# Biot number

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It is named after the eighteenth century French physicist Jean-Baptiste Biot (1774–1862), and 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.

In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 signal 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.

## Definition

The Biot number is defined as:

${\displaystyle \mathrm {Bi} ={\frac {L_{C}h}{\ k}}}$

where:

• h = film coefficient or heat transfer coefficient or convective heat transfer coefficient
• LC = characteristic length, which is commonly defined as the volume of the body divided by the surface area of the body, such that ${\displaystyle {\mathit {L_{C}}}={\frac {V_{\mathrm {body} }}{A_{\mathrm {surface} }}}}$
• kb = thermal conductivity of the body

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. For systems where it is much less than one, the interior of the sphere may be presumed always to have the same temperature, although this temperature may be changing, as heat passes into the sphere from the surface. The equation to describe this change in (relatively uniform) 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 (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.

## 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 (or inapplicability) 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 (also called lumped system analysis).[1] Typically this type of analysis leads to simple exponential heating or cooling behavior ("Newtonian" cooling or heating) since the amount of thermal energy (loosely, amount of "heat") 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 (a "thermally thick" substance) 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[2] .[3] Often such problems are too difficult to be done except numerically, with the use of a computer model of heat transfer.

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,

${\displaystyle {T-T_{\infty } \over T_{0}-T_{\infty }}=e^{\mathrm {-BiFo} }}$

## Mass transfer analogue

An analogous version of the Biot number (usually called the "mass transfer Biot number", or ${\displaystyle \mathrm {Bi} _{m}}$) is also used in mass diffusion processes:

${\displaystyle \mathrm {Bi} _{m}={\frac {h_{m}L_{C}}{D_{AB}}}}$

where:

## References

1. ^ Incropera, Frank P.; David P. DeWitt; Theodore L. Bergman; Adrienne S. Lavine (2007). Fundamentals of Heat and Mass Transfer (6th ed.). John Wiley & Sons. pp. 260–261. ISBN 978-0-471-45728-2.
2. ^ "EXACT". Exact Analytical Conduction Toolbox. University of Nebraska. January 2013. Retrieved 24 January 2015.
3. ^ Cole, Kevin D.; et al. (April 2014). "Intrinsic Verification and a Heat Conduction Database". Int. J. Thermal Science. 78: 36.