Dimensionless groups

Dimensionless groups

A dimensionless group is any combination of dimensional or dimensionless quantities possessing zero overall dimensions. Dimensionless groups are frequently encountered in engineering studies of complicated processes or as similarity criteria in model studies. A typical dimensionless group is the Reynolds number (a dynamic similarity criterion), NRe = VDρ/μ. Since the dimensions of the quantities involved are velocity V: [L/Θ]; characteristic dimension D: [L]; density ρ: [M/L3]; and viscosity μ: [M/LΘ] (with M, L, and Θ as the fundamental units of mass, length, and time), the Reynolds number reduces to a dimensionless group and can be represented by a pure number in any coherent system of units. See Dynamic similarity

Many important problems in applied science and engineering are too complicated to permit completely theoretical solutions to be found. However, the number of interrelated variables involved can be reduced by carrying out a dimensional analysis to group the variables as dimensionless groups. See Dimensional analysis

The advantages of using dimensionless groups in studying complicated phenomena include:

  • 1.  A significant reduction in the number of “variables” to be investigated; that is, each dimensionless group, containing several physical variables, may be treated as a single compound “variable,” thereby reducing the number of experiments needed as well as the time required to correlate and interpret the experimental data.
  • 2.  Predicting the effect of changing one of the individual variables in a process (which it may be impossible to vary much in available equipment) by determining the effect of varying the dimensionless group containing this parameter (this must be done with some caution, however).
  • 3.  Making the results independent of the scale of the system and of the system of units being used.
  • 4.  Simplifying the scaling-up or scaling-down of results obtained with models of systems by generalizing the conditions which must exist for similarity between a system and its model.
  • 5.  Deducing variation in importance of mechanisms in a process from the numerical values of the dimensionless groups involved; for instance, an increase in the Reynolds number in a flow process indicates that molecular (viscous) transfer mechanisms will be less important relative to transfer by bulk flow (“inertia” effects), since the Reynolds number is known to represent a measure of the ratio of inertia forces to viscous forces. See Froude number, Knudsen number, Mach number, Reynolds number

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Inserting the dimensionless groups from Table 2 into Eq.
Table 2 constructs dimensionless groups from the dimensional quantities defined in Table 3.
The dimensionless groups obtained from the literature review for gravity displacements are listed in Table 1.
New correlations based on dimensionless groups for the prediction of flooding in narrow passages are proposed and found to be in good agreement with the available data.
Plotting experimental data in the form of dimensionless groups on logarithmic coordinates, representing the plotted values by a straight line, and thereby inferring an algebraic correlating equation in the form of a power-law have been characteristic practices in chemical engineering for over a century.
The common dimensionless groups (see (1a)-(1c)) are the particle-diameter-based Reynolds number [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
In recent years the dimensionless groups, as demonstrated in detail here, have grown in significance and importance in contemporary mathematical and computer modeling as well as the traditional fields of physical modeling.
We have represented this data in terms of the dimensionless groups, a dimensionless throughput Q* versus a dimensionless pressure gradient, (dp/dx)* where
The design tool presented in this paper is based on correlation of dimensionless groups that are comprised of variables describing the design of geothermal heat pump systems.