Mass flux transfer relationships

Here Jv is the volumetric flow rate of fluid per unit surface area (the volume flux), and Js is the mass flux for a dissolved solute of interest. The driving forces for mass transfer are expressed in terms of the pressure gradient (AP) and the osmotic pressure gradient (All). The osmotic pressure (n) is related to the concentration of dissolved solutes (c) for dilute ideal solutions, this relationship is given by... 

The results demonstrate that for liquid mass fluxes, which do not lead into the direct proximity of the state of saturation of air, a stoichiometric relationship does not play an unimportant role. Rather, the mass transfer area of the fluidized bed is the limiting factor of the absorption. Over-stoichiometric operation does not lead to any improved separation of the SO2 (Figs. 16.14 and 16.15), the reason being the permanent destruction of the liquid film by particle-particle collisions and the consequent production of inactive reactants. 

Manipulation of Equations (26.53)-(26.56) leads to a number of important mass transfer relationships. When the molar flux, current density, and electroneutrality equations are combined... 

Jensen and Tang also give relationships for in-line bundles. Application of Eqs. 15.195-15.202 requires a knowledge of local quality within the bundle. If the mass flux is known, then this can be obtained very simply from a heat balance, but if the mass flux is unknown and has to be calculated (as in kettle reboilers), then recourse must be had to methodologies of the type described by Brisbane et al. [213] and Whalley and Butterworth [214]. As in the case of heat transfer coefficient, simple methods have been developed for prediction of critical heat flux in tube bundles by Palen and coworkers (Palen [215], Palen and Small [224]), who relate the single tube critical heat flux (calculated by the correlations given previously on p. 15.63-15.65) by a simple bundle correction factor O/, as follows ... 

If the concentration of gas in water (Q ) is higher than CJK, then the movement of gas from water to atmosphere occurs. The flux of gas can be described using a first-order mass transfer relationship as follows ... 

Mass transfer properties of species are affected by various parameters. These are dependent on the membrane characteristics and the species. Simon et al. [115] explained theoretically how the coupling of various parameters affects the effective separation efficiency of membranes. Mass transport was found to be strongly affected by these coupling effects that involve coupling between flows of one character and forces of another as well as coupling between the flows of various solutes. These effects can affect significantly the efficiency of the membrane separation. The relationships among the mass fluxes, forces, and their interdependency have also... 

All three models propose a linear relationship between the vapour flux and the vapour pressure difference, so Kunz et al. (1996) proposed a general expression for describing the mass flux. This is related through the overall mass transfer coefficient, which accounts for all three resistances (feed, membrane and stripping) of water vapour transport to the driving force. 

Another analogous relationship is that of mass transfer, represented by Pick s law of diffusion for mass flux, J, of a dilute component, I, into a second fluid, 2, which is proportional to the gradient of its mass concentration, mi. Thus we have, J = p Du Vmt, where the constant Z)/2 is the binary diffusion coefficient and p is density. By using similar solutions we can find generalized descriptions of diffusion of electrons, homogeneous illumination, laminar flow of a liquid along a spherical body (assuming a low-viscosity, non-compressible and turbulent-lree fluid) or even viscous flow applied to the surface tension of a plane membrane. 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

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

The Fourier law gives the rate at which heat is transferred by conduction through a substance without mass transfer. This states that the heat flow rate per unit area, or heat flux, is proportional to the temperature gradient in the direction of heat flow. The relationship between heat flux and temperature gradient is characterized by the thermal conductivity which is a property of the substance. It is temperature dependent and is determined experimentally. 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

If external diffusion dominates the overall rate, the process obviously reduces the observed enzyme activity. The flux N through the stagnating film at the surface can be expressed as in Eq. (5.54), where 8 signifies the thickness of the stagnating layer and ks is the mass transfer coefficient of the respective solute ks can be estimated by the simple relationship of Eq. (5.55). 

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

Chemical equilibrium constant for dimerization Liquid-liquid distribution ratio Liquid flow rate Number of equilibrium stages Number of relationships Number of design variables Minimum number of equilibrium stages Number of phases Number of repetition variables Number of variables Rate of mass transfer Molar flux... 

A comprehensive difference model was developed by Madireddi et al. [71] to predict membrane fouling in commercial spiral-wound membranes with various spacers. This is a useful paper for experimental studies on the effect of flow channel thickness on flux and fouling. Avlonitis et al. [72] presented an analytical solution for the performance of spiral-wound modules with seawater as the feed. In a key finding they showed that it was necessary to incorporate the concentration and pressure of the feed into the correlation for the mass transfer coefficient. In a similar study, Boudinar et al. [73] developed the following relationship for calculating mass transfer coefficients in channels equipped with a spacer ... 

This leads to the fact that the mass transfer coefficient, and thus the permeate flux, is related to velocity according to the relationship ... 

Let us now discuss more precisely the relationship between current and mass transfer at the electrode surface. From Eq. (134) it is seen that the flux of electroactive species is given, in the absence of migration, by Eq. (140), at the electrode surface (x = 0). 

Mass-Transfer Units The mass-transfer unit concept follows directly from mass-transfer coefficients. The choice of one or the other as a basis for analyzing a given application often is one of preference. Colburn [Ind. Eng. Chem., 33(4), pp. 450-467 (1941)] provides an early review of the relationship between the height of a transfer unit and volumetric mass-transfer coefficients (k a). From a differential material balance and application of the flux equations, the required contacting height of an extraction column is related to the height of a transfer unit and the number of transfer units... 

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