Dimensionless analysis

Specific Speed. A review of the dimensionless analysis (qv) as related to pumps can be found in Reference 14. One of these nondimensional quantities is called the specific speed. The universal dimensionless specific speed, Q, is defined as in equation 9 ... 

Recently, Razumovskid441 studied the shape of drops, and satellite droplets formed by forced capillary breakup of a liquid jet. On the basis of an instability analysis, Teng et al.[442] derived a simple equation for the prediction of droplet size from the breakup of cylindrical liquid jets at low-velocities. The equation correlates droplet size to a modified Ohnesorge number, and is applicable to both liquid-in-liquid, and liquid-in-gas jets of Newtonian or non-Newtonian fluids. Yamane et al.[439] measured Sauter mean diameter, and air-entrainment characteristics of non-evaporating unsteady dense sprays by means of an image analysis technique which uses an instantaneous shadow picture of the spray and amount of injected fuel. Influences of injection pressure and ambient gas density on the Sauter mean diameter and air entrainment were investigated parametrically. An empirical equation for the Sauter mean diameter was proposed based on a dimensionless analysis of the experimental results. It was indicated that the Sauter mean diameter decreases with an increase in injection pressure and a decrease in ambient gas density. It was also shown that the air-entrainment characteristics can be predicted from the quasi-steady jet theory. 

This is no longer the case when competition involves reactions with different orders, as in Scheme 2.17. Unlike the preceding case, the C and D concentration profiles do not have the same shape. Appropriate dimensionless analysis (see Section 6.2.8), where the space variable is normalized toward the reaction layer thickness, leads to the dimensionless parameter... 

Using this dimensionless analysis approach, the fraction absorbed can be predicted from Eq. 2.21 on the basis of two dimensionless variables as follows ... 

Application of the usual procedures of dimensionless analysis to Eq. (8) gives... 

Obviously liquid residence time is not an appropriate parameter to describe pore diffusion effects in fluidized bed adsorption. This may be elucidated by assessing particle side transport by a dimensionless analysis. Hall et al. [73] described pore diffusion during adsorption by a dimensionless transport number Np according to Eq. (17), De denoting the effective pore diffusion coefficient in case of hindered transport in the adsorbent pores and Ue the... 

Fig. 29. Electrodeposition of Ag from 0.017 M AgCN + 0.92 M KCN + 0.11 M K2CO3 solution dimensionless analysis of experimental potentiostatic current transients (/, and tm are the current and time corresponding to the maximum on the current transient curve, respectively). Upper curve calculated for the instantaneous nucleation mechanism lower curve, for the progressive nucleation mechanism. Different symbols/experimental points relating to different potentials [136], Reproduced by permission of The Electrochemical Society, Inc.

In this case, the average heat transfer coefficient for the pipe will depend on both and D. If dimensionless analysis is applied to this problem, assuming for simplicity that the buoyancy force effects are negligible, then one possible result would be ... 

Consideration will now be given to what is basically the same question that was earlier dealt with by using dimensionless analysis. The problem is, essentially, that of determining under what conditions the flow and temperature distributions about... 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

KolmogorofTs theory Brian et al.,9 Elenkov et al.,26 and Middleman84 used Kolmogoroff s theory of local isotropic turbulence in an attempt to correlate the effective relative velocity with some macroscopic variables, such as stirrer speed and particle diameter. From the dimensionless analysis of agitated slurry reactors,45,84 they suggested a correlation... 

Dimensionless analysis of the general problem of particle motion under equilibrium conditions gives a unique relationship between two dimensionless groups, drag coefficient (C, ) and Reynolds number Re) where ... 

The gist of dimensionless analysis is as follows For any two dynamically similar systems, all the dimensionless numbers necessary to describe the process have the same numerical value. °° Once a process is expressed in terms of dimensionless variables, we are magically transferred to a world where there is no space and no time. Therefore, there is no scale and, consequently, there are no scale-up problems. The process is characterized solely by numerical values of the dimensionless variables (numbers). In other words, dimensionless representation of the process is scale-invariant. 

