Fouriers Law of Conduction

Finally, Let us consider a boundary node—for example, the boundary of a flat plate—now subject to convective heat transfer to an ambient at temperature T. The discretized balance of thermal energy and the Fourier law of conduction applied to a Ax /2-thick boundary difference system (Fig. 4.23) yields for 9 = T — To,... 

This law on the diffusion of momentum and the Fourier law of conduction (on the diffusion of heat) are special cases of diffusion phenomena. 

The model we present here is very simple, although it contains all the ingredients needed for the description of long range order. It consists of a system obeying the Fourier law of conduction of heat and maintained out of equilibrium by appropriate boundary conditions. We shall use a Langevin approach in order to calculate the temperature fluctuations in the steady state. These calculated fluctuations differ from those of, even local, equilibrium. However small the constraint is, there will appear a correlation between fluctuations taking place everywhere in the sys-tem[ 2c,5]-... 

The rate of heat transferred by conduction in the x-direction through a finite cross-sectional area A is, according to the Fourier Law of Conduction, proportional to the temperature difference ... 

In this paper we give a brief review of the relation between microscopic dynamical properties and the Fourier law of heat conduction as well as the connection between anomalous conduction and anomalous diffusion. We then discuss the possibility to control the heat flow. 

The Fourier law of heat conduction relates the flux of heat q per unit area, as a result of a temperature gradient, such that... 

Fourier s Law of Conduction. Experiments have shown that at steady state, the heat flux qy, which is the rate of heat transfer, Q, per unit area, A, through a material due to conduction, is proportional to the temperature gradient in the direction of heat flow, y in this case. 

Notice the similarity between Eq. (11-1) and the Fourier law of heat conduction. 

A physical interpretation of Equation (35) is possible if one notes that it is mathematically analogous to Fourier law of heat conduction. The constant factor in the right-hand side plays the role of thermal conductivity, and the local incident radiation GA(r) plays the role of temperature. In that sense, differences in the latter variable among neighboring regions in the medium drive the diffusion of radiation toward the less radiated zone. Note that the more positive the asymmetry parameter, the higher the conductivity that is, forward scattering accelerates radiation diffusion while backscatter-ing retards it. 

It is important to emphasize that thermodynamic force Xq is a vector, whereas Xq is its Cartesian component corresponding to the Cartesian coordinate i of heat flux Jq. The centuries old practice states the well known relationships between heat fluxes and temperature gradients, which are expressed by the Fourier law of heat conduction... 

Pick s law of diffusion and Fourier s law of conduction are usually represented by second order ordinary differential equations (ODEs). In this chapter, we describe how one can obtain analytical solutions for linear boundary value problems using Maple and the matrix exponential. 

Fourier s law of conduction (Step 3) in light of Eq. (2.1) gives the governing equation (Step 4) as... 

The second fact that is implicit in macroscopic or continuum laws is the idea of local thermodynamic equilibrium. For example, when we write the Fourier law of heat conduction, it is inherently assumed that one can define a temperature at any point in space. This is a rather severe assumption since temperature can be defined only under thermodynamic equilibrium. The question that we might ask is the following. If there is thermodynamic equilibrium in a system, then why should there be any net transport of energy Thus, we implicitly resort to the concept of local thermodynamic equilibrium, where we assume that thermodynamic equilibrium can be defined over a volume which is much smaller than the overall size of the system. What happens when the size of the object becomes on the order of this volume Obviously, the macroscopic or continuum theories break down and new laws based on nonequilibrium thermodynamics need to be formulated. This chapter focuses on developing more generalized theories of transport which can be used for nonequilibrium conditions. This involves going to the root of the macroscopic or continuum theories. 

Now consider the next larger length and timescales or , and x or xr. When L , r and t x, xr, transport is ballistic in nature and local thermodynamic equilibrium cannot be defined. This transport is nonlocal in space. One has to resort to time-averaged statistical particle transport equations. On the other hand, if L , , and t x, xr, then approximations of local thermodynamic equilibrium can be assumed over space although time-dependent terms cannot be averaged. The nonlocality is in time but not in space. When both L , r and t x, xr, statistical transport equations in full form should be used and no spatial or temporal averages can be made. Finally, when both L , , and t x, xr, local thermodynamic equilibrium can be applied over space and time leading to macroscopic transport laws such as the Fourier law of heat conduction. 

This is the Fourier law of heat conduction with the thermal conductivity being k = Cvi 3. Note that we have not made any assumption of the type of energy carrier and, hence, this is a universal law for all energy carriers. The only assumption made is that of local thermodynamic equilibrium such that the energy density u at any location is a function of the local temperature. 

The first term containing /0 drops out since the integral over all the directions becomes zero. Equation 8.18 is the Fourier law of heat conduction with the integral being the thermal conductivity k. If one assumes that the relaxation time and velocity are independent of particle energy, then the integral becomes... 

Kinetic theory is introduced and developed as the initial step toward understanding microscopic transport phenomena. It is used to develop relations for the thermal conductivity which are compared to experimental measurements for a variety of solids. Next, it is shown that if the time- or length scale of the phenomena are on the order of those for scattering, kinetic theory cannot be used but instead Boltzmann transport theory should be used. It was shown that the Boltzmann transport equation (BTE) is fundamental since it forms the basis for a vast variety of transport laws such as the Fourier law of heat conduction, Ohm s law of electrical conduction, and hyperbolic heat conduction equation. In addition, for an ensemble of particles for which the particle number is conserved, such as in molecules, electrons, holes, and so forth, the BTE forms the basis for mass, momentum, and energy conservation equa-... 

We can treat mass transfer in a manner somewhat similar to that used in heat transfer with Fourier s law of conduction. However, an important difference is that in molecular mass transfer one or more of the components of the medium is moving. In heat transfer by conduction the medium is usually stationary and only energy in the form of heat is being transported. This introduces some differences between heat and mass transfer that will be discussed in this chapter. 

The viscous dissipation is a specific power consumption, i.e., power per unit volume. In S.I. units it is expressed in Watts per cubic meter [W/m ]. The heat fiux away from the polymer melt is determined by the heat fiux from the melt to the barrel and screw. If the screw is neutral the heat fiux to the screw is usually small and can be assumed to be negligible. If screw cooling is used, this assumption will not be correct. The heat fiux (heat flow per unit cross-sectional area) for cooling the polymer melt is determined by Fourier s law of conductive heat transport ... 

Equation (7.51) can be interpreted either algebraically or electrically as was done for Newton s Law of cooling in convection or for Fourier s Law of conduction through composite materials. The electrical analogy for Equation (7.51) can be made by comparing ( 5, —/,) to the driving potential difference and Q, as the electrical current. 

Now the rate of heat flow or the heat current Jq = dQ/dt is given by the laws of heat conduction. For example according to the Fourier law of heat conduction, Jq = a(r 1 — T2), in which a is the coefficient of heat conductivity. Note that the thermod3mamic flow Jq is driven by the thermodynamic force F = (1/7 2 — 1/Ti). For the rate of entropy production we have from (3.5.2) that... 

For anisotropic solids the heat conductivity k is a tensor of the second rank. The empirical Fourier law of heat conduction is then written as... 

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