Numerical solution of heat conduction

Douglas, J. Rachford, H.H. On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Soc. 82 (1956) 421-439... 

C.1 Numerical Solution of Heat Conduction in a Slab. Do Example 5.5 for the case in which the boundary conditions are given in terms of a heat transfer coefficient and the thermal coefficients are temperature dependent (see Table 5.6). Compare the results to those obtained in Example 5.5 in which constant thermal coefficients were used. 

Generally, the Crank-Nicolson method is used for numerical solution of heat conduction problems that are parabolic in nature. This method solves a mesh with mesh size h in x-direction and mesh size k in y-direction (time direction). It calculates the values of u at six points as shown in Figure 1.17. 

The numerical solution of problem (1) by means of iteration schemes can be done using the alternating direction scheme for the heat conduction... 

Only a finite difference numerical solution can give exact results for conduction. However, often the following approximation can serve as a suitable estimation. For the unsteady case, assuming a semi-infinite solid under a constant heat flux, the exact solution for the rate of heat conduction is... 

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. 

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. 

Alifanov O, Artyukhin E (1976) Regularized numerical solution of nonlinear inverse heat-conduction problem. J Engin Phys 29 934-948... 

Cranck, J., and P. Nicolson A Practical Method for Numerical Evaluation of Solutions of P.D.E. of Heat Conduction Type, Proc. Camb. Phil. Soc., vol. 43, p. 50, l947. 

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)]. 

In this work, microscale evaporation heat transfer and capillary phenomena for ultra thin liquid film area are presented. The interface shapes of curved liquid film in rectangular minichannel and in vicinity of liquid-vapor-solid contact line are determined by a numerical solution of simplified models as derived from Navier-Stokes equations. The local heat transfer is analyzed in term of conduction through liquid layer. The data of numerical calculation of local heat transfer in rectangular channel and for rivulet evaporation are presented. The experimental techniques are described which were used to measure the local heat transfer coefficients in rectangular minichannel and thermal contact angle for rivulet evaporation. A satisfactory agreement between the theory and experiments is obtained. 

The above phenomena me physically miomalous and can be remedied through the introduction of a hyperbolic equation based on a relaxation model for heat conduction, which accounts for a finite thermal propagation speed. Recently, considerable interest has been generated toward the hyperbolic heat conduction (HHC) equation and its potential applications in engineering and technology. A comprehensive survey of the relevant literature is available in reference [6]. Some researchers dealt with wave characteristics and finite propagation speed in transient heat transfer conduction [3], [7], [8], [9] and [10]. Several analytical and numerical solutions of the HHC equation have been presented in the literature. 

The numerical solution of a transient heat conduction problem is of particular importance when temperature dependent material properties or bodies with... 

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]. 

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... 

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 steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

A dimensional analysis cannot be made unless enough is known about the physics of the situation to decide what variables are important in the problem and what physical laws would be involved in a mathematical solution if one were possible. Sometimes this condition is fairly easily met the fundamental differential equations of fluid flow, for example, combined with the laws of heat conduction and diffusion, suffice to establish the dimensions and dimensionless groups appropriate to a large number of chemical engineering problems. Dimensional analysis, however, does not yield a numerical equation, and experiment is needed to complete the solution to the problem. 

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. 

The calculation of drying processes requires a knowledge of a number of characteristics of drying techniques, such as the characteristics of the material, the coefficients of conductivity and transfer, and the characteristics of shrinkage. In most cases these characteristics cannot be calculated by analysis, and it is emphasized in the description of mathematical models of the physical process that the so-called global conductivity and transfer coefficients, which reflect the total effect on the partial processes, must frequently be interpreted as experimental characteristics. Consequently, these characteristics can be determined only by adequate experiments. With experimental data it is possible to apply analytical or numerical solutions of simultaneous heat and mass transfer to practical calculations. 

For creeping flow (0 < Re < 1), the solutions of the conduction-convection equation with flow held of the Hadamard-Rybczynski or Stokes are given by numerical integration [1]. The numerical results show that the concentration contours are not symmetrical (Figure 5.1 and Figure 5.2) and that the how inside and outside the sphere largely inhuences heat or mass transfer. In the case of a sphere with weak viscosity ratio, the heat or mass transfer is facilitated. 

Equation (6.109) is known as Fourier s law of heat conduction. As noted above, there will be a potential contribution as well. Solution to the integral equation, Eq. (6.95), for the function A w) gives numerical values of the thermal conductivity from Eq. (6.110)." ... 

Examples of a simultaneous numerical solution of molar and energy balance equations for the gas bulk and catalyst particles are introduced in Figure 5.27, in which the concentration and temperature profiles in a methanol synthesis reactor are analyzed. The methanol synthesis reaction, CO -F 2H2 CH3OH, is a strongly exothermic and diffusion-limited reaction. This implies that concentration gradients emerge in the catalyst particles, whereas the heat conductivity of the particles is so good that the catalyst particles are practically isothermal. 

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