View source for Thermal conductance and resistance

{{Short description|Materials' resistance to heat transfer}}
{{Use American English|date = January 2019}}
{{more footnotes needed|date=January 2015}}
{{Infobox physical quantity|name=Thermal conductance|symbols={{mvar|G}}|unit=watt per kelvin (W/K)|baseunits=kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅K<sup>−1</sup>|dimension=<math> \mathsf L^2 \mathsf M \mathsf T^{-3}\mathsf \Theta^{-1}</math>}}
{{Infobox physical quantity|name=Thermal resistance|symbols={{mvar|R}}|unit=kelvin per watt (K/W)|baseunits=kg<sup>−1</sup>⋅m<sup>−2</sup>⋅s<sup>3</sup>⋅K|dimension=wikidata}}
In [[heat transfer]], [[thermal engineering]], and [[thermodynamics]]''', thermal conductance''' and '''thermal resistance''' are fundamental concepts that describe the ability of materials or systems to conduct [[heat]] and the opposition they offer to the [[heat current]]. The ability to manipulate these properties allows engineers to control [[temperature gradient]], prevent [[thermal shock]], and maximize the efficiency of [[thermal systems]]. Furthermore, these principles find applications in a multitude of fields, including [[materials science]], [[mechanical engineering]], [[electronics]], and [[energy management]]. Knowledge of these principles is crucial in various scientific, engineering, and everyday applications, from designing efficient [[temperature control]], [[thermal insulation]], and [[Thermal management (electronics)|thermal management]] in industrial processes to optimizing the performance of [[electronic devices]].

'''Thermal conductance''' (''G'') measures the ability of a material or system to conduct heat. It provides insights into the ease with which heat can pass through a particular system. It is measured in units of [[watt]]s per [[kelvin]] (W/K). It is essential in the design of [[heat exchangers]], [[Thermal efficiency|thermally efficient materials]], and various engineering systems where the controlled movement of heat is vital.

Conversely, '''thermal resistance''' (''R'') measures the opposition to the heat current in a material or system. It is measured in units of kelvins per watt (K/W) and indicates how much [[temperature]] difference (in kelvins) is required to transfer a unit of heat current (in watts) through the material or object. It is essential to optimize the [[building insulation]], evaluate the efficiency of electronic devices, and enhance the performance of [[heat sinks]] in various applications.

Objects made of insulators like rubber tend to have very high resistance and low conductance, while objects made of conductors like metals tend to have very low resistance and high conductance. This relationship is quantified by [[Thermal resistivity|resistivity]] or [[Thermal conductivity|conductivity]]. However, the nature of a material is not the only factor as it also depends on the [[size]] and [[shape]] of an object because these properties are [[Intensive and extensive properties|extensive rather than intensive]]. The relationship between thermal conductance and resistance is analogous to that between [[electrical conductance and resistance]] in the domain of electronics.

'''Thermal insulance''' ([[R-value (insulation)|''R''-value]]) is a measure of a material's resistance to the heat current. It quantifies how effectively a material can resist the transfer of heat through conduction, convection, and radiation. It has the units [[square metre]] kelvins per watt (m<sup>2</sup>⋅K/W) in [[SI units]] or [[square foot]] [[degree Fahrenheit]]–[[hour]]s per [[British thermal unit]] (ft<sup>2</sup>⋅°F⋅h/Btu) in [[imperial units]]. The higher the thermal insulance, the better a material insulates against heat transfer. It is commonly used in construction to assess the insulation properties of materials such as walls, roofs, and insulation products.

==Practical applications==
{{Prose|date=October 2023|section}}
Thermal conductance and resistance have several practical applications in various fields:

