Fourier s equation

The energy densities of laser beams which are conventionally used in the production of thin films is about 10 — 10 Jcm s and a typical subsU ate in the semiconductor industry is a material having a low drermal conductivity, and drerefore dre radiation which is absorbed by dre substrate is retained near to dre surface. Table 2.8 shows dre relevant physical properties of some typical substrate materials, which can be used in dre solution of Fourier s equation given above as a first approximation to dre real situation. 

An upper limit of die heat losses tluough the reaction container wall, usually in the form of a cylindrical ciiicible with an increasing diameter from bottom to top, by assuming that die whole reaction mixture achieves the hnal reaction teiiiperamre immediately, and heat losses occur dirough the ciiicible refractory walls by conduction. The solution of Fourier s equation... 

Solutions to Fourier s equation are in the form of infinite series but are often more conveniently expressed in graphical form. In the solution the following dimensionless groups are used. 

Fig. 5.23 show the solution to Fourier s equation in terms of the temperature gradient at the centre line of section considered and the Fourier Number for... 

From a microscopic standpoint, thermal conduction refers to energy being handed down from one atum or molecule in the next one. In a liquid or gas, ihese particles change their position continuously even withoul visible movemeni and they transport energy also in this way. From a macroscopic or continuum viewpoint, thermal conduction is quantitatively described by Fourier s equation, which states that the heat flux q per unit time and unit area through an area element arbitrarily located in the medium is proportional to the drop in temperature, -grad T. per unit length in the direction normal to the area and to a transport property k characteristic of the medium and called thermal conductivity ... 

The time dependence of the temperature of the pyroelectric material can be related to the spatial dependence of the temperature by means of Fourier s equation for heat... 

Hence, the thermal conductivity may be determined by a rearrangement of Fourier s equation (section 8.1.2) ... 

Heat transfer by conduction through walls follows the basic relation given by Fourier s equation [Eq. (21)], which states that the rate of heat flow, Q, is proportional to the temperature gradient, dT/dx, and... 

If steady-state conditions exist (flow rates, temperatures, composition, fluid properties, pressures), Fourier s equation applies to macro-systems in which energy is transferred across a heat exchanger or an evaporator surface ... 

The thermal conductivity, n, of a substance is defined as the rate of heat transfer by conduction across a unit area, through a layer of unit thickness, under the influence of a unit temperature difference, the direction of heat transmission being normal to the reference area. Fourier s equation for steady conduction may be written as... 

Localization. The reduction scheme of variables used in space is very general and is found in other energy varieties (see for instance the Fourier equation in the thermal domain—case study G2 Fourier s Equation of Heat Transfer in Chapter 10). The scheme relies on the scalar nature of state variables (at the global level) and uses a sequence of spatial operators that are the contra-gradient, the curl (also called rotational), and the divergence. 

Case Study G2 Fourier s Equation (Heat Transfer).440... 

Spatially reduced Ohm s law in electrodynamics, Fourier s equation of heat transfer, Fick s law for diffusion, and Newton s law in hydrodynamics are among the subjects treated and their comparison is enlightening. The various transfers tackled in this chapter are stationary diffusion,... 

From this equation, the original Fourier s equation is retrieved by substituting the entropic flow density Js times the temperature Tby the heat flow density. 

With this lineic density, Newton s law as in Equation G4.1 is merely written as a dissipative relationship in a spatially reduced form, similar to the resistivity or conductivity relationships in many domains (see case studies G1 Reduced Ohm s Law and G2 Fourier s Equation of Heat Transfer for instance)... 

The thermocouples are assumed to be ideal, that is, with an invariable coupling (without Thomson effect) and without dissipation, that is, no Joule effect or heat transfer according to Fourier s equation. The potential difference of each thermocouple is proportional to the temperature difference through a constant Seebeck coefficient ... 

In this section Fourier s equation (4.1-2). will be used to obtain equations for onedimensional steady-state conduction of heat through some simple geometries. For a flat slab or wall where the cross-sectional area A and k in Eq. (4.1-2) are constant, we obtained Eq. (4.1-10), which we rewrite as... 

In the case where there is a multilayer wall of more than one material present as shown in Fig. 4.3-1, we proceed as follows. The temperature profiles in the three materials A, B, and C are shown. Since the heat flow g must be the same in each layer, we can write Fourier s equation for each layer as... 

Suppose that two plane solids A and B are placed side by side in parallel, and the direction of heat flow is perpendicular to the plane of the e.xposed surface of each solid. Then the total heat flow is the sum of the heat flow through solid A plus that through B. Writing Fourier s equation for each solid and summing. 

Substituting Fourier s equation for conduction and the convection equation,... 

For a stationary state Fourier s equation (12.15) reduces to Laplace s equation... 

The heat conduction rate through the outer surface (A ) of the cladding is given by Fourier s equation as... 

This expression is identical to Equation 2.18d, which had been derived by means of a shell balance. Its counterpart for heat conduction, known as Fourier s equation, is given by... 

Authoritative compilahons of solutions to Pick s and Fourier s equations can be found in the classical monographs ... 

---