Most web speed and length measurements are based on measuring the RPMs of a non-slipping roller of a known circumference.
V = 2pnr (V = speed in fpm, p = 3.1415, n = RPMs, r = radius in feet)
V = wr (V = speed in inches/s, w = radians per sec, r = radius in inches)
To calculate length, simply multiply speed times time.
L = 2pnrt (L = length in feet, p = 3.1415, n = RPMs, r = radius in feet, t = time in minutes)
L = wrt (L = length in inches, w = radians per sec, r = radius in inches, t = time in seconds)
These calculations assume the web speed and the roller surface speed are 1:1, no slip, no strain.
Besides slip on the monitoring rollers, there are two other sources of error in speed and length measurement – strain from tension and strain from curvature. (I’ve told you before that strain is the key to web handling.)
The difference between a tensioned and untensioned webs is strain. Any speed or length measurement made while the web is tensioned will overestimate the untensioned length of material by the percent strain of the web. If you measure PET at 0.2% and compare it to a sample unrolled on the warehouse floor, you will see you have lost 0.2% from your measured length. For a 5000 ft roll, you will lose 10 feet. If you are running PE or PP at 1% strain, you will lose 50 feet out of 5000.
I usually recommend using an existing roller to monitor web speed, often a driven roller that already has an encoder or tachometer. However, measuring speed on a wrapped roller will have a small error due to the through-thickness or Z-direction strain variation of a curved web. When a web conforms around a roller the outer surface of the web elongates and the inner layer contracts. The difference in strain from the outer to inner surface is equal to the web thickness over the roller radius.
De = t/r (De = change in strain, t = web thickness, r = roller radius)
The center of the web will be at average strain, but the inner surface where you are measuring speed will be at a lower strain equal to half this value shown above. For thicker webs on smaller rollers, this is a significant error. If you measure a 10 mil web using a 4-inch diameter roller, the t/2r is 0.010/4 or 0.025 or 2.5%. Over 5000 ft, this would be a 125 foot error! For thinner webs and larger roller, this is a big deal. For a 1-mil web on a 10-inch diameter roller, t/2r = 0.001 / (2)(5) = 0.0001 or 1/100 of 1%. Over 5000 feet, this would be only a 6-inch error.
To avoid the curved web measurement error, use a tally wheel style roller that rides on the flat web between rollers.