381:
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.
4932:, 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.
3949:
1291:
1184:
1075:
5425:
377:
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.
4899:, 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
1030:
4113:. 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
2294:
373:. 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.
4976:
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
356:
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
4964:
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
422:
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.
4980:
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
495:
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
368:
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
217:
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
135:
Conduction is the main mode of heat transfer for solid materials because the strong inter-molecular forces allow the vibrations of particles to be easily transmitted, in comparison to liquids and gases. Liquids have weaker inter-molecular forces and more space between the particles, which makes the
435:
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
380:
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
360:
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
322:
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
309:
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
147:
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
417:
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
330:. 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
105:
A region with greater thermal energy (heat) corresponds with greater molecular agitation. Thus when a hot object touches a cooler surface, the highly agitated molecules from the hot object bump the calm molecules of the cooler surface, transferring the microscopic kinetic energy and causing the
75:
is heat transfer through stationary matter by physical contact. (The matter is stationary on a macroscopic scaleâwe know there is thermal motion of the atoms and molecules at any temperature above absolute zero.) Heat transferred between the electric burner of a stove and the bottom of a pan is
376:
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
4956:
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.
40:
is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy until an object has the same kinetic energy throughout.
874:
364:
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.
3923:
142:
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.
49:, is a property that relates the rate of heat loss per unit area of a material to its rate of change of temperature. Essentially, it is a value that accounts for any property of the material that could change the way it conducts heat. Heat spontaneously flows along a
4977:
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.
4786:
318:
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).
2130:
1951:
4652:
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:
89:
occurs when microwaves, infrared radiation, visible light, or another form of electromagnetic radiation is emitted or absorbed. An obvious example is the warming of the Earth by the Sun. A less obvious example is thermal radiation from the human
3173:
749:
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
2123:
4270:
436:
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 vacuum, which is physically inadmissible within the framework of relativity.
3397:
393:
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
384:
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
3045:
3285:
3505:
106:
colder part or object to heat up. Mathematically, thermal conduction works just like diffusion. As temperature difference goes up, the distance traveled gets shorter or the area goes up thermal conduction increases:
3721:
4656:
1823:
3603:
1349:
1133:
4386:
4879:(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
4009:
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:
1243:
4922:
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.
1856:
1478:
326:
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
4323:, 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.
1025:{\displaystyle {\frac {\partial T}{\partial t}}=\alpha \left({\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}+{\frac {\partial ^{2}T}{\partial z^{2}}}\right)}
4861:
2728:
4049:
1644:
4111:
4321:
3050:
2030:
4650:
2556:
864:
1751:
1406:
635:
2608:
2035:
1696:
4458:
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
4155:
3290:
2934:
4150:
1314:
1207:
1098:
3714:
136:
vibrations of particles harder to transmit. Gases have even more space, and therefore infrequent particle collisions. This makes liquids and gases poor conductors of heat.
664:
546:
4925:
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.
2889:
1287:
4563:
2491:
2440:
2289:{\displaystyle {\frac {\Delta Q}{\Delta t}}={\frac {A\,(-\Delta T)}{{\frac {\Delta x_{1}}{k_{1}}}+{\frac {\Delta x_{2}}{k_{2}}}+{\frac {\Delta x_{3}}{k_{3}}}+\cdots }}.}
4615:
4589:
2400:
2364:
1851:
595:
1571:
1504:
723:
4816:
2941:
2765:
2302:
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
4436:
3665:
3638:
2849:
2829:
2802:
2674:
2642:
1157:
246:
57:
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,
3178:
4534:
4508:
4482:
361:
point inside the system no longer change. Once this happens, transient conduction is ended, although steady-state conduction may continue if heat flow continues.
1180:
4069:
1593:
1546:
1524:
1377:
1066:
796:
772:
747:
690:
569:
82:
is the heat transfer by the macroscopic movement of a fluid. This type of transfer takes place in a forced-air furnace and in weather systems, for example.
401:
Occasionally, transient conduction problems may be considerably simplified if regions of the object being heated or cooled can be identified, for which
4915:
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".
190:
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
1763:
405:
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
323:
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.
5194:
contains a variety of transient expressions for heat conduction, along with algorithms and computer code for obtaining precise numerical values.
467:. It is known as "second sound" because the wave motion of heat is similar to the propagation of sound in air.this is called Quantum conduction
4339:
3402:
151:
The inter-molecular transfer of energy could be primarily by elastic impact, as in fluids, or by free-electron diffusion, as in metals, or
5097:
1427:
4390:
The Biot number increases as the
Fourier number decreases. There are five steps to determine a temperature profile in terms of time.
2685:
331:
4973:
The principle of thermal conductivity of gases can also be used to measure the concentration of a gas in a binary mixture of gases.
4012:
2769:
The rules for combining resistances and conductances (in series and parallel) are the same for both heat flow and electric current.
1606:
4074:
4903:. By calculating the heat transfer coefficient from this Biot number, one can find a liquid medium suitable for the application.
1970:
3918:{\displaystyle {\dot {Q}}=4k\pi {\frac {T_{1}-T_{2}}{1/{r_{1}}-1/{r_{2}}}}=4k\pi {\frac {(T_{1}-T_{2})r_{1}r_{2}}{r_{2}-r_{1}}}}
1319:
1103:
3510:
2515:
810:
2298:
For heat conduction from one fluid to another through a barrier, it is sometimes important to consider the conductance of the
1967:
are the same for all layers. In a multilayer partition, the total conductance is related to the conductance of its layers by:
1715:
600:
5344:
5310:
5276:
5242:
5215:
2568:
357:
source or sink of heat suddenly introduced within an object, causing temperatures near the source or sink to change in time.
301:
is a quantity derived from conductivity, which is a measure of its ability to exchange thermal energy with its surroundings.
1659:
1212:
4781:{\displaystyle T(x,t)-T_{i}={\frac {T_{i}\Delta X}{2{\sqrt {\pi \alpha t}}}}\exp \left(-{\frac {x^{2}}{4\alpha t}}\right)}
3970:
210:
is caused by the interaction of heat flux and electric current. Heat conduction within a solid is directly analogous to
5406:
2323:
2319:
1424:
for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as
5395:
Liqiu Wang, Xuesheng Zhou, Xiaohao Wei, 'Heat
Conduction: Mathematical Models and Analytical Solutions' Springer 2008
5392:
Jan Taler, Piotr Duda, 'Solving Direct and
Inverse Heat Conduction Problems' Springer-Verlag Berlin Heidelberg 2005
5375:
5167:
5050:
4876:
4821:
4116:
3996:
3978:
17:
5429:
5411:
John H Lienhard IV and John H Lienhard V, 'A Heat
Transfer Textbook', Fifth Edition, Dover Pub., Mineola, NY, 2019
5025:
119:
Thermal conduction (power) is the heat per unit time transferred some distance â between the two temperatures.
4287:
2326:. Ideally, the formulae for conductance should produce a quantity with dimensions independent of distance, like
5435:
5389:
F. Dehghani, CHNG2801 â 'Conservation and
Transport Processes: Course Notes', University of Sydney, Sydney 2007
4620:
5449:
4990:
3974:
144:
1384:
5000:
488:
1946:{\displaystyle R={\frac {1}{U}}={\frac {\Delta x}{k}}={\frac {A\,(-\Delta T)}{\frac {\Delta Q}{\Delta t}}}.}
5480:
5475:
5020:
4912:
432:
511:
139:
31:
4887:, creating a very hard and strong product. To achieve this, it is necessary to quench at the "nose" (or
5015:
2897:
503:
419:
4275:
3168:{\displaystyle {\dot {Q}}\int _{r_{1}}^{r_{2}}{\frac {1}{r}}\,dr=-2k\pi \ell \int _{T_{1}}^{T_{2}}dT}
801:
For many simple applications, Fourier's law is used in its one-dimensional form, for example, in the
4919:
1955:
Resistance is additive when several conducting layers lie between the hot and cool regions, because
1297:
1190:
1081:
5465:
3959:
3677:
1033:
406:
294:
148:
properties. Interfaces often contribute significantly to the observed properties of the materials.
647:
529:
3963:
2854:
1260:
199:
4539:
2457:
2408:
5470:
5386:
H. S. Carslaw and J. C. Jaeger 'Conduction of heat in solids' Oxford University Press, USA 1959
5073:
4594:
4568:
2777:
Conduction through cylindrical shells (e.g. pipes) can be calculated from the internal radius,
2369:
2333:
2118:{\displaystyle {\frac {1}{U}}={\frac {1}{U_{1}}}+{\frac {1}{U_{2}}}+{\frac {1}{U_{3}}}+\cdots }
1830:
869:
574:
5334:
5300:
5266:
5232:
4790:
Splat cooling is a fundamental concept that has been adapted for practical use in the form of
1553:
1486:
705:
597:. The heat flux density is the amount of energy that flows through a unit area per unit time.
389:
for the analytical approach). However, most often, because of complicated shapes with varying
5132:
4995:
4801:
4265:{\displaystyle {\frac {T-T_{f}}{T_{i}-T_{f}}}=\exp \left({\frac {-hAt}{\rho C_{p}V}}\right).}
2310:âbut when dealing with thin high-conductance barriers it can sometimes be quite significant.
