Heat conduction equation properties

Cellular models (Kottke and Niiler, 1988 Astapchik et al, 1993 Hwang et ai, 1997) represent another direction for modeling the combustion synthesis process. For these models, the reaction medium is tessellated into a discrete matrix of cells, where the temperature and effective properties are assumed to be uniform throughout the cell. The interactions between the cells are based on various rules, often a discretized form of the heat conduction equation. 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

In the application of the heat conduction equation in its general form (2.8) a series of simplifying assumptions are made, through which a number of special differential equations, tailor made for certain problems, are obtained. A significant simplification is the assumption of constant material properties A and c. The linear partial differential equations which emerge in this case are discussed in the next section. Further simple cases are... 

The heat conduction equation for bodies with constant material properties... 

In the derivation of the heat conduction equation in (2.8) we presumed an incompressible body, g = const. The temperature dependence of both the thermal conductivity A and the specific heat capacity c was also neglected. These assumptions have to be made if a mathematical solution to the heat conduction equation is to be obtained. This type of closed solution is commonly known as the exact solution. The solution possibilities for a material which has temperature dependent properties will be discussed in section 2.1.4. 

This equation offers a clear interpretation of the thermal diffusivity a and the heat conduction equation itself. According to (2.14) the change in the temperature with time d d/dt at each point in the conductive body is proportional to the thermal diffusivity. This material property, therefore, has an effect on how quickly the temperature changes. As Table 2.1 shows, metals do not only have high thermal conductivities, but also high values for the thermal diffusivity, which imply that temperatures change quickly in metals. 

If the temperature dependence of the material properties A = A( ) and c = c(i9) cannot be neglected then the heat conduction equation (2.8) must be the starting point for the solution of a conduction problem. We have a non-linear problem, that can only be solved mathematically in exceptional cases. With... 

As the heat conduction equation derived in 2.1.1 shows, the only material property which has an effect on the steady state temperature field, dfi/dt = 0, is the thermal conductivity A = A( ). Assuming that A is constant,... 

For the introduction and explanation of the method we will discuss the case of transient, geometric one-dimensional heat conduction with constant material properties. In the region x0 < x < xn the heat conduction equation... 

If A and eg change with temperature, cf. section 2.1.4, a closed solution to the heat conduction equation cannot generally be found, which only leaves the possibility of using a numerical solution method. We will show how temperature dependent properties are accounted for by using the example of the plate, m = 0 in (2.274). The transfer of the solution to a cylinder or sphere (m = 1 or 2 respectively) is... 

We will assume that the material properties are constant. The heat conduction equation for planar, transient temperature fields with heat sources has the form... 

Another feature of the present theory is that it provides a formalism for deducing a complete mathematical representation of a phenomenon. Such a representation consists, typically, of (1) Balance equations for extensive properties (such as the "equations of change" for mass, energy and entropy) (2) Thermokinematic functions of state (such as pv = RT, for simple perfect gases) (3) Thermokinetic functions of state (such as the Fourier heat conduction equation = -k(T,p)VT) and (4) The auxiliary conditions (i.e., boundary and/or initial conditions). The balances are pertinent to all problems covered by the theory, although their formulation may differ from one problem to another. Any set of... 

Small-scale experiments w ere conducted with ambient temperature CO2 [ ], and an experimental run-time coefficient was determined. This coefficient is based on the one-dimensional steady-state heat conduction equation with constant material properties ... 

The temperature field in the mold is governed by the three-dimensional transient heat conduction equation with constant properties. The equation is... 

Another interesting property of Eq. (4.233) is that it is identical in form, mathematically, to the integral solution of the heat-conduction equation. Thus the reaction-rate function R v Bn) must also satisfy the differential equation... 

For the design and analysis of fixed-bed catalytic reactors as well as the determination of catalyst efficiency under nonisothermal conditions, the effective thermal conductivity of the porous pellet must be known. A collection of thermal conductivity data of solids published by the Thermophysical Properties Research Centre at Purdue University [ ] shows "a disparity in data probably greater than that of any other physical property ". Some of these differences naturally can be explained, as no two samples of solids, especially porous catalysts, can be made completely identical. However, the main reason is that the assumed boundary conditions for the Fourier heat conduction equation... 

Green s identities for a 2D Laplace s equation (heat conduction) Here, we will demonstrate how to develop Green s identities for a two-dimensional heat conduction problem, which for a material with constant properties is described by the Laplace equation for the temperature, i.e.,... 

