Dynamic similitude

For modeling, the similitude laws governing modeling must be followed. The topics of dynamic similitude and theory of models are discussed in most textbooks on fluid mechanics,and only the resulting equations are discussed here. 

A dynamic similitude embodies both hydrodynamic and thermodynamic similitudes. A hydrodynamic similitude requires a similar flow pattern and similar veloc-... 

In financial matters, a similar technique, ratio analysis, is used to judge whether a company is healthy witli respect to industry standards. In ratio analysis most of die quantities compared are quantities of money, so the ratio of one quantity of money to another quantity of money is, by definition, a new dimensionless ratio (like die Reynolds number or lift coefficient), and these financial ratios can be used to compare businesses in similar industries or businesses (in engineering, such comparison is referred to as dynamic similitude). 

It is recognized that a phenomenon which occurs in an apparatus or a plant at different scales (various geometrical dimensions) presents the same evolution for all scales only if the conditions of the geometric similarity (geometric similitude), material similarity (material similitude), dynamic similarity (dynamic similitude) are respected and if the phenomenon shows the same initial state in all cases. The parametric description of a phenomenon occurring at laboratory and prototype scales is given in Table 6.7. In this case, we consider that the initial state of the phenomenon is identical for both scales. 

Two geometrically similar systems are called kinematically similar if they have the same ratio of velocities between the corresponding system points. Two kinematically similar systems are dynamically similar when they have the same ratio of forces between the corresponding points. Dynamic similitude for wet granulation would imply that the wet mass flow patterns in the bowl are similar. 

This study was done on Collette Oral Mixers (8, 25, 75, and 600 L) and followed the accepted—and by now, standard—methodology developed earlier. The problem with the scale-up in the Oral mixers was the lack of geometric similitude there was significant distortion factor between the bowl geometries at different scales. In addition, the researchers had to take into account the lack of dynamic similitude because of... 

Streeter, V.L., and Wylie, E.B. (1983). Fluid Mechanics, First SI Metric Edition, McGraw-Hill, New York, Chapter 4, Dimensional analysis and dynamic similitude. 

The Reynolds number is a dimensionless number used in fluid dynamics to determine dynamic similitude [4] and is genetically defined as... 

Safoniuk et al. (1999) presented a method of scaling three-phase fluidized beds based upon achieving geometric and dynamic similitude with the aid of the Buckingham Pi theorem, using the following dimensionless groups ... 

Sanchez et al. (2008) tested the scale-up method for three-phase fluidized bed hydrodynamics proposed by Safoniuk et al. (1999), based upon the principles of dynamic similitude. The authors matched several systems operated at pressures in the range of 0.79-15.6MPa (prototype) with model systems operated at atmospheric pressure. Experiments were carried out to test this technique by comparing global phase holdups, and flow regime in systems where the five dimensionless groups were matched operated at significantly different pressures. 

Physical Properties of Fluids and Solids Used in Geometric and Dynamic Similitude Scaling Method... 

In scaling-down of EBR, it has to be assured that hydrodynamic characteristics be identical, maintaining phase holdups unchanged. The key parameters in scaling-down are catalyst particle diameter, reactor diameter, as well as liquid and gas superficial velocities. For scaling-up, methods based on geometric and dynamic similitude are very useful. 

If the process takes place along the z-axis, then we can write that X = z. Considering now that and e j2 are the average or mean probabilities for the process evolution with +v or -v states at the z position, we can observe a similitude between system (4.106) and Eqs. (4.31) and (4.32) that describe the model explained in the preceding paragraphs. The solution of the system (4.106) [4.5] is given in Eq. (4.107). It shows that the process evolution after a random movement depends not only on the system state when the change occurs but also on the movement dynamics ... 

Because all the dimensional material and dynamic multiplexes are reported only for the LM or for the prototype, we can combine these r q dimensional multiplexes and the characteristic geometric parameters (so we have r q-tl dimensional terms) to formulate new dimensionless independent multiplexes. It is not difficult to observe that these independent dimensionless multiplexes are the dimensionless pi groups that characterize the evolution of the phenomenon. If III, n2,.ris represent the dimensionless groups that characterize the evolution of the phenomenon in the laboratory device or in the prototype, we transform the similitude postulate into the next new statement ... 

Fortunately, numerical modeling despite its many limitations associated with grid resolution, choice of turbulence model, or assignment of boundary conditions is not intrinsically limited by similitude or scale constraints. Thus, in principle, it should be possible to numerically simulate all aspects of fires within canopies for which realistic models exist for combustion, radiation, fluid properties, ignition sources, pyrolysis, etc. In addition it should be possible to examine all interactions of fire properties individually, sequentially and combined to evaluate nonlinear effects. Thus, computational fluid dynamics may well provide a greater understanding of the behavior of small, medium, and mass fires in the future. 

On the basis of experimental data, with the use of similitude theory, a relationship has been established between the drag coefficient Cx and the Reynolds number Re [273]. The particle drag coefficient Cx is the ratio of the dragF r to the product of the particle midsection S by the dynamic pressure... 

In summary, we have correlated the data from two experiments to generate an approximate similitude equation. The same equation provides reasonable agreement with both experiments. In addition, an entirely different similitude equation has been generated by correlating a set of hydro-dynamic calculations (as though they were experiments) for both TNT and thermite steam explosions. This independent equation predicted impulses that were about 60% of the measured values. Part of this under-prediction could be due to internal reflections and superposition of shock waves in the experiments. Another part could be due to experimental efficiencies higher than those assxamed in the set of calculations. In either case, this analysis procedure implies that the experimental explosions had efficiencies that could have been as much as 50-100% of theoretical maximums. Many more experiments are required, however, before these efficiencies could be considered confirmed. 

Application of chemical similitude to reactions leads to a number of interesting conclusions. It appears that for all types of reactions, temperatures or the temperature-time profile must be the same in the large and small units. In homogeneous systems the reaction times and the initial concentrations must be the same in heterogeneous ones, the product of the reaction time and the interfacial area per unit volume of the reactor must be the same for both. Both dynamic and chemical similarity ordinarily are rarely achieved simultaneously e.g., if reaction times are preserved, the Reynolds numbers which also involve linear or mass velocities, cannot also be preserved. 

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