Conduction is the process by which
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 al ...
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
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 ...
'', and is denoted .
Heat spontaneously flows along a
temperature gradient
A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature gradient is a dimensional quantity expressed in units of degree ...
(i.e. from a hotter body to a colder body). For example, heat is conducted from the
hotplate
A hot plate is a portable self-contained tabletop small appliance cooktop
A cooktop (American English), stovetop (American English) or hob (British English), is a device commonly used for cooking that is commonly found in kitchens and used t ...
of an electric stove to the bottom of a saucepan in contact with it. In the absence of an opposing external driving energy source, within a body or between bodies,
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 ...
differences decay over time, and
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
is approached, temperature becoming more uniform.
In conduction, the heat flow is within and through the body itself. In contrast, in 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) is ...
, the transfer is often between bodies, which may be separated spatially. Heat can also be transferred by a combination of conduction and radiation. In
solids
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ri ...
, conduction is mediated by the combination of vibrations and collisions of molecules, propagation and collisions of
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
s, and diffusion and collisions of
free electrons. In
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es and
liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
s, conduction is due to the collisions and
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
of molecules during their
random motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
Photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alway ...
in this context do not collide with one another, and so heat transport by
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
is conceptually distinct from heat conduction by microscopic diffusion and collisions of material particles and phonons. But the distinction is often not easily observed unless the material is semi-transparent.
In the engineering sciences, heat transfer includes the processes of
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) is ...
,
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
, and sometimes
mass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration, ...
. Usually, more than one of these processes occurs in a given situation.
Overview
On a microscopic scale, conduction occurs within a body considered as being stationary; this means that the kinetic and potential energies of the bulk motion of the body are separately accounted for.
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 ...
diffuses as rapidly moving or vibrating atoms and
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s interact with neighbouring particles, transferring some of their microscopic kinetic and potential energies, these quantities being defined relative to the bulk of the body considered as being stationary. Heat is transferred by conduction when adjacent atoms or molecules collide, or as several
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s move backwards and forwards from atom to atom in a disorganized way so as not to form a macroscopic electric current, or as photons collide and scatter. Conduction is the most significant means of heat transfer within a solid or between solid objects in
thermal contact
In heat transfer and thermodynamics, a thermodynamic system is said to be in thermal contact with another system if it can exchange energy through the process of heat. Perfect thermal isolation is an idealization as real systems are always in therm ...
. Conduction is greater in solids because the network of relatively close fixed spatial relationships between atoms helps to transfer energy between them by vibration.
Thermal contact conductance
In physics, thermal contact conductance is the study of heat conduction between solid or liquid bodies in thermal contact. The thermal contact conductance coefficient, h_c, is a property indicating the thermal conductivity, or ability to conduct he ...
is the study of heat conduction between solid bodies in contact. A temperature drop is often observed at the interface between the two surfaces in contact. This phenomenon is said to be a result of a thermal contact resistance existing between the contacting surfaces.
Interfacial thermal resistance
Interfacial thermal resistance, also known as thermal boundary resistance, or Kapitza resistance, is a measure of resistance to thermal flow at the interface between two materials. While these terms may be used interchangeably, Kapitza resistance ...
is a measure of an interface's resistance to thermal flow. This thermal resistance differs from contact resistance, as it exists even at atomically perfect interfaces. Understanding the thermal resistance at the interface between two materials is of primary significance in the study of its thermal properties. Interfaces often contribute significantly to the observed properties of the materials.
The inter-molecular transfer of energy could be primarily by elastic impact, as in fluids, or by free-electron diffusion, as in metals, or
phonon vibration, as in insulators. In
insulators
Insulator may refer to:
* Insulator (electricity), a substance that resists electricity
** Pin insulator, a device that isolates a wire from a physical support such as a pin on a utility pole
** Strain insulator, a device that is designed to work ...
, the heat flux is carried almost entirely by
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
vibrations.
