Dimensional analysis base quantities

The choice of physical variables to be included in the dimensional analysis must be based on an understanding of the nature of the phenomenon being studied although, on occasions there may be some doubt as to whether a particular quantity is relevant or not. 

Dimensional analysis, often referred to as the II-theorem is based on the fact that every system that is governed by m physical quantities can be reduced to a set of m - n mutually independent dimensionless groups, where n is the number of basic dimensions that are present in the physical quantities. The II-theorem was introduced by Buckingham [1] in 1914 and is therefore known as the Buckingham II-theorem. The II-theorem is a procedure to determine dimensionless numbers from a list of variables or physical quantities that are related to a specific problem. This is best illustrated by an example problem. 

Various laws restrict the values of the process quantities. These laws may represent either fundamental, empirical physicochemical relationships or experimentally identified equations (from statistical modelling or from dimensional analysis particularizations). In contrast to statistical or dimensional analysis based models [1.14], which are used to fix the behaviour of signal transformers, the models of transport phenomena are used to represent generalized phases and elementary phase connections. Here, the model equations reveal all the characterizing attributes given in the description of a structure. They include balance equations, constitutive equations and constraints. 

Careful measurements and the proper use of significant figures, along with correct calculations, will yield accurate numerical results. But to be meaningful, the answers also must be expressed in the desired units. The procedure we use to convert between units in solving chemistry problems is called dimensional analysis (also called the factor-label method). A simple technique requiring little memorization, dimensional analysis is based on the relationship between different units that express the same physical quantity. For example, by definition 1 in = 2.54 cm (exactly). This equivalence enables us to write a conversion factor as follows ... 

Unit for temperature The SI unit system defines a base unit for temperature kelvin with the dimension ( ). The physical origin of this definition is the thermodynamic temperature scale where the temperature unit is defined as being proportional to the kinetic mean energy of the molecules in a system of matter. Therefore, for dimensional analysis it is admissible to use the same unit for temperature quantities as for energy, if this is expedient. In this case, the dimension for temperature is (L MT ). 

We now show that the algebraic realization of the one-dimensional Morse potential can be adopted as a starting point for recovering this same problem in a conventional wave-mechanics formulation. This will be useful for several reasons (1) The connection between algebraic and conventional coordinate spaces is a rigorous one, which can be depicted explicitly, however, only in very simple cases, such as in the present one-dimensional situation (2) for traditional spectroscopy it can be useful to know that boson operators have a well-defined differential operator counterpart, which will be appreciated particularly in the study of transition operators and related quantities and (3) the one-dimensional Morse potential is not the unique outcome of the dynamical symmetry based on U(2). As already mentioned, the Poschl-Teller potential, being isospectral with the Morse potential in the bound-state portion of the spectrum, can be also described in an algebraic fashion. This is particularly apparent after a detailed study of the differential version of these two anharmonic potential models. Here we limit ourselves to a brief description. A more complete analysis can be found elsewhere [25]. As a... 

However, the void area fraction is equivalent to the void volume fraction, based on equation (21-76) and the definition of intrapellet porosity Sp at the bottom of p. 555. Effectiveness factor calculations in catalytic pellets require an analysis of one-dimensional pseudo-homogeneous diffusion and chemical reaction in a coordinate system that exploits the symmetry of the macroscopic boundary of a single pellet. For catalysts with rectangular symmetry as described above, one needs an expression for the average diffusional flux of reactants in the thinnest dimension, which corresponds to the x direction. Hence, the quantity of interest at the local level of description is which represents the local... 

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