Conduction, thermal Fourier equation

The simplest model of this kind can be represented by one in which an isolated particle surrounded by gas is in contact with or in the vicinity of the heating surface for a certain time, during which the heat transfer between the particle and the heating surface takes place by transient conduction, as shown in Fig. 12.4. In terms of the model, the Fourier equation of thermal conduction can be expressed as... 

At this point we retrace our development slightly to introduce a different conceptual viewpoint for Fourier s law. The heat-transfer rate may be considered as a flow, and the combination of thermal conductivity, thickness of material, and area as a resistance to this flow. The temperature is the potential, or driving, function for the heat flow, and the Fourier equation may be written... 

This is referred to as the Fourier equation regarding the thermal conduction. 

Thermal conductance and conductivities are derived from the Fourier equation. The total amount of heat a material conducts is directly proportional to the surface area, the time of contact, and the temperature gradient and is inversely proportional to the thickness of the sample, according to ... 

The analytical method for determination of the thermal conductivity is presented below. The heat flow from the line heat source (a heating wire) of infinite length and infinitely small diameter imbedded in an infinite homogeneons medium can be expressed by the Fourier equation ... 

The thermal diffusivity can also be measured directly by employing transient heat conduction. The basic differential equation (Fourier heat conduction equation) governing heat conduction in isotropic bodies is used in this method. A rectangular copper box filled with grain is placed in an ice bath (0°C), and the temperature at its center is recorded [44]. The solution of the Fourier equation for the temperature at the center of a slab is used ... 

The temperature distribution in the solid polymer sample can well be apvproximated by the Fourier equation for transient heat conduction within a medium of constant thermal diffusivity, i.e. 

The heat transfer obeys the Fourier equation which expresses the proportionality of the thermal energy flow, classically called heat flux, with the temperature gradient between two points in space, the proportionality coefficient being the product of the thermal conductivity cby the area>l on which the heat flow is counted. 

The Seebeck effect corresponds to the electricity production from a difference of temperature. This effect can be reversible and is the inverse of the Peltier effect, which is the phenomenon of conversion of electric energy into thermal energy (heat). These effects can be superimposed onto the dissipative processes of transport by conduction of electric charges (Joule effect) and to the transport of heat (Fourier equation) which are both irreversible processes. 

These considerations were based on the conduction of electrodynamical energy. Now, one turns attention to the thermal energy. The conduction of heat in the same circuit is possible when all the materials have a nonzero thermal conductance then the Fourier equation allows the following equations to be written ... 

Thermal transport in a solid (or in a stagnant fluid gas or Hquid) occurs by conduction, and is described in terms of the Fourier equation here expressed in a differential form as... 

The line-source technique is a transient method capable of very fast measurements. A line source of heat is located at the center of the sample being tested as shown in Fig. 4. The whole is at a constant initial temperature. During the course of the measurement, a known amount of heat is produced by the line source, resulting in a heat wave propagating radially into the sample. The rate of heat propagation is related to the thermal diffusivity of the polymer. The temperature rise of the line somce varies linearly with the logarithm of time. Starting with the Fourier equation, it is possible to develop a relationship which can be used directly to calculate the thermal conductivity of the sample from the slope of the linear portion of the curve ... 

Figure 17.1 A simple thermal gradient maintained by a constant flow of heat. In the stationary state, the entropy current Js,out — diS/dt + The stationary state can be obtained either as a solution of the Fourier equation for heat conduction or by using the theorem of minimum entropy production. Both lead to a temperature T(x) that is a linear function of the position x...

The heat conducted per unit time through the cladding on one surface is given by the Fourier equation as k AiT — r )/c where is the thermal conductivity of the cladding. Equating this to the rate of heat conduction into the cladding, as given by equation (6.20), we obtain the relation... 

Since thermal conductivity values are usually substituted into some particular solution of the Fourier equation, the thermal conductivity values are given in the form of a coefficient k that will give reasonable results when used in the Fourier equation. The apparent mean thermal conductivity values T were calculated from the following equation ... 

