooling describes the rate of convective heat loss from a body as being proportional to the difference in temperature between the object and its surroundings. This principle forms the basis of many transient heat transfer calculations, where the temperature difference drives the cooling process, and it leads to the formulation of a first-order differential equation that governs the evolution of the object's temperature over time.

Surface Area to Volume Ratio

The surface area to volume ratio is a critical factor in determining the rate of cooling of an object. A higher ratio implies that there is more surface area available for convective heat transfer relative to the volume (and thus the thermal energy content) of the object, leading to faster cooling. This concept is important in various applications, including the design of cooling systems and understanding how geometric dimensions impact thermal performance.

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