Conduction equation transfer

This type of heat transfer may be described by an equation that is simi lar to the conduction equation. The rate of flow of heat is proportional to the tcmpcraiure difference betw een the hot and cold liquid, and the heat transtei area. It is expressed ... 

Wlien a temperature difference exists in or across a body, an energy transfer occurs from the high-tem-perature region to the low-temperature region. This heat transfer, q, which can occur in gases, liquids, and solids, depends on a change m temperature, AT, over a distance. Ax (i.e., AT/z)ix) and a positive constant, k, which is called the thermal conductivity of the material. In equation form, the rate of conductive heat transfer per unit area is written as... 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

There is also another key parameter linked to the choice of the material for the reactor. First, the choice is obviously determined by the reactive medium in terms of corrosion resistance. However, it also has an influence on the heat transfer abilities. In fact, the heat transport depends on the effusivity relative to the material, deflned by b = (XpCp) the effusivity b appears in the unsteady-state conduction equation. 

Yoder showed that radiation heat transfer and axial conduction heat transfer in the tube wall have a negligible effect on predicting wall temperatures. The following equations were used by Yoder and Rohsenow (1980) as well as previous investigators such as Bennett et al. (1967b), Hynek (1969), and Groeneveld (1972). 

The first type of model considers the heat transfer surface to be contacted alternately by gas bubbles and packets of closely packed particles. This leads to a surface renewal process whereby heat transfer occurs primarily by transient conduction between the heat transfer surface and the particle packets during their time of residence at the surface. Mickley and Fairbanks (1955) provided the first analysis of this renewal mechanism. Treating the particle packet as a pseudo-homogeneous medium with solid volume fraction, e, and thermal conductivity (kpa), they solved the transient conduction equation to obtain the following expression for the average heat transfer coefficient due to particle packets,... 

For the analysis, a steady-state fire was assumed. A series of equations was thus used to calculate various temperatures and/or heat release rates per unit surface, based on assigned input values. This series of equations involves four convective heat transfer and two conductive heat transfer processes. These are ... 

The initial stage of boiling is usually controlled by the heat transfer from the ground. This is especially true for a spill of liquid with a normal boiling point below ambient temperature or ground temperature. The heat transfer from the ground is modeled with a simple onedimensional heat conduction equation, given by... 

To study the effects due to droplet heating, one must determine the temperature distribution T(r, t) within the droplet. In the absence of any internal motion, the unsteady heat transfer process within the droplet is simply described by the heat conduction equation and its boundary conditions... 

Conduction is the rate of heat transfer through a medium without mass transfer. The basic rate of conduction heat-transfer equation is Fourier s law ... 

In addition to the similarity between the heat conduction equation and the diffusion equation, erosion is often described by an equation similar to the diffusion equation (Culling, 1960 Roering et al., 1999 Zhang, 2005a). Flow in a porous medium (Darcy s law) often leads to an equation (Turcotte and Schubert, 1982) similar to the diffusion equation with a concentration-dependent diffu-sivity. Hence, these problems can be treated similarly as mass transfer problems. 

Let us consider heat transfer in a closed vessel filled with fluid. All of the vessel s walls are impermeable to the fluid. Some of the walls are thermally insulated, the rest are maintained at different fixed temperatures. It is the heat transfer between these regions heated to different temperatures which interests us. We will assume that the fluid is incompressible. We multiply both sides of the thermal conduction equation... 

The bulk conductivity a depends on the concentration of charge carriers and on their mobility, either of which can be modulated by exposure to the gas. The first prerequisite of such an interaction is the penetration of the analyte to the interior of the layer. The second is the ability of the gas to form a charge-transfer complex with the selective layer. This process then constitutes secondary doping, which affects the overall conductivity. For a mixed semiconductor, the overall conductivity is determined by the combined contribution from the holes (p) and electrons (n), as given by the general conductivity equation. 

The first three terms on the left hand side are the net convective, radiative and conductive heat transfers, whose expressions are reported in Equations (7.6-7.8). The fourth term is the heat generated by the chemical/electrochemical reactions (m-T As) and by Joule effect, while the last term is the electrical power generated in the slice. 

