Dimensionless groups of quantities

Dimensionless groups of quantities or numerics occupy a unique position in physics. The magnitude of a numeric is independent of the units in which its component physical properties are measured provided only that these are measured in consistent units. The numerics form a distinct class of entities, which, though being dimensionless, cannot be manipulated as pure numbers. They do not follow the usual rules of addition and multiplication since they have only a meaning if they are related to a specific phenomenon. 

The laws of physics may all be expressed as relations between numerics and are in their simplest form when thus expressed. The use of dimensionless expressions is of particular value in dealing with phenomena too complicated for a complete treatment in terms of the fundamental transport equations of mass, energy and angular momentum. Most of the physical problems in the process industry are of this complicated nature and the combination of variables in the form of dimensionless groups can always be regarded as a safe start in the investigation of new problems. 

A complete physical law expressed as an equation between numerics is independent of the size of the system. Therefore dimensionless expressions are of great importance in problems of change of scale. When two systems exhibit similarity, one of them, and usually the smaller system, can be regarded as the model . Two systems are dynamically similar when the ratio of every pair of forces or rates in one system is the same as the corresponding ratio in the other. The ratio of any pair of forces or rates constitutes a dimensionless quantity. Corresponding dimensionless quantities must have the same numerical value if dynamical similarity holds. 

The value of dimensionless groups has long been recognised. As early as 1873, Von Helmholtz derived groups now called the Reynolds and Froude numbers , although Weber (1919) was the first to name these numerics. 

The standardised notation of numerics is a two-letter abbreviation of the name of the investigator after whom the numeric is named Xy 

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