Dimensionless numbers, from similarity

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

Reliable scale-up of the desired operating conditions from the model to the full-scale plant. According to the theory of models, two processes may be considered similar to one another if they take place under geometrically similar conditions and all dimensionless numbers, which describe the process, have the same numerical value. 

The theory of models requires that in scale-up from a model (index m) to industrial scale (index x), not only must the geometric similarity be ensured but also all dimensionless numbers describing the problem must retain the same numerical values (rii = idem). This means that in scale-up of boats or ships, for example, the dimensionless numbers governing the hydrodynamics here... 

In geometrically similar systems there is complete similarity if all necessary dimensionless criteria derived either from differential equations or by using the pi theorem are equal. In complex precipitation processes such complete similarity is impossible. Moreover, because we want to obtain identical not similar products from the systems differing in scale, we usually want to reproduce the product quality (particle size, particle morphology), mixture composition, and structure of the suspension on a larger scale. We thus use limited similarity, which means that we lose several degrees of freedom (we cannot manipulate particle size, solution composition, viscosity, and diffu-sivity), and we obtain this way a reduced number of similarity criteria. 

It is important to notice the similarity between Eqs. 1.1,1.6, and 1.20. The heat conduction equation, Eq. 1.1, describes the transport of energy the diffusion law, Eq. 1.6, describes the transport of mass and the viscous shear equation, Eq. 1.20, describes the transport of momentum across fluid layers. We note also that the kinematic viscosity v, the thermal diffusivity a, and the diffusion coefficient D all have the same dimensions L2/f. As shown in Table 1.10, a dimensionless number can be formed from the ratio of any two of these quantities, which will give relative speeds at which momentum, energy, and mass diffuse through the medium. 

Hence, the dimensional scaling factor for convective mass transfer is the same in each mass balance. Similarly, dimensional scaling factors for all of the independent chemical reactions do not change from one mass balance to the next. However, when the r - - 2 dimensional scaling factors in the mass balance for component i are divided by the dimensional scaling factor for component i s rate of diffusion (i.e., i.mixCAo/L ), one obtains r - -1 dimensionless numbers... 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

Overall chemical similarity is achieved not merely by the application of a few dimensionless numbers, but results from careful application of rigorous mathematical models. 

We also met the Stokes number in Qiapter 9, where it was one of the dimensionless numbers used in the scale up of gas cyclones for the separation of particles from gases. There are obvious similarities between the collection of particles in a gas cyclone and collection of particles in the airways of the respiratory system. The Stokes number we met in Chapter 13, describing collision between granules, is not readily comparable with the one used here.)... 

If similarity theory is applied correctly, the dimensionless numbers and characteristics that are determined are independent of the scale. This makes it possible, by using appropriate scale-up rules, to specify the operating parameters for industrial-scale systems from the results of tests carried out on models. Computational fluid dynamics (CFD) can be used to visualize the impeller system at its full scale, thus contributing to solving scale-up problems. 

From the dimensionless energy equation it follows that thermal similarity can be attained if the dimensionless numbers of Graez and Brinkmann are the same for both sizes of machines. 

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

This is similar to the analysis obtained by Ainsley and Smith (see Chhabra, 1992) using the slip line theory from soil mechanics, which results in a dimensionless group called the plasticity number ... 

Fig. 9 shows comparisons of CFD results with experimental data at a Reynolds number of 986 at three of the different bed depths at which experiments were conducted. The profiles are plotted as dimensionless temperature versus dimensionless radial position. The open symbols represent points from CFD simulation the closed symbols represent the points obtained from experiment. It can be seen that the CFD simulation reproduces the magnitude and trend of the experimental data very well. There is some under-prediction in the center of the bed however, the shapes of the profiles and the temperature drops in the vicinity of the wall are very similar to the experimental case. More extensive comparisons at different Reynolds numbers may be found in the original reference. This comparison gives confidence in interstitial CFD as a tool for studying heat transfer in packed tubes. 

This method can be easily used to show the logic behind the scale-up from original R D batches to production-scale batches. Although scale-of agitation analysis has its limitations, especially in mixing of suspension, non-Newtonian fluids, and gas dispersions, similar analysis could be applied to these systems, provided that pertinent system variables were used. These variables may include superficial gas velocity, dimensionless aeration numbers for gas systems, and terminal settling velocity for suspensions. 

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