Diffusion and Mass Transfer

Let the moles and mass of component A per unit volume of mixture be cA and pA respectively. Then the mole fraction of A is cAfe or xA, and the mass fraction is pAfp or wA. 

Consider a nomniform fluid mixture with n components that is experiencing bulk motion. The statistical mean velocity of component i in the x direction with respect to stationary cooidinates is The molal average velocity of the mixture in the x direction Is defined as 

The corresponding definiticns of molal fluxes in the x direction for component i are as follows.  

Analogous expressions of course may be written for the mass fluxes. Relationship between the various fluxes are obtained in the following manner. To relate molal fluxes Ib and A4, consider Eqs. (7.1-1), (7.1-3), and (7.1-4)  

1-2 Steady-State Equimolal Counterdiffusion and Unimolal Unidirectional Diffusion 

Analogous expressions of course may be written for the mass fluxes. Relationships between the various fluxes are obtain in the following manner. 

1-2 Steady-State Equimolai Counterdiffusion and Unimoiai Uradirectionai Diffusion 

The mass transfer coefficients for parallel and cross air flow, respectively, are given below  

Chapters 2 and 3 dealt with momentum transfer and rheological equations of state and their applications to polymer processing. In this chapter and the next one we are concerned with the other two transfers mass and heat. A substantial number of polymer processes involve changes of composition of the component materials through mass diffusion and convection methods. In many cases these changes of composition do not necessarily involve chemical reactions. We describe some of these polymer processes below. 

Polymer Processing Principles and Design, Second Edition. Donald G. Baird and Dimitris I. Collias. 2014 John Wiley Sons, Inc. Published 2014 by John Wiley Sons, Inc. 

We do not intend in this chapter to present an extensive analysis of mass transfer concepts but, rather, to summarize the basics of mass transfer as required in the design and analysis of polymer processing operations. In this regard, we give only an extensive overview of the estimation techniques for the diffusivity, solubility, and permeability of solvents in polymers. The laws of diffusional mass transfer, as well as the relationships for convective mass transfer, remain the same as applied to any material. The books by Perry and Chilton (1973), Reid et al. (1977), and Brandmp and Immergut (1989) provide an extensive overview of experimental data and formulas for the calculation of diffusivity, solubility, and permeability of various polymer systems. 

This chapter is organized as follows. In Section 4.1 we describe the fundamentals of mass transfer, such as the various definitions for concentrations and velocities. Pick s first law of diffusion, and the microscopic mass balance principle. 

The transfer of heat in a liquid may occur by means of conduction, convection, diffusion, or radiation. Heat transport by way of convection and diffusion will be considered in the appropriate sections, and since heat transport through radiation is small (essentially only occurring at very high temperatures), we consider here only heat transport due to conduction. 

If in a liquid there is a temperature gradient VT, then according to the Fourier law, the heat transport is proportional to VT and in the opposite direction to VT  

The coefficient k is called the heat conductivity factor. In parallel with this, the factor of thermal diffusivity is frequently used in practice (sometimes also called the coefficient of heat conductivity) 

The dimensions of a are the same as those of the diffusion coefficient and of the kinematic viscosity, therefore the process of heat transport due to conduction can be treated as the diffusion of heat with the diffusion coefficient a, bearing in mind that the transport mechanisms of diffusion and heat conductivities are identical. The coefficient of heat conductivity of gases increases with temperature. For the majority of liquids the value of k decreases with increasing T. Polar liquids, such as water, are an exception. For these, the dependence k(T) shows a maximum value. As well as the coefficient of viscosity, the coefficient of heat conductivity also shows a weak pressure-dependence. 

Heat flux may also occur as a result of a concentration gradient (the Dufour effect). However, as a rule, it is negligibly small, and so hereafter we shall neglect this effect. 

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Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

The total column dispersion is due to the combined effects of flow dispersion, longitudinal diffusion and mass transfer. 

Amino-functionalized SNT catalyst was prepared by using APTES and SNT having 1-demensional channel. The catalysts could be used as showing nanoreactors by high catalytic activities for the base-catalyzed reactions due to rapid diffusion and mass transfer of substrates into SNT channel. 

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

Finally, the liquid-phase diffusivities and mass-transfer coefficients are related, as a consequence of equation 9.2-7, by... 

