The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by , , or .
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal conductivity. For instance, metals typically have high thermal conductivity and are very efficient at conducting heat, while the opposite is true for insulating materials like Styrofoam. Correspondingly, materials of high thermal conductivity are widely used in heat sink applications and materials of low thermal conductivity are used as thermal insulation. The reciprocal of thermal conductivity is called thermal resistivity.
The defining equation for thermal conductivity is , where is the heat flux, is the thermal conductivity, and is the temperature gradient. This is known as Fourier's Law for heat conduction. Although commonly expressed as a scalar, the most general form of thermal conductivity is a second-rank tensor. However, the tensorial description only becomes necessary in materials which are anisotropic.
Consider a solid material placed between two environments of different temperatures. Let be the temperature at and be the temperature at , and suppose . A possible realization of this scenario is a building on a cold winter day: the solid material in this case would be the building wall, separating the cold outdoor environment from the warm indoor environment.
According to the second law of thermodynamics, heat will flow from the hot environment to the cold one in an attempt to equalize the temperature difference. This is quantified in terms of a heat flux , which gives the rate, per unit area, at which heat flows in a given direction (in this case the x-direction). In many materials, is observed to be directly proportional to the temperature difference and inversely proportional to the separation:^{[1]}
The constant of proportionality is the thermal conductivity; it is a physical property of the material. In the present scenario, since heat flows in the minus x-direction and is negative, which in turn means that . In general, is always defined to be positive. The same definition of can also be extended to gases and liquids, provided other modes of energy transport, such as convection and radiation, are eliminated.
For simplicity, we have assumed here that the does not vary significantly as temperature is varied from to . Cases in which the temperature variation of is non-negligible must be addressed using the more general definition of discussed below.
Thermal conduction is defined as the transport of energy due to random molecular motion across a temperature gradient. It is distinguished from energy transport by convection and molecular work in that it does not involve macroscopic flows or work-performing internal stresses.
Energy flow due to thermal conduction is classified as heat and is quantified by the vector , which gives the heat flux at position and time . According to the second law of thermodynamics, heat flows from high to low temperature. Hence, it reasonable to postulate that is proportional to the gradient of the temperature field , i.e.
where the constant of proportionality, , is the thermal conductivity. This is called Fourier's law of heat conduction. In actuality, it is not a law but a definition of thermal conductivity in terms of the independent physical quantities and .^{[2]}^{[3]} As such, its usefulness depends on the ability to determine for a given material under given conditions. Note that itself usually depends on and thereby implicitly on space and time. An explicit space and time dependence could also occur if the material is inhomogeneous or changing with time.^{[4]}
In some solids, thermal conduction is anisotropic, i.e. the heat flux is not always parallel to the temperature gradient. To account for such behavior, a tensorial form of Fourier's law must be used:
where is symmetric, second-rank tensor called the thermal conductivity tensor.^{[5]}
An implicit assumption in the above description is the presence of local thermodynamic equilibrium, which allows one to define a temperature field .
In engineering practice, it is common to work in terms of quantities which are derivative to thermal conductivity and implicitly take into account design-specific features such as component dimensions.
For instance, thermal conductance is defined as the quantity of heat that passes in unit time through a plate of particular area and thickness when its opposite faces differ in temperature by one kelvin. For a plate of thermal conductivity , area and thickness , the conductance is , measured in W⋅K^{−1}.^{[6]} The relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance.
Thermal resistance is the inverse of thermal conductance.^{[6]} It is a convenient measure to use in multicomponent design since thermal resistances are additive when occurring in series.^{[7]}
There is also a measure known as the heat transfer coefficient: the quantity of heat that passes in unit time through a unit area of a plate of particular thickness when its opposite faces differ in temperature by one kelvin. In ASTM C168-15, this area-independent quantity is referred to as the "thermal conductance".^{[8]} The reciprocal of the heat transfer coefficient is thermal insulance. In summary, for a plate of thermal conductivity , area and thickness , we have
The heat transfer coefficient is also known as thermal admittance in the sense that the material may be seen as admitting heat to flow.
An additional term, thermal transmittance, quantifies the thermal conductance of a structure along with heat transfer due to convection and radiation. It is measured in the same units as thermal conductance and is sometimes known as the composite thermal conductance. The term U-value is also used.
Finally, thermal diffusivity combines thermal conductivity with density and specific heat:^{[9]}
As such, it quantifies the thermal inertia of a material, i.e. the relative difficulty in heating a material to a given temperature using heat sources applied at the boundary.^{[10]}
In the International System of Units (SI), thermal conductivity is measured in watts per meter-kelvin (W/(m⋅K)). Some papers report in watts per centimeter-kelvin (W/(cm⋅K)).
In imperial units, thermal conductivity is measured in BTU/(h⋅ft⋅°F).^{[note 1]}^{[11]}
The dimension of thermal conductivity is M^{1}L^{1}T^{−3}Θ^{−1}, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ).
Other units which are closely related to the thermal conductivity are in common use in the construction and textile industries. The construction industry makes use of units such as the R-value (resistance) and the U-value (transmittance). Although related to the thermal conductivity of a material used in an insulation product, R- and U-values are dependent on the thickness of the product.^{[note 2]}
Likewise the textile industry has several units including the tog and the clo which express thermal resistance of a material in a way analogous to the R-values used in the construction industry.
