Calculation of the external thermal resistance of buried cables through conformal transformation

Abstract

The calculation of the outside thermal resistance of a buried cable system related to uniform, nonuniform and temperature-dependent resistivity of soil can be made, if a suitable transformation of the thermal field is automatically drawn and the resistance is computed on the transformed field. A suitable conformal transformation is analysed-giving rise to maps of the transformed field. On the maps, the calculation of resistances involves simple operations even when the soil is nonuniform and of temperature-dependent resistivity, provided some approximations are introduced. The errors incurred by such approximations are slight and can be checked, when necessary, by direct computation of the thermal field. The calculation of resistances on the maps can also be carried out by means of automatic design programs. LIST OF SYMBOLSc = cable circumference D = cable diameter g = thermal resistivity hj = constant proportional to power loss of source i = general numerical suffix kj, l n j = lengths read on transformed maps m ' = number of source pairs p = vertical distance of cable centre from x axis q = horizontal distance of cable centre from y axis R = thermal resistance r = modulus of co-ordinate z Ru R 3 = thermal resistance for lateral cables R 2 = thermal resistance for centre cable 5 = position of cable centre (q -jp) T = temperature T c = isotherm at which soil resistivity changes (normally 50°C) T Q = ambient temperature Ti = cable surface temperature W = cable power loss Y o = earth surface in transformed plane Yj_ = cable surface in transformed plane z = complex co-ordinate (x + jy) in original plane Z = complex co-ordinate (X + jY) in transformed plane zj, zi = co-ordinates of point sources in original plane 6 = displacement of point source 6 = argument of co-ordinate z

The Effect of Drying around Power Cables on the Vadose Zone Temperature

“…One example of a system where the one‐dimensional case is realized is that of a slab of soil that, at its top surface (ground surface), has a uniform temperature T a and a uniform effective saturation S E,a , and it has a uniform planar heat source at the bottom surface. We shall focus on this case, as the solution of the Laplace equation for steady‐state heat transfer between a cylindrical isothermal surface (buried cable) and an isothermal plane (ground surface) is obtained by conformal mapping from the solution of this case (Carslaw and Jaeger, 1959; Luoni et al, 1972).…”

Natural and forced-cooling of HV underground cables: UK practice

“…The use of stabilised backfill materials, having dried out TRs similar to those of the indigenous soils, has simplified winter rating calculations but still leaves the problem of summer ratings. The calculation of external thermal resistances of cables buried in soils having nonuniform and temperature-dependent thermal resistivities may be tackled mathematically using conformal transformation techniques [120]. However, all calculation methods for regions of variable TRs involve limiting assumptions, and so it is often necessary to use an alternative method to obtain realistic ratings.…”