The ultimate width of a peak is determined by the total amount of diffusion occurring during movement of the solute through the system, and on the rate of mass transfer between the two phases. These effects are shown diagrammatically in Figure 4.14. Both diffusion and mass transfer effects are inter-dependent and complex, being made up of a number of contributions from different sources. Because they are kinetic effects, their influence on efficiency is determined by the rate at which the mobile phase... 

Effects of diffusion and mass transfer on peak width, (a) Concentration profiles of a solute at the beginning of a separation, (b) Concentration profiles of a solute after passing some distance through the system. 

Diffusion and mass transfer effects cause the dimensions of the separated spots to increase in all directions as elution proceeds, in much the same way as concentration profiles become Gaussian in column separations (p. 86). Multiple path, molecular diffusion and mass transfer effects all contribute to spreading along the direction of flow but only the first two cause lateral spreading. Consequently, the initially circular spots become progressively elliptical in the direction of flow. Efficiency and resolution are thus impaired. Elution must be halted before the solvent front reaches the opposite edge of the plate as the distance it has moved must be measured in order to calculate the retardation factors (Rf values) of separated components (p. 86). 

Fig. 14.2. van Deemter plot showing contributions of eddy diffusion, molecular diffusion and mass transfer to the rate of band broadening. Picture courtesy of Prof. Harold McNair. 

In classical kinetic theory the activity of a catalyst is explained by the reduction in the energy barrier of the intermediate, formed on the surface of the catalyst. The rate constant of the formation of that complex is written as k = k0 cxp(-AG/RT). Photocatalysts can also be used in order to selectively promote one of many possible parallel reactions. One example of photocatalysis is the photochemical synthesis in which a semiconductor surface mediates the photoinduced electron transfer. The surface of the semiconductor is restored to the initial state, provided it resists decomposition. Nanoparticles have been successfully used as photocatalysts, and the selectivity of these reactions can be further influenced by the applied electrical potential. Absorption chemistry and the current flow play an important role as well. The kinetics of photocatalysis are dominated by the Langmuir-Hinshelwood adsorption curve [4], where the surface coverage PHY = KC/( 1 + PC) (K is the adsorption coefficient and C the initial reactant concentration). Diffusion and mass transfer to and from the photocatalyst are important and are influenced by the substrate surface preparation. 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

PHASE EQUILIBRIA, MOLECULAR DIFFUSION AND MASS TRANSFER... 

The phase equilibria of the most important compounds will be described in the following section. In the sections thereafter, we will treat mass transport in melt-phase polycondensation, as well as in solid-state polycondensation, and discuss the diffusion and mass transfer models that have been used for process simulation. 

DIFFUSION AND MASS TRANSFER IN MEL T-PHASE POL YCONDENSA TION... 

Hence, the following chapters will start from homogeneous reactions and proceed to diffusion and mass transfer, and then to heterogeneous reactions. 

The terms diffusion and mass transfer seem to be used more or less interchangeably in the literature. The tendency seems to be to use mass transfer as the general term embracing all mechanisms of transport of a chemical species and to reserve diffusion for mass transport by molecular motion within a single phase. 

In reading the literature on the fluid mechanics of diffusion, one encounters numerous difficulties because of the diversity of reference frames and definitions used by authors in various fields. Frequently more time is spent in translating from one system of notation to another than is spent in the actual study of the physics of the problem. It is hoped that the glossaries of terminology and symbols given here will be of use to those whose fields of research require a familiarity with the literature on diffusion. This exposition should emphasize the extreme importance of giving lucid definitions in any discussion of diffusion and mass transfer. 

Surface effects and adsorption equilibria thus will significantly influence the course of photoelectrochemical transformations since they will effectively control the movement of reagents from the electrolyte to the photoactivated surface as well as the desorption of products (avoiding overreaction or complete mineralization). The stability and accessibility toward intermolecular reaction of photogenerated intermediates will also be controlled by the photocatalyst surface. Since diffusion and mass transfer to and from the photocatalyst surface will also depend on the solvent and catalyst pretreatment, detailed quantitative descriptions will be difficult to transfer from one experiment to another, although qualitative principles governing these events can be easily recognized. 

In order to calculate the multicomponent diffusion matrices [D], the binary diffusivities in both phases should be known. The him thickness representing an important model parameter is estimated via the mass transfer coefficients (57,83). The binary diffusivities and mass transfer coefficients were calculated from the correlations summarized in Table 3. 

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