# Units, Dimensions, and Dimensionless Numbers

Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number the number of tick marks to the units that the standard represents, they conceptually are referring to a dimensionful quantity. Properties A dimensionless quantity has no physical unit associated with it. A dimensionless proportion has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the SI system of units or the imperial system of units.

This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent. Example The power consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter , and the speed of the stirrer.

For example, specific gravity is a simple way to express the relative masses or weights of equal volumes of various materials. The specific gravity is defined as the ratio of the weight of a volume of the substance to the weight of an equal volume of water.

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If the density of water, that is the mass of unit volume of water, is known, then if the specific gravity of some substance is determined, its density can be calculated from the following relationship:. Perhaps the most important attribute of a dimensionless ratio, such as specific gravity, is that it gives an immediate sense of proportion.

This sense of proportion is very important to food technologists as they are constantly making approximate mental calculations for which they must be able to maintain correct proportions. For example, if the specific gravity of a solid is known to be greater than 1 then that solid will sink in water. The fact that the specific gravity of iron is 7.

## Dimensionless Numbers | Symscape

Another advantage of a dimensionless ratio is that it does not depend upon the units of measurement used, provided the units are consistent for each dimension. Dimensionless ratios are employed frequently in the study of fluid flow and heat flow.

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They may sometimes appear to be more complicated than specific gravity, but they are in the same way expressing ratios of the unknown to the known material or fact. These dimensionless ratios are then called dimensionless numbers and are often called after a prominent person who was associated with them, for example Reynolds number , Prandtl number , and Nusselt number , and these will be explained in the appropriate section. When evaluating dimensionless ratios, all units must be kept consistent. For this purpose, conversion factors must be used where necessary. Precision of Measurement. Every measurement necessarily carries a degree of precision, and it is a great advantage if the statement of the result of the measurement shows this precision.

The statement of quantity should either itself imply the tolerance, or else the tolerances should be explicitly specified. For example, a quoted weight of Where there is doubt, it is better to express the limits explicitly as The temptation to refine measurements by the use of arithmetic must be resisted. For example, if the surface of a rectangular tank is measured as 4. A more reasonable answer would be 28 m 3.

Multiplication of quantities in fact multiplies errors also. In process engineering, the degree of precision of statements and calculations should always be borne in mind.

## DIMENSIONLESS GROUPS

Every set of data has its least precise member and no amount of mathematics can improve on it. Only better measurement can do this.

A large proportion of practical measurements are accurate only to about 1 part in In some cases factors may well be no more accurate than 1 in 10, and in every calculation proper consideration must be given to the accuracy of the measurements. Electronic calculators and computers may work to eight figures or so, but all figures after the first few may be physically meaningless. For much of process engineering three significant figures are all that are justifiable.

Dimensions It has been found from experience that everyday engineering quantities can all be expressed in terms of a relatively small number of dimensions. Therefore 7. Unit Operations in Food Processing. Mach twice speed sound, with like meters are treated exactly same ascoefficients variables, body, acceleration Units Converter This appendix lists those decisions of CGPM and CIPM that bear directly upon definitions units SI, dependent field variables characterize moving stationary fluids scalars.

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