Dimensional similitude

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... 

One of the limitations of dimensional similitude is that it shows no direct quantitative information on the detailed mechanisms of the various rate processes. Employing the basic laws of physical and chemical rate processes to mathematically describe the operation of the system can avert this shortcoming. The resulting mathematical model consists of a set of differential equations that are too complex to solve by analytical methods. Instead, numerical methods using a computerized simulation model can readily be used to obtain a solution of the mathematical model. 

While the simplest method for scaleup of a chemical reactor is to start with a small working unit that provides the desired results and then progressively build larger units, this method is usually too expensive and time-consuming. Consequently, more semiempirical approaches, such as dimensional similitude, are taken. Other approaches include mathematical modeling or a combined dimensionless similitude/mathematical modeling method but are not involved in the scaleup example discussed here. 

Dimensional similitude is based on principles of similarity and uses dimensionless ratios of the physical and chemical parameters that govern the working model to design the scaled-up prototype. This method is followed when the prototypical unit is expected to be dimensionally similar to the small-scale unit. The degree of success in using dimensional similitude as an approach to scaleup depends mainly on the extent to which the physical parameters necessary to achieve a desired result influence the process. 

Frequently, complete dimensional similitude cannot be reached between two reactors with widely different scales as a result, this method is usually limited in practice to relatively simple chemical reaction systems. For a cleaning vessel in which the reaction rate is very fast and the process is governed by the physical rates of the process, e.g., mass transfer, heat transfer, etc., the dimensionless groups describing the process consist of fluid mechanic and thermodynamic quantities. The cleaning process can usually be a relatively simple mechanism to describe, making dimensional similitude more easily achievable. 

For R —> 0 ( point particles), theories of particle and molecular diffusion are equivalent. Schmidt numbers for particle diffusion are much larger than unity, often of the same order of magnitude as for molecular diffusion in liquids. The principle of dimensional similitude tells us that the results of diflusion experiments with liquids can be used to predict rates of diffusion of point particles in gases, at the same Reynolds number. 

Safoniuk, M, Grace JR, Hackman L, McKnight CA. Use of dimensional similitude for scale-up of hydrodynamics in three-phase fluidized beds. Chem Eng Sci 54 4961-4966, 1999. 

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. 

Complete the following table to satisfy dimensional similitude [Eq. (7.2)]. 

Complete the following table to satisfy dimensional similitude. 

It is desired to estimate the power developed in a large Kaplan turbine that will have a rotor diameter of 20 ft by building a 1 20 scale model and operating it under conditions of dimensional similitude, such that the measured power for the model may be used to predict the horsepower of the turbine. The prototype, of course, will operate with water, and the model will also be tested with water. 

Therefore, the initial velocity of the model car should be 52.5/5 = 10.5 mph. When the values of and are determined as indicated above, then it follows from the principle of dimensional similitude that ... 

Sanchez, J.L., Ruiz, R.S., Alonso, R, Ancheyta, J. 2008. Evaluation of the hydrodynamics of high-pressure ebullated beds based on dimensional similitude. Catal. Today 130 519-526. 

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