Heat conduction numerical methods

Conduct analytical, numerical, and experimental studies on a variety of heat conduction enhancement methods (metal foams, screens, wires, wools). 

Nour-Omid, B., 1987. Lanezos method for heat conduction analysis. Int. J. Numer. Methods Eng. 24, 251-262. 

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. 

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. 

Lotkin (L10) gives a scheme for numerical integration of the heat conduction equation in a finite ablating slab, using unequal subdivisions in both space and time variables. Near the melting surface it is advantageous to choose rather small integration steps. Stability characteristics of the method are established. 

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... 

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. 

J. Crank and P. Nicolson, A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-conducting Type, Proc. Cambridge. Philos. Soc., 43,50-67 (1947). 

The balance between conduction and diffusion still operates for a much larger isolated wet object, provided radiation is excluded. This is the basis of the wet bulb thermometer method for measuring humidity. The actual rate of evaporation now is not as simply determined and is influenced by wind. The wet bulb temperature is almost independent of wind condition, owing to a convenient accident. Heat conduction is a diffusion process, and the diffusion coefficient for water vapor in air (0.24 sq. cm./sec.) is numerically close to the diffusion coefficient of temperature in air (thermal conductivity/specific heat = 0.20 sq. cm./sec.). Hence, the exact way in which each molecular diffusion process merges into the more rapid eddy diffusion process is not important because no matter how complex the transition is, it must be quantitatively similar for the two processes. 

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. 

Physical situations that involve radiation with conduction and convection are fairly common. Examples include automobile radiators and heat transfer in the furnace of boilers and incinerators. The energy equations become more complex as they comprise both temperature differences coming from convection and temperature derivatives coming from conduction. Hence, there are no classical methods of solution, but numerical methods and specific methods for particular problems. 

A tube has diameters of 4 mm and S mm and a thermal conductivity 20 W/m2 °C. Heat is generated uniformly in the tube at a rate of 500 MW/m3 and the outside surface temperature is maintained at 100°C. The inside surface may be assumed to be insulated. Divide the tube wall into four nodes and calculate the temperature at each using the numerical method. Check with an analytical solution. 

A concrete slab 15 cm thick has a thermal conductivity of 0.87 W/m - °C and has one face insulated and the other face exposed to an environment. The slab is initially uniform in temperature at 300°C, and the environment temperature is suddenly lowered to 90°C. The heat-transfer coefficient is proportional to the fourth root of the temperature difference between the surface and environment and has a value of 11 W/m2 °C at time zero. The environment temperature increases linearly with time and has a value of 200°C after 20 min. Using the numerical method, obtain the temperature distribution in the slab after 5, 10, 15, and 20 min. 

The above derivation for LMTD involves two important assumptions (1) the fluid specific heats do not vary with temperature, and (2) the convection heat-transfer coefficients are constant throughout the heat exchanger. The second assumption is usually the more serious one because of entrance effects, fluid viscosity, and thermal-conductivity changes, etc. Numerical methods must normally be employed to correct for these effects. Section 10-8 describes one way of performing a variable-properties analysis. 

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 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. 

Specification of necessary information/data related to flow process under consideration. Once a suitable grid is generated, the user has to specify the necessary information concerning the physicochemical properties of fluids such as molecular viscosity, density, conductivity etc. for the solution of model equations. If the process under consideration involves chemical reactions, all the other necessary data about reaction kinetics (and stoichiometry, heat of reaction etc.) need to be supplied. In addition to system-specific data, specification of boundary conditions on the edges/external surfaces of the solution domain is a further crucial aspect of the solution process. It is also necessary to provide all the information related to the numerical method selected... 

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. 

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