This page calculates the isothermal temperature a rectangular volume embedded in a semi-infinite medium with an applied isothermal temperature.
The rectangular volume temperature Tb is calculated as:
Tb = Ti + q / (k x sf)
Where Ti is the isothermal temperature applied to the medium surrounding the rectangular volume, q is the heat flow, sf is the shape factor, and k is the conductivity of the medium.
The rectangular volume shape factor is calculated as:
sf = 1.685 L [ log ( 1 + ( d / W ) ) ]-0.59 x ( d / H)-0.078
Where W is the rectangular volume width, L is the rectangular volume length, H is the rectangular volume height, and d is the rectangular volume depth.
Holman, J.P., Heat Transfer, 7th ed., McGraw Hill Book Company, New York, 1990. p 74 - 83