Oranges of 2.5 -in-diameter (k=0.26 Btu / h ·ft

Step 1:u000ALetu0027s identify the given information:u000Au002D Orange diameter u003D 2.5 in u003D 2.5/12 u003D 0.2083 ftu000Au002D

Oranges of 2.5 -in-diameter $\left(k=0.26 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.$ and $\alpha=1.4 \times 10^{-6} \mathrm{ft}^2 / \mathrm{s}$ ) initially at a uniform temperature of $78^{\circ} \mathrm{F}$ are to be cooled by refrigerated air at $25^{\circ} \mathrm{F}$ flowing at a velocity of $1 \mathrm{ft} / \mathrm{s}$. The average heat transfer coefficient between the oranges and the air is experimentally determined to be $4.6 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^2$. ${ }^{\circ} \mathrm{F}$. Determine how long it will take for the center temperature of the oranges to drop to $40^{\circ} \mathrm{F}$. Also, determine if any part of the oranges will freeze during this process.

Key Concepts

Transient Heat Conduction in Solids

This concept deals with how temperature within a solid object changes over time due to internal energy conduction and external conditions. In problems such as cooling of a spherical object, the unsteady heat transfer equation is used to predict the spatial and temporal temperature distribution, taking into account the material’s thermal properties and initial as well as boundary conditions.

Spherical Geometry and Its Analytical Solutions

When the object under consideration is a sphere, its unique geometry must be taken into account in the analysis. Analytical solutions for transient conduction in spherical coordinates often involve series solutions or eigenvalue approaches, capturing how the temperature varies from the center to the surface over time. This is crucial for determining the time required for the internal temperature to reach a specified value.

Convective Heat Transfer and Boundary Conditions

The convective heat transfer coefficient represents the rate at which heat is transferred from the surface of the object to the surrounding fluid. It sets the boundary condition for the conduction problem and is central to calculating the cooling rate under forced convection. Understanding this concept is essential in linking the external environment’s influence with the internal transient conduction process.

Biot Number and the Lumped Capacitance Method

The Biot number is a dimensionless parameter that compares the internal conductive resistance to the convective resistance at the surface of the object. It helps determine whether the temperature within the object can be assumed uniform (lumped system analysis) or if spatial gradients must be considered. This assessment is key in selecting an appropriate analytical or numerical method for solving the transient heat transfer problem.

Phase Change and Freezing Conditions

This concept involves evaluating whether the object's temperature will drop below the material’s freezing point during the cooling process. Understanding phase change is important because if any region of the object reaches the freezing temperature, latent heat effects and the physical change of state must be considered. In heat transfer analyses, ensuring conditions avoid or predict phase change is critical for proper interpretation of the cooling process.

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