Dimensionless numbers, table listing

The dimensionless source term essentially represents the ratio of generation to convection. For various generation terms, several additional dimensionless numbers may be defined. For example, if the generation of momentum due to gravitational forces is considered, a dimensionless number, called as the Froude number (Fr), is defined as the ratio of convection to gravitational factors. The dimensionless numbers discussed here along with other dimensionless numbers are listed in Table 2.1 together with their physical interpretation. 

The correlations derived from the analytical models, numerical modeling, and experimental results are listed in Table 4.21. The dimensionless numbers used to describe the droplet deformation... 

Tables 4.2 to 4.4 list several dimensionless numbers that are used in various areas of engineering. This list can be helpful in performing a dimensional analysis, to help interpret results that are sometimes difficult to discern from the variety of dimensionless numbers that can result during such an undertaking.

A list of dimensionless numbers that can arise in the analysis of convective heat transfer is given in Table 1.2. 

If such a calculation is carried out for a real three-dimensional crystal, the result is a series (such as that just given in brackets) whose value sums to a dimensionless number that depends upon the crystal structure. That number is called the Madelung constant, M, and its value is independent of the unitcell dimensions. Table 21.5 lists the values of the Madelung constant for several crystal structures. The lattice energy is again the opposite of the total potential energy. Expressed in terms of the Madelung constant, it is... 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

The dynamical response of two-phase flows can be commonly characterized successfully in terms of the dimensionless numbers [2]. Table 1 lists some force-related dimensionless numbers. These dimensionless numbers demonstrate competing phenomena of forces buoyancy, gravitational, inertial, viscous and interfacial forces. The Grashof number (buoyancy to viscous forces), the Bond number (gravitational to interfacial forces) and the... 

A dimensionless group not listed in Table 5.1 is the so-called Wall Sherwood Number that represents the resistance to mass transfer through a fluid in laminar flow divided by the resistance within the tubular wall. It is used to gauge the relative importance of these resistances in industrial membrane processes as well as those occurring in living organisms (see Chapter 8). 

Many different dimensionless numbers can be obtained by simply dividing the terms in any dimensional consistent equation, as seen in the Appendix B. The most connnonly used dimensionless munbers are listed in Table B.4. 

In Table 1.4, the characteristic time-scales for selected operations are listed. The rate constants for surface and volume reactions are denoted by and respectively. Furthermore, the Sherwood number Sh, a dimensionless mass-transfer coefficient and the analogue of the Nusselt number, appears in one of the expressions for the reaction time-scale. The last column highlights the dependence of z p on the channel diameter d. Apparently, the scale dependence of different operations varies from dy f to (d ). Owing to these different dependences, some op-... 

Examples of dimensionless groups that specify ratios of transport mechanisms are listed next in Table II and depend on the size and shape of the domain. The Peclet numbers for heat (Pet) and solute (Pes) and momentum (Re) transport are ratios of scales for convective to diffusive transport and depend on the magnitudes of the velocity field and the length scale for the diffusion gradient. Boundary layers form at large Peclet numbers (Pet or Pes) or Reynolds numbers (Re). The fonnation of a boundary layer at a large Re is particularly important in crystal growth from the melt, because the low... 

The several variables on which fluid flow depends may be gathered into a smaller number of dimensionless groups, of which the Reynolds number and friction factor are of particular importance. They are defined and written in the common kinds of units also in Table 6.1. Other dimensionless groups occur less frequently and will be mentioned as they occur in this chapter a long list is given in Perry s Chemical Engineers Handbook, p. 6 9). 

A number of excellent compilations of dimensionless group correlations are available in the literature (see Levich [12], Selman and Tobias [14], and Poulson [17]). These compilations are not reproduced here, but the correlations for a number of common hydrodynamic geometries are listed in Table 1. Because these equations are empirical in nature, they are valid only for the employed ranges in Re and Sc. 

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. 

It will also be noticed from the equations that a number of dimensionless groups ate involved. These were calculated using the parameter values listed above and are summarized in Table 19.6. Mass transfer coefficients for three-phase systems cannot easily be estimated (although an approximate equation is given later on). In the absence of this information, simulation studies were confined to a very high Biot number (100), so that film transfer resistances could be ignored. 

A number of authors have studied the ratio of viscous to capillary forces and the effect of the ratio or dimensionless group on oil recovery. Table 2 lists several of these groups which have been described by various authors (11,15,21,38-51). 

The gN values are dimensionless and characteristic of the type of nucleus. yN is the nuclear gyromagnetic ratio. Some values are listed in Table 1, together with the corresponding quantum numbers I and Mj, in which the quantum number Mj allows for the space-quantization of... 

Using the results derived for the Poiseuille flow in straight channels, the hydraulic resistance 7 ijyd for six different cross-sections has been listed in Table 2.3. These results are all valid for the special case of a translation invariant (straight) channel. The vanishing of the nonlinear term (v S7)v in the N-S equation has been used in the derivation. The analysis is carried out using the approximation of low dimensionless Reynolds number and high aspect ratio of microchannel. 

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