Diffusion 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. 

Theoretical considerations show that bacterial adhesion to solid, liquid, or sorbed substrates is a powerful mechanism to improve substrate mass transfer. According to Fick s law, the diffusive mass flux of a substrate towards the cell surface J is strongly affected by the space coordinate in direction of the transport ... 

In general, the diffusive mass-flux vector (kg/m2 s) is given by... 

In the foregoing discussion the diffusive mass fluxes are written in terms of the diffusion velocities, which in turn are determined from gradients of the concentration, temperature, and pressure fields. Such explicit evaluation of the diffusion velocities requires the evaluation of the multicomponent diffusion coefficients from the binary diffusion coefficients. 

Consider the net mass flow through the cylindrical differential element illustrated in Fig. 3.6. The following analysis makes no explicit reference to the scalar product of the flux vector and the outward normal, j ndA. Rather, it is based on a more direct observation of how mass diffuses into and out of the control volume. It is presumed that the spatial components of j are resolved into spatial components that are normal to the control-volume faces, jk,z, jk,r, and jk,e Further it is presumed that a positive value for a spatial component of jk means that the corresponding flux is in the direction of the positive coordinate. The components of the diffusive mass flux are presumed to be continuous and differentiable throughout the fluid. Therefore the flux components can be expanded in a first-order Taylor series to express the local variations in the flux. The net mass of species k that crosses the control surfaces diffusively is given by the incoming minus the outgoing mass transport. Consider, for example, transport in the radial direction ... 

It is important to recognize that the control-volume face areas may vary from one side of the control volume to the other. The radial face area depends on r as dA r) = rdddz. Hence, on one side of the control volume, the radial face area is rdOdz, and on the other side, it is (/ + dr)dOdz. By analogous procedures, the net circumferential and axial diffusive mass fluxes are... 

Determine the profiles of the diffusive mass flux by ordinary diffusion (i.e., jk,-)-Plot the profiles for the major species, as well as the net mass flux by ordinary diffusion. How do the magnitudes and the directions of the species diffusive mass fluxes compare with the net convective mass flux Discuss the results in the context of the solution profiles. 

Determine the thermal-difffusive mass-flux profiles for all the species. Discuss the results in the context of the solution profiles. Which species have have important contributions to thermal-diffusive mass flux In what regions is the thermal diffusion important What is the direction of the thermal diffusion for the various species How does thermal diffusion compare in magnitude to ordinary diffusion for this problem ... 

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... 

In addition to the reference scales and nondimensional variables used for the Navier-Stokes equations, new scaling parameters must be introduced to nondimensionalize the temperature and diffusive mass flux. In a mixture-averaged setting... 

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... 

Evaluate the four multicomponent diffusion velocities k using Eq. 12.166. Verify that the sum of the diffusive mass fluxes is zero,... 

Despite the fact that steady-state solutions are the principal concern here, the transient terms are retained to facilitate the hybrid solution algorithm as discussed in Chapter 15 [159]. Alternative formulations for the diffusive mass flux, jkiZ, were introduced briefly in Section 3.5.2, and are discussed in more depth later in this chapter. 

The initial diffusive mass flux density can be calculated from Equation (6.31a). 

Here is the (diffusive) mass flux of species A (mass transfer by diffusion per unit time and per unit area normal to the direction of mass transfer, in kg/s m ) and is the (diffusive) molar flux (in kmol/s m ). The mass flux of a species at a location is propoitional to the density of the mixture at that location. Note that p = Px + Pb density and C = Q + is the molar concentration of the binary mixture, and in general, they may vary throughout the mixture. Therefore, pd(pjp) dp or Cd(C /C) + dC - But in the special case of constant mixture density p or constant molar concentration C, the relations above simplify to... 

In general, the diffusive mass flux is composed of diffusion due to concentration gradients (chemical potential gradients), diffusion due to thermal effects (Soret diffusion) and diffusion due to pressure and external forces. It is possible to include the full multicomponent model for concentration gradient driven diffusion (Taylor and Krishna, 1993 Bird, 1998). In most cases, in the absence of external forces, it is... 

For binary mixtures the diffusive mass flux jc of species c is normally approximated by Pick s first law ... 

Consistent Diffusive Mass Flux Closiu es for the Governing Reactive Flow Equations... 

In this case the dusty gas diffusive mass flux can be written as ... 

Solving this flow model for the velocity the pressure is calculated from the ideal gas law. The temperature therein is obtained from the heat balance and the mixture density is estimated from the sum of the species densities. It is noted that if an inconsistent diffusive flux closure like the Wilke equation is adopted (i.e., the sum of the diffusive mass fluxes is not necessarily equal to zero) instead, the sum of the species mass balances does not exactly coincide with the mixture continuity equation. 

The net diffusive mass flux for each phase still vanishes for binary systems as A s using Pick s law, whereas for dilute pseudobinary systems the latter relationship is only approximate. 

Here j, is a diffusive mass flux of species i, with units of mass/(area x time), as mentioned before. Substituting Eq. 1.60 in, for example, Eq. 1.54, we obtain the energy equation for a mixture ... 

Time-averaged particle mass flux in general can be expressed in terms of the time-averaged particle velocity Up, time-averaged mass concentration CJ (phase density), and particle diffusive mass flux m (covariance of particle mass concentration and velocity) as... 

However, for a fully developed gas-solid pipe flow, the particle diffusive mass flux is usually negligibly small compared with the particle mass flux (Zhu et al., 1991a). Hence, the isokinetic sampling of a gas-solid suspension flow in principle is able to yield the particle mass concentration provided that the particle velocity can be determined. This principle has served as the most primary method for the calibration of measuring systems on particle mass concentration. 

Jn = the diffusion mass flux of the Ic-chem-ical species in f-direction ... 

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