Metals (e.g., copper, platinum, gold, etc.) are usually good
conductors of thermal energy. This is due to the way that metals bond chemically:
metallic bond
Metallic bonding is a type of chemical bonding that arises from the electrostatic attractive force between conduction electrons (in the form of an electron cloud of delocalized electrons) and positively charged metal ions. It may be descri ...
s (as opposed to
covalent
A covalent bond is a chemical bond that involves the sharing of electrons to form electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs. The stable balance of attractive and repulsive forces between atoms ...
or
ionic bonds
Ionic bonding is a type of chemical bonding that involves the electrostatic attraction between oppositely charged ions, or between two atoms with sharply different electronegativities, and is the primary interaction occurring in ionic compounds. ...
) have free-moving electrons that transfer thermal energy rapidly through the metal. The ''electron fluid'' of a
conductive
In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric current is gener ...
metallic solid conducts most of the heat flux through the solid. Phonon flux is still present but carries less of the energy. Electrons also conduct
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
through conductive solids, and the
thermal
A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
and
electrical conductivities of most metals have about the same ratio. A good electrical conductor, such as
copper
Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkis ...
, also conducts heat well.
Thermoelectricity
The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. A thermoelectric device creates a voltage when there is a different temperature on each side. Conversely, when ...
is caused by the interaction of heat flux and electric current. Heat conduction within a solid is directly analogous to
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
of particles within a fluid, in the situation where there are no fluid currents.
In gases, heat transfer occurs through collisions of gas molecules with one another. In the absence of convection, which relates to a moving fluid or gas phase, thermal conduction through a gas phase is highly dependent on the composition and pressure of this phase, and in particular, the mean free path of gas molecules relative to the size of the gas gap, as given by the
Knudsen number
The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is named ...
.
To quantify the ease with which a particular medium conducts, engineers employ 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 ...
, also known as the conductivity constant or conduction coefficient, ''k''. In
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 ...
, ''k'' is defined as "the quantity of heat, ''Q'', transmitted in time (''t'') through a thickness (''L''), in a direction normal to a surface of area (''A''), due to a temperature difference (Δ''T'')
... Thermal conductivity is a material ''
property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
'' that is primarily dependent on the medium's
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
, temperature, density, and molecular bonding.
Thermal effusivity
In thermodynamics, a material's thermal effusivity, thermal inertia or thermal responsivity is a measure of its ability to exchange thermal energy with its surroundings. It is defined as the square root of the product of the material's thermal co ...
is a quantity derived from conductivity, which is a measure of its ability to exchange thermal energy with its surroundings.
Steady-state conduction
Steady-state conduction is the form of conduction that happens when the temperature difference(s) driving the conduction are constant, so that (after an equilibration time), the spatial distribution of temperatures (temperature field) in the conducting object does not change any further. Thus, all partial derivatives of temperature ''concerning space'' may either be zero or have nonzero values, but all derivatives of temperature at any point ''concerning time'' are uniformly zero. In steady-state conduction, the amount of heat entering any region of an object is equal to the amount of heat coming out (if this were not so, the temperature would be rising or falling, as thermal energy was tapped or trapped in a region).
For example, a bar may be cold at one end and hot at the other, but after a state of steady-state conduction is reached, the spatial gradient of temperatures along the bar does not change any further, as time proceeds. Instead, the temperature remains constant at any given cross-section of the rod normal to the direction of heat transfer, and this temperature varies linearly in space in the case where there is no heat generation in the rod.
In steady-state conduction, all the laws of direct current electrical conduction can be applied to "heat currents". In such cases, it is possible to take "thermal resistances" as the analog to
electrical resistance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
s. In such cases, temperature plays the role of voltage, and heat transferred per unit time (heat power) is the analog of electric current. Steady-state systems can be modeled by networks of such thermal resistances in series and parallel, in exact analogy to electrical networks of resistors. See
purely resistive thermal circuits for an example of such a network.
Transient conduction
During any period in which temperatures changes ''in time'' at any place within an object, the mode of thermal energy flow is termed ''transient conduction.'' Another term is "non-steady-state" conduction, referring to the time-dependence of temperature fields in an object. Non-steady-state situations appear after an imposed change in temperature at a boundary of an object. They may also occur with temperature changes inside an object, as a result of a new source or sink of heat suddenly introduced within an object, causing temperatures near the source or sink to change in time.
When a new perturbation of temperature of this type happens, temperatures within the system change in time toward a new equilibrium with the new conditions, provided that these do not change. After equilibrium, heat flow into the system once again equals the heat flow out, and temperatures at each point inside the system no longer change. Once this happens, transient conduction is ended, although steady-state conduction may continue if heat flow continues.
