A bar of gold (A u) is in thermal contact with a bar of...

A bar of gold $(A u)$ is in thermal contact with a bar of silver $(\mathrm{Ag})$ of the same length and area (Fig. $P 20.49) .$ One end of the compound bar is maintained at $80.0^{\circ} \mathrm{C},$ and the opposite end is at $30.0^{\circ} \mathrm{C} .$ When the energy transfer reaches steady state, what is the temperature at the junction? (FIGURE CANNOT COPY)

Key Concepts

Steady-State Heat Conduction

Steady-state heat conduction is a condition in which the temperature distribution in a system does not change with time because the amount of heat entering each segment of the material is equal to the amount leaving it. In composite systems, achieving steady state ensures that the same heat flow is maintained across all interfaces, which is critical to determining the temperature at the junction between different materials.

Thermal Resistance in Series

When multiple materials are in thermal contact, each material can be thought of as offering a thermal resistance to heat flow. In a series configuration, the overall heat transfer is influenced by the sum of the individual resistances. This concept is crucial for analyzing the temperature drop across each segment and determining the interface temperature between different materials in a composite system.

Fourier's Law

Fourier's Law states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient and the area through which the heat flows. This fundamental principle is used to determine the heat flow through materials in conduction problems, especially in situations with a steady-state temperature distribution.

Thermal Conductivity

Thermal conductivity is a property of a material that indicates its ability to conduct heat. High thermal conductivity means that the material can transfer heat more efficiently. In problems involving different materials, such as metals with different conductivities, comparing their thermal conductivities is essential to understanding how temperature differences are managed along the composite system.

Transcript

00:01 Consider two metal bars of the same length and cross -sectional area, but one is made of gold and one is made of silver, that are in formal contact with each other and insulated on the sides.

00:11 The end of the gold bar is maintained at a temperature t .s.

00:15 Hot of 80 .0 degrees celsius, while the silver bar is maintained at a temperature t's of cold of 30 .0 degrees celsius.

00:25 And we want to know what is the temperature at the junction point.

00:30 T sub j and we're going to do this using just the law of formal radiation which tells us that the power is given by the thermal conductivity k times the cross -sectional area a which again is the same for both of our bars and then just the temperature gradient d t over d x and for us for these bars, this temperature gradient is just going to be the hot temperature minus the cold temperature of each individual bar over their length, which again is the same length for both bars.

01:12 So let's go ahead and write these power equations for each bar.

01:16 So our power for gold is o.

01:19 But first we're going to need to figure out what these caves are.

01:22 So let's go into table 20 .3, which will give us our thermal conductivities, and our kappa her goal is going to be 314, and these come in watts per meter degree celsius, and our k4 silver, is going to be 427 watts.

01:52 Per meter degree celsius.

01:55 So let's go ahead and write these.

01:58 So our for gold, this is going to be again, that 314 watts per meter degree celsius, our thermal conductivity.

02:08 We can go ahead and pull out these constants between both of them, the a and the l.

02:15 And put that right here as an a over l.

02:20 And those units are meters squared over meters.

02:25 So this is just in meters.

02:27 And finally, just the change in temperatures.

02:31 So for our block of gold, we're going from this temperature hot to the junction point...