For heat transfer purposes, an egg can be considered to be a \(5.5-\mathrm{cm}\)-diameter sphere having the properties of water. An egg that is initially at \(8^{\circ} \mathrm{C}\) is dropped into the boiling water at \(100^{\circ} \mathrm{C}\). The heat transfer coefficient at the surface of the egg is estimated to be \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the egg is considered cooked when its center temperature reaches \(60^{\circ} \mathrm{C}\), determine how long the egg should be kept in the boiling water. Solve this problem using analytical one-term approximation method (not the Heisler charts).

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations to compare the rate of heat conduction within an object to the rate of heat transfer across the object's boundary surface.

The formula for the Biot number is given by:
\[\begin{equation}Bi = \frac{h \cdot L_c}{k}\end{equation}\]
where \(h\) is the heat transfer coefficient, \(L_c\) is the characteristic length, and \(k\) is the thermal conductivity of the material. In the case of the egg, since it is assumed to have the properties of water, \(k\) is taken as the thermal conductivity of water. A high Biot number signifies that the conduction resistance within the object is significant compared to the convective heat transfer at the surface. Conversely, a low Biot number indicates that the internal resistance to conduction is small relative to the surface convection resistance.

In the solved example, a Biot number of 12.28 suggests a considerable internal conduction resistance and necessitates the use of methods suitable for objects with non-negligible internal temperature gradients.

The Fourier number (Fo) is another dimensionless number that arises in the analysis of heat conduction. It is used to estimate the time scale of heat conduction relative to the time scale of heat storage.

The Fourier number can be calculated by:
\[\begin{equation}Fo = \frac{\alpha \cdot t}{L_c^2}\end{equation}\]
where \(\alpha\) is thermal diffusivity, \(t\) is time, and \(L_c\) is the characteristic length. Since \(\alpha\) represents the ability of the material to conduct thermal energy relative to its ability to store it, the Fourier number effectively compares heat conduction over time against the thermal mass of the object.

In the context of the egg-cooking problem, the Fourier number gauges how quickly the egg reaches the desired temperature considering its size and the thermal properties of water, which in this situation allows us to deduce the necessary cooking time for the egg.

Thermal diffusivity (\(\alpha\)) is a physical property that measures a material's ability to conduct heat relative to its ability to store thermal energy. It is defined as the ratio of the material's thermal conductivity (\(k\)) to its density (\(\rho\)) and specific heat capacity (\(c_p\)). The formula is given by:
\[\begin{equation}\alpha = \frac{k}{\rho \cdot c_p}\end{equation}\]
High thermal diffusivity means that heat spreads through the material rapidly, whereas low thermal diffusivity means that heat is absorbed and released more slowly. In practical applications, such as cooking an egg, understanding thermal diffusivity helps in determining how long to apply heat for the object to reach a specific temperature.

Thermal diffusivity is a critical factor in calculating the time, as shown in the exercise, where it is used to estimate how long the egg must be kept in boiling water for it to cook adequately.

The analytical one-term approximation method is a mathematical approach used to approximate the temperature distribution in an object experiencing heat transfer. This method is highly effective for simple geometries and boundary conditions, where an exact solution is complex to obtain.

To apply this method, one starts with the temperature ratio (\(\Theta\)) which is a normalized measure of the temperature difference at a point in the object. The technique involves determining the first term in a series solution to provide a reasonably accurate estimate of the temperature:
\[\begin{equation}\Theta = \frac{T - T_{\infty}}{T_i - T_{\infty}}\end{equation}\]
Here, \(T\) is the temperature at a point within the object, \(T_{\infty}\) is the ambient temperature, and \(T_i\) is the initial temperature.

In our egg-cooking scenario, using the analytical one-term approximation method allows for the calculation of the cooking time without resorting to more complex numerical methods or graphical solutions such as the Heisler charts. Through this method, we can efficiently obtain an approximation of how long it takes for the egg's center to reach the desired temperature.