Diffusive boundary mass flux

For the control volume, the heat flux at the boundary is given as if = hc(T — T. ). The diffusion mass flux supplying the reaction is given as m" = hm(yFj00 — yF ), where from heat and mass transfer principles hm — hc/cv. Let Vand S be the volume and surface area of the control volume. The reaction rate per unit volume is given as m " — AYf E ilRT] for the fuel in this problem. 

Let us just consider the piloted ignition case. Then, at Tpy a sufficient fuel mass flux is released at the surface. Under typical fire conditions, the fuel vapor will diffuse by turbulent natural convection to meet incoming air within the boundary layer. This will take some increment of time to reach the pilot, whereby the surface temperature has continued to rise. 

In this book we limit our treatment to dilute solutions so that the diffusional mass flux is small. In this way the existence of diffusion does not appreciably alter the fluid motion, so that the velocity and stress boundary conditions can be considered to be unaltered. Treatments of diffusion with high mass fluxes appear elsewhere (B3, S3, S4). 

For thick membranes and sufficiently fast reaction kinetics at the membrane boundaries ( equilibrium domain ), diffusion through the membrane interior is rate limiting and we obtain for the mass flux of ions ... 

The approach pursued in this and the next chapter is focused on the common mathematical characteristics of boundary processes. Most of the necessary mathematics has been developed in Chapter 18. Yet, from a physical point of view, many different driving forces are responsible for the transfer of mass. For instance, air-water exchange (Chapter 20), described as either bottleneck or diffusive boundary, is controlled by the turbulent energy flux produced by wind and water currents. The nature of these and other phenomena will be discussed once the mathematical structure of the models has been developed. 

The mathematics of diffusion at flat wall boundaries has been derived in Section 18.2 (see Fig. 18.5a-c). Here, the well-mixed system with large diffusivity corresponds to system B of Fig. 18.5 in which the concentration is kept at the constant value Cg. The initial concentration in system A, CA, is assumed to be smaller than Cg. Then the temporal evolution of the concentration profile in system A is given by Eq. 18-22. According to Eq. 18-23 the half-concentration penetration depth , x1/2, is approximatively equal to (DAt)m. The cumulative mass flux from system B into A at time t is equal to (Eq. 18-25) ... 

Figure 19.15 (a) Concentration profile at a diffusive boundary between two different phases. At the interface the instantaneous equilibrium between CAB and CB/A is expressed by the partition coefficient KB/A. The hatched areas show the integrated mass exchange after time t MA (t) = MB (<). (b) As before, but the size of KB/A causes a net mass flux in the opposite direction, that is from system B into system A. 

All the necessary tools to develop kinetic models for air-water exchange have been derived already in Chapters 18 and 19. However, we don t yet understand in detail the physical processes which act at the water surface and which are relevant for the exchange of chemicals between air and water. In fact, we are not even able to clearly classify the air-water interface either as a bottleneck boundary, a diffusive boundary, or even something else. Therefore, for a quantitative description of mass fluxes at this interface, we have to make use of a mixture of theoretical concepts and empirical knowledge. Fortunately, the former provide us with information which is independent of the exact nature of the exchange process. As it turned out, the different flux equations which we have derived so far (see Eqs. 19-3, 19-12, 19-57) are all of the form ... 

The species boundary condition at the stagnation surface follows from the fact that the diffusive mass flux in the fluid is balanced by a heterogeneous chemical reaction rate on the surface. In general, this can involve multiple and complex surface reactions and complex descriptions of the molecular diffusion. Here, however, we restrict attention to a single species that is dilute in a carrier gas and a single first-order surface reaction. Under these circumstances the surface reaction rate (mass of Y consumed per unit surface area) is given... 

Heterogeneous reactions at a gas-surface interface affect the mass and energy balance at the interface, and thus have an important influence on the boundary conditions in a chemically reacting flow simulation. The convective and diffusive mass fluxes of gas-phase species at the surface are balanced by the production (or destruction) rates of gas-phase species by surface reactions. This relationship is... 