If changes in external temperatures or internal heat generation changes are too rapid for the equilibrium of temperatures in space to take place, then the system never reaches a state of unchanging temperature distribution in time, and the system remains in a transient state.
An example of a new source of heat "turning on" within an object, causing transient conduction, is an engine starting in an automobile. In this case, the transient thermal conduction phase for the entire machine is over, and the steady-state phase appears, as soon as the engine reaches steady-state
operating temperature
An operating temperature is the allowable temperature range of the local ambient environment at which an electrical or mechanical device operates. The device will operate effectively within a specified temperature range which varies based on the de ...
. In this state of steady-state equilibrium, temperatures vary greatly from the engine cylinders to other parts of the automobile, but at no point in space within the automobile does temperature increase or decrease. After establishing this state, the transient conduction phase of heat transfer is over.
New external conditions also cause this process: for example, the copper bar in the example steady-state conduction experiences transient conduction as soon as one end is subjected to a different temperature from the other. Over time, the field of temperatures inside the bar reaches a new steady-state, in which a constant temperature gradient along the bar is finally set up, and this gradient then stays constant in time. Typically, such a new steady-state gradient is approached exponentially with time after a new temperature-or-heat source or sink, has been introduced. When a "transient conduction" phase is over, heat flow may continue at high power, so long as temperatures do not change.
An example of transient conduction that does not end with steady-state conduction, but rather no conduction, occurs when a hot copper ball is dropped into oil at a low temperature. Here, the temperature field within the object begins to change as a function of time, as the heat is removed from the metal, and the interest lies in analyzing this spatial change of temperature within the object over time until all gradients disappear entirely (the ball has reached the same temperature as the oil). Mathematically, this condition is also approached exponentially; in theory, it takes infinite time, but in practice, it is over, for all intents and purposes, in a much shorter period. At the end of this process with no heat sink but the internal parts of the ball (which are finite), there is no steady-state heat conduction to reach. Such a state never occurs in this situation, but rather the end of the process is when there is no heat conduction at all.
The analysis of non-steady-state conduction systems is more complex than that of steady-state systems. If the conducting body has a simple shape, then exact analytical mathematical expressions and solutions may be possible (see
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 t ...
for the analytical approach). However, most often, because of complicated shapes with varying
thermal conductivities within the shape (i.e., most complex objects, mechanisms or machines in engineering) often the application of approximate theories is required, and/or numerical analysis by computer. One popular graphical method involves the use of
Heisler Chart
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 suppleme ...
s.
Occasionally, transient conduction problems may be considerably simplified if regions of the object being heated or cooled can be identified, for which
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 ...
is very much greater than that for heat paths leading into the region. In this case, the region with high conductivity can often be treated in the
lumped capacitance model
The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...
, as a "lump" of material with a simple thermal capacitance consisting of its aggregate
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 capacity i ...
. Such regions warm or cool, but show no significant temperature ''variation'' across their extent, during the process (as compared to the rest of the system). This is due to their far higher conductance. During transient conduction, therefore, the temperature across their conductive regions changes uniformly in space, and as a simple exponential in time. An example of such systems is those that follow
Newton's law of cooling
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 q ...
during transient cooling (or the reverse during heating). The equivalent thermal circuit consists of a simple capacitor in series with a resistor. In such cases, the remainder of the system with a high thermal resistance (comparatively low conductivity) plays the role of the resistor in the circuit.
Relativistic conduction
The theory of
relativistic heat conduction
Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. In special (and general) relativity, the usual heat equation for non-relativistic heat conducti ...
is a model that is compatible with the theory of special relativity. For most of the last century, it was recognized that the Fourier equation is in contradiction with the theory of relativity because it admits an infinite speed of propagation of heat signals. For example, according to the Fourier equation, a pulse of heat at the origin would be felt at infinity instantaneously. The speed of information propagation is faster than the speed of light in a vacuum, which is physically inadmissible within the framework of relativity.
