Minimum distance between two reels

[SteveNY123]

Dear All,
Not sure if this is the right area of the forum. If not please let me know.

I tried googling everywhere but could not find a formula I can use in Excel to solve this problem.

PROBLEM TO SOLVE:
With reference to the attached picture, the system is a classic two-reel system where some tape is wound back and forth between said two reels - practically like in a classic magnetic tape compact cassette used for music in the 90's. Because that the tape is wound back and forth between the two reels (i.e. from one to the other then back to the first), the overall diameter of each of the two reels including the tape, varies at all time.

Because of the limited space in the enclosure I need to position the reels as close to each other as possible but with a gap E which is defined as the minimum in the worst case condition (not sure at what ratio of tape winding that is... I don't think that is 50/50...?!). In other words I need to find out the minimum distance A between the two reels centres allowing a minimum gap E in the worst case scenario.

WHAT I'M ASKING:
An equation I can put in excel with the following parameters as input/output (with reference to the drawing above)

Output:
- An equation I can put into Excel (or already in excel if available) providing the following as output:
- A = minimum distance between reels centres
- Generally know at what percentage of tape wound to one side is the worst case scenario

Inputs:
- B = Core diameter (O.D.) - generally as guidelines between 18 and 30mm
- K = Tape thickness - generally as guidelines between 4 and 10 micron
- L = Tape length (not labelled on the drawing) - generally as guideline anything between 400 and 1500 meters
- (NOTE: the above 3 inputs should allow the calculation at any time of a roll diameter C assuming a length L is entirely wound onto that reel of core diameter B - but cannot find the equation - just calculators, see links at the bottom)
- E = minimum gap to be maintained between the tape "C" and "D" or, in the worst case scenario between "C" and "B" (if the tape is all wound on one reel)

- General assumptions:
- considering the tape length and thickness (see dimensions guidelines in the above points) distances M and G are negligible and for this purpose they can even be considered to be 0 or set to a realistic fixed value like 100mm (or again if you feel like it, also added as input parameter).
- Because of the above point the roll F can be disregarded as if it is not on the system

Thank you very much for all the help!

Steve

[Guest]

Roughly speaking the worst case scenario, the distance between the two reals is as small as possible, happens when the reels each contain half the tape, assuming F is irrelevant. The proof is long and boring and so I'm not going to go over it.

Roughly if n is the number of times the tape is wound around the real (it is actually a slight over-estimate just to be safe), d is the amount of tape, t the tape thickness, b the reel radius, then
d = 2 pi (n-1) b + pi (n-1) (n-2) t

d = 0 when n=1
d = 2 pi b when n=2
d = 2 pi b + 2 pi (b+t) when n=3
d = 2 pi b + 2 pi (b+t) + 2 pi (b+2t) when n=4
Each time we wrap around the contribution to d slightly increases because the radius increases.

Given d you can use the quadratic formula to solve for n. From n you can work out the radius is b + n t

[SteveNY123]

Thank you very much