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Q. 1.61

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An Introduction to Thermal Physics
Found in: Page 40
An Introduction to Thermal Physics

An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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Short Answer

Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases by 20∘C per kilometer of depth and the thermal conductivity of the rock is 2.5W/m⋅K. What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of the earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is 6400km.)

Rate of heat conduction and the rate at which the earth loses heat via conduction

QΔt=0.05W

Qtotal Δt=2.573×1013

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Calculation of ratio

Because the rock that makes up the Earth's crust has a thermal conductivity and there is a temperature differential between a point underground and the Earth's surface, the Earth loses energy through heat conduction. We've calculated the following using Schroeder's values:ΔT=20∘K per Δx=1000m and kt=2.5W∘K−1, the rate of heat conduction in an area of 1m2 is, therefore:

localid="1651746825640" QΔt=ktAΔTΔx=2.5×1×201000=0.05 W/m2

localid="1651746839506" QΔt=0.05 W/m2

Step 2: Calculation of heat

Even if the rate of heat loss for a square metre is fairly low, if we assume that this figure applies to the entire Earth, the total heat loss is:

Qtotal Δt= Loss per meter square × Total area of earth

Qtotal Δt=0.05×4πr2=0.05×4π6400×1032

Qtotal Δt=2.573×1013 W

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