Mass flux, definition

Again, it is mentioned that the binary mass flux definitions given above are commonly used also for pseudo-binary systems. In these particular cases the above relationships are only approximate. 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

It should be noted that the derivative is negative, so that at certain conditions the denominator of Eq. (15-51) can be zero, resulting in an infinite pressure gradient. This condition corresponds to the speed of sound, i.e., choked flow. For a nonflashing liquid and an ideal gas mixture, the corresponding maximum (choked) mass flux G follows directly from the definition of the speed of sound ... 

However, the present results already allow a conclusion to be drawn with respect to the solids mass flux mc loss which is lost from the cyclone as a result of attrition inside the cyclone. From Eq. (23) it follows withw = -0.5 and with the definition of Ra c... 

The definition of the flame speed as the mass flux through the flame per unit area of the flame divided by the unbumed gas density po is useful for turbulent nonstationary and oblique flames as well. 

The emission index in general is defined as the mass of pollutant emitted per unit mass of fuel consumed. In quasi-steady diffusion flames, this is the ratio of the mass flux of pollutant out of the flame to the mass rate of consumption of fuel per unit flame area. Depending on the application, it may be more desirable to consider only the flux of pollutant to the air or the sum of the pollutant flux to both air and fuel. The latter definition is selected here, and a pollutant balance for the flame then enables the emission index to be expressed as the ratio of the mass rate of production of pollutant per unit area to the mass rate of consumption of fuel per unit area. In terms of the mass rate of production of species i per unit volume cDj, the mixture fraction, and the magnitude of its gradient VZ, the mass rate of production of species i per unit area is... 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

The quantity kr is so defined that when hr is positive the component A moves to the cold region, and when kr is negative A moves to the hot region. This definition of kr is in agreement with the accepted definitions of previous workers, and in particular with the text books of Chapman and Cowling (C3) and Grew and Ibbs (G12). In the gas phase many authors prefer to write the mass flux in terms of the thermal diffusion factor a, which is defined as... 

Because of the definition of the ji, only v — 1 of these mass fluxes are independent (since 2ji = 0). It has been shown by Curtiss and Hirschfelder (C12) in their development of the kinetic theory of multicomponent gas mixtures that the foregoing relations may be put into an alternate form 12... 

The rate constant of dilution is calculated from the definition of concentration, C as mass flux per unit time, m, divided by volume flux of water, Q ... 

Along a streamline (i.e., a line of constant ty), = 0. Equation 3.24 requires that mass flux cannot cross a streamline, since along the streamline dm = 0, and by definition, rh is the mass flow crossing the line (area). Equation 3.24 also requires that the mass flow rate between any two streamlines is related simply to the difference of the stream function on the two streamlines... 

Equating the two representations of the wall mass flux (Eqs. 5.31 and 5.32) provides the operational definition for the mass-transfer coefficient,... 

The mass-transfer coefficients, by definition, are equal to the ratios of the molar mass flux to the concentration driving forces. The mass-transfer coefficients are related to each other as follows ... 

Here f is the mass flux density and c(x, t) is the concentration of a solute, continuously distributed in the spatial field x. For this general anisotropic case B is the positive definite diffusion matrix.1... 

The mass flux of a solute can be related to a mass transfer coefficient which gathers both mass transport properties and hydrodynamic conditions of the system (fluid flow and hydrodynamic characteristics of the membrane module). The total amount transferred of a given solute from the feed to the receiving phase can be assumed to be proportional to the concentration difference between both phases and to the interfacial area, defining the proportionality ratio by a mass transfer coefficient. Several types of mass transfer coefficients can be distinguished as a function of the definition of the concentration differences involved. When local concentration differences at a particular position of the membrane module are considered the local mass transfer coefficient is obtained, in contrast to the average mass transfer coefficient [37]. 

It is the calculation of these fluxes (particularly the molar ones), which is our main concern. However, before getting down to business we need to define a few more fluxes in particular, the dijfusion flux, which is the flux of species i relative to the flux of the mixture as a whole. The definition of this flux raises the first of our problems, which mixture velocity are we going to use We have already introduced two, v and u, and there are others that we have not discussed yet. The literature on diffusion would be a good deal simpler if there were only one way to define diffusion fluxes. For each choice of reference velocity there are at least two different diffusion fluxes that we could define, mass fluxes and molar fluxes. 

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. 

The design of a complete set of governing equations for the description of reactive flows requires that the combined fluxes are treated in a convenient way. In principle, several combined flux definitions are available. However, since the mass fluxes with respect to the mass average velocity are preferred when the equation of motion is included in the problem formulation, we apply the species mass balance equations to a (/-component gas system with q — independent mass fractions Wg and an equal number of independent diffusion fluxes js. However, any of the formulations derived for the multicomponent mass diffusion flux can be substituted into the species mass balance (1.39), hence a closure selection optimization is required considering the specified restrictions for each constitutive model and the computational efforts needed to solve the resulting set of model equations for the particular problem in question. 

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