Dimensionless groups interpretation

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

It is also possible to interpret similarity in terms of dimensionless groups and variables in a proeess model. Kline [3] expressed that the numerieal values of all the dimensionless groups should remain eon-stant during seale-up. However, in praetiee this has been shown to be impossible [4],... 

In liquid-liquid extraction using wetted-wall columns, analysis is possible only by dimensionless groups (75) for the core fluid, flowing up inside the tube, k varies as approximately D and for the fluid falling down the inner walls, varies as Systems studied include phenol-kerosene-water, acetic acid-methylisobutylketone-water, and uranyl nitrate between water and organic solvents (7S, 80-82) interfacial resistances of the order 100 sec.cm." are observed in the last system. These resistances are interpreted as being caused by a rather slow third-order interfacial exchange of of solvent molecules (S) coordinated about each UOa" ion ... 

According to Bisio (62), scale-up can be achieved by maintaining the dimensionless groups characterizing the phenomena of interest constant from small scale to large scale. However, for complex phenomena, this may not be possible. Alternatively, dimensionless numbers can be weighted so that the untoward influence of unwieldy variables can be minimized. On the other hand, this camouflaging of variables could lead to an inadequate characterization of a process and a false interpretation of laboratory or pilot plant data. 

The effects of the many variables that bear on the magnitudes of individual heat transfer coefficients are represented most logically and compactly in terms of dimensionless groups. The ones most pertinent to heat transfer are listed in Table 8.8. Some groups have ready physical interpretations that may assist in selecting the ones appropriate to particular heat transfer processes. Such interpretations are discussed for example by GrOber et al. (1961, pp. 193-198). A few are given here. 

Since the meaning of the dimensionless groups is not immediately clear, we shall attempt to provide an interpretation here ... 

This method of compiling a complete set of dimensionless numbers makes it clear that the numbers formed in this way cannot contain numerical values or any other constant. These appear in dimensionless groups only when they are established and interpreted as ratios on the basis of known physical interrelations. Examples ... 

This dimensionless group, introduced for conciseness in rate correlations, has no simple physical interpretation. It is the product of several ratios A/kT represents the ratio of the characteristic London interaction energy to the thermal energy of the particle, R is the aspect ratio, while the Peclet number may be considered as the ratio of a characteristic energy for drag losses to the thermal energy possessed by the particle. This interpretation for the Peclet number becomes evident by using the relation D = tnkT to write... 

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

Each dimensionless group provides a physical interpretation, which can be helpful in assessing its influence in a particular application. 

The physical interpretation of each dimensionless group is not an easy task. Because each dimensionless group can present various physical interpretations, the study of each particular dimensionless pi term has to be carefully carried out. 

The case of the Reynolds number discussed above sho vs that the physical interpretation of one dimensionless group is not unique. Generally, the interpretation of dimensionless groups used in the flo v area in terms of different energies involved in the process, can be obtained starting vith the Bernoulli flovr equation. The relationship existing between the terms of this equation introduces one dimensionless group. 

For certain flow regimes, it is possible to reduce the number of dimensionless groups necessary to characterize a system by properly combining them. This further simplifies data collection and interpretation in several cases of considerable practical importance as shown in the sections that follow,... 

SIGNIFICANCE OF DIMENSIONLESS GROUPS. The three dimensionless groups in Eq. (9.14) may be given simple interpretations. Consider the group nDgpjp. Since the impeller tip speed H2 equals tcD r,... 

INTERPRETATION OF DIMENSIONLESS GROUPS. The relationships among several of the common dimensionless groups can be made clearer by considering them as/ratios of various arbitrarily defined fluxes—that is, rates of flow per unit area. The fluxes are ... 

In Table 1.10 those dimensionless groups that appear frequently in the heat and mass transfer literature have been listed. The list includes groups already mentioned above as well as those found in special fields of heat transfer. Note that, although similar in form, the Nus-selt and Biot numbers differ in both definition and interpretation. The Nusselt number is defined in terms of thermal conductivity of the fluid the Biot number is based on the solid thermal conductivity. 

Several dimensionless groups appear in the non-dimensional equations these are given in Table 5.4. These non-dimensional groups serve to highlight the relative importance of the myriad parameters in the problem the interpretation of these groups in terms of ratios of physical forces are also given in Table 5.4. These parameters will be referred to as appropriate in the following sections. 

One may interpret this dimensionless group ijay/K (or ijaE/K) as the ratio of viscous forces tending to deform (and break up) droplets to the interfacial tension forces tending to hold them together. It is most appropriately called a Taylor number. 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

The exponents of the dimensionless groups Ga and adp in Eqs 20 and 24 are very close. The exponent of the Reynolds number in Eq. 24 may be interpreted on the basis of Eq. 22. This yields a mean value of close to 0.7 It should also be noted that other correlations of specific of trickling... 

A descriptive interpretation of these dimensionless groups of parameters may be helpful to better understand the specifics under conditions of high pressures. 

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