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In one dimensional cylinder and two dimensional Cartesian geometries, transport kernels in integral transfer equation and boundary terms which are surface integrals, involve integrals of modified Bessel and Bickley functions, respectively. Traditionally these integrals are evaluated numerically. In this note, analytical properties, series and asymptotic expansions of these integrals are reported. In addition series expansion of generalized exponential integral function is derived and reported.

The entropy generation during transient laminar natural convection in a square enclosure is numerically investigated. Two different cases are considered. The enclosure is heated either completely or partially from the left side wall and cooled from the opposite side wall. The bottom and the top of the enclosure are assumed as insulated. The Boussinesq approximation is used in the natural convection modelling. The solutions are obtained from quiescent conditions proceeded through the transient up to the steady-state. The calculations are made for the Prandtl numbers 0.01 and 1.0 and Rayleigh numbers between 102-108. The entropy generation and the active places triggering the entropy generation are obtained for each case after the flow and thermal characteristics are determined. It is found that the active sites in the completely heated case are at the left bottom corner of the heated wall and the right top corner of the cooled wall at the same magnitudes. In the case of partial heating, however, the active site is observed at the top corner of the heated section especially at lower Pr and Ra values

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