heat conduction
(thermodynamics) The flow of thermal energy through a substance from a higher-to a lower-temperature region.
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(thermodynamics) The flow of thermal energy through a substance from a higher-to a lower-temperature region.
The flow of thermal energy through a substance from a higher- to a lower-temperature region. Heat conduction occurs by atomic or molecular interactions. Conduction is one of the three basic methods of heat transfer, the other two being convection and radiation. See also Convection (heat); Heat radiation; Heat transfer.
Steady-state conduction is said to exist when the temperature at all locations in a substance is constant with time, as in the case of heat flow through a uniform wall. Examples of essentially pure transient or periodic heat conduction and simple or complex combinations of the two are encountered in the heat-treating of metals, air conditioning, food processing, and the pouring and curing of large concrete structures. Also, the daily and yearly temperature variations near the surface of the Earth can be predicted reasonably well by assuming a simple sinusoidal temperature variation at the surface and treating the Earth as a semi-infinite solid. The widespread importance of transient heat flow in particular has stimulated the development of a large variety of analytical solutions to many problems. The use of many of these has been facilitated by presentation in graphical form.
For an example of the conduction process, consider a gas such as nitrogen which normally consists of diatomic molecules. The temperature at any location can be interpreted as a quantitative specification of the mean kinetic and potential energy stored in the molecules or atoms at this location. This stored energy will be partly kinetic because of the random translational and rotational velocities of the molecules, partly potential because of internal vibrations, and partly ionic if the temperature (energy) level is high enough to cause dissociation. The flow of energy results from the random travel of high-temperature molecules into low-temperature regions and vice versa. In colliding with molecules in the low-temperature region, the high temperature molecules give up some of their energy. The reverse occurs in the high-temperature region. These processes take place almost instantaneously in infinitesimal distances, the result being a quasi-equilibrium state with energy transfer. The mechanism for energy flow in liquids and solids is similar to that in gases in principle, but different in detail.
The transfer of heat from the warmer to the cooler of two solid bodies that are in contact. In the human body, the rate of conduction depends on the temperature gradient between the skin and the material with which the skin is in contact, and on the thermal properties of the material. See also clo unit.
Heat conduction or thermal conduction is the spontaneous transfer of thermal energy through matter, from a region of higher temperature to a region of lower temperature, and hence acts to even out temperature differences.
The thermal energy, in the form of continuous random motion of the particles of the matter, is transferred by the same forces that act to support the structure of matter, so can be said to move by 'physical' contact between the particles.
It should be noted that heat can also be transferred by Thermal radiation and/or convection, and often more than one of these processes occur in a particular situation.
The law of heat conduction, also known as Fourier's law, states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area at right angles, to that gradient, through which the heat is flowing:

where
The above differential equation, when integrated for a simple linear situation (see diagram), where uniform temperature across equally sized end surfaces and perfectly insulated sides exist, gives the heat flow rate between the end surfaces as:

where
This law forms of the basis for the derivation of the heat equation.
Writing

where U is the conductance.
Fourier's law can also be stated as:

The reciprocal of conductance is resistance, R, given by:

and it is resistance which is additive when several conducting layers lie between the hot and cool regions, because A and Q are the same for all layers. In a multilayer partition, the total conductance is related to the conductance of its layers by:

So, when dealing with a multilayer partition, the following formula is usually used:

When heat is being conducted from one fluid to another through a barrier, it is sometimes important to consider the conductance of the thin film of fluid which remains stationary next to the barrier. This thin film of fluid is difficult to quantify, its characteristics depending upon complex conditions of turbulence and viscosity, but when dealing with thin high-conductance barriers it can sometimes be quite significant.
A related principle, Newton's law of cooling, states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. The law is

This form of heat loss principle is sometimes not very precise; an accurate formulation may require analysis of heat flow, based on the (transient) heat transfer equation in a nonhomogeneous, or else poorly conductive, medium. The following simplification may be applied so long as it is permitted by the Biot number, which relates surface conductance to interior thermal conductivity in a body. If this ratio permits, it shows that the body has relatively high internal conductivity, such that (to good approximation) the entire body is at same uniform temperature as it is cooled from the outside, by the environment. If this is the case, then it is easy to derive from these conditions the behavior of exponential decay of temperature of a body. In such cases, the entire body is treated as lumped capacitance heat reservoir, with total heat content which is proportional to simple total heat capacity, and the temperature of the body. If T(t) is the temperature of such a body at time t, and Tenv is the temperature of the environment around the body, then

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

Here, T(t) is the temperature at time t, and T(0) is the initial temperature at zero time, or t = 0. For example, simplified climate models may use Newtonian cooling instead of a full (and computationally expensive) radiation code to maintain atmospheric temperatures.

where
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