Fourier’s law for heat 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. 

Modeling diffusive transport requires appropriate constitutive relationships, such as Fourier s law for heat conduction or Fick s law for species diffusion. It is important to... 

Entropy flux in the absence of a net particle flow is equivalent to Jq/T where Jq is the heat flux. Thus, Eq. (6.9.4) is a formulation of Fourier s Law for heat conduction, Jq — —kVT, thereby identifying the thermal conductivity associated with the transport of charge carriers as... 

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. 

Fourier s law for heat conduction can be written as follows for constant p andc, ... 

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. 

Following the same procedure we can introduce the extended Fourier s law for heat conduction and the extended Hooke s law for deformation, which will not be... 

This result for the most likely change in moment is equivalent to Fourier s law of heat conduction. To see this take note of the fact that in the steady state the total rate of change of moment is zero, E = 0, so that the internal change is... 

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 the simplest level, as Griskey (1) notes, Pick s law of diffusion for mass transfer and Fourier s law of heat conduction characterize mass and heat transfer, respectively, as vectors, i.e., they have magnitude and direction in the three coordinates, x, y, and z. Momentum or flow, however, is a tensor which is defined by nine components rather than three. Hence, its more complex characterization at the simplest level, in accordance with Newton s law, is... 

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... 

Thermal conductivity is the intensive property of a material that indicates its ability to conduct heat. For one-dimensional heat flow in the x-direction the steady state heat transfer can be described by Fourier s law of heat conduction ... 

To obtain a general relation for Fourier s law of heat conduction, consider a medium in which the temperature distribution is three-dimensional. Fig. 2-8 shows an isothermal surface in that medium. The heat flux vector at a point P on this surface must be perpendicular to the surface, and it must point in the direction of decreasing temperature. If n is Ihe normal of the isothermal surface at point the rate of heat conduction at that point can be expressed by Fourier s law as... 

Consider a plane wall of thickness L and average thermal conductivity k. The Isvo surfaces of the wall are maintained at constant temperatures of r, and T2. For one-dimensional steady heat conduction through the wall, we have 7(.v). Then Fourier s law of heat conduction for the wall can be expressed 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. 

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. 

Numerical solutions to simple thermal energy transport problems in the absence of radiative mechanisms require that the viscosity fi, density p, specific heat Cp, and thermal conductivity k are known. Fourier s law of heat conduction states that the thermal conductivity is constant and independent of position for simple isotropic fluids. Hence, thermal conductivity is the molecular transport property that appears in the linear law that expresses molecular transport of thermal energy in terms of temperature gradients. The thermal diffusivity a is constructed from the ratio of k and pCp. Hence, a = kjpCp characterizes diffusion of thermal energy and has units of length /time. 

The Landauer expression for heat transfer (Equation 12.13) assumes the absence of inelastic scattering processes in the system, and the two opposite phonon flows of different temperatures are out of equilibrium with each other. This leads to an anomalous transport of heat, where (classically) the energy flux is proportional to the temperature difference, Tl - T, rather than to the temperature gradient VT, as asserted by the Fourier s law of heat conductivity. 

Foams, in addition to being useful as cushioning, can be used to provide thermal insulation for products. A frozen product, for example, might be packaged with ice (or dry ice or gel packs) to provide cooling, and encased in a foam container to help reduce the conduction of heat from the surroundings into the container. Often the temperature inside and outside the container can be regarded as relatively constant, and the heat transfer process can considered essentially one-dimensional. In such cases, Fourier s law of heat conduction reduces to its one-dimensional steady-state form ... 

---