There are several ways to measure thermal conductivity; each is suitable for a limited range of materials. Broadly speaking, there are two categories of measurement techniques: steady-state and transient. Steady-state techniques infer the thermal conductivity from measurements on the state of a material once a steady-state temperature profile has been reached, whereas transient techniques operate on the instantaneous state of a system during the approach to steady state. Lacking an explicit time component, steady-state techniques do not require complicated signal analysis (steady state implies constant signals). The disadvantage is that a well-engineered experimental setup is usually needed, and the time required to reach steady state precludes rapid measurement.
In comparison with solid materials, the thermal properties of fluids are more difficult to study experimentally. This is because in addition to thermal conduction, convective and radiative energy transport are usually present unless measures are taken to limit these processes. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity.^{[12]}^{[13]}
The thermal conductivities of common substances span at least four orders of magnitude. Gases generally have low thermal conductivity, and pure metals have high thermal conductivity. For example, under standard conditions the thermal conductivity of copper is over 10000 times that of air.
Of all materials, allotropes of carbon, such as graphite and diamond, are usually credited with having the highest thermal conductivities at room temperature.^{[14]} The thermal conductivity of natural diamond at room temperature is several times higher than that of a highly conductive metal such as copper (although the precise value varies depending on the diamond type).^{[15]}
Thermal conductivities of selected substances are tabulated here; an expanded list can be found in the list of thermal conductivities. These values should be considered approximate due to the uncertainties related to material definitions.
Substance | Thermal conductivity (W·m^{−1}·K^{−1}) | Temperature (°C) |
---|---|---|
Air^{[16]} | 0.026 | 25 |
Styrofoam^{[17]} | 0.033 | 25 |
Water^{[18]} | 0.6089 | 26.85 |
Concrete^{[18]} | 0.92 | – |
Copper^{[18]} | 384.1 | 18.05 |
Natural diamond^{[15]} | 895–1350 | 26.85 |
The effect of temperature on thermal conductivity is different for metals and nonmetals. In metals, heat conductivity is primarily due to free electrons. Following the Wiedemann–Franz law, thermal conductivity of metals is approximately proportional to the absolute temperature (in kelvins) times electrical conductivity. In pure metals the electrical conductivity decreases with increasing temperature and thus the product of the two, the thermal conductivity, stays approximately constant. However, as temperatures approach absolute zero, the thermal conductivity decreases sharply.^{[19]} In alloys the change in electrical conductivity is usually smaller and thus thermal conductivity increases with temperature, often proportionally to temperature. Many pure metals have a peak thermal conductivity between 2 K and 10 K.
On the other hand, heat conductivity in nonmetals is mainly due to lattice vibrations (phonons). Except for high quality crystals at low temperatures, the phonon mean free path is not reduced significantly at higher temperatures. Thus, the thermal conductivity of nonmetals is approximately constant at high temperatures. At low temperatures well below the Debye temperature, thermal conductivity decreases, as does the heat capacity, due to carrier scattering from defects at very low temperatures.^{[19]}
When a material undergoes a phase change (e.g. from solid to liquid), the thermal conductivity may change abruptly. For instance, when ice melts to form liquid water at 0 °C, the thermal conductivity changes from 2.18 W/(m⋅K) to 0.56 W/(m⋅K).^{[20]}
Even more dramatically, the thermal conductivity of a fluid diverges in the vicinity of the vapor-liquid critical point.^{[21]}
Some substances, such as non-cubic crystals, can exhibit different thermal conductivities along different crystal axes, due to differences in phonon coupling along a given crystal axis. Sapphire is a notable example of variable thermal conductivity based on orientation and temperature, with 35 W/(m⋅K) along the C-axis and 32 W/(m⋅K) along the A-axis.^{[22]} Wood generally conducts better along the grain than across it. Other examples of materials where the thermal conductivity varies with direction are metals that have undergone heavy cold pressing, laminated materials, cables, the materials used for the Space Shuttle thermal protection system, and fiber-reinforced composite structures.^{[23]}
When anisotropy is present, the direction of heat flow may not be exactly the same as the direction of the thermal gradient.
In metals, thermal conductivity approximately tracks electrical conductivity according to the Wiedemann–Franz law, as freely moving valence electrons transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals. Highly electrically conductive silver is less thermally conductive than diamond, which is an electrical insulator, but due to its orderly array of atoms it is conductive of heat via phonons.
The influence of magnetic fields on thermal conductivity is known as the thermal Hall effect or Righi–Leduc effect.
Air and other gases are generally good insulators, in the absence of convection. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which obstruct heat conduction pathways. Examples of these include expanded and extruded polystyrene (popularly referred to as "styrofoam") and silica aerogel, as well as warm clothes. Natural, biological insulators such as fur and feathers achieve similar effects by trapping air in pores, pockets or voids, thus dramatically inhibiting convection of air or water near an animal's skin.
Low density gases, such as hydrogen and helium typically have high thermal conductivity. Dense gases such as xenon and dichlorodifluoromethane have low thermal conductivity. An exception, sulfur hexafluoride, a dense gas, has a relatively high thermal conductivity due to its high heat capacity. Argon and krypton, gases denser than air, are often used in insulated glazing (double paned windows) to improve their insulation characteristics.
