Similitude

For the optimum design of a production-scale fermentation system (prototype), we must translate the data on a small scale (model) to the large scale. The fundamental requirement for scale-up is that the model and prototype should be similar to each other. 

Two kinds of conditions must be satisfied to insure similarity between model and prototype. They are  

Geometric similarity of the physical boundaries The model and the prototype must be the same shape, and all linear dimensions of the model must be related to the corresponding dimensions of the prototype by a constant scale factor. 

Dynamic similarity of the flow fields The ratio of flow velocities of corresponding fluid particles is the same in model and prototype as well as the ratio of all forces acting on corresponding fluid particles. When dynamic similarity of two flow fields with geometrically similar boundaries is achieved, the flow fields exhibit geometrically similar flow patterns. 

The first requirement is obvious and easy to accomplish, but the second is difficult to understand and also to accomplish and needs explanation. For example, if forces that may act on a fluid element in a fermenter during agitation are the viscosity force Fv, drag force on impeller F0, and gravity force FG, each can be expressed with characteristic quantities associated with the agitating system. According to Newton s equation of viscosity, viscosity force is 

Many engineering problems are so complex that a complete analytical solution is impractical, either due to lack of time or because the general problem cannot be resolved into components for which known solutions exist. Even if a problem can be divided into subproblems that can be solved, the composite solution will sometimes be inadequate due to important interactions among the subproblems. In such cases, it often proves expedient to study the performance of a physical model of the full-scale prototype. One of the most important uses of dimensional analysis lies in the design of engineering models and the interpretation of their performance. 

A true model of an engineering situation is one for which all Pi quantities associated with the problem are equal for both model and prototype. When this is the case, a condition of dimensional similitude is said to exist. In designing a model, certain variables may be assigned arbitrary values, but the number of these must not exceed the number of dimension-ally independent quantities. The remaining variables must then be modeled so that all Pi quantities are the same for model and protot5q)e in order that a true model pertains. 

Usually, scale models are used for which all dimensions stand in a fixed ratio and all angles are identical in model and prototype. Such a model is said to be geometrically similar to the prototype. When geometrical similitude is established, one of the arbitrary scales is fixed. 

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The nucleophilic carbon of ketomethylene compounds can react with anhydrobases of different species in a basic medium. This reaction presents a narrow similitude with -CHj attack. The resulting dye, neut-rodimethine cyanine either mesomethyl-substituted or not. varies with the nature of the anhydro base (Scheme 30) (53. 54). 

Rearrangement to an open chain imine (165) provides an intermediate whose acidity toward lithiomethylthiazole (162) is rather pronounced. Proton abstraction by 162 gives the dilithio intermediate (166) and regenerates 2-methylthiazole for further reaction. During the final hydrolysis, 166 affords the dimer (167) that could be isolated by molecular distillation (433). A proof in favor of this mechanism is that when a large excess of butyllithium is added to (161) at -78°C and the solution is allowed to warm to room temperature, the deuterolysis affords only dideuterated thiazole (170), with no evidence of any dimeric product. Under these conditions almost complete dianion formation results (169), and the concentration of nonmetalated thiazole is nil. (Scheme 79). This dimerization bears some similitude with the formation of 2-methylthia-zolium anhydrobase dealt with in Chapter DC. Meyers could confirm the independence of the formation of the benzyl-type (172) and the aryl-type... 

Analysis of a method of maximizing the usefiilness of smaH pilot units in achieving similitude is described in Reference 67. The pilot unit should be designed to produce fully developed large bubbles or slugs as rapidly as possible above the inlet. UsuaHy, the basic reaction conditions of feed composition, temperature, pressure, and catalyst activity are kept constant. Constant catalyst activity usuaHy requires use of the same particle size distribution and therefore constant minimum fluidization velocity which is usuaHy much less than the superficial gas velocity. Mass transport from the bubble by diffusion may be less than by convective exchange between the bubble and the surrounding emulsion phase. 

