A 65-kg beef carcass (k=0.47 W / m · ^∘ C. and α=0.13 ×10^-6 m^2 / s ) initially at a uniform

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A 65-kg beef carcass (k=0.47 W / m · ^∘ C. and α=0.13...

A $65-\mathrm{kg}$ beef carcass $\left(k=0.47 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right.$ and $\alpha=0.13 \times 10^{-6} \mathrm{~m}^2 / \mathrm{s}$ ) initially at a uniform temperature of $37^{\circ} \mathrm{C}$ is to be cooled by refrigerated air at $-10^{\circ} \mathrm{C}$ flowing at a velocity of $1.2 \mathrm{~m} / \mathrm{s}$. The average heat transfer coefficient between the carcass and the air is $22 \mathrm{~W} / \mathrm{m}^2$. ${ }^{\circ} \mathrm{C}$. Treating the carcass as a cylinder of diameter 24 cm and height 1.4 m and disregarding heat transfer from the base and top surfaces, determine how long it will take for the center temperature of the carcass to drop to $4^{\circ} \mathrm{C}$. Also, determine if any part of the carcass will freeze during this process.

Key Concepts

Fourier Number

The Fourier number is a dimensionless grouping that represents the ratio of heat conduction rate to the rate of thermal energy storage in the material. It is defined as ?t/L², where ? is the thermal diffusivity, t is time, and L is a characteristic length. The Fourier number provides insight into how quickly temperature changes propagate within the object, helping to estimate the cooling time required for the temperature at a specific location to drop to a desired level.

Eigenfunction Solutions in Cylindrical Coordinates

For transient conduction problems in bodies with non?simple geometries, such as a cylinder, the solution often involves representing the temperature field as a series expansion in eigenfunctions (or modes). These solutions typically use Bessel functions to handle the radial coordinate. The dominant (first) term in the series often provides a good approximation for calculating the center temperature evolution over time.

Phase Change and Freezing Point Considerations

When cooling food products, it is important to determine whether the temperature in any part of the object drops below its freezing point, leading to phase change and potential quality issues. This concept involves considering the cooling boundary conditions along with the temperature distribution developed through conduction. In this context, assessing the likelihood of freezing involves comparing the local temperatures reached during cooling to the known freezing point of the material.

Convective Boundary Conditions

In heat transfer problems involving an object exposed to a fluid environment, the surface of the object loses or gains heat by convection. The convective boundary condition relates the heat flux at the surface to the difference between the object’s surface temperature and the ambient fluid temperature, using the convective heat transfer coefficient. This condition is crucial because it determines how effectively the external cooling environment influences the temperature distribution within the object.

Transient Conduction

This concept involves the study of how temperature within an object changes with time due to internal heat conduction. It is governed by the heat equation, which takes into account the material’s thermal properties and describes the evolution of temperature fields in time and space. In problems like cooling a carcass, solving the transient conduction equation helps determine how the internal (center) temperature decreases over time.

Biot Number

The Biot number is a dimensionless parameter that compares the internal thermal resistance of a body to the external convective resistance. Its value helps determine whether temperature gradients within the object are significant. A small Biot number (typically less than 0.1) indicates that the object can be approximated as having a uniform temperature (lumped capacitance approach), while larger values require a more detailed spatial analysis of the temperature field.

Thermal Properties of Materials

Understanding material properties such as thermal conductivity and thermal diffusivity is essential in heat transfer analysis. Thermal conductivity indicates how well a material conducts heat, while thermal diffusivity reflects the rate at which temperature disturbances propagate through the material. These properties directly affect the rate at which the carcass cools and are crucial parameters in the transient conduction analysis.

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