Stefan flux

To deal with this more complex problem, we follow Sekerka et al. [2] and Sekerka and Wang [3] and first establish a general analysis that allows for these changes of volume. The previous melting problem was solved by first obtaining independent solutions to the diffusion equation in each phase and then coupling them via the Stefan flux condition at the interface, similar approach can be employed for the present problem. To accomplish this, it is necessary to identify suitable frames for analyzing the diffusion in each phase and then to find the relations between them necessary to construct the Stefan condition. 

Gas phase mass transfer fluxes (Stefan flux in the gas phase is negligible as long as the vapor phase mole fractions are below say 20 %, which means either moderate temperatures and/or high sweep gas flow rates) are ... 

Liquid phase mass transfer fluxes (Stefan flux not negligible) fulfil the relationship... 

Figure 4.24 shows the reactive arheotrope trajectories according to Eq. (83) for various amounts of the liquid phase mass transfer resistance - that is, for various values of Kiiq and a low sweep gas flow rate G (at large NTt/ -values). As a result, the reactive arheotropic composition X, 02 is shifted to larger values as the liquid phase mass transfer resistance becomes more important - that is, as the value of Kuq decreases. Note that the interface liquid concentrations are in equilibrium with the vapor phase bulk concentrations. Therefore, gas phase mass transfer resistances cannot have any influence on the position of the reactive arheotrope compositions. On the other hand, liquid phase mass transfer resistances do have an effect, though the value of all binary hiq have been set equal. Again, this effect results from the competition between the diffusion fluxes and the Stefan flux in the liquid phase. 

Eventually, there is an essential difference between batch distillation processes and column distillation processes. Whilst in most adiabatic column distillation processes Stefan fluxes are usually very small (unless inert gases are present, as in hydrogenation or oxidation processes), the batch distillation process just lives from... 

The constitutive Maxwell-Stefan flux equations are the same as those presented in the last section (Section 8.3) because in this case to maintain the constant pressure of the closed system the sum of all fluxes must be zero, the same requirement as that in the Loschmidt tube. The flux equations are given by eqs. (8.3-7) with the matrix B given by eq. (8.3-6). 

As discussed in Section 8.5, the correct driving force for mass transfer in nonisothermal, non-isobaric conditions is the partial pressure gradient. Written in terms of these gradients (Appendix 8.5) the constitutive Maxwell-Stefan flux equations are ... 

The difference in the flow rate as calculated by Eqs. (1-149) and (1-152), is the factor c/(c — c,), which is due to the additional superseding one-directional diffusion ( Stefan flux ). The amount of mass flux transferred by diffusion is therefore larger... 

Multicomponent Mass Diffusion Flux Models C.9 The Mass Based Maxwell-Stefan Flux Model... 

Effective mass based diffusion coefftcient in the explicit expression for the Maxwell-Stefan flux (m /s)... 

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. 

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 ... 

Stefan-Maxwell Equations Following Eq. (5-182), a simple and intuitively appeahng flux equation for apphcations involving N components is... 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

Equation 10.30 is known as Stefan s Law(3). Thus the bulk flow enhances the mass transfer rate by a factor Cj/Cjj, known as the drift factor. The fluxes of the components are given in Table 10.1. 

The solar flux can be calculated via Stefan s law from the observed surface temperature of the Sun, and the level of radiation at a known distance is calculated via the inverse square law (Figure 7.6). 

The measurement of an enthalpy change is based either on the law of conservation of energy or on the Newton and Stefan-Boltzmann laws for the rate of heat transfer. In the latter case, the heat flow between a sample and a heat sink maintained at isothermal conditions is measured. Most of these isoperibol heat flux calorimeters are of the twin type with two sample chambers, each surrounded by a thermopile linking it to a constant temperature metal block or another type of heat reservoir. A reaction is initiated in one sample chamber after obtaining a stable stationary state defining the baseline from the thermopiles. The other sample chamber acts as a reference. As the reaction proceeds, the thermopile measures the temperature difference between the sample chamber and the reference cell. The rate of heat flow between the calorimeter and its surroundings is proportional to the temperature difference between the sample and the heat sink and the total heat effect is proportional to the integrated area under the calorimetric peak. A calibration is thus... 

Since interaction phenomena due to simultaneous diffusion of several components play an important role, the Maxwell-Stefan theory has been selected to describe the mass transfer processes. The general form of the flux expressions can be represented by (Taylor and Krishna, 1993)... 

III) Mass transfer is described by a modified Maxwell-Stefan theory by assuming n.4 l and dAgnA l (the absorption flux is sufficiently small). In this case eq. (22) can be reduced to the following explicit expression for the dimensionless absorption flux by linearization of the exponential terms (see Appendix A) ... 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

DISCUSSION AND CONCLUSIONS In this study a general applicable model has been developed which can predict mass and heat transfer fluxes through a vapour/gas-liquid interface in case a chemical reaction occurs in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. A film model has been adopted which postulates the existence of a well-mixed bulk and stagnant zones where the principal mass and heat transfer resistances are situated. Due to the mathematical complexity the equations have been solved numerically by a finite-difference technique. In this paper (Part I) the Maxwell-Stefan theory has been compared with the classical theory due to Pick for isothermal absorption of a pure gas A in a solvent containing component B. Component A is allowed to react by a unimolecular chemical reaction or by a bimolecular chemical reaction with... 

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