Solids temperature gradient

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". 

Maxwell considered the motion of a gas in the neighborhood of a plane solid wall, in che presence of a temperature gradient. In particular, when Che velocity field is one dimensional and everywhere parallel to the wall, and the temperature gradient is parallel to the velocity field, he found that... 

FIG. 5-1 Temperature gradients for steady heat conduction in series through three solids. 

FIG. 5-6 Temperature gradients for a steady flow of heat by conduction and convection from a warmer to a colder fluid separated by a solid wall. 

The definition of the heat-transfer coefficient is arbitrary, depending on whether bulk-fluid temperature, centerline temperature, or some other reference temperature is used for ti or t-. Equation (5-24) is an expression of Newtons law of cooling and incorporates all the complexities involved in the solution of Eq. (5-23). The temperature gradients in both the fluid and the adjacent solid at the fluid-solid interface may also be related to the heat-transfer coefficient ... 

In most circumstances, it can be assumed diat die gas-solid reaction proceeds more rapidly diaii die gaseous transport, and dierefore diat local equilibrium exists between die solid and gaseous components at die source and sink. This implies diat die extent and direction of die transport reaction at each end of die temperature gradient may be assessed solely from diermodynamic data, and diat die rate of uansport across die interface between die gas and die solid phases, at bodi reactant and product sites, is not rate-determining. Transport of die gaseous species between die source of atoms and die sink where deposition takes place is die rate-determining process. 

An important mixing operation involves bringing different molecular species together to obtain a chemical reaction. The components may be miscible liquids, immiscible liquids, solid particles and a liquid, a gas and a liquid, a gas and solid particles, or two gases. In some cases, temperature differences exist between an equipment surface and the bulk fluid, or between the suspended particles and the continuous phase fluid. The same mechanisms that enhance mass transfer by reducing the film thickness are used to promote heat transfer by increasing the temperature gradient in the film. These mechanisms are bulk flow, eddy diffusion, and molecular diffusion. The performance of equipment in which heat transfer occurs is expressed in terms of forced convective heat transfer coefficients. 

Conduction is the heat transfer due to spatial temperature differences (temperature gradient) without any macroscopic material movement. Conduction is important in solids and depends essentially on the materia properties (Fig. 1 1.27). 

Conduction is heat transfer through a solid nonporous barrier when a temperature difference exists across the barrier. The thermal transfer capability of the specific barrier or wall material, known as thermal conductivity, determines the temperature gradient that will exist through the material. 

The most important driving forces for the motion of ionic defects and electrons in solids are the migration in an electric field and the diffusion under the influence of a chemical potential gradient. Other forces, such as magnetic fields and temperature gradients, are commonly much less important in battery-type applications. It is assumed that the fluxes under the influence of an electric field and a concentration gradient are linearly superimposed, which... 

A temperature gradient is applied to the compound to be crystallized in which the compound can exist as a liquid or as liquid and solid. 

Fig. 3 The annual development of the seasonal thermocline (expressed as temperature gradients) (a) for a selected set of years when hypolimnetic withdrawal resulted in a deep thermocline and an extensive metalimnion, and (b) for a selected set of years when epilimnetic water flowed through the intermediate outlet inducing the development of a shallower thermocline. The solid black line shows the daily development of the withdrawal depth and the gray areas the bottom of the reservoir. Modified from Moreno-Ostos et al. [37]...

The wall boundary condition applies to a solid tube without transpiration. The centerline boundary condition assumes S5anmetry in the radial direction. It is consistent with the assumption of an axis5Tnmetric velocity profile without concentration or temperature gradients in the 0-direction. This boundary condition is by no means inevitable since gradients in the 0-direction can arise from natural convection. However, it is desirable to avoid 0-dependency since appropriate design methods are generally lacking. 

Thermal desorption of solid traps by microwave energy is unsuitable for thermally labile compounds. In microwave thermal analysis [431] the (solid) sample is heated directly via interactions of the microwaves with the sample, providing more even heating and reduction of temperature gradients in comparison to heating with electrical furnaces. By passing air over a microwave-heated volatile sample evolved gases may be collected [432]. 

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