Similarities kinematic

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. 

Thermal similarity is achieved in the ACR by providing a temperature profile which can be held geometrically similar when scaled. The temperature profile drives the ACR chemical kinetics and is a combined result of the heat transfer attributable to cracking and the heat effects caused by the bulk fluid movement. Thus, true thermal similarity in the ACR can only be achieved in conjunction with chemical and kinematic similarity. Kinematic similarity in the ACR is made possible during scale-up by forcing geometrically similar velocity profiles. The ACR temperature, pressure, and velocity profiles are governed by compressible gas dynamics so that an additional key scale parameter is the Mach number. 

A distinct similarity exists between reactions involving the exchange of a vd W bond and recombination reactions of vdW molecules, since in both the bond that breaks is a weak one. Therefore similar kinematic constraints may be expected. 

The viscosity index (ASTM D-2270, IP 226) is a widely used measure of the variation in kinematic viscosity due to changes in the temperature of petroleum between 40°C and 100°C (104°F and 212°F). For crude oils of similar kinematic viscosity, the higher the viscosity index the smaller is the effect of temperature on its kinematic viscosity. The accuracy of the calculated viscosity index is dependent only on the accuracy of the original viscosity determination. 

Regardless of the strategy utilised in the miniaturisation process, it should be noted that miniaturisation does not necessarily involve simple downscaling of the dimensions of the flow system and related parameters. For this reason, the concept of similarity for scaling and modelling the behaviour of fluids has been proposed based on three cornerstones geometric similarity, kinematic similarity and dynamic similarity [113]. 

During the first normal-coordinate studies of ferrocene, the authors treated the Cp rings as separate entities [50] therefore, the effect of the changing reduced mass was not introduced in this treatment. The revision of the normal-coordinate analysis of ferrocene by Brunvoll and co-workers, based on the treatment of the ferrocene complex as a whole, has demonstrated that similar kinematic coupling effects play a significant role in the normal vibrational frequencies for ferrocene as well [43c]. This study also concluded that the differences between the frequencies of ferrocene and those of the free Cp- ring are... 

As in conventional chemical reactors, similarity criteria are also employed in the scale-up of electrochemical reactors. Apart from similarities such as geometric similarity, kinematic similarity, and chemical similarity, electrical similarity is a unique criterion in the scale-up of electrochemical cells. It is defined as the condition where geometrically, kinematically, and chemically similar cells have identical cell voltages and current distributions inside the cells. It is also important to note that, although the principles of similarity criteria are the same for chemical and electrochemical reactors, suitable modifications have to be made to obtain electrochemical similarity. 

However, there should be a joint that operates like an elbow joint where one would expect to see an elbow joint, and the various limb segments should be of a size consistent with a normal human being, i.e., any replacement should have similar kinematics and Idnetics. With regard to the issue of hand size, artificial hands are usually selected to be smaller than their physiological counterparts. This is so because artificial hands are perceived to be larger than they i ly are, probably due to their rigid structure and essentially static appearance. 

Mechanical similarity comprises three subsimilarities, which are static similarity, kinematic similarity, and dynamic similarity. Static similarity demands that two geometrically similar objects have relative deformation for a constant applied stress. 

Along the lines of similar kinematic assumptions as with Poiseuille flows, and using incompressibility, one is naturally inclined to have a unidirectional flow along the Ox direction, in the form ... 

Geometric similarity is similarity of shape. The requirement is that any ratio of length in one model to the corresponding length in another model is eveiywhere the same. This ratio is referred to as the scale factor. Geometric similarity is the first requirement of physical similarity. Kinematic similarity is similarity of motion and requires similarity of both length and time interval. 

With certain products which exhibit gel-like behavior, exercise care that flow time measurements are made at sufficiently high temperatures for such materials to flow freely, so that similar kinematic viscosity results are obtained in viscometers of different capillary diameters. 

Steady-state, laminar, isothermal flow is assumed. For a given viscometer with similar fluids and a constant pressure drop, the equation reduces to 77 = Kt or, more commonly, v = r /p = Ct where p is the density, V the kinematic viscosity, and C a constant. Therefore, viscosity can be determined by multiplying the efflux time by a suitable constant. 

Kinematic similarity. Two vessels are kinematically similar if they are first geometrically similar and have the same ratio of velocities in corresponding positions of the vessel... 

Kinematic similarity exists between two systems of different sizes when they are geometrieally similar and when the ratios of veloeities between eoiresponding points in one system are equal to tliose in tlie otlier. 

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. 

Having established that these assumptions are reasonable, we need to consider the relationship between the parameters of the actual offset jet and the equivalent wall jet that will produce the same (or very similar) flow far downstream of the nozzle. It can be shown that the ratio of the initial kinematic momentum per unit length of nozzle of the wall jet to the offset jet,, and the ratio of the two nozzle heights,, depend on the ratio D/B, where D is the offset distance betw een the jet nozzle and the surface of the tank, and h, is the nozzle height of the offset jet. The relationship, which because of the assumptions made in the analysis is not valid at small values of D/hj, is shown in Fig 10.72. 

Kinematic similarity requires geometric similarity and requires corresponding points in the system to have the same velocity ratios and move in the same direction between the new system and the model. 

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... 

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

Kinematic similarity exists in two geometrically similar units when the velocities at corresponding points have a constant ratio. Also, the paths of fluid motion (flow patterns) must be alike. 

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. 

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. 

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