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 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 (much smaller than 1) 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.

Definition

The Biot number is defined as:

${\displaystyle \mathrm {Bi} ={\frac {h}{k}}L}$

where:

• ${\displaystyle {k}}$ is the thermal conductivity of the body [W/(m·K)]
• ${\displaystyle {h}}$ is a convective heat transfer coefficient [W/(m²·K)]
• ${\displaystyle {L}}$ is a characteristic length [m] of the geometry considered.

The characteristic length in most of relevant problems becomes the heat characteristic length, i.e. the ratio between the body volume and the heated (or cooled) surface of the body:

${\displaystyle {\mathit {L}}={\frac {V}{A_{\mathrm {Q} }}}}$

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 (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 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 (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. 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.[4]

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:

${\displaystyle \mathrm {Bi} _{m}={\frac {k_{c}}{D}}L}$

where:

• ${\displaystyle {k_{c}}}$ : convective mass transfer coefficient (analogous to the h of the heat transfer problem)
• ${\displaystyle D}$ : mass diffusivity (analogous to the k of heat transfer problem)
• ${\displaystyle {L}}$ : characteristic length

See also

• Convection
• Fourier number
• Heat conduction

References

1. Incropera, Frank P.; DeWitt, David P.; Bergman, Theodore L.; Lavine, Adrienne S. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). John Wiley & Sons. pp. 260–261. ISBN 978-0-471-45728-2. OCLC 288958608.
2. "EXACT". Exact Analytical Conduction Toolbox. University of Nebraska. January 2013. Retrieved 24 January 2015.
3. Cole, Kevin D.; Beck, James V.; Woodbury, Keith A.; de Monte, Filippo (2014). "Intrinsic verification and a heat conduction database". International Journal of Thermal Sciences. 78: 36–47. doi:10.1016/j.ijthermalsci.2013.11.002. ISSN 1290-0729.
4. Delgado, Mónica; Lázaro, Ana; Mazo, Javier; Zalba, Belén (January 2012). "Review on phase change material emulsions and microencapsulated phase change material slurries: Materials, heat transfer studies and applications". Renewable and Sustainable Energy Reviews. 16 (1): 253–273. doi:10.1016/j.rser.2011.07.152. ISSN 1364-0321.