91Ó°ÊÓ

The Biot Number (\( \text{Bi} \)) is a dimensionless quantity that helps in analyzing transient heat conduction. It demonstrates how heat transfers in a material between its surface and its interior.

For the lumped capacitance method to be applicable, the Biot Number must be less than 0.1. This condition indicates that the temperature difference within the object is negligible compared to the temperature difference between the object and its surrounding environment. Essentially, the object behaves as if it has a uniform temperature.

In our case, the Biot Number is calculated using the formula:
\[\text{Bi} = \frac{h L_c}{k}\]
where \( h \) is the convection coefficient, \( L_c \) is the characteristic length, and \( k \) is the thermal conductivity.

• Convection coefficient \( (h) = 8 \frac{\text{W}}{\text{m}^2\cdot\text{K}} \).
• Thermal conductivity \( (k) = 0.3 \frac{\text{W}}{\text{m}\cdot\text{K}} \).
• Characteristic length (for a long cylinder):
  \( L_c = \frac{D}{4} = \frac{0.03}{4} \) m.

Therefore, with a correctly measured Biot Number, we can confidently use the lumped capacitance method to predict how temperature changes over time.

Transient heat conduction deals with how heat moves through materials until they reach thermal equilibrium. Unlike steady-state conduction, where temperature remains constant over time, transient conduction involves changes in temperature.

The lumped capacitance method is a simplified approach used to analyze these transient processes when the Biot Number is small. It assumes that the entire object reaches equilibrium almost simultaneously, allowing us to simplify calculations.

• We track how the temperature \( T(t) \) of the rod changes over time as it cools.
• The equation for this method is:
  \[\frac{T(t)-T_\infty}{T_0-T_\infty} = e^{-\frac{h A_s}{\rho V c_p}t}\]
• \( T_\infty \) is the ambient temperature.

This formula provides the fraction of heat loss as exponential decay, revealing that the rod's temperature approaches the surrounding temperature gradually and predictably.

Therefore, transient heat conduction using the lumped capacitance method helps us estimate how a heated rod's temperature diminishes over a set duration, like the 3-minute window in this problem.

Newton's Law of Cooling highlights the rate at which an exposed body changes temperature through convection. Specifically, it states that the rate of heat transfer between the body and its environment is proportional to the difference in temperature between them.

Using the lumped capacitance method, Newton’s Law is expressed in energy balance terms. It helps us describe how the rod loses temperature due to exposure:

• Initial temperature \( T_0 \)
• Surrounding or ambient temperature \( T_\infty \)
• The temperature of the rod at any time \( T(t) \)

From the original equation:
\[\frac{T(t)-T_\infty}{T_0-T_\infty} = e^{-\frac{h A_s}{\rho V c_p}t}\]
we can isolate \( T(t) \) by rearranging the equation, giving us a clear calculation for the moment the rod might reach or maintain desired temperatures during cooling. This approach is particularly beneficial to predict temperatures before they cool beyond useful limits, ensuring optimal conditions for the rod's intended use.