Diffusion mass transfer coefficient species

Mass transfer coefficients are the basis for models where the dissolved species are transported by a combination of diffusive and advective processes. The diffusive mass transfer coefficient ko, m/sec) is based on boundary layer theory. The basic premise of boundary layer theory is that, for laminar ffow, the ffuid velocity adjacent to a solid surface is zero (the no slip condition ) and the velocity increases as a parabolic function of distance away from the surface until it matches the velocity of the bulk fluid (Figure 7.5). This means that there is a thin layer of fluid with a thickness of 5d (m) adjacent to the surface that is effectively static. The rate of mass transport through this layer is limited by the diffusion rate of the dissolved species. The diffusional boundary layer is much thinner than the velocity boundary layer. For laminar flow past a flat surface, the thickness of the diffusional boundary layer is related to the thickness of the velocity boundary layer (Sy) by the Schmidt number, which compares the fluid viscosity to the diffusivity (Probstein, 1989). 

The diffusion mass transfer coefficient of species i in the gas channel is denoted by and is found via [95] ... 

R is rate of reaction per unit area, a is interfacial area per unit volume, S is solubiHty of solute in continuous phase, D is diffusivity of solute, k is rate constant, kj is mass-transfer coefficient, is concentration of reactive species, and Z is stoichiometric coefficient. When Dk is considerably greater (10 times) than Ra = aS Dk. 

However, flow generated by a cylinder rotating at high speed was subsequently used by others, and in particular by King and co-workers (K3, K4a), to demonstrate that dissolution and electrochemical corrosion may both be transport limited. The dependence of the mass-transfer coefficient on the rotation rate and on the diffusivity of the dissolving species was established by correlation of experimental data (see Table VII, System 43). 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

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. 

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

C] represents the concentration of species Ain the liquid and x represents the mole fraction of the diffusing species in the liquid. Note that we have introduced kL, the liquid phase mass transfer coefficient. The cap ( ) indicates average value. 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

An important performance characteristic of passive samplers that operate in the TWA regime is the diffusion barrier that is inserted between the sampled medium and the sorption phase. This barrier is intended to control the rate of mass transfer of analyte molecules to the sorption phase. It is also used to define the selectivity of the sampler and prevent certain classes (e.g., polar or nonpolar compounds) of analytes, molecular sizes, or species from being sequestered. The resistance to mass transfer in a passive sampler is, however, seldom caused by a single barrier (e.g., a polymeric membrane), but equals the sum of the resistances posed by the individual media (e.g., aqueous boundary layer, biofilm, and membrane) through which analyte diffuses from the bulk water phase to the sorption phase.19 The individual resistances are equal to the reciprocal value of their respective mass transfer coefficients and are additive. They are directly proportional to the thickness of the barrier... 

Turning our attention to surface phenomena rather than diffusion, we recognize that species will transit across the boundary layer and may be created or destroyed in this passage due to chemical reactions which will proceed at finite rates (homogeneous gas phase reactions). Upon impacting the surface, they may adsorb and then decompose, leaving a solid thin film. This will be a heterogeneous surface reaction which will have a characteristic chemical reaction rate. One way to describe this phenomena is in terms of a mass transfer" coefficient. The mass flux can be expressed in terms of this coefficient, as follows ... 

Diffusion of the species to where the reaction occurs (described by a mass transfer coefficient kd - see Chapter 5). 

Transport to the electrode surface as described in Chapter 5 assumes that this occurs solely and always by diffusion. In hydrodynamic systems, forced convection increases the flux of species that reach a point corresponding to the thickness of the diffusion layer from the electrode. The mass transfer coefficient kd describes the rate of diffusion within the diffusion layer and kc and ka are the rate constants of the electrode reaction for reduction and oxidation respectively. Thus for the simple electrode reaction O + ne-— R, without complications from adsorption,... 