The thermal conductivity through bulk materials in porous or granular form is governed by the type of gas in the gaseous phase, and its pressure.^{[24]} At lower pressures, the thermal conductivity of a gaseous phase is reduced, with this behaviour governed by the Knudsen number, defined as , where is the mean free path of gas molecules and is the typical gap size of the space filled by the gas. In a granular material corresponds to the characteristic size of the gaseous phase in the pores or intergranular spaces.^{[24]}
The thermal conductivity of a crystal can depend strongly on isotopic purity, assuming other lattice defects are negligible. A notable example is diamond: at a temperature of around 100 K the thermal conductivity increases from 10,000 W·m^{−1}·K^{−1} for natural type IIa diamond (98.9% ^{12}C), to 41,000 for 99.9% enriched synthetic diamond. A value of 200,000 is predicted for 99.999% ^{12}C at 80 K, assuming an otherwise pure crystal.^{[25]}
The atomic mechanisms of thermal conduction vary among different materials, and in general depend on details of the microscopic structure and atomic interactions. As such, thermal conductivity is difficult to predict from first-principles. Any expressions for thermal conductivity which are exact and general, e.g. the Green-Kubo relations, are difficult to apply in practice, typically consisting of averages over multiparticle correlation functions.^{[26]} A notable exception is a dilute gas, for which a well-developed theory exists expressing thermal conductivity accurately and explicitly in terms of molecular parameters.
In a gas, thermal conduction is mediated by discrete molecular collisions. In a simplified picture of a solid, thermal conduction occurs by two mechanisms: 1) the migration of free electrons and 2) lattice vibrations (phonons). The first mechanism dominates in pure metals, and the second mechanism dominates in non-metallic solids. In liquids, by contrast, the precise microscopic mechanisms of thermal conduction are poorly understood.^{[27]}
In a simplified model of a dilute monatomic gas, molecules are modeled as rigid spheres which are in constant motion, colliding elastically with each other and with the walls of their container. Consider such a gas at temperature and with density , specific heat and molecular mass . Under these assumptions, an elementary calculation yields for the thermal conductivity
where is a numerical constant of order , is the Boltzmann constant, and is the mean free path, which measures the average distance a molecule travels between collisions.^{[28]} Since is inversely proportional to density, this equation predicts that thermal conductivity is independent of density for fixed temperature. The explanation is that increasing density increases the number of molecules which carry energy but decreases the average distance a molecule can travel before transferring its energy to a different molecule: these two effects cancel out. For most gases, this prediction agrees well with experiments at pressures up to about 10 atmospheres.^{[29]} On the other hand, experiments show a more rapid increase with temperature than (here is independent of ). This failure of the elementary theory can be traced to the oversimplified "elastic sphere" model, and in particular to the fact that the interparticle attractions, present in all real-world gases, are ignored.
To incorporate more complex interparticle interactions, a systematic approach is necessary. One such approach is provided by Chapman–Enskog theory, which derives explicit expressions for thermal conductivity starting from the Boltzmann equation. The Boltzmann equation, in turn, provides a statistical description of a dilute gas for generic interparticle interactions. For a monatomic gas, expressions for derived in this way take the form
where is an effective particle diameter and is a function of temperature whose explicit form depends on the interparticle interaction law.^{[30]}^{[31]} For rigid elastic spheres, is independent of and very close to . More complex interaction laws introduce a weak temperature dependence. The precise nature of the dependence is not always easy to discern, however, as is defined as a multi-dimensional integral which may not be expressible in terms of elementary functions. An alternate, equivalent way to present the result is in terms of the gas viscosity , which can also be calculated in the Chapman-Enskog approach:
where is a numerical factor which in general depends on the molecular model. For smooth spherically symmetric molecules, however, is very close to , not deviating by more than for a variety of interparticle force laws.^{[32]} Since , , and are each well-defined physical quantities which can be measured independent of each other, this expression provides a convenient test of the theory. For monatomic gases, such as the noble gases, the agreement with experiment is fairly good.^{[33]}
For gases whose molecules are not spherically symmetric, the expression still holds. In contrast with spherically symmetric molecules, however, varies significantly depending on the particular form of the interparticle interactions: this is a result of the energy exchanges between the internal and translational degrees of freedom of the molecules. An explicit treatment of this effect is difficult in the Chapman-Enskog approach. Alternately, the approximate expression has been suggested by Eucken, where is the heat capacity ratio of the gas.^{[32]}^{[34]}
The entirety of this section assumes the mean free path is small compared with macroscopic (system) dimensions. In extremely dilute gases this assumption fails, and thermal conduction is described instead by an apparent thermal conductivity which decreases with density. Ultimately, as the density goes to the system approaches a vacuum, and thermal conduction ceases entirely. For this reason a vacuum is an effective insulator.
