# 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 thermal conduction (where it is guaranteed by Fourier's law) as the thermal conductivity of most materials is only weakly dependent on temperature, but it is only very approximately true in conditions of convective heat transfer. Newton's Law is more closely followed for forced air or pumped fluid cooling, where the velocity of the fluid does not vary with temperature. Finally, in the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature changes, and a more accurate description is given by Planck's Law.

## Contents

Sir Isaac Newton did not originally state his law in the above form in 1701, when it was originally formulated. 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. [1]

When stated in terms of temperature differences, Newton's law (with several further simplifying assumptions, such as a low Biot number and temperature-independent heat capacity) results in a simple differential equation for temperature-difference as a function of time. This equation has a solution that specifies a simple negative exponential rate of temperature-difference decrease, over time. This characteristic time function for temperature-difference behavior, is also associated with Newton's law of cooling.

## Relationship to mechanism of cooling

Convection-cooling is sometimes called "Newton's law of cooling." This use is based on a work by Sir Isaac Newton published anonymously in 1701 as "Scala graduum Caloris. Calorum Descriptiones & signa." in Philosophical Transactions, volume 22, issue 270. [2] [3]

When the heat transfer coefficient is independent, or relatively independent, of the temperature difference between object and environment, Newton's law is followed. This independence is sometimes the case, but is not generally so. Newton's law is most closely obeyed in purely conduction-type cooling, but becomes a function of the temperature difference in natural convective heat transfer.In this case, Newton's law only approximates the result when the temperature difference is relatively small. Newton himself realized this limitation. (though the law holds well for forced air and pumped liquid cooling, where the fluid velocity does not rise with increasing temperature difference)

A correction to Newton's law concerning convection for larger temperature differentials by including an exponent, was made in 1817 by Dulong and Petit. [4] (These men are better-known for their formulation of the Dulong–Petit law concerning the molar specific heat capacity of a crystal.)

Another situation which also does not obey Newton's law, is radiative heat transfer, being better described by Planck's law as varying with the 4th power of absolute temperature.

## Heat transfer version of the law

The heat-transfer version of Newton's law, which (as noted) requires a constant heat transfer coefficient, states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings.

The rate of heat transfer in such circumstances is derived below: [5]

Newton's cooling law in convection is a restatement of the differential equation given by Fourier's law:

${\displaystyle {\frac {dQ}{dt}}=-h\cdot A\cdot (T(t)-T_{\text{env}})=-h\cdot A\Delta T(t),}$

where

${\displaystyle Q}$ is the thermal energy (SI unit: joule),
${\displaystyle h}$ is the heat transfer coefficient (assumed independent of T here) (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 and interior (since these are the same in this approximation) (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 thermal gradient 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 can be analyzed using the presumption that Newton's law will hold.

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 rather low compared to turbulent flows; this is due to turbulent flows having a thinner stagnant fluid film layer on the heat transfer surface. [6] However, note that Newton's law breaks down if the flow transitions between laminar or turbulent flow, since this changes the heat transfer coefficient h, which is assumed constant in solving the equation.

### The Biot number

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

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

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 ${\displaystyle L_{C}=V_{\text{body}}/A_{\text{surface}}}$,
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 at 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 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 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, 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.

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 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.

## Rate-of-change of temperature-difference version of the law

As noted in the section above, accurate formulation for temperatures may require analysis based on changing heat transfer coefficients at different temperatures, a situation frequently found in free-convection situations, and which precludes accurate use of Newton's law.

As noted above, Newton's law behavior when stated in terms of temperature change in the body, also requires that internal heat conduction within the object be large in comparison to the loss/gain of heat by surface transfer (conduction and/or convection), which is the condition where the Biot number is less than about 0.1. This allows the presumption of a single "temperature" inside the body (as a function of time) to make sense, as 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).

Assumption of rapid internal conduction also allows use of the so-called lumped capacitance model. In this model, the amount of thermal energy in the body is calculated by assuming a constant heat capacity and thus thermal energy in the body is assumed to be a linear function of the body's temperature.

