A "dimension" can be measured or derived. The "fundamental dimensions" (length, time, mass, temperature, amount) are distinct and are sufficient to define all the others. We also use many derived dimensions (velocity, volume, density, etc.) for convenience.
"Units" can be counted or measured. Measured units are specific values of dimensions defined by law or custom. Many different units can be used for a single dimension, as inches, miles, centimeters, furlongs, and versts are all units used to measure the dimension length.
It is always good practice to attach units to all numbers in an engineering calculation. Doing so
different dimensions: length, temperature -- so cannot be added
same dimension: length, different units -- can add
Values may be combined; units combine in similar fashion.
4.5 is a "dimensionless" quantity (in this case a pure number)
Trigonometric functions can only have angular units (radians, degrees). All other functions and function arguments, including exponentiation, powers, etc., must be dimensionless.
is OK; but is meaningless.
is never defined.
Every valid equation must be "dimensionally homogeneous" (a.k.a. dimensionally consistent). All additive terms must have the same dimension.
Consider an equation from physics that describes the position of a moving object:
When we say a quantity is dimensionless, we mean one of two things. First, it may just be a number like we get when counting.
EXAMPLES: Dimensionless Numbers
Second -- and of particular interest in engineering -- are combinations of variables where all the dimension/units have "canceled out" so that the net term has no dimension.
These are often called "dimensionless groups" or "dimensionless numbers" and often have special names and meanings. Most of these have been found using techniques of "dimensional analysis" -- a way of examining physical phenomena by looking at the dimensions that occur in the problem without considering any numbers.
Probably the most common dimensionless group used in chemical engineering is the "Reynolds Number", given by . This describes the ratio of inertial forces to viscous forces (or convective momentum transport to molecular momentum transport) in a flowing fluid. It thus serves to indicate the degree of turbulence. Low Reynolds numbers mean the fluid flows in "lamina" (layers), while high values mean the flow has many turbulent eddies.
Modified: 6/1/94; 8/25/95, 7/29/96; 5/17/2004
Copyright 2004 by R.M. Price -- All Rights Reserved