Thies, Stefan

For oxidation of solid carbon to produce CO (either by 02 or C02), two moles of gaseous combustion products are produced per mole of gaseous reactant, resulting in a net gas flow away from the particle surface. This Stefan flow reduces the rates of heat and, especially, mass transfer from the boundary layer to the particle surface (akin to drag reduction on a flat surface with... 

To illustrate the solution procedure of this Stefan-Maxwell approach, we apply it to the experimental results of Duncan and Toor (1962). The two well-mixed bulbs used have volumes of 7.8 x 10 and 7.86 x 10 m, respectively, and the capillary tube has a length and a diameter of 0.086 m and 0.00208 m, respectively. The operating conditions are 35 and 101.3 kPa. Duncan and Toor (1962) used a ternary system of hydrogen, nitrogen and carbon dioxide in their experiment. We use the following numerical values to denote the three components used in their experiment 1-hydrogen 2-nitrogen and 3-carbon dioxide. 

This is known as the Stefan-Boltzmaim law of radiation. If in this calculation of total energy U one uses the classical equipartition result = k T, one encounters the integral f da 03 which is infinite. This divergence, which is the Rayleigh-Jeans result, was one of the historical results which collectively led to the inevitability of a quantum hypothesis. This divergence is also the cause of the infinite emissivity prediction for a black body according to classical mechanics. 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

There are n Stefan-Maxwell relations in an n-component mixture, but they are not independent since each side of (2.16) yields zero on summing over r from 1 to n. Physically this is not surprising, since they describe only momentum exchange between pairs of species, and say nothing about the total momentum of the mixture. In order to complete the determination of the fluxes N.... N the Stefan-Maxwell relations must be supple-I n... 

This is an explicit solution of the Stefan-Maxwell equations for the diffusion fluxes. The species flux vectors are then given by... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

Thermal Emission Laws. AH bodies emit infrared radiation by virtue of their temperature. The total amount of radiation is governed by Kirchhoff s law, which states that a body at thermal equiUbrium, ie, at the same temperature as its surroundings, must emit as much radiation as it absorbs at each wavelength. An absolutely blackbody, one that absorbs all radiation striking it, must therefore emit the most radiation possible for a body at a given temperature. The emission of this so-called blackbody is used as the standard against which all emission measurements are compared. The total radiant emittance, M., for a blackbody at temperature Tis given by the Stefan-Boltzmaim law,... 

Conduction with Change of Phase A special type of transient problem (the Stefan problem) involves conduction of heat in a material when freezing or melting occurs. The liquid-solid interface moves with time, and in addition to conduction, latent heat is either generated or absorbed at the interface. Various problems of this type are discussed by Bankoff [in Drew et al. (eds.). Advances in Chemical Engineering, vol. 5, Academic, New York, 1964]. 

The other mechanism appears in scrubbers. When water vapor diffuses from a gas stream to a cold surface and condenses, there is a net hydrodynamic flow of the noncondensable gas directed toward the surface. This flow, termed the Stefan flow, carries aerosol particles to the condensing surface (Goldsmith and May, in Davies, Aero.sol Science, Academic, New York, 1966) and can substantially improve the performance of a scrubber. However, there is a corresponding Stefan flow directed away from a surface at which water is evaporating, and this will tend to repel aerosol particles from the surface. 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

I By Radiation (Wr) This heat loss is related to the difference of the fourth power of the absolute temperatures and the emissivity of the enclosure, and is represented by the Stefan-Boltzmann law expressed by (see Dwight ci al.. 1940)... 

Where, the diffusivity D for the transfer of one gas in another is not known and experimental determination is not practicable, it is necessary to use one of the many predictive procedures. A commonly used method due to Gilliland 6 is based on the Stefan-Maxwell hard sphere model and this takes the form ... 

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

Some Paracelsian alchemists, especially Heinrich Khun rath (ca. 1560-1605) and Stefan Michelspacher (active ca. 1615-23), were objects of persecution on the part of hoth Lutheran and Catholic authorities. Khunrath was an alchemist from Saxony, the heartland of the Reformation, but his theological stance was characteristic of the second generation of Protestants who felt that Luther s work had been left incomplete and that another religious reform was essential. In Khunrath s ideas this would take the form of a Lutheranism that could accommodate an autonomous personal piety. To express their Lutheran piety intellectually the alchemists employed the terms of Paracelsian theosophy, while they found an emotive outlet in the mystical experience of the power and grace of the Holy Spirit. They felt themselves to be inspired (literally breathed ) by the Spirit, a force that they identified with alchemical pneuma. Khunrath called himself an enthusiast, hlled with the presence of the divine. 

In this context, see also Urszula Szulakowska, The Apocalyptic Eucharist and Religious Dissidence in Stefan Michelspacher s CabalaP Aries. Journal for the Study of Western Esotericism, 3 (2003) 200—23. 

The author thanks Stefan Vogtner for numerical studies of expansions of 1/r in a Gaussian basis which have challenged the present analytic investigation. Discussions with Christoph van Wiillen and Wim Klopper on this subject have been very helpful. 

The total quantity of radiation emitted by a black body can be calculated by integrating the curves of Figure 3.19. This has been supplemented by experimental data. The result is the Stefan-Boltzmann law, which is given by... 

A qualitative interference that can be drawn from Stefan s law pertains to the effect of high absolute temperatures on the quantities of heat radiated. This aspect is of great practical importance. As the temperature of a body is raised above that of its surroundings, the amount of heat it can radiate to them increases at a phenomenal rate. 

Acknowledgements I thank Prof. Dr. Stefan Schulz for the invitation to write this review. Prof. Wilhelm Boland is gratefully acknowledged for support during the preparation of this manuscript. I thank Emily Wheeler, Dr. Martin Heil, Janine Rattke, Dr. Anne Busch, Dr. Uli Lion, Thomas Wichard, Sven Adolph, Theresa Wiesemeier, Christoph Beckmann, and Johann Pohnert for helpful discussion during the preparation of this manuscript. The Max-Planck-Society is acknowledged for funding. 

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