Similarity thermal

Geometrically similar systems are thermally similar when the corresponding temperature differences bear a constant ratio to one another and when the systems if moving are also kinematically similar. For multiphase reactors, an additional requirement will be that of hydrodynamic regime and turbulence similarity since the turbulence structure decides the rate of heat transfer. 

In many industrially important situations, it is impossible to maintain geometric, mechanical (kinematic/hydrodynamic and turbulence similarities), and thermal similarities simultaneously. Consider a stirred tank reactor with heat exchange only through a jacket on its external surface. The jacket heat transfer area to vessel volume ratio is proportional to (l/T). Evidently, with scale-up, this ratio decreases, and it is difficult to maintain the same heat transfer area per unit volume as in the small-scale unit. Additional heat transfer area is required to cater to the extra heat load resulting from increase in reactor volume. This area can be provided in the form of a coil inside the reactor or an external heat exchanger circuit. In both cases, the flow patterns are significantly different than the model contactor used in bench-scale studies and kinematic similarity is violated. This is the classic dilemma of a chemical engineer it is impossible to preserve the different types of similarities simultaneously. 

Rate of Formation of the Solute / Solvent Mixture Rate of Bulk Flow 

Rate of Formation of the Solvent / Solute Mixture Rate of Molecular Diffusion 

In both model and prototype, the formation time for the solvent/ solute mixture will be of the same rate or reaction order, and this requirement fixes the relative velocities in continuous-flow systems. These velocities are incompatible with the velocities necessary for kinematic similarity except at very low or very high velocities. 1 

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Nitropyrazoles rearrange to 4-nitropyrazoles in H2SO4 and to 3-nitropyrazoles thermally. Similar rearrangements are known for 7V-nitro-l,2,4-triazoles. 

For thermal similarity, that is the same temperature in both systems ... 

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

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. 

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)... 

Heat flow, whether by radiation, conduction, convection, or the bulk transfer of matter, introduces temperature as another variable. Thus, for systems in motion, thermal similarity requires kinematic similarity. Thermal similarity is described by... 

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. 

He knows that he may not vary the temperature T0 and dp if he does not want to risk influencing the chemical course of the reaction. Consequently, as already mentioned, geometric similarity is inevitably violated during scale-up on account of dp/d Z idem. Damkohler is therefore prepared to waive adherence to L/d = idem as well. However, he points out that this will necessarily lead to consequences for heat transfer behaviour. In this case, he uses the hypothesis that thermal similarity is guaranteed if the ratio of IV to III (heat conduction through the tube wall to heat removal by convection) is kept equal ... 

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. 

Each of these four states is dependent on the other three. Mechanical similarity is impossible to achieve if the model and prototype are not geometrically similar. The requirements for chemical and thermal similarity are then impossible to achieve without the requirements for mechanical similarity being met. 

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]. 

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. 

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. 

Chemical and biochemical similarities are associated with transformations due to chemical or biochemical reactions within the system. According to Johnstone and Thrings (1957), Geometrically and thermally similar systems are chemically similar when corresponding concentration dijferences bear a constant ratio to one another and when the systems, if moving, are kinematically similar."... 

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