A bar of gold is in thermal contact with a bar of silver of the same length and area (Fig. P20.4

A bar of gold is in thermal contact with a bar of silver of...

Key Concepts

Thermal Conduction

Thermal conduction is the process by which heat energy is transmitted through collisions between neighboring molecules or electrons. In solids, it is governed by how efficiently energy is transferred from high temperature regions to low temperature regions, and this efficiency is quantified by the material’s thermal conductivity.

Fourier's Law of Heat Conduction

Fourier's Law provides the mathematical framework for conduction, stating that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the area through which the heat passes, and inversely proportional to the distance. This law is essential for calculating how temperature differences drive energy flow in a material or series of materials.

Steady-State Heat Transfer

Steady-state heat transfer occurs when the temperature distribution within the system remains constant over time, meaning the amount of heat entering any part of the system is equal to the heat leaving that part. In such conditions, the heat flux is uniform throughout the medium, allowing for the analysis of temperature profiles using constant parameters.

Composite Conductor

A composite conductor involves multiple materials with different thermal conductivities joined together. In such systems, each segment contributes its own thermal resistance to the overall flow of heat. The analysis involves connecting thermal resistances in series, similar to resistors in an electrical circuit, to understand how heat is distributed across the materials.

Thermal Resistance

Thermal resistance quantifies how a material opposes the flow of heat, analogous to electrical resistance in circuits. It depends on the material’s dimensions and thermal conductivity. In composite systems, the overall temperature drop is the sum of the temperature drops across individual materials, each scaled by its thermal resistance.

Continuity of Heat Flux

In a series of materials forming a composite conductor, the principle of continuity of heat flux dictates that the same amount of heat per unit time must pass through each section. This continuity condition is crucial for determining interface temperatures, as it requires that the heat leaving one material is the same as the heat entering the next.

Transcript

00:01 So here in this problem we are given that a gold bar is in thermal contact with the silver bar and the temperature towards the left side we have 80 degrees celsius and towards the right side we have 30 degree celsius and we have to calculate this intermediate temperature t so let us draw the schematic thermal circuit corresponding to this bar so here this one is thermal resistance for au and this one is thermal resistance for ag and here the temperature is 80 degrees celsius and here the temperature is 30 degrees celsius and we are given that the length and the area of a u and ag is same so we can take it as l and a so this resistance the thermal resistance for au this is for gold will be equals to l over k so k we have to take it for gold times a and thermal resistance for ag we have l over ka so k we have to take it for a times a now since the body's in thermal contact so the value of q that passes through here this is q and this is also q and this is also q remains the same so q we know it is equals to delta t over rth so if you write q for the gold au it should be equals to this is 80 degree minus let this temperature be t here let temperature be t here so it it minus t over rth we have l over this is k a so k will go in the numerator so it becomes k of a u times area similarly q for ag it will be equals to t minus this one is 30 degrees celsius divide by the same l over k ag times a so l is in the denominator and this one comes in the numerator so it becomes k of ag times a so we can compare both these two equations and we'll get a t minus t divide by l times k a g a u times a is equals to t minus 30 degrees celsius over l times k of a times area so we can see that this area and area will cancel out similarly this l and l will cancel out now we have the value of k of au its standard value and similarly the standard of standard value of ag that is thermal conductivity we know we can plug the values here to calculate the temperature so the temperature t the expression for temperature t comes out to be as k of a u times 80 degree celsius plus this is k of a g times 30 degrees celsius divide by k of ag plus k of a u now we're going to plug the values so this one is 314 watt per meter kelvin times 80 plus this this value we have 427 it's standard value times 30 degrees celsius divide by this one is 314 plus 427 so from here we'll get the answer as nearly 51 .2 degrees celsius.

03:48 So this is the intermediate temperature t at the junction...

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