G. Murphy, Similitude in Engineerings The Ronald Press Co., New York, 1950. 

S. J. Kline, Similitude and Approximation Theory McGraw-HiU Book Co., New York, 1965. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

If using dimensionless parameters obtained through dimensional analysis, or similitude, the data will eollapse into a single eurve. In this ease, the useful parameters are redueed flow, q, redueed power. 

The utility of K or any elastic plastic fracture mechanics (EPFM) parameter to describe the mechanical driving force for crack growth is based on the ability of that parameter to characterize the stress-strain conditions at the crack tip in a maimer which accounts for a variety of crack lengths, component geometries and loading conditions. Equal values of K should correspond to equal crack tip stress-strain conditions and, consequently, to equivalent crack growth behavior. In such a case we have mechanical similitude. Mechanical similitude implies equivalent crack tip inelastic zones and equivalent elastic stress fields. Fracture mechanics is... 

The few studies which addressed the fracture mechanics behavior of very small cracks generally revealed a quantitative departure from behavior determined at longer crack lengths. This result may be attributed to a departure from perfect mechanical and microstructural similitude between long and small cracks. 

One of tlie limitations of dimensional similitude is tliat it shows no dueet quantitative information on tlie detailed meehanisms of the various rate proeesses. Employing the basie laws of physieal and eheiTtieal rate proeesses to matliematieally deseiibe tlie operation of tlie system ean avert this shorteoiTung. The resulting matliematieal model eonsists of a set of differential equations tliat are too eomplex to solve by analytieal metliods. Instead, numerieal methods using a eomputerized simulation model ean readily be used to obtain a solution of tlie matliematieal model. 

A combination of dimensional similitude and the mathematical modeling technique can be useful when the reactor system and the processes make the mathematical description of the system impossible. This combined method enables some of the critical parameters for scale-up to be specified, and it may be possible to characterize the underlying rate of processes quantitatively. 

For example, Flamilton et al. [10] employed dimensional similitude in eombination with mathematieal modeling in the design of a pilot plant and in evaluating the results to provide the basis for seale-up to a eommereial seale plant involving a reaetion of the type... 

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. 

Exact geometric similitude between the model and the real-world prototype... 

Equation (14.91) contains only the mass flow ratio /u as a characteristic number of the mechanics of similitude of the mixture. All the other irnpor rant factors, such as particle size, solid density, etc., are contained in the additional pressure-loss coefficient of the solid particles, A, which is determined separately for each material. 

Skoglund, V. J., Similitude Theory and Applications, International Textbook Company, 1967. 

Zl. Zenkevitch, B. A., Similitude relations for critical heat loading in forced liquid flow At. Energ., USSR, 4, 74 (1958) [English transl. Merte, H., WAPD-AD-Th-539 (1959)]. 

Benzylic, allylic and propargylic positions enhance the cathodic cleavage rate of C— heteroatom bonds as, for example, in the reduction of benzylic and allylic halides or alcohols56. Similar activated sulphones, due to their acidity, are in a class apart. Figure 8 shows the similitude between the cathodic behaviour of an allylic sulphone and its isomer, i.e., the corresponding vinylic sulphone when the electrolyses are run in an aprotic solvent. However, in the presence of an excess of proton donor, discrepancies appear. 

Sauter mean, as in dSM, Sauter mean diameter subcooled condition superheated condition transition boiling, or Taylor bubble crossflow due to droplet deposition a group of thermodynamic similitude... 

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

Fig. 31. Example of neuroactive indole alkaloids from plants. Note the similitude of chemical structure of harmine, harmaline, and serotonin.

A technique which can assist in the scale-up of commercial plants designs is the use of scale models. A scale model is an experimental model which is smaller than the hot commercial bed but which has identical hydrodynamic behavior. Usually the scale model is fluidized with air at ambient conditions and requires particles of a different size and density than those used in the commercial bed. The scale model relies on the theory of similitude, sometimes through use of Buckingham s pi theorem, to design a model which gives identical hydrodynamic behavior to the commercial bed. Such a method is used in the wind tunnel testing of small model aircraft or in the towing tank studies of naval vessels. 

For all three limiting cases identified above, similitude can be obtained by maintaining constant values for the dimensionless parameters,... 

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