The exact mechanisms of thermal conduction are poorly understood in liquids: there is no molecular picture which is both simple and accurate. An example of simple but very rough theory is that of Bridgman, in which a liquid is ascribed a local molecular structure similar to that of a solid, i.e. with molecules located approximately on a lattice. Elementary calculations then lead to the expression
where is the Avogadro constant, is the volume of a mole of liquid, and is the speed of sound in the liquid. This is commonly called Bridgman's equation.^{[35]}
For metals at low temperatures the heat is carried mainly by the free electrons. In this case the mean velocity is the Fermi velocity which is temperature independent. The mean free path is determined by the impurities and the crystal imperfections which are temperature independent as well. So the only temperature-dependent quantity is the heat capacity c, which, in this case, is proportional to T. So
with k_{0} a constant. For pure metals such as copper, silver, etc. l is large, so the thermal conductivity is high. At higher temperatures the mean free path is limited by the phonons, so the thermal conductivity tends to decrease with temperature. In alloys the density of the impurities is very high, so l and, consequently k, are small. Therefore, alloys, such as stainless steel, can be used for thermal insulation.
Heat transport in both amorphous and crystalline dielectric solids is by way of elastic vibrations of the lattice (phonons). This transport mode is limited by the elastic scattering of acoustic phonons at lattice defects. These predictions were confirmed by the experiments of Chang and Jones on commercial glasses and glass ceramics, where the mean free paths were limited by "internal boundary scattering" to length scales of 10^{−2} cm to 10^{−3} cm.^{[36]}^{[37]}
The phonon mean free path has been associated directly with the effective relaxation length for processes without directional correlation. If V_{g} is the group velocity of a phonon wave packet, then the relaxation length is defined as:
where t is the characteristic relaxation time. Since longitudinal waves have a much greater phase velocity than transverse waves,^{[38]} V_{long} is much greater than V_{trans}, and the relaxation length or mean free path of longitudinal phonons will be much greater. Thus, thermal conductivity will be largely determined by the speed of longitudinal phonons.^{[36]}^{[39]}
Regarding the dependence of wave velocity on wavelength or frequency (dispersion), low-frequency phonons of long wavelength will be limited in relaxation length by elastic Rayleigh scattering. This type of light scattering from small particles is proportional to the fourth power of the frequency. For higher frequencies, the power of the frequency will decrease until at highest frequencies scattering is almost frequency independent. Similar arguments were subsequently generalized to many glass forming substances using Brillouin scattering.^{[40]}^{[41]}^{[42]}^{[43]}
Phonons in the acoustical branch dominate the phonon heat conduction as they have greater energy dispersion and therefore a greater distribution of phonon velocities. Additional optical modes could also be caused by the presence of internal structure (i.e., charge or mass) at a lattice point; it is implied that the group velocity of these modes is low and therefore their contribution to the lattice thermal conductivity λ_{L} (_{L}) is small.^{[44]}
Each phonon mode can be split into one longitudinal and two transverse polarization branches. By extrapolating the phenomenology of lattice points to the unit cells it is seen that the total number of degrees of freedom is 3pq when p is the number of primitive cells with q atoms/unit cell. From these only 3p are associated with the acoustic modes, the remaining 3p(q − 1) are accommodated through the optical branches. This implies that structures with larger p and q contain a greater number of optical modes and a reduced λ_{L}.
From these ideas, it can be concluded that increasing crystal complexity, which is described by a complexity factor CF (defined as the number of atoms/primitive unit cell), decreases λ_{L}.^{[45]} This was done by assuming that the relaxation time τ decreases with increasing number of atoms in the unit cell and then scaling the parameters of the expression for thermal conductivity in high temperatures accordingly.^{[44]}
Describing of anharmonic effects is complicated because exact treatment as in the harmonic case is not possible and phonons are no longer exact eigensolutions to the equations of motion. Even if the state of motion of the crystal could be described with a plane wave at a particular time, its accuracy would deteriorate progressively with time. Time development would have to be described by introducing a spectrum of other phonons, which is known as the phonon decay. The two most important anharmonic effects are the thermal expansion and the phonon thermal conductivity.
Only when the phonon number ‹n› deviates from the equilibrium value ‹n›^{0}, can a thermal current arise as stated in the following expression
where v is the energy transport velocity of phonons. Only two mechanisms exist that can cause time variation of ‹n› in a particular region. The number of phonons that diffuse into the region from neighboring regions differs from those that diffuse out, or phonons decay inside the same region into other phonons. A special form of the Boltzmann equation
states this. When steady state conditions are assumed the total time derivate of phonon number is zero, because the temperature is constant in time and therefore the phonon number stays also constant. Time variation due to phonon decay is described with a relaxation time (τ) approximation
which states that the more the phonon number deviates from its equilibrium value, the more its time variation increases. At steady state conditions and local thermal equilibrium are assumed we get the following equation
Using the relaxation time approximation for the Boltzmann equation and assuming steady-state conditions, the phonon thermal conductivity λ_{L} can be determined. The temperature dependence for λ_{L} originates from the variety of processes, whose significance for λ_{L} depends on the temperature range of interest. Mean free path is one factor that determines the temperature dependence for λ_{L}, as stated in the following equation
where Λ is the mean free path for phonon and denotes the heat capacity. This equation is a result of combining the four previous equations with each other and knowing that for cubic or isotropic systems and .^{[46]}
At low temperatures (< 10 K) the anharmonic interaction does not influence the mean free path and therefore, the thermal resistivity is determined only from processes for which q-conservation does not hold. These processes include the scattering of phonons by crystal defects, or the scattering from the surface of the crystal in case of high quality single crystal. Therefore, thermal conductance depends on the external dimensions of the crystal and the quality of the surface. Thus, temperature dependence of λ_{L} is determined by the specific heat and is therefore proportional to T^{3}.^{[46]}
Phonon quasimomentum is defined as ℏq and differs from normal momentum because it is only defined within an arbitrary reciprocal lattice vector. At higher temperatures (10 K < T < Θ), the conservation of energy and quasimomentum , where q_{1} is wave vector of the incident phonon and q_{2}, q_{3} are wave vectors of the resultant phonons, may also involve a reciprocal lattice vector G complicating the energy transport process. These processes can also reverse the direction of energy transport.