#[[Building insulation]]: Understanding thermal resistance helps in designing energy-efficient buildings with effective insulation materials to reduce heat transfer.
#[[Electronics cooling]]: Thermal resistance is crucial for designing heat sinks and thermal management systems in electronic devices to prevent overheating.  Calculating thermal conductance is crucial for designing effective heat sinks and cooling systems in electronic devices.
#[[Automotive design]]: Automotive engineers use thermal resistance to optimize the cooling system and prevent overheating in engines and other vehicle components. Evaluating thermal resistance helps in designing engine components and automotive cooling systems.
#[[Cookware and bakeware|Cookware]] design: Thermal conductance is important for designing cookware to ensure even heat distribution and cooking efficiency. Assessing thermal conductance is important in designing cookware for even heat distribution.
#[[Heat exchanger]]s: In industries like HVAC and chemical processing, heat exchangers use thermal conductance to efficiently transfer heat between fluids.
#[[Aerospace]]: In spacecraft and aircraft, thermal resistance and conductance are critical for managing temperature variations in extreme environments. Designing spacecraft and aviation systems require considerations of thermal conductance and resistance to manage temperature extremes.
#[[Cryogenics]]: Understanding thermal properties is vital for the design of cryogenic systems used in superconductors and medical applications.
#Energy efficiency: In the energy sector, thermal resistance and conductance play a role in designing efficient heat exchangers for power plants and energy-efficient appliances.
#[[Medical device]]s: Thermal management is crucial for medical equipment like [[magnetic resonance imaging]] (MRI) machines and laser systems to maintain precise operating temperatures. Ensuring proper thermal management is crucial for the safety and performance of medical devices and laser systems.
#[[Food processing]]: The food industry uses knowledge of thermal conductance to optimize processes like [[pasteurization]] and cooking and design equipment for food processing, such as ovens and refrigeration units.
#[[Materials science]]: Researchers use thermal conductance data to develop new materials for various applications, including energy storage and advanced coatings.
#[[Environmental science]]: Thermal resistance is considered in climate studies to understand heat transfer in Earth's atmosphere and oceans. Evaluating thermal resistance is useful in studying soil temperature profiles for environmental and agricultural research.
#[[Heating, ventilation, and air conditioning]] (HVAC): Understanding thermal resistance aids in optimizing heating, ventilation, and air conditioning systems for better energy efficiency.
#Thermal packaging: Ensuring proper thermal conductance and resistance is crucial for protecting sensitive goods during transport.
#[[Solar energy]] systems: Understanding thermal resistance is important in the design of solar collectors and thermal energy storage systems.
#[[Manufacturing]] processes: Controlling thermal conductance is essential in processes like welding, heat treatment, and metal casting.
#[[Geothermal energy]]: Assessing thermal conductance is important in geothermal heat exchangers and energy production.
#[[Thermography|Thermal imaging]]: Infrared cameras and thermal imaging devices use principles of thermal conductance to detect temperature variations.

== Absolute thermal resistance ==
Absolute thermal resistance is the [[temperature]] difference across a structure when a unit of [[heat]] energy flows through it in unit [[time]].  It is the reciprocal of [[thermal conductivity|thermal conductance]]. The [[SI]] unit of absolute thermal resistance is kelvins per watt (K/W) or the equivalent [[degrees Celsius]] per watt (°C/W) – the two are the same since the intervals are equal: Δ''T'' = 1&nbsp;K = 1&nbsp;°C.

The thermal resistance of materials is of great interest to electronic engineers because most electrical components generate heat and need to be cooled. Electronic components malfunction or fail if they overheat, and some parts routinely need measures taken in the design stage to prevent this.

== Analogies and nomenclature ==
{{main|analogical models|Onsager reciprocal relations}}

Electrical engineers are familiar with [[Ohm's law]] and so often use it as an analogy when doing calculations involving thermal resistance. Mechanical and structural engineers are more familiar with [[Hooke's law]] and so often use it as an analogy when doing calculations involving thermal resistance.

