Numerical solutions to heat conduction problems

Two methods are available for the numerical solution of initial-boundary-value problems, the finite difference method and the finite element method. Finite difference methods are easy to handle and require little mathematical effort. In contrast the finite element method, which is principally applied in solid and structure mechanics, has much higher mathematical demands, it is however very flexible. In particular, for complicated geometries it can be well suited to the problem, and for such cases should always be used in preference to the finite difference method. We will limit ourselves to an introductory illustration of the difference method, which can be recommended even to beginners as a good tool for solving heat conduction problems. The application of the finite element method to these problems has been described in detail by G.E. Myers [2.52]. Further information can be found in D. Marsal [2.53] and in the standard works [2.54] to [2.56]. 

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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. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Instead of starting with a rigorous and mathematical development of the finite element technique, we proceed to present the finite element method through a solution of onedimensional applications. To illustrate the technique, we will first find a numerical solution to a heat conduction problem with a volumetric heat source... 

We will discuss the solution of steady-state and unsteady-state heat conduction problems in this chapter, using the finite-difference method.. The finite-difference method comprises the replacement of the governing equations and corresponding boundary conditions by a set of algebraic equations. The discussion here is not meant to be exhaustive in its mathematical rigor. The basics are presented, and the solution of the finite-difference equations by numerical methods are discussed. The solution of convection problems using the finite-difference method is discussed in a later chapter. 

Eq. 4-19 is called the characteristic equation or eigenfunction, and its roots are called the characteristic values or eigenvalues. The characteristic equation is implicit in this case, and thus the characteristic values need to be determined numerically. Then it follows that there arc an infinite number of solutions of the form Ae cos(AX), and the solution of this linear heat conduction problem is a linear combination of them,... 

As a result of this many solutions to the heat conduction equation can be transferred to the analogous mass diffusion problems, provided that not only the differential equations but also the initial and boundary conditions agree. Numerous solutions of the differential equation (2.342) can be found in Crank s book [2.78]. Analogous to heat conduction, the initial condition prescribes a concentration at every position in the body at a certain time. Timekeeping begins with this time, such that... 

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. 

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, ... 

Comini G, Del Guidice S, Lewis RW, ZienMewicz (1974) Finite element solution of nonlinear heat conduction problems with special reference to phase change. Int J Numer Meth Eng 8 613-624... 

The first type, which includes, for example, the problem of strong explosion or propagation of heat in a medium with nonlinear thermal conductivity [3], is characterized by the fact that the exponents are found from physical considerations, from the conservation laws and their dimensionality. In addition, the exponents turn out to be rational numbers. The task of the calculation is to find the dimensionless functions by integration of ordinary differential equations. After this the problem is completely solved, since the numerical constants are determined by normalizing the solution to the conserved quantity (the total energy released in these examples). 

The solutions of conductivity problems shown in the previous sections were obtained for zero-order kinetics. When the approximation by zero-order kinetics is not justified, which is the case, especially for autocatalytic reactions, a numerical solution is required. Here the use of finite elements is particularly efficient. The geometry of the container is described by a mesh of cells and the heat balance is established for each of these cells (Figure 13.5). The problem is then solved by iterations. As an example, a sphere can be described by a succession of concentric shells (like onion skins). In each cell, a mass and a heat balance are established. This gives access to the temperature profile if one considers the temperature of the different cells, or the temperature and conversion may be obtained as a function of time. 

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. 

An immense number of analytical solutions for conduction heat-transfer problems have been accumulated in the literature over the past 100 years. Even so, in many practical situations the geometry or boundary conditions are such that an analytical solution has not been obtained at all, or if the solution has been developed, it involves such a complex series solution that numerical evaluation becomes exceedingly difficult. For such situations the most fruitful approach to the problem is one based on finite-difference techniques, the basic principles of which we shall outline in this section. 

There is a myriad of analytical solutions for steady-state conduction heat-transfer problems available in the literature. In this day of computers most of these solutions are of small utility, despite their exercise in mathematical facilities. This is not to say that we cannot use the results of past experience to anticipate answers to new problems. But, most of the time, the problem a person wants to solve can be attacked directly by numerical techniques, except when there is an easier way to do the job. As a summary, the following suggestions are offered ... 

In Section S-3 we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Many heat transfer problems encountered in practice can be approximated as being one-dimensional, but this is not always the case. Sometimes we need to consider heat transfer in other directions as well when the variation of temperature in other directions is significani. In this section we consider the numerical formulation and solution of two-dimensional steady lieat conduclion in rectangular coordinates using the finite difference method. The approach presented below can be extended to three-dimensional cases. 

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