Mass transfer flux expressions

The diffusive MTC. The mass-transfer flux expression common throughout this book is used here to describe the chemodynamics transport across a bed-sediment layer of... 

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... 

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 ... 

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... 

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... 

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

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... 

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... 

Where Ti is the dried layer temperature, T2 is the frozen layer temperature, is the mass transfer flux of the water vapour, Cj is the bound water and H is the sublimation interface. The different parameters of the model are presented in [12], In this work, we use a simplified equation to describe the dynamic of the mass flux based on the diffusion equations of Evans. The equation is given by the following expression ... 

To calculate [A ], which is the value needed to compute mass transfer fluxes, we may avoid computing [Sh]. Equations 8.4.23 or 8.4.30 and 8.4.31 may be used directly as written to compute the multicomponent mass transfer coefficients with the eigenvalues of [A ] computed from the appropriate expression in Section 9.2 as described above. 

That is, the net rate of mass generation in phase k is given by the sum of the component mass transfer fluxes for binary mixtures using Fick s law. For a two-phase system the interfacial mass jump balance (3.143) is expressed as Fvg + Fyi = 0. 

The driving force is usually defined as a concentration difference between the interface and the bulk phase. For gases, however, it is more common to use the pressure difference as the driving force. We can thus formulate two different definitions of the mass transfer flux, and then derive an expression for the relation between the concentration based and the pressure based mass transfer coefficients. 

Although the definitions of mass transfer coefficients expressed in Eq, (2,4-1) are must commonly used, an alternative definition originally employed by Colburn and Drew1 is useful ueder conditions of large convective flow in the direction of transport. The flux across a transfer surface at position I is given as... 

Equation (2,4-3) is applicable regardless of the magnitude of the flux so it may he employed to demonstrate the effect of high mass flux. For any ralio of NJNB as NA + NB - 0, the mass transfer coefficients kr and k approach DjJCth. If D CIl is desigaaled as fc°. the coefficient under low transfer tales, an expression for the effect of total mass transfer flux on the ratio A can be derived,... 

In its simplest form, the penetration ihcoty assumes that a fluid of initial composition is brought into contact with en juteiface at a fixed composition xA, for a time i. For short contact rimes the composition far from the interface (j - ) remains at Jtj. If bulk How is neglected (dilute solution or low transfer runs), solution of the unsteady-stale diffusion equation provides an expression for the average mass transfer flux and coefficient for a contact time 6. 

When we deal with situations which do not involve either diffusion of only one substance or equimolar counterdiffusion, or if mass-transfer rates are large, F-type coefficients should be used. The general approach is the same, although the resulting expressions are more cumbersome than those developed above. Thus, in a situation like that shown in Figures 3.3 to 3.5, the mass-transfer flux is... 

The mass-transfer performance, expressed as a CO2 flux through the membrane, is shown in Figure 4.14 for the... 

The mass transfer flux, j, in the first extraction period, is controlled by the convection mass transfer and depends on solute concentration in the solid phase. It is expressed in Equation 5.6, whereas the flux inside the particles, which depends on the solute diffusion from the interior of the solid to the surface, is expressed by Equation 5.7 ... 

The fllm theory is the simplest model for interfacial mass transfer. In this case it is assumed that a stagnant fllm exists near the interface and that all resistance to the mass transfer resides in this fllm. The concentration differences occur in this film region only, whereas the rest of the bulk phase is perfectly mixed. The concentration at the depth I from the interface is equal to the bulk concentration. The mass transfer flux is thus assumed to be caused by molecular diffusion through a stagnant fllm essentially in the direction normal to the interface. It is further assumed that the interface has reached a state of thermodynamic equilibrium. The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. Consider a gas-liquid interface, as sketched in Fig. 5.16. The mathematical problem is to formulate and solve the diffusion flux equations determining the fluxes on both sides of the interface within the two films. The resulting concentration profiles and flux equations can be expressed as ... 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

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