A long rod, insulated to prevent heat loss along its sides,...

A long rod, insulated to prevent heat loss along its sides, is in perfect thermal contact with boiling water (at atmospheric pressure) at one end and with an ice-water mixture at the other (Fig. 17.31$)$ . The rod consists of a $1.00-\mathrm{m}$ section of copper (one end in boiling water) joined end to end to a length $L_{2}$ of steel (one end in the ice-water mixture). Both sections of the rod have crosssectional areas of 4.00 $\mathrm{cm}^{2}$ . The temperature of the copper-steel junction is 65.0" Cafter a steady state has been set up. (a) How much heat per second flows from the boiling water to the ice-water mixture? (b) What is the length $L_{2}$ of the steel section?

Key Concepts

Steady-State Heat Transfer

Steady-state heat transfer occurs when the temperature distribution within a system does not change with time, meaning that the heat entering any component equals the heat leaving it. In such a scenario, the temperature at the interfaces is constant and the heat flow remains constant throughout the conduction path. This concept simplifies the analysis of composite systems because it allows the assumption of a constant heat flow across each section.

Thermal Conduction

Thermal conduction is the process by which heat energy is transferred through a material due to a temperature gradient. It occurs when kinetic energy is passed from one molecule to another within the material, and it is described by properties like thermal conductivity. This concept is central to understanding how heat flows in solids, particularly in rods or bars that connect two reservoirs at different temperatures.

Fourier's Law

Fourier's Law quantitatively describes heat transfer by conduction. It states that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the area through which the heat flows. The law provides the relation Q/t = kA(?T)/L, where k is the thermal conductivity, A is the cross-sectional area, ?T is the temperature difference, and L is the length over which the temperature difference occurs.

Composite Conductor Systems

In many practical situations, the path for heat conduction involves more than one material, each with its own thermal conductivity. In composite conductor systems, the overall heat transfer is determined by the series combination of each section’s thermal resistance, since the same amount of heat must pass through each segment. Understanding how to sum these resistances is crucial in predicting the total heat flow through systems that are made up of different materials.

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