Fourier’s equation for heat

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... 

Note the similarity between this equation and Fourier s equations for heat flow ... 

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. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Fourier s law for heat flow rate and Ohm s law for charge flowrate (i.e., electrical current). For three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (Qv/e) = (volumetric charge density/permittivity) and (QG//0 = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m 2) and (K m 2). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

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... 

This equation is analogous to Fourier s law for heat transfer by conduction ... 

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. 

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,... 

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

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. 

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

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. 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

Fourier s equation for one dimensional steady state heat... 

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

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... 

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 ... 

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. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

Constitutive equations, which quantitatively describe the physical properties of the fluids. The most important constitutive equations used in this book are the Newton s viscosity law, the Fourier s law of heat conduction, and the Pick s law of mass diffusion. The equation of state and more empirical relations for the physical properties of the fluid mixture also belong to this group of equations. 

How powerful the life that is instinct in a true mathematical model can be seen from the Fourier s theory of heat conduction, where the mathematical equations are fecund of all manner of purely mathematical developments. At the other end of the scale, a model can cease to be a model by becoming too large and too detailed a simulation of a situation whose natural line of development is to the particular rather chan the general. It ceases to have a life of its own by becoming dependent for its vitality on its physical realization. (The emphasis is ours). Maynard Smith... 

In these equations T is the temperature, p the mass density, iua the mass fraction of species A. and o,v the. r-component of the fluid velocity vector. The parameter k is the thermal conductivity, D the diffusion coefficient for species A. and / the fluid viscosity from experiment the values of these parameters are all greater than or equal to zero (this is. in fact, a requirement for the system to evolve toward equilibrium). Equation 1.7-2 is known as Fourier s law of heat conduction, Eq. 1.7-." is called Pick s first law of diffusion, and Eq-. 1.7-4 is Newton s law of viscosity. 

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