Quantum conduction
Second sound
Second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave-like motion, rather than by the more usual mechanism of diffusion. Its presence leads to a very high thermal conductivity. It is known as "second sound" because t ...
is a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
phenomenon in which
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 conduction, ...
occurs by
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
-like motion, rather than by the more usual mechanism of
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. Heat takes the place of pressure in normal sound waves. This leads to a very high
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 ...
. It is known as "second sound" because the wave motion of heat is similar to the propagation of sound in air.
Fourier's law
The law of heat conduction, also known as
Fourier's law, states that the rate 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 conduction, ...
through a material is
proportional to the negative
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 gradi ...
in the temperature and to the area, at right angles to that gradient, through which the heat flows. We can state this law in two equivalent forms: the integral form, in which we look at the amount of energy flowing into or out of a body as a whole, and the differential form, in which we look at the flow rates or
fluxes of energy locally.
Newton's law of cooling
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 q ...
is a discrete analogue of Fourier's law, while
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
is the electrical analogue of Fourier's law and
Fick's laws 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 equ ...
is its chemical analogue.
Differential form
The differential form of Fourier's law of thermal conduction shows that the local
heat flux
Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time. In SI its units are watts per square metre (W/m2). It has both a ...
density
is equal to the product of
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 ...
and the negative local temperature gradient
. The heat flux density is the amount of energy that flows through a unit area per unit time.
where (including the
SI units)
*
is the local heat flux density,
W/m
2,
*
is the material's
conductivity
Conductivity may refer to:
*Electrical conductivity, a measure of a material's ability to conduct an electric current
**Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution
**Ionic conductivity (solid state), elec ...
, W/(m·
K),
*
is the temperature gradient, K/m.
The thermal conductivity
is often treated as a constant, though this is not always true. While the thermal conductivity of a material generally varies with temperature, the variation can be small over a significant range of temperatures for some common materials. In
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
materials, the thermal conductivity typically varies with orientation; in this case
is represented by a second-order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
. In non-uniform materials,
varies with spatial location.
For many simple applications, Fourier's law is used in its one-dimensional form, for example, in the direction:
In an isotropic medium, Fourier's law leads to
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 t ...
with a
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...
famously known as
heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectru ...
.
Integral form
By integrating the differential form over the material's total surface
, we arrive at the integral form of Fourier's law:
:
where (including the
SI units):
*
is the amount of heat transferred per unit time (in W),
*
is an
oriented surface area element (in m
2).
The above
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, an ...
, when
integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as
where
*
is the time interval during which the amount of heat
flows through a cross-section of the material,
*
is the cross-sectional surface area,
*
is the temperature difference between the ends,
*
is the distance between the ends.
This law forms the basis for the derivation of 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 t ...
.
Conductance
Writing
where is the conductance, in W/(m
2 K).
Fourier's law can also be stated as:
The reciprocal of conductance is resistance,
is given by:
Resistance is additive when several conducting layers lie between the hot and cool regions, because and are the same for all layers. In a multilayer partition, the total conductance is related to the conductance of its layers by:
or equivalently
So, when dealing with a multilayer partition, the following formula is usually used:
For heat conduction from one fluid to another through a barrier, it is sometimes important to consider the conductance of the
thin film
A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ap ...
of fluid that remains stationary next to the barrier. This thin film of fluid is difficult to quantify because its characteristics depend upon complex conditions of
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
and
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
—but when dealing with thin high-conductance barriers it can sometimes be quite significant.
Intensive-property representation
The previous conductance equations, written in terms of
extensive properties
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
, can be reformulated in terms of
intensive properties
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
. Ideally, the formulae for conductance should produce a quantity with dimensions independent of distance, like
Ohm's Law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
for electrical resistance,
, and conductance,
.
From the electrical formula:
, where ''ρ'' is resistivity, ''x'' is length, and ''A'' is cross-sectional area, we have
, where ''G'' is conductance, ''k'' is conductivity, ''x'' is length, and ''A'' is cross-sectional area.
For heat,
where is the conductance.
Fourier's law can also be stated as:
analogous to Ohm's law,
or
The reciprocal of conductance is resistance, ''R'', given by:
analogous to Ohm's law,
The rules for combining resistances and conductances (in series and parallel) are the same for both heat flow and electric current.