{| class="wikitable"
! type
! structural analogy<ref>
Tony Abbey.
"Using FEA for Thermal Analysis".
Desktop Engineering magazine.
2014 June.
p. 32.
</ref>
! hydraulic analogy
! thermal
! electrical analogy<ref>
[http://sound.whsites.net/heatsinks.htm "The Design of Heatsinks"]. {{Archive url|url=https://web.archive.org/web/20160905231346/http://sound.whsites.net/heatsinks.htm|date=2016-09-05}}
</ref>
|-
|[[quantity]]
|[[impulse (physics)|impulse]] <math>J</math> [N·s]
|[[volume]] <math>V</math> [m<sup>3</sup>]
|[[heat]] <math>Q</math> [J]
|[[electric charge|charge]] <math>q</math> [C]
|-
|[[potential]]
|[[displacement (vector)|displacement]] <math>X</math> [m]
|[[pressure]] <math>P</math> [N/m<sup>2</sup>]
|[[temperature]] <math>T</math> [K]
|[[Electric potential|potential]] <math>V</math> [V = J/C]
|-
|[[flux]]
|load or [[force]] <math>F</math> [N]
|flow rate <math>Q</math> [m<sup>3</sup>/s]
|[[heat transfer rate]] <math>\dot{Q}</math> [W = J/s]
|[[current (electricity)|current]] <math>I</math> [A = C/s]
|-
|[[flux density]]
|[[stress (mechanics)|stress]] <math> \sigma </math> [Pa = N/m<sup>2</sup>]
|[[velocity]] <math>\mathbf{v}</math> [m/s]
|[[heat flux]] <math>\mathbf{q}</math> [W/m<sup>2</sup>]
|[[current density]] <math>\mathbf{j}</math> [C/(m<sup>2</sup>·s) = A/m<sup>2</sup>]
|-
|resistance
|[[flexibility]] ([[rheology]] defined) [1/Pa]
|[[fluid resistance]] <math>R</math> [...]
|thermal resistance <math>R</math> [K/W]
|[[electrical resistance]] <math>R</math> [&Omega;]
|-
|conductance
|... <math>...</math> [Pa]
|[[fluid conductance]] <math>G</math> [...]
|[[thermal conductivity|thermal conductance]] <math>G</math> [W/K]
|[[electrical conductance]] <math>G</math> [S]
|-
|[[resistivity]]
|[[flexibility]] <math>1/k</math> [m/N]
|[[fluid resistivity]]
|[[thermal resistivity]] [(m·K)/W]
|[[electrical resistivity]] <math>\rho</math> [&Omega;·m]
|-
|conductivity
|[[stiffness]] <math>k</math> [N/m]
|[[fluid conductivity]]
|[[thermal conductivity]] <math>k</math> [W/(m·K)]
|[[electrical conductivity]] <math>\sigma</math> [S/m]
|-
|lumped element linear model
|[[Hooke's law]] <math>\Delta X = F/k </math>
|[[Hagen–Poiseuille equation]] <math>\Delta P = Q R </math>
|[[Newton's law of cooling]] <math>\Delta T = \dot{Q} R</math>
|[[Ohm's law]] <math>\Delta V = IR</math>
|-
|distributed linear model
|... <math>...</math>
|... <math>...</math>
|[[Fourier's law]] <math> \mathbf{q} = -  k \boldsymbol{\nabla} T </math>
|[[Ohm's law]] <math>\mathbf{J} = \sigma \mathbf{E} = -\sigma \boldsymbol{\nabla} V</math>
|}

=== Explanation from an electronics point of view ===

==== Equivalent thermal circuits ====
[[File:Equivalient thermal circuit 2.png|thumb|The diagram shows an equivalent thermal circuit for a semiconductor device with a [[heat sink]]:{{ubl
 | <math>\dot{Q}</math> is the power dissipated by the device.
 | <math>T_{\rm J}</math> is the [[junction temperature]] in the device.
 | <math>T_{\rm C}</math> is the temperature at its case.
 | <math>T_{\rm H}</math> is the temperature where the heat sink is attached.
 | <math>T_{\rm amb}</math> is the ambient air temperature.
 | <math>R_{\theta \rm JC}</math> is the device's absolute thermal resistance from junction to case.
 | <math>R_{\theta \rm CH}</math> is the absolute thermal resistance from the case to the heatsink.
 | <math>R_{\theta \rm HA}</math> is the absolute thermal resistance of the heat sink.
}}]]
The heat flow can be modelled by analogy to an electrical circuit where heat flow is represented by current, temperatures are represented by voltages, heat sources are represented by constant current sources, absolute thermal resistances are represented by resistors and thermal capacitances by capacitors.

The diagram shows an equivalent thermal circuit for a semiconductor device with a [[heat sink]].

==== Example calculation ====
{{unreferenced section|date=January 2015}}
{| class="toccolours collapsible collapsed" style="text-align:left"
!Example
|-
|
Consider a component such as a silicon transistor that is bolted to the metal frame of a piece of equipment.  The transistor's manufacturer will specify parameters in the datasheet called the ''absolute thermal resistance from junction to case'' (symbol: <math>R_{\theta \rm JC}</math>), and the maximum allowable temperature of the semiconductor junction (symbol: <math>T_{J\rm max}</math>).  The specification for the design should include a maximum temperature at which the circuit should function correctly.  Finally, the designer should consider how the heat from the transistor will escape to the environment: this might be by convection into the air, with or without the aid of a [[heat sink]], or by conduction through the [[printed circuit board]].  For simplicity, let us assume that the designer decides to bolt the transistor to a metal surface (or [[heat sink]]) that is guaranteed to be less than <math>\Delta T_{\rm HS}</math> above the ambient temperature. Note: T<sub>HS</sub> appears to be undefined.

