Calculating the Length of String on a Reel

Date: 12/07/2001 at 10:25:54
From: Larry Schworer
Subject: Calculate string on a reel

What is the formula for calculating the length of wire that can go on 
a reel (spindel) where the reels may have different diameters, hub 
diameters, and widths, and the wires may have different diameters?

Once a reel and wire combination are selected they do not change for 
that reel combination.

I wish to write a program to make this calculation.

Thanks,
La

Date: 12/07/2001 at 12:06:15
From: Doctor Peterson
Subject: Re: Calculate string on a reel

Hi, Larry.

The answer will be similar to that for a similar question we have 
answered many times, about the amount of material (such as carpet) on 
a roll:

   How Much Carpet is Left?
   http://mathforum.org/dr.math/problems/gillette.03.27.99.html   

We find the length by equating the volume of material on the roll to 
its volume laid out flat.

There is always a caveat in this calculation, however: we have to 
assume that the material retains the same thickness as it bends around 
the roll, and is not either compressed or wound loosely. The 
calculation is very sensitive to the actual thickness of the material 
on the roll.

The same is even more true in your case. The wire may be wound in 
different ways that will incorporate different amounts of air space 
between turns, so it is hard to be sure how much of the volume on the 
reel consists of wire. The best thing to do would be to measure an 
actual reel of known length in order to determine this ratio and 
adjust the formula, rather than blindly trust theory.

Having said that, let's look at the problem. For simplicity, I'm going 
to pretend that the wire has a square cross section, since then there 
would be no need for space between turns. Your measurements will 
determine how closely this corresponds to reality.

Suppose the reel has hub diameter Di, outside diameter Do, and width 
W. Then the volume of wire on it is

    V = pi(Do^2 - Di^2)W/4

Suppose you have a wire with length L and square cross-section with 
side Dw (the diameter of the wire). Its volume is

    V = LDw^2

Setting these equal, we get

    LDw^2 = pi(Do^2 - Di^2)W/4

and we can solve to find

    L = pi(Do^2 - Di^2)W/(4Dw^2)

Now suppose you measure an actual reel, and the actual length is X 
times this calculated length (let's hope not too far from 1). Then you 
can take X as the typical correction factor, and multiply any 
calculated length by X to estimate the actual length when you don't 
know it ahead of time. This correction factor may well be more or less 
constant for all reels wound by the same method.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/