Mass-average velocity

The first equahty (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity while the equahty in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity These must be used with consistent flux expressions for fixed coordinates and for Nc components, such as ... 

Eor a dilnte aqneons solntion the mass average velocity is determined from the equation of motion for a Newtonian flnid, the Navier-Stokes eqnation,... 

Leaving aside the difficult question of whether this model holds for multiphase flows, we still have the problem of determining in terms of the computed properties of the flow. The reader should appreciate that choosing an effective viscosity for a multiphase flow is much more complicated than just adding a turbulence model as done in single-phase turbulent flows. Indeed, even for a case involving two fluids (e.g., two immiscible liquids) for which the molecular viscosities are constant, the choice of the effective viscosities is not obvious. For example, even if the mass-average velocity defined by... 

As mentioned above, this approach treats each phase as a constituent to a mixture. Thus, all parameters are mixture parameters and must be averaged, usually by the saturation. Unlike the models mentioned at the end of the previous section, the models here use capillary phenomena. Furthermore, although the mixture moves at a mass-average velocity, interfacial drag between the phases and other conditions allow each separate phase velocity to be determined. The liquid-phase velocity is found by 9... 

In these equations pt is the mass density (g. cm.-3) of the fth chemical species, fc is the rate of production of the fth chemical species by chemical reaction (g. cm.-3 sec.-1), and Fi is the external body force per unit mass acting on the ith species. The velocity v is the local mass average velocity (that velocity measured by a Pitot tube), p is the over-all density of the fluid, and U is the local thermodynamic internal energy (per unit mass) of the mixture. The j, are the fluxes of the various chemical species in g. cm.-2 sec.-1 with respect to the local mass average velocity, v. It should be noted that 2j, = 0, 2/c,- = 0, and = p these relations are used in deriving the over-all equation of continuity [Eq. (4)] by adding up the individual equations of continuity given in Eq. (24). 

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. 

There are also many ways of expressing the velocity of a chemical species present in a flow system. We do not concern ourselves here with the instantaneous velocity of the individual molecules of a species, but rather with the average macroscopic velocities with which the species travel. These may be measured from a stationary coordinate system, but for flow systems the velocities of individual species are frequently measured from a coordinate frame moving with (a) the mass-average velocity of the stream, (b) the molar average velocity of the stream, or (c) the velocity of one particular component. The mass- and the molar-average velocity are defined in Table II and the notation for the various velocities of an individual species is given. 

Ua = average velocity of species A with respect to a stationary coordinate system Va = average velocity of species A with respect to the local mass-average velocity v (that is, vA = Ua - v)... 

Molar flux of A with respect to mass-average velocity (see Table III) (38)... 

With reasonable assumptions [178], vo can be identified as the local mass-average velocity, pi is the local value of species i density, and T is the local temperature. 

The mass diffusion velocity of species k relative to the mass average velocity will be denoted V, which is defined by... 

Fick s first law of diffusion relates the diffusive flux of species k to its mass fraction or mole fraction gradient. For a binary mixture of species j and k, the mass flux of species k relative to the mass average velocity is, related to the mass fraction gradient of k as... 

To calculate the mixture-averaged diffusion coefficient relating the mass flux with respect to the mass-average velocity V in terms of the mass fraction gradient, write an expression analogous to Eq. 12.158 ... 

L flux of compound i relative to mass average velocity... 

The magnitude of the forced fluid velocity, U, relative to the body. The manner in which this is defined will depend on the nature of the problem as indicated in Fig. 1.14. In the case of external flows, the undisturbed ffeestream velocity is usually the most convenient to use for Uy whereas in the case of internal flows it is usually more convenient to take U as the mean or mass average velocity in the duct. 

Equations (2.53) and (2.54) are called Fick s first law of diffusion, and indicate that mass flows from a high to a low concentration region. It is valid for any binary fluid or solid solution, provided that j, is defined as the mass flow relative to the mixture mass average velocity v defined by... 

Equation (2.59) is the mass flow in terms of mass average velocity v, and using the diffusion velocity (v, - v), we obtain... 

If the velocity of the observer is the same as the mass average velocity of the fluid v with components vx, vy, and vz, then the rate of temperature change is given by... 

The mass flow of component i, p,v is a vector showing the flow of a component relative to a motionless coordinate system. On the other hand, diffusion flow shows the transport of a component relative to a coordinate system moving at the reference velocity vr. The diffusion flow relative to the center-of-mass velocity v (or mass average velocity) is... 

The balance equation for the kinetic energy is obtained by scalar multiplication of the momentum balance, Eq. (3.97), and the mass average velocity, and is given by... 

Equation (3.133) can be applied to a fluid element moving with the mass average velocity v. After replacing the differential operators with substantial time derivative operators inEq. (3.133), we have... 

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