1417:
370:
327:
2733:
5005:
4414:
3643:
3616:
2834:
2807:
2780:
2647:
2613:
1836:
1138:
693:
549:
464:
402:
390:
314:
may either be zero or have nonzero values, but all derivatives of temperature at any point
260:
252:
224:
195:
187:
50:
3392:{\displaystyle R_{c}={\frac {\Delta T}{\dot {Q}}}={\frac {\ln(r_{2}/r_{1})}{2\pi k\ell }}}
8:
5441:
5207:
Science of Sintering: New Directions for Materials Processing and Microstructural Control
4795:
4513:
4487:
4461:
1653:
182:) have free-moving electrons that transfer thermal energy rapidly through the metal. The
62:
1162:
409:, as a "lump" of material with a simple thermal capacitance consisting of its aggregate
5136:
4940:
4054:
1600:
1578:
1531:
1509:
1362:
1051:
781:
757:
732:
675:
554:
448:
298:
156:
5402:
5401:
Latif M. Jiji, Amir H. Danesh-Yazdi, 'Heat Conduction' Springer, Fourth Edition 2024
5371:
5340:
5306:
5272:
5238:
5211:
5173:
5163:
5140:
5116:
4961:
2127:
So, when dealing with a multilayer partition, the following formula is usually used:
289:
207:
754:
materials, the thermal conductivity typically varies with orientation; in this case
463:. Heat takes the place of pressure in normal sound waves. This leads to a very high
53:(i.e. from a hotter body to a colder body). For example, heat is conducted from the
5128:
4929:
4791:
476:
251:
To quantify the ease with which a particular medium conducts, engineers employ the
191:
522:
The differential form of Fourier's law of thermal conduction shows that the local
5445:
5205:
5155:
5045:
5040:
4888:
214:
of particles within a fluid, in the situation where there are no fluid currents.
167:
3040:{\displaystyle {\dot {Q}}=-kA_{r}{\frac {dT}{dr}}=-2k\pi r\ell {\frac {dT}{dr}}}
5098:"5.6 Heat Transfer Methods â Conduction, Convection and Radiation Introduction"
5030:
4896:
219:
202:
of most metals have about the same ratio. A good electrical conductor, such as
175:
5459:
5177:
4936:
4455:
3933:
3718:
Solving in a similar manner as for a cylindrical shell (see above) produces:
3280:{\displaystyle {\dot {Q}}=2k\pi \ell {\frac {T_{1}-T_{2}}{\ln(r_{2}/r_{1})}}}
1701:
1353:
484:
480:
456:
452:
410:
395:
386:
343:
171:
100:
5412:
5298:
3671:
2327:
1649:
507:
444:
218:
path of gas molecules relative to the size of the gas gap, as given by the
4441:
Compared required to point to trace specified Biot number on the nomogram.
3500:{\textstyle {\dot {Q}}=2\pi k\ell r_{m}{\frac {T_{1}-T_{2}}{r_{2}-r_{1}}}}
5035:
4892:
1409:
1037:
352:
at any place within an object, the mode of thermal energy flow is termed
179:
68:
Every process involving heat transfer takes place by only three methods:
58:
5398:
W. Kelly, 'Understanding Heat Conduction' Nova Science Publischer, 2010
5117:"Effective Thermal Conductivity of Submicron Powders: A Numerical Study"
1648:
With a simple 1-D steady heat conduction equation which is analogous to
4884:
2303:
1048:
By integrating the differential form over the material's total surface
751:
170:
of thermal energy. This is due to the way that metals bond chemically:
5204:
III, H. Palmour; Spriggs, R. M.; Uskokovic, D. P. (11 November 2013).
5010:
4880:
4872:
2307:
2299:
1356:
transferred by conduction (in W), time derivative of the transferred
523:
497:
460:
255:, also known as the conductivity constant or conduction coefficient,
211:
3948:
3667:, can be calculated in a similar manner as for a cylindrical shell.
4900:
4327:
2851:, and the temperature difference between the inner and outer wall,
1421:
492:
54:
5299:
Rajiv Asthana; Ashok Kumar; Narendra B. Dahotre (9 January 2006).
413:. Such regions warm or cool, but show no significant temperature
5424:
3613:
The conduction through a spherical shell with internal radius,
775:
697:
203:
160:
152:
128:ÎT is the difference in temperature from one side to the other.
1818:{\displaystyle {\frac {\Delta Q}{\Delta t}}=UA\,(-\Delta T).}
166:
Metals (e.g., copper, platinum, gold, etc.) are usually good
5153:
4965:
be correlated with the database to identify the gas sample.
4960:
The working of this instrument is by principle based on the
4935:
These differences are among the defining characteristics of
3605:. It is important to note that this is the log-mean radius.
1357:
667:
131:â is the length of the path the heat has to be transferred.
4336:
coordinate and the Fourier number, which is calculated by
3598:{\textstyle r_{m}={\frac {r_{2}-r_{1}}{\ln(r_{2}/r_{1})}}}
1344:{\displaystyle \mathbf {q} \cdot \mathrm {d} \mathbf {S} }
1128:{\displaystyle \mathbf {q} \cdot \mathrm {d} \mathbf {S} }
5292:
4928:
For example, the material of the wall must not undergo a
4381:{\displaystyle {\textit {Fo}}={\frac {\alpha t}{L^{2}}}.}
459:-like motion, rather than by the more usual mechanism of
2318:
The previous conductance equations, written in terms of
5230:
5191:
4895:, the time it takes for the material to quench, or the
1250:
638:
5210:. Springer Science & Business Media. p. 164.
4891:) of the TTT diagram. Since materials differ in their
3513:
3405:
1301:
1238:{\displaystyle \nabla T\cdot \mathrm {d} \mathbf {S} }
1194:
1085:
4875:
is a transient heat transfer process in terms of the
4824:
4804:
4659:
4623:
4597:
4571:
4542:
4516:
4490:
4464:
4438:
to relative temperature with the boundary conditions.
4417:
4342:
4290:
4158:
4119:
4077:
4057:
4015:
3724:
3680:
3646:
3619:
3293:
3181:
3053:
2944:
2900:
2857:
2837:
2810:
2783:
2736:
2688:
2650:
2616:
2571:
2518:
2460:
2411:
2372:
2336:
2133:
2038:
1973:
1859:
1833:
1766:
1718:
1662:
1609:
1581:
1556:
1534:
1512:
1506:
is the time interval during which the amount of heat
1489:
1430:
1387:
1365:
1322:
1300:
1263:
1215:
1193:
1165:
1141:
1106:
1084:
1054:
877:
813:
784:
760:
735:
708:
678:
650:
603:
577:
557:
532:
227:
27:
Process by which heat is transferred within an object
5336:
Handbook of Residual Stress and Deformation of Steel
4071:, is introduced in this formula, and is measured in
2313:
5203:
1700:This law forms the basis for the derivation of the
1473:{\displaystyle Q=-k{\frac {A\Delta t}{L}}\Delta T,}
1068:, we arrive at the integral form of Fourier's law:
868:In an isotropic medium, Fourier's law leads to the
4855:
4810:
4780:
4644:
4609:
4583:
4557:
4528:
4502:
4476:
4430:
4380:
4315:
4264:
4144:
4105:
4063:
4043:
3917:
3708:
3659:
3632:
3597:
3499:
3391:
3279:
3167:
3039:
2928:
2883:
2843:
2823:
2796:
2759:
2722:
2668:
2636:
2602:
2550:
2485:
2434:
2394:
2358:
2288:
2117:
2024:
1945:
1845:
1817:
1745:
1690:
1638:
1587:
1565:
1540:
1518:
1498:
1472:
1400:
1371:
1343:
1308:
1281:
1237:
1201:
1174:
1151:
1127:
1092:
1060:
1024:
858:
790:
766:
741:
717:
684:
658:
629:
589:
563:
540:
240:
65:is approached, temperature becoming more uniform.
2482:
2391:
2355:
5457:
5332:
5158:; Incropera, Frank P.; Dewitt, David P. (2011).
4951:
510:is the electrical analogue of Fourier's law and
348:During any period in which temperatures changes
4906:
4856:{\displaystyle \alpha ={\frac {k}{\rho C_{p}}}}
4397:Determine which relative depth matters, either
4330:. A nomogram has a relative temperature as the
3927:
2723:{\displaystyle R={\frac {\Delta T}{\dot {Q}}},}
1573:is the temperature difference between the ends,
506:is a discrete analogue of Fourier's law, while
297:, temperature, density, and molecular bonding.
279:), in a direction normal to a surface of area (
5302:Materials Processing and Manufacturing Science
5231:Sam Zhang; Dongliang Zhao (19 November 2012).