Cylindrical shells
Conduction through cylindrical shells (e.g. pipes) can be calculated from the internal radius,
, the external radius,
, the length,
, and the temperature difference between the inner and outer wall,
.
The surface area of the cylinder is
When Fourier's equation is applied:
and rearranged:
then the rate of heat transfer is:
the thermal resistance is:
and
, where
. It is important to note that this is the log-mean radius.
Spherical
The conduction through a spherical shell with internal radius,
, and external radius,
, can be calculated in a similar manner as for a cylindrical shell.
The
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
of the sphere is:
Solving in a similar manner as for a cylindrical shell (see above) produces:
Transient thermal conduction
Interface heat transfer
The heat transfer at an interface is considered a transient heat flow. To analyze this problem, the Biot number is important to understand how the system behaves. The Biot number is determined by:
The heat transfer coefficient
, is introduced in this formula, and is measured in
. If the system has a Biot number of less than 0.1, the material behaves according to Newtonian cooling, i.e. with negligible temperature gradient within the body. If the Biot number is greater than 0.1, the system behaves as a series solution. The temperature profile in terms of time can be derived from the equation
which becomes
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 measured in
, and represents the transfer of heat at an interface between two materials. This value is different at every interface and is an important concept in understanding heat flow at an interface.
The series solution can be analyzed with a
nomogram
A nomogram (from Greek , "law" and , "line"), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. ...
. A nomogram has a relative temperature as the coordinate and the Fourier number, which is calculated by
The Biot number increases as the Fourier number decreases. There are five steps to determine a temperature profile in terms of time.
# Calculate the Biot number
# Determine which relative depth matters, either ''x'' or ''L''.
# Convert time to the Fourier number.
# Convert
to relative temperature with the boundary conditions.
# Compared required to point to trace specified Biot number on the nomogram.
Thermal conduction applications
Splat cooling
Splat cooling Splat quenching is a metallurgical, metal morphing technique used for forming metals with a particular crystal structure by means of extremely rapid quenching, or cooling.
A typical technique for splat quenching involves casting molten metal by pou ...
is a method for quenching small droplets of molten materials by rapid contact with a cold surface. The particles undergo a characteristic cooling process, with the heat profile at
for initial temperature as the maximum at
and
at
and
, and the heat profile at
for
as the boundary conditions. Splat cooling rapidly ends in a steady state temperature, and is similar in form to the Gaussian diffusion equation. The temperature profile, with respect to the position and time of this type of cooling, varies with:
Splat cooling is a fundamental concept that has been adapted for practical use in the form of
thermal spraying
Thermal spraying techniques are coating processes in which melted (or heated) materials are sprayed onto a surface. The "feedstock" (coating precursor) is heated by electrical (plasma or arc) or chemical means (combustion flame).
Thermal sprayi ...
. The
thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI ...
coefficient, represented as
, can be written as
. This varies according to the material.
Metal quenching
Metal
quenching
In materials science, quenching is the rapid cooling of a workpiece in water, oil, polymer, air, or other fluids to obtain certain material properties. A type of heat treating, quenching prevents undesired low-temperature processes, such as pha ...
is a transient heat transfer process in terms of the
time temperature transformation (TTT). It is possible to manipulate the cooling process to adjust the phase of a suitable material. For example, appropriate quenching of steel can convert a desirable proportion of its content of
austenite
Austenite, also known as gamma-phase iron (γ-Fe), is a metallic, non-magnetic allotrope of iron or a solid solution of iron with an alloying element. In plain-carbon steel, austenite exists above the critical eutectoid temperature of 1000 K ...
to
martensite
Martensite is a very hard form of steel crystalline structure. It is named after German metallurgist Adolf Martens. By analogy the term can also refer to any crystal structure that is formed by diffusionless transformation.
Properties
Mart ...
, creating a very hard and strong product. To achieve this, it is necessary to quench at the "nose" (or
eutectic) of the TTT diagram. Since materials differ in their
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 ...
s, the time it takes for the material to quench, or the
Fourier number, varies in practice.
In steel, the quenching temperature range is generally from 600 °C to 200 °C. To control the quenching time and to select suitable quenching media, it is necessary to determine the Fourier number from the desired quenching time, the relative temperature drop, and the relevant Biot number. Usually, the correct figures are read from a standard
nomogram
A nomogram (from Greek , "law" and , "line"), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. ...