Given all this information, the designer can construct a model of the heat flow from the semiconductor junction, where the heat is generated, to the outside world.  In our example, the heat has to flow from the junction to the case of the transistor, then from the case to the metalwork.  We do not need to consider where the heat goes after that, because we are told that the metalwork will conduct heat fast enough to keep the temperature less than <math>\Delta T_{\rm HS}</math> above ambient: this is all we need to know.

Suppose the engineer wishes to know how much power can be put into the transistor before it overheats.  The calculations are as follows.

:Total absolute thermal resistance from junction to ambient = <math>R_{\theta \rm JC}+R_{\theta \rm B}</math>

where <math>R_{\theta \rm B}</math> is the absolute thermal resistance of the bond between the transistor's case and the metalwork.  This figure depends on the nature of the bond - for example, a thermal bonding pad or [[thermal grease|thermal transfer grease]] might be used to reduce the absolute thermal resistance.

: Maximum temperature drop from junction to ambient = <math>T_{J\rm max} - (T_{\rm amb} + \Delta T_{\rm HS})</math>.

We use the general principle that the temperature drop <math>\Delta T</math> across a given absolute thermal resistance <math>R_{\theta}</math> with a given heat flow <math>\dot{Q}</math> through it is:
:<math>\Delta T = \dot{Q} R_\theta \,</math>.

Substituting our own symbols into this formula gives:
:<math>T_{J\rm max} - (T_{\rm amb} + \Delta T_{\rm HS}) = \dot{Q}_{\rm max} (R_{\theta \rm JC} + R_{\theta \rm B} + R_{\theta \rm HA})\,</math>,

and, rearranging,
:<math>
   \dot{Q}_{\rm max} = 
   { 
     { T_{J\rm max}-(T_{\rm amb} + \Delta T_{\rm HS}) } \over { R_{\theta \rm JC} + R_{\theta \rm B}+R_{\theta \rm HA} }
   }
</math>

The designer now knows <math>\dot{Q}_{\rm max}</math>, the maximum power that the transistor can be allowed to dissipate, so they can design the circuit to limit the temperature of the transistor to a safe level.

Let us substitute some sample numbers:
: <math>T_{J\rm max} = 125 \ ^{\circ}\text{C}</math> (typical for a silicon transistor)
: <math>T_{\rm amb} = 21 \ ^{\circ}\text{C}</math> (a typical specification for commercial equipment)
: <math>R_{\theta \rm JC} = 1.5 \ ^{\circ}\text{C}/\text{W} \,</math> (for a typical [[TO-220]] package{{citation needed|date=January 2015}})
: <math>R_{\theta \rm B} = 0.1 \ ^{\circ}\text{C}/\text{W} \,</math>  (a typical value for an [[elastomer]] heat-transfer pad for a TO-220 package{{citation needed|date=January 2015}})
: <math>R_{\theta \rm HA} = 4 \ ^{\circ}\text{C}/\text{W} \,</math>  (a typical value for a heatsink for a TO-220 package{{citation needed|date=January 2015}})

The result is then:
: <math>\dot{Q} = {{125\ ^{\circ}\text{C}-(21\ ^{\circ}\text{C})} \over {1.5\ ^{\circ}\text{C}/\text{W}+0.1\ ^{\circ}\text{C}/\text{W}+4\ ^{\circ}\text{C}/\text{W}}} = 18.6 \ \text{W} </math>

This means that the transistor can dissipate about 18 watts before it overheats.  A cautious designer would operate the transistor at a lower power level to increase its [[Reliability engineering|reliability]].

This method can be generalized to include any number of layers of heat-conducting materials, simply by adding together the absolute thermal resistances of the layers and the temperature drops across the layers.
|}

===Derived from Fourier's law for heat conduction===

From [[Heat conduction#Fourier's law|Fourier's law]] for [[heat conduction]], the following equation can be derived, and is valid as long as all of the parameters (x and k) are constant throughout the sample.

