Dimensional analysis essentials

Short Summary of the Essentials of Dimensional Analysis and Scale-Up... 

Scale-up of the tableting process in the pharmaceutical industry is still an empirical process. Dimensional analysis, a powerful method that has been successfully used in other applications, can provide a solid scientific basis for tableting scale-up. It is a method for producing dimensionless numbers that completely describe the process. The analysis should be carried out before the measurements are made, because dimensionless numbers essentially condense the frame in which the measurements are performed and evaluated. It can be applied even when the equations governing the process are not known. 

Few calculations of three-dimensional convection in CZ melts (or other systems) have been presented because of the prohibitive expense of such simulations. Mihelcic et al. (176) have computed the effect of asymmetries in the heater temperature on the flow pattern and showed that crystal rotation will eliminate three-dimensional convection driven by this mechanism. Tang-born (172) and Patera (173) have used a spectral-element method combined with linear stability analysis to compute the stability of axisymmetric flows to three-dimensional instabilities. Such a stability calculation is the most essential part of a three-dimensional analysis, because nonaxisymmetric flows are undesirable. 

It is usually impossible to determine all the essential facts for a given fluid flow by pure theory, and hence much dependence must be placed on experimental investigations. The number of tests to be made can be greatly reduced by a systematic program based on dimensional analysis and specifically on the laws of similitude or similarity, which means certain relations by which test data can be applied to other cases. 

The first important advantage of using dimensional analysis exists in the essential compression of the statement. The second important advantage of its use is related to the safeguarding of a secure scale-up. This will be convincingly shown in the next two examples. 

After a review of the customary calculation methods [97] for both essential classes of dryers (convective and contact dryers), the dimensioning methods for spray dryers [98, 99], fluidized and spouted bed dryers [100, 101, 102], cascading rotary dryers [103], pneumatic conveying dryers [104], conductive-heating agitated dryers [105] and layer dryers [106] were presented. They all confirmed the initially made conclusion that the scaling up of dryers is still made today without dimensional analysis and the model theory based thereupon. 

This revolution is the application of the methods of asymptotic analysis to problems in transport phenomena. Perhaps more important than the detailed solutions enabled by asymptotic analysis is the emphasis that it places on dimensional analysis and the development of qualitative physical understanding (based upon the m. thematical structure) of the fundamental basis for correlations between dependent and independent dimensionless groups [cf. 7-10]. One major simplification is the essential reduction in detailed geometric considerations, which determine the magnitude of numerical coefficients in these correlations, but not the form of the correlations. Unlike previous advances in theoretical fluid mechanics and transport phenomena, in these developments chemical engineers have played a leading role. 

It is also possible to derive the Reynolds number by dimensional analysis. This represents a more analytical, but less intuitive, approach to defining the condition of similar fluid flow and is essentially independent of particular shape. In this approach, variables in the Navier-Stokes equation (relative particle-fluid velocity, a characteristic dimension of the particle, fluid density, and fluid viscosity) are combined to yield a dimensionless expression. Thus... 

It can be shown, essentially by dimensional analysis (9), that the lowest order gradient-corrected exchange-correlation functional has the form... 

A possible solution to the above problems would be the triple-dimensional analysis by using GC x GC coupled to TOFMS. Mass spectrometric techniques improve component identification and sensitivity, especially for the limited spectral fragmentation produced by soft ionization methods, such as chemical ionization (Cl) and field ionization (FI). The use of MS to provide a unique identity for overlapping components in the chromatogram makes identification much easier. Thus MS is the most recognized spectroscopic tool for identification of GC X GC-separated components. However, quadru-pole conventional mass spectrometers are unable to reach the resolution levels required for such separations. Only TOFMS possess the necessary speed of spectral acquisition to give more than 50 spectra/sec. This area of recent development is one of the most important and promising methods to improve the analysis of essential oil components. 

The maximum dissipation rates for the dispersion of droplets and gas bubbles with different stirrer types such as turbine and pitched-blade stirrers, Lightnin A 310 and Cheminer HE 3 was determined using ID-LDA measurements, and the resulting turbulent fluctuating velocities were calculated using a model based on dimensional analysis. The essential parameters were identified using statistical analysis supported by sensitivity analysis. It was found that in the relationship e oc the stirrer speed dependence with n is correct, but that for the stirrer diameter with was set too low. The number of baffles was of secondary importance [608]. 

Short Summary of the Essentials of the Dimensional Analysis and Scale-Up In short, the advantages made possible by correct and timely use of dimensional analysis are as follows ... 

Dimensional homogeneity is fundamental to equations relating variables in the description of natural processes. The recognition of this basic attribute is the substance of dimensional analysis, which results in the reduction of relevant parameters to the essential minimum. Physical quantities comprise combinations of one or more basic dimensions. Table 2.2 shows some commonly used physical quantities in engineering expressed in terms of basic dimensions (mass, M length, L time, T, temperature, 0). 

Always remember that the units in a calculation must combine to give the proper units in the answer. Thus, in this example, grams cancel to leave the proper unit, moles/liter, or molarity. Using units in the calculation to check if the final units are proper is called dimensional analysis. Accurate use of dimensional analysis is essential to properly setting up computations. 

This is as far as dimensional analysis will bring us. We now need to add some physical insight. We note first that x m /v is essentially a Reynolds number. Typical values of m and V for the atmosphere are 100 cm s and 0.1 cm s, respectively. Thus... 

The GCxGC resolution advantage is known to improve the efficiency of enantioselective essential oil analysis (in contrast to one-dimensional analysis). In a single temperature-programmed analysis, the individual antipodes of optically active components can be separated and are effectively free from matrix interferences. The enantiomeric compositions of a number of monoterpene hydrocarbons and oxygenated monoterpenes in Australian tea tree Melaleuca alternifolia), including sabinene, a-pinene, (3-phellandrene, limonene, trans-sabinene hydrate, ds-sabinene hydrate, linalool, terpinen-4-ol, and a-terpineol shown in Figure 7,... 

It is evident from this model that the lap joint peels in essentially the same way as the T joint, except for the extra elastic stretching of the material caused by the higher force required. If we assume that the force F depends on the work of adhesion W in the same manner as the peeling joint, but also on the elastic modulus E of the strips and on the size L of the strips, then it is obvious from a simple dimensional analysis that the force must be given by an equation of the kind... 

Those engineering disciplines concerned with fluid flow, such as aeronautical, civil, and mechanical, have used dimensional analysis to good effect for the past hundred years. Their success is largely attributable to the fact that fluid flow requires only three fundamental dimensions and generates a limited number of dimensionless parameters. Those engineering disciplines can still use the Rayleigh indices method, which is, essentially, a hand calculation, to derive the dimensionless parameters. 

Insightful understanding of the granulation processes is essential for the identification of key variables and parameters for the dimensional analysis and scale-up considerations. While development of definitive mathematical models for the granulation processes is incomplete, the scaling approaches recommended in this chapter help reduce uncertainty during new product development and transfer to industrial sites. 

In concentrating on the details of the formalism above, we should not lose track of the basic ideas of the similarity approach, wherein (1) insight is nsed to reduce a problem to an essential and idealized formulation (2) dimensional analysis is used to identify nondimensional dependencies and (3) empiricism is used to determine the form of the functions of the relevant dimensionless nnm-bers. Unfortunately, for most processes we wish to parameterize this three-step recipe is not easy to follow. More often than not our insights are not sufficiently developed for us to arrive at compelling simplifications. Even when they are compelling, however, the empirical step often involves measuring functions that involve more than one variable and are not readily accessible to measurement. 

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