A $30-$m$-$long, $10-$cm$-$diameter hot$-$water pipe of a district heating system is buried in the soil $50$ cm below the ground surface. The outer surface temperature of the pipe is $80$C$.$ Taking the surface temperature of the earth to be $10$C and the thermal conductivity of the soil at that location to be $0\mathrm{.}9$ W$/$m $$C$.$

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Investigate the rate of heat loss from the pipe. The shape factor for this configuration is given by

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where z is the distance of the pipe from the ground surface, and D is the diameter of the pipe.