Scaling Models

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

Cochran, L. S., and J. E. Cerniak. 1992. Full- and model-scale cladding pressures on the Texas Tech University experimental building. /. Wind Engineering and Industrial Aerodynamics, vols. 41-44, pp. 1589-1600. 

Important conclusions can be drawn from the general modeling Eq. (13.79). The equation shows that the required prototype flow rates are directly proportional to the model flow rates. For scaling, the equation shows that the prototype flow rate has a strong dependence on the accuracy of the model scale (5/3 power). Both of these parameters are easy to establish accurately. The flow rate is rather insensitive (varies as the 1/3 powet) to the changes in the model and prototype heat flow tates, densities, and temperatures. This is desirable because an inaccuracy in the estimate of the model variable will have a rather small effect on the tesulting ptototype flow rate. 

E. J. Mackay and K. S. Sorbie. Modelling scale inhibitor squeeze treatments in high crossflow horizontal wells. J Can Petrol Technol, 39(10) 47-51, October 1998. 

This paper reviews NSE measurements on polymer melts, networks and solutions, published since the first successful NSE study on polymers [16] was performed in 1978. The experimental observations are discussed in the framework of related microscopic models, scaling predictions or other theoretical approaches. 

The particle diameters in the model scale by the same factor as the bed diameter, by the ratio of the kinematic viscosities to the two-thirds power. Equating the Froude number and rearranging,... 

Vilan A (2007) Analyzing molecular current-voltage characteristics with the Simmons tunneling model scaling and linearization. J Phys Chem C 111 4431-4444... 

Andersson, B., "Model Scale Compartment Fire Tests With Wall Lining Materials", Report LUTVDG/(TVBB-3041), Department of Fire Safety Engineering, Lund University,... 

Both Hartree-F ock and density functional models actually formally scale as the fourth power of the number of basis functions. In practice, however, both scale as the cube or even lower power. Semi-empirical models appear to maintain a cubic dependence. Pure density functional models (excluding hybrid models such as B3LYP which require the Hartree-F ock exchange) can be formulated to scale linearly for sufficiently large systems. MP2 models scale formally as the fifth power of the number of basis functions, and this dependence does not diminish significantly with increasing number of basis functions. 

It is obvious that this conclusion is wrong Dimensional analysis is a method based on logical and mathematical fundamentals (2,6). If relevant parameters cannot be listed because they are unknown, one cannot blame the method. The only solution is to perform the model measurements with the same material system and to change the model scales. 

Is one model scale sufficient or should tests be carried out in models of different sizes One model scale is sufficient if the relevant numerical values of the dimensionless numbers necessary to describe the problem (the so-called process point in the pi space describing the operational condition of the technical plant) can be adjusted by choosing the appropriate process parameters or physical properties of the model material system. If this is not possible, the process characteristics must be determined in models of different sizes, or the process point must be extrapolated from experiments in technical plants of different sizes. 

If 0 1, the correlation function Y(r,t) becomes practically stationary. Indeed, in terms of the auto-model (scaled) variable 77 = r/ o we get... 

This qualitative analysis of length scales can also guide discussion of appropriate modeling scales and techniques. For the properties nearest the interface, details of the electronic structure are required, whereas to describe the solvent near the interface, an atomic scale description is needed. Finally for... 

The "correlative" multi-scale CFD, here, refers to CFD with meso-scale models derived from DNS, which is the way that we normally follow when modeling turbulent single-phase flows. That is, to start from the Navier-Stokes equations and perform DNS to provide the closure relations of eddy viscosity for LES, and thereon, to obtain the larger scale stress for RANS simulations (Pope, 2000). There are a lot of reports about this correlative multi-scale CFD for single-phase turbulent flows. Normally, clear scale separation should first be distinguished for the correlative approach, since the finer scale simulation need clear specification of its boundary. In this regard, the correlative multi-scale CFD may be viewed as a "multilevel" approach, in the sense that each span of modeled scales is at comparatively independent level and the finer level output is interlinked with the coarser level input in succession. 

According to our knowledge of the pertinent pi-space (Fig. 1), Re = idem implies Eu = idem. The numerical value of the Euler number EuM, measured in the model-scale at the given ReM value, therefore corresponds to that of the full-scale plant. This then allows us to determine the numerical value of Ap, in the industrial plant from the numerical value of EuM in the model and the given operational parameters ... 

However, in the transition from model to full-scale, a complete similarity cannot be achieved. This is because in using the same material system ReH = p v L/H = idem, v /L = idem cannot be ensured at the same time. It is recommended to use the same material system, but to change the model scale. An exception to this is represented by pure hydrodynamic processes in the creeping flow region (p irrelevant) at steady-state and isothermal conditions. Here mechanical similarity can be obtained in spite of constant physical properties see Example 26 Single-screw machines. 

Occasionally, one could have read about the failures of the Theory of Similarity or of its limits. However, this criticism has arisen when, due to some physical reasons, a complete similarity could not be achieved (see e.g. remarks of Damkohler [113] on p. 183) or the scale-up criterion could not have been worked out with certainty because the measuring conditions did not allow it (false model scale, wanted sensitivity of the target quantity, non-availability of the model material system, ignorance about relevant physical properties, such as in foams and sludges, etc.). 

Model Scale and the State of Flow Problems Concerning Mini Plants... 

In section 7.3 (Experimental Techniques for Scale-up), it has already been mentioned that the model size depends on the scale factor p = 1X/1M and on the measuring accuracy attainable in the tests. At p = 10 a measuring accuracy of 10 % will possibly not suffice and one will therefore have to chose a larger model scale to reduce the scale factor p. 

Measuring accuracy often depends on model scale. This is demonstrated with two examples taken from the area of stirring technology. 

Compared with this, so-called surface aeration brings about a comparatively modest oxygen uptake. Therefore, it is important to very accurately measure and/or to use a larger model scale. The latter is advisable because the diameters of the full-sized surface aerators amount to d-r 3 m and, therefore, the scale factor surpasses i= 10, even if a relatively large laboratory stirrer diameter of dM = 0.3 m is used. 

In case of a realistic scale factor of p = 10, this means that the efficiency of the full-scale device would only be 32 % of that of model scale. 

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