4044:{\displaystyle {\textit {Bi}}={\frac {hL}{k}}}
1639:{\displaystyle R={\frac {1}{k}}{\frac {L}{A}}}
1526:flows through a cross-section of the material,
159:, the heat flux is carried almost entirely by
5234:Aeronautical and Aerospace Materials Handbook
4106:{\displaystyle \mathrm {\frac {J}{m^{2}sK}} }
2678:The reciprocal of conductance is resistance,
1827:The reciprocal of conductance is resistance,
122:Îș is the thermal conductivity of the material
5224:
4316:{\displaystyle \mathrm {\frac {W}{m^{2}K}} }
571:and the negative local temperature gradient
293:that is primarily dependent on the medium's
5370:, American Institute of Physics, New York,
5264:
4326:The series solution can be analyzed with a
3977:. Unsourced material may be challenged and
2025:{\displaystyle R=R_{1}+R_{2}+R_{3}+\cdots }
125:A is the cross-sectional area of the object
5326:
4645:{\displaystyle -\infty \leq x\leq \infty }
4484:for initial temperature as the maximum at
3938:
475:The law of heat conduction, also known as
426:
304:
4863:. This varies according to the material.
4132:
3997:Learn how and when to remove this message
3105:
2590:
2551:{\displaystyle U={\frac {kA}{\Delta x}},}
2481:
2390:
2354:
2163:
1903:
1796:
1675:
859:{\displaystyle q_{x}=-k{\frac {dT}{dx}}.}
5305:. ButterworthâHeinemann. pp. 158â.
1746:{\displaystyle U={\frac {k}{\Delta x}},}
1401:{\displaystyle \mathrm {d} \mathbf {S} }
630:{\displaystyle \mathbf {q} =-k\nabla T,}
287:) ". Thermal conductivity is a material
5258:
2603:{\displaystyle {\dot {Q}}=U\,\Delta T,}
337:
14:
5458:
5160:Fundamentals of heat and mass transfer
5133:10.4028/www.scientific.net/AMM.846.500
1691:{\displaystyle \Delta T=R\,{\dot {Q}}}
5444:by Jeff Bryant based on a program by
4345:
4018:
2772:
2565:Fourier's law can also be stated as:
1760:Fourier's law can also be stated as:
439:
283:), due to a temperature difference (Î
267:is defined as "the quantity of heat,
5339:. ASM International. pp. 322â.
5192:Exact Analytical Conduction Toolbox
5162:(7th ed.). Hoboken, NJ: Wiley.
5068:
5066:
3975:adding citations to reliable sources
3942:
2938:When Fourier's equation is applied:
2894:The surface area of the cylinder is
1548:is the cross-sectional surface area,
517:
5114:
4408:Convert time to the Fourier number.
3175:then the rate of heat transfer is:
310:partial derivatives of temperature
194:through conductive solids, and the
24:
4866:
4707:
4639:
4627:
4604:
4578:
4552:
4307:
4298:
4293:
4133:
4097:
4094:
4085:
4080:
3310:
2698:
2591:
2536:
2322:, can be reformulated in terms of
2248:
2216:
2184:
2170:
2145:
2137:
1930:
1922:
1910:
1882:
1803:
1778:
1770:
1731:
1663:
1557:
1490:
1461:
1449:
1389:
1332:
1226:
1216:
1116:
1001:
987:
964:
950:
927:
913:
889:
881:
709:
618:
581:
334:for an example of such a network.
25:
5492:
5417:
5063:
5051:General equation of heat transfer
2929:{\displaystyle A_{r}=2\pi r\ell }
2454:is cross-sectional area, we have
2314:Intensive-property representation
1603:of the 1-D homogeneous material:
1599:One can define the (macroscopic)
1595:is the distance between the ends.
774:is represented by a second-order
725:is the temperature gradient, K/m.
332:purely resistive thermal circuits
61:differences decay over time, and
5423:
4450:
3947:
1757:is the conductance, in W/(m K).
1394:
1337:
1324:
1289:
1231:
1182:
1121:
1108:
1073:
1043:
666:is the local heat flux density,
652:
605:
534:
470:
5360:
5121:Applied Mechanics and Materials
4911:One statement of the so-called
4877:time temperature transformation
4445:
4145:{\displaystyle q=-h\,\Delta T,}
5450:Wolfram Demonstrations Project
5197:
5184:
5147:
5108:
5090:
4991:List of thermal conductivities
4946:
4675:
4663:
4051:The heat transfer coefficient
3864:
3838:
3589:
3561:
3369:
3341:
3271:
3243:
2176:
2164:
1916:
1904:
1809:
1797:
1707:
1309:{\displaystyle \scriptstyle S}
1202:{\displaystyle \scriptstyle S}
1093:{\displaystyle \scriptstyle S}
798:varies with spatial location.
145:Interfacial thermal resistance
13:
1:
5056:
5001:Convection diffusion equation
4968:
4952:Thermal conductivity analyzer
3709:{\displaystyle A=4\pi r^{2}.}
2405:From the electrical formula:
1410:oriented surface area element
418:systems is those that follow
5237:. CRC Press. pp. 304â.
5026:ChurchillâBernstein equation
5021:Relativistic heat conduction
4913:zeroth law of thermodynamics
4907:Zeroth law of thermodynamics
4798:coefficient, represented as
3928:Transient thermal conduction
3608:
778:. In non-uniform materials,
659:{\displaystyle \mathbf {q} }
541:{\displaystyle \mathbf {q} }
433:relativistic heat conduction
45:, frequently represented by
7:
5271:. Springer. pp. 174â.
4984:
3287:the thermal resistance is:
2884:{\displaystyle T_{2}-T_{1}}
2330:for electrical resistance,
1282:{\displaystyle {\dot {Q}}=}
548:is equal to the product of
483:), states that the rate of
206:, also conducts heat well.
140:Thermal contact conductance
110:Thermal conduction (power)=
94:
32:Conduction (disambiguation)
10:
5497:
5368:A Survey of Thermodynamics
4591:, and the heat profile at
4558:{\displaystyle x=-\infty }
3931:
2486:{\displaystyle G=kA/x\,\!}
2435:{\displaystyle R=\rho x/A}
514:is its chemical analogue.
341:
98:
76:transferred by conduction.
29:
5333:George E. Totten (2002).
5268:Drop-Surface Interactions
5115:Dai; et al. (2015).
4610:{\displaystyle t=\infty }
4584:{\displaystyle x=\infty }
4394:Calculate the Biot number
4276:heat transfer coefficient
2509:is cross-sectional area.
2395:{\displaystyle G=I/V\,\!}
2359:{\displaystyle R=V/I\,\!}
1846:{\displaystyle {\big .}R}
729:The thermal conductivity
590:{\displaystyle -\nabla T}
200:electrical conductivities
2730:analogous to Ohm's law,
2610:analogous to Ohm's law,
1566:{\displaystyle \Delta T}
1499:{\displaystyle \Delta t}
718:{\displaystyle \nabla T}
512:Fick's laws of diffusion
407:lumped capacitance model
5442:Newton's Law of Cooling
5102:Douglas College Physics
5016:Fick's law of diffusion
4939:. In a sense, they are
4811:{\displaystyle \alpha }
3939:Interface heat transfer
3640:, and external radius,
2804:, the external radius,
504:Newton's law of cooling
427:Relativistic conduction
420:Newton's law of cooling
305:Steady-state conduction
275:) through a thickness (
271:, transmitted in time (
155:, as in insulators. In
5154:Bergman, Theodore L.;
4857:
4812:
4782:
4646:
4611:
4585:
4559:
4530:
4504:
4478:
4432:
4382:
4317:
4266:
4146:
4107:
4065:
4045:
3919:
3710:
3661:
3634:
3599:
3501:
3393:
3281:
3169:
3041:
2930:
2885:
2845:
2825:
2798:
2761:
2760:{\displaystyle R=V/I.}
2724:
2670:
2638:
2604:
2552:
2487:
2436:
2396:
2360:
2290:
2119:
2026:
1947:
1847:
1819:
1747:
1692:
1640:
1589:
1567:
1542:
1520:
1500:
1474:
1402:
1373:
1345:
1310:
1283:
1239:
1203:
1176:
1153:
1129:
1094:
1062:
1036:famously known as the
1026:
860:
792:
768:
743:
719:
686:
660:
631:
591:
565:
542:
487:through a material is
391:thermal conductivities
328:electrical resistances
242:
5438:â Thermal-FluidsPedia
4996:Electrical conduction
4981:to that of nitrogen.
4858:
4813:
4783:
4647:
4612:
4586:
4560:
4531:
4505:
4479:
4433:
4431:{\displaystyle T_{i}}
4383:
4318:
4267:
4147:
4108:
4066:
4046:
3920:
3711:
3662:
3660:{\displaystyle r_{2}}
3635:
3633:{\displaystyle r_{1}}
3600:
3502:
3394:
3282:
3170:
3042:
2931:
2886:
2846:
2844:{\displaystyle \ell }
2826:
2824:{\displaystyle r_{2}}
2799:
2797:{\displaystyle r_{1}}
2762:
2725:
2671:
2669:{\displaystyle I=VG.}
2639:
2637:{\displaystyle I=V/R}
2605:
2553:
2488:
2437:
2397:
2361:
2291:
2120:
2027:
1948:
1848:
1820:
1748:
1693:
1641:
1590:
1568:
1543:
1521:
1501:
1475:
1418:differential equation
1403:
1374:
1346:
1311:
1284:
1249:where (including the
1240:
1204:
1177:
1154:
1152:{\displaystyle {}={}}
1130:
1095:
1063:
1027:
861:
793:
769:
744:
720:
687:
661:
637:where (including the
632:
592:
566:
543:
371:operating temperature
354:transient conduction.
