An experiment is to be conducted to determine heat transfer...

An experiment is to be conducted to determine heat transfer coefficient on the surfaces of tomatoes that are placed in cold water at $7^{\circ} \mathrm{C}$. The tomatoes $\left(k=0.59 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right.$, $\alpha=0.141 \times 10^{-6} \mathrm{~m}^2 / \mathrm{s}, \rho=999 \mathrm{~kg} / \mathrm{m}^3, c_p=3.99 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$ ) with an initial uniform temperature of $30^{\circ} \mathrm{C}$ are spherical in shape with a diameter of 8 cm . After a period of 2 hours, the temperatures at the center and the surface of the tomatoes are measured to be $10.0^{\circ} \mathrm{C}$ and $7.1^{\circ} \mathrm{C}$, respectively. Using analytical one-term approximation method (not the Heisler charts), determine the heat transfer coefficient and the amount of heat transfer during this period if there are eight such tomatoes in water.

Key Concepts

Heat Transfer Coefficient

The heat transfer coefficient quantifies the convective heat transfer between a solid surface and a fluid. It describes the rate at which heat is transferred from the surface to the fluid (or vice versa) per unit area per unit temperature difference. This concept is central in determining how efficiently a body can exchange heat with its surroundings and is influenced by both the properties of the fluid and the nature of the flow near the surface.

Transient Heat Conduction

Transient heat conduction refers to the time-dependent process by which temperature within a body changes due to internal conduction and external thermal interactions. It involves solving partial differential equations that account for spatial and temporal variations in temperature, and is essential for analyzing non-steady-state cooling or heating processes in solids.

Analytical One-Term Approximation Method

The analytical one-term approximation method is a simplified approach to solving the governing equations of transient heat conduction. By retaining only the dominant term of the eigenfunction expansion in the solution, it provides a practical and reasonably accurate estimation of temperature distributions and heat transfer rates in objects. This technique is particularly useful when more complex methods (such as full series solutions or numerical simulations) are not feasible.

Biot Number

The Biot number is a dimensionless parameter that compares the rate of internal conduction within a body to the rate of convective heat transfer at its surface. It helps determine whether the temperature within the object can be assumed uniform (lumped system analysis) or if spatial variations must be taken into account. A low Biot number generally indicates that the lumped capacitance method is valid, while a high Biot number suggests significant internal temperature gradients.

Energy Balance in Heat Transfer

An energy balance in heat transfer involves accounting for the energy stored, gained, or lost by a system over time. It typically relates the amount of heat transferred to a change in temperature, mass, and the specific heat capacity of the material. This concept is fundamental in quantifying the total energy exchange during transient processes.

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