Editing Biot number (section)

==Applications==
The value of the Biot number can indicate the applicability (or inapplicability) of certain methods of solving transient heat transfer problems. For example, a Biot number smaller than about 0.1 implies that heat conduction inside the body offers much lower thermal resistance than the heat convection at the surface, so that temperature [[gradient]]s are negligible inside of the body (such bodies are sometimes labeled "thermally thin"). In this situation, the simple [[lumped-capacitance model]] may be used to evaluate a body's transient temperature variation. The opposite is also true: a Biot number greater than about 0.1 indicates that thermal resistance within the body is not negligible, and more complex methods are need in analyzing heat transfer to or from the body (such bodies are sometimes called "thermally thick"). 

=== Heat conduction for finite Biot number===

When the Biot number is greater than 0.1 or so, the [[heat equation]] must be solved to determine the time-varying and spatially-nonuniform temperature field within the 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.<ref>{{cite web |url=http://exact.unl.edu |publisher=University of Nebraska |date=January 2013 |title=EXACT |website=Exact Analytical Conduction Toolbox |access-date=24 January 2015}}</ref><ref name="ColeBeck2014">{{cite journal|last1=Cole|first1=Kevin D.|last2=Beck|first2=James V.|last3=Woodbury|first3=Keith A.|last4=de Monte|first4=Filippo|title=Intrinsic verification and a heat conduction database|journal=International Journal of Thermal Sciences|volume=78|year=2014|pages=36–47|issn=1290-0729|doi=10.1016/j.ijthermalsci.2013.11.002|bibcode=2014IJTS...78...36C }}</ref>
Often such problems are too difficult to be done except numerically, with the use of a computer model of heat transfer.

=== Heat conduction for Bi ≪ 1 ===
As noted, a Biot number smaller than about 0.1 shows that the conduction resistance inside a body is much smaller than heat convection at the surface, so that temperature [[gradient]]s are negligible inside of the body. In this case, the [[lumped-capacitance model]] of transient heat transfer can be used. (A Biot number less than 0.1 generally indicates less than 3% error will be present when using the lumped-capacitance model.<ref>{{cite journal |last1=Ostorgorsky |first1=Aleks G. |title=Simple Explicit Equations for Transient Heat Conduction in Finite Solids |journal=Journal of Heat Transfer |date=January 2009 |volume=131 |issue=1 |page=011303 |doi=10.1115/1.2977540}}</ref>)

The simplest type of lumped capacity solution, for a step change in fluid temperature, shows that a body's temperature decays exponentially in time ("Newtonian" cooling or heating) because the [[internal energy]] of the body is directly proportional to the temperature of the body, and the difference between the body temperature and the fluid temperature is linearly proportional to rate of heat transfer into or out of the body. Combining these relationships with the [[First law of thermodynamics]] leads to a simple first-order linear differential equation. The corresponding lumped capacity solution can be written 
:<math>\frac{T - T_\infty}{T_0 - T_\infty} = e^{-t/\tau}</math>
in which <math>\tau = \frac{\rho c_p V}{h A_Q}</math> is the [[Time_constant#Thermal_time_constant|thermal time constant]] of the body, <math>\rho</math> is the [[mass density]] (kg/m<sup>3</sup>), and <math>c_p</math> is [[specific heat capacity]] (J/kg-K).

The study of heat transfer in micro-encapsulated phase-change slurries is an application where the Biot number is useful. 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 thermal gradients within the dispersed phase are negligible.<ref>{{Cite journal|last1=Delgado|first1=Mónica|last2=Lázaro|first2=Ana|last3=Mazo|first3=Javier|last4=Zalba|first4=Belén|date=January 2012|title=Review on phase change material emulsions and microencapsulated phase change material slurries: Materials, heat transfer studies and applications|journal=Renewable and Sustainable Energy Reviews|volume=16|issue=1|pages=253–273|doi=10.1016/j.rser.2011.07.152|bibcode=2012RSERv..16..253D |issn=1364-0321}}</ref>