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This page calculates the isothermal temperature a rectangular volume embedded in a semi-infinite medium with an applied isothermal temperature.

The rectangular volume temperature T_b is calculated as:

T_b = T_i + q / (k x sf)

Where T_i 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.

References

Holman, J.P., Heat Transfer, 7th ed., McGraw Hill Book Company, New York, 1990. p 74 - 83