Heat Equation and Earths Internal Heating

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Heat EquationandEarths Internal Heating

• Chris Guggino

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Overview

• Earths Interior
• Heat Equation
• Integral Transform
• Solution
• Problems

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Earths Interior

• 3 Layers
• Core - Mostly Iron, some Nickel
• - Outer core is liquid, inner core is
  solid
• Mantle - Medium density rocks, Iron/Magnesium
  silicates
• - Most of Planet mass is in the Mantle
• Crust - Low density rock, Granite/Basalt

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Earths Three Heat Sources
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Heat Equation with Spherical Coordinates
When dealing with a sphere this can be reduced to
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Boundary Conditions

• At rb, boundary condition of third kind

At t 0 Initial condition is function of r
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Simplifying More
Defining new variable UrT simplifies the problem
more
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Jump Conditions

• Split up problem into parts
• Multiple solutions
• Solutions at boundaries must be equal
• Heat flux at boundaries must be equal

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Integral Transform

• Find the Kernal by solving eigenproblem

Where Bms are roots of
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Finding roots for Kernal
Initial guess function
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General Solution after Integral Transform
Where
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Some Specifics
Generation Term
Initial Condition
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Code

• function T earth(r,t)
• format long
• n 65
• m 11
• h2 .16
• k2 1
• b 64
• H2 h2/k2 - 1/b
• k 9807696
• c 935.33
• rho 5.5151021
• alpha k/(crho)
• alph pi/(4b)
• T 0
• for i 1m

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Problems

• Solution methods
• Other Earth Factors
• Numerics

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QUESTIONS?