Boundary conditions mass transfer, diffusion

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

The kinetic equations for the volume phase of the solid body are equations of the diffusion type (63). Much attention has been given to them in the literature [154,155], therefore here will be reminded only those aspects of the theory of mass transfer for which the lattice-gas model has been used. These are problems involved in the construction of expressions for the diffusion the coefficients and boundary conditions of the diffusion equations. 

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. 

Convection mass transfer coefficients are often used as convective boundary conditions for gas diffusion in a stationary media. However, while applying mass transfer correlations to describe mass species transport from the electrode-gas diffusion layer to gas flow stream in the channel, it is assumed that species mass transport rate at the wall is small and does not alter the hydrod5mamic, thermal, and concentration boundary layers like in boundary layers with wall suction or blowing. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

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] ... 

The boundary conditions for this early dissolution model included saturated solubility for HA at the solid surface (Cha ) with sink conditions for both HA and A at the outer boundary of a stagnant film (Cha = Ca = 0). Since diffusion is the sole mechanism for mass transfer considered and the process occurs within a hypothesized stagnant film, these types of models are colloquially referred to as film models. Applying the simplifying assumption that the base concentration at the solid surface is negligible relative to the base concentration in the bulk solution (CB CB(o)), it is possible to derive a simplified scaled expression for the relative flux (N/N0) from HPWH s original expressions ... 

Mass transfer phenomena usually are very effective on distance scales much larger than the dimensions of the cell wall and the double layer dimensions. Thicknesses of steady-state diffusion layers1 in mildly stirred systems are of the order of 10 5 m. Thus, one may generally adopt a picture where the local interphasial properties define boundary conditions while the actual mass transfer processes take place on a much larger spatial scale. 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

For some biological systems, the species that eventually crosses the cell membrane has travelled through different media, each one with its own mass transfer characteristics. As an example, we deal with the case where the two media are the bulk solution and the cell wall (with the separation surface parallel to the cell membrane) with diffusion as the only relevant mass transfer phenomenon in each medium. Apart from having different parameters in the differential equations in each medium (due to the unequal diffusion coefficients), we need to impose two new boundary conditions at the separating plane which we denote as a. The first boundary condition follows from the continuity of the material flux ... 

Four strategies are generally employed to demonstrate mass transfer limitation in aquatic systems. Most commonly, measured uptake rates are simply compared with calculated maximal mass transfer rates (equation (17)) (e.g. [48,49]). Uptake rates can also be compared under different flow conditions (e.g. [52,55,56,84]), or by varying the biomass under identical flow conditions (e.g. [85]). Finally, several recent, innovative experiments have demonstrated diffusion boundary layers using microsensors [50,51]. Of the documented examples of diffusion limitation, three major cases have been identified ... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

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. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

It is worth noting that the derivation outlined above is more generally applicable since, for many types of diffusional mass transfer (spherical, cylindrical, bounded, etc.), it is possible to rewrite the original differential equations in the time domain in terms of new variables in such a way that the second diffusion law is of the same form as eqn. (19b), with appropriate formulation of the boundary conditions [22, 75]. However, in finding the inverse transforms, difficulties may arise because of the more complex meaning of the time domain variables. 

---