: <math> R_{\theta} = \frac{\Delta x}{A k} = \frac{\Delta x r}{A}</math>

where:
* <math>R_{\theta}</math> is the absolute thermal resistance (K/W) across the thickness of the sample
* <math>\Delta x</math> is the thickness (m) of the sample (measured on a path parallel to the heat flow) 
* <math>k</math> is the thermal conductivity (W/(K·m)) of the sample 
* <math>r</math> is the thermal resistivity (K·m/W) of the sample 
* <math>A</math> is the cross-sectional area (m<sup>2</sup>) perpendicular to the path of heat flow.

In terms of the temperature gradient across the sample and [[heat flux]] through the sample, the relationship is:

: <math> R_{\theta} = \frac{\Delta x}{A \phi_q}\frac{\Delta T}{\Delta x} = \frac{\Delta T}{q}</math>

where:
* <math>R_\theta</math> is the absolute thermal resistance (K/W) across the thickness of the sample,
* <math>\Delta x</math> is the thickness (m) of the sample (measured on a path parallel to the heat flow),
* <math>\phi_q</math> is the [[heat flux]] through the sample ([[Watt|W]]·m<sup>−2</sup>),
* <math>\frac{\Delta T}{\Delta x}</math> is the temperature gradient ([[Kelvin|K]]·m<sup>−1</sup>) across the sample,
* <math>A</math> is the cross-sectional area (m<sup>2</sup>) perpendicular to the path of heat flow through the sample,
* <math>\Delta T</math> is the temperature difference ([[Kelvin|K]]) across the sample,
* <math>q</math> is the [[rate of heat flow]] ([[Watt|W]]) through the sample.

=== Problems with electrical resistance analogy ===
A 2008 review paper written by Philips researcher Clemens J. M. Lasance notes that: "Although there is an analogy between heat flow by conduction (Fourier's law) and the flow of an electric current (Ohm’s law), the corresponding physical properties of thermal conductivity and electrical conductivity conspire to make the behavior of heat flow quite unlike the flow of electricity in normal situations. [...] Unfortunately, although the electrical and thermal differential equations are analogous, it is erroneous to conclude that there is any practical analogy between electrical and thermal resistance. This is because a material that is considered an insulator in electrical terms is about 20 orders of magnitude less conductive than a material that is considered a conductor, while, in thermal terms, the difference between an "insulator" and a "conductor" is only about three orders of magnitude. The entire range of thermal conductivity is then equivalent to the difference in electrical conductivity of high-doped and low-doped silicon."<ref name="Lasance"/>

== Measurement standards ==
{{expand section|date=January 2015}}

The junction-to-air thermal resistance can vary greatly depending on the ambient conditions.<ref name="TongLai2013">{{cite book|author1=Ho-Ming Tong|author2=Yi-Shao Lai|author3=C.P. Wong|title=Advanced Flip Chip Packaging|url=https://archive.org/details/advancedflipchip00lanz|url-access=limited|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4419-5768-9|pages=[https://archive.org/details/advancedflipchip00lanz/page/n463 460]–461}}</ref> (A more sophisticated way of expressing the same fact is saying that junction-to-ambient thermal resistance is not Boundary-Condition Independent (BCI).<ref name="Lasance">{{Cite journal | doi = 10.1080/01457630701673188| title = Ten Years of Boundary-Condition- Independent Compact Thermal Modeling of Electronic Parts: A Review| journal = Heat Transfer Engineering| volume = 29| issue = 2| pages = 149–168| year = 2008| last1 = Lasance | first1 = C. J. M. |bibcode = 2008HTrEn..29..149L | s2cid = 121803741| doi-access = free}}</ref>) [[JEDEC]] has a standard (number JESD51-2) for measuring the junction-to-air thermal resistance of electronics packages under [[natural convection]] and another standard (number JESD51-6) for measurement under [[forced convection]].

A JEDEC standard for measuring the junction-to-board thermal resistance (relevant for [[surface-mount technology]]) has been published as JESD51-8.<ref name="Shabany2011">{{cite book|author=Younes Shabany|title=Heat Transfer: Thermal Management of Electronics|year=2011|publisher=CRC Press|isbn=978-1-4398-1468-0|pages=111–113}}</ref>