243:
241:{\displaystyle K_{n}}
5432:at Wikimedia Commons
5265:Martin Eein (2002).
5006:R-value (insulation)
4822:
4818:, can be written as
4802:
4657:
4621:
4595:
4569:
4540:
4514:
4488:
4462:
4415:
4340:
4288:
4156:
4117:
4075:
4055:
4013:
3971:improve this section
3722:
3678:
3644:
3617:
3511:
3403:
3291:
3179:
3051:
2942:
2898:
2855:
2835:
2808:
2781:
2734:
2686:
2648:
2614:
2569:
2562:is the conductance.
2516:
2458:
2409:
2370:
2334:
2324:intensive properties
2320:extensive properties
2131:
2036:
1971:
1857:
1831:
1764:
1716:
1660:
1607:
1579:
1554:
1532:
1510:
1487:
1428:
1385:
1363:
1320:
1298:
1261:
1213:
1191:
1163:
1139:
1104:
1082:
1052:
1034:fundamental solution
875:
811:
782:
758:
733:
706:
676:
648:
601:
575:
555:
550:thermal conductivity
530:
465:thermal conductivity
451:phenomenon in which
403:thermal conductivity
338:Transient conduction
261:thermal conductivity
253:thermal conductivity
225:
51:temperature gradient
43:Thermal conductivity
30:For other uses, see
5481:Transport phenomena
5476:Physical quantities
5366:Bailyn, M. (1994).
5156:Lavine, Adrienne S.
4796:thermal diffusivity
4529:{\displaystyle T=0}
4503:{\displaystyle x=0}
4477:{\displaystyle t=0}
3158:
3094:
2366:, and conductance,
1654:electric resistance
500:of energy locally.
479:(compare Fourier's
63:thermal equilibrium
5078:energyeducation.ca
5074:"Energy Education"
4943:of heat transfer.
4853:
4808:
4778:
4642:
4607:
4581:
4555:
4526:
4500:
4474:
4428:
4378:
4313:
4262:
4142:
4103:
4061:
4041:
3915:
3706:
3674:of the sphere is:
3657:
3630:
3595:
3497:
3389:
3277:
3165:
3130:
3066:
3037:
2926:
2881:
2841:
2821:
2794:
2773:Cylindrical shells
2757:
2720:
2666:
2634:
2600:
2548:
2483:
2432:
2392:
2356:
2286:
2115:
2022:
1943:
1843:
1815:
1743:
1688:
1636:
1601:thermal resistance
1585:
1563:
1538:
1516:
1496:
1470:
1398:
1369:
1341:
1306:
1305:
1279:
1235:
1199:
1198:
1175:{\displaystyle -k}
1172:
1149:
1125:
1090:
1089:
1058:
1022:
856:
788:
764:
739:
715:
692:is the material's
682:
656:
627:
587:
561:
538:
449:quantum mechanical
440:Quantum conduction
299:Thermal effusivity
238:
38:Thermal conduction
18:Fourier's law
5428:Media related to
5346:978-1-61503-227-3
5312:978-0-08-046488-6
5278:978-3-211-83692-7
5244:978-1-4398-7329-8
5217:978-1-4899-0933-6
4962:Wheatstone bridge
4851:
4771:
4732:
4729:
4373:
4347:
4311:
4284:, is measured in
4253:
4204:
4101:
4064:{\displaystyle h}
4039:
4020:
4007:
4006:
3999:
3913:
3821:
3734:
3593:
3495:
3415:
3387:
3327:
3325:
3275:
3191:
3103:
3063:
3035:
2994:
2954:
2715:
2713:
2581:
2543:
2501:is conductivity,
2281:
2272:
2240:
2208:
2152:
2107:
2087:
2067:
2047:
1938:
1937:
1892:
1874:
1785:
1738:
1685:
1634:
1624:
1588:{\displaystyle L}
1541:{\displaystyle A}
1519:{\displaystyle Q}
1459:
1372:{\displaystyle Q}
1273:
1061:{\displaystyle S}
1015:
978:
941:
896:
851:
791:{\displaystyle k}
767:{\displaystyle k}
742:{\displaystyle k}
685:{\displaystyle k}
564:{\displaystyle k}
518:Differential form
208:Thermoelectricity
85:Heat transfer by
16:(Redirected from
5488:
5427:
5379:
5364:
5358:
5357:
5355:
5353:
5330:
5324:
5323:
5321:
5319:
5296:
5290:
5289:
5287:
5285:
5262:
5256:
5255:
5253:
5251:
5228:
5222:
5221:
5201:
5195:
5188:
5182:
5181:
5151:
5145:
5144:
5112:
5106:
5105:
5094:
5088:
5087:
5085:
5084:
5070:
4930:phase transition
4862:
4860:
4859:
4854:
4852:
4850:
4849:
4848:
4832:
4817:
4815:
4814:
4809:
4792:thermal spraying
4787:
4785:
4784:
4779:
4777:
4773:
4772:
4770:
4759:
4758:
4749:
4733:
4731:
4730:
4719:
4713:
4706:
4705:
4695:
4690:
4689:
4651:
4649:
4648:
4643:
4616:
4614:
4613:
4608:
4590:
4588:
4587:
4582:
4564:
4562:
4561:
4556:
4535:
4533:
4532:
4527:
4509:
4507:
4506:
4501:
4483:
4481:
4480:
4475:
4437:
4435:
4434:
4429:
4427:
4426:
4387:
4385:
4384:
4379:
4374:
4372:
4371:
4362:
4354:
4349:
4348:
4335:
4322:
4320:
4319:
4314:
4312:
4310:
4306:
4305:
4292:
4283:
4271:
4269:
4268:
4263:
4258:
4254:
4252:
4248:
4247:
4234:
4220:
4205:
4203:
4202:
4201:
4189:
4188:
4178:
4177:
4176:
4160:
4151:
4149:
4148:
4143:
4112:
4110:
4109:
4104:
4102:
4100:
4093:
4092:
4079:
4070:
4068:
4067:
4062:
4050:
4048:
4047:
4042:
4040:
4035:
4027:
4022:
4021:
4002:
3995:
3991:
3988:
3982:
3951:
3943:
3924:
3922:
3921:
3916:
3914:
3912:
3911:
3910:
3898:
3897:
3887:
3886:
3885:
3876:
3875:
3863:
3862:
3850:
3849:
3836:
3822:
3820:
3819:
3818:
3817:
3807:
3796:
3795:
3794:
3784:
3775:
3774:
3773:
3761:
3760:
3750:
3736:
3735:
3727:
3715:
3713:
3712:
3707:
3702:
3701:
3666:
3664:
3663:
3658:
3656:
3655:
3639:
3637:
3636:
3631:
3629:
3628:
3604:
3602:
3601:
3596:
3594:
3592:
3588:
3587:
3578:
3573:
3572:
3553:
3552:
3551:
3539:
3538:
3528:
3523:
3522:
3506:
3504:
3503:
3498:
3496:
3494:
3493:
3492:
3480:
3479:
3469:
3468:
3467:
3455:
3454:
3444:
3442:
3441:
3417:
3416:
3408:
3398:
3396:
3395:
3390:
3388:
3386:
3372:
3368:
3367:
3358:
3353:
3352:
3333:
3328:
3326:
3318:
3316:
3308:
3303:
3302:
3286:
3284:
3283:
3278:
3276:
3274:
3270:
3269:
3260:
3255:
3254:
3235:
3234:
3233:
3221:
3220:
3210:
3193:
3192:
3184:
3174:
3172:
3171:
3166:
3157:
3156:
3155:
3145:
3144:
3143:
3104:
3096:
3093:
3092:
3091:
3081:
3080:
3079:
3065:
3064:
3056:
3047:and rearranged:
3046:
3044:
3043:
3038:
3036:
3034:
3026:
3018:
2995:
2993:
2985:
2977:
2975:
2974:
2956:
2955:
2947:
2935:
2933:
2932:
2927:
2910:
2909:
2890:
2888:
2887:
2882:
2880:
2879:
2867:
2866:
2850:
2848:
2847:
2842:
2830:
2828:
2827:
2822:
2820:
2819:
2803:
2801:
2800:
2795:
2793:
2792:
2766:
2764:
2763:
2758:
2750:
2729:
2727:
2726:
2721:
2716:
2714:
2706:
2704:
2696:
2675:
2673:
2672:
2667:
2643:
2641:
2640:
2635:
2630:
2609:
2607:
2606:
2601:
2583:
2582:
2574:
2561:
2557:
2555:
2554:
