Consider a 1000-W iron whose base plate is made of 0.5...

Consider a 1000-W iron whose base plate is made of 0.5 -cm-thick aluminum alloy 20211 -T6 ( $\rho=2770 \mathrm{~kg} / \mathrm{m}^3$, $c_p=875 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, \alpha=7.3 \times 10^{-5} \mathrm{~m}^2 / \mathrm{s}$ ). The base plate has a surface area of $0.03 \mathrm{~m}^2$. Initially, the iron is in thermal equilibrium with the ambient air at $22^{\circ} \mathrm{C}$. Taking the heat transfer coefficient at the surface of the base plate to be $12 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$ and assuming 85 percent of the heat generated in the resistance wires is transferred to the plate, determine how long it will take for the plate temperature to reach $140^{\circ} \mathrm{C}$. Is it realistic to assume the plate temperature to be uniform at all times?

Key Concepts

Thermal Mass

Thermal mass is a measure of an object's ability to store thermal energy, determined by its mass and specific heat capacity. It is a crucial factor in transient thermal analysis because a higher thermal mass means that more energy is required to change the object's temperature by a given amount.

Lumped Capacitance Method

This method assumes that the object being heated or cooled has a uniform temperature throughout its volume at any given time, meaning there are no internal temperature gradients. This simplification enables us to use an ordinary differential equation to model the transient temperature response, provided that the Biot number is sufficiently small.

Convective Heat Transfer

Convective heat transfer is the mechanism by which heat is exchanged between the surface of a solid and the surrounding fluid, typically described by Newton's law of cooling. The rate of heat transfer depends on the heat transfer coefficient, the surface area, and the temperature difference between the solid surface and the fluid.

Heat Transfer Coefficient

The heat transfer coefficient quantifies the effectiveness of convective heat transfer at the interface between a solid and a fluid. It serves as a proportionality constant in Newton's law of cooling, linking the heat flux from the surface to the temperature difference driving the heat transfer.

Thermal Diffusivity

Thermal diffusivity characterizes how quickly heat spreads through a material and is defined as the ratio of the thermal conductivity to the product of density and specific heat capacity. It plays a significant role in determining the time scales over which temperature changes occur within a material.

Assumption of Uniform Temperature

Assuming a uniform temperature throughout a body simplifies thermal analysis by neglecting internal temperature gradients. However, this assumption is only realistic when the internal resistance to heat conduction is much smaller than the resistance to heat transfer at the surface, often evaluated using the Biot number.

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