### Temperature function-of-time solution in terms of object heat capacity

Given that the body is treated as a lumped capacitance thermal energy reservoir with a ${\displaystyle Q}$ (total thermal energy content) which is proportional to ${\displaystyle C}$ (simple total heat capacity) and ${\displaystyle T}$ (the temperature of the body), then ${\displaystyle Q=CT}$. It is expected that the system will experience exponential decay in the temperature difference of body and surroundings as a function of time. This is proven in the following sections:

From the definition of heat capacity ${\displaystyle C}$ comes the relation ${\displaystyle C=dQ/dT}$. Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time)

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

This expression may be used to replace ${\displaystyle dQ/dt}$ in the first equation which begins this section, above. Then, if ${\displaystyle T(t)}$ is the temperature of such a body at time ${\displaystyle t}$, and ${\displaystyle T_{\text{env}}}$ is the temperature of the environment around the body:

${\displaystyle {\frac {dT(t)}{dt}}=-r(T(t)-T_{\text{env}})=-r\Delta T(t),}$

where ${\displaystyle r=hA/C}$ is a positive constant characteristic of the system, which must be in units of ${\displaystyle {\text{time}}^{-1}}$, and is therefore sometimes expressed in terms of a characteristic time constant ${\displaystyle \tau }$ given by

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

Thus, in thermal systems, ${\displaystyle \tau =C/(hA)}$. (The total heat capacity ${\displaystyle C}$ of a system may be further represented by its mass-specific heat capacity ${\displaystyle c_{P}}$ multiplied by its mass ${\displaystyle m}$, so that the time constant ${\displaystyle \tau }$ is also given by ${\displaystyle mc_{P}/(hA)}$).

The solution of this differential equation, by standard methods of integration and substitution of boundary conditions, gives

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

If ${\displaystyle \Delta T(t)}$ is defined as ${\displaystyle T(t)-T_{\text{env}}}$, where ${\displaystyle \Delta T(0)}$ is the initial temperature difference at time 0, then the Newtonian solution is written as

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

This same solution is more immediately apparent if the initial differential equation is written in terms of ${\displaystyle \Delta T(t)}$, as a single function of time to be found, or "solved for":

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

## 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 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 conduction is the transfer of heat in internal energy by microscopic collisions of particles and movement of electrons within a body. The microscopically colliding particles, that include molecules, atoms and electrons, transfer disorganized microscopic kinetic and potential energy, jointly known as internal energy. Conduction takes place in all phases of solids, liquids, gases and waves. The rate at which energy is conducted as heat between two bodies is a function of the temperature difference temperature gradient between the two bodies and the properties of the conductive interface through which the heat is transferred.

In physics and mathematics, the heat equation is a partial differential equation that describes how the distribution of some quantity evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is a special case of the diffusion equation.

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.

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.

A heat current is a kinetic exchange rate between molecules, relative to the material in which the kinesis occurs. It is defined as , where is heat and is time.

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 watts. 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.

Sol-air temperature (Tsol-air) is a variable used to calculate cooling load of a building and determine the total heat gain through exterior surfaces. It is an improvement over:

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. 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 of 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.

In thermodynamics, heat is energy in transfer to or from a thermodynamic system, by mechanisms other than thermodynamic work or transfer of matter. The mechanisms include conduction, through direct contact of immobile bodies, or through a wall or barrier that is impermeable to matter; or radiation between separated bodies; or friction due to isochoric mechanical or electrical or magnetic or gravitational work done by the surroundings on the system of interest, such as Joule heating due to an electric current driven through the system of interest by an external system, or through a magnetic stirrer. When there is a suitable path between two systems with different temperatures, heat transfer occurs necessarily, immediately, and spontaneously from the hotter to the colder system. Thermal conduction occurs by the stochastic (random) motion of microscopic particles. In contrast, thermodynamic work is defined by mechanisms that act macroscopically and directly on the system's whole-body state variables; for example, change of the system's volume through a piston's motion with externally measurable force; or change of the system's internal electric polarization through an externally measurable change in electric field. The definition of heat transfer does not require that the process be in any sense smooth. For example, a bolt of lightning may transfer heat to a body.

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.

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. History of Newton's cooling law Archived 2015-06-14 at the Wayback Machine
2. Anonymous (March–April 1701), "Scala graduum Caloris. Calorum Descriptiones & signa.", Philosophical Transactions, 22 (270): 824–829, doi:10.1098/rstl.1700.0082, JSTOR   102813
3. 824 829; ed. Joannes Nichols, Isaaci Newtoni Opera quae exstant omnia, vol. 4 (1782), 403 407.
4. Whewell, William (1866). History of the Inductive Sciences from the Earliest to the Present Times.
5. Louis C. Burmeister, (1993) “Convective Heat Transfer”, 2nd ed. Publisher Wiley-Interscience, p. 107. ISBN   0-471-57709-X, 9780471577096.
6. “Convection Heat Transfer”, Engineers Edge, 2009, Accessed 20.03.2009.
7. Frank Incropera; Theodore L. Bergman; David DeWitt; Adrienne S. Lavine (2007). Fundamentals of Heat and Mass Transfer (6th ed.). John Wiley & Sons. pp. 260–261. ISBN   978-0-471-45728-2.