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In the study of
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
, Newton's law of cooling is a
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narro ...
which states that
The rate of
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
loss of a body is directly proportional to the difference in the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
s 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. As such, it is equivalent to a statement that the
heat transfer coefficient In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
, which mediates between heat losses and temperature differences, is a constant. This condition is generally met in
heat conduction Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flows along a te ...
(where it is guaranteed by
Fourier's law Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its '' thermal conductivity'', and is denoted . Heat spontaneously flows along a t ...
) as the
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
of most materials is only weakly dependent on temperature. In
convective heat transfer Convection (or convective heat transfer) is the transfer of heat from one place to another due to the movement of fluid. Although often discussed as a distinct method of heat transfer, convective heat transfer involves the combined processes o ...
, 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. Finally, in the case of heat transfer by
thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) i ...
, 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 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 thermal resistances ''ins ...
and a temperature-independent
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
) results in a simple
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
expressing temperature-difference as a function of time. The solution to that equation describes an exponential decrease of temperature-difference over time. This characteristic decay of the temperature-difference is also associated with Newton's law of cooling.


Historical background

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
published his work on cooling anonymously in 1701 as "Scala graduum Caloris. Calorum Descriptiones & signa" in ''
Philosophical Transactions ''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...
'', volume 22, issue 270. Newton did not originally state his law in the above form in 1701. Rather, using today's terms, Newton noted after some mathematical manipulation that ''the rate of temperature change'' of a body is proportional to the difference in temperatures between the body and its surroundings. This final simplest version of the law, given by Newton himself, was partly due to confusion in Newton's time between the concepts of heat and temperature, which would not be fully disentangled until much later. In 2020, Maruyama and Moriya repeated Newton's experiments with modern apparatus, and they applied modern data reduction techniques. In particular, these investigators took account of thermal radiation at high temperatures (as for the molten metals Newton used), and they accounted for buoyancy effects on the air flow. By comparison to Newton's original data, they concluded that his measurements (from 1692 to 1693) had been "quite accurate".


Relationship to mechanism of cooling

Convection cooling is sometimes said to be governed by "Newton's law of cooling." When the
heat transfer coefficient In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
is independent, or relatively independent, of the temperature difference between object and environment, Newton's law is followed. The law holds well for forced air and pumped liquid cooling, where the fluid velocity does not rise with increasing temperature difference. Newton's law is most closely obeyed in purely conduction-type cooling. However, the heat transfer coefficient is a function of the temperature difference in natural convective (buoyancy driven) heat transfer. In that case, Newton's law only approximates the result when the temperature difference is relatively small. Newton himself realized this limitation. A correction to Newton's law concerning convection for larger temperature differentials by including an exponent, was made in 1817 by Dulong and Petit. (These men are better-known for their formulation of the Dulong–Petit law concerning the molar specific heat capacity of a crystal.) Another situation that does not obey Newton's law is radiative heat transfer. 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.


Mathematical formulation of Newton's law

The statement of Newton's law used in the heat transfer literature puts into mathematics the idea that ''the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings''. For a temperature-independent heat transfer coefficient, the statement is: \dot = h A \left(T(t) - T_\text\right) = h A \, \Delta T(t), where * \dot is the rate of heat transfer out of the body (SI unit:
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James ...
), * h is the
heat transfer coefficient In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
(assumed independent of ''T'' and averaged over the surface) (SI unit: W/m2⋅K), * A is the heat transfer surface area (SI unit: m2), * T is the temperature of the object's surface (SI unit: K), * T_\text is the temperature of the environment; i.e., the temperature suitably far from the surface (SI unit: K), * \Delta T(t) = T(t) - T_\text is the time-dependent temperature difference between environment and object (SI unit: K). The heat transfer coefficient ''h'' depends upon physical properties of the fluid and the physical situation in which convection occurs. Therefore, a single usable heat transfer coefficient (one that does not vary significantly across the temperature-difference ranges covered during cooling and heating) must be derived or found experimentally for every system that is to be analyzed. Formulas and correlations are available in many references to calculate heat transfer coefficients for typical configurations and fluids. For laminar flows, the heat transfer coefficient is usually smaller than in turbulent flows because turbulent flows have strong mixing within the
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
on the heat transfer surface. Note the heat transfer coefficient changes in a system when a transition from laminar to turbulent flow occurs.


Simplified formulation

By non-dimensionalizing, the differential equation becomes \dot = r \left(T_\text - T(t)\right), where * \dot is the rate of heat loss (SI unit: K/second), * T is the temperature of the object's surface (SI unit: K), * T_\text is the temperature of the environment; i.e., the temperature suitably far from the surface (SI unit: K), * r is the coefficient of heat transfer (SI unit: 1/second). Solving the initial-value problem using separation of variables gives T(t) = T_\text + (T(0)-T_\text)e^.