. By calculating the heat transfer coefficient from this Biot number, one can find a liquid medium suitable for the application.
Zeroth law of thermodynamics
One statement of the so-called
zeroth law of thermodynamics
The zeroth law of thermodynamics is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fowl ...
is directly focused on the idea of conduction of heat. Bailyn (1994) writes that "the zeroth law may be stated: All diathermal walls are equivalent".
[Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics, New York, , page 23.]
A
diathermal wall is a physical connection between two bodies that allows the passage of heat between them. Bailyn is referring to diathermal walls that exclusively connect two bodies, especially conductive walls.
This statement of the "zeroth law" belongs to an idealized theoretical discourse, and actual physical walls may have peculiarities that do not conform to its generality.
For example, the material of the wall must not undergo a phase transition, such as evaporation or fusion, at the temperature at which it must conduct heat. But when only thermal equilibrium is considered and time is not urgent, so that the conductivity of the material does not matter too much, one suitable heat conductor is as good as another. Conversely, another aspect of the zeroth law is that, subject again to suitable restrictions, a given diathermal wall is indifferent to the nature of the heat bath to which it is connected. For example, the glass bulb of a thermometer acts as a diathermal wall whether exposed to a gas or a liquid, provided that they do not corrode or melt it.
These differences are among the defining characteristics 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 conduction, ...
. In a sense, they are
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of heat transfer.
Thermal conduction instruments
Thermal conductivity analyzer
Thermal conduction property of any gas under standard conditions of pressure and temperature is a fixed quantity. This property of a known reference gas or known reference gas mixtures can, therefore, be used for certain sensory applications, such as the thermal conductivity analyzer.
The working of this instrument is by principle based on the Wheatstone bridge containing four filaments whose resistances are matched. Whenever a certain gas is passed over such network of filaments, their resistance changes due to the altered thermal conductivity of the filaments and thereby changing the net voltage output from the Wheatstone Bridge. This voltage output will be correlated with the database to identify the gas sample.
Gas sensor
The principle of thermal conductivity of gases can also be used to measure the concentration of a gas in a binary mixture of gases.
Working: if the same gas is present around all the Wheatstone bridge filaments, then the same temperature is maintained in all the filaments and hence same resistances are also maintained; resulting in a balanced Wheatstone bridge. However, If the dissimilar gas sample (or gas mixture) is passed over one set of two filaments and the reference gas on the other set of two filaments, then the Wheatstone bridge becomes unbalanced. And the resulting net voltage output of the circuit will be correlated with the database to identify the constituents of the sample gas.
Using this technique many unknown gas samples can be identified by comparing their thermal conductivity with other reference gas of known thermal conductivity. The most commonly used reference gas is nitrogen; as the thermal conductivity of most common gases (except hydrogen and helium) are similar to that of nitrogen.
See also
*
List of thermal conductivities
In heat transfer, the thermal conductivity of a substance, ''k'', is an intensive property that indicates its ability to conduct heat. For most materials, the amount of heat conducted varies (usually non-linearly) with temperature.
Thermal con ...
*
Electrical conduction
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
*
Convection diffusion equation
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convect ...
*
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 per ...
*
Heat pipe
A heat pipe is a heat-transfer device that employs phase transition to transfer heat between two solid interfaces.
At the hot interface of a heat pipe, a volatile liquid in contact with a thermally conductive solid surface turns into a vapor ...
*
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 eq ...
*
Relativistic heat conduction
Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. In special (and general) relativity, the usual heat equation for non-relativistic heat conducti ...
*
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 wit ...
References
*Dehghani, F 2007, CHNG2801 – Conservation and Transport Processes: Course Notes, University of Sydney, Sydney
* John H Lienhard IV and John H Lienhard V, 'A Heat Transfer Textbook', Fifth Edition, Dover Pub., Mineola, NY, 201
External links
*
Heat conduction– Thermal-FluidsPedia
Newton's Law of Coolingby Jeff Bryant based on a program by
Stephen Wolfram
Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...
,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
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{{DEFAULTSORT:Thermal Conduction
Heat conduction
Heat transfer
Physical quantities
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bg:Топлопроводимост
Transport phenomena