Therefore, these processes are also known as Umklapp (U) processes and can only occur when phonons with sufficiently large q-vectors are excited, because unless the sum of q_{2} and q_{3} points outside of the Brillouin zone the momentum is conserved and the process is normal scattering (N-process). The probability of a phonon to have energy E is given by the Boltzmann distribution . To U-process to occur the decaying phonon to have a wave vector q_{1} that is roughly half of the diameter of the Brillouin zone, because otherwise quasimomentum would not be conserved.
Therefore, these phonons have to possess energy of , which is a significant fraction of Debye energy that is needed to generate new phonons. The probability for this is proportional to , with . Temperature dependence of the mean free path has an exponential form . The presence of the reciprocal lattice wave vector implies a net phonon backscattering and a resistance to phonon and thermal transport resulting finite λ_{L},^{[44]} as it means that momentum is not conserved. Only momentum non-conserving processes can cause thermal resistance.^{[46]}
At high temperatures (T > Θ), the mean free path and therefore λ_{L} has a temperature dependence T^{−1}, to which one arrives from formula by making the following approximation and writing . This dependency is known as Eucken's law and originates from the temperature dependency of the probability for the U-process to occur.^{[44]}^{[46]}
Thermal conductivity is usually described by the Boltzmann equation with the relaxation time approximation in which phonon scattering is a limiting factor. Another approach is to use analytic models or molecular dynamics or Monte Carlo based methods to describe thermal conductivity in solids.
Short wavelength phonons are strongly scattered by impurity atoms if an alloyed phase is present, but mid and long wavelength phonons are less affected. Mid and long wavelength phonons carry significant fraction of heat, so to further reduce lattice thermal conductivity one has to introduce structures to scatter these phonons. This is achieved by introducing interface scattering mechanism, which requires structures whose characteristic length is longer than that of impurity atom. Some possible ways to realize these interfaces are nanocomposites and embedded nanoparticles/structures.
Specific thermal conductivity is a material property used to compare the heat-transfer ability of different materials to each other. Absolute thermal conductivity, however, is a component property used to compare the heat-transfer ability of different components to each other. Components, as opposed to materials, take into account size and shape, including basic properties such as thickness and area, instead of just material type. In this way, thermal-transfer ability of components of the same physical dimensions, but made of different materials, may be compared and contrasted, or components of the same material, but with different physical dimensions, may be compared and contrasted.
In component datasheets and tables, since actual, physical components with distinct physical dimensions and characteristics are under consideration, thermal resistance is frequently given in absolute units of or , since the two are equivalent. However, thermal conductivity, which is its reciprocal, is frequently given in specific units of . It is therefore often-times necessary to convert between absolute and specific units, by also taking a component's physical dimensions into consideration, in order to correlate the two using information provided, or to convert tabulated values of material thermal conductivity into absolute thermal resistance values for use in thermal resistance calculations. This is particularly useful, for example, when calculating the maximum power a component can dissipate as heat, as demonstrated in the example calculation here.
"Thermal conductivity λ is defined as ability of material to transmit heat and it is measured in watts per square metre of surface area for a temperature gradient of 1 K per unit thickness of 1 m".^{[47]} Therefore, specific thermal conductivity is calculated as:
where:
= temperature difference (K, or °C) = 1 K during measurement
Absolute thermal conductivity, on the other hand, has units of or , and can be expressed as
where = absolute thermal conductivity (W/K, or W/°C).
Substituting for into the first equation yields the equation which converts from absolute thermal conductivity to specific thermal conductivity:
Solving for , we get the equation which converts from specific thermal conductivity to absolute thermal conductivity:
Again, since thermal conductivity and resistivity are reciprocals of each other, it follows that the equation to convert specific thermal conductivity to absolute thermal resistance is:
The thermal conductivity of T-Global L37-3F thermal conductive pad is given as 1.4 W/(mK). Looking at the datasheet and assuming a thickness of 0.3 mm (0.0003 m) and a surface area large enough to cover the back of a TO-220 package (approx. 14.33 mm x 9.96 mm [0.01433 m x 0.00996 m]),^{[48]} the absolute thermal resistance of this size and type of thermal pad is:
This value fits within the normal values for thermal resistance between a device case and a heat sink: "the contact between the device case and heat sink may have a thermal resistance of between 0.5 up to 1.7 ºC/W, depending on the case size, and use of grease or insulating mica washer".^{[49]}
In an isotropic medium the thermal conductivity is the parameter k in the Fourier expression for the heat flux
where is the heat flux (amount of heat flowing per second and per unit area) and the temperature gradient. The sign in the expression is chosen so that always k > 0 as heat always flows from a high temperature to a low temperature. This is a direct consequence of the second law of thermodynamics.