The heat transport due to conduction and that due to radiation are not readily separable from the experimental data. Curve A of Fig. 4 shows the measured temperature distribution through a typical sample containing 29 shields per inch. Curve B shows the temperature distribution expected if each sheet of aluminum foil were a floating radiation shield. These results were obtained from Fig. 1. Curve C shows the temperature distribution througji an ideal sample, whose thermal conductivity would be independent of temperature. The observed result is probably a combination of radiation heat transfer and the change in thermal conductivity of the insulation with temperature. The thermal conductivity of most disordered dielectrics is approximately proportional to the first power of the temperature, but the temperature dependence of multiple contacts is not well understood. The fact that the temperature distribution for a sample of this type can be accounted for by a temperature-dependent thermal conductivity is sufficient justification for using Eq. (3), a particular solution of the Fourier equation, rather than Eq. (1), the heat flux equation for radiant heat transport, to represent our results. 

Mpemba paradox arises intrinsically from heating and undercoordination induced 0 H-0 bond relaxation. Heat emission proceeds at a rate depending on the initial energy storage, and the skin supersolidity creates the gradients of density, specific heat, and thermal conductivity for heat conduction in Fourier s equation of fluid thermodynamics. 

To verify the prediction on the presence of thermal conductivity gradient due to the joint effect of heating and skin supersolidity, one needs to solve the one-dimensional nonlinear Fourier equation [41] numerically by introducing the supersolid skin [17] in a tube container. Considering a one-dimensional approach, water in a cylindrical tube can be divided into the bulk (B) and the skin (S) region along the x-axial direction and put the tube into a drain of constant temperature 0 °C. The other end is open to the drain without the skin. The heat transfer in the partitioned fluid follows this transport equation and the associated initial and boundary conditions... 

For a flat slab specimen of thickness h (m), the thermal conductivity X (W m K ) is the heat flux per unit of temperature gradient in the direction perpendicular to an isothermal surface and is a material constant defined by the one-dimensional Fourier equation X = Qh/A(Ti — Tq) where Q is the time rate of the heat flow (W), A the area (m ) on a selected isothermal surface, and T and T2 the temperatures of the hot and cold surfaces, respectively, and h the sample thickness (m). Several steady-state or transient methods are available to measure the thermal conductivity of organic and inorganic materials." The guarded-hotplate (ASTM F433), heat-flow (ASTM C518), and Colora ther-moconductometer methods are accurate for thick plastic samples with thermal conductivity of 0.1—10 W m K . In electronics, steady-state techniques... 

In this chapter, we briefly describe fundamental concepts of heat transfer. We begin in Section 20.1 with a description of heat conduction. We base this description on three key points Fourier s law for conduction, energy transport through a thin film, and energy transport in a semi-infinite slab. In Section 20.2, we discuss energy conservation equations that are general forms of the first law of thermodynamics. In Section 20.3, we analyze interfacial heat transfer in terms of heat transfer coefficients, and in Section 20.4, we discuss numerical values of thermal conductivities, thermal diffusivities, and heat transfer coefficients. 

The analogies between heat and mass transfer are reflected in the equations used to describe them. Thermal conduction is described by Fourier s law, which in one dimension is... 

Fourier s law for thermal conduction An equation describing the relationship between the rate of heat flux and the temperature gradient. See Eq. (23). 

To use Fourier s law of heat conduction, a thermal balance must first be constructed. The energy balance is performed over a thin element of the material, x to x + Ax in a rectangular coordinate system. The energy balance is shown in equation 13 ... 

The energy densities of laser beams which are conventionally used in the production of thin films is about 103 — 104Jcm 2s, and a typical substrate in the semiconductor industry is a material having a low thermal conductivity, and therefore the radiation which is absorbed by the substrate is retained near to the surface. Table 2.8 shows the relevant physical properties of some typical substrate materials, which can be used in the solution of Fourier s equation given above as a first approximation to the real situation. 

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