To use the various criteria given in the previous section, some experimental data on the reacting system are necessary. These are the effective diffusivity of the key species in the pores of the catalyst, the heat and mass transfer coefficients at the fluid-solid interface, and the effective thermal conductivity of the catalyst. The accuracy of some of these parameters, which are usually obtained from known correlations, may sometimes be subject to question. For example, under labo-... 

Let us consider the electrode kinetics associated with charge transfer from an n-type semiconductor particle to an electrode. As indicated by Albery et al. [164], the crucial difference between the electrochemistry of a colloidal particle and an ordinary electrochemically active solution phase species is the number of electrons transferred from the particle to the electrode may be large and will depend upon the potential of the electrode. Fig. 9.5 shows the model for an encounter of a particle with an electrode used by Albery and co-workers. kD is the mass-transfer coefficient for the transport of the particles to the electrode surface. In the simplest case, wherein it is assumed that the lifetime of the transferable electrons (majority carriers of thermal or photonic origin) is greater than the time taken by a particle to traverse the ORDE diffusion layer, this is given by... 

For sufficiently large electrodes with a small vibration amplitude, aid < 1, a solution of the hydrodynamic problem is possible [58, 59]. As well as the periodic flow pattern, a steady secondary flow is induced as a consequence of the interaction of viscous and inertial effects in the boundary layer [13] as shown in Fig. 10.10. It is this flow which causes the enhancement of mass-transfer. The theory developed by Schlichting [13] and Jameson [58] applies when the time of oscillation, w l is small in comparison with the time taken for a species to diffuse across the hydrodynamic boundary layer (thickness SH= (v/a>)ln diffusion timescale 8h/D), i.e., when v/D t> 1. Re needs to be sufficiently high for the calculation to converge but sufficiently low such that the flow does not become turbulent. Experiment shows that, for large diameter wires (radius, r, — 1 cm), the condition is Re 2000. The solution Sh = 0.746Re1/2 Sc1/3(a/r)1/6, where Sh (the Sherwood number) = kmr/D and km is the mass-transfer coefficient,... 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Kl — physical mass transfer coefficient, CL = liquid flow rate, D = diffusion coefficient of absorbing species. 

The mass transfer coefficient, jK, , for a sphere can be determined from the Sherwood number, Ns (= Q2Rq/Aib> where is the molecular diffusion coefficient of the solvent, species A, in the drying gas, species B), and the following engineering correlation [15] ... 

In the third simulation example, we carried out an analysis of some of the aspects that characterize the case of the mass transfer of species through a membrane which is composed of two layers (the separative and the support layers) with the same thickness but with different diffusion coefficients of each entity or species. To answer this new problem the early model has been modified as follows (i) the term corresponding to the source has been eliminated (u) different conditions for bottom and top surfaces have been used for example, at the bottom surface, the dimensionless concentration of species is considered to present a unitary value while it is zero at the top surface (iii) a new initial condition is used in accordance with this case of mass transport through a two-layer membrane (iv) the values of the four thermal diffusion coefficients from the original model are replaced by the mass diffusion coefficients of each entity for both membrane layers (v) the model is extended in order to respond correctly to the high value of the geometric parameter 1/L. 

The first necessary condition [6.21] to be taken into account in all particularization cases is the use of general mixing parameters (factors related to the geometry of agitation, the properties of liquid media, the type of agitators and rotation speed) as well as the use of the specific factors of the studied application. For example, in the case of suspended solid dissolution, we can consider the mass transfer coefficient for dissolving suspended solids, the mean dimension of the suspended solid particles, and the diffusion coefficient of the dissolved species in the liquid. 

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)... 

Solution of a reactor problem in the mass transfer limit requires an estimation of the appropriate mass transfer coefficient. Fortunately, mass transfer correlations have been developed to aid the determination of mass transfer coefficients. For example, the Sherwood number, Sh, relates the mass transfer coefficient of a species A to its diffusivity and the radius of a catalyst particle, Rp ... 

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