In the one-dimensional case q = H/A with H the amount of heat flowing per second through a surface with area A and the temperature gradient is dT/dx so
In case of a thermally insulated bar (except at the ends) in the steady state H is constant. If A is constant as well the expression can be integrated with the result
where T_{H} and T_{L} are the temperatures at the hot end and the cold end respectively, and L is the length of the bar. It is convenient to introduce the thermal-conductivity integral
The heat flow rate is then given by
If the temperature difference is small k can be taken as constant. In that case
Boron arsenide is a chemical compound involving boron and arsenic, usually with a chemical formula BAs. Other boron arsenide compounds are known, such as the subarsenide B12As2. Chemical synthesis of cubic BAs is very challenging and its single crystal forms usually have defects.
ConductivityConductivity 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), electrical conductivity due to ions moving position in a crystal lattice
Hydraulic conductivity, a property of a porous material's ability to transmit water
Thermal conductivity, an intensive property of a material that indicates its ability to conduct heat
Heat fluxHeat flux or thermal flux, sometimes also referred to as heat flux density or heat flow rate intensity is a flow of energy per unit of area per unit of time. In SI its units are watts per square metre (W⋅m^{−2}). It has both a direction and a magnitude, and so it is a vector quantity. To define the heat flux at a certain point in space, one takes the limiting case where the size of the surface becomes infinitesimally small.
Heat flux is often denoted , the subscript q specifying heat flux, as opposed to mass or momentum flux. Fourier's law is an important application of these concepts.
Heat sinkA heat sink (also commonly spelled heatsink) is a passive heat exchanger that transfers the heat generated by an electronic or a mechanical device to a fluid medium, often air or a liquid coolant, where it is dissipated away from the device, thereby allowing regulation of the device's temperature at optimal levels. In computers, heat sinks are used to cool central processing units or graphics processors. Heat sinks are used with high-power semiconductor devices such as power transistors and optoelectronics such as lasers and light emitting diodes (LEDs), where the heat dissipation ability of the component itself is insufficient to moderate its temperature.
A heat sink is designed to maximize its surface area in contact with the cooling medium surrounding it, such as the air. Air velocity, choice of material, protrusion design and surface treatment are factors that affect the performance of a heat sink. Heat sink attachment methods and thermal interface materials also affect the die temperature of the integrated circuit. Thermal adhesive or thermal grease improve the heat sink's performance by filling air gaps between the heat sink and the heat spreader on the device. A heat sink is usually made out of copper or aluminium. Copper is used because it has many desirable properties for thermally efficient and durable heat exchangers. First and foremost, copper is an excellent conductor of heat. This means that copper's high thermal conductivity allows heat to pass through it quickly. Aluminium heat sinks are used as a low-cost, lightweight alternative to copper heat sinks, and have a lower thermal conductivity than copper.
Helium analyzerA Helium analyzer is an instrument used to identify the presence and concentration of helium in a mixture of gases. In Technical diving where breathing gas mixtures known as Trimix comprising oxygen, helium and nitrogen are used, it is necessary to know the fraction of helium in the mixture to reliably calculate decompression schedules for dives using that mixture.
List of thermal conductivitiesIn heat transfer, the thermal conductivity of a substance, k, is an intensive property that indicates its ability to conduct heat.
Thermal conductivity is often measured with laser flash analysis. Alternative measurements are also established.
Mixtures may have variable thermal conductivities due to composition. Note that for gases in usual conditions, heat transfer by advection (caused by convection or turbulence for instance) is the dominant mechanism compared to conduction.
This table shows thermal conductivity in SI units of watts per metre-kelvin (W·m−1·K−1). Some measurements use the imperial unit BTUs per foot per hour per degree Fahrenheit (1 BTU h−1 ft−1 F−1 = 1.728 W·m−1·K−1).
Oxygen-free copperOxygen-free copper (OFC) or oxygen-free high thermal conductivity (OFHC) copper is a group of wrought high conductivity copper alloys that have been electrolytically refined to reduce the level of oxygen to .001% or below.
PellistorA pellistor is a solid-state device used to detect gases which are either combustible or which have a significant difference in thermal conductivity to that of air. The word "pellistor" is a combination of pellet and resistor.
Pirani gaugeThe Pirani gauge is a robust thermal conductivity gauge used for the measurement of the pressures in vacuum systems. It was invented in 1906 by Marcello Pirani.Marcello Stefano Pirani was a German physicist working for Siemens & Halske which was involved in the vacuum lamp industry. In 1905 their product was tantalum lamps which required a high vacuum environment for the filaments. The gauges that Pirani was using in the production environment were some fifty McCloud gauges, each filled with 2kg of mercury in glass tubes.Pirani was aware of the gas thermal conductivity investigations of Kundt and Warburg (1875) published thirty years earlier and the work of Smoluchowski (1898). In 1906 he described his "directly indicating vacuum gauge" that used a heated wire to measure vacuum by monitoring the heat transfer from the wire by the vacuum environment.
