Molar flux mass transfer

The major part of the next few chapters are devoted to methods of estimating the low flux mass transfer coefficients k and [A ] and of calculating the high flux coefficients k and [/c ]. In practical applications we will need these coefficients to calculate the diffusion fluxes 7, and the all important molar fluxes N. The are needed because it is these fluxes that appear in the material balance equations for particular processes (Chapters 12-14). Thus, even if we know (or have an estimate of) the diffusion fluxes 7 we cannot immediately calculate the molar fluxes because all n of these fluxes are independent, whereas only n — 1 of the J I are independent. We need one other piece of information if we are to calculate the N. Usually, the form of this additional relationship is dictated by the context of the particular mass transfer process. The problem of determining the knowing the 7 has been called the bootstrap problem. Here, we consider its solution by considering some particular cases of practical importance. 

The elements of the matrix of low flux mass transfer coefficients may be computed using Eqs. 10.4.25 and 10.4.30. This requires the matrix of Fick diffusion coefficients in the mass average velocity reference frame. This matrix can be computed with the help of Eqs. 4.2.2, from which [D] is obtained in the molar average velocity reference frame, and Eqs. 3.2.11, which allows transformation to the mass average reference velocity frame. Thus, we need the Maxwell-Stefan diffusivities of the three binary pairs in the vapor phase and the molar masses of the three components. 

We require the density of the vapor mixture in order to calculate the low flux mass transfer coefficients. The molar density of the vapor may be estimated using the ideal gas law and, since the system is almost isothermal, may safely be assumed to be nearly constant. The mass density, however, is likely to vary considerably between the bulk and interface, since the molar masses of the three components in the vapor phase cover such a wide range. The mass density should, therefore, be evaluated with the average molar mass... 

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

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

The difference in mole fractions is most significant in the case of S02 where this difference is 15% of the bulk phase level. This result indicates that external mass transfer limitations are indeed significant, and that this difference should be taken into account in the analysis of kinetic data from this system. Note that there is a difference in nitrogen concentration between the bulk fluid and the external surface because there is a change in the number of moles on reaction, and there is a net molar flux toward... 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

It is often convenient to express the molar flux for mass transfer in terms of a mass transfer coefficient, and in these circumstances Eq. (10) can be written as... 

Fig. 9. The molar flux of component A at the vapour-liquid interface (°) and at the boundary between mass transfer film and liquid bulk (S) as function of reaction rate constant in case (a) the mass transfer coefficients are equal and (b) the mass transfer coefficients are different.

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]. 

For mass transfer in the gas phase, the molar flux of a particular component N (in kmolm s ) is related to the concentration difference in the gas phase AC, expressed in terms of molar concentration (kmol m ), by... 

A gas component A in air is absorbed into water at latm and 20 °C. The Henry s law constant of A for this system is 1.67 X 10 Pa m kmol h The liquid film mass transfer coefficient and gas film coefficient I(q are 2.50x10 and 3.00 X10" ms respectively, (i) Determine the overall coefficient of gas-liquid mass transfer (ms ). (ii) When the bulk concentrations of A in the gas phase and liquid phase are 1.013 X 10 Pa and 2.00 kmol m , respectively, calculate the molar flux of A. 

The mass-transfer coefficients, by definition, are equal to the ratios of the molar mass flux to the concentration driving forces. The mass-transfer coefficients are related to each other as follows ... 

Mass Transfer Rate. The molar flux, NA = /cl(( cioiis — cc10h8 ), assuming the concentration of naphthalene in the bulk of the liquid phase is negligible, i.e. cc10hSl = 0, becomes... 

N0 Molar flux for mass transfer between two phases, component i. stage/ Sec. 4.2.13. 

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... 

In these equations cj and cf are the molar densities of the superscripted phases, yj is the mole fraction in the bulk vapor phase, xf is the mole fraction in the bulk liquid phase, and xj and yj are the mole fractions of species i at the phase interface. Also is the total molar flux in phase p, and kj and are the mass-transfer coefficients for... 

Although diffusion of reacting species can be written in terms of the diffusivity and boundary layer thickness, the magnitude of 8 is unknown. Therefore, the mass-transfer coefficient is normally used. That is, the average molar flux from the bulk fluid to the solid surface is —x direction in Figure 6.2.1)... 

The movement of a contaminant near the surface of a part to the bulk fluid is governed by mass transport mechanisms. The molar flux of a species A from a surface may be expressed in terms of a composition driving force and a mass transfer coefficient ... 

In a binary flow system with interfacial fluid composition xao and bulk fluid composition x b, the local mass-transfer coefficient k corrected for the interfacial total molar flux cvq + Nbo] satisfies... 

Here is the (diffusive) mass flux of species A (mass transfer by diffusion per unit time and per unit area normal to the direction of mass transfer, in kg/s m ) and is the (diffusive) molar flux (in kmol/s m ). The mass flux of a species at a location is propoitional to the density of the mixture at that location. Note that p = Px + Pb density and C = Q + is the molar concentration of the binary mixture, and in general, they may vary throughout the mixture. Therefore, pd(pjp) dp or Cd(C /C) + dC - But in the special case of constant mixture density p or constant molar concentration C, the relations above simplify to... 

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