Uniform heat flow in a semi-infinite medium disturbed by a body of different thermal conductivity

Summary The approximate solution for the disturbance of a uniform heat flow in a homogeneous semi-infinite medium by an object of different thermal conductivity buried in it is generally used in the interpretation of heat flow anomalies on the ocean bottom. In order to know the accuracy of the approximate solution, a comparison between the approximate solution and the exact solution is given in the case of a very long horizontal cylinder in a semi-infinite medium. The computed results indicate that the two solutions agree to within 10% whend>1.3 and 0.5<<2, whered is ratio of the depth to the radiusR₀ of the cylinder and is the factor of the contrast of the thermal conductivities between the medium and the body. As for the cases when 1 and 1, the same accuracy can be obtained only whend>2. A similar approach is also applied to the case of a spherical conductor in a semi-infinite medium by using a bispherical harmonic solution. The results of both the bipolar solution and the bispherical solution show that when 1 andd1, the vertical thermal gradient at the surface of the semi-infinite medium is always positive and tends to zero, but a negative vertical gradient may be obtained for the approximate solutions.