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Heat Equation and Earth s Internal Heating

Overview. Earth's InteriorHeat EquationIntegral TransformSolutionProblems. Earth's Interior. 3 LayersCore - Mostly Iron, some Nickel - Outer core is liquid, inner core is solidMantle - Medium density rocks, Iron/Magnesium silicates - Most of Planet mass is in the MantleC

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Heat Equation and Earth s Internal Heating

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    1. Heat Equation and Earth’s Internal Heating Chris Guggino

    2. Overview Earth’s Interior Heat Equation Integral Transform Solution Problems

    3. Earth’s 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

    4. Earth’s Three Heat Sources

    5. Heat Equation with Spherical Coordinates

    6. Boundary Conditions At r=b, boundary condition of third kind

    7. Simplifying More

    8. Jump Conditions Split up problem into parts Multiple solutions Solutions at boundaries must be equal Heat flux at boundaries must be equal

    9. Integral Transform Find the Kernal by solving eigenproblem

    10. Finding roots for Kernal

    11. General Solution after Integral Transform

    12. Some Specifics

    13. 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 = 5.515*10^21; alpha = k/(c*rho); alph = pi/(4*b); T = 0; for i = 1:m alph = alph + i*pi/(2*b); B = root(alph); Q1 = @(r1) r1.*sin(B*r1); K = quadl(Q1,0,b); A = K*sqrt(2)*sqrt((B^2+H2^2)/(b*(B^2+H2^2)+H2)); Q2 = @(t1) exp(alpha*B^2*.t1).*A.*2.523.*10^12.*(1/2).^t1; A2 = quadl(Q2,0,t)/(rho*c); Q3 = @(r2) r2.*(6750 - 100.*r2).*sqrt(2).*sqrt((B^2+H2^2)/(b*(B^2+H2^2)+H2)).*sin(B.*r2); A3 = quadl(Q3,0,b); T = T + (1/r)*exp(-alpha*B^2*t)*r*abs(sin(B*r))*sqrt(2)*sqrt((B^2+H2^2)/(b*(B^2+H2^2)+H2))*(A2 + A3); end

    14. Problems Solution methods Other Earth Factors Numerics

    15. QUESTIONS?

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