Vector mass-flux

The mass flux vector is also the sum of four components j (l), the mass flux due to a concentration gradient (ordinary diffusion) jYp), the mass flux associated with a gradient in the pressure (pressure diffusion) ji(F), the mass flux associated with differences in external forces (forced diffusion) and j,-(r), the mass flux due to a temperature gradient (the thermal diffusion effect or the Soret effect). The mass flux contributions may then be summarized ... 

Notation for Mass-Flux Vectors in Binaby Systems... 

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

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

It is often problematic to divide by Xk, since a divide by zero would occur in regions of a flow that do not contain some species components. Therefore the ratio of molecular weights provides a better numerical implementation of the mass flux vector. 

Recognizing the terms in the parenthesis on the right-hand side as the divergence of the mass-flux vector and dV = rdrdOdz, it can be seen that the procedure has recovered the Gauss divergence theorem (Eq. 2.29). That is,... 

Up to now, the mass-continuity equations (e.g., Eq. 3.124) have been written in terms of the mass-flux vector j, which is a function of the species composition field. As noted in Section 3.5.2, different levels of theory can used to specify the functional relationship between flux and composition gradients, and mass flux can also depend on temperature or... 

In this equation it still remains to write out the components of the mass-flux vector (e.g., jkZ) in terms of the appropriate composition (and possibly temperature) gradients, Section 3.5.2. Moreover the dissipation function contributes a lot of terms that must be written out in cylindrical coordinates, Eq. 3.201. 

In these one-dimensional equations, the independent variable is the spatial coordinate z, and the dependent variables are the temperature T and the species mass fractions Yk. The continuity equation is satisfied exactly by m" = pu, which is a constant. Other variables are the z component of the mass-flux vector jkz, the molar production rate of species by chemical reaction 6)k, the thermal conductivity A, the species enthaplies hk, and the molecular weights Wk. The diffusion fluxes are determined as... 

In the preceding discussion and in what follows below we have introduced to the bar symbol to emphasize that we refer to a specific (i.e. per unit mass) quantity. Note that Jk is a mass flux vector although Jk has a strange appearance this quantity is dimensionally correct Jk - pkvk, whence. Jk -pk. vk is the rate of transport of potential energy density. 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

Mass flux vector of species i due to molecular transport, kg/(m s)... 

In (1.25), the terms inside the brackets can be reformulated by use of vector and tensor notations. By comparing the terms inside the brackets with the mathematical definitions of the nabla or del operator, the vector product between this nabla operator and the mass flux vector we recognize that these... 

It is possible to justify several alternative definitions of the multicomponent diffusivities. Even the multicomponent mass flux vectors themselves are expressed in either of two mathematical forms or frameworks referred to as the generalized Fick- and Maxwell-Stefan equations. 

Due to this difficulty it is preferable to transform the Fickian diffusion problem in which the mass-flux vector, js, is expressed in terms of the driving force, ds, into the corresponding Maxwell-Stefan form where is given as a linear function of jg. The key idea behind this procedure is that one intends to rewrite the Fickian diffusion problem in terms of an alternative set of diffusivities (i.e., preferably the known binary diffusivities) which are less concentration dependent than the Fickian diffusivities. 

For binary systems the Fickian mass flux vector reduces to ... 

It is necessary to associate mathematical quantities with each type of momentum transfer rate process that is contained in the vector force balance. The fluid momentum vector is expressed as p, which is equivalent to the overall mass flux vector. This is actually the momentum per unit volume of fluid because mass is replaced by density in the vectorial representation of fluid momentum. Mass is an extrinsic property that is typically a linear function of the size of the system. In this respect, mv is a fluid momentum vector that changes magnitude when the mass of the system increases or decreases. This change in fluid momentum is not as important as the change that occurs when the velocity vector is affected. On the other hand, fluid density is an intrinsic property, which means that it is independent of the size of the system. Hence, pv is the momentum vector per unit volume of fluid that is not affected when the system mass increases or decreases. The total fluid momentum within an arbitrarily chosen control volume V is... 

If the interface is stationary, or if it translates without accelerating, then a steady-state force balance given by equation (8-180) states that the sum of all surface-related forces acting on the interface must vanish. Body forces are not an issue because the system (i.e., the gas-liquid interface) exhibits negligible volume. The total mass flux vector of an adjacent phase relative to a mobile interface is... 

This is a surface-related phenomenon based on the mass flux vector of component i and the surface area across which this flux acts. Relative to a stationary reference frame, p, v, is the mass flux vector of component i with units of mass of species i per area per time. It is extremely important to emphasize that p, v, contains contributions from convective mass transfer and molecular mass transfer. The latter process is due to diffusion. When one considers the mass of component i that crosses the surface of the control volume due to mass flux, the species velocity and the surface velocity must be considered. For example. Pi (Vr — Vsurface) is the mass flux vector of component i with respect to the surface... 

Sect. 6 the hydrodynamic equation of continuity for each species and a formal expression for the mass flux vector of each species Sect. 7 the hydrodynamic equation of motion for the liquid mixture and a formal expression for the stress tensor Sect. 8 the energy equation for the liquid and a formal expression for the heat flux vector... 

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