The Biot number is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth centuryFrench physicistJean-Baptiste Biot, and gives a simple index of the ratio of the heat transfer resistances inside of a body 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. In general, problems involving small Biot numbers are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 indicate more difficult problems due to non-uniformity of temperature fields within the object. It should not be confused with Nusselt number, which employs the thermal conductivity of the fluid and hence is a comparative measure of conduction and convection, both in the fluid. The Biot number has a variety of applications, including transient heat transfer and use in extended surface heat transfer calculations.
The characteristic length in most of relevant problems becomes the heat characteristic length, i.e. the ratio between the body volume and the heated surface of the body: Here, Q for heat is used to denote that the surface to be considered is only the portion of the total surface through which the heat Q passes. 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, 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. For systems where it is much less than one, the interior of the sphere may be presumed to be a uniform temperature, although this temperature may be changing, as heat passes into the sphere from the surface. The equation to describe this change in temperature inside the object, is simple exponential one described in Newton's law of cooling. 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 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.
Applications
Values of the Biot number smaller than 0.1 imply that the heat conduction inside the body is much faster than the heat convection away from its surface, and temperature gradients are negligible inside of it. This can indicate the applicability of certain methods of solving transient heat transfer problems. For example, a Biot number less than 0.1 typically indicates less than 5% error will be present when assuming a lumped-capacitance model of transient heat transfer. Typically this type of analysis leads to simple exponential heating or cooling behavior since the amount of thermal energy in the body is directly proportional to its temperature, which in turn determines the rate of heat transfer into or out of it. This leads to a simple first-order differential equation which describes heat transfer in these systems. Having a Biot number smaller than 0.1 labels a substance as "thermally thin," and temperature can be assumed to be constant throughout the material's volume. The opposite is also true: A Biot number greater than 0.1 indicates that one cannot make this assumption, and more complicated heat transfer equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body. Analytic methods for handling these problems, which may exist for simple geometric shapes and uniform material thermal conductivity, are described in the article on the heat equation. Examples of verified analytic solutions along with precise numerical values are available. Often such problems are too difficult to be done except numerically, with the use of a computer model of heat transfer. The heat transfer study of micro-encapsulated Phase-change slurry is one application where the Biot number comes in handy; for the dispersed phase of the micro-encapsulated Phase-change slurry, the micro-encapsulated Phase-change material itself, the Biot number is calculated to be below 0.1 and so it can be assumed that there are no thermal gradient within the dispersed phase. Together with the Fourier number, the Biot number can be used in transient conduction problems in a lumped parameter solution, which can be written as,