2549:
2544:
2542:
2534:
2526:
2497:is conductance,
2492:
2490:
2489:
2484:
2477:
2446:is resistivity,
2441:
2439:
2438:
2433:
2428:
2401:
2399:
2398:
2393:
2386:
2365:
2363:
2362:
2357:
2350:
2295:
2293:
2292:
2287:
2282:
2280:
2273:
2271:
2270:
2261:
2260:
2259:
2246:
2241:
2239:
2238:
2229:
2228:
2227:
2214:
2209:
2207:
2206:
2197:
2196:
2195:
2182:
2179:
2158:
2153:
2151:
2143:
2135:
2124:
2122:
2121:
2116:
2108:
2106:
2105:
2093:
2088:
2086:
2085:
2073:
2068:
2066:
2065:
2053:
2048:
2040:
2032:or equivalently
2031:
2029:
2028:
2023:
2015:
2014:
2002:
2001:
1989:
1988:
1966:
1960:
1952:
1950:
1949:
1944:
1939:
1936:
1928:
1920:
1919:
1898:
1893:
1888:
1880:
1875:
1867:
1852:
1850:
1849:
1844:
1839:
1838:
1824:
1822:
1821:
1816:
1786:
1784:
1776:
1768:
1756:
1752:
1750:
1749:
1744:
1739:
1737:
1726:
1697:
1695:
1694:
1689:
1687:
1686:
1678:
1645:
1643:
1642:
1637:
1635:
1627:
1625:
1617:
1594:
1592:
1591:
1586:
1572:
1570:
1569:
1564:
1547:
1545:
1544:
1539:
1525:
1523:
1522:
1517:
1505:
1503:
1502:
1497:
1479:
1477:
1476:
1471:
1460:
1455:
1444:
1407:
1405:
1404:
1399:
1397:
1392:
1378:
1376:
1375:
1370:
1351:
1350:
1348:
1347:
1342:
1340:
1335:
1327:
1316:
1315:
1313:
1312:
1307:
1293:
1292:
1288:
1286:
1285:
1280:
1275:
1274:
1266:
1245:
1244:
1242:
1241:
1236:
1234:
1229:
1209:
1208:
1206:
1205:
1200:
1186:
1185:
1181:
1179:
1178:
1173:
1158:
1156:
1155:
1150:
1148:
1143:
1135:
1134:
1132:
1131:
1126:
1124:
1119:
1111:
1100:
1099:
1097:
1096:
1091:
1077:
1076:
1067:
1065:
1064:
1059:
1031:
1029:
1028:
1023:
1021:
1017:
1016:
1014:
1013:
1012:
999:
995:
994:
984:
979:
977:
976:
975:
962:
958:
957:
947:
942:
940:
939:
938:
925:
921:
920:
910:
897:
895:
887:
879:
865:
863:
862:
857:
852:
850:
842:
834:
823:
822:
806:
797:
795:
794:
789:
773:
771:
770:
765:
748:
746:
745:
740:
724:
722:
721:
716:
691:
689:
688:
683:
665:
663:
662:
657:
655:
636:
634:
633:
628:
608:
596:
594:
593:
588:
570:
568:
567:
562:
547:
545:
544:
539:
537:
491:to the negative
312:concerning space
247:
245:
244:
239:
237:
236:
192:electric current
153:phonon vibration
48:
21:
5496:
5495:
5491:
5490:
5489:
5487:
5486:
5485:
5466:Heat conduction
5456:
5455:
5446:Stephen Wolfram
5436:Heat conduction
5430:Heat conduction
5420:
5383:
5382:
5365:
5361:
5351:
5349:
5347:
5331:
5327:
5317:
5315:
5313:
5297:
5293:
5283:
5281:
5279:
5263:
5259:
5249:
5247:
5245:
5229:
5225:
5218:
5202:
5198:
5189:
5185:
5170:
5152:
5148:
5113:
5109:
5096:
5095:
5091:
5082:
5080:
5072:
5071:
5064:
5059:
5046:Heat Conduction
5041:False diffusion
4987:
4971:
4954:
4949:
4920:diathermal wall
4909:
4869:
4867:Metal quenching
4844:
4840:
4836:
4831:
4823:
4820:
4819:
4803:
4800:
4799:
4760:
4754:
4750:
4748:
4744:
4740:
4718:
4714:
4701:
4697:
4696:
4694:
4685:
4681:
4658:
4655:
4654:
4622:
4619:
4618:
4596:
4593:
4592:
4570:
4567:
4566:
4541:
4538:
4537:
4515:
4512:
4511:
4489:
4486:
4485:
4463:
4460:
4459:
4453:
4448:
4422:
4418:
4416:
4413:
4412:
4367:
4363:
4355:
4353:
4344:
4343:
4341:
4338:
4337:
4331:
4301:
4297:
4296:
4291:
4289:
4286:
4285:
4279:
4243:
4239:
4235:
4221:
4219:
4215:
4197:
4193:
4184:
4180:
4179:
4172:
4168:
4161:
4159:
4157:
4154:
4153:
4118:
4115:
4114:
4088:
4084:
4083:
4078:
4076:
4073:
4072:
4056:
4053:
4052:
4028:
4026:
4017:
4016:
4014:
4011:
4010:
4003:
3992:
3986:
3983:
3968:
3952:
3941:
3936:
3930:
3906:
3902:
3893:
3889:
3888:
3881:
3877:
3871:
3867:
3858:
3854:
3845:
3841:
3837:
3835:
3813:
3809:
3808:
3803:
3790:
3786:
3785:
3780:
3776:
3769:
3765:
3756:
3752:
3751:
3749:
3726:
3725:
3723:
3720:
3719:
3697:
3693:
3679:
3676:
3675:
3651:
3647:
3645:
3642:
3641:
3624:
3620:
3618:
3615:
3614:
3611:
3583:
3579:
3574:
3568:
3564:
3554:
3547:
3543:
3534:
3530:
3529:
3527:
3518:
3514:
3512:
3509:
3508:
3488:
3484:
3475:
3471:
3470:
3463:
3459:
3450:
3446:
3445:
3443:
3437:
3433:
3407:
3406:
3404:
3401:
3400:
3373:
3363:
3359:
3354:
3348:
3344:
3334:
3332:
3317:
3309:
3307:
3298:
3294:
3292:
3289:
3288:
3265:
3261:
3256:
3250:
3246:
3236:
3229:
3225:
3216:
3212:
3211:
3209:
3183:
3182:
3180:
3177:
3176:
3151:
3147:
3146:
3139:
3135:
3134:
3095:
3087:
3083:
3082:
3075:
3071:
3070:
3055:
3054:
3052:
3049:
3048:
3027:
3019:
3017:
2986:
2978:
2976:
2970:
2966:
2946:
2945:
2943:
2940:
2939:
2905:
2901:
2899:
2896:
2895:
2875:
2871:
2862:
2858:
2856:
2853:
2852:
2836:
2833:
2832:
2815:
2811:
2809:
2806:
2805:
2788:
2784:
2782:
2779:
2778:
2775:
2746:
2735:
2732:
2731:
2705:
2697:
2695:
2687:
2684:
2683:
2649:
2646:
2645:
2626:
2615:
2612:
2611:
2573:
2572:
2570:
2567:
2566:
2559:
2535:
2527:
2525:
2517:
2514:
2513:
2505:is length, and
2473:
2459:
2456:
2455:
2450:is length, and
2424:
2410:
2407:
2406:
2382:
2371:
2368:
2367:
2346:
2335:
2332:
2331:
2316:
2266:
2262:
2255:
2251:
2247:
2245:
2234:
2230:
2223:
2219:
2215:
2213:
2202:
2198:
2191:
2187:
2183:
2181:
2180:
2159:
2157:
2144:
2136:
2134:
2132:
2129:
2128:
2101:
2097:
2092:
2081:
2077:
2072:
2061:
2057:
2052:
2039:
2037:
2034:
2033:
2010:
2006:
1997:
1993:
1984:
1980:
1972:
1969:
1968:
1962:
1956:
1929:
1921:
1899:
1897:
1881:
1879:
1866:
1858:
1855:
1854:
1835:
1834:
1832:
1829:
1828:
1777:
1769:
1767:
1765:
1762:
1761:
1754:
1730:
1725:
1717:
1714:
1713:
1710:
1677:
1676:
1661:
1658:
1657:
1626:
1616:
1608:
1605:
1604:
1580:
1577:
1576:
1555:
1552:
1551:
1533:
1530:
1529:
1511:
1508:
1507:
1488:
1485:
1484:
1445:
1443:
1429:
1426:
1425:
1393:
1388:
1386:
1383:
1382:
1364:
1361:
1360:
1336:
1331:
1323:
1321:
1318:
1317:
1299:
1296:
1295:
1294:
1290:
1265:
1264:
1262:
1259:
1258:
1257:
1230:
1225:
1214:
1211:
1210:
1192:
1189:
1188:
1187:
1183:
1164:
1161:
1160:
1159:
1147:
1142:
1140:
1137:
1136:
1120:
1115:
1107:
1105:
1102:
