# BIOT NUMBER

The Biot Number is a dimensionless group named after J. B. Biot who, in 1804, analysed the interaction between conduction in a solid and convection at its surface. The numerical value of Biot Number (Bi) is a criterion which gives a direct indication of the relative importance of conduction and convection in determining the temperature history of a body being heated or cooled by convection at its surface. Bi should always be enumerated at the outset to identify transient conduction problems which may be treated simply as lumped parameter problems, for which Bi < 0.1 and for which it is seldom necessary to solve the conduction equation, i.e., convection is the rate controlling process.

Figure 1 may be used to impute physical meaning to Bi, and shows the limiting steady-state temperature distributions in two plates [(a) high λ and (b) low λ] which have cooled from an initial uniform temperature T1 as a result of exposure to a cooling flow at T2 along one face.

For both plates, a heat balance at the cooled surface can be constructed by using Fourier’s Law for the conduction flux in the solid at the surface and Newton’s Law of Cooling for the convective loss at the surface:

(1) This heat balance can be rearranged to produce the dimensionless Bi:

(2) from which it can be seen that Bi may be considered to be the ratio of the resistance to heat transfer presented by the conduction Rcond and the convection Rconv processes.

Figure 1 exemplifies the temperature distribution in a low Bi system (e.g., a steel plate (λ = 35 W/mK) of 5 cm thickness cooling in air [(α = 10 W/m2K) giving Bi = 0.0023)] where, because the resistance to heat flow within the solid is small relative to the resistance presented by the convection processes at the surface, the internal temperature distribution in the solid is relatively uniform. Generally, in bodies of simple geometry, e.g., plates, cylinders, spheres, the error introduced by the assumption of uniform body temperature will be less than 5% when the internal resistance is less than 10% of the external resistance, i.e., when the Bi < 0.1. Figure 1. Comparison of fully developed temperature profiles in two plates cooled by the same fluid.