In the study of heat transfer, Newton's law of cooling is a physical law which states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment.
The law is frequently qualified to include the condition that the temperature difference is small and the nature of heat transfer mechanism remains the same.
The thermal conductivity of most materials is only weakly dependent on temperature, so the constant heat transfer coefficient condition is generally met.
In convective heat transfer, Newton's Law is followed for forced air or pumped fluid cooling, where the properties of the fluid do not vary strongly with temperature, but it is only approximately true for buoyancy-driven convection, where the velocity of the flow increases with temperature difference.
In the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature differences.
When stated in terms of temperature differences, Newton's law (with several further simplifying assumptions, such as a low Biot number and a temperature-independent heat capacity) results in a simple differential equation expressing temperature-difference as a function of time.
Isaac Newton published his work on cooling anonymously in 1701 as "Scala graduum Caloris.
The law holds well for forced air and pumped liquid cooling, where the fluid velocity does not rise with increasing temperature difference.
In that case, Newton's law only approximates the result when the temperature difference is relatively small.
A correction to Newton's law concerning convection for larger temperature differentials by including an exponent, was made in 1817 by Dulong and Petit.
[5] (These men are better-known for their formulation of the Dulong–Petit law concerning the molar specific heat capacity of a crystal.)
Radiative cooling is better described by the Stefan–Boltzmann law in which the heat transfer rate varies as the difference in the 4th powers of the absolute temperatures of the object and of its environment.
where In global parameters by integrating on the surface area the heat flux, it can be also stated as:
Formulas and correlations are available in many references to calculate heat transfer coefficients for typical configurations and fluids.
[6] Note the heat transfer coefficient changes in a system when a transition from laminar to turbulent flow occurs.
where The physical significance of Biot number can be understood by imagining the heat flow from a hot metal sphere suddenly immersed in a pool to the surrounding fluid.
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.
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).
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.
This condition allows the presumption of a single, approximately uniform temperature inside the body, which varies in time but not with position.
The condition of low Biot number leads to the so-called lumped capacitance model.
The lumped capacitance solution that follows assumes a constant heat transfer coefficient, as would be the case in forced convection.
For free convection, the lumped capacitance model can be solved with a heat transfer coefficient that varies with temperature difference.
[8] A body treated as a lumped capacitance object, with a total internal energy of
, may be expressed by Newton's law of cooling, and where no work transfer occurs for an incompressible material.
The temperature difference between the body and the environment decays exponentially as a function of time.