Fourier's law of heat conduction

The basis of conduction heat transfer is Fourier’s Law. This law involves the idea that the heat flux is. Thermal conductivity, k, a property of proportional to the temperature gradient in any direction materials that is temperature dependent, is the constant of proportionality.
Heat always conducts from warmer objects to cooler objects. The composition of a material effects its conduction rate. If a copper rod and an iron rod are joined together end to end, and the ends placed in heat sources, the heat will conduct through the copper end more quickly than the iron end because copper has a K value of 92, whereas, iron has a K value of 11.
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 rate, in time, 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:


In many systems the area A is a function of the distance in the direction n. One important extension is heat conduction equation.that this can be combined with the first law of thermodynamics to yield the For constant thermal conductivity, this is given as



In this equation, a is the thermal diffusivity and is the internal heat generation per unit volume.qG
Some problems, typically steady-state, one-dimensional formulations where only the heat flux is desired,can be solved simply from Equation (4.1.1). Most conduction analyses are performed with Equation(4.1.2). In the latter, a more general approach, the temperature distribution is found from this equationand appropriate boundary conditions. Then the heat flux, if desired, is found at any location using Equation (4.1.1). Normally, it is the temperature distribution that is of most importance. For example,it may be desirable to know through analysis if a material will reach some critical temperature, like its melting point. Less frequently the heat flux is desired.
While there are times when it is simply desired to understand what the temperature response of a structure is, the engineer is often faced with a need to increase or decrease heat transfer to some specificlevel. Examination of the thermal conductivity of materials gives some insight into the range of possibilities that exist through simple conduction. Of the more common engineering materials, pure copper exhibits one of the higher abilities to conduction heat with a thermal conductivity approaching 400 W/m2 K. Aluminum, also considered to be a good conductor, has a thermal conductivity a little over half that of copper. To increase the heat transfer above
values possible through simple conduction, more-involved designs are necessary that incorporate a variety of other heat transfer modes like convection and phase change.
Decreasing the heat transfer is accomplished with the use of insulations. A separate discussion of these follows.

Insulations
The term thermal insulation can refer to materials used to reduce the rate of heat transfer, or the methods and processes used to reduce heat transfer. Thermal insulation is the method of preventing heat from escaping a container or from entering the container. In other words, thermal insulation can keep an enclosed area such as a building warm, or it can keep the inside of a container cold. Heat is transferred from one material to another by conduction, convection and/or radiation. Insulators are used to minimize the transfer of heat energy. In home insulation, the R-value is an indication of how well a material insulates. The major types of insulation are associated with the major types of heat transfer.

Insulations are used to decrease heat flow and to decrease surface temperatures. These materials are loose fill, batt, and rigid. Even a gas, like air, can be a good found in a variety of forms, typically insulator if it can be kept from moving when it is heated or cooled. A vacuum is an excellent insulator.Usually, though, the engineering approach to insulation is the addition of a low-conducting material to the surface. While there are many chemical forms, costs, and maximum operating temperatures of common forms of insulations, it seems that when a higher operating temperature is required, many times the thermal conductivity and cost of the insulation will also be higher. Loose-fill insulations include such materials as milled aluminasilica (maximum operating temperature and thermal conductivities in the range of 0.1 to 0.2 W/m K) and perlite (maximum operating of 1260 C and thermal conductivities in the range of 0.05 to 1.5 W/m K). Batt-type temperature of 980 insulations include one of the more common types — glass fiber. This type of insulation comes in avariety of densities, which, in turn, have a profound affect on the thermal conductivity. Rigid insulations show conducttivities for glass fiber insulations can range from about 0.03 to 0.06 W/m a very wide range of forms and performance characteristics. For example, a rigid insulation in foam form, polyurethane, is very lightweight, shows a very low thermal conductivity (about 0.02 W/m K),C. Rigid insulations in refractory form but has a maximum operating temperature only up to about 120 show quite different characteristics. For example, high-alumina brick is quite dense, has a thermal k, but can remain operational to temperatures around 1760C. Many conductivity of about 2 W/m insulations are characterized in the book edited by Guyer (1989).Often, commercial insulation systems designed for high-temperature operation use a layered approach.Temperature tolerance may be critical. Perhaps a refractory is applied in the highest temperature region,an intermediate-temperature foam insulation is used in the middle section, and a high-performance, low-temperature insulation is used on the outer side near ambient conditions.Analyses can be performed including the effects of temperature variations of thermal conductivity.However, the most frequent approach is to assume that the thermal conductivity is constant at some
temperature between the two extremes experienced by the insulation.