A JEDEC standard for measuring the junction-to-case thermal resistance (JESD51-14) is relatively newcomer, having been published in late 2010; it concerns only packages having a single heat flow and an exposed cooling surface.<ref name="LasancePoppe2013">{{cite book|author1=Clemens J.M. Lasance|author2=András Poppe|title=Thermal Management for LED Applications|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4614-5091-7|page=247}}</ref><ref>{{Cite web | url=http://www.electronics-cooling.com/2013/02/experiment-vs-simulation-part-3-jesd51-14/ | title=Experiment vs. Simulation, Part 3: JESD51-14| date=2013-02-22}}</ref><ref>{{Cite conference | doi = 10.1109/STHERM.2011.5767204| chapter = Transient dual interface measurement — A new JEDEC standard for the measurement of the junction-to-case thermal resistance| title = 2011 27th Annual IEEE Semiconductor Thermal Measurement and Management Symposium| pages = 222| year = 2011| last1 = Schweitzer | first1 = D. | last2 = Pape | first2 = H. | last3 = Chen | first3 = L. | last4 = Kutscherauer | first4 = R. | last5 = Walder | first5 = M. | isbn = 978-1-61284-740-5}}</ref>

==Resistance in composite wall==

===Resistances in series===
When resistances are in series, the total resistance is the sum of the resistances:

<math>R_{\rm tot} = R_A + R_B + R_C + ...</math>

===Parallel thermal resistance===
[[File:Parallel Thermal Resistance.jpg|thumb|Parallel thermal resistance in composite walls]]
Similarly to electrical circuits, the total thermal resistance for steady state conditions can be calculated as follows.

The total thermal resistance

{{NumBlk|:|<math> { {1 \over R_{\rm tot}} = {1 \over R_B} + {1 \over R_C} }</math>|1}}

Simplifying the equation, we get

{{NumBlk|:|<math> {R_{\rm tot} = {R_B R_C \over R_B + R_C} } </math>|2}}
With terms for the thermal resistance for conduction, we get

{{NumBlk|:|<math> {R_{t,\rm cond} = {L \over (k_b + k_c)A}}</math>|3}}

===Resistance in series and parallel===
[[File:Equivalent thermal circuits for series-parallel composite wall.jpg|thumb|Equivalent thermal circuits for series-parallel composite wall]]
It is often suitable to assume one-dimensional conditions, although the heat flow is multidimensional. Now, two different circuits may be used for this case. For case (a) (shown in picture), we presume [[isothermal]] surfaces for those normal to the x- direction, whereas for case (b) we presume [[adiabatic]] surfaces parallel to the x- direction. We may obtain different results for the total resistance <math> {R_{tot} }</math> and the actual corresponding values of the heat transfer are bracketed by <math> {q} </math>.  When the multidimensional effects becomes more significant, these differences are increased with increasing <math> {| k_f - k_g |} </math>.<ref name="Incropera, Dewitt, Bergman, Lavine 2013">{{cite book |last=Incropera, Dewitt, Bergman, Lavine |first= Frank P., David P., Theodore L., Adrienne S.|date=2013 |title=Principles of Heat and Mass Transfer|publisher=John Wiley & Sons; 7th Edition, Interna edition|isbn=978-0470646151}}</ref>

=== Radial systems ===
Spherical and cylindrical systems may be treated as one-dimensional, due to the [[temperature gradient]]s in the radial direction. The standard method can be used for analyzing radial systems under steady state conditions, starting with the appropriate form of the heat equation, or the alternative method, starting with the appropriate form of [[Fourier's law]]. For a hollow cylinder in steady state conditions with no heat generation, the appropriate form of heat equation is <ref name="Incropera, Dewitt, Bergman, Lavine 2013"/>

{{NumBlk|:|<math>{ {1 \over r}{d\over dr} \left(kr{dT \over dr} \right) = 0 }</math>|4}}

Where <math> {k} </math> is treated as a variable. Considering the appropriate form of Fourier's law, the physical significance of treating <math> {k} </math> as a variable becomes evident when the rate at which energy is conducted across a cylindrical surface, this is represented as

{{NumBlk|:|<math>{ q_r = -kA {dT \over dr} = -k(2\pi rL){dT \over dr}}</math>|5}}

Where <math> {A = 2\pi rL} </math> is the area that is normal to the direction of where the heat transfer occurs. Equation 1 implies that the quantity <math> {kr(dT/dr)} </math> is not dependent of the radius <math> {r} </math>, it follows from equation 5 that the heat transfer rate, <math> {q_r} </math> is a constant in the radial direction.

