Consider a plate of thickness 1 in , a long cylinder of...

Consider a plate of thickness 1 in , a long cylinder of diameter 1 in , and a sphere of diameter 1 in , all initially at $400^{\circ} \mathrm{F}$ and all made of bronze $\left(k=15.0 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.$ and $\alpha=0.333 \mathrm{ft}^2 / \mathrm{h}$ ). Now all three of these geometries are exposed to cool air at $75^{\circ} \mathrm{F}$ on all of their surfaces, with a heat transfer coefficient of $7 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^2 \cdot{ }^{\circ} \mathrm{F}$. Determine the center temperature of each geometry after 5, 10, and 30 min . Explain why the center temperature of the sphere is always the lowest.

Key Concepts

Dimensionless Analysis (Fourier Number)

The Fourier number is another dimensionless parameter that characterizes the transient heat conduction process, representing the ratio of diffusive transport rate to the storage rate of thermal energy. It provides a means of scaling time and geometry to predict temperature changes in different bodies, allowing comparisons across objects with different shapes and sizes under similar thermal conditions.

Transient Heat Conduction

This concept involves studying how temperature within a body varies with time as heat is transferred within it and out of its surfaces. When objects are suddenly exposed to a boundary condition different from their initial state, the temperature distribution changes over time, and analytical or numerical solutions are used to predict the center and overall temperature evolution.

Geometric Effects on Heat Transfer

The geometry of an object—its shape and size—strongly influences how heat is conducted internally and convected externally. Different shapes, such as plates, cylinders, and spheres, have different surface area to volume ratios and thermal path lengths, which in turn affect the rate at which heat is lost or gained and the temperature gradients that develop within them.

Surface Area to Volume Ratio

This ratio is critical in transient conduction because it governs how much surface is available for convective heat transfer relative to the amount of material that must be heated or cooled. A high surface area to volume ratio allows for faster heat exchange with the environment, which is why the sphere, with its relatively high ratio compared to the other shapes, tends to reach a lower center temperature more quickly.

Biot Number

The Biot number is a dimensionless parameter that compares the rate of conductive heat transfer within a body to the convective heat transfer across its surface. A low Biot number justifies the use of lumped capacitance methods, while higher values necessitate more detailed spatial analysis. Understanding the Biot number in each geometry helps in assessing how internal temperature gradients develop during the cooling process.

Please give Ace some feedback