The Biot number

The Biot number, a dimensionless quantity, is defined for a body as \text = \frac, where * ''h'' = film coefficient or
heat transfer coefficient In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat (i.e., the temperature difference, ). ...
or convective heat transfer coefficient, * ''L''C =
characteristic length In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by ...
, which is commonly defined as the volume of the body divided by the surface area of the body, such that L_ = V_\text / A_\text, * ''k''b =
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
of the body. 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. The heat flow experiences two resistances: the first outside the surface of the sphere, and the second within the solid metal (which is influenced by both the size and composition of the sphere). The ratio of these resistances is the dimensionless Biot number. If the thermal resistance at 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 the simple exponential one described in Newton's law of cooling expressed in terms of temperature difference (see below). 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. In this case, temperature 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 at the fluid/sphere boundary, even with a much smaller sphere. In this case, again, the Biot number will be greater than one. 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
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
s 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). Typically, this type of analysis leads to simple exponential heating or cooling behavior ("Newtonian" cooling or heating) since the internal energy of 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 Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
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 The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
, are described in the article on the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
.


Application of Newton's law transient cooling

Simple solutions for transient cooling of an object may be obtained when the internal thermal resistance within the object is small in comparison to the resistance to heat transfer away from the object's surface (by external conduction or convection), which is the condition for which the Biot number is less than about 0.1. This condition allows the presumption of a single, approximately uniform temperature inside the body, which varies in time but not with position. (Otherwise the body would have many different temperatures inside it at any one time.) This single temperature will generally change exponentially as time progresses (see below). The condition of low Biot number leads to the so-called lumped capacitance model. In this model, the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
(the amount of thermal energy in the body) is calculated by assuming a constant
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat cap ...
. In that case, the internal energy of the body is a linear function of the body's single internal temperature. 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.


First-order transient response of lumped-capacitance objects

A body treated as a lumped capacitance object, with a total
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
of U (in joules), is characterized by a single uniform internal temperature, The heat capacitance, of the body is C = dU/dT (in J/K), for the case of an incompressible material. The internal energy may be written in terms of the temperature of the body, the heat capacitance (taken to be independent of temperature), and a reference temperature at which the internal energy is zero: Differentiating U with respect to time gives: \frac = C \, \frac. Applying the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant am ...
to the lumped object gives where the rate of heat transfer out of the body, may be expressed by Newton's law of cooling, and where no work transfer occurs for an incompressible material. Thus, \frac = -\frac (T(t) - T_\text) = -\frac\ \Delta T(t), where the time constant of the system is The heat capacitance C may be written in terms of the object's
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
, c (J/kg-K), and mass, m (kg). The time constant is then When the environmental temperature is constant in time, we may define The equation becomes \frac = \frac = -\frac \Delta T(t). The solution of this differential equation, by integration from the initial condition, is \Delta T(t) = \Delta T(0) \, e^. where \Delta T(0) is the temperature difference at time 0. Reverting to temperature, the solution is T(t) = T_\text + (T(0) - T_\text) \, e^. The temperature difference between the body and the environment decays exponentially as a function of time.


See also

*
Thermal transmittance Thermal transmittance is the rate of transfer of heat through matter. The thermal transmittance of a material (such as insulation or concrete) or an assembly (such as a wall or window) is expressed as a U-value. The thermal insulance of a struct ...
* List of thermal conductivities * Convection diffusion equation *
R-value (insulation) In the context of construction, the R-value is a measure of how well a two-dimensional barrier, such as a layer of insulation, a window or a complete wall or ceiling, resists the conductive flow of heat. R-value is the temperature difference pe ...
* Heat pipe *
Fick's law of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion ...
* Relativistic heat conduction *
Churchill–Bernstein equation In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier–St ...
* Fourier number *
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 thermal resistances ''ins ...
*
False diffusion False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection–diffusion equations. The more accurate central difference scheme can be used for the convection term, but for grids w ...
* Mpemba effect


References

See also: *Dehghani, F 2007, CHNG2801 – Conservation and Transport Processes: Course Notes, University of Sydney, Sydney


External links


Heat conduction
- Thermal-FluidsPedia
Newton's Law of Cooling
by Jeff Bryant based on a program by Stephen Wolfram, Wolfram Demonstrations Project.
''A Heat Transfer Textbook, 5/e''
free ebook. {{DEFAULTSORT:Newton's law of cooling Equations of physics Heat conduction Heat transfer Isaac Newton History of physics Scientific observation Experimental physics