A father and son conducted the following simple experiment on a hot dog, which measured 12.5 cm in length and 2.2 cm in diameter. They inserted one food thermometer into the midpoint of the hot dog, and another one was placed just under the skin of the hot dog. The temperatures of the thermometers were monitored until both thermometers read 20°C, which is the ambient temperature. The hot dog was then placed in 94°C boiling water, and after exactly 2 minutes, they recorded the center temperature and the skin temperature of the hot dog to be 59°C and 88°C, respectively. Assuming the following properties for the hot dog: rho = 980 kg/m3 and cp = 3900 J/kg·K, and using transient temperature charts, determine (a) the thermal diffusivity of the hot dog, (b) the thermal conductivity of the hot dog, and (c) the convection heat transfer coefficient.

To determine the thermal diffusivity, thermal conductivity, and convection heat transfer coefficient of the hot dog, we need to make certain assumptions and apply heat transfer principles, including transient conduction and convection.

(a) **Thermal diffusivity (α)**: The thermal diffusivity is a measure of the rate at which heat diffusion occurs within a material. It can be calculated from the transient temperature response of the material. However, using transient temperature charts (like Heisler charts) requires non-dimensional time and temperature parameters. We don't have enough information to provide an exact value, but I can describe the methodology used to determine it:

You would compare the time it took for the hot dog to change in temperature to the characteristic time for heat conduction (Fourier's number). Since we have a cylinder, the characteristic length used is the radius of the cylinder (half the diameter). First, you calculate the Fourier number:

\[ \text{Fo} = \frac{\alpha t}{(r/2)^2} \]

Where: - \( \alpha \) is the thermal diffusivity (what we are trying to find), - \( t \) is the time (120 seconds), - \( r \) is the diameter of the hot dog (2.2 cm).

However, since \( \alpha \) is unknown, you would typically look at a Heisler chart for a cylinder, find the dimensionless temperature at the center of the cylinder (\( \Theta \)) that corresponds to the ratio of the temperature change at the center of the object to the overall temperature change, and read the dimensionless time (Fourier number) from the chart. You then rearrange the Fourier equation to solve for \( \alpha \).

(b) **Thermal conductivity (k)**: Once thermal diffusivity is known, thermal conductivity can be determined using the relationship:

\[ k = \alpha \cdot \rho \cdot c_p \]

Where \( \rho \) is the density and \( c_p \) is the specific heat capacity given.

(c) **Convective heat transfer coefficient (h)**: To find the heat transfer coefficient, you'd use Newton's law of cooling:

\[ Q = hA(T_s - T_\infty) \]

Where: - \( Q \) is the heat transfer rate, - \( h \) is the convective heat transfer coefficient ***(what we are trying to find)***, - \( A \) is the surface area through which heat is being transferred, - \( T_s \) is the surface temperature of the hot dog, - \( T_\infty \) is the temperature of the boiling water.

In this experiment, you'd have to use the initial and final temperatures to calculate the energy change and thus the rate of heat transfer. Using the density and specific heat capacity given, you could estimate the heat transfer into the hot dog, but without specific times and a thorough analysis, we cannot provide an exact value for \( h \). The temperature of the skin and the center would be used in conjunction with the transient models for a cylinder to estimate \( h \).

Extra: In heat transfer analysis, the thermal diffusivity of materials represents their ability to conduct thermal energy relative to their ability to store thermal energy. It plays a crucial role in transient heat transfer situations, where temperature gradients exist within a material and change over time. The larger the thermal diffusivity, the faster the heat spreads throughout the material.

The thermal conductivity, on the other hand, is a property that indicates how well a material can conduct heat. It does not account for the material's density or specific heat capacity.

The convective heat transfer coefficient is a measure of the convective heat transfer between a surface and a fluid flowing over it. It is influenced by numerous factors, including the properties of the fluid, the flow conditions, and the nature of the surface.

For students, it's important to understand that these are all properties that describe how heat moves within and between materials, and they greatly affect how we can apply materials in real-world situations for thermal management, such as in cooking, industrial processes, and electronics cooling.