The dimensionless analysis of equations is an analytic mathematical technique of frequent use in engineering. Indeed, this technique allows one to concentrate all the a priori independent parameters resulting in an identical effect into a single dimensionless parameter. To illustrate the principle and operational interest of the method, let us consider a general second-order homogeneous reaction, such as that in Eq. (160), the reactant concentration being at the beginning (t = 0) of the experiment ... 

One obtains the pseudo-rate constant k = ko[A]"[B]. .. Thus dimensionless analysis affords the differential equation in Eq. (211), which describes the concentration profile of the P species when A, is given as in Eq. (212). 

All these methods stem from the fact that for a given reaction sequence, such as that in Eqs. (209) and (210), which involve a single rate-determining step, all the kinetics and thus the shape and location of the voltammogram depends only on the dimensionless rate constant parameter X in Eq. (212). As a result, any modification of the experimental conditions that keep X constant does not modify the dimensionless voltammogram. Thus quantitative information on the chemical mechanism [Eq. (210) may be a succession of chemical steps] is obtained without mathematical derivation, but only from dimensionless analysis (compare Chapter 2). 

The simplification in Eqs. (2) and (3) is equivalent to saying that k is much smaller than the mass transfer rate D/5, which is of the order of 5s. From Chapter 1, this implies that the R/P redox couple gives a chemically reversible cyclic voltammogram at a scan rate corresponding to the same mass transfer rate. From a dimensionless analysis and Table 5 in Chapter 1, this corresponds to a scan rate v such as Fv/RT D/5", that is, v of the order 0.lVs Thus, observation for identical or close experimental conditions of a chemically reversible cyclic voltammogram at v 0.1 V s" for the redox couple of interest is sufficient to prove the validity of the assumptions leading to the homogeneous equivalent rate laws in Eqs. (7) and (8),... 

In this paper we describe a model of a cup plater with a peripheral continuous contact and passive elements that shape the potential field. The model takes into account the ohmic drop in the electrolyte, the charge-transfer overpotential at the electrode surface, the ohmic drop within the seed layer, and the transient effect of the growing metal film as it plates up (treated as a series of pseudo-steady time steps). Comparison of experimental plated thickness profiles with thickness profile evolution predicted by the model is shown. Tool scale-up for 300 mm wafers was also simulated and compared with results from a dimensionless analysis. 

Based on this dimensionless analysis, it was attempted to scale-up the cup plater for 300 mm wafers. All the dimensions in the cup plater were scaled-up 1.5 times. If one substitutes the parameters in G0,G,and War, then it turns out that the nonunuformity N,is proportional to the wafer radius raised to the 1.78 power ... 

Each of these is a ratio of a convective transfer rate to the corresponding diffusion rate of transfer. Dimensionless analysis indicates that, for fixed geometry and constant properties, the Nusselt number and the Sherwood number depend on the Reynolds number (forced convection), Rayleigh number (natural convection), flow characteristics, Prandtl number (heat transfer), and Schmidt number (mass transfer). 

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... 

Dimensionless analysis in mathematical model of gas emission in coal particles... 

Qin, Y.P. et al. 1998. Dimensionless analysis on heat dissipation of rock surrounding in coal face. Journal of China Coal Society 23(1) 62-66. 

Outside of the capillary dynamic approach, scaling arguments and dimensionless analysis can provide valuable insight into fluid physics on the micro-/nanoliter scale. An exhaustive review of these approaches has been conducted for a variety of microfluidic techniques [1]. However, for this article, we will strictly consider scaling metrics and dimensionless numbers pertinent to the surface-directed approach. 

A drug release mechanism from compressed cellulose ether matrices using dimensionless analysis has been discussed (17). A non-Fickian mechanism could be dismissed when calculating the Deborah and the Swelling interface numbers from relaxation, penetrant diffusion, swelling and drug diffusivity data (17). 

The analysis of a beam on an elastic foundation is governed by exactly the same equation as the pile under lateral loading. The analysis of this problem requires the beam equation for the link between lateral loading and lateral deflection of an elastic beam - this is another topic for reinforcement through duplication. The beam equation is a fourth order ordinary differential equation so that a certain amount of mathematical confidence is required for its solution. These are again problems which lend themselves to dimensionless analysis - and, indeed, it is through reduction of the governing equations to their dimensionless form that the appreciation of the importance of relative stiffnesses of soil and structure can be obtained. 

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