Steady heat conduction numerical methods

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. 

In this chapter we will deal with steady-state and transient (or non steady-state) heat conduction in quiescent media, which occurs mostly in solid bodies. In the first section the basic differential equations for the temperature field will be derived, by combining the law of energy conservation with Fourier s law. The subsequent sections deal with steady-state and transient temperature fields with many practical applications as well as the numerical methods for solving heat conduction problems, which through the use of computers have been made easier to apply and more widespread. 

We speak of steady-state heat conduction when the temperature at every point in a thermally conductive body does not change with time. Some simple cases, which are of practical importance, have already been discussed in the introductory chapter, namely one dimensional heat flow in flat and curved walls, cf. section 1.1.2. In the following sections we will extend these considerations to geometric one-dimensional temperature distributions with internal heat sources. Thereafter we will discuss the temperature profiles and heat release of fins and we will also determine the fin efficiency first introduced in section 1.2.3. We will also investigate two- and three-dimensional temperature fields, which demand more complex mathematical methods in order to solve them, so that we are often compelled to make use of numerical methods, which will be introduced in section 2.4.6. 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

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 usually time varying and eventually reaches a steady state. This process is represented by parabolic partial differential equations with known initial conditions and boundary conditions at two ends. Both linear and nonlinear parabolic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear parabolic partial differential equations and numerical solutions for nonlinear parabolic partial differential equations based on the numerical method of lines. 

In previous sections of this chapter we discussed steady-state heat conduction in one direction. In many cases, however, steady-state heat conduction is occurring in two directions i.e., two-dimensional conduction is occurring. The two-dimensional solutions are more involved and in most cases analytical solutions are not available. One important approximate method to solve such problems is to use a numerical method discussed in detail in Section 4.15. Another important approximate method is the graphical method, which is a simple method that can provide reasonably accurate answers for the heat-transfer rate. This method is particularly applicable to systems having Isothermal boundaries. 

Numerical calculation methods for unsteady-state heat conduction are similar to numerical methods for steady state discussed in Section 4.15. The solid is subdivided into sections or slabs of equal length and a fictitious node is placed at the center of each section. Then a heat balance is made for each node. This method differs from the steady-state method in that we have heat accumulation in a node for unsteady-state conduction. 

Derivation of method for steady state. In Fig. 6.6-1 a two-dimensional solid shown with unit thickness is divided into Squares. The numerical methods for steady-state molecular diffusion are very similar to those for steady-state heat conduction discussed in Section 4.15. Hence, only a brief summary will be given here. The solid inside of a square is imagined to be concentrated at the center of the square at c and is called a node, which is connected to the adjacent nodes by connecting rods through which the mass diffuses. 

Derive steady-state and nonsteady-state mass and energy balances for a catalyst monolith channel in which several chemical reactions take place simultaneously. External and internal mass transfer limitations are assumed to prevail. The flow in the chaimel is laminar, but radial diffusion might play a role. Axial heat conduction in the solid material must be accounted for. For the sake of simplicity, use cylindrical geometry. Which numerical methods do you recommend for the solution of the model ... 

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. 

In order to identify EPHs of the cell or electrode reactions from the experimental information, there had been two principal approaches of treatments. One was based on the heat balance under the steady state or quasi-stationary conditions [6,11, 31]. This treatment considered all heat effects including the characteristic Peltier heat and the heat dissipation due to polarization or irreversibility of electrode processes such as the so-call heats of transfer of ions and electron, the Joule heat, the heat conductivity and the convection. Another was to apply the irreversible thermodynamics and the Onsager s reciprocal relations [8, 32, 33], on which the heat flux due to temperature gradient, the component fluxes due to concentration gradient and the electric current density due to potential gradient and some active components transfer are simply assumed to be directly proportional to these driving forces. Of course, there also were other methods, for instance, the numerical simulation with a finite element program for the complex heat and mass flow at the heated electrode was also used [34]. 

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