Determine the effect of a glacial epoch on the surface geothermal gradient as follows. At the start of the glacial epoch t = −τ , the subsurface temperature is T0 +βy. The surface is y = 0, and y increases downward. During the period of glaciation the surface temperature drops to 284 Heat Transfer
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T0 − ∆T0. At the end of the glacial period, t = 0, the surface temperature again rises to T0. Find the subsurface temperature T (y, t) and the surface heat flow for t > −τ . If the last glaciation began at 13,000 year BP and ended 8000 year BP and ∆T0 = 20 K (κ = 1 mm2 s−1, k = 3.3 W m−1 K−1), determine the effect on the present surface heat flow. HINT: Use the idea of superposition to combine the elementary solutions to the heat conduction equation in such a way as to develop the solution of this problem without having to solve a differential equation again. Problem 4.35 One technique for measuring the thermal conductivity of sediments involves the insertion of a very thin cylinder, or needle, heated by an internal heater wire at a known and constant rate, into the sediments. A small thermistor inside the needle measures the rise of temperature T with time t. After the heater has been on for a short time, measurements of T show a linear growth with ln t, T = c1 ln t+ c2.
The sediment conductivity can be deduced from the slope of a T versus ln t plot, c1, with the aid of a theoretical formula you can derive as follows. Consider the temperature field due to an infinite line source that emits Q units of heat per unit time and per unit length for times t > 0 in an infinite medium initially at temperature T0. Determine T (r, t) by solving Equation (4–70) subject to the appropriate initial and boundary conditions.
HINT: A similarity solution with the similarity variable η = r2/4κt works.