Dimensional analysis derived quantities

The rise velocities of bubbles were derived by Peebles and Garber (1953). Using the techniques of dimensional analysis as was used in the derivation of the power dissipation for rotational mixers, they discovered that the functionality of the rise velocities of bubbles can be described in terms of three dimensionless quantities Gi = gn lpiO, G2 = g f) Vi,Yp]la, and Re = IpiVbflp. Re is a Reynolds number g is the acceleration due to gravity p is the absolute viscosity of fluid p is the mass density of fluid a is the surface tension of fluid and f is the average radius of the bubbles. To give G, and G2 some names, call G, the Peebles number and G2 the Garber number. 

Thermodynamics is a quantitative subject. It allows us to derive relations between the values of numerous physical quantities. Some physical quantities, such as a mole fraction, are dimensionless the value of one of these quantities is a pure number. Most quantities, however, are not dimensionless and their values must include one or more units. This chapter reviews the SI system of units, which are the preferred units in science applications. The chapter then discusses some useful mathematical manipulations of physical quantities using quantity calculus, and certain general aspects of dimensional analysis. 

The study of the basaltic dykes in evaporites demonstrates that dissolution and precipitation of phosphate minerals is a key process for the control of REE mobility and REE fractionation. In the present case, all REE found in secondary apatite in the basalt and in the salt are derived from the dissolution of primary magmatic apatite during basalt corrosion. This loss of REE from the basalt to the salt was not sufficient to lower significantly the REE concentrations of the basalt and it could only be detected by the analysis of the salt. The absolute quantity of REE transferred from the basalt into the salt, however, cannot be quantified because we have no three-dimensional control on the REE concentrations around the basalt apophy sis. 

The flow velocities in flame systems are such that transport processes (diffusion and thermal conduction) make appreciable contributions to the overall flows, and must be considered in the analysis of the measured profiles. Indeed, these processes are responsible for the propagation of the flame into the fresh gas supporting it, and the exponential growth zone of the shock tube experiments is replaced by an initial stage of the reaction where active centres are supplied by diffusion from more reacted mixture sightly further downstream. The measured profiles are related to the kinetic reaction rates by means of the continuity equations governing the one-dimensional flowing system. Let Wi represent the concentration (g. cm" ) of any quantity i at distance y and time t, and let F,- represent the overall flux of the quantity (g. cm". sec ). Then continuity considerations require that the sum of the first distance derivative of the flux term and the first time derivative of the concentration term be equal to the mass chemical rate of formation q,- of the quantity, i.e. 

Geometrical descriptors are derived from the three-dimensional representations and include the principal moments of inertia, molecular volume, solvent-accessible surface area, and cross-sectional areas. Since conformational analysis (see Conformational Analysis 1 Conformational Analysis 2 and Conformational Analysis 3) often requires calculation of atomic charges, these routines can also produce electronic descriptors. Electronic descriptors characterize the molecular structures with such quantities as LUMO and HOMO energies, bond orders, partial atoim c charges, etc. Hybrid descriptors combine aspects of several of these descriptor types. The design and implementation of new descriptors is one important aspect of on-going research in the area of QSPR. 

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