Conduction, heat computer solutions

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 ... 

Investigate the computer routines that are available at your computer center for solution of conduction heat-transfer problems. 

The diffusion cloud chamber has been widely used in the study of nucleation kinetics it is compact and produces a well-defined, steady supersaturation field. The chamber is cylindrical in shape, perhaps 30 cm in diameter and 4 cm high. A heated pool of liquid at the bottom of the chamber evaporates into a stationary carrier gas, usually hydrogen or helium. The vapor diffuses to the top of the chamber, where it cools, condenses, and drains back into the pool at the bottom. Because the vapor is denser than the carrier gas, the gas density is greatest at the bottom of the chamber, and the system is stable with respect to convection. Both diffusion and heat transfer are one-dimensional, with transport occurring from the bottom to the top of the chamber. At some position in the chamber, the temperature and vapor concentrations reach levels corre.sponding to supersaturation. The variation in the properties of the system are calculated by a computer solution of the onedimensional equations for heat conduction and mass diffusion (Fig. 10.2). The saturation ratio is calculated from the computed local partial pressure and vapor pressure. 

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, ... 

In the following discussion, attention is focused on the application of thermal-hydraulic system codes. Under this category codes like APROS, ATHLET, CATHARE, RELAP5 and TRAC are included, all based upon the solution of a main system of six partial differential equations. Two main fields, one per each of the two phases liquid and steam are considered and coupling is available with the solution of the conduction heat transfer equations within solids interfaced with the fluid phases. A one-dimensional solution for the characteristics of the fluid is achieved in the direction of the fluid motion in time dependent conditions. It should be emphasized that more sophisticate models are also available including three-dimensional solutions and multi-field approaches in two and multiphase fluids. However the present qualification level of those sophisticated computational tools is questionable as well as their actual need in the design or in the safety applications. 

McAdams (Heat Transmission, 3d ed., McGraw-HiU, New York, 1954) gives various forms of transient difference equations and methods of solving transient conduction problems. The availabihty of computers and a wide variety of computer programs permits virtually routine solution of complicated conduction problems. 

The numerical approaches to the solution of the Laplace equation usually demand access to minicomputers with fast processing capabilities. Numerical methods of this sort are essential when the electrolyte is unconfined, as for an off-shore rig or a submarine hull. However, where the electrolyte is confined, as within essentially cylindrical equipment such as pipework and heat-exchangers, or for restricted electrolyte depths, a simpler modelling procedure may be adopted in the case of electrolytes of good conductivity, such as sea-water . This simpler procedure enables computation to be carried out on small, desk-top microcomputers. 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

Let us return to our discussion of the prediction of ignition time by thermal conduction models. The problem reduces to the prediction of a heat conduction problem for which many have been analytically solved (e.g. see Reference [13]). Therefore, we will not dwell on these multitudinous solutions, especially since more can be generated by finite difference analysis using digital computers and available software. Instead, we will illustrate the basic theory to relatively simple problems to show the exact nature of their solution and its applicability to data. 

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. 

The prospective applications ofmolecular assemblies seem so wide that their limits are difficult to set. The sizes of electronic devices in the computer industry are close to their lower limits. One simply cannot fit many more electronic elements into a cell since the walls between the elements in the cell would become too thin to insulate them effectively. Thus further miniaturization of today s devices will soon be virtually impossible. Therefore, another approach from bottom up was proposed. It consists in the creation of electronic devices of the size of a single molecule or of a well-defined molecular aggregate. This is an enormous technological task and only the first steps in this direction have been taken. In the future, organic compounds and supramolecular complexes will serve as conductors, as well as semi- and superconductors, since they can be easily obtained with sufficient, controllable purity and their properties can be fine tuned by minor adjustments of their structures. For instance, the charge-transfer complex of tetrathiafulvalene 21 with tetramethylquinodimethane 22 exhibits room- temperature conductivity [30] close to that of metals. Therefore it could be called an organic metal. Several systems which could serve as molecular devices have been proposed. One example of such a system which can also act as a sensor consists of a basic solution of phenolophthalein dye 10b with P-cyciodextrin 11. The purple solution of the dye not only loses its colour upon the complexation but the colour comes back when the solution is heated [31]. 

For this edition examples and problems oriented toward numerical (computer-generated) solutions have been expanded for both steady state and transient conduction in Chapters 3 and 4. New convection correlations have been added in Chapters 5, 6, and 7, and summary tables have been provided for convenience of the reader. New examples have also been provided in the radiation, convection, and heat exchanger material and over 250 new problems have been added throughout the book. Over 200 of the previous problems have been restated so that they are new for student work. In addition, all problems have been reorganized to follow the sequence of chapter topics. A total of over 850 problems is provided. 

Equation (5-2) considers the thermal conductivity to be variable. If k is expressed as a function of temperature, Eq. (5-2) is nonlinear and difficult to solve analytically except for certain special cases. Usually in complicated systems numerical solution by means of computer is possible. A complete review of heat conduction has been given by Davis and Akers [Chem. Eng., 67(4), 187, (5), 151 (I960)] and by Davis [Chem. Eng., 67(6), 213, (7), 135 (8), 137 (I960)]. 

The orthogonal collocation method using piecewise cubic Her-mite polynomials has been shown to give reasonably accurate solutions at low computing cost to the elliptic partial differential equations resulting from the inclusion of axial conduction in models of heat transfer in packed beds. The method promises to be effective in solving the nonlinear equations arising when chemical reactions are considered, because it allows collocation points to be concentrated where they are most effective. 

So far we Kave mostly considered relatively simple heat conduction problems Involving simple geoineiries with simple boundary conditions because only such simple problems can be solved analytically. But many problems encountered in practice involve complicated geometries with complex boundary conditions or variable properties, and cannot be solved analytically. In such cases, sufficiently accurate approximate solutions can be obtained by computers using a numerical method. 

In the finite difference method, the derivatives dfl/dt, dfl/dx and d2d/dx2, which appear in the heat conduction equation and the boundary conditions, are replaced by difference quotients. This discretisation transforms the differential equation into a finite difference equation whose solution approximates the solution of the differential equation at discrete points which form a grid in space and time. A reduction in the mesh size increases the number of grid points and therefore the accuracy of the approximation, although this does of course increase the computation demands. Applying a finite difference method one has therefore to make a compromise between accuracy and computation time. 

Electric networks have been used to describe radiation heat transfer. Because electric networks have commonly available solutions, this analogy is useful. It also permits the use of an analog computer for solving complex problems. Similarly, conduction systems have been studied using small analog models made of various materials, including conducting paper. 

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