1101:
1083:
1080:
1079:
1078:
1074:
1072:
1053:
1050:
1049:
1046:
1008:
1004:
1000:
990:
986:
985:
983:
971:
967:
963:
953:
949:
948:
946:
934:
930:
926:
916:
912:
911:
909:
908:
904:
888:
880:
878:
876:
873:
872:
843:
835:
833:
818:
814:
812:
809:
808:
802:
783:
780:
779:
759:
756:
755:
734:
731:
730:
707:
704:
703:
677:
674:
673:
651:
649:
646:
645:
604:
602:
599:
598:
576:
573:
572:
556:
553:
552:
533:
531:
528:
527:
520:
473:
442:
429:
346:
340:
316:concerning time
307:
232:
228:
226:
223:
222:
174:(as opposed to
113:
111:
103:
97:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5494:
5484:
5483:
5478:
5473:
5468:
5454:
5453:
5439:
5433:
5419:
5418:External links
5416:
5415:
5414:
5409:
5407:978-3031437397
5399:
5396:
5393:
5390:
5387:
5381:
5380:
5359:
5345:
5325:
5311:
5291:
5277:
5257:
5243:
5223:
5216:
5196:
5183:
5168:
5146:
5107:
5089:
5061:
5060:
5058:
5055:
5054:
5053:
5048:
5043:
5038:
5033:
5031:Fourier number
5028:
5023:
5018:
5013:
5008:
5003:
4998:
4993:
4986:
4983:
4970:
4967:
4953:
4950:
4948:
4945:
4908:
4905:
4897:Fourier number
4868:
4865:
4847:
4843:
4839:
4835:
4830:
4827:
4807:
4776:
4769:
4766:
4763:
4757:
4753:
4747:
4743:
4739:
4736:
4728:
4725:
4722:
4717:
4712:
4709:
4704:
4700:
4693:
4688:
4684:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4641:
4638:
4635:
4632:
4629:
4626:
4606:
4603:
4600:
4580:
4577:
4574:
4554:
4551:
4548:
4545:
4525:
4522:
4519:
4499:
4496:
4493:
4473:
4470:
4467:
4452:
4449:
4447:
4444:
4443:
4442:
4439:
4425:
4421:
4409:
4406:
4395:
4377:
4370:
4366:
4361:
4358:
4352:
4309:
4304:
4300:
4295:
4261:
4257:
4251:
4246:
4242:
4238:
4233:
4230:
4227:
4224:
4218:
4214:
4211:
4208:
4200:
4196:
4192:
4187:
4183:
4175:
4171:
4167:
4164:
4152:which becomes
4141:
4138:
4135:
4131:
4128:
4125:
4122:
4099:
4096:
4091:
4087:
4082:
4060:
4038:
4034:
4031:
4025:
4005:
4004:
3955:
3953:
3946:
3940:
3937:
3932:Main article:
3929:
3926:
3909:
3905:
3901:
3896:
3892:
3884:
3880:
3874:
3870:
3866:
3861:
3857:
3853:
3848:
3844:
3840:
3834:
3831:
3828:
3825:
3816:
3812:
3806:
3802:
3799:
3793:
3789:
3783:
3779:
3772:
3768:
3764:
3759:
3755:
3748:
3745:
3742:
3739:
3733:
3730:
3705:
3700:
3696:
3692:
3689:
3686:
3683:
3654:
3650:
3627:
3623:
3610:
3607:
3591:
3586:
3582:
3577:
3571:
3567:
3563:
3560:
3557:
3550:
3546:
3542:
3537:
3533:
3526:
3521:
3517:
3491:
3487:
3483:
3478:
3474:
3466:
3462:
3458:
3453:
3449:
3440:
3436:
3432:
3429:
3426:
3423:
3420:
3414:
3411:
3385:
3382:
3379:
3376:
3371:
3366:
3362:
3357:
3351:
3347:
3343:
3340:
3337:
3331:
3324:
3321:
3315:
3312:
3306:
3301:
3297:
3273:
3268:
3264:
3259:
3253:
3249:
3245:
3242:
3239:
3232:
3228:
3224:
3219:
3215:
3208:
3205:
3202:
3199:
3196:
3190:
3187:
3164:
3161:
3154:
3150:
3142:
3138:
3133:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3102:
3099:
3090:
3086:
3078:
3074:
3069:
3062:
3059:
3033:
3030:
3025:
3022:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2992:
2989:
2984:
2981:
2973:
2969:
2965:
2962:
2959:
2953:
2950:
2925:
2922:
2919:
2916:
2913:
2908:
2904:
2878:
2874:
2870:
2865:
2861:
2840:
2831:, the length,
2818:
2814:
2791:
2787:
2774:
2771:
2756:
2753:
2749:
2745:
2742:
2739:
2719:
2712:
2709:
2703:
2700:
2694:
2691:
2665:
2662:
2659:
2656:
2653:
2633:
2629:
2625:
2622:
2619:
2599:
2596:
2593:
2589:
2586:
2580:
2577:
2547:
2541:
2538:
2533:
2530:
2524:
2521:
2480:
2476:
2472:
2469:
2466:
2463:
2431:
2427:
2423:
2420:
2417:
2414:
2389:
2385:
2381:
2378:
2375:
2353:
2349:
2345:
2342:
2339:
2315:
2312:
2285:
2279:
2276:
2269:
2265:
2258:
2254:
2250:
2244:
2237:
2233:
2226:
2222:
2218:
2212:
2205:
2201:
2194:
2190:
2186:
2178:
2175:
2172:
2169:
2166:
2162:
2156:
2150:
2147:
2142:
2139:
2114:
2111:
2104:
2100:
2096:
2091:
2084:
2080:
2076:
2071:
2064:
2060:
2056:
2051:
2046:
2043:
2021:
2018:
2013:
2009:
2005:
2000:
1996:
1992:
1987:
1983:
1979:
1976:
1942:
1935:
1932:
1927:
1924:
1918:
1915:
1912:
1909:
1906:
1902:
1896:
1891:
1887:
1884:
1878:
1873:
1870:
1865:
1862:
1842:
1837:
1814:
1811:
1808:
1805:
1802:
1799:
1795:
1792:
1789:
1783:
1780:
1775:
1772:
1742:
1736:
1733:
1729:
1724:
1721:
1709:
1706:
1684:
1681:
1674:
1671:
1668:
1665:
1633:
1630:
1623:
1620:
1615:
1612:
1597:
1596:
1584:
1574:
1562:
1559:
1549:
1537:
1527:
1515:
1495:
1492:
1469:
1466:
1463:
1458:
1454:
1451:
1448:
1442:
1439:
1436:
1433:
1414:
1413:
1396:
1391:
1380:
1368:
1339:
1334:
1330:
1326:
1304:
1278:
1272:
1269:
1247:
1246:
1233:
1228:
1224:
1221:
1218:
1197:
1171:
1168:
1146:
1123:
1118:
1114:
1110:
1088:
1057:
1045:
1042:
1020:
1011:
1007:
1003:
998:
993:
989:
982:
974:
970:
966:
961:
956:
952:
945:
937:
933:
929:
924:
919:
915:
907:
903:
900:
894:
891:
886:
883:
855:
849:
846:
841:
838:
832:
829:
826:
821:
817:
787:
763:
738:
727:
726:
714:
711:
701:
681:
671:
654:
626:
623:
620:
617:
614:
611:
607:
586:
583:
580:
560:
536:
519:
516:
496:flow rates or
472:
469:
441:
438:
431:The theory of
428:
425:
396:Heisler Charts
342:Main article:
339:
336:
306:
303:
235:
231:
220:Knudsen number
184:electron fluid
172:metallic bonds
133:
132:
129:
126:
123:
120:
109:
108:
96:
93:
92:
91:
83:
77:
26:
9:
6:
4:
3:
2:
5493:
5482:
5479:
5477:
5474:
5472:
5471:Heat transfer
5469:
5467:
5464:
5463:
5461:
5451:
5447:
5443:
5440:
5437:
5434:
5431:
5426:
5422:
5421:
5413:
5410:
5408:
5404:
5400:
5397:
5394:
5391:
5388:
5385:
5384:
5377:
5376:0-88318-797-3
5373:
5369:
5363:
5348:
5342:
5338:
5337:
5329:
5314:
5308:
5304:
5303:
5295:
5280:
5274:
5270:
5269:
5261:
5246:
5240:
5236:
5235:
5227:
5219:
5213:
5209:
5208:
5200:
5193:
5187:
5179:
5175:
5171:
5169:9780470501979
5165:
5161:
5157:
5150:
5142:
5138:
5134:
5130:
5126:
5122:
5118:
5111:
5104:. 2016-08-22.
