Conduction, heat analytic approximation

As indicated in Table 7.10, only in the last decade have models considered all three phenomena of heat transfer, fluid flow, and hydrate dissociation kinetics. The rightmost column in Table 7.10 indicates whether the model has an exact solution (analytical) or an approximate (numerical) solution. Analytic models can be used to show the mechanisms for dissociation. For example, a thorough analytical study (Hong and Pooladi-Danish, 2005) suggested that (1) convective heat transfer was not important, (2) in order for kinetics to be important, the kinetic rate constant would have to be reduced by more than 2-3 orders of magnitude, and (3) fluid flow will almost never control hydrate dissociation rates. Instead conductive heat flow controls hydrate dissociation. 

Precise solution of the multidimensional problem of heat conductivity by analytical methods is very complicated and laborious. Therefore, an approximate finite difference method was developed based on the differential heat conductivity equation and boundary conditions. In this method, the temperature of the vulcanized section of the covering fragment was subdivided into elementary volumes of unit thickness because it is necessary to define the temperature field of the vulcanized. 

The idea that the reaction proceeds principally at a temperature which is close to Tx also permits us to find closed analytical expressions for the velocity of flame propagation. The heat released during the reaction is partly spent on heating the reacting gas itself and partly carried away by heat conduction to neighboring elements of the gas volume. If the temperature of the zone in which the reaction effectively occurs is already close to Tlt the amount of the heat spent on heating the reacting gas up to its final temperature beyond the flame front, Tu is small compared to the total released heat of reaction. Approximately, we can consider that all the heat from the reaction zone is carried away by conduction. The Fourier equation... 

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

The preceding examples discuss the heat-conduction problem without melt removal in a semi-infinite solid, using different assumptions in each case regarding the thermophysical properties of the solid. These solutions form useful approximations to problems encountered in everyday engineering practice. A vast collection of analytical solutions on such problems can be found in classic texts on heat transfer in solids (10,11). Table 5.1 lists a few well-known and commonly applied solutions, and Figs. 5.5-5.8 graphically illustrate some of these and other solutions. 

The thermophysical properties, such as glass transition, specific heat, melting point, and the crystallization temperature of virgin polymers are by-and-large available in the literature. However, the thermal conductivity or diffusivity, especially in the molten state, is not readily available, and values reported may differ due to experimental difficulties. The density of the polymer, or more generally, the pressure-volume-temperature (PVT) diagram, is also not readily available and the data are not easily convertible to simple analytical form. Thus, simplification or approximations have to be made to obtain a solution to the problem at hand. 

Using the one-term approximation, the solutions of onedimensional transient heat conduction problems are expressed analytically as... 

An experiment is to be conducted to determine heat transfer coefficient on the surfaces of tomatoes that are placed in cold water at 7°C, The tomatoes (k = 0,59 W/m °C, a = 0,141 X lQ- mVs, p = 999 kg/m , 3,99 kJ/kg °C) with an initial uniform temperature of 30°C are spherical in. shape willr a diameter of 8 cm. After a period of 2 hours, the temperatures at the center and the surface of the tomatoes arc measured to be 10.0°C and 7.1 C, respectively. Using analytical one-term approximation method (not the Hcisler charts), determine the heat transfer coefficient and the amount of heat transfer during this period if there are eight such tomatoes in water. 

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. 

Even in. simple geometries, heat transfer problems cannot be. solved analytically if the thermal conditions are not sufficiently simple. For example, the consideration of the variation of thermal conductivity with temperature, the variation of the heat transfer coefficient over the surface, or the radiation heat transfer on the surfaces can make it impossible to obtain au analytical. solution. Therefore, analytical solutions are limited to problems that are simple or can be simplified with rea.sonable approximations. 

Conjugated eonduetion-convection problems are among the elassieal formulations in heat transfer that still demand exact analytical treatment. Since the pioneering works of Perelman (1961) [14] and Luikov et al. (1971) [15], such class of problems continuously deserved the attention of various researchers towards the development of approximate formulations and/or solutions, either in external or internal flow situations. For instance, the present integral transform approach itself has been applied to obtain hybrid solutions for conjugated conduction-convection problems [16-21], in both steady and transient formulations, by employing a transversally lumped or improved lumped heat conduction equation for the wall temperature. 

In the equation the rate of rise of temperature dT/bt depends upon the balance between heat conduction (given by the first term on the right-hand side) and heat evolution (which involves the reaction rate parameter A and exothermicity parameter Q). C is the heat capacity per unit volume, k the thermal conductivity, E the activation energy, and k the Boltzmann constant. The equation cannot be solved analytically, but approximate solutions were first described for gas-phase reactions [49] and later for decomposing solids [50]. 

The determination of the thermal conductivity of grain is based on the comparison of the temperature history data obtained by using the line heat source probe with the approximate analytical and numerical methods [35,54]. The analytical method has the advantage of being quick in calculating thermal conductivity. This method, however, requires a perfect line source and a small diameter tube holding the line heat source. In reality, this requirement is difficult to meet. Therefore, a time-correction procedure has been introduced [52,54,56]. Another objection to the analytical method is that it cannot easily be used to calculate the temperature distribution in the heated grain and to compare it with the measured one. Such a comparison can be easily accomplished by a numerical method, where the estimated accuracy for thermal conductivity is determined and the thermal conductivity of the device is taken into account [54]. 

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. 

The general field of problems described above, except in some special areas, may be treated by the well-known methods and analytical models of mathematical physics. It has already been noted that the most general description of the neutron population usually starts with a neutron-balance relation of the Boltzmann type. The Boltzmann equation was developed in connection with the study of nonuniform gas mixtures, and the application to the neutron problem represents a considerable simplification of the general gas problem. (Whereas in gas problems all the particles are in motion, in reactor problems only the neutrons are in motion. ) The fundamental equation of reactor physics, then, is already a familiar one from the kinetic theory. Further, many of the most useful neutron models obtained from approximations to the Boltzmann equation reduce to familiar forms, such as the heat-conduction, Helmholtz, and telegraphist s equations. These simplifications result from the elimination of various independent variables in the... 

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