Fourier-Kirchhoff equation

The simulator used was a DISMOL, described previously by Batistella and Maciel (2). All explanations of the equations used, the solution methods, and the routine of solution are described in Batistella and Maciel (5). DISMOL is a simulator that permits changes in feed composition, feed temperaturethe evaporation rate, as well as feed flow rate. The effective rate of surface evaporation is obtained from the kinetic theory of gases. The liquid film thickness is obtained by mass balance and geometry of the evaporator. The temperature in the liquid obeys the Fourier-Kirchhoff equation. The solution of the velocity profile requires knowledge of the viscosity and the liquid film thickness over the evaporator. The solution for the temperature and the concentration profiles requires knowledge of the velocity profiles, which determine the convective heat and mass fluxes. 

Equation (1.29) is called the Fourier-Kirchhoff equation. When the process takes place under isobaric conditions, i.e. 

When the initial and boundary conditions are known, the physical problem of heat conduction is to find adequate solutions of the Fourier-Kirchhoff equation or the Fourier equation. 

Linear differential equation of first order called the heat balance equation of a simple body, has found wide application in calorimetry and thermal analysis as mathematical models used to elaborate various methods for the determination of heat effects. It is important to define the conditions for correct use of this equation, indicating all simplifications and limitations. They can easily be recognized from the assumption made to transform the Fourier-Kirchhoff equation into the heat balance equation of a simple body. 

Let us consider [1, 6, 8, 17] that the heat transfer process takes place under isobaric conditions, without mass exchange and that the thermal parameters of the body are constant. The Fourier-Kirchhoff equation can then be written as... 

Equations (1.99) and (1.100) are commonly known as the heat balance equations of a simple body. From the above considerations, it is clear that these heat balance equations and the Fourier-Kirchhoff equation Piq. (1.87)] are equivalent to each other when ... 

A real calorimeter is composed of many parts made from materials with different heat conductivities. Between these parts one can expect the existence of heat resistances and heat bridges. To describe the heat transfer in such a system, a new mathematical model of the calorimeter was elaborated [8, 19] based on the assumption that constant temperatures are ascribed to particular parts of the calorimeter. In the system discussed, temperature gradients can occur a priori. Before defining the general heat balance equation, let us consider the particular solutions of the equation of conduction of heat for a rod and sphere. For a body treated as a rod in which the process takes place under isobaric conditions, without mass exchange, the Fourier-Kirchhoff equation may be written as... 

Various classifications of calorimeters have been presented [75-83]. The classification given here [84] is based on the assumption that the calorimeter is a dynamic object in which heat is generated. Calorimeters are graded by applying the criteria of the temperature conditions under which the measurement was made. As an initial basis for further considerations, the Fourier - Kirchhoff equation (Eq. (1.29)) has been used in the following form ... 

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 Fourier-Kirchhoff differential equation and the equation of conduction of heat describe the transfer of heat in general form. In order to obtain the particular solutions of these equations, it is necessary to determine the initial and boundary conditions. 

Applying the Fourier transform to both sides of this equation, we can obtain the corresponding Kirchhoff formula in the frequency domain. According to the convolution theorem (Arfken and Weber, 1995), the Fourier transform of the convolution of two functions is equal to the product of spectra of these functions. Therefore we obtain ... 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

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