A 2-cm-diameter plastic rod has a thermocouple inserted to...

A 2-cm-diameter plastic rod has a thermocouple inserted to measure temperature at the center of the rod. The plastic rod $\left(\rho=1190 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=1465 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $k=0.19$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ ) was initially heated to a uniform temperature of $70^{\circ} \mathrm{C}$ and allowed to be cooled in ambient air at $25^{\circ} \mathrm{C}$. After $1388 \mathrm{~s}$ of cooling, the thermocouple measured the temperature at the center of the rod to be $30^{\circ} \mathrm{C}$. Determine the convection heat transfer coefficient for this process. Solve this problem using the analytical one-term approximation method.

Key Concepts

Biot Number

The Biot number is a dimensionless parameter defined as the ratio of the conduction resistance within a solid to the convective resistance at its surface. It serves as an indicator of whether the temperature within the solid can be assumed uniform (lumped system analysis) or if spatial gradients must be considered, and is crucial when applying analytical solutions in transient heat transfer problems.

Thermal Diffusivity

Thermal diffusivity is a material property defined as the ratio of thermal conductivity to the product of density and specific heat capacity. It represents how quickly a material can respond to changes in temperature, influencing the rate at which heat diffuses through a solid during transient conduction.

One-Term Approximation Method

The one-term approximation method is an analytical technique used in transient conduction problems to simplify the infinite series solution by retaining only the dominant term. This approach yields a good estimate for the temperature field when the conditions justify the dominance of the first eigenvalue, thereby facilitating the calculation of parameters such as the convective heat transfer coefficient.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient (h) quantifies the rate of heat transfer between a solid surface and a fluid in motion. It is a key parameter in boundary conditions for transient conduction problems, linking the convective loss from the surface to the temperature difference between the surface and the surrounding ambient fluid.

Transient Heat Conduction

Transient heat conduction deals with the analysis of how temperature within a solid changes over time as a result of thermal energy exchange. It typically involves solving the heat equation with initial and boundary conditions, capturing how internal energy storage and spatial temperature gradients evolve during processes like cooling or heating.

Please give Ace some feedback