Heat conduction 124 Subject

As an example of this method, the thermal stress distribution in a 2-D rectangular elastic body under steady-state heat conduction subject to the first kind of displacement boundary condition is shown. 

The rate of heat conduction is further complicated by the effect of sunshine onto the outside. Solar radiation reaches the earth s surface at a maximum intensity of about 0.9 kW/ m. The amount of this absorbed by a plane surface will depend on the absorption coefficient and the angle at which the radiation strikes. The angle of the sun s rays to a surface (see Figure 26.1) is always changing, so this must be estimated on an hour-to-hour basis. Various methods of reaching an estimate of heat flow are used, and the sol-air temperature (see CIBSE Guide, A5) provides a simplification of the factors involved. This, also, is subject to time lag as the heat passes through the surface. 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

Mills and Gilchrist (270) analysed the heat transfer that occurs when closed cell foams are subjected to impact, to predict the effect on the uniaxial compression stress-strain curve. Transient heat conduction from the hot compressed gas to the cell walls occurs on the 10 ms... 

The equations for the diffusion profile can be obtained from the heat-conduction equations of Carslaw and Jaeger [Ref. 3, Eq. 9.4 (10)] by using the substitutions we have indicated. The subject is discussed from a different approach by Adams, Quan, and Balkwell (I). The profiles can then be integrated over the volume of the sphere to obtain the uptake as a function of time. 

Thermal conductivity, X, is the property of a material of transmitting heat when subjected to a temperature difference. It is defined as the heat flow, Q, through unit thickness of material, x, of unit cross section, A, when the temperatures on each side differ by unity. In terms of the parameters shown in Figure 5.5, the heat flow is... 

Taking account of the boundary conditions, this equation can be integrated by elementary methods at each given instant and in a given layer. This determines the function late stage the plane field may be represented in the form H2 = curl (n ), where n = (0,0,1). After this the function (which now coincides with the vector potential component Az) is also subject to an equation of the heat conduction type. Consequently, H2 decays asymptotically. 

Particles suspended in a nonuniform gas may be subject to absorption or loss of heat or material by diffusional transport. If the particle is suspended without motion in a stagnant gas, heat or mass transfer to or from the body can be estimated from heat conduction or diffusion theory. One finds that the net rate of transfer of heat to the particle surface in a gas is... 

The advantage of Eq. 3.42 over Eq. 3.35, other than simplicity, is that the standard textbooks on diffusion [14, 15] and on heat conduction [18] (subject to an identical mathematical law) are replete with solutions to this fundamental equation. One of the most important solutions is the Gaussian function, which we will describe in some detail in Chapter 5. 

C Consider one-dimensional heat conduction through a large plane wall with no heat generation that is perfectly irtsu-lated on one side and is subjected to convection and radiation on the other side. It is claimed that under steady conditions, the temperaltire in a plane wall must be uniform (the same everywhere). Do you agree with this claim Why ... 

Consider a short cyUnder of radius r<, and height H in which heat is generated at a constant rate of Heat is lost from the cylindrical surface at r = r by convection to the surrounding medium at temperature with a heat transfer coefficient of /i. The bottom surface of the cylinder at z = 0 is insulated, while the top surface at z — is subjected to uniform heat flux Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not. solve. 

CoWsider a plane wall of thickness 2L initially at a uniform temperature of T , as shown in Fig. 4—1 In. At lime t = 0, the wall is immersed in a fluid at temperature 7 and is subjected to convection heal transfer from both sides with a convection coefficient of h. The height and the widlh of the wall are large relative to its thickness, and thus heat conduction in the wall can be approximated to be one-dimensional. Also, there is thermal symmetry about the inidplane passing through.x = 0, and thus the temperature distribution must be symmetrical about tlie midplane. Therefore, the value of temperature at any -.T value in - A "S. t 0 at any time t must be equal to the value at f-.r in 0 X Z, at the same time. This means we can formulate and solve the heat conduction problem in the positive half domain O x L, and then apply the solution to the other half. 

This completes the analysts for the solution of one-dimensional transient heat conduction problem in a plane wall. Solutions in other geometries such as a long cylinder and a sphere can be determined using the same approach. The results for all three geometries arc summarized in Table 4—1. The solution for the plane wall is also applicable for a plane wall of thickness L whose left surface at, r = 0 is insulated and the right surface at.t = T. is subjected to convection since this is precisely the mathematical problem we solved. 

Consider steady heat conduction in a plane wall whose left. surface (node 0) is maintained at 30°C while the right surface (node 8) is subjected to a heat flux of 1200 W/mT Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heal generation. Also obtain the finite dif-... 

C Consider transient onc-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall ate subjected to specified heat flux, express the stability criterion for this problem in its simplest form. 

Consider a solid plane wall (medium B) of area A, thickness L, and density p. The wall is subjected on both sides to different concentrations of a species A to which it is permeable. The boundary surfaces at.t = 0 and x - L are located within the solid adjacent to the interfaces, and the mass fractions of A at those surfaces are maintained at and 2. respectively, at all times (Fig. 14-19). The mass fraction of species A in the wall varies in the. v-direction only and can be expressed as >v (.t). Therefore, mass transfer through the wall in this case can be modeled as steady and one-dimensional. Here we determine the rate of mass diffusion of species A through the wall using a similar approach to that used in Chapter 3 for heat conduction. 

This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

Heat waves have been theoretically studied in a very thin film subjected to a laser heat source and a sudden symmetric temperature change at two side walls. The non-Fourier, hyperbolic heat conduction equation is solved using a numerical technique based on MacCormak s predictor-corrector scheme. Results have been obtained for ftie propagation process, magnitude and shape of thermal waves and the range of film ftiickness Mid duration time wiftiin which heat propagates as wave. 

Yuen, W. W. and Lee, S. C. (1989) Non-Fourier Heat Conduction in a Semi-Infinite Solid Subjected to Oscillatory Surface Thermal Disturb ices, Journal of Heat Transfer, Vol. Ill, pp. 178-181. 

Tan, Z. M., and Yang, W.-J. (1997) Non-Fourier Heat Conduction in a Thin Film Subjected to a Sudden Temperature Change on Two Sides, Journal of Non-Equilibrium Thermodynamics, Vol. 22, pp. 75-87. 

Thermal Properties.—The thermal qualities of refractories, specific heat, conductivity and expansion are determined according to the established physical methods. It is evident that these properties are of considerable practical importance. The data available, however, on these subjects are quite meager, especially if it is considered that the structure of the manufactured product, irrespective of its chemical nature, is of paramount importance. Furthermore, these properties are subject to change with temperature and comparatively few constants are at hand to illustrate the character of these relations. It is known that the specific heat and thermal conductivity increase with temperature but the fundamental laws governing these changes have not been established. Furthermore, it must be realized that the structure of all these materials is certain to undergo physical changes which affect the thermal qualities. 

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. 

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