Consider the behavior of upstream diffusion as illustrated by the solutions in Fig. 16.8. Despite the fact that the unbumed reagents are only methane and air, it may be observed in the lower left-hand panel that significant levels of H2 are present at the inlet boundary. Had the burner-face boundary condition been specified as a fixed composition of methane and air (instead of the mass-flux fractions as in Eq. 16.100), the solution in the vicinity of the burner would have been different, since the H2 fraction would have vanished at the boundary. The influence of upstream diffusion, and hence the need for the mass-flux-fraction boundary condition, is increasingly important at low flow rates or low pressure. In either case relatively strong diffusive mass transport can cause reaction-product species to diffuse to the inlet boundary. 

Mass diffusion between grain boundaries in a polycrystal can be driven by an applied shear stress. The result of the mass transfer is a high-temperature permanent (plastic) deformation called diffusional creep. If the mass flux between grain boundaries occurs via the crystalline matrix (as in Section 16.1.3), the process is called Nabarro-Herring creep. If the mass flux is along the grain boundaries themselves via triple and quadjunctions (as in Sections 16.1.1 and 16.1.2), the process is called Coble creep. 

The development of the miscibility gap for W < 0 and the antiphases ( Tjeq) for W > 0 have entirely different kinetic implications. For decomposition, mass flux is necessary for the evolution of two phases with differing compositions. Furthermore, interfaces between these two phases necessarily develop. The evolution of ordered phases from disordered phases (i.e., the onset of nonzero structural order parameters) can occur with no mass flux macroscopic diffusion is not necessary. Because the 77+q-phase is thermodynamically equivalent to the 7/iq-phase, the development of 77+q-phase in one material location is simultaneous with the evolution of r lq-phase at another location. The impingement of these two phases creates an antiphase domain boundary. These interfaces are regions of local heterogeneity and increase the free energy above the homogeneous value given by Eq. 17.14. The kinetic implications of macroscopic diffusion and of the development of interfaces are treated in Chapter 18. 

When deposition is controlled by diffusion. Equation (28) shows that variations in boundary layer thickness, 5, influence the mass flux due to diffusion and thereby, the deposition rate. In practical CVD reactors, boundary layer thicknesses can vary so that thickness uniformity of deposition can be poor unless this phenomena is recognized and corrected. 

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 ... 

In the case of thermal decomposition of a mineral, there is only the solid B on the left-hand side of equation (5.26). These thermal decompositions can also be treated by the same rate limiting steps as given previously. Although the product layer is often porous, it can produce a slower rate of either heat conduction or diffusion than the boundary layer. As a result fluid-solid reactions occur at a sharply defined reaction interface, at a position r within the particle of size R. The mass flux associated with boundary layer mass transfer is given by... 

Note that species transfer at the surface (y - 0) is by diffusion only because of the no-slip boundary condition, and mass flux of spede.s A at the surface can be expressed by Pick s law as (Fig. 14—42)... 

Stemling and Scriven wrote the interfacial boundary conditions on nonsteady flows with free boundary and they analyzed the conditions for hydrodynamic instability when some surface-active solute transfer occurs across the interface. In particular, they predicted that oscillatory instability demands suitable conditions cmcially dependent on the ratio of viscous and other (heat or mass) transport coefficients at adjacent phases. This was the starting point of numerous theoretical and experimental studies on interfacial hydrodynamics (see Reference 4, and references therein). Instability of the interfacial motion is decided by the value of the Marangoni number, Ma, defined as the ratio of the interfacial convective mass flux and the total mass flux from the bulk phases evaluated at the interface. When diffusion is the limiting step to the solute interfacial transfer, it is given by... 

Convective mass and heat transfer to a plate in a longitudinal flow of a non-Newtonian fluid was considered in [443]. By solving the corresponding problem in the diffusion boundary layer approximation (at high Peclet numbers), we arrive at the following expression for the dimensionless diffusion flux ... 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

It is only necessary to consider diffusional fiux across the lateral surface because axial diffusion is insignificant at high mass transfer Peclet numbers. The generalized quasi-macroscopic mass balance for one-dimensional fluid flow through a straight channel with arbitrary cross section and nonzero mass flux at the lateral boundaries is... 

Typical examples are solid catalyzed reactions or wall reactions occurring in free radical chemistry. Usually reacting surfaces are covered by a boundary layer of the fluid. Then, if is of no surprise fhat the fluxes can be expressed in terms of the diffusive fluxes exclusively. In any mass balance, we usually have mass fluxes expressed in terms of V N,. From standard definitions (Bird etal, 2002, p. 537) ... 

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