R-value (insulation)In building and 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 conductive flow of heat. R-values measure the thermal resistance per unit of a barrier's exposed area. The greater the R-value, the greater the resistance, and so the better the thermal insulating properties of the barrier. R-values are used in describing effectiveness of insulating material and in analysis of heat flow across assemblies (such as walls, roofs, and windows) under steady-state conditions. Heat flow through a barrier is driven by temperature difference between two sides of the barrier, and the R-value quantifies how effectively the object resists this drive: The temperature difference divided by the R-value and then multiplied by the surface area of the barrier gives the total rate of heat flow through the barrier, as measured in watts or in BTUs per hour.
As long as the materials involved are dense solids in direct mutual contact, R-values are additive; for example, the total R-value of an barrier composed of several layers of material is the sum of the R-values of the individual layers. Note that the R-value is the building industry term for what is in other contexts called "thermal resistance per unit area." It is sometimes denoted RSI-value if the SI (metric) units are used. An R-value can be given for a material (e.g. for polyethylene foam), or for an assembly of materials (e.g. a wall or a window). In the case of materials, it is often expressed in terms of R-value per unit length (e.g. per inch of thickness). The latter can be misleading in the case of low-density building thermal insulations, for which R-values are not additive: their R-value per inch is not constant as the material gets thicker, but rather usually decreases.
The units of an R-value (see below) are usually not explicitly stated, and so it is important to decide from context which units are being used: an R-value expressed in I-P (inch-pound) units is about 5.68 times larger than when expressed in SI units, so that, for example, a window that is R-2 in I-P units has an RSI of 0.35 (since 2/5.68 = 0.35). For R-values there is no difference between US customary units and imperial units. As far as how R-values are reported, all of the following mean the same thing: "this is an R-2 window"; "this is an R2 window"; "this window has an R-value of 2"; "this is a window with R = 2" (and similarly with RSI-values, which also include the possibility "this window provides RSI 0.35 of resistance to heat flow").
The more a material is intrinsically able to conduct heat, as given by its thermal conductivity, the lower its R-value. On the other hand, the thicker the material, the higher its R-value. Sometimes heat transfer processes other than conduction (namely, convection and radiation) significantly contribute to heat transfer within the material. In such cases, it is useful to introduce an "apparent thermal conductivity", which captures the effects of all three kinds of processes, and to define the R-value in general as . This comes at a price, however: R-values that include non-conductive processes may no longer be additive and may have significant temperature dependence. In particular, for a loose or porous material, the R-value per inch generally depends on the thickness, almost always so that it decreases with increasing thickness (polyisocyanurate ("polyso") being an exception; its R-value/inch increases with thickness). For similar reasons, the R-value per inch also depends on the temperature of the material, usually increasing with decreasing temperature (polyso again being an exception); a nominally R-13 fiberglass batt may be R-14 at −12 °C (10 °F) and R-12 at 43 °C (109 °F). Nevertheless, in construction it is common to treat R-values as independent of temperature. Note that an R-value may not account for radiative or convective processes at the material's surface, which may be an important factor for some applications.^{[citation needed]}
The R-value is the reciprocal of the thermal transmittance (U-factor) of a material or assembly. The U.S. construction industry prefers to use R-values, however, because they are additive and because bigger values mean better insulation, neither of which is true for U-factors.
Soil thermal propertiesThe thermal properties of soil are a component of soil physics that has found important uses in engineering, climatology and agriculture. These properties influence how energy is partitioned in the soil profile. While related to soil temperature, it is more accurately associated with the transfer of energy (mostly in the form of heat) throughout the soil, by radiation, conduction and convection.
The main soil thermal properties are:
Volumetric heat capacity, SI Units: J∙m−3∙K−1
Thermal conductivity, SI Units: W∙m−1∙K−1
Thermal diffusivity, SI Units: m2∙s−1
Thermal conductionThermal conduction is the transfer of heat (internal energy) by microscopic collisions of particles and movement of electrons within an organ. 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 including 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 medium through which the heat is transferred.
Heat spontaneously flows from a hotter to a colder body. For example, heat is conducted from the hotplate of an electric stove to the bottom of a saucepan in contact with it. In the absence of an external driving energy source to the contrary, within a body or between bodies, temperature differences decay over time, and thermal equilibrium 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, the transfer is often between bodies, which may be separated spatially. Also possible is transfer of heat by a combination of conduction and thermal radiation. In convection, internal energy is carried between bodies by a moving material carrier. In solids, conduction is mediated by the combination of vibrations and collisions of molecules, of propagation and collisions of phonons, and of diffusion and collisions of free electrons. In gases and liquids, conduction is due to the collisions and diffusion of molecules during their random motion. Photons in this context do not collide with one another, and so heat transport by electromagnetic radiation 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, convection, and sometimes mass transfer. Usually, more than one of these processes occurs in a given situation. The conventional symbol for thermal conductivity is k.
