Power dimensional analysis

Other dimensional systems have been developed for special appHcations which can be found in the technical Hterature. In fact, to increase the power of dimensional analysis, it is advantageous to differentiate between the lengths in radial and tangential directions (13). In doing so, ambiguities for the concepts of energy and torque, as well as for normal stress and shear stress, are eliminated (see Ref. 13). 

If using dimensionless parameters obtained through dimensional analysis, or similitude, the data will eollapse into a single eurve. In this ease, the useful parameters are redueed flow, q, redueed power. 

There are nine variables and three primary dimensions, and therefore by Buekingham s theorem. Equation 7-1 ean be expressed by (9-3) dimensionless groups. Employing dimensional analysis. Equation 7-1 in terms of the three basie dimensions (mass M, length L, and time T) yields Power = ML T. ... 

If a simple pump is considered, it is possible to state that there must be a working relation between the power input, and the flow rate, pressure rise, fluid properties, and size of the machine. If a dimensional analysis is performed, it can be shown that a working relation may exist... 

Dimensional analysis is a very powerful tool in the analysis of problems involving a large number of variables. However, there are many pitfalls for the unwary, and the technique should never be used without a thorough understanding of the underlying basic principles of the physical problem which is being analysed. 

From a dimensional analysis, obtain a relation between the power and the four variables. The power consumption is found, experimentally, to be proportional to the square of the speed of rotation. By what factor would the power be expected to increase if the impeller diameter were doubled ... 

In the Rayleigh method of carrying out a dimensional analysis the dependent variable is assumed to be proportional to the product of the independent variables raised to different powers. By equating dimensions, the number of independent dimensionless groups and one set of their possible forms can be obtained. By way of illustration two examples may be considered. 

There are also several possibilities for the temporal distribution of releases. Although some releases, such as those stemming from accidents, are best described as instantaneous release of a total amount of material (kg per event), most releases are described as rates kg/sec (point source), kg/sec-m (line source), kg/sec-m (area source). (Note here that a little dimensional analysis will often indicate whether a factor or constant in a fate model has been inadvertently omitted.) The patterns of rates over time can be quite diverse (see Figure 3). Many releases are more or less continuous and more or less uniform, such as stack emissions from a base-load power plant. Others are intermittent but fairly regular, or at least predictable, as when a coke oven is opened or a chemical vat... 

It is important to realize that the process of dimensional analysis only replaces the set of original (dimensional) variables with an equivalent (smaller) set of dimensionless variables (i.e., the dimensionless groups). It does not tell how these variables are related—the relationship must be determined either theoretically by application of basic scientific principles or empirically by measurements and data analysis. However, dimensional analysis is a very powerful tool in that it can rovide a direct guide for... 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

Three repeating variables are selected, as it x, l and p, to form the II s from the other variables. Let IIj = w, lb, pc, p. Therefore by dimensional analysis IIi must have no dimensions. Equating powers for each dimension gives... 

It can be shown by dimensional analysis [Holland and Chapman] that the power number Po can be related to the Reynolds number for mixing ReM, and the Froude number for mixing FrM, by the equation... 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

Conventional dimensional analysis employs single length and time scales. Correlations are thus obtained for the mass or heat transfer coefficients in terms of the minimum number of independent dimensionless groups these can generally be represented by power functions such as... 

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. 

The bust line of Eq. (7.9) follows since dimensionless quantities can depend only on dimensionless combinations of their variables. As a result of this simple dimensional analysis in the continuous chain model we have found the two parameter theory instead of the three independent parameters ,3A.nt physical observables involve only the two combinations Rq, z. Perturbation theory now proceeds in powers of 2. Thus the continuous chain limit gives a precise meaning to the simple argument presented in Chap. 6. 

This transformation leaves both jRq and 2 = Pen 2 (Eq. (7.10)) invariant. It just expresses naive dimensional analysis in the continuous chain model. The power of the RG-approach lies in the fact that we can construct nontrivial realizations. These take into account more than just the leading n-dependence of each order of perturbation theory and therefore obey the condition of invariance of the macroscopic observables up to much smaller corrections. 

Power consumption by agitation is a function of physical properties, operating condition, and vessel and impeller geometry. Dimensional analysis provides the following relationship ... 

The power consumption by impeller P in geometrically similar fermenters is a function of the diameter Dl and speed N of impeller, density p and viscosity p. of liquid, and acceleration due to gravity g. Determine appropriate dimensionless parameters that can relate the power consumption by applying dimensional analysis using the Buckingham-Pi theorem. 

Unfortunately, many of the structures that we have described have been determined using two-dimensional X-ray crystal analysis only. Although in favourable cases this method can be extremely powerful, and indeed until about ten years ago was almost the only practicable method, yet its accuracy and reliability are far below what can now be achieved by complete three-dimensional analysis. With the increasing availability of high-speed computational facilities and more accurate means of data collection, three-dimensional refinement of the more important structures could now be undertaken. A great field of interesting and important work therefore awaits the modern crystallographer, and we may soon expect results of an accuracy sufficient to provide critical tests of diverse theoretical treatments. 

From a dimensional analysis, obtain a relation between the power and the four variables. 

The fact that the analytical presentations of the pi-relationships encountered in engineering literature often take the shape of power products does not stem from certain laws inherent to dimensional analysis. It can be simply explained by the engineer s preference for depicting test results in double-logarithmic plots. Curve sections which can be approximated as straight lines are then analytically expressed as power products. Where this proves less than easy, the engineer will often be satisfied with the curves alone, cf. Fig 1. 

In stirring, distinction is made between micro- and macro-mixing. Micro-mixing concerns the state of flow in the tiniest eddies. It is determined by the kinematic viscosity, v, of the liquid and by the dissipated power per unit of mass, = P/pV. Correspondingly, the so-called Kolmogorov s micro-scale k. of the turbulence is laid down as being k = (v3/e)1 4. (By the way, this equation is clearly derived from dimensional analysis )... 

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