A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end (Fig. 19–35). The copper end is placed in a furnace maintained at a constant temperature of 205°C. The aluminum end is placed in an ice bath held at a constant temperature of 0.0°C. Calculate the temperature at the point where the two rods are joined.

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Understand the problem: Heat flows through the copper and aluminum rods until thermal equilibrium is reached at the junction. The heat transfer rate through both rods must be equal at the junction since the system is in steady state.

Write the heat transfer equation for each rod using Fourier's law of heat conduction: \( Q/t = k \cdot A \cdot (T_{hot} - T_{junction}) / L \) for the copper rod and \( Q/t = k \cdot A \cdot (T_{junction} - T_{cold}) / L \) for the aluminum rod, where \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, \( L \) is the length, and \( T \) represents temperature.

Set the heat transfer rates equal to each other because the heat flow through the copper rod equals the heat flow through the aluminum rod at the junction: \( k_{Cu} \cdot (T_{hot} - T_{junction}) = k_{Al} \cdot (T_{junction} - T_{cold}) \).

Rearrange the equation to solve for \( T_{junction} \): \( T_{junction} = \frac{k_{Cu} \cdot T_{hot} + k_{Al} \cdot T_{cold}}{k_{Cu} + k_{Al}} \).

Substitute the given values: \( T_{hot} = 205 \ \degree C \), \( T_{cold} = 0 \ \degree C \), \( k_{Cu} = 385 \ \text{W/m·K} \), and \( k_{Al} = 205 \ \text{W/m·K} \). Simplify the expression to find \( T_{junction} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Thermal Conductivity

Thermal conductivity is a material property that indicates how well a substance can conduct heat. It is defined as the amount of heat that passes through a unit area of the material per unit time for a temperature difference of one degree. Different materials have different thermal conductivities; for instance, copper has a higher thermal conductivity than aluminum, meaning it can transfer heat more efficiently.

Heat Transfer

Heat transfer is the process by which thermal energy moves from one object or substance to another due to a temperature difference. In this scenario, heat flows from the hot end of the copper rod to the cold end of the aluminum rod. The rate of heat transfer depends on the thermal conductivities of the materials and the temperature gradient between them.

Steady-State Condition

A steady-state condition occurs when the temperature at any point in a system does not change over time, despite the ongoing heat transfer. In this problem, once the system reaches steady-state, the temperature at the junction of the copper and aluminum rods will stabilize, allowing us to calculate the temperature at that point based on the heat conducted through each rod.