# Newton's law of cooling

Last updated

Newton's law of cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. 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, which mediates between heat losses and temperature differences, is a constant. This condition is generally met in heat conduction (where it is guaranteed by Fourier's law) as the thermal conductivity of most materials is only weakly dependent on temperature. 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. Finally, in the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature differences.

## Contents

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. 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 published his work on cooling anonymously in 1701 as "Scala graduum Caloris. Calorum Descriptiones & signa." in Philosophical Transactions, volume 22, issue 270. [1] [2]

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. [3]

In 2020, Shigenao and Shuichi repeated Newton's experiments with modern apparatus, and they applied modern data reduction techniques. [4] 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 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. [5] (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:

${\displaystyle {\dot {Q}}=h\cdot A\cdot (T(t)-T_{\text{env}})=h\cdot A\cdot \Delta T(t),}$

where

${\displaystyle {\dot {Q}}}$ is the rate of heat transfer out of the body (SI unit: watt),
${\displaystyle h}$ is the heat transfer coefficient (assumed independent of T and averaged over the surface) (SI unit: W/m2-K),
${\displaystyle A}$ is the heat transfer surface area (SI unit: m2),
${\displaystyle T}$ is the temperature of the object's surface (SI unit: K),
${\displaystyle T_{\text{env}}}$ is the temperature of the environment; i.e. the temperature suitably far from the surface (SI unit: K),
${\displaystyle \Delta T(t)=T(t)-T_{\text{env}}}$ 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 on the heat transfer surface. [6] Note the heat transfer coefficient changes in a system when a transition from laminar to turbulent flow occurs.

### The Biot number

The Biot number, a dimensionless quantity, is defined for a body as

${\displaystyle {\text{Bi}}={\frac {hL_{\rm {C}}}{k_{\rm {b}}}},}$

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 ${\displaystyle L_{\rm {C}}=V_{\text{body}}/A_{\text{surface}}}$,
kb = thermal conductivity 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 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). [7] 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 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.

## 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 amount of thermal energy in the body) is calculated by assuming a constant heat capacity. 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. [8]

### First-order transient response of lumped-capacitance objects

A body treated as a lumped capacitance object, with a total internal energy of ${\displaystyle U}$ (in joules), is characterized by a single uniform internal temperature, ${\displaystyle T(t)}$. The heat capacitance, ${\displaystyle C}$, of the body is ${\displaystyle C=dU/dT}$ (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: ${\displaystyle U=C(T-T_{\text{ref}})}$.

Differentiating ${\displaystyle U}$ with respect to time gives:

${\displaystyle {\frac {dU}{dt}}=C\,{\frac {dT}{dt}}.}$

Applying the first law of thermodynamics to the lumped object gives ${\textstyle {\frac {dU}{dt}}=-Q}$, where the heat transfer out of the body, ${\displaystyle Q}$, may be expressed by Newton's law of cooling, and where no work transfer occurs for an incompressible material. Thus,

${\displaystyle {\frac {dT(t)}{dt}}=-{\frac {hA}{C}}(T(t)-T_{\text{env}})=-{\frac {1}{\tau }}\Delta T(t),}$

where the time constant of the system is ${\displaystyle \tau =C/(hA)}$. The heat capacitance ${\displaystyle C}$ may be written in terms of the object's specific heat capacity, ${\displaystyle c}$ (J/kg-K), and mass, ${\displaystyle m}$ (kg). The time constant is then ${\displaystyle \tau =mc/(hA)}$.

When the environmental temperature is constant in time, we may define ${\displaystyle \Delta T(t)=T(t)-T_{\text{env}}}$. The equation becomes

${\displaystyle {\frac {dT(t)}{dt}}={\frac {d\Delta T(t)}{dt}}=-{\frac {1}{\tau }}\Delta T(t).}$

The solution of this differential equation, by integration from the initial condition, is

${\displaystyle \Delta T(t)=\Delta T(0)\,e^{-t/\tau }.}$

where ${\displaystyle \Delta T(0)}$ is the temperature difference at time 0. Reverting to temperature, the solution is

${\displaystyle T(t)=T_{\text{env}}+(T(0)-T_{\text{env}})\,e^{-t/\tau }.}$

The temperature difference between the body and the environment decays exponentially as a function of time.

## Related Research Articles

In fluid dynamics, the Nusselt number (Nu) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.

The Grashof number (Gr) is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number. It's believed to be named after Franz Grashof. Though this grouping of terms had already been in use, it wasn't named until around 1921, 28 years after Franz Grashof's death. It's not very clear why the grouping was named after him.

The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by , , or .

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection.

Thermal insulation is the reduction of heat transfer between objects in thermal contact or in range of radiative influence. Thermal insulation can be achieved with specially engineered methods or processes, as well as with suitable object shapes and materials.

Thermal conduction is the transfer of internal energy by microscopic collisions of particles and movement of electrons within a body. The colliding particles, which include molecules, atoms and electrons, transfer disorganized microscopic kinetic and potential energy, jointly known as internal energy. Conduction takes place in all phases: solid, liquid, and gas.

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 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.

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 conduction, thermal convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system.

The lumped-element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems, mechanical multibody systems, heat transfer, acoustics, etc.

