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There is an area of thin mantle at the bottom of the ocean which is...

There is an area of thin mantle at the bottom of the ocean which is substantially raising the temperature of the sea bed due to a breach of molten material. Use the heat equation to analytically solve for the transient temperature profile within the sea given a fixed Dirichlet boundary condition at the bottom of the ocean. The initial 1D temperature versus depth profile, at the instant when the ocean bed breach occurs, can be represented by a piece-wise function: the temperature is a constant Tα = 25◦C from sea-level d0 = 0m to a depth of d1 = 500m; there is then a sharp transition to a constant temperature of Tβ = 5◦C from d1 to the ocean floor d2 = 2600m; finally, the exposed mantle causes the temperature at d2 to be fixed at Tγ = 460◦C. The thermal diffusivity of sea-water can be approximated to κt = 1.6 × 10−2m2/s.

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4. By expressing the temperatures and depths as variables (rather than actual values), show that the unknown Fourier coefficient can be expressed as (5%): .d2 .d2 Bn = K1 x sin(px) dx + K2 sin(px) dx (1) do d1 2(Ta - Ty) 2(TB - Ta) where K1 = and K2 = d2

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