An egg, which may be modeled as a 4 cm-diameter sphere with the thermal properties of water, is

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An egg, which may be modeled as a 4 cm-diameter sphere with...

An egg, which may be modeled as a $4 \mathrm{~cm}$-diameter sphere with the thermal properties of water, is initially at $5^{\circ} \mathrm{C}$ and is immersed in boiling water. Determine the temperature at the center of the egg after (i) 4 minutes. (ii) 7 minutes. Take the outside heat transfer coefficient as $1200 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$.

An egg, which may be modeled as a $4 \mathrm{~cm}$-diameter sphere with the thermal properties of water, is initially at $5^{\circ} \mathrm{C}$ and is immersed in boiling water. Determine the temperature at the center of the egg after
(i) 4 minutes.
(ii) 7 minutes.
Take the outside heat transfer coefficient as $1200 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$.

Key Concepts

Analytical Methods for Heat Conduction in Spherical Coordinates

When modeling heat conduction in spherical objects, the governing differential equations are expressed in spherical coordinates. Analytical techniques, such as separation of variables and series solutions, are used to solve the transient conduction equation with corresponding boundary conditions. These methods allow for a detailed analysis of temperature distributions within spherical geometries, capturing the effects of both internal conduction and external convection.

Convective Boundary Conditions

Convective boundary conditions describe the heat transfer interaction between a solid's surface and its surrounding fluid. They are quantified by the heat transfer coefficient, which represents the efficiency of convective heat transfer at the interface. In conduction problems with convection at the boundaries, Newton’s law of cooling is applied, whereby the heat flux at the surface is proportional to the difference between the surface temperature and the ambient fluid temperature.

Transient Heat Conduction

This concept involves the study of temperature changes within a body over time due to heat transfer. In transient heat conduction, the temperature at any point in the material changes as a function of both space and time. This requires solving the unsteady heat conduction equation, often using techniques such as separation of variables or numerical methods, to capture the evolution of temperature from an initial condition to eventually reaching thermal equilibrium.

Biot Number

The Biot number is a dimensionless parameter that helps determine whether a lumped system analysis is valid for a heat transfer problem. It is defined as the ratio of the internal thermal resistance within a body to the external thermal resistance due to convection. A small Biot number indicates that temperature gradients within the solid are negligible (making lumped capacitance analysis valid), while a larger Biot number means that internal conduction effects are significant and a spatially varying temperature profile must be considered.

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