5103:
5099:
5093:
5079:
5075:
5069:
5067:
5062:
5052:
5049:
5047:
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4988:
4982:
4978:
4974:
4966:
4963:
4958:
4944:
4942:
4938:
4937:heat transfer
4933:
4931:
4926:
4923:
4921:
4916:
4914:
4904:
4902:
4898:
4894:
4890:
4886:
4882:
4878:
4874:
4864:
4845:
4841:
4837:
4833:
4828:
4825:
4805:
4797:
4793:
4788:
4774:
4767:
4764:
4761:
4755:
4751:
4745:
4741:
4737:
4734:
4726:
4723:
4720:
4715:
4710:
4702:
4698:
4691:
4686:
4682:
4678:
4672:
4669:
4666:
4660:
4636:
4633:
4630:
4624:
4601:
4598:
4575:
4572:
4549:
4546:
4543:
4523:
4520:
4517:
4497:
4494:
4491:
4471:
4468:
4465:
4457:
4456:Splat cooling
4451:Splat cooling
4440:
4423:
4419:
4410:
4407:
4404:
4400:
4396:
4393:
4392:
4391:
4388:
4375:
4368:
4364:
4359:
4356:
4350:
4334:
4329:
4324:
4302:
4282:
4277:
4272:
4259:
4255:
4249:
4244:
4240:
4236:
4231:
4228:
4225:
4222:
4216:
4212:
4209:
4206:
4198:
4194:
4190:
4185:
4181:
4173:
4169:
4165:
4162:
4139:
4136:
4129:
4126:
4123:
4120:
4089:
4058:
4036:
4032:
4029:
4023:
4001:
3998:
3990:
3980:
3976:
3972:
3966:
3965:
3961:
3956:This section
3954:
3950:
3945:
3944:
3935:
3934:Heat equation
3925:
3907:
3903:
3899:
3894:
3890:
3882:
3878:
3872:
3868:
3859:
3855:
3851:
3846:
3842:
3832:
3829:
3826:
3823:
3814:
3810:
3804:
3800:
3797:
3791:
3787:
3781:
3777:
3770:
3766:
3762:
3757:
3753:
3746:
3743:
3740:
3737:
3731:
3728:
3716:
3703:
3698:
3694:
3690:
3687:
3684:
3681:
3673:
3668:
3652:
3648:
3625:
3621:
3606:
3584:
3580:
3575:
3569:
3565:
3558:
3555:
3548:
3544:
3540:
3535:
3531:
3524:
3519:
3515:
3489:
3485:
3481:
3476:
3472:
3464:
3460:
3456:
3451:
3447:
3438:
3434:
3430:
3427:
3424:
3421:
3418:
3412:
3409:
3383:
3380:
3377:
3374:
3364:
3360:
3355:
3349:
3345:
3338:
3335:
3329:
3322:
3319:
3313:
3304:
3299:
3295:
3266:
3262:
3257:
3251:
3247:
3240:
3237:
3230:
3226:
3222:
3217:
3213:
3206:
3203:
3200:
3197:
3194:
3188:
3185:
3162:
3159:
3152:
3148:
3140:
3136:
3131:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3100:
3097:
3088:
3084:
3076:
3072:
3067:
3060:
3057:
3031:
3028:
3023:
3020:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2990:
2987:
2982:
2979:
2971:
2967:
2963:
2960:
2957:
2951:
2948:
2936:
2923:
2920:
2917:
2914:
2911:
2906:
2902:
2892:
2876:
2872:
2868:
2863:
2859:
2838:
2816:
2812:
2789:
2785:
2770:
2767:
2754:
2751:
2747:
2743:
2740:
2737:
2717:
2710:
2707:
2701:
2692:
2689:
2681:
2676:
2663:
2660:
2657:
2654:
2651:
2631:
2627:
2623:
2620:
2617:
2597:
2594:
2587:
2584:
2578:
2575:
2563:
2545:
2539:
2531:
2528:
2522:
2519:
2510:
2508:
2504:
2500:
2496:
2478:
2474:
2470:
2467:
2464:
2461:
2453:
2449:
2445:
2429:
2425:
2421:
2418:
2415:
2412:
2403:
2387:
2383:
2379:
2376:
2373:
2351:
2347:
2343:
2340:
2337:
2329:
2325:
2321:
2311:
2309:
2305:
2301:
2296:
2283:
2277:
2274:
2267:
2263:
2256:
2252:
2242:
2235:
2231:
2224:
2220:
2210:
2203:
2199:
2192:
2188:
2173:
2167:
2160:
2154:
2148:
2140:
2125:
2112:
2109:
2102:
2098:
2094:
2089:
2082:
2078:
2074:
2069:
2062:
2058:
2054:
2049:
2044:
2041:
2019:
2016:
2011:
2007:
2003:
1998:
1994:
1990:
1985:
1981:
1977:
1974:
1965:
1959:
1953:
1940:
1933:
1925:
1913:
1907:
1900:
1894:
1889:
1885:
1876:
1871:
1868:
1863:
1860:
1853:is given by:
1840:
1825:
1812:
1806:
1800:
1793:
1790:
1787:
1781:
1773:
1758:
1740:
1734:
1727:
1722:
1719:
1705:
1703:
1702:heat equation
1698:
1682:
1679:
1672:
1669:
1666:
1655:
1652:for a simple
1651:
1646:
1631:
1628:
1621:
1618:
1613:
1610:
1602:
1582:
1575:
1560:
1550:
1535:
1528:
1513:
1493:
1483:
1482:
1481:
1467:
1464:
1456:
1452:
1446:
1440:
1437:
1434:
1431:
1423:
1419:
1411:
1381:
1366:
1359:
1355:
1354:thermal power
1328:
1302:
1276:
1270:
1267:
1256:
1255:
1254:
1252:
1222:
1219:
1195:
1169:
1166:
1144:
1112:
1086:
1071:
1070:
1069:
1055:
1044:Integral form
1041:
1039:
1035:
1018:
1009:
1005:
996:
991:
980:
972:
968:
959:
954:
943:
935:
931:
922:
917:
905:
901:
898:
892:
884:
871:
870:heat equation
866:
853:
847:
844:
839:
836:
830:
827:
824:
819:
815:
805:
799:
785:
777:
761:
753:
736:
712:
702:
699:
695:
679:
672:
669:
644:
643:
642:
640:
624:
621:
615:
612:
609:
584:
578:
558:
551:
525:
515:
513:
509:
505:
501:
499:
494:
490:
486:
485:heat transfer
482:
481:heat equation
478:
477:Fourier's law
471:Fourier's law
468:
466:
462:
458:
454:
453:heat transfer
450:
446:
437:
434:
424:
421:
416:
412:
411:heat capacity
408:
404:
399:
397:
392:
388:
387:heat equation
382:
378:
374:
372:
366:
362:
358:
355:
351:
345:
344:Heat equation
335:
333:
329:
324:
320:
317:
313:
302:
300:
296:
292:
291:
286:
282:
278:
274:
270:
266:
262:
258:
254:
249:
233:
229:
221:
215:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
164:
162:
158:
154:
149:
146:
141:
137:
130:
127:
124:
121:
118:
117:
116:
107:
102:
101:Heat equation
88:
84:
81:
78:
74:
71:
70:
69:
66:
64:
60:
56:
52:
44:
39:
33:
19:
5367:
5362:
5350:. Retrieved
5335:
5328:
5316:. Retrieved
5301:
5294:
5282:. Retrieved
5267:
5260:
5248:. Retrieved
5233:
5226:
5206:
5199:
5186:
5159:
5149:
5124:
5120:
5110:
5101:
5092:
5081:. Retrieved
5077:
4979:
4975:
4972:
4959:
4955:
4934:
4927:
4924:
4917:
4910:
4893:Biot numbers
4870:
4789:
4454:
4446:Applications
4402:
4398:
4389:
4332:
4325:
4280:
4273:
4008:
3993:
3984:
3969:Please help
3957:
3717:
3672:surface area
3669:
3612:
2937:
2893:
2776:
2768:
2682:, given by:
2679:
2677:
2564:
2511:
2506:
2502:
2498:
2494:
2451:
2447:
2443:
2404:
2317:
2297:
2126:
1963:
1957:
1954:
1826:
1759:
1711:
1699:
1647:
1598:
1415:
1248:
1047:
867:
803:
800:
728:
694:conductivity
521:
502:
489:proportional
474:
445:Second sound
443:
430:
414:
400:
383:
379:
375:
367:
363:
359:
353:
349:
347:
325:
321:
315:
311:
308:
288:
284:
280:
276:
272:
268:
264:
256:
250:
216:
183:
165:
163:vibrations.
150:
138:
134:
114:
104:
86:
79:
72:
67:
42:
37:
36:
5127:: 500â505.
5036:Biot number
4947:Instruments
1708:Conductance
1038:heat kernel
807:direction:
752:anisotropic
180:ionic bonds
59:temperature
5460:Categories
5378:, page 23.
