A person puts a few apples into the freezer at -15^∘ C to cool them quickly for guests who are a

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A person puts a few apples into the freezer at -15^∘ C to...

A person puts a few apples into the freezer at $-15^{\circ} \mathrm{C}$ to cool them quickly for guests who are about to arrive. Initially, the apples are at a uniform temperature of $20^{\circ} \mathrm{C}$, and the heat transfer coefficient on the surfaces is $8 \mathrm{~W} / \mathrm{m}^2$, ${ }^{\circ} \mathrm{C}$. Treating the apples as 9 -cm-diameter spheres and taking their properties to be $\rho=840 \mathrm{~kg} / \mathrm{m}^3, c_p=3.81 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, k=$ $0.418 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$, and $\alpha=1.3 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}$, determine the center and surface temperatures of the apples in 1 h . Also, determine the amount of heat transfer from each apple.

Key Concepts

Newton’s Law of Cooling (Convective Boundary Condition)

This law provides the relationship between the surface heat flux and the temperature difference between the object’s surface and the surrounding fluid. It is expressed through a heat transfer coefficient and is essential for setting the boundary conditions in transient heat conduction problems where convective cooling or heating is involved.

Energy Balance in Transient Processes

An energy balance involves accounting for the heat transfer into or out of a system over time to determine changes in the internal energy. In transient heat conduction problems, this approach is used to calculate the total amount of heat lost or gained by the object, providing insight into the cooling or heating process over a specified duration.

Thermal Diffusivity

Thermal diffusivity is a material property that combines thermal conductivity, density, and specific heat capacity to indicate how quickly temperature changes occur within a material. It is a key factor in predicting the rate of transient thermal response and is used in defining the Fourier Number, an important non-dimensional time parameter in heat conduction problems.

Transient Heat Conduction

This concept involves the study of temperature changes within a material over time as heat is conducted. It requires solving the unsteady heat conduction equation with appropriate initial and boundary conditions to understand how internal temperatures evolve during a cooling or heating process.

Conduction in Spherical Coordinates

When dealing with problems involving spherical objects, the heat conduction equation must be formulated in spherical coordinates to account for the geometry. Solutions often involve series expansions or eigenfunction methods that reflect the radial symmetry of the body, which is critical for accurately determining temperature distributions within spheres.

Biot Number

The Biot number is a dimensionless parameter that compares the internal thermal resistance of a body to the external thermal resistance due to convection. It helps determine whether the temperature gradients within the object are significant, guiding the decision to use either a lumped system analysis (if the Biot number is very small) or a distributed parameter model for transient conduction.

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