Dimensional analysis, worked example

We can use percentages as factors in working with dimensional analysis. For example if an elementary school class is 40% girls and 60% boys, we can tell how many children are in a class with 48 boys ... 

How does principal component analysis work Consider, for example, the two-dimensional distribution of points shown in Figure 7a. This distribution clearly has a strong linear component and is closer to a one-dimensional distribution than to a full two-dimensional distribution. However, from the one-dimensional projections of this distribution on the two orthogonal axes X and Y you would not know that. In fact, you would probably conclude, based only on these projections, that the data points are homogeneously distributed in two dimensions. A simple axes rotation is all it takes to reveal that the data points... 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

In this example, the development of a new flotation technique for waste water treatment is discussed. Dimensional analysis has been used since the onset of this work. It concerns the so-called Induced Air Flotation, IAF, which is an alternative to the well-known Dissolved Air Flotation, DAF, the latter being already discussed in Example 7. 

The previous calculation is an example of the use of the factor label method, also called dimensional analysis, in which a quantity is multiplied by a factor equal or equivalent to 1. The units inclnded in the factor are the labels. In the previous example, 9 is equivalent to 1 hour (h), and the calculation changes the number of hours worked to the equivalent number of dollars. To use the factor label method, first put down the given quantity, then multiply by a conversion factor (a rate or ratio) that will change the units given to the units desired for the answer. The factor may be a known constant or a value given in the problem. 

The Reynolds number is useful for characterizing flow of a Newtonian fluid through a tube. The variables in dimensional analysis can be arranged to suggest a valid ratio when none exists. Therefore, established dimensionless variables should be used where applicable. A number of these dimensionless variables have been proven to work in scaleup applications. Examples include the Reynolds, Nusselt, Grashof, and Sherwood numbers, all of which are completely described in Perry s Chemical Engineers Handbook. 

Real-World Reading Link If you have ever had a job, one of the first things you probably did was figure out how much you would earn per week. If you make 10 dollars per hour and work 20 hours per week, how much money will you make Performing this calculation is an example of dimensional analysis. 

Summarizing, it has been shown that dimensional analysis can serve as a motivation to become familiar with data, to turn them in various ways. Further, dimensional analysis should give creativity in as far as making curious how some connections on several fields of chemistry could be. The handling of data and plotting should not be a cumbersome task with the advent of spreadsheet calculators. A few examples have been given to illustrate the value of dimensional analysis and also how to deal with the laws in a critical way. It could not be the goal to establish some relationships without critical interpretation of them. Numerous further examples that are not quite obvious can be worked out in this way. 

The only type of work involved in most chemical and physical changes is pressure-volume work. From dimensional analysis we can see that the product of pressure and volume is work. Pressure is the force exerted per unit area, where area is distance squared, d volume is distance cubed, i Thus, the product of pressure and volume is force times distance, which is work. An example of a physical change (a phase change) in which the system expands and thus does work as it absorbs heat is shown in Figure 15-6. Even if the book had not been present, the expanding system pushing against the atmosphere would have done work for the expansion. 

However, it shall be emphasized that a versatile tool such as the dimensional analysis can only be learnt in one way, namely through training and independent work with solution of practical tasks. May these few examples serve as inspiration ... 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

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