Thermal conductivity detectorThe thermal conductivity detector (TCD), also known as a katharometer, is a bulk property detector and a chemical specific detector commonly used in gas chromatography. This detector senses changes in the thermal conductivity of the column effluent and compares it to a reference flow of carrier gas. Since most compounds have a thermal conductivity much less than that of the common carrier gases of helium or hydrogen, when an analyte elutes from the column the effluent thermal conductivity is reduced, and a detectable signal is produced.
Thermal diffusivityIn 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 derived unit of m²/s. Thermal diffusivity is usually denoted α but a,h,κ, K, and D are also used. The formula is:
where
Together, can be considered the volumetric heat capacity (J/(m³·K)).
As seen in the heat equation,
one way to view thermal diffusivity is as the ratio of the time derivative of temperature to its curvature, quantifying the rate at which temperature concavity is "smoothed out". In a sense, thermal diffusivity is the measure of thermal inertia. In a substance with high thermal diffusivity, heat moves rapidly through it because the substance conducts heat quickly relative to its volumetric heat capacity or 'thermal bulk'.
Thermal diffusivity is often measured with the flash method. It involves heating a strip or cylindrical sample with a short energy pulse at one end and analyzing the temperature change (reduction in amplitude and phase shift of the pulse) a short distance away.
Thermal insulationThermal insulation is the reduction of heat transfer (i.e. the transfer of thermal energy between objects of differing temperature) 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.
Heat flow is an inevitable consequence of contact between objects of different temperature. Thermal insulation provides a region of insulation in which thermal conduction is reduced or thermal radiation is reflected rather than absorbed by the lower-temperature body.
The insulating capability of a material is measured as the inverse of thermal conductivity (k). Low thermal conductivity is equivalent to high insulating capability (Resistance value). In thermal engineering, other important properties of insulating materials are product density (ρ) and specific heat capacity (c).
Thermodynamics of nanostructuresAs the devices continue to shrink further into the sub-100 nm range following the trend predicted by Moore’s law, the topic of thermal properties and transport in such nanoscale devices becomes increasingly important. Display of great potential by nanostructures for thermoelectric applications also motivates the studies of thermal transport in such devices. These fields, however, generate two contradictory demands: high thermal conductivity to deal with heating issues in sub-100 nm devices and low thermal conductivity for thermoelectric applications. These issues can be addressed with phonon engineering, once nanoscale thermal behaviors have been studied and understood.
Thermoelectric generatorA thermoelectric generator (TEG), also called a Seebeck generator, is a solid state device that converts heat flux (temperature differences) directly into electrical energy through a phenomenon called the Seebeck effect (a form of thermoelectric effect). Thermoelectric generators function like heat engines, but are less bulky and have no moving parts. However, TEGs are typically more expensive and less efficient.Thermoelectric generators could be used in power plants in order to convert waste heat into additional electrical power and in automobiles as automotive thermoelectric generators (ATGs) to increase fuel efficiency. Another application is radioisotope thermoelectric generators which are used in space probes, which has the same mechanism but use radioisotopes to generate the required heat difference.
Thermoelectric materialsThermoelectric materials show the thermoelectric effect in a strong or convenient form.
The thermoelectric effect refers to phenomena by which either a temperature difference creates an electric potential or an electric potential creates a temperature difference. These phenomena are known more specifically as the Seebeck effect (converting temperature to current), Peltier effect (converting current to temperature), and Thomson effect (conductor heating/cooling). While all materials have a nonzero thermoelectric effect, in most materials it is too small to be useful. However, low-cost materials that have a sufficiently strong thermoelectric effect (and other required properties) could be used in applications including power generation and refrigeration. A commonly used thermoelectric material in such applications is bismuth telluride (Bi2Te3).
Thermoelectric materials are used in thermoelectric systems for cooling or heating in niche applications, and are being studied as a way to regenerate electricity from waste heat.
Uranium disilicideUranium disilicide is an inorganic chemical compound of uranium in oxidation state +4. It is a silicide of uranium.
There has been recent interest in using uranium disilicide as an alternative to uranium dioxide for
fuel in nuclear reactors. . Advantages are higher
percentage of uranium and higher thermal conductivity. A direct replacement of UO2 with U3Si2
should enable a reactor to generate more energy from a set of fuel rods and also provide more "coping time" in the case of a LOCA (Loss of Cooling Accident).
The development of uranium disilicide, uranium nitride, or other high thermal conductivity uranium compound may be critical for the performance of "Accident Tolerant Fuel", a
development effort mandated by the US Department of Energy. This is due to zircalloy having a higher thermal
conductivity than all replacement materials being developed. In particular, SIC-SiC CMC (link), which has several superior material properties to zircalloy for this application, has about five times lower thermal conductivity (varies due to the manufacturing methods used for the fiber and for the matrix) than zircalloy.(refs on SiC-SiC and zircalloy). The lower thermal conductivity means that a reactor using fuel rods with SiC-SiC CMC cladding and conventional UO2 fuel will have to either: 1) Run at a lower power output to keep the fuel the same temperature, or 2) Run with the same power, with the fuel hotter, which means the reactor has less coping time (time to fix what is wrong before something fails). The alternative, enabled by U3Si2 which has about five times better thermal conductivity than UO2 , is expected to be a fuel rod capable of equal power output, slightly better energy output, and longer coping time.
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