A watermelon initially at 35^∘ C is to be cooled by dropping it into a lake at 15^∘ C. After 4 h

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A watermelon initially at 35^∘ C is to be cooled by dropping...

A watermelon initially at $35^{\circ} \mathrm{C}$ is to be cooled by dropping it into a lake at $15^{\circ} \mathrm{C}$. After 4 h and 40 min of cooling, the center temperature of watermelon is measured to be $20^{\circ} \mathrm{C}$. Treating the watermelon as a $20-\mathrm{cm}$-diameter sphere and using the properties $k=0.618 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}, \alpha=0.15 \times$ $10^{-6} \mathrm{~m}^2 / \mathrm{s}, \rho=995 \mathrm{~kg} / \mathrm{m}^3$, and $c_p=4.18 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, determine the average heat transfer coefficient and the surface temperature of watermelon at the end of the cooling period.

Key Concepts

Transient Heat Conduction in Solids

Transient heat conduction addresses how temperature within a body changes over time due to spatial and temporal gradients in temperature. In problems involving cooling of solids, such as a sphere, the time-dependent temperature distribution is solved using methods like separation of variables and eigenfunction expansions, which capture the internal resistance to heat flow.

Biot Number

The Biot number is a dimensionless parameter that compares the internal thermal resistance of a body to the external convective heat transfer resistance. It helps determine whether an object can be treated using lumped capacitance methods (if the Biot number is small) or if a full transient conduction analysis is required. This concept is important in understanding the temperature gradients inside a cooling object.

Thermal Properties

Thermal properties such as thermal conductivity, thermal diffusivity, density, and specific heat capacity characterize how a material conducts and stores heat. These properties directly influence the rate of temperature change within a body and are essential inputs in modeling both conduction and convection processes during transient thermal analysis.

Newton's Law of Cooling

Newton's law of cooling relates the rate of heat loss of a body to the difference in temperature between the body and its surrounding environment. This law is fundamental in determining the convective heat transfer from the surface of an object by linking the surface temperature, the ambient temperature, and the heat transfer coefficient.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient is a parameter that quantifies the rate at which heat is transferred between a surface and a fluid in motion. In scenarios where an object is immersed in a cooling fluid, this coefficient is critical to determining the rate of convective cooling and is used in conjunction with the surface area and temperature difference in energy balance equations.

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