## Consistent Units

The conclusion that dimensionless numerical values are universal is valid only if a consistent system of units is used for all quantities in a given equation. If such is not the case, then the numerical quantities may include conversion factors relating the different units. For example, the velocity (V) of a fluid flowing in a pipe can be related to the volumetric flow rate (Q) and the internal pipe diameter (D) by any of the following equations:

although the dimensions of V (i.e., L/t) are the same as those for Q/D2 (i.e., L3/tL2 = L/t), it is evident that the numerical coefficient is not universal despite the fact that it must be dimensionless. This is because a consistent system of units is not used except in Eq. (2-10). In each equation, the units of V are ft/s. However, in Eq. (2-7), Q is in ft3/s, whereas in Eq. (2-8), Q is in gallons per minute (gpm), and in Eq. (2-9) it is in barrels per hour (bbl/hr), with D in inches in each case. Thus, although the dimensions are consistent, the units are not, and thus the numerical coefficients include unit conversion factors. Only in Eq. (2-10) are all the units assumed to be from the same consistent system (i.e., Q in ft3/s and D in ft) so that the factor 4/^ is both dimensionless and unitless and is thus universal. It is always advisable to write equations in a universally valid from to avoid confusion; i.e., all quantities should be expressed in consistent units.

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