Diffusion similarity 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. 

At first sight mathematically these problems are not completely new similar ones have been met in different fields and one can count one s weapons before attacking the formulation and add the necessary complements examples of diffusion of heat or matter (diffusion-absorptions) (diffusion-reactions) [51], heat transfer with reactions c r propagation of vivid heat ( chaleur vive ), diffusion of heat through media including... 

Specimens of NR ABS/(Octa -I- AO) heat-treated at 350°-400°C developed brittleness of connected pores, whereas VO ABS (Octa -1- AO -I- EPDM), similarly treated, was tougher with large elongated pores about twice the size of the non-treated specimen. Such behavior suggests an intumescent effect of EPDM, i.e. the development of a thick porous surface layer, inhibiting the diffusion of flammable products of plastic degradation towards the gas phase and heat transfer into the plactic mass. 

In this section, we focus on diffusive mass transfer. The mathematical description of mass transfer is similar to that of heat transfer. Furthermore, heat transfer may also play a role in heterogeneous reactions such as crystal growth and melting. Heat transfer, therefore, will be discussed together with mass transfer and examples may be taken from either mass transfer or heat transfer. 

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. 

While our primary interest in this text is internal flow, there are certain similarities with the classic aerodynamics-motivated external flows. Broadly speaking, the stagnation flows discussed in Chapter 6 are classified as boundary layers where the outer flow that establishes the stagnation flow has a principal flow direction that is normal to the solid surface. Outside the boundary layer, there is typically an outer region in which viscous effects are negligible. Even in confined flows (e.g., a stagnation-flow chemical-vapor-deposition reactor), it is the existence of an inviscid outer region that is responsible for some of the relatively simple correlations of diffusive behavior in the boundary layer, like heat and mass transfer to the deposition surface. 

The volume change in these gels is not due to ionic effects, but rather to a thermodynamic phenomenon a lower critical solution temperature (LCST). The uncrosslinked polymer which makes up the gel is completely miscible with water below the LCST above the LCST, water-rich and polymer-rich phases are formed. Similarly, the gel swells to the limit of its crosslinks below the LCST, and collapses above the LCST to form a dense polymer-rich phase. Hence, the kinetics of swelling and collapse are determined mostly by the rate of water diffusion in the gel, but also by the heat transfer rate to the gel. 

The convective diffusion equation is analogous to equations commonly used in dealing with heat and mass transfer. Similarly, if migration can be neglected in a multicomponent solution, then the convective diffusion equation can be applied to each species,... 

Earlier studies of intracrystalline diffusion in zeolites were carried out almost exclusively by direct measurement of sorption rates but the limitations imposed by the intrusion of heat transfer and extra-crystalline mass transfer resistances were not always fully recognized. As a result the reported diffu-sivities showed many obvious inconsistencies such as differences in diffusivity between adsorption and desorption measurements(l-3), diffusivities which vary with fractional uptake (4) and large discrepancies between the values measured in different laboratories for apparently similar systems. More recently other experimental techniques have been applied, including chromatography and NMR methods. The latter have proved especially useful and have allowed the microdynamic behaviour of a number of important systems to be elucidated in considerable detail. In this paper the advantages and limitations of some of the common experimental techniques are considered and the results of studies of diffusion in A, X and Y zeolites, which have been the subject of several detailed investigations, are briefly reviewed. 

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. 

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

SEM microphotogaphs and EDAX scans of the cross section and outer surface of the slag deposit, illustrated in Figure 10, indicate the chemistry of the deposit is not uniform. The bulk of the fused material is rich in silica, low in iron, and virtually depleted of potassium. The outermost layers, no more than 2 to 3p thick, are very rich in iron and frequently also rich in calcium. On occasion, the outer surface is covered with small particulate, several microns in diameter, or undissolved cubic or octahedral crystals whose origin is pyrites. Similar formations have been observed in full-scale operation. The evidence indicates deposits form under axial symmetric flow conditions in the furnace by the fluxing action at the heat transfer surface of small particles, <8p in diameter, of decidedly different chemical composition and mineral source. Migration of the fly ash to the surface is by means of eddy diffusion, thermophoresis, or Brownian motion. 

In many catalytic systems multiple reactions occur, so that selectivity becomes important. In Sec. 2-10 point and overall selectivities were evaluated for homogeneous well-mixed systems of parallel and consecutive reactions. In Sec. 10-5 we saw that external diffusion and heat-transfer resistances affect the selectivity. Here we shall examiineHEieHnfiuence of intrapellet res ahces on selectivity. Systems with first-order kinetics at isothermal conditions are analyzed analytically in Sec. 11-12 for parallel and consecutive reactions. Results for other kinetics, or for nonisothermal conditions, can be developed in a similar way but require numerical solution. ... 

FLOW INSIDE PIPES. Correlations for mass transfer to the inside wall of a pipe are of the same form as those for heat transfer, since the basic equations for diffusion and conduction are similar. For laminar flow, the Sherwood number shows the same trends as the Nusselt number, with a limiting value of 3.66 for very long tubes and a one-third-power dependence on flow rate for short tubes. The solution shown in Figure 12.2 can be used for Agh if the Graetz number is based on the dilfusivity or on the Schmidt number as follows ... 

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