One of the simplified heat transfer models of two-phase flows is the pseudocontinuum one-phase flow model, in which it is assumed that (1) local thermal equilibrium between the two phases exists (2) particles are evenly distributed (3) flow is uniform and (4) heat conduction is dominant in the cross-stream direction. Therefore, the heat balance leads to a single-phase energy equation which is based on effective gas-solid properties and averaged temperatures and velocities. For an axisymmetric flow heated by a cylindrical heating surface at rw, the heat balance equation can be written as... 

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential 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. 

It is correct to state that according to the Theory of Gases energy can be expressed as temperature. However, this is advantageous and reasonable only if the physical process is governed by molecular events. For macroscopic interrelations like the Boussinesq s problem, the molecular nature of the gas is irrelevant. Here, the microscopic parameters are replaced by mean values of the macroscopic ones, these appearing in measurable physical properties such as specific heat and heat conductivity. To equate energy with temperature as Riabouchinsky did, introduced irrelevant physics to the problem. See also the remarks of L.I. Sedov [48, p. 40+]. 

For one-dimensional, time dependent heat conduction with constant properties, we can select a slab for convenience of illustration. The typical problem is governed by the energy equation ... 

The kinetic equations serve as a bridge between the microscopic domain and the behavior of macroscopic irreversible processes through the description of hydrodynamics in terms of intermolecular collisions. Hydrodynamics can specify a large number of nonequilibrium states by a small number of reproducible properties such as the mass, density, velocity, and energy density of a fluid conserved during the collision of molecules. Therefore, the hydrodynamic equations can describe a wide range of relaxation processes of nonequilibrium states to equilibrium state. We call such processes decay processes represented by phenomenological equations, such as Fourier s law of heat conduction. The decay rates are determined by the transport coefficients. Reliable transport coefficients provide microscopic and macroscopic information, and validate the results of molecular dynamics. 

Equation (2.26) for heat conduction and Eq. (2.3) for momentum transfer are similar, and the flow is proportional to the negative of the gradient of a macroscopic variable the coefficient of proportionality is a physical property characteristic of the medium and dependent on the temperature and pressure. In a three-dimensional transport, Eqs. (2.27) and (2.15) differ because the heat flow is a vector with three components, and the momentum flow t is a second-order tensor with nine components. 

The equations governing the steady state, quasi-one-dimensional flow of a reacting gas with negligible transport properties can easily be obtained from equations (l-19)-(l-22). When transport by diffusion is negligible 0 and Dtj 0 for ij = 1,..., N the diffusion velocities, of course, vanish [FJ 0 for / = 1,..., N, see equation (1-14)]. If, in addition, transport by heat conduction is negligible (A 0) and = 0, then the heat flux q vanishes [see equation (1-15)]. Finally, in inviscid flow 0 and K 0), equations (1-16)-(1-18) show that all diagonal elements of the pressure tensor reduce to the hydrostatic pressure, pu = pjj — P33 = P-The steady-state forms of equations (1-20), (l-21a), and (1-22) then become... 

The basis for the solution of mass diffusion problems, which go beyond the simple case of steady-state and one-dimensional diffusion, sections 1.4.1 and 1.4.2, is the differential equation for the concentration held in a quiescent medium. It is known as the mass diffusion equation. As mass diffusion means the movement of particles, a quiescent medium may only be presumed for special cases which we will discuss first in the following sections. In a similar way to the heat conduction in section 2.1, we will discuss the derivation of the mass diffusion equation in general terms in which the concentration dependence of the material properties and chemical reactions will be considered. This will show that a large number of mass diffusion problems can be described by differential equations and boundary conditions, just like in heat conduction. Therefore, we do not need to solve many new mass diffusion problems, we can merely transfer the results from heat conduction to the analogue mass diffusion problem. This means that mass diffusion problem solutions can be illustrated in a short section. At the end of the section a more detailed discussion of steady-state and transient mass diffusion with chemical reactions is included. 

Derive the differential equation for the temperature field = (r, t), that appears in a cylinder in transient, geometric one-dimensional heat conduction in the radial direction. Start with the energy balance for a hollow cylinder of internal radius r and thickness Ar and execute this to the limit Ar — 0. The material properties A and c depend on internal heat sources are not present. 

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