Boundary layer separation model

Boundary layer separation models. In this class of model, the critical heat flux phenomenon is considered to be analogous to the phenomenon of boundary layer separation from a permeable plate through which gas is flowed in a direction normal to the flow over the plate. This mechanism was initially suggested by Kutateladze and Leontiev [312] and was further developed by Tong [313] and others and more recently by Celata et al. [314], This method of prediction leads to an equation of the form... 

The prediction of turbulent boundary-layer separation by MVF methods has not been very successful. Indeed, it may be appropriate to identify turbulent separation in terms of the turbulence near the wall, and this will require use of a more sophisticated model (i ITE or MRS), quite possibly in their full (rather than boundary-layer) form. 

The first example is related to thermal discharges with electron temperature deviating from the temperature of heavy particles, which can take place, in particular, in boundary layers separating plasma from electrodes and walls. In this case, the two-temperature statisties and thermodynamies can be developed (Boulos et al., 1994). These models assume that partition functions depend on two temperatures. Electron temperature determines the partition functions related to ionization processes, whereas chemical processes are determined by the temperature of heavy particles. The partition functions can then be applied to calculate thermodynamic functions, composition, and properties. An example of such a calculation of composition in two-temperature Ar plasma is given in Fig. 3-2. 

Fluid boundary layer separation at vascular bifurcations or curves (as found in the carotid and coronary arteries) may be considered 2D if evaluating centerline flow (Steinman and Ethier, 1994). Along this plane, the secondary flows brought on by the vessel cross-sectional curvature will not affect the flow patterns. These models may be used to evaluate boundary layer separation in the carotid artery, the coronaries, and graft anastomoses. 

Another possible source of nonideal behavior and large pressure fluctuations would be boundary layer separation caused by the interaction with the reflected shock wave. Boundary layer separation and bifurcated reflected shock waves are observed under certain conditions in shock tubes with nonreactive flows. Mark formulated a simple model that predicts the occurrence of bifurcation shock bifurcation and boundary layer separation will occur when the pressure jump across the reflected shock exceeds the maximum stagnation pressure possible in the cold boundary layer fluid. Numerical calculation for the present situation reveals bifurcation would not be expected when the detonation first reflects. This is a situation peculiar to detonations and is due to the much lower reflected-shock pressure ratio relative to that which would be produced by reflecting a shock wave of comparable strength. Consideration of the reflected shock motion at later times indicates that bifurcation would not occur until after the shock had reflected from the far end of the tube. 

The SST model uses the standard k-e model in the bulk flow and incorporates the transport of turbulent shear stress while using the k-(0 model in the boundary layer. This model can therefore be used over a wider range of operating parameters than the standard k-(0 model, such as the flow around curved bodies and flows with separating boundary layers. Prieske et al. (2007) examined the relationship between circulation velocity using the SST model and aeration flow rate in a pilot-scale MBR. 

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

Phase-averaged values of 4 in a plane midway between two baffles of a stirred tank have been plotted in Fig. 1 (from Hartmann et al., 2004a) for two different SGS models (Smagorinsky and Voke, respectively) in LES carried out in a LB approach. The highest values, i.e., the strongest deviations from isotropy, occur in the impeller zone, in the boundary layers along wall and bottom of the tank, and at the separation points at the vessel wall from which the anisotropy is advected into the bulk flow. In the recirculation loops, the turbulent flow is more or less isotropic. 

The formation of concentration gradients caused by the flow of ions through a single cationic membrane is shown in Figure 10.8. As in the treatment of concentration polarization in other membrane processes, the resistance of the aqueous solution is modeled as a thin boundary layer of unstirred solution separating the... 

Concentration polarization can dominate the transmembrane flux in UF, and this can be described by boundary-layer models. Because the fluxes through nonporous barriers are lower than in UF, polarization effects are less important in reverse osmosis (RO), nanofiltration (NF), pervaporation (PV), electrodialysis (ED) or carrier-mediated separation. Interactions between substances in the feed and the membrane surface (adsorption, fouling) may also significantly influence the separation performance fouling is especially strong with aqueous feeds. 

Schuette and McCreery [34] demonstrated that with decreasing wire diameter there was a significant increase in current enhancement and modulation depth. This approached 100% modulation for a wire of diameter, d = 25 pm vibrated at 160 Hz. They showed that in these circumstances, for low Re numbers, the limiting current strictly followed the wire velocity and used [6] an empirical power-law correlation of mass-transfer coefficient to flow velocity /lim = /min(l + A/ cos(ft>.f)f) with s 0.7. They also noted that the frequency and amplitude dependence of the mean current, and the modulation depth, was linked to whether the flow was strictly laminar or not. Flow modelling indicated that for Re > 5 where Re = u dlv, there was separation of the boundary layer at the wire surface, when aid 1. For Re > 40 the flow pattern became very irregular. Under these circumstances, a direct relation between velocity and current should be lost, and they indeed showed that the modulation depth decreased steeply with increase of wire diameter, down to 10% for 0.8 mm diameter wire. 

DePaolo, 1979 O Nions et al., 1979), divide the mantle into two convectively isolated layers with a boundary at 670 km. Such models incorporate the degassing of the upper mantle reservoir to the atmosphere. In order to explain the high OIB He/ He ratios, the underlying gas-rich reservoir is isolated from the degassing upper mantle. Therefore, these layered mantle models can be considered to incorporate two separate systems the upper mantle-atmosphere and the lower mantle. There is no interaction between these two systems, and the lower mantle is completely isolated except for a minor flux to OIB that marks its existence. It is further assumed that the mantle was initially uniform in noble gas and parent isotope concentrations, so that both systems had the same starting conditions. Note that various modifications to this basic scheme have been proposed, and are discussed below (Allegre et a/., 1983, 1986). 

According to the assumptions in Section 6.2.1, the liquid phase concentration changes only in axial direction and is constant in a cross section. Therefore, mass transfer between liquid and solid phase is not defined by a local concentration gradient around the particles. Instead, a general mass transfer resistance is postulated. A common method describes the (external) mass transfer mmt i as a linear function of the concentration difference between the concentration in the bulk phase and on the adsorbent surface, which are separated by a film of stagnant liquid (boundary layer). This so-called linear driving force model (LDF model) has proven to be sufficient in... 

The simple one-dimensional models of multicomponent condensation and cocurrent separation processes described in this chapter are well able to model the performance of a wetted-wall column operated by Modine (1963) and a vertical tube condenser operated by Sardesai (1979). The results obtained with the one-dimensional model are probably good enough for design purposes it is doubtful if a more sophisticated boundary layer analysis could yield any better results. 

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