Dimensionless Group Model

Svarovskys model has its origins in Reitema s optimum design under these circumstances Equation (8.36) reduces to  

Reitema s optimum design used a LIde ratio of 5, hence 3.5tc 

Thus if the feed flow rate to such a cyclone is altered the new diaracteristic velocity can be calculated iom Equation (8.38), Reynolds number Rom (8.39), Euler number ftom (8.42) and pressure drop from (8.40). The new cut size can be calculated from Equation (8.37) after using (8.41) to give the new Stokes-50 number. However, Equations (8.41) and (8.42) are vaUd only for cyclones of Reitema s optimum geometry. 

Svarovsky propoi d that there is a general relation between the Stokes and Euler numbers for all geometries, thus  

The factor in the final column in Table 8.4 is called the running cost criterion where  

---

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

Turbomachines can be compared with each other by dimensional analysis. This analysis produces various types of geometrically similar parameters. Dimensional analysis is a procedure where variables representing a physical situation are reduced into groups, which are dimensionless. These dimensionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis as used in turbomachines can be employed to (1) compare data from various types of machines—it is a useful technique in the development of blade passages and blade profiles, (2) select various types of units based on maximum efficiency and pressure head required, and (3) predict a prototype s performance from tests conducted on a smaller scale model or at lower speeds. 

The objectives are not realized when physical modeling are applied to complex processes. However, consideration of the appropriate differential equations at steady state for the conservation of mass, momentum, and thermal energy has resulted in various dimensionless groups. These groups must be equal for both the model and the prototype for complete similarity to exist on scale-up. 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

It is also possible to interpret similarity in terms of dimensionless groups and variables in a proeess model. Kline [3] expressed that the numerieal values of all the dimensionless groups should remain eon-stant during seale-up. However, in praetiee this has been shown to be impossible [4],... 

A differential equation deseribing the material balanee around a seetion of the system was first derived, and the equation was made dimensionless by appropriate substitutions. Seale-up eriteria were then established by evaluating the dimensionless groups. A mathematieal model was further developed based on the kineties of the reaetion, deseribing the effeet of the proeess variables on the eonversion, yield, and eatalyst aetivity. Kinetie parameters were determined by means of both analogue and digital eomputers. 

This allows the pressure, which has significance only when the fluid experiencing it is named, to be replaced by a dimensionless group of properties which has significance for all fluids. Thus, the first requirement in constructing a model of a system is that the inlet pressures should be selected so that the density ratio of the two phases in the model is the same as that in the system... 

The ratio pjpv is assumed to be the same for both the modeling fluid and the fluid of interest. This step establishes the corresponding pressures, but it must be noted that it is subject to the condition that there will not be another important dimensionless group of fluid properties, as was discussed earlier. 

Plotting the data as In Cai versus t will give - (k x) as the slope and permit calculating k. It should be noted that calculating k from data involves solving the equations, but establishing (k x) as the only important dimensionless group requires only the dimensionless model. 

Annular flow. In annular flow, as mentioned in Section 3.4.6.1, modeling of the interfacial shear remains empirical. For adiabatic two-phase flow, Asali et al. (1985) suggested that the friction factor, fjfs, is dependent on a dimensionless group for the film thickness, 8+, as defined in Eq. (3-136), and the gas Reynolds number, Rec ... 

Blumenkrantz and Taborek (1971) applied the density effect model of Boure to predict instability in natural-circulation systems in thermosiphon reboil-ers used in the petrochemical industry. An important conclusion of their work was that similarity analysis in terms of the model s dimensionless groups can be used to extrapolate threshold stability data from one fluid to another. 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

Writing the model in dimensionless form, the degree of axial dispersion of the liquid phase will be found to depend on a dimensionless group vL/D or Peclet number. This is completely analogous to the case of the tubular reactor with axial dispersion (Section 4.3.6). 

---