Heat conduction process

The differential equations of fluid dynamics express conservation of mass, conse rvation of momentum, conservation of energy and an equation of state. For an adiabatic reversible process, viscosity and heat conduction processes are absent and the equations are 2.1.1 to 2.1.13, inclusive... 

VI. Veinik, A. L., Approximate Calculation of Heat Conduction Processes. Gose-nergoizdat, Moscow, USSR, 1959. (In Russian.)... 

So, the reaction rate is uniquely defined by the corresponding affinity, since J >eq becomes constant due to uniform concentration when a system is in the vicinity of global equilibrium with fast diffusion and heat conduction processes. Comparing Eq. (9.120) with Eq. (9.124), the coefficient L is defined by... 

Reconsider the brick wall already discussed. The temperature at any point on the wall at a specified time also depends on the condition of the wall at the beginning of the heat conduction process. Such a condition, which is usually specified at time t = 0, is called the initial condition, which is a mathematical expression for the temperature distribution of the medium initially. Note that we need only one initial condition for a heat conduction problem regardless of the dimension since the conduction equation is first order in time (it involves the first derivative of temperature with respect to time). 

The three bodies — plate, very long cylinder and sphere — shall have a constant initial temperature d0 at time t = 0. For t > 0 the surface of the body is brought into contact with a fluid whose temperature ds d0 is constant with time. Heat is then transferred between the body and the fluid. If s < i90, the body is cooled and if i9s > -i90 it is heated. This transient heat conduction process runs until the body assumes the temperature i9s of the fluid. This is the steady end-state. The heat transfer coefficient a is assumed to be equal on both sides of the plate, and for the cylinder or sphere it is constant over the whole of the surface in contact with the fluid. It is independent of time for all three cases. If only half of the plate is considered, the heat conduction problem corresponds to the unidirectional heating or cooling of a plate whose other surface is insulated (adiabatic). 

The equations, (2.171) for the temperature distribution in the plate as well as (2.173) and (2.174) for the released heat have been repeatedly evaluated and illustrated in diagrams, cf. [2.34] and [2.35]. In view of the computing technology available today the direct evaluation of the relationships given above is advantageous, particularly in simulation programs in which these transient heat conduction processes appear. The applications of the relationships developed here is made easier, as for large values of t+ only the first term of the infinite series is required, cf. section 2.3.4.5. Special equations for very small t+ will be derived in section 2.3.4.6. In addition to these there are also approximation equations which are numerically more simple than the relationships derived here, see [2.74]. 

The local boundary conditions can be split into three groups, just as in heat conduction processes. 

We have already considered steady-state one-dimensional diffusion in the introductory sections 1.4.1 and 1.4.2. Chemical reactions were excluded from these discussions. We now want to consider the effect of chemical reactions, firstly the reactions that occur in a catalytic reactor. These are heterogeneous reactions, which we understand to be reactions at the contact area between a reacting medium and the catalyst. It takes place at the surface, and can therefore be formulated as a boundary condition for a mass transfer problem. In contrast homogeneous reactions take place inside the medium. Inside each volume element, depending on the temperature, composition and pressure, new chemical compounds are generated from those already present. Each volume element can therefore be seen to be a source for the production of material, corresponding to a heat source in heat conduction processes. 

Initial Region. From a different perspective, with no methane flowing in a long tube the reactive potential of the nitrogen jet is seen to persist to some appreciable extent almost indefinitely, but decays as the heat flow decays, presumably through heat conduction processes. As the methane flow is introduced, some of this potential to react (PR for brevity) is now used up by the methane, while the balance decays by the heat conduction process. This gives rise to the initial region. ... 

Plateau Region. Because mixing is not instantaneous at the point of mixing, some of the PR is still used up by heat conduction processes at the equivalence point. As the methane flow is further increased, the equivalent amount mixes closer and closer to the slot so that less of the residual PR is lost to heat conduction. This latter process probably gives rise to the observed residual slope in the plateau region of the titration curves. 

Surface wettability is a significant factor in determining the physical and chemical properties of materials. Superhydrophobic surfaces have found application in a variety of settings, including self-cleaning surfaces, prevention of snow sticking, oxidation and heat conduction processes and others [1-3], Considering the... 

Consider one-dimensional heat conduction in a large slab of width L with constant temperatures Tj and T2 at surfaces x = 0 and x = L and internal heat generation, q, as shown in Figure 22.6. Assume constant thermal conductivity and heat conduction process is steady. The conduction equation is... 

A body is called thin if one (or more) of its characteristic dimensions is much smaller than the others. A thin rod and a thin plate are examples of such bodies. We consider a boundary value problem describing a heat conduction process in a thin rod, where the ratio e of the thickness of the rod to its length is a small parameter. To simplify the presentation, we consider the problem for a planar rod, that is, in the thin rectangle (OsjfSl) X (0[Pg.166]

Equation (1.1), which describes the heat conductivity process, is a partial differential equation of the parabolic type. When the coefficients of this equation do not depend on time, the solution can be independent of time as well. In this case, the term with the partial derivative with respect to time is absent from the equation. Such a differential equation is an equation of the elliptic type. 

Advanced learners should also study transient heat conduction processes... 

Yu, Q. H. (2006). Study on the heat conduction process of roadbeds in permafrost region and new control methods. Ph. D. thesis. Cold Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences [in Chinese with English abstract]. 

Thermal conduction in the solid phase is a key factor, as already mentioned in section 1.2.4. The heat conduction process is accounted for by Fourier s law in the heat balance equation which is thus a second order partial differential equation. An efficient numerical technique is required to avoid "numerical conduction" because the solid temperature gradient is very sharp at the light-off point (see section 3.1). There is no study of Ais numerical problem in the literature. However, Eigenberger (1972) studied the consequences of heat conduction on steady-state multiplicity. He showed that the conduction process is responsible for a reduction of the number of steady state solutions. In the example studied by Eigenberger, the steady-state solution is close to the "highest steady state" (i.e., steady state with the temperature maximum close to reactor inlet) without conduction because "the temperature maximum moves to the front of the reactor, driven by the backward conduction of heat". 

---