Similarity: geometric, kinematic, dynamic

Chemical similarity exists between systems that show geometric, kinematic, dynamic, and thermal similarity, if concentration differences between corresponding points in the two systems have a constant ratio to one another. 

In most cases, scale-up by similarity is not always fully achieved. A process may be geometrically similar, but not thermally similar. Depending on the type of process involved, one or several kinds of similarities may be required. These may be geometric, kinematic, dynamic, thermal, kinetic or chemical similarities. 

The step of scaling up a reactor from pilot plant to industrial scale is an issue where much empiricism is still used and where expensive and time-consuming experimental programs are usually required. Complete geometric, kinematic, dynamic, chemical, and thermal similarity cannot be simultaneously achieved in a scale up procedure, and so some differences should be allowed at some point [251]. 

Various methods of scale-up have been proposed all based on geometric similarity between the laboratory equipment and the full-scale plant. It is not always possible to have the large and small vessels geometrically similar, although it is perhaps the simplest to attain. If geometric similarity is achievable, dynamic and kinematic similarity cannot often be predicted at the same time. For these reasons, experience and judgment are relied on with aspects to scale-up. 

For similarity in two mixing systems, it is important lo achieve geometric kinematic and dynamic similarity,... 

Thus, the ratios of the various forces occurring in mixing vessels can be expressed as the above dimensionless groups which, in turn, serve as similarity parameters for scale-up of mixing equipment. It can be shown that the existence of geometric and dynamic similarities also ensures kinematic similarity. 

Dynamic. similarity exists if the systems are similar geometrically and kinematically, and moreover, the ratios of forces between corresponding points are equal in both systems (similar field of forces). 

Thermal similarity exists if, in addition to geometric, kinematic, and dynamic similarity, temperature differences between corresponding points in both systems have a constant ratio. 

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. 

Equality of all groups in Equation 7-13 assures similarity between systems of different sizes. The types of similarity are geometric, kinematic, and dynamic. The last three terms of Equation 7-13 represent the conditions for geometric similarity, which require that all corresponding dimensions in systems of different sizes have the same ratio to each other. For geometric similarity, Equation 7-13 becomes... 

Homologous pumps— Pumps that are similar. Similarities are estabfished dynamically, kinematically, or geometrically. 

In tableting applications, the process scale-up involves different speeds of production in what is essentially the same unit volume (die cavity in which the compaction takes place). Thus, one of the conditions of the theory of models (similar geometric space) is met. However, there are still kinematic and dynamic parameters that need to be investigated and matched for any process transfer. 

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. 

There are three parameters that should be kept similar in scaling tumble blenders. These include geometric, kinematic, and dynamic similarity. This approach is valid for tumble blending without the use of shear (such as an intensifier bar), for low-cohesion materials. Cohesive materials have poor flow and hence blend less efficiently at smaller scales therefore, this technique may overestimate the blending time for scaling up, and underestimate the time when scaling down. 

In order to simulate the action of a flow on an adhering particle we must maintain geometric, kinematic (velocity ratio), and dynamic (mass and force) similarity. We cannot choose these conditions in detail. We can only note that for river-type flows the Froude criterion Fr = v /gZ ensures d5mamic similarity and the Reynolds criterion Re = vd/ v kinematic similarity. Since the forces of gravity prevail over viscosity, the decisive factor in simulating river flows will be the Froude criterion. 

Dynamic similarity is similarity of forces. Since there may be several kinds of forces acting on a fluid particle, it is usually impossible to satisfy dynamic similarity for all of them simultaneously. The justification for comparing observations from one model flow system to another is that the fluid behaviour in both systems is similar thus implying kinematic similarity. Geometric similarity alone does not imply dynamic similarity. The requirement for kinematic similarity is to have both geometric and dynamic similarity. This pa-oduoes geometric similarity of flow patterns and it is this which is of paime importance in this study. 

The dynamic response used to describe fluid motion in the system is bulk velocity. Kinematic similarity exists with geom.etric similarity in turbulent agitation [32]. To duplicate a velocity in the kinematically similar system, the kno m velocity must be held constant, for example, the velocity of the tip speed of the impeller must be constant. Ultimately, the process result should be duplicated in the scaled-up design. Therefore, the geometric similarity goes a long way in achieving this for some processes, and the achievement of dynamic and/or kinematic similarity is sometimes not that essential. 

Because the most common impeller type is the turbine, most scale-up published studies have been devoted to that unit. Almost all scale-up situations require duplication of process results from the initial scale to the second scaled unit. Therefore, this is the objective of the outline to follow, from Reference [32]. The dynamic response is used as a reference for agitation/mixer behavior for a defined set of process results. For turbulent mixing, kinematic similarity occurs with geometric similarity, meaning fixed ratios exist between corresponding velocities. 

Dynamic similarity requires geometric and kinematic similarity in addition lo force ratios at corresponding points being equal, involving properties of gravitation, surface tension, viscosity and inertia [8, 21]. With proper and careful application of this principle scale-up from test model lo large scale systems is often feasible and quite successful. Tables 5-... 

Similarity for scale up, 312, 313 Dynamic, 313 Geometric, 312, 313 Kinematic, 313 Turbulence, 323 Mixing heat transfer... 

Kinematic and dynamic similarities both require geometrical similarity, so the corresponding positions 1 and 2 can be identified in the two systems. Some of the various types of forces that may arise during mixing or agitation will now be formulated. 

Systems (e.g. laboratory installations and full-scale plants) behave similarly, i.e. are similar, if geometric similarity, kinematic similarity, dynamic similarity, thermal similarity, and chemical similarity are preserved. 

Dynamic similarity exists between two systems when, in addition to being geometrically and kinematically similar, the ratios of forces between corresponding points in one system are equal to those in the other. 

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)... 

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