Dimensionless groups relevant

TABLE 5.1 Some Dimensionless Groups Relevant to Mnltiphase Systems ... 

Brinkman number n. A dimensionless group relevant for heat transfer in flowing viscous liquids such as polymer melts. It is defined by Nfir = pV /kAT, in which p is the viscosity, V the velocity, k the thermal conductivity, and AT is the difference in temperature between the stream and the confining wall. The number represents the ratio of the rates of heat generation and heat conduction. Shenoy AV (1996) Thermoplastics melt rheology and processing. Marcel Dekker, New York. 

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

Liquid flows down an inclined surface as a film. On what variables will the thickness of the liquid film depend Obtain the relevant dimensionless groups. It may be assumed that the surface is sufficiently wide for edge effects to be negligible. 

A glass particle settles under the action of gravity in a liquid. Upon which variables would you expect the terminal velocity of the particle to depend Obtain a relevant dimensionless grouping of the variables. The falling velocity is found to be proportional to the square of the particle diameter when other variables are kept constant. What will be the effect of doubling the viscosity of the liquid What does this suggest about the nature of the flow ... 

A spherical particle settles in a liquid contained in a narrow vessel. Upon what variables would you expect the falling velocity of the particle to depend Obtain the relevant dimensionless groups. 

Obtain relevant dimensionless groups for this problem. 

It must be emphasized that dimensional analysis is used to find the minimum number of dimensionless groups of all the variables known to be relevant to the description of a... 

A current example of a problem that can be simplified through segregation of its components by physical scale is the deposition of on-chip interconnects onto a wafer. Takahashi and Gross have analyzed the scaling properties of interconnect fabrication problems and identified the relevant control parameters for the different levels of pattern scale [135], They define several dimensionless groups which determine the type of problem that must be solved at each level. 

The principle of similarity [Holland (1964), Johnstone and Thring (1957)] together with the use of dimensionless groups is the essential basis of scale-up. The types of similarity relevant to liquid mixing systems together with their definitions are listed as follows. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

Another way to derive the dimensionless groups needed for scale-up purposes has been proposed by Edgeworth and Thring (1957). The general concept is that the differential equations that describe the process could be used, since they are known for most processes that are relevant to chemical engineering. The main problem is that many of these equations cannot be easily integrated or are very difficult to be handled, in general. 

While the Deborah number is often used to compare the time for deformation with the time of observation in experiments, it also inspires us to identify and formulate other dimensionless groups that compare the various characteristic times and forces relevant in colloidal phenomena. We discuss some of the important ones. 

In addition to the Peclet number, one can also define other dimensionless groups that compare either relevant time scales or energies of interaction. Using some of the concepts previewed in Section 4.7c and Table 4.4, one can define an electrostatic group (in terms of the zeta potential f and relative permittivity cr of the liquid) as... 

Single screw extruder. Let us take the case of a single screw extruder section that works well when dispersing a liquid additive within a polymer matrix. The single screw extruder was already discussed in the previous section. However, the effect of surface tension, which is important in dispersive mixing, was not included in that analysis. Hence, if we also add surface tension as a relevant physical quantity, it would add one more column on the dimensional matrix. To find the additional dimensionless group associated with surface tension, as, and size of the dispersed phase, R, two new columns to the matrix in eqn. (4.32) must be added resulting in ... 

Thus, the equations describing the thermal stability of batch reactors are written, and the relevant dimensionless groups are singled out. These equations have been used in different forms to discuss different stability criteria proposed in the literature for adiabatic and isoperibolic reactors. The Semenov criterion is valid for zero-order kinetics, i.e., under the simplifying assumption that the explosion occurs with a negligible consumption of reactants. Other classical approaches remove this simplifying assumption and are based on some geometric features of the temperature-time or temperature-concentration curves, such as the existence of points of inflection and/or of maximum, or on the parametric sensitivity of these curves. 

Table 2 Relevant dimensionless groups used in boiling heat transfer in miorosoale configurations...

Velocity and temperature fields are therefore only similar when also the dimensionless groups or numbers concur. These numbers contain geometric quantities, the decisive temperature differences and velocities and also the properties of the heat transfer fluid. The number of dimensionless quantities is notably smaller than the total number of all the relevant physical quantities. The number of experiments is significantly reduced because only the functional relationship between the dimensionless numbers needs to be investigated. Primarily, the values of the dimensionless numbers are varied rather than the individual quantities which make up the dimensionless numbers. 

Figure 2.30 Summary of experimental and simulated data for coumarin-induced leveling during nickel deposition in a semicircular groove. The data are shown as a function of the dimensionless group p along with the relevant experimental parameters. In this instance (unlike...

A dilute gas mixture is assumed to behave as a continuum when the mean free path of the molecules is much smaller than the characteristic dimensions of the problem geometry. A relevant dimensionless group of variables, the Knudsen number Kn, is defined as [47, 30, 31] ... 

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