Thermal shock is a type of rapidly transient mechanical load. By definition, it is a mechanical load caused by a rapid change of temperature of a certain point. It can be also extended to the case of a thermal gradient, which makes different parts of an object expand by different amounts. This differential expansion can be more directly understood in terms of strain, than in terms of stress, as it is shown in the following. At some point, this stress can exceed the tensile strength of the material, causing a crack to form. If nothing stops this crack from propagating through the material, it will cause the object's structure to fail.

The heat transfer coefficient or film coefficient, or film effectiveness, in thermodynamics and in mechanics is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat :

The rate of heat flow is the amount of heat that is transferred per unit of time in some material, usually measured in watt. Heat is the flow of thermal energy driven by thermal non-equilibrium, so that 'heat flow' is a redundancy. Heat must not be confused with stored thermal energy, and moving a hot object from one place to another must not be called heat transfer. But, in spite of all these remarks, it is common in normal parlance to say ‘heat flow’, to talk of ‘heat content’, etc.

In physics and engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient heat conduction. Conceptually, it is the ratio of diffusive or conductive transport rate to the quantity storage rate, where the quantity may be either heat or matter (particles). The number derives from non-dimensionalization of the heat equation or Fick's second law and is used along with the Biot number to analyze time dependent transport phenomena.

Convection 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 of conduction and advection. Convection is usually the dominant form of heat transfer in liquids and gases.

Natural convection is a type of flow, of motion of a liquid such as water or a gas such as air, in which the fluid motion is not generated by any external source but by some parts of the fluid being heavier than other parts. In most cases this leads to natural circulation, the ability of a fluid in a system to circulate continuously, with gravity and possible changes in heat energy. The driving force for natural convection is gravity. For example if there is a layer of cold dense air on top of hotter less dense air, gravity pulls more strongly on the denser layer on top, so it falls while the hotter less dense air rises to take its place. This creates circulating flow: convection. As it relies on gravity, there is no convection in free-fall (inertial) environments, such as that of the orbiting International Space Station. Natural convection can occur when there are hot and cold regions of either air or water, because both water and air become less dense as they are heated. But, for example, in the world's oceans it also occurs due to salt water being heavier than fresh water, so a layer of salt water on top of a layer of fresher water will also cause convection.

Thermal resistance is a heat property and a measurement of a temperature difference by which an object or material resists a heat flow. Thermal resistance is the reciprocal of thermal conductance.

Heisler charts are a graphical analysis tool for the evaluation of one-dimensional transient conductive heat transfer in thermal engineering. They are a set of two charts per included geometry introduced in 1947 by M. P. Heisler which were supplemented by a third chart per geometry in 1961 by H. Gröber. Heisler charts permit evaluation of the central temperature for transient heat conduction through an infinitely long plane wall of thickness 2L, an infinitely long cylinder of radius ro, and a sphere of radius ro. Each aforementioned geometry can be analyzed by three charts which show the midplane temperature, temperature distribution, and heat transfer.

In engineering, physics and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics, it places a heavy emphasis on the commonalities between the topics covered. Mass, momentum, and heat transport all share a very similar mathematical framework, and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others.

CFD stands for computational fluid dynamics. As per this technique, the governing differential equations of a flow system or thermal system are known in the form of Navier–Stokes equations, thermal energy equation and species equation with an appropriate equation of state. In the past few years, CFD has been playing an increasingly important role in building design, following its continuing development for over a quarter of a century. The information provided by CFD can be used to analyse the impact of building exhausts to the environment, to predict smoke and fire risks in buildings, to quantify indoor environment quality, and to design natural ventilation systems.

The convection–diffusion equation describes the flow of heat, particles, or other physical quantities in situations where there is both diffusion and convection or advection. For information about the equation, its derivation, and its conceptual importance and consequences, see the main article convection–diffusion equation. This article describes how to use a computer to calculate an approximate numerical solution of the discretized equation, in a time-dependent situation.

## References

1. Anonymous (March–April 1701), "Scala graduum Caloris. Calorum Descriptiones & signa.", Philosophical Transactions, 22 (270): 824–829, doi:, JSTOR   102813
2. 824 829; ed. Joannes Nichols, Isaaci Newtoni Opera quae exstant omnia, vol. 4 (1782), 403 407.
3. History of Newton's cooling law Archived 2015-06-14 at the Wayback Machine
4. Maruyama, Shigenao; Moriya, Shuichi (2021). "Newton's Law of Cooling: Follow up and exploration". International Journal of Heat and Mass Transfer. 164: 120544. doi:. Retrieved November 15, 2020.
5. Whewell, William (1866). History of the Inductive Sciences from the Earliest to the Present Times.
6. Lienhard, John H., IV; Lienhard, John H., V (2019). "Laminar and turbulent boundary layers". A Heat Transfer Textbook (5th ed.). Mineola, NY: Dover Publications. pp. 271–347. ISBN   9780486837352.
7. Frank Incropera; Theodore L. Bergman; David DeWitt; Adrienne S. Lavine (2007). (6th ed.). John Wiley & Sons. pp.  260–261. ISBN   978-0-471-45728-2.
8. Lienhard, John H., IV; Lienhard, John H., V (2019). A Heat Transfer Textbook (5th ed.). Mineola, NY: Dover Publications. pp. 419–420. ISBN   9780486837352.