Characteristic dimensionless parameters

In a series of studies, summarized in Ref 18, Merzhanov and co-workers examined conditions under which thermal explns may be quenched. This theoretical analysis provided two characteristic dimensionless parameters which determine the conditions for quenching. These are Kcr obtained from inflection points of 0M vs k plots (0M is the max value of 6), and a quantity e = [(6m/k) (deM/dK)] =Kcr,whichisa measure of the uncertainty in the determination of the critical conditions. Both CI and are functions of 0 and 7, which (as well as k and 0) are defined as follows ... 

Using characteristic values is a widespread and helpful means. An example for material characteristics is the specific energy input based measured in kWh/kg (see Section 6.5.1), while the available torque/axis distance measured in Nm/m3 is a machine characteristic. Dimensionless parameters (with the unit 1) play an important role for scale down/scale up considerations. The Reynolds number is a known dimensionless parameter, determining whether flow is laminar or turbulent. However, it is of little importance for extruder scale downs/scale ups. 

As in the analysis of catalytic reactions we make the modelling equations dimensionless and establish the characteristic dimensionless parameter combinations associated with the LPCVD process. By defining ... 

The transition between both globular structures and the coil state turns out to be the first order phase transition. The relation between the free energies of structures A and B depends on the temperature and on the characteristic dimensionless parameter /7a. The latter dependence is manifested already at the temperatures which are much lower than the coil-globule transition temperature, i.e. formally at A > l13. In this region, the following simple result can be obtained ... 

Hence, the semidilute solutions with a high value of the characteristic dimensionless parameter cS have the tricritical properties if and only if the other dimensionless parameter c. So turns out to be. small to satisfy inequality 190. 

In the qualitative sense one can arrive at a similar conclusion as above regarding the bed by examining the magnitudes of characteristic dimensionless parameters (Table 27.8). Observe that Sw/h = 0.0005 from water. The Mach number Mao is about 0.14, which means that wave speed in water would be about seven times that in mud, a manifestation of the significant role of mud as an energy dissipater. The very low value (0.002) of r]"/rim, the ratio of loss due to mud elasticity to mud viscosity, corroborates the inference that the muck was more like a fluid than a solid. 

One such study by Hoftyzer [29] proposed a characteristic dimensionless parameter Iz/y to characterize this dispersion. 1 is the lateral dispersion coefficient, z is the axial distance down the bed required to achieve the equilibrated condition, and y is the radial distance for which this equilibrated uniform flow is found. The dispersion coefficient can be quantified by using the data from several workers [24], [52], [72], [89] as shown by Koros [35]. Thus, a value can be obtained v ich gives the com-... 

The Weber number becomes important at conditions of high relative velocity between the injected Hquid and surrounding gas. Other dimensionless parameters, such as the Ohnesorge ((We /Re), Euler (AP/Pj y i)y and Taylor (Re/ We) numbers, have also been used to correlate spray characteristics. These parameters, however, are not used as often as the Reynolds and Weber numbers. 

Eqs. (74-78) contain two dimensional parameters, and D, and two dimensionless parameters, A and e. This means that any characteristic length scale i and growth velocity v of the possible structures can be presented in the form... 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

Several conclusions result from the preceding equations They reveal that the dimensionless parameter of the system is the fragmentation number (Fa) with the characteristic strength given by... 

The blast scaling law which is almost universally used to predict characteristics of blast waves from explosions at high altitude is that of Sachs (Reference 10). Sachs law states that dimensionless overpressure and dimensionless impulse can be expressed as unique functions of a dimensionless scaled distance, where the dimensionless parameters include quantities which define the ambient atmospheric conditions prior to the explosion. 

Deactivation of the Mediator Deactivation of the mediator is a commonly encountered event in the practice of homogeneous catalysis. Among the various ways of deactivating the mediator, the version sketched in Scheme 2.10 is particularly important in view of its application to the determination of the redox characteristics of transient free radicals (see Section 2.7.2).14 The current-potential responses are governed by three dimensionless parameters, 2ei = /F)(ke Cjl/v), which measures the effect of the rate-determining... 

It is obvious that we could name all the geometric parameters indicated in the sketch. They were all the geometric parameters of the stirrer and of the vessel, especially its diameter D and the liquid height H. In cases of complex geometry, such a procedure would compulsorily deflect from the problem. It is therefore advisable to introduce only one characteristic geometric parameter, knowing that all the others can be transformed into dimensionless geometric numbers by division with this one. 

Reynolds (Rl, R2) was one of the earlier investigators to appreciate the random nature of turbulence. The dimensionless parameter bearing his name is widely used as a measure of the physical characteristics of steady, uniform flow. Such a measure is essentially macroscopic and does not describe the local or transient behavior at a point in the stream. In recent years much effort has been devoted to understanding the basic mechanism of momentum transport by turbulence. The early work of Prandtl (P6), Taylor (Tl), Karmdn (Kl), and Howarth (K4) laid a basis for the statistical theory of turbulence which is apparently in reasonable agreement with experiment. More recently Onsager (03), Corrsin (C6), and Kolmogoroff (K10) extended the statistical theory of turbulence to describe the available experimental data in terms of kinetic-energy... 

In order to estimate the dimensionless parameters D and to in (6.3.9), (6.3.11) assume the following reasonable values for the parameters of the system (for the filter we adopt values characteristic for the Nuclepore membrane employed in [7]-[9]). 

A dimensionless variable is the ratio of that variable to a characteristic quantity of the same dimensions (e.g., r = tiff). A dimensionless parameter is the ratio of combinations of characteristic quantities having the same dimensions [Da = k/(l/6)]. We shall go into more detail on the question of rendering equations dimensionless later (See pp. 28-33), but two principles are of sufficient importance that they are worth reiterating.3... 

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