Local Instantaneous Transport Equations

Although it is appealing from a scientific point of view to regard the interface as a 3D region of finite thickness, the computational difficulties involved considering the implicit numerical grid resolution requirements make the ap -plication of this concept infeasible. 

Furthermore, adopting the macro scale viewpoint several alternative modeling approaches are proposed in the literature for deriving the jump conditions. Moreover, coinciding results are achieved from independent thermodynamical and mechanical derivations, since the surface tension is equivalently defined from energy and force considerations [150]. The relevant concepts can be briefly summarized as follows. In a series of papers a general balance principle in which the interface is represented by a 2D dividing surface of no thickness 

The 2D dividing surface model was originally proposed by Gibbs [83] (p 219). 

Consider a joint material control volume (CV) containing two phases with phase index k = 1,2) and an interface with area A/(t) moving with the velocity V/, as illustrated in Fig. 3.3. The macroscopic volume occupied by the CV is denoted by V = Vi t) + and A = Ai(t) - - A2(t) is the closed 

With the use of the excess property relations, the first term on the LHS of (3.51) is written as  

Consider a joint material control volume (CV) containing two phases with phase index k(= 1,2) and an interface with area Ai(t) moving with the velocity v/, as illustrated in Fig. 3.3. The macroscopic volume occupied by the CV is denoted by V = Vi t) + Viit), and A = Ai(t) - - A2(0 is the closed macroscopic surface of the CV. Hence it follows that for each phase k separately, an arbitrary non-material control volume Vk t) bounded by a closed surface having partly an external CV surface Ak t) and partly an interface surface A/(f) is employed. lj t) is the line formed by the intersection of A/() with A(t). It is emphazised that the interface is not a material surface because mass transfer may occur between the volumes Vi(t) and 1/2(0 through A/(t). 

The molecular (diffusive) fluxes of the property V through the boundaries of the macroscopically discontinuous CV depicted by V are rewritten in a similar manner, in which we are led to define the surface excess lineal flux of V through A/ [68]  

For orthogonal coordinate systems, a set of unit vectors ia = a a I can be defined. Hence, at each point F on A/ one can define a set of orthonormal basis vectors (ii, i2, n/), where ii and i2 are lying in the tangent plane and n/ is the unit normal to A/. In this frame the vector c associated with points in the surface can be expressed by 

Differentiating ti -ti = 1, with respect to arc length si we find ti -dti/dsi =0 which implies that the vector di jds is perpendicular to the tangent vector ti. Therefore, if we divide di /ds by the length dti/ds, we obtain aunit vector which is orthogonal to ti. 

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Therefore, as basis we formulate the generalized form of the local instantaneous transport equation for the mixture mass, component mass, momentum, energy and entropy in a fixed control volume (CV) on microscopic scales, as illustrated in Fig. 1.1. 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

In scalar mixing studies and for infinite-rate reacting flows controlled by mixing, the variance of inert scalars is of interest since it is a measure of the local instantaneous departure of concentration from its local instantaneous mean value. For non-reactive flows the variance can be interpreted as a departure from locally perfect mixing. In this case the dissipation of concentration variance can be interpreted as mixing on the molecular scale. The scalar variance equation (1.462) can be derived from the scalar transport equation... 

In summary, these models supply interesting information but still rely on experimental fitting to predict the initiation times correctly. Among their weaknesses, there is lack of precise data on the local chemistry of concentrated solutions and lack of prediction of the effect of the potential of the free surfaces. The use of a unidimensional transport equation and the assumption of instantaneous equilibrium of the hydrolysis and solubility reactions are also questionable. 

This additional Eq. (18) was discretized at the same resolution as the flow equations, typical grids comprising 1203 and 1803 nodes. At every time step, the local particle concentration is transported within the resolved flow field. Furthermore, the local flow conditions yield an effective 3-D shear rate that can be used for estimating the local agglomeration rate constant /10. Fig. 10 (from Hollander et al., 2003) presents both instantaneous and time-averaged spatial distributions of /i0 in vessels agitated by two different impellers color versions of these plots can be found in Hollander (2002) and in Hollander et al. (2003). 

For KjpCpD — 1, the relation between concentration and temperature. (9.9), is independent of the nature of the flow, either laminar or turbulent. It applies to both the instantaneous and time-averaged concentration and temperature fields, but only in regions in which condensation has not yet occurred. When the equations of transport for the jet flow are reduced to the form used in turbulent flow, the molecular diffusivity and thermal diffusivity are usually neglected in comparison with the turbulent diffusivities. This is acceptable for studies of gross transport and the time-averaged composition and temperature. However, this frequently made assumption is not correct for molecular scale processes like nucleation and condensation, which depend locally on the molecular transport properties. 

One consequence of the continuum approximation is the necessity to hypothesize two independent mechanisms for heat or momentum transfer one associated with the transport of heat or momentum by means of the continuum or macroscopic velocity field u, and the other described as a molecular mechanism for heat or momentum transfer that will appear as a surface contribution to the macroscopic momentum and energy conservation equations. This split into two independent transport mechanisms is a direct consequence of the coarse resolution that is inherent in the continuum description of the fluid system. If we revert to a microscopic or molecular point of view for a moment, it is clear that there is only a single class of mechanisms available for transport of any quantity, namely, those mechanisms associated with the motions and forces of interaction between the molecules (and particles in the case of suspensions). When we adopt the continuum or macroscopic point of view, however, we effectively spht the molecular motion of the material into two parts a molecular average velocity u = (w) and local fluctuations relative to this average. Because we define u as an instantaneous spatial average, it is evident that the local net volume flux of fluid across any surface in the fluid will be u n, where n is the unit normal to the surface. In particular, the local fluctuations in molecular velocity relative to the average value (w) yield no net flux of mass across any macroscopic surface in the fluid. However, these local random motions will generally lead to a net flux of heat or momentum across the same surface. 

The problem of how to fulfill the average condition (13) both locally and instantaneously cannot be solved completely in a description of the electrolyte solution based on Eqs. (11). These equations, however, give the important information on the time that is needed by the system to return to the local electrostatic equilibrium (Debye relaxation time) after perturbation. The ionic representation of electrolyte solutions is not generally necessary since most of the transport processes in electrolyte solutions take place with a characteristic time larger than the Debye relaxation time. 

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