[[File:Hollow cylinder with convective surface conditions.jpg|thumb|Hollow cylinder with convective surface conditions in thermal conduction]]
In order to determine the temperature distribution in the cylinder, equation 4 can be solved applying the appropriate [[boundary condition]]s. With the assumption that <math> {k } </math> is constant

{{NumBlk|:|<math> { T(r) = C_1\ln r + C_2 } </math>|6}}
Using the following boundary conditions, the constants <math> {C_1} </math> and <math> {C_2} </math> can be computed
:<math> { T(r_1) = T_{s,1} } </math>          and          <math> { T(r_2) = T_{s,2} } </math>

The general solution gives us

:<math> { T_{s,1} = C_1 \ln r_1 + C_2 }</math>          and          <math> { T_{s,2} = C_1 \ln r_2 + C_2 } </math>
Solving for <math> {C_1} </math> and <math> {C_2} </math> and substituting into the general solution, we obtain
{{NumBlk|:|<math> { T(r) = {T_{s,1} - T_{s,2} \over {\ln(r_1/r_2) } } \ln \left( {r \over r_2} \right) + T_{s,2} } </math>|7}}

The logarithmic distribution of the temperature is sketched in the inset of the thumbnail figure.
Assuming that the temperature distribution, equation 7, is used with Fourier's law in equation 5, the heat transfer rate can be expressed in the following form

:<math> {\dot{Q}_r = {2\pi Lk (T_{s,1} - T_{s,2}) \over \ln(r_2/r_1)} } </math>

Finally, for radial conduction in a cylindrical wall, the thermal resistance is of the form
:<math> { R_{t,\mathrm{cond}}  = {\ln(r_2/r_1) \over 2\pi Lk}               }</math> such that <math> {r_2 > r_1} </math>

== See also ==
* [[Thermal engineering]]
* [[Thermal design power]]
* [[Safe operating area]]

== References ==
{{reflist}}10. K Einalipour, S. Sadeghzadeh'', F. Molaei. “Interfacial thermal resistance engineering for polyaniline (C3N)-graphene heterostructure”,'' ''The Journal of Physical Chemistry,'' 2020. DOI:[[doi:10.1021/acs.jpcc.0c02051|10.1021/acs.jpcc.0c02051]]
*Michael Lenz, Günther Striedl, Ulrich Fröhler (January 2000) [http://www.infineon.com/dgdl/smdpack.PDF?folderId=db3a304412b407950112b417b3e623f4&fileId=db3a304412b407950112b417b42923f5 Thermal Resistance, Theory and Practice]. [[Infineon|Infineon Technologies AG]], [[Munich]], [[Germany]].
*Directed Energy, Inc./IXYSRF (March 31, 2003) [https://ixysrf.com/wp-content/uploads/2017/09/thermal_resistance_power_dissipation.pdf  R Theta And Power Dissipation Technical Note]. [http://ixysrf.com/ Ixys RF], Fort Collins, Colorado. Example thermal resistance and power dissipation calculation in semiconductors.

== Further reading ==
There is a large amount of literature on this topic. In general, works using the term "thermal resistance" are more engineering-oriented, whereas works using the term [[thermal conductivity]] are more [pure-]physics-oriented. The following books are representative, but may be easily substituted.

* {{cite book|editor=Terry M. Tritt|title=Thermal Conductivity: Theory, Properties, and Applications|year=2004|publisher=Springer Science & Business Media|isbn=978-0-306-48327-1}}
* {{cite book|author=Younes Shabany|title=Heat Transfer: Thermal Management of Electronics|year=2011|publisher=CRC Press|isbn=978-1-4398-1468-0}}
* {{cite book|author=Xingcun Colin Tong|title=Advanced Materials for Thermal Management of Electronic Packaging|year=2011|publisher=Springer Science & Business Media|isbn=978-1-4419-7759-5}}

== External links ==
* Guoping Xu (2006), [https://cseweb.ucsd.edu/classes/wi06/cse291-b/slide/let9/thermal.pdf Thermal Management for Electronic Packaging], Sun Microsystems
* [http://www.electronics-cooling.com/2012/09/update-on-jedec-thermal-standards/ Update on JEDEC Thermal Standards]
*The importance of [https://web.archive.org/web/20141202091212/http://tempsand.com.au/services/tempsand-soil-test-soil-thermal-conductivity-resistivity/ Soil Thermal Resistivity] for power companies

[[Category:Heat conduction]]
[[Category:Electronic engineering]]