Mass transfer fluxes

The solute concentrations very close to the interface, and are assumed to be in equiUbrium, in the absence of any slow interfacial reaction. According to the linear distribution law, Cg. = thus from equation 14 the mass-transfer flux can be expressed in terms of an overall... 

From the basic mass-transfer flux relationship for species A (Sec. 5),... 

J mass-transfer flux relative to molar average velocity, mol/(m -s) ... 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

The special case of a dry atmosphere, under the one-dimensional steady assumptions with a low mass transfer flux, can be written as (from Equations (6.29) and (6.30))... 

Mass-transfer flux, 10 752 Mass-transfer mechanisms, 9 102, 103, 109-110... 

If there is a difference in concentration of species A between two locations 1 and 2 in a flowing fluid, the mass transfer flux Ja of species A is given by the expression... 

We included the term r = 0 to indicate that there is no reaction in the gas phase. The mass transfer rates obviously have opposite signs, and we have to multiply the mass transfer flux by [areaA olume], where the volume is that occupied by that phase. Note that the mass transfer term after dividing out becomes proportional to R. Since the reactor volume is proportional to R while the surface area for mass transfer is proportional to R, the falling film column obviously becomes less efficient for larger reactor sizes. This is a fundamental problem with the falling film reactor in that small tubes give high mass transfer rates but low total production of product. 

Thus, the mass transfer flux of solute in feed, hquid membrane, and stripping phases can be expressed by the following equations ... 

Figure 8.1 shows, in graphical terms, the concentration gradients of a diffusing solute in the close vicinity and inside of the dialyzer membrane. As discussed in Chapter 6, the sharp concentration gradients in liquids close to the surfaces of the membrane are caused by the hquid film resistances. The solute concentration within the membrane depends on the solubility of the solute in the membrane, or in the liquid in the minute pores of the membrane. The overall mass transfer flux of the solute J(kmol h m" ) is given as... 

For a flat membrane, the mass transfer fluxes through the two hquid films on the membrane surfaces and through the membrane should be equal to... 

The mass transfer flux N out of one phase is the product of the film coefficient and the concentration gradient in the film, and is equal to the flux into the second phase ... 

Closely related to the diffusion layer term is the mass transfer coefficient m,. In a general way, this coefficient is the proportionality constant between the mass transfer flux and the concentration difference between the electrode surface and the bulk of the solution. From the current expression given by Eq. (1.181), one can write... 

SimpleBox is a multimedia mass balance model of the so-called Mackay type. It represents the environment as a series of well-mixed boxes of air, water, sediment, soil, and vegetation (compartments). Calculations start with user-specified emission fluxes into the compartments. Intermedia mass transfer fluxes and degradation fluxes are calculated by the model on the basis of user-specified mass transfer coefficients and degradation rate constants. The model performs a simultaneous mass balance calculation for all the compartments, and produces steady-state concentrations in... 

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... 

Fig. 9.25. Mass transfer fluxes of SO2 and CO2 in the flue gas along the column (adapted from Ref. [70]).

The removal of the acid components H2S and CO2 from gases by means of alkanolamine solutions is a well-established process. The description of the H2S and CO2 mass transfer fluxes in this process, however, is very complicated due to reversible and, moreover, interactive liquid-phase reactions hence the relevant penetration model based equations cannot be solved analytically [6], Recently we, therefore, developed a numerical technique in order to calculate H2S and CO2 mass transfer rates from the model equations [6]. 

If the amine depletion in the penetration zone in a simultaneous absorption situation is negligible, the mass transfer fluxes are independent of each other and the respective enhancement factors may be obtained easily from analytical solutions of single gas mass transfer models. 

We define linear stationary system with regular fluxes as the nonequilibrium system in which (1) the values of measured parameters do not change with time and (2) parameters of all interior heat/mass transfer fluxes (their values and directions) do not change with time. 

The first feature concerns the structure of the terms in Eq. (A.33). Each term can be viewed as the product between a generalized (driving) force Xk and a generalized flux Jk. The first term in Eq. (A.33) has the temperature gradient as a force and heat transfer rate as a flux. The second term has a composition gradient and a mass transfer flux. The third term has affinity as a force (indicative of the distance away from chemical equilibrium) and reaction rate as the flux. The fourth term is already a composite related to pressure drop and fluid flow. Equation (A.33) can therefore be written compactly as... 

In the equations given in Table 5.2 the surface temperature is not known. The surface temperature is obtained by making the heat and mass transfer fluxes equal as given in equation (5.32). This is equivalent to equating the expressions for the values of t for heat and mass transfer... 

At steady state, both the mass transfer flux and the heat transfer flux are balanced according to... 

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