Fourier equation for heat conduction

The rate of heat transfer through a substance by conduction is given by the Fourier equation for heat conduction ... 

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...

In 1822 Fourier derived an equation for heat conduction. Realizing that the process of transferring heat by induction is analogous to the process of diffusion and that both are due to random molecular motion, in 1855 Pick adapted the Fourier equation to describe diffusion. 

Fourier s law is the fundamental differential equation for heat transfer by conduction ... 

Fick first recognized the analogy among diffusion, heat conduction, and electrical conduction and described diffusion on a quantitative basis by adopting the mathematical equations of Fourier s law for heat conduction or Ohm s law for electrical conduction [1], Fick s first law relates flux of a solute to its concentration gradient, employing a constant of proportionality called a diffusion coefficient or diffu-sivity ... 

This section deals with problems involving diffusion and heat conduction. Both diffusion and heat conduction are described by similar forms of equation. Pick s Law for diffusion has already been met in Section 1.2.2 and the similarity of this to Fourier s Law for heat conduction is apparent. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

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... 

The macroscopic phenomenological equation for heat flow is Fourier s law, by the mathematician Jean Baptiste Joseph Fourier (1768-1830). It appeared in his 1811 work, Theorie analytique de la chaleur (The analytic theory of heart). Fourier s theory of heat conduction entirely abandoned the caloric hypothesis, which had dominated eighteenth century ideas about heat. In Fourier s heat flow equation, the flow of heat (heat flux), q, is written as ... 

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-... 

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 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... 

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 transfer of heat by conduction also follows this basic equation and is written as Fourier s law for heat conduction in fluids or solids. 

You are likely already familiar with many of the simple direct force/flux pair relationships that are used to describe mass, charge, and heat transport—they include Pick s first law (diffusion). Ohm s law (electrical conduction), Fourier s law (heat conduction), and Poiseuille s law (convection). These transport processes are summarized in Table 4.1 using molar flux quantities. As this table demonstrates. Pick s first law of diffusion is really nothing more than a simplification of Equation 4.7 for... 

Introducing the Fourier s Law for heat conduction, qj = —KdTjdxj, yields the equation of internal energy in the form... 

On the other hand, the energy balance due to convection and conduction results from an energy balance in cylindrical coordinates. In the limit and using Fourier equation for the heat flux, q, we obtain Equation 4.36 ... 

Algebraic equations (14.3) correspond to constitutive equations, which are generally based on physical and chemical laws. They include basic definitions of mass, energy, and momentum in terms of physical properties, like density and temperature thermodynamic equations, through equations of state and chonical and phase equilibria transport rate equations, such as Pick s law for mass transfer, Fourier s law for heat conduction, and Newton s law of viscosity for momentum transfer chemical kinetic expressions and hydraulic equations. 

Example 3.2 (Heat conduction problem, Fourier s law and heat capacity). The dissipative energy equation (3.50) defines the governing equation of heat conduction. If force effects are not accounted for, we can ignore the first term of the r.h.s. of (3.50) to obtain... 

The heat transport in the particle is described by the basic equation for heat transport by conduction, Fourier s law ... 

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... 

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. 

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