Diffusion comparison with heat transfer

Gohrbandt s data for camphor spheres (40, 97) afford comparison of rates with diffusion controlling and with heat transfer controlling. Extrapolation to low temperatures of the heat transfer portion indicates sufficient heat transfer but inadequate diffusion. Similarly, extrapolation to high temperatures of the diffusion portion indicates sufficient diffusional driving force but inadequate heat transfer to maintain the surface temperature. 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

In accordance with the usual process conditions, the initial temperature of the reactive mixture To and the upper cap temperature Tw are constant during filling, and the temperature of the insert Ti equals the ambient temperature (20°C). The model takes into account that during filling the temperature of the insert increases due to heat transfer from the reactive mix. It is assumed that the thermal properties and density of both the reactive mass and the insert are constant. It is reasonable to neglect molecular diffusion, because the coefficient of diffusion is very small 264 therefore, the diffusion term is negligible in comparison with the other terms in the mass balance equation. 

In order to minimize external (bed) diffusion resistance and maximize the heat transfer rate it is desirable to use a very small adsorbent sample with the crystals spread as thinly as possible over the balance pan or within the containing vessel. To minimize the effect of non-linearities, such as the strong concentration dependence of the diffusivity, measurements should be made differentially over small concentration changes. Variation of the step size and comparison of adsorption and desorption curves provide simple tests for linearity of the system. The large differences between adsorption and desorption diffusivities, reported in some of the earlier work, have been shown to be due to the concentration dependence of the diffusivity(8) and in differential measurements under similar conditions no such anomaly was observed. 

Gondim, R.R., Cotta, R.M., Santos, C.A.C., and Mat, M. (2003) Internal Transient Forced Convection with Axial Diffusion Comparison of Solutions Via Integral Transforms, ICHMT International Symposium on Transient Convective Heat And Mass Transfer in Single and Two-Phase Flows, Cesme, Tmkey, August 17 - 22. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

To illustrate the effects, corrections to the diffusion coefficient data were made using an empirically chosen value for the heat transfer coefficient. When these results were converted to solvent mobilities, the extrapolation as a function of the volume fraction of solvent appeared to be consistent with the self-diffusion coefficient for toluene. However, this comparison is dependent on the method used to reduce the diffusion data. 

Eddy Diffusivity Models. The mean velocity data described in the previous section provide the bases for evaluating the eddy diffusivity for momentum (eddy viscosity) in heat transfer analyses of turbulent boundary layers. These analyses also require values of the turbulent Prandtl number for use with the eddy viscosity to define the eddy diffusivity of heat. The turbulent Prandtl number is usually treated as a constant that is determined from comparisons of predicted results with experimental heat transfer data. 

For optically thick aerogels, e.g., for most organic, opacified or carbon aerogels, radiative heat transfer is described by the diffusion of photons. The photons interact within short distances in comparison to the macroscopic dimension of the aerogel with its solid backbone. A corresponding solution to the diffusion equation for photons can be derived in analogy to the diffusion of phonons by way of ... 

We begin this chapter with a comparison of the mechanisms responsible for mass and heat transfer. The mathematical similarities suggested by these mechanisms are discussed in Section 21.1, and the physical parallels are explored in Section 21.2. The similar mechanisms of mass and heat transfer are the basis for the analysis of drying, both of solids and of sprayed suspensions. However, the detailed models differ, as shown by the examples in Section 21.3. In Section 21.4, we outline cooling-tower design as an example based on mass and heat transfer coefficients. Finally, in Section 21.5, we describe thermal diffusion and effusion. 

A comparison of the interactive film models that use the Chilton-Colburn analogy to obtain the heat and mass transfer coefficients with the turbulent eddy diffusivity models. 

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