Fourier equation of heat flow

On the left of Fig. 4.55 the changes of the temperatures are illustrated for a standard DSC experiment. The data were calculated with K represented by Cj/20 J K s and using the Fourier equation of heat flow (see Sect. 4.3.5). The temperature of the constantan body, T, which is controlled by the heater starts at time... 

Cp/20 in J K s and that the Fourier equation of heat flow is valid. At steady state, the heat capacity is ... 

Standard techniques of vector analysis allow to equate the heat flow into the volume V to the heat flow across its surface. This operation leads to the linear and homogeneous Fourier differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity in m s, which is equal to the thermal conductivity k divided by the density and specific heat capacity. The Laplacian operator is + d dy + d ld-z, where x, y, and z are the space coordinates. 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

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 ... 

Heat transfer by conduction through walls follows the basic relation given by Fourier s equation [Eq. (21)], which states that the rate of heat flow, Q, is proportional to the temperature gradient, dT/dx, and... 

In the region ou tside the boundary layer, where the fluid may be assumed to have no viscosity, the mathematical solution takes on the form known as potential flow. This flow is analogous to the flow of heat in a temperature field or to the flow of charge in an electrostatic field. The basic equations of heat conduction (Fourier s law) are... 

Molecular diffusion is the mechanism of transfer of a substance either through a fluid which is motionless or, if the fluid is in laminar flow, in a direction perpendicular to the velocity of the fluid. The phenomenon has been studied from many points of view7ffequently conflicting, the most important of which are those of Fick and of Maxwell-Stefan. Fick (7) applied the well-known Fourier equation for rate of heat flow to the problem of diffusion. Unfortunately the mechanism of the two processes is not identical, since in the penetration of a liquid by a diffusing solute there will necessarily be displacement of the liquid and consequent volume changes arising for which the Fourier equation does not account. As an approxi-... 

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. 

The relation between the heat energy, expressed by the heat flux q, and its intensity, expressed by temperature T, is the essence of the Fourier law, the general character of which is the basis for analysis of various phenomena of heat considerations. The analysis is performed by use of the heat conduction equation of Fourier-Kirchhoff. Let us derive this equation. To do this, we will consider the process of heat flow by conduction from a solid body of any shape and volume V located in an environment of temperature To(t) [5,6]. 

The guarded hot-plate test is based on the principle of heat flow across a large flat sample at steady state, in which the heat flow is unidirectional through the thickness. A temperature diflerential across the sample provides the driving force for heat transfer, following the Fourier equation ... 

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...

An alternative metlrod of solution to these analytical procedures, which is particularly useful in computer-assisted calculations, is the finite-difference technique. The Fourier equation describes the accumulation of heat in a thin slice of the heated solid, between the values x and x + dx, resulting from the flow of heat tlirough the solid. The accumulation of heat in the layer is the difference between the flux of energy into the layer at x = x, J and the flux out of the layer at x = x + dx, Jx +Ox- Therefore the accumulation of heat in the layer may be written as... 

Payer80 states that the UNSAT-H model was developed to assess the water dynamics of arid sites and, in particular, estimate recharge fluxes for scenarios pertinent to waste disposal facilities. It addresses soil-water infiltration, redistribution, evaporation, plant transpiration, deep drainage, and soil heat flow as one-dimensional processes. The UNSAT-H model simulates water flow using the Richards equation, water vapor diffusion using Fick s law, and sensible heat flow using the Fourier equation. 

UNSAT-H uses the Richards equation, Fick s law, and the Fourier equation to estimate the flow of soil-water, vapor, and heat. This may be the strongest part of the model because these are the most rigorous, currently known, theoretical methods for estimating these parameters. 

The calorimeter response to a unit impulse must therefore be determined. This may be achieved by solving the Fourier equation [Eq. (23)] for a theoretical model of a heat-flow calorimeter and for this particular heat evolution. 

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... 

Physical situations that involve radiation with other modes of heat transfer are fairly common. If conduction enters the problem, the Fourier conduction law states that the heat flow depends upon the temperature gradient, thus introducing derivatives of the first power of the temperature. If convection matters, the heat flow depends roughly on the first power of the temperatures, the exact power depends on the type of flow. For instance, natural convection depends on a temperature difference between the 1.25 and l. 4 power. Physical properties that are temperature dependent introduce more temperature dependencies. This all means that the governing equations are highly nonlinear. 

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 objective of any heat-transfer analysis is usually to predict heat flow or the temperature which results from a certain heat flow. The solution to Eq. (3-1) will give the temperature in a two-dimensional body as a function of the two independent space coordinates x and y. Then the heat flow in the x and y directions may be calculated from the Fourier equations... 

---