Nondimensional parameters

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. ... 

To determine the deterioration in component performance and efficiency, the values must be corrected to a reference plane. These corrected measurements will be referenced to different reference planes depending upon the point, which is being investigated. Corrected values can further be adjusted to a transposed design value to properly evaluate the deterioration of any given component. Transposed data points are very dependent on the characteristics of the components performance curves. To determine the characteristics of these curves, raw data points must be corrected and then plotted against representative nondimensional parameters. It is for this reason that we must evaluate the turbine train while its characteristics have not been altered due to component deterioration. If component data were available from the manufacturer, the task would be greatly reduced. 

Numerical simulations by Hooke and Venner et al. [57,58] for small amplitude wavy surfaces show that the amplitude attenuation of waviness in an EHL contact depends on the operating conditions and the orientation of the roughness pattern. They found that the attenuation appeared to be governed by a single nondimensional parameter the ratio of the wavelength to the length of the inlet pressure sweep at the entrance to the conjunction, it can be evaluated by... 

The concept of turbulent flame stretch was introduced by Karlovitz long ago in [15]. The turbulent Karlovitz number (Ka) can be defined as the ratio of a turbulent strain rate (s) to a characteristic reaction rate (to), which has been commonly used as a key nondimensional parameter to describe the flame propagation rates and flame quenching by turbulence. For turbulence s >/ />, where the dissipation rate e and u, L and v... 

For analytical solutions, it is more convenient to work with nondimensional forms of the diffusion equations. We choose the following nondimensional substitutions. The time coordinate / is replaced by the nondimensional parameter /, and a is the root radius ... 

These substitutions replace the eight dimensional parameters in the original equations by the four nondimensional parameters above. The parameter Pe is the Peclet number (37) and shows the relative importance of convection compared to diffusion. The advantage of this formulation becomes obvious when typical parameter values are substituted into the equations. 

System Specification. Any reverse osmosis system may be specified in terms of three nondimensional parameters y 6 nd... 

Since G and Ch appear only as a product, G = GCh is used as the relevant nondimensional parameter governing heat release in the present studies. The other governing parameters for this flow are Re and Pr, along with the precise strength and distribution of the source term in the energy equation. 

Working with the equations of motion and conservation of mass for both fluid and particles, we have the following nondimensional parameters (Glicksman, 1984) ... 

The fu st term is a modified Archimedes number, while the second one is the Froude number based on particle size. Alternatively, the first term can be substituted by the Reynolds number. To attain complete similar behavior between a hot bed and a model at ambient conditions, the value of each nondimensional parameter must be the same for the two beds. When all the independent nondimensional parameters are set, the dependent parameters of the bed are fixed. The dependent parameters include the fluid and particle velocities throughout the bed, pressure distribution, voidage distribution of the bed, and the bubble size and distribution (Glicksman, 1984). In the region of low Reynolds number, where viscous forces dominate over inertial forces, the ratio of gas-to-solid density does not need to be matched, except for beds operating near the slugging regime. 

In liquids, the mean free path is typically of the order of 10 10 m. Hence the Knudsen effect is not important (i.e., diffusing molecules collide with solvent molecules long before they typically arrive at a pore wall). However, diffusion is affected by a different mechanism, the viscous drag caused by the pore walls. This is known as the Renkin effect (Renkin, 1954). In essence, the ratio of pore diffusivity in the liquid-filled pore space and diffusivity in the free liquid, D(pore/D,free, is a function of the nondimensional parameter... 

Dynamic similarity is achieved when the values of the nondimensional parameters are the same at geometrically similar locations. 

Now we introduce a few nondimensional parameters concentration (G), real (X), imaginary (Y), and complex (Z) frequencies ... 

The nondimensional parameter /) (positive for exothermic reactions) is a measure of nonisothermal effects and is called the heat generation function. It represents the ratio between the rate of heat generation due to the chemical reaction and the heat flow by thermal conduction. Nonisothermal effects may become important for increasing values of /3, while the limit (3 - 0 represents an isothermal pellet. Table 9.1 shows the values of [3 and some other parameters for exothermic catalytic reactions. For any interior points within the pore where the reactant is largely consumed, the maximum temperature difference for an exothermic reaction becomes... 

Normalized variables and nondimensional parameters may be identified from equation (114) [73], and an iterative method of solution may be devised [50] and applied [73] to obtain an approximate formula for — Po o Po o- This formula exhibits the attributes of the burning velocity for the heterogeneous regime that were discussed in Section 11.6.1. Thus a more formal development leads to the qualitative results that have been inferred from physical reasoning. It is instructive to have seen a representative sequence of approximations involved in a formal development. The iterative method [50] has been applied successfully [77] with fewer approximations, through use of techniques for numerical integration, to derive... 

Both Eq. (35) and the boundary conditions contain only one nondimensional parameter S ... 

Variation of Dh affects the following nondimensional parameters the electrokinetic diameter, and the strength of the viscous forces in the ratio of electrical to viscous forces. The... 

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