Equation Fourier

The extension of Newton s law of cooling leads to Fourier s law of heat conduction. The heat flux is given by 

Again we find the fiux not to be proportional to the force. We insert the flux into the balance expression 

To see if the entropy production rate has an extremum for the stationary state corresponding to a constant heat flux Jo we evaluate the functional derivative with respect to the temperature field 

For a stationary state Fourier s equation (12.15) reduces to Laplace s equation 

We insert the stationarity condition Jo = —kVTst (r) into (16, 17) to obtain 

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The analytical methods for solving the Fourier equation, in which Q and T are functions of the spatial co-ordinate and time, include a change of variables by combination, and in the more general case the use of Laplace U ansforms. 

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... 

The thickness of die solid is then divided into thin slices, and the separate differentials at tire mth slice in tire Fourier equation can be expressed in terms of the functions... 

Fouriers equation for non-steady heat flow in one dimension, x, is... 

In an isotropic medium, as for normal liquids, the Fourier equation holds ... 

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 both calculations, the boundary conditions are linear with respect to 0 and its first-order derivatives. The solution of the Fourier equation, with respect to the space variables, may be developed in a series of orthogonal functions, winch are exponential with respect to the time variable [for the solution of similar problems, see (45)]- The time-dependance of the temperature distribution along a single space variable r, resulting from a unit pulse, is therefore given by... 

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

In a solid wall such as Figure 8.1(a), the variation of temperature with time and position is represented by the one-dimensional Fourier equation... 

The idea that the reaction proceeds principally at a temperature which is close to Tx also permits us to find closed analytical expressions for the velocity of flame propagation. The heat released during the reaction is partly spent on heating the reacting gas itself and partly carried away by heat conduction to neighboring elements of the gas volume. If the temperature of the zone in which the reaction effectively occurs is already close to Tlt the amount of the heat spent on heating the reacting gas up to its final temperature beyond the flame front, Tu is small compared to the total released heat of reaction. Approximately, we can consider that all the heat from the reaction zone is carried away by conduction. The Fourier equation... 

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 heat flux can be expressed in terms of temperature gradient by the Fourier equation ... 

An image of the particle is obtained, the profile of that image is converted to a set of x,y pairs, a process known as digitizing. The x,y set is then converted to polar coordinates, (R,0). The curve in the R,0 space is converted to a Fourier equation, the coefficients of which are extracted, and then mathematically transformed to morphic terms which themselves constitute the shape features of the particle. A sample usually consists of 100 particles, upwards of 150 profile points are extracted from each particle, giving a total of 15,000 x,y points per analysis. Once the morphic terms are obtained, the data analysis can be carried out in order to facilitate the... 

The shape and texture features are invariant, unique and unequivocal. They are associated with indications of physical meaning. The used form of the Fourier equation is... 

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... 

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