Partial differential equations heat conduction problem

Chapter 4 eoncerns differential applications, which take place with respect to both time and position and which are normally formulated as partial differential equations. Applications include diffusion and conduction, tubular chemical reactors, differential mass transfer and shell and tube heat exchange. It is shown that such problems can be solved with relative ease, by utilising a finite-differencing solution technique in the simulation approach. 

Let us consider the semi-infinite (thermally thick) conduction problem for a constant temperature at the surface. The governing partial differential equation comes from the conservation of energy, and is described in standard heat transfer texts (e.g. Reference [13]) ... 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

If the thermal conductivity k and the product pCp are temperature independent, Eq. 5.3-1 reduces for homogeneous and isotropic solids to a linear partial differential equation, greatly simplifying the mathematics of solving the class of heat transfer problems it describes.1... 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem,... 

The governing partial differential equations for heat transfer to a SD in a 303 x 406 can with R = 4.2 cm and L = 10.7 cm, and no head space were solved and the details are given in Yang (1997) and Yang and Rao (1998a, 1998b). Because the SD exhibited non-Newtonian behavior after the initial gelatinization temperature was reached, the problem could not be approximated to simple heat transfer by conduction. 

The formulation of heat conduction problems for the determination of the one-dimj .nsional transient temperature distribution in a plane wall, a cylinder, or a sphefeTesults in a partial differential equation whose solution typically involves irtfinite series and transcendental equations, wliicli are inconvenient to use. Bijt the analytical soluliop provides valuable insight to the physical problem, hnd thus it is important to go through the steps involved. Below we demonstrate the solution procedure for the case of plane wall. 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

In the application of the heat conduction equation in its general form (2.8) a series of simplifying assumptions are made, through which a number of special differential equations, tailor made for certain problems, are obtained. A significant simplification is the assumption of constant material properties A and c. The linear partial differential equations which emerge in this case are discussed in the next section. Further simple cases are... 

The Laplace transformation has proved an effective tool for the solution of the linear heat conduction equation (2.110) with linear boundary conditions. It follows a prescribed solution path and makes it possible to obtain special solutions, for example for small times or at a certain position in the thermally conductive body, without having to determine the complete time and spatial dependence of its temperature field. An introductory illustration of the Laplace transformation and its application to heat conduction problems has been given by H.D. Baehr [2.25]. An extensive representation is offered in the book by H. Tautz [2.26]. The Laplace transformation has a special importance for one-dimensional heat flow, as in this case the solution of the partial differential equation leads back to the solution of a linear ordinary differential equation. In the following introduction we will limit ourselves to this case. 

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 distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

Derivation of the method. Since the advent of the fast digital computers, solutions to many complex two-dimensional heat-conduction problems by numerical methods are readily possible. In deriving the equations we can start with the partial differential equation (4.15-5). Setting up the finite difference of d T/dx, ... 

Equation (3.7) has the same form as the partial differential equation for heat conduction. The mathematical solutions which are known for a great variety of heat-conduction problems can be applied in the theory of diffusion. 

Thermal conduction in the solid phase is a key factor, as already mentioned in section 1.2.4. The heat conduction process is accounted for by Fourier s law in the heat balance equation which is thus a second order partial differential equation. An efficient numerical technique is required to avoid "numerical conduction" because the solid temperature gradient is very sharp at the light-off point (see section 3.1). There is no study of Ais numerical problem in the literature. However, Eigenberger (1972) studied the consequences of heat conduction on steady-state multiplicity. He showed that the conduction process is responsible for a reduction of the number of steady state solutions. In the example studied by Eigenberger, the steady-state solution is close to the "highest steady state" (i.e., steady state with the temperature maximum close to reactor inlet) without conduction because "the temperature maximum moves to the front of the reactor, driven by the backward conduction of heat". 

On the basis of their initial and boundary conditions, partial differential equations may be further classified into initial-value or boundary-value problems. In the first case, at least one of the independent variables has an open region. In the unsteady-state heat conduction problem, the time variable has the range 0 r >, where no condition has been specified at r = eo therefore, this is an initial-value problem. When the region is closed for all independent variables and conditions are specified at all boundaries, then the problem is of the boundary-value type. An example of this is the three-dimensional steady-state heat conduction problem described by the equation... 

In chemical engineering, solutions of partial differential equations are often required, particularly for the problems associated with heat conduction. Different numerical methods are used for solving different types of partial differential equations. 

In this book, the main results of research are presented in terms of equations and figures. In all cases, the theoretical treatment is presented in a didactic manner, so that the readers not familiar with terms can, nevertheless, easily understand its development. The assumptions for which calculations are made and the solutions obtained, are clearly presented. It is true that the problems of heat transfer by conduction, as well as those of diffusion of liquid through a rubber, are not easy to understand. This is the reason why the solutions of the differential equations with partial derivatives that express these transfers have been deeply explained. On the other hand, in order to facilitate the reader s generalizing the results, the figures are drawn by using dimensionless numbers as coordinates as often as possible, leading to master curves of value in various applications. Thus, by introducing the typical values of their problems, readers can obtain the particular result of the actual problem put before them. 

Here, the concentration at x = 0 always remains constant contrary to the previous example, where a fixed concentration is introduced once. This problem is analogous to transient conduction in semi-infinite solid with constant surface temperature boundary condition. The detailed solution procedure can be found in regular heat transfer book. The solution for the above problem can be obtained by using f The governing equation in partial differential form... 

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