Variable physical properties dimensional analysis

TREATMENT OE VARIABLE PHYSICAL PROPERTIES BY DIMENSIONAL ANALYSIS... 

Treatment of Variable Physical Properties by Dimensional Analysis... 

Treatment of Variable Physical Properties by Dimensional Analysis The pi-set of eight remodeled pi-numbers now reads ... 

Spray Correlations. One of the most important aspects of spray characterization is the development of meaningful correlations between spray parameters and atomizer performance. The parameters can be presented as mathematical expressions that involve Hquid properties, physical dimensions of the atomizer, as well as operating and ambient conditions that are likely to affect the nature of the dispersion. Empirical correlations provide useful information for designing and assessing the performance of atomizers. Dimensional analysis has been widely used to determine nondimensional parameters that are useful in describing sprays. The most common variables affecting spray characteristics include a characteristic dimension of atomizer, d Hquid density, Pjj Hquid dynamic viscosity, ]ljj, surface tension. O pressure, AP Hquid velocity, V gas density, p and gas velocity, V. ... 

Although most physical properties (e.g., viscosity, density, heat conductivity and capacity, and surface tension) must be regarded as variable, it is of particular value that viscosity can be varied by many orders of magnitude under certain process conditions (5,11). In the following, dimensional analysis will be applied exemplarily to describe the temperature dependency of the viscosity and the viscosity of non-Newtonian fluids (pseudoplastic and viscoelastic, respectively) as influenced by the shear stress. 

Engineers commonly use dimensionless ratios such as the Reynolds number and the lift coefficient to help understand complex experimental data, organize equations and model building, and relate model testing in a wind tunnel to that of a prototype flight. This kind of analysis is called dimensional analysis because it uses the dimensional nature of important variables to derive dimensionless parameters that determine the scaling properties of a physical system. 

Along with the methods of similarity theory, Ya.B. extensively used and enriched the important concept of self-similarity. Ya.B. discovered the property of self-similarity in many problems which he studied, beginning with his hydrodynamic papers in 1937 and his first papers on nitrogen oxidation (25, 26). Let us mention his joint work with A. S. Kompaneets [7] on selfsimilar solutions of nonlinear thermal conduction problems. A remarkable property of strong thermal waves before whose front the thermal conduction is zero was discovered here for the first time their finite propagation velocity. Independently, but somewhat later, similar results were obtained by G. I. Barenblatt in another physical problem, the filtration of gas and underground water. But these were classical self-similarities the exponents in the self-similar variables were obtained in these problems from dimensional analysis and the conservation laws. 

A stream of droplets of liquid is formed rapidly at an orifice submerged in a second, immiscible liquid. What physical properties would be expected to influence the mean size of droplet formed Using dimensional analysis obtain a functional relation between the variables. 

The affinity laws express the mathematical relationship between the variables involved in a pump s performance. By applying the principles of dimensional analysis on the physical properties affecting pump operation, the relationship is ... 

As already suggested, there are many empirical conelations, and also theoretical equations with supporting experimental data in the literature. In general, all are different. However, the correlation of Zwietering , who carried out a dimensional analysis of the important variables and covered experimentally a very wide range of impeller types, sizes and off-bottom clearances, vessel sizes and physical properties, is broadly similar to a large number of them. The equation is... 

Dimensional analysis is commonly applied to complex materials and processes to assess the significance of some phenomenon or property regime. It enables analysis of situations that cannot be described by an equation however, there is no a priori guarantee that a dimensionless analysis will be physically meaningful. Dimensionless quantities are combinations of variables that lack units (i.e., pure numbers), used to categorize the relationship of physical quantities and their interdependence in order to anticipate the behavior. Several dimensionless quantities relevant to polymer rheology and processing are... 

Dimensional Analysis The flow phenomena in an ejector are complex, and hence, it is not possible to predict the flow rate of the secondary fluid a priori. In conventional chemical engineering, such phenomena are treated by a dimensional analysis of the dependent and independent variables based on the Buckingham % theorem. In the present case, the dependent variable is the rate of entrainment of the secondary fluid, and the independent variables are the physical properties, ejector configuration, and operating parameters. The latter are defined mainly by the flow rate of the primary fluid. Ben Brahim et al. (1984) gave the following separate dependences for the primary and secondary fluids ... 

This step is one of the most difficult and is, of course, extremely important because all pertinent variables have to be included in the analysis. The term variable includes any physical quantity, dimensional and apparently non-dimensional constant that plays a role in the phenomenon under investigation. The determination of the variables must take into account practical knoivledge of the problem as ivell as the physical laivs governing the phenomenon. Variables typically include the parameters that are necessary not only to describe the geometry of the system (such as the diameter of the pipe in the example beloiv), but also to define the fluid properties (such as the density, viscosity, thermal capacity, thermal conductivity of the fluid, the diffusion coefficient for one species in the working fluid, etc.) as well as to indicate the external effects that influence the system (such as the driving pressure drop in the further discussed cases). 

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