Fundamentals of Heat Conduction

In 1807, Fourier presented his work to the most qualified doctoral committee in history, including Laplace, Lagrange, and Monge. Poisson was also involved. Lagrange was critical so Fourier s degree was delayed. 

In his 1807 paper, Fourier used the experiments of Biot to argue that the heat flux q should be proportional to the temperature gradient VT  

To calculate heat fluxes or temperature profiles, we make energy balances and then combine these with Fourier s law. The ways in which this is done are best seen in terms of two examples heat conduction across a thin film and into a semi-infinite slab. The choice of these two examples is not casual. As for diffusion, they bracket most of the other problems, and so provide limits for conduction. 

1 Steady Heat Conduction Across a Thin Film 

As a first example, consider a thin solid membrane separating two well-stirred fluids, as shown schematically in Fig. 20.1-1. Because one fluid is hotter than the other, energy will be conducted from left to right across the thin film. To find the amormt of conduction, we make a steady-state energy balance on a thin layer located between z and z + Az  

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The first step in the process is to relate heat flow to a temperature gradient, just as a diffusive flux can be related to a concentration gradient. The fundamental law of heat conduction was proposed by Jean Fourier in 1807 and relates the heat flux (q) to the temperature gradient ... 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

The Taylor dispersion problem is closely related to that discussed in the previous section, but also differs from it in some important fundamental respects. In the preceding problem, we assumed that the fluid was initially at a constant temperature upstream of z = 0 and that there was a constant heat flux into (or out of) the tube for all z > 0. In that case, the system has a steady-state temperature distribution at large times, and it was that steady-state problem that we solved. In the present case, there is no steady state. If the velocity were uniform across the tube instead of having the parabolic form (3 220), the temperature pulse that is initially at z = 0 would simply propagate downstream with the uniform velocity of the fluid, gradually spreading in the axial direction because of the action of heat conduction (i.e., the diffusion of heat). After a time /, the pulse would have moved downstream by a distance Uf, and the temperature pulse would have spread out over a distance of 0(s/(K tt)). Even in this simple case, there is clearly no steady state. The temperature distribution continues to evolve for all time.21... 

A dimensional analysis cannot be made unless enough is known about the physics of the situation to decide what variables are important in the problem and what physical laws would be involved in a mathematical solution if one were possible. Sometimes this condition is fairly easily met the fundamental differential equations of fluid flow, for example, combined with the laws of heat conduction and diffusion, suffice to establish the dimensions and dimensionless groups appropriate to a large number of chemical engineering problems. Dimensional analysis, however, does not yield a numerical equation, and experiment is needed to complete the solution to the problem. 

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-... 

Fourier s Law of Heat Conduction n The fundamental equation for steady heat flow through solids. It is... 

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. 

There are three fundamental types of heat transfer conduction, convection, and radiation. All three types may occur at the same time, and it is advisable to consider the heat transfer by each type in any particular case. 

Commercial dryers differ fundamentally by the methods of heat transfer employed (see classification of diyers, Fig. 12-45). These industrial-diyer operations may utihze heat transfer by convection, conduction, radiation, or a combination of these. In each case, however, heat must flow to the outer surface and then into the interior of the solid. The single exception is dielectric and microwave diying, in which high-frequency electricity generates heat internally and produces a high temperature within the material and on its surface. 

Natural phenomena are striking us every day by the time asymmetry of their evolution. Various examples of this time asymmetry exist in physics, chemistry, biology, and the other natural sciences. This asymmetry manifests itself in the dissipation of energy due to friction, viscosity, heat conductivity, or electric resistivity, as well as in diffusion and chemical reactions. The second law of thermodynamics has provided a formulation of their time asymmetry in terms of the increase of the entropy. The aforementioned irreversible processes are fundamental for biological systems which are maintained out of equilibrium by their metabolic activity. 

Radiation is fundamentally different from conduction as it describes the transfer of heat between two substances that are not in contact with each other. Like conduction, radiation is an independent form of heat transfer. Ignoring the conflicts of wave and quantum theory, radiation, refers to the transmission of electromagnetic energy through space. 

In addition to the temperature dependence of the properties such as strength and modulus, which we will discuss individually for each material class, there are two fundamental topics that are often described in the context of heat transfer properties or thermodynamics of materials—for example, thermal conductivity or specific heat—but are related more to mechanical properties because they involve dimensional changes. These two properties, thermoelasticity and thermal expansion, are closely related, but will be described separately. 

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