Time to reach specific temperature (related to Biot and Fourier numbers)

Description

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. Gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface.
The physical significance of Biot number can be understood by imagining the heat flow from a small hot metal sphere suddenly immersed in a pool, to the surrounding fluid. The heat flow experiences two resistances: the first within the solid metal (which is influenced by both the size and composition of the sphere), and the second at the surface of the sphere. If the thermal resistance of the fluid/sphere interface exceeds that thermal resistance offered by the interior of the metal sphere, the Biot number will be less than one.
In contrast, the metal sphere may be large, causing the characteristic length to increase to the point that the Biot number is larger than one. Now, thermal gradients within the sphere become important, even though the sphere material is a good conductor. Equivalently, if the sphere is made of a thermally insulating (poorly conductive) material, such as wood or styrofoam, the interior resistance to heat flow will exceed that of the fluid/sphere boundary, even with a much smaller sphere. In this case, again, the Biot number will be greater than one.

The Fourier number (Fo) or Fourier modulus, is a dimensionless number that characterizes heat conduction. it is the ratio of diffusive/conductive transport rate by the quantity storage rate and arises from non-dimensionalization of the heat equation. The transported quantity is usually either heat or matter (particles).

The shown equation, derived with the product of the Biot and Fourier numbers, can be used to estimate the time for the object to reach a specific temperature.

Related formulas

Variables

ttime (s)
ρdensity (kg/m3)
cpspecific heat capacity (J/K*kg)
Vvolume of the object (m3)
hconvective heat transfer coefficient for the surrounding fluid (W/m2*K)
Asurface area (m2)
T0initial temperature (K)
Tinfinitytemperature of the bulk fluid (K)
Ttemperature of the object at time t (K)