Principles of similarity

The most important stages of similarity in process and bioprocess engineering are (a) geometric similarity, (b) mechanical similarity, (c) thermal similarity, and (d) chemical and biochemical similarity. 

These reactors/fermenters will be geometrically similar if for example  

Where H is the height and D the diameter of the reactor/fermenter. As expressed by Johnstone and Thrings (1957), two bodies are geometrically similar when to every point in the one body there exists a corresponding point in the other.  

Mechanical similarity includes static, kinematic, and dynamic similarities. In addition, mechanical similarity is an extension of geometric similarity. 

Thermal similarity is related to heat flows. Temperature differences between two points at the same instant in time in the model must be equal to the temperature differences at the corresponding points at the same instant in the prototype. 

When scaling the process, the model and prototype must have Reynolds numbers in the same regime (i.e., laminar or turbulent flow) in order to achieve similar results. Ideally, the two reactors would have nearly identical or identic Reynolds numbers. In order to satisfy this requirement for scaleup, the model and prototype must be similar to each other with respect to reactor design, fluid flow, and physical dimensions. According to the principle of similarity, for every process condition and point in the model, there must be a corresponding condition and point in the prototype. This prindple is applied to the scaling process by observing similar requirements for several other dimensionless ratios or variables that must be treated in a similar fashion. 

The process of scaleup can be simplified by dividing the principle of similarity into four separate states or categories  

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Figure 10.3-16. The principle of similarity searches. The query (target, precursor) as well as the catalog compound are transformed by the criterion maximum oxidation state". Since the transformation for both compounds results in the samie transformed structure, the catalog compound is presented to the user as a suitable starting material. The comparison of the structure is performed by a hashcode algorithm.

The principle of similarity [Holland (1964), Johnstone and Thring (1957)] together with the use of dimensionless groups is the essential basis of scale-up. The types of similarity relevant to liquid mixing systems together with their definitions are listed as follows. 

In order to extrapolate, use is made of what is known as the extended principle of similarity, where equations of the form... 

Detonation Principle of Similarity, Its Application in Shock Waves and Scaling Effects. See "Detonation Shock Waves Principle of Similarity, Its Application and Scaling Effects in ... 

Normal Reflection of Shock and Rarefaction Waves (82-4) Types of Interaction (86) Normal Reflection of Rarefaction (86-7) Normal Refraction of Shock and Rarefaction Waves (87-8) Head-on Collisions (88-9) Oblique Intersections (89 90) Oblique Interactions (90-1) Spherical Shock Waves (97-8) Distinction Between Shock and Detonation Fronts (163-66) Application to Solid Explosives (166-68) Principle of Similarity and Its Application to Shock Waves (307-10) Effects of Ionization in the Shock Front (387-90)... 

Detonation, Shock Wave Principle of Similarity, Its Application and Sea ling... 

Another application of the principle of similarity is to the calculation of maximum quantities of explosives and numbers of persons allowed in the rooms of any building used for expl operations. The rules to be followed are outlined in the US Army "AMC Safety Manual , AMCR 385-224 (June 1964) The principle of similarity is also used in evaluation of structures for ammunition manufacture and for calculation of safe distances between the buildings manufg or storing expls, distances to roads, inhabited area, etc... 

Irrespective of the approach taken to scale-up, the scaling of unit operations and manufacturing processes requires a thorough appreciation of the principles of similarity. Process similarity is achieved between two processes when they accomplish the same process objectives by the same mechanisms and produce the same product to the required specifications. Johnstone and Thring (56) stress the importance of four types of similarity in effective process translation (1) geometric similarity (2) mechanical (static, kinematic, and dynamic) similarity (3) thermal similarity and (4)... 

The important concept for scale-up is the principle of similarity (1-6). When scaling up any mixer/granulator (e.g., planetary mixer, high-speed mixer, pelletizing dish, etc.), the following three types of similarity need to be considered geometric, kinematic, and dynamic. Two systems are geometrically similar when the ratio of the linear dimensions of the small-scale and scaled-up system are constant. 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

The principle of similarity requires geometrically similar systems to be compared at equal values of the appropriate dimensionless groups, called similarity criteria. However, in many cases, this requirement is not possible to be fulfilled, since it may be either impractical or impossible to be achieved. 

The classical principle of similarity can then be expressed by an equation of the form... 

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. 

The scaleup process is not just a matter of plugging values into prescribed equations, nor can exact scaleup criteria be obtained from generalized correlations for certain types of equipment. Instead, in a good scaleup approach, all variables that describe the process are determined desired process conditions and the magnitude of the scaleup taken into account and a design selected. Scaleup designs are based almost exclusively on the principle of similarity. 

The above example illustrates a general principle of similarity analysis geometrical similarity corresponds to a topological equivalence [108]. In order to formulate the above principle in more precise terms, we have to specify a general framework applicable to most molecular shape similarity problems. The (Pyfysimilarity concept [108] provides such a general framework. 

ABSTRACT A novel reactor configuration has been developed in our laboratory which addresses the heat transfer limitations usually encountered in vacuum pyrolysis technology. In order to scale-up this reactor to an industrial scale, a systematic study on the heat transfer, the chemical reactions and the movement of the bed of particles inside the reactor has been carried out over the last ten years. Two different configurations of moving and stirred bed pilot units have been used to scale-up a continuous feed vacuum pyrolysis reactor, in accordance with the principle of similarity. A dynamic model for the reactor scale-up was developed, which includes heat transfer, chemical kinetics and particle flow mechanisms. Based on the results of the experimental and theoretical studies, an industrial vacuum pyrolysis reactor, 14.6 m long and 2.2 m in diameter, has been constructed and operated. The operation of the pyrolysis reactor has been successful, with the reactor capacity reaching the predicted feed rate of 3000 kg/h on a biomass feedstock anhydrous basis. 

In a properly designed industrial scale reactor, feedstock conversion is achieved at a certain throughput capacity. In order to scale-up the reactor, heat and mass transport phenomena must be studied. This includes heat transfer phenomena, feedstock conversion kinetics and the movement of particles inside the reactor. In this work, both experimental and theoretical studies were carried out to investigate these phenomena. Two different configurations of moving and stirred bed reactors, the batch scale rotative and a continuous feed Process Development Unit (PDU), have been used to generate the data in accordance with the principle of similarity. A dynamic model to scale-up the reactor was then tested. 

Two different configurations of moving and stirred bed reactors, the batch scale rotative and PDU reactors, have been used for the tests in accordance with the principle of similarity. 

Apart from equifinality, another important property of these operator networks is the periodic character of the changes in the properties and nature of the clusters constituting them. This peculiarity is clearly revealed when such networks are transformed into periodic tables grouped, for instance, in accordance with the principle of similarity of formal electric charge for corresponding clusters, as shown in Fig. 4.10 for even (represented by carbon) and for odd (represented by phosphorus) elements. Such tables of... 

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