This chapter describes the fundamental principles of heat and mass transfer in gas-solid flows. For most gas-solid flow situations, the temperature inside the solid particle can be approximated to be uniform. The theoretical basis and relevant restrictions of this approximation are briefly presented. The conductive heat transfer due to an elastic collision is introduced. A simple convective heat transfer model, based on the pseudocontinuum assumption for the gas-solid mixture, as well as the limitations of the model applications are discussed. The chapter also describes heat transfer due to radiation of the particulate phase. Specifically, thermal radiation from a single particle, radiation from a particle cloud with multiple scattering effects, and the basic governing equation for general multiparticle radiations are discussed. The discussion of gas phase radiation is, however, excluded because of its complexity, as it is affected by the type of gas components, concentrations, and gas temperatures. Interested readers may refer to Ozisik (1973) for the absorption (or emission) of radiation by gases. The last part of this chapter presents the fundamental principles of mass transfer in gas-solid flows. 

Solve the one-dimensional, unsteady conductive heat transfer equation for a homogeneous solid sphere. (Hint (1) Let = R T so that... 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

The simplest technique is to use separate numerical solvers for the fluid and solid phases and to exchange information through the boundary conditions. The use of separate solvers allows a flexible gridding inside the solid phase, which is required because of the three orders of magnitude difference in thermal conductivities between the solid and gas. It is also easy to include various physical phenomena such as charring and moisture transfer. Quite often, ID solution of the heat conduction equation on each wall cell is sufficiently accurate. This technique is implemented as an internal subroutine in FDS. 

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. 

Equation (5.2) indicates that the time vapor is advected downwind will increase as zt (height of vapor containment box) increases, as this increases the time to fill the vapor box. This additional time allows for the effects of decreasing rates of conductive heat transfer from the dike floor for cryogenic materials, or decreasing convective mass transfer for materials with boiling points that are higher than ambient temperatures, to take effect. 

In a system with homogeneous reactions (e.g. reactive absorption), mass and heat transfer is described by the following convective diffusion and convective heat conduction equations (Kenig, 2000) ... 

The theoretical model assumes a line heat source dissipating heat radially into an infinite solid, initially at uniform temperature. The fundamental heat conduction equation in cylindrical coordinates, assuming uniform radial heat transfer, is [3] ... 

The left-hand side of the equation represents the convective heat transfer, the first bracketed term on the right-hand side represents the conductive heat transfer, and the second term represents the viscous energy dissipation owing to friction in the fluid. 

To analyze a transient heat-transfer problem, we could proceed by solving the general heat-conduction equation by the separation-of-variables method, similar to the analytical treatment used for the two-dimensional steady-state problem discussed in Sec. 3-2. We give one illustration of this method of solution for a case of simple geometry and then refer the reader to the references for analysis of more complicated cases. Consider the infinite plate of thickness 2L shown in Fig. 4-1. Initially the plate is at a uniform temperature T, and at time zero the surfaces are suddenly lowered to T = T,. The differential equation is... 

When strongly exothermic or endothermic reactions occur in the catalyst pellet, the temperature cannot be regarded as uniform throughout the catalyst particle. To describe heat transfer through a catalyst particle it is usually considered to be homogeneous. So, the heat flow is described by the conventional heat conduction equation used for isotropic solids or stagnant fluids. When chemical reactions take place, the energy balance equation is... 

Similar equations apply to cylindrical and spherical coordinate systems. Finite difference, finite volume, or finite element methods are generally necessary to solve (5-15). Useful introductions to these numerical techniques are given in the General References and Sec. 3. Simple forms of (5-15) (constant k, uniform S) can be solved analytically. See Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966, p. 180, and Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. For problems involving heat flow between two surfaces, each isothermal, with all other surfaces being adiabatic, the shape factor approach is useful (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 164). 

One-Dimensional Conduction Lumped and Distributed Analysis The one-dimensional transient conduction equations in rectangular (b = 1), cylindrical (b = 2), and spherical (b = 3) coordinates, with constant k, initial uniform temperature 7), S = 0, and convection at the surface with heat-transfer coefficient h and fluid temperature 77, are... 

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