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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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1. Heat EquationandEarth’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?