Diffusional heat transfer

Suppose we take a sample of bone-dry air at some temperature, Ti, and directly contact it with water until it becomes saturated at the same temperature. The water vapor that enters into the air contains with it its latent heat of vaporization. The vapor pressure of water out of the liquid will be greater than it is in the saturated air, causing vaporization to occur and subsequently increasing the humidity of the air-water-vapor mixture. The process of vaporization ends when the vapor pressure of the water in the air becomes equal to that of the liquid. At this condition the air is saturated. During the air saturation process, isothermal conditions for the water can be maintained if heat is supplied to replace the heat lost from it to the gas as latent heat of vaporization. Thus, heat transfer during the saturation of a gas with a liquid can be accomplished without a temperature differential (although this is rarely encountered). This type of heat transfer phenomenon, better known as diffusional heat transfer, is different from conduction, convection or radiation. 

Total Heat Diffusional Heat Transfer Transfer = Heat Transfer + By Per Unit Rate Convection... 

Since the temperature of the emitter is generally known (preselected or readily determined in an actual operation), the absorptivity value Cr is the unknown. This absorptivity is partly a measure of the ability of radiant heat to penetrate the body of a solid particle (or a moisture film) instantly as compared with diffusional heat transfer by conduction. Such instant penetration greatly reduces processing time and case-hardening effects. Moisture release and other mass transfer, however, still progress by diffusional means. 

Traditionally, production of metallic glasses requites rapid heat removal from the material (Fig. 2) which normally involves a combination of a cooling process that has a high heat-transfer coefficient at the interface of the Hquid and quenching medium, and a thin cross section in at least one-dimension. Besides rapid cooling, a variety of techniques are available to produce metallic glasses. Processes not dependent on rapid solidification include plastic deformation (38), mechanical alloying (7,8), and diffusional transformations (10). 

Although they are termed homogeneous, most industrial gas-phase reactions take place in contact with solids, either the vessel wall or particles as heat carriers or catalysts. With catalysts, mass diffusional resistances are present with inert solids, the only complication is with heat transfer. A few of the reactions in Table 23-1 are gas-phase type, mostly catalytic. Usually a system of industrial interest is liquefiea to take advantage of the higher rates of liquid reactions, or to utihze liquid homogeneous cat ysts, or simply to keep equipment size down. In this section, some important noncatalytic gas reactions are described. 

An immobilized enzyme-carrier complex is a special case that can employ the methodology developed for evaluation of a heterogeneous cat ytic system. The enzyme complex also has external diffusional effects, pore diffusional effects, and an effectiveness factor. When carried out in aqueous solutions, heat transfer is usually good, and it is safe to assume that isothermal conditions prevail for an immobihzed enzyme complex. 

In order for diffusional limitations to be negligible, the effectiveness factor must be close to 1, i.e. nearly complete catalyst utilization, which requires that the Thiele modulus is suffieiently small (< ca. 0.5), see Figure 3.32. Therefore, the surface-over-volume ratio must be as large as possible (particle size as small as possible) from a diffusion (and heat-transfer) point of view. There are many different catalyst shapes that have different SA/V ratios for a given size. 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

Values for the average vapor-transfer coefficient from the gas phase to the airway epithelium can also be estimated from heat-transfer data in straight, curved, or bifurcating cylindrical tubes by using the analogy between heat transfer and mass transfer. Such an approach has been used by Yeh to predict the diffusional deposition of small particles in the conducting airways. 

Metal particles, most commonly aluminum particles, are also known as additives for propellants and pyrolants that increase the combustion temperature and hence also the specific impulse. However, a heat-transfer process from the high-tempera-ture gas to the aluminum particles is required to melt the particles and then a subsequent diffusional process of oxidizer fragments toward each aluminum particle... 

The most advanced technique of quick freezing is by pouring the material onto a freezing belt. Before drying, the material is granulated or sliced to improve heat and diffusional mass transfer. These operations are conducted in cold rooms at about —46°C. 

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. 

The system is heterogeneous, external mass and heat transfer between the pellet and the bulk gas is negligible, but the intraparticle diffusional resistance is considerable. 

Models for phenomena such as heat conduction, fluid flow, and diffusional mass transfer are also based on Laplace s equation. Consequently, many solutions to the potential distribution problems or the analogous problems in other fields are available. The current distribution can be obtained from the potential distribution through Ohm s law [Eq. (22)]. 

Although multiplicities of the effectiveness factor have also been detected experimentally, these are of minor importance practically, since for industrial processes and catalysts, Prater numbers above 0.1 are less common. On the contrary, effectiveness factors above unity in real systems are frequently encountered, although the dominating part of the overall heat transfer resistance normally lies in the external boundary layer rather than inside the catalyst pellet. For mass transfer the opposite holds the dominating diffusional resistance is normally located within the pellet, whereas the interphase mass transfer most frequently plays a minor role (high space velocity). 

This methods depends on the implicit assumption that the uptake rate is controlled entirely by intracrystalline diffusion in an isothermal system, with all other resistances to either mass or heat transfer negligible. This is a valid approximation if diffusion is sufficiently slow or if the zeolite crystals are sufficiently large but the dominance of intracrystalline diffusional resistance should not be assumed without experimental verification. In many practical systems, particularly with small commercial zeolite crystals, the external heat and mass transfer resistances are in fact dominant. A detailed discussion of such effects has been given by Lee and Ruthven(5-7). 

In the chromatographic method a pulse or step change in sor-bate concentration is introduced into the carrier stream at the inlet of a packed adsorption column and the diffusional time constant is determined from the dispersion of the response signal at the column outlet. Since heat transfer in a packed bed is much faster than in a closed system the chromatographic method may, in principle, be used to follow somewhat faster sorption processes. 

Rapid equilibration limited to the transport to the surface. The only rate effects to be noted are enthalpic equilibration (exothermic heat transfer) and/or diffusional over very short distances. The exception would be for small micropores if they exist in the inner reaches of a bulk form, such as monolithic zeolites. 

While the above criteria are useful for diagnosing the effects of transport limitations on reaction rates of heterogeneous catalytic reactions, they require knowledge of many physical characteristics of the reacting system. Experimental properties like effective diffusivity in catalyst pores, heat and mass transfer coefficients at the fluid-particle interface, and the thermal conductivity of the catalyst are needed to utilize Equations (6.5.1) through (6.5.5). However, it is difficult to obtain accurate values of those critical parameters. For example, the diffusional characteristics of a catalyst may vary throughout a pellet because of the compression procedures used to form the final catalyst pellets. The accuracy of the heat transfer coefficient obtained from known correlations is also questionable because of the low flow rates and small particle sizes typically used in laboratory packed bed reactors. 

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