Mixed equilibrium-dynamic modeling

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

In general, r4 must be computed from a dynamic model for Hfle. /). However, for fully developed scalar fields (equilibrium mixing), the mixing time can be approximated from a model spectrum for ( c). 

Assuming a local equilibrium at the interface, the adsorption isotherm can be used at any t to describe the relationship between F and coi, which is needed to solve Eq. (4.94). This assumption also allows to used Eq. (4.91). So far, no theories describe the adsorption dynamics at liquid-liquid interfaces when a local equilibrium cannot be assumed. For liquid-vapour systems, some models are available to describe this situation, often called mixed adsorption dynamics. 

Since we assume the atmosphere and ocean are in chemical equilibrium, and the input from rivers and burial in the sediments are small compared to the other fluxes, the entire dynamics of the model is reduced to the rate of surface-deep mixing and the sinldng of particles. (For simplicity, DOC transport is not considered in this simple model.) One can see that for a steady state to be achieved the flux of carbon to the surface ocean must equal the sinking flux of particles. The mean residence time for deep water is that determined by natural measurements (see Chapter 6) 500-1000 y. 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

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