5083:2024-08-19
5057:References
4969:Gas sensor
4941:symmetries
4885:martensite
2512:For heat,
2304:turbulence
1422:integrated
1416:The above
455:occurs by
188:conductive
168:conductors
157:insulators
99:See also:
80:Convection
73:Conduction
5178:713621645
5141:114611104
5011:Heat pipe
4881:austenite
4873:quenching
4838:ρ
4826:α
4806:α
4765:α
4746:−
4738:
4724:α
4721:π
4708:Δ
4679:−
4640:∞
4637:≤
4631:≤
4628:∞
4625:−
4605:∞
4579:∞
4553:∞
4550:−
4357:α
4237:ρ
4223:−
4213:
4191:−
4166:−
4134:Δ
4127:−
3958:does not
3900:−
3852:−
3833:π
3798:−
3763:−
3747:π
3732:˙
3691:π
3609:Spherical
3559:
3541:−
3482:−
3457:−
3431:ℓ
3425:π
3413:˙
3384:ℓ
3378:π
3339:
3323:˙
3311:Δ
3241:
3223:−
3207:ℓ
3204:π
3189:˙
3132:∫
3128:ℓ
3125:π
3116:−
3068:∫
3061:˙
3015:ℓ
3009:π
3000:−
2961:−
2952:˙
2924:ℓ
2918:π
2869:−
2839:ℓ
2711:˙
2699:Δ
2592:Δ
2579:˙
2537:Δ
2419:ρ
2328:Ohm's law
2308:viscosity
2300:thin film
2278:⋯
2249:Δ
2217:Δ
2185:Δ
2171:Δ
2168:−
2146:Δ
2138:Δ
2113:⋯
2020:⋯
1931:Δ
1923:Δ
1911:Δ
1908:−
1883:Δ
1804:Δ
1801:−
1779:Δ
1771:Δ
1732:Δ
1683:˙
1664:Δ
1650:Ohm's law
1558:Δ
1491:Δ
1462:Δ
1450:Δ
1438:−
1329:⋅
1271:˙
1223:⋅
1217:∇
1167:−
1113:⋅
1002:∂
988:∂
965:∂
951:∂
928:∂
914:∂
902:α
890:∂
882:∂
828:−
710:∇
619:∇
613:−
582:∇
579:−
524:heat flux
508:Ohm's law
461:diffusion
415:variation
212:diffusion
87:radiation
4985:See also
4901:nomogram
4889:eutectic
4411:Convert
4328:nomogram
3987:May 2013
3507:, where
2493:, where
2442:, where
1712:Writing
1253:units):
526:density
493:gradient
290:property
176:covalent
95:Overview
55:hotplate
3979:removed
3964:sources
1420:, when
1412:(in m).
1379:(in J),
1352:is the
1032:with a
696:, W/(m·
641:units)
350:in time
196:thermal
115:Where:
5405:
5374:
5343:
5309:
5275:
5241:
5214:
5176:
5166:
5139:
4871:Metal
4794:. The
2558:where
1753:where
1480:where
1408:is an
776:tensor
498:fluxes
204:copper
161:phonon
112:ÎșAÎT/â
5352:7 May
5318:7 May
5284:7 May
5250:7 May
5137:S2CID
447:is a
295:phase
259:. In
186:of a
90:body.
5403:ISBN
5372:ISBN
5354:2013
5341:ISBN
5320:2013
5307:ISBN
5286:2013
5273:ISBN
5252:2013
5239:ISBN
5212:ISBN
5190:The
5174:OCLC
5164:ISBN
4617:for
4565:and
4510:and
4274:The
3962:any
3960:cite
3670:The
3399:and
2306:and
1961:and
1358:heat
457:wave
198:and
5129:doi
5125:846
4883:to
4735:exp
4536:at
4401:or
4210:exp
3973:by
2644:or
670:/m,
178:or
5462::
5448:,
5172:.
5135:.
5123:.
5119:.
5100:.
5076:.
5065:^
4918:A
4346:Fo
4278:,
4019:Bi
3556:ln
3336:ln
3238:ln
2891:.
2402:.
1704:.
1656::
1251:SI
1040:.
700:),
639:SI
398:.
263:,
248:.
5452:.
5356:.
5322:.
5288:.
5254:.
5220:.
5180:.
5143:.
5131::
5086:.
4846:p
4842:C
4834:k
4829:=
4775:)
4768:t
4762:4
4756:2
4752:x
4742:(
4727:t
4716:2
4711:X
4703:i
4699:T
4692:=
4687:i
4683:T
4676:)
4673:t
4670:,
4667:x
4664:(
4661:T
4634:x
4602:=
4599:t
4576:=
4573:x
4547:=
4544:x
4524:0
4521:=
4518:T
4498:0
4495:=
4492:x
4472:0
4469:=
4466:t
4424:i
4420:T
4405:.
4403:L
4399:x
4376:.
4369:2
4365:L
4360:t
4351:=
4333:y
4308:K
4303:2
4299:m
4294:W
4281:h
4260:.
4256:)
4250:V
4245:p
4241:C
4232:t
4229:A
4226:h
4217:(
4207:=
4199:f
4195:T
4186:i
4182:T
4174:f
4170:T
4163:T
4140:,
4137:T
4130:h
4124:=
4121:q
4098:K
4095:s
4090:2
4086:m
4081:J
4059:h
4037:k
4033:L
4030:h
4024:=
4000:)
3994:(
3989:)
3985:(
3981:.
3967:.
3908:1
3904:r
3895:2
3891:r
3883:2
3879:r
3873:1
3869:r
3865:)
3860:2
3856:T
3847:1
3843:T
3839:(
3830:k
3827:4
3824:=
3815:2
3811:r
3805:/
3801:1
3792:1
3788:r
3782:/
3778:1
3771:2
3767:T
3758:1
3754:T
3744:k
3741:4
3738:=
3729:Q
3704:.
3699:2
3695:r
3688:4
3685:=
3682:A
3653:2
3649:r
3626:1
3622:r
3590:)
3585:1
3581:r
3576:/
3570:2
3566:r
3562:(
3549:1
3545:r
3536:2
3532:r
3525:=
3520:m
3516:r
3490:1
3486:r
3477:2
3473:r
3465:2
3461:T
3452:1
3448:T
3439:m
3435:r
3428:k
3422:2
3419:=
3410:Q
3381:k
3375:2
3370:)
3365:1
3361:r
3356:/
3350:2
3346:r
3342:(
3330:=
3320:Q
3314:T
3305:=
3300:c
3296:R
3272:)
3267:1
3263:r
3258:/
3252:2
3248:r
3244:(
3231:2
3227:T
3218:1
3214:T
3201:k
3198:2
3195:=
3186:Q
3163:T
3160:d
3153:2
3149:T
3141:1
3137:T
3122:k
3119:2
3113:=
3110:r
3107:d
3101:r
3098:1
3089:2
3085:r
3077:1
3073:r
3058:Q
3032:r
3029:d
3024:T
3021:d
3012:r
3006:k
3003:2
2997:=
2991:r
2988:d
2983:T
2980:d
2972:r
2968:A
2964:k
2958:=
2949:Q
2921:r
2915:2
2912:=
2907:r
2903:A
2877:1
2873:T
2864:2
2860:T
2817:2
2813:r
2790:1
2786:r
2755:.
2752:I
2748:/
2744:V
2741:=
2738:R
2718:,
2708:Q
2702:T
2693:=
2690:R
2680:R
2664:.
2661:G
2658:V
2655:=
2652:I
2632:R
2628:/
2624:V
2621:=
2618:I
2598:,
2595:T
2588:U
2585:=
2576:Q
2560:U
2546:,
2540:x
2532:A
2529:k
2523:=
2520:U
2507:A
2503:x
2499:k
2495:G
2479:x
2475:/
2471:A
2468:k
2465:=
2462:G
2452:A
2448:x
2444:Ï
2430:A
2426:/
2422:x
2416:=
2413:R
2388:V
2384:/
2380:I
2377:=
2374:G
2352:I
2348:/
2344:V
2341:=
2338:R
2284:.
2275:+
2268:3
2264:k
2257:3
2253:x
2243:+
2236:2
2232:k
2225:2
2221:x
2211:+
2204:1
2200:k
2193:1
2189:x
2177:)
2174:T
2165:(
2161:A
2155:=
2149:t
2141:Q
2110:+
2103:3
2099:U
2095:1
2090:+
2083:2
2079:U
2075:1
2070:+
2063:1
2059:U
2055:1
2050:=
2045:U
2042:1
2017:+
2012:3
2008:R
2004:+
1999:2
1995:R
1991:+
1986:1
1982:R
1978:=
1975:R
1964:Q
1958:A
1941:.
1934:t
1926:Q
1917:)
1914:T
1905:(
1901:A
1895:=
1890:k
1886:x
1877:=
1872:U
1869:1
1864:=
1861:R
1841:R
1813:.
1810:)
1807:T
1798:(
1794:A
1791:U
1788:=
1782:t
1774:Q
1755:U
1741:,
1735:x
1728:k
1723:=
1720:U
1680:Q
1673:R
1670:=
1667:T
1632:A
1629:L
1622:k
1619:1
1614:=
1611:R
1583:L
1561:T
1536:A
1514:Q
1494:t
1468:,
1465:T
1457:L
1453:t
1447:A
1441:k
1435:=
1432:Q
1395:S
1390:d
1367:Q
1338:S
1333:d
1325:q
1303:S
1277:=
1268:Q
1232:S
1227:d
1220:T
1196:S
1170:k
1145:=
1122:S
1117:d
1109:q
1087:S
1056:S
1019:)
1010:2
1006:z
997:T
992:2
981:+
973:2
969:y
960:T
955:2
944:+
936:2
932:x
923:T
918:2
906:(
899:=
893:t
885:T
854:.
848:x
845:d
840:T
837:d
831:k
825:=
820:x
816:q
804:x
786:k
762:k
737:k
713:T
698:K
680:k
668:W
653:q
625:,
622:T
616:k
610:=
606:q
585:T
559:k
535:q
285:T
281:A
277:L
273:t
269:Q
265:k
257:k
234:n
230:K
47:k
34:.
20:)
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