Stefan

J. Rant -J. Stefan Inst.. J. Stade - BAM, Germany. M. Balasko - KFKI, Hungary. M. Kaling, FUJI, Germany... 

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. 

Boltzmaim showed that the energy density emided per second from a unit surface of a black body is a7 where T is the temperature and a is the Stefan-Boltzmaim constant, but it takes statistical mechanics to produce the fonnula... 

Arnold Neumaier, Stefan Dallwig, Waltraud Huyer, and Hermann Schichl ... 

Siegmar Braun, Hans-Otto Kalinowski, Stefan Berger... 

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... 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

Equations (5.11), of which only n-1 are independent, may be regarded as a generalization of the Stefan-Maxwel1 relations. 

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 ... 

Spectral bandwidth of excita- Stefan-Boltzmann constant a... 

Stefan s law states that the total energy / radiated by a blackbody per unit time and area (power per unit area) varies as the fourth power of the absolute temperature ... 

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,... 

The total energy radiating from a blackbody of unit area is given by the Stefan-Boltzman 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]. 

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