Mass transfer/transport boundary conditions

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

To address media-specific problems, single-media models for air, surface water, groundwater and soil pollution have been developed and used by different disciplines. Although these models generally provide detailed description of the pollutant distribution in space and time and incorporate mass transfer from other media as boundary conditions, they are not capable of characterizing the total environmental impact of a pollutant release. Multimedia models have been, therefore, developed to predict the concentration of chemicals in multiple environmental media simultaneously with consideration of chemical transport and transformation within and among media [1],... 

Although mass transfer across the water-air interface is difficult in terms of its application in a sewer system, it is important to understand the concept theoretically. The resistance to the transport of mass is mainly expected to reside in the thin water and gas layers located at the interface, i.e., the two films where the gradients are indicated (Figure 4.3). The resistance to the mass transfer in the interface itself is assumed to be negligible. From a theoretical point of view, equilibrium conditions exist at the interface. Because of this conceptual understanding of the transport across the air-water boundary, the theory for the mass transport is often referred to as the two-film theory (Lewis and Whitman, 1924). 

Mass transport phenomena usually are effective on distance scales much larger than cell wall and double layer dimensions. Thicknesses of steady-state diffusion layers in mildly stirred systems are in the order of 10 5m. Thus one may generally adopt a picture where the local interphasial properties define the boundary conditions, while the actual mass transfer processes take place on a much larger spatial scale. 

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

Table 5.1 shows that, with the boundary conditions present in most environmental flows (i.e., the Earth s surface, ocean top and bottom, river or lake bottom), turbulent flow would be the predominant condition. One exception that is important for interfacial mass transfer would be very close to an interface, such as air-solid, solid-liquid, or air-water interfaces, where the distance from the interface is too small for turbulence to occur. Because turbulence is an important source of mass transfer, the lack of turbulence very near the interface is also significant for mass transfer, where diffusion once again becomes the predominant transport mechanism. This will be discussed further in Chapter 8. 

What makes the fabrication of composite materials so complex is that it involves simultaneous heat, mass, and momentum transfer, along with chemical reactions in a multiphase system with time-dependent material properties and boundary conditions. Composite manufacturing requires knowledge of chemistry, polymer and material science, rheology, kinetics, transport phenomena, mechanics, and control systems. Therefore, at first, composite manufacturing was somewhat of a mystery because very diverse knowledge was required of its practitioners. We now better understand the different fundamental aspects of composite processing so that this book could be written with contributions from many composite practitioners. 

The overall mass-transfer rates on both sides of the membrane can only be calculated when we know the convective velocity through the membrane layer. For this, Equation 14.2 should be solved. Its solution for constant parameters and for first-order and zero-order reaction have been given by Nagy [68]. The differential equation 14.26 with the boundary conditions (14.28a) to (14.28c) can only be solved numerically. The boundary condition (14.28c) can cause strong nonlinearity because of the space coordinate and/or concentration-dependent diffusion coefficient [40, 57, 58] and transverse convective velocity [11]. In the case of an enzyme membrane reactor, the radial convective velocity can often be neglected. Qin and Cabral [58] and Nagy and Hadik [57] discussed the concentration distribution in the lumen at different mass-transport parameters and at different Dm(c) functions in the case of nL = 0, that is, without transverse convective velocity (not discussed here in detail). 

Mathematical models for mass transfer at the NAPL-water interface often adopt the assumption that thermodynamic equilibrium is instantaneously approached when mass transfer rates at the NAPL-water interface are much faster than the advective-dispersive transport of the dissolved NAPLs away from the interface [28,36]. Therefore, the solubility concentration is often employed as an appropriate concentration boundary condition specified at the interface. Several experimental column and field studies at typical groundwater velocities in homogeneous porous media justified the above equilibrium assumption for residual NAPL dissolution [9,37-39]. 

Heat and mass transfer processes always proceed with finite rates. Thus, even when operating under steady state conditions, more or less pronounced concentration and temperature profiles may exist across the phase boundary and within the porous catalyst pellet as well (Fig. 2). As a consequence, the observable reaction rate may differ substantially from the intrinsic rate of the chemical transformation under bulk fluid phase conditions. Moreover, the transport of heat or mass inside the porous catalyst pellet and across the external boundary layer is governed by mechanisms other than the chemical reaction, a fact that suggests a change in the dependence of the effective rate on the operating conditions (i.e concentration and temperature). 

When the concentration boundary layer is sufficiently thin the mass transport problem can be solved under the approximation that the solution velocity within the concentration boundary layer varies linearly with distance away from the surface. This is called the L6v que approximation (8, 9] and is satisfactory under conditions where convection is efficient compared with diffusion. More accurate treatments of mass transfer taking account of the full velocity profile can be obtained numerically [10, 11] but the Ldveque approximation has been shown to be valid for most practical electrodes and solution velocities. Using the L vSque approximation, the local value of the concentration boundary layer thickness, 8k, (determined by equating the calculated flux to the flux that would be obtained according to a Nernstian diffusion layer approximation that is with a linear variation of concentration across the boundary layer) is given by equation (10.6) [12]. 

From the above outline, the mass-transport problem is seen to consist of coupled boundary value problems (in gas and aqueous phase) with an interfacial boundary condition. Cloud droplets are sufficiently sparse (typical separation is of order 100 drop radii) that drops may be treated as independent. For cloud droplets (diameter 5 ym to 40 pm) both gas- and aqueous-phase mass-transport are dominated by molecular diffusion. The flux across the interface is given by the molecular collision rate times an accommodation coefficient (a 1) that represents the fraction of collisions leading to transfer of material across the interface. Magnitudes of mass-accommodation coefficients are not well known generally and this holds especially in the case of solute gases upon aqueous solutions. For this reason a is treated as an adjustable parameter, and we examine the values of a for which interfacial mass-transport limitation is significant. Values of a in the range 10 6 to 1 have been assumed in recent studies (e.g.,... 

As noted in Section 2, when the electron-transfer kinetics are slow relative to mass transport (rate determining), the process is no longer in equilibrium and does not therefore obey the Nernst equation. As a result of the departure from equilibrium, the kinetics of electron transfer at the electrode surface have to be considered when discussing the voltammetry of non-reversible systems. This is achieved by replacement of the Nernstian thermodynamic condition by a kinetic boundary condition (36). 

The general approach for modelling catalyst deactivation is schematically organised in Figure 2. The central part are the mass balances of reactants, intermediates, and metal deposits. In these mass balances, coefficients are present to describe reaction kinetics (reaction rate constant), mass transfer (diffusion coefficient), and catalyst porous texture (accessible porosity and effective transport properties). The mass balances together with the initial and boundary conditions define the catalyst deactivation model. The boundary conditions are determined by the axial position in the reactor. Simulations result in metal deposition profiles in catalyst pellets and catalyst life-time predictions. 

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