A hot dog can be considered to be a \(12-\mathrm{cm}-\mathrm{long}\) cylinder whose diameter is \(2 \mathrm{~cm}\) and whose properties are \(\rho=980 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.9 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.76 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\alpha=\) \(2 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\). A hot dog initially at \(5^{\circ} \mathrm{C}\) is dropped into boiling water at \(100^{\circ} \mathrm{C}\). The heat transfer coefficient at the surface of the hot dog is estimated to be \(600 \mathrm{~W} / \mathrm{m}^{2}\). K. If the hot dog is considered cooked when its center temperature reaches \(80^{\circ} \mathrm{C}\), determine how long it will take to cook it in the boiling water.

Understanding the Biot Number (Bi) is crucial in the context of unsteady-state heat transfer, as it determines the applicability of certain simplifications in heat transfer analysis. The Biot Number is a dimensionless quantity that compares the resistance to heat conduction within an object to the resistance to heat transfer across the boundary (convective heat transfer).
In our exercise involving the cooking of a hot dog, the Biot Number is calculated using the formula:
\[\begin{equation} Bi = \frac{hL_c}{k} \end{equation}\]
Here,

• \(h\) is the heat transfer coefficient (600 W/m²K),
• \(L_c\) is the characteristic length, which for a cylindrical object such as a hot dog, is half of its diameter (0.01 m),
• and \(k\) is the thermal conductivity of the hot dog (0.76 W/mK).

Biot Number Importance:

• A low Biot Number (Bi < 0.1) implies that the temperature gradient within the object is negligible, and the 'lumped capacitance method' can be used.
• A higher Biot Number indicates that the temperature varies significantly within the object, and a more detailed analysis is required.

For our hot dog, the Biot Number calculation yields 7.89, which is larger than 0.1, suggesting that the temperature distribution within the hot dog cannot be assumed uniform. Consequently, we must lean towards using unsteady-state heat transfer equations for an accurate approximation of the cooking time.

When analyzing heat transfer in cylindrical objects, like our hot dog example, it's essential to apply the equations that account for the shape's geometry. Cylindrical coordinates, as opposed to Cartesian coordinates, are better suited for these problems since they align with the object's symmetry.
\[\begin{equation} T(t) - T_i = (T_∞ - T_i)[1 - 2 * \sum (\exp(-\lambda_n²\alpha t) * (\lambda_n * \sin(\lambda_n) - \cos(\lambda_n)) / (\lambda_n² + 1))] \end{equation}\]
The heat conduction in cylindrical coordinates is described with a longer and more complicated expression than in Cartesian coordinates. The equation uses a sum of terms involving eigenvalues (\(lambda_n\)) — these are determined from the boundary conditions of the object and are related to the 'shape factor' in conduction problems.
Challenges and Solutions:

• Add complexity: Cylindrical problems often involve infinite series and Bessel functions, which can be challenging to solve analytically.
• Approximation: In practice, using the first term (\(n = 1\)) of the series provides an approximation that can simplify the solution while still offering an accurate enough result for practical purposes like cooking a hot dog.

In the context of our hot dog, setting up the unsteady-state heat transfer equation in cylindrical coordinates allows us to quantify how the temperature at the center changes over time when the hot dog is dropped into boiling water.

Time-dependent (or transient) heat transfer problems can be quite intricate and generally require elaborate mathematical methods to solve. Since exact solutions are often not feasible due to the complexity, we use approximate solutions, such as truncating an infinite series to its first term for practical purposes.
In our hot dog scenario, we seek to determine how long it will take for the center of the hot dog to reach 80°C. We approximate the time-dependent part of the heat transfer equation by considering only the first term in the series. This leads to a vastly simplified equation:
\[\begin{equation} t = \left[\frac{(T(t)-T_i)}{(T_∞-T_i)} - 2 * \frac{(\lambda_1 * \sin(\lambda_1) - \cos(\lambda_1))}{(\lambda_1²+1)}\right] / (\lambda_1²\alpha) \end{equation}\]
This approach is effective because the higher-order terms in the solution typically diminish quickly. Therefore, by using only the dominant first term, we can achieve a reasonable approximation of the actual time needed for the hot dog to cook.
To finalize our calculation, we replace the variables with the provided values – initial temperature, desired center temperature, water temperature, the first root of the eigenvalue equation, and the thermal diffusivity – and solve for \(t\). The result is approximately 208.24 seconds, which gives us a practical estimate of the cooking time. This method demonstrates how an approximate solution can provide useful and actionable results, especially in the context of engineering and physics problems where detail to the second decimal is not strictly necessary.