# Packing Efficiency of Three and Four Parallel Cells

#### CaptainBrock

##### Newly Enlightened
This post analyzes the packing efficiency of three parallel cylindrical cells in a tube, versus four of the same cells in a tube.
Packing efficiency is defined as the percentage of total volume of the containing tube occupied by the cells. One might expect three cells in a triangular arrangement to have a greater packing efficiency than four cells in a square pattern because hexagonal packing in a large field is known to be more efficient than square packing.
Let's see what happens when the cells are packed into a cylindrical tube. In the following analysis, only the two-dimensional cross-sectional area is considered, and cells are represented by unit circles with a radius of one.

Diagram of three cells packed in a cylindrical tube:

Computation of packing efficiency for three cells in a tube:

Diagram of four cells packed in a cylindrical tube:

Computation of packing efficiency for four cells in a tube:

The result was surprising to me. The four cells occupy the space within an enclosed tube more completely than three cells do! The four cells use 68.6% of the internal space, while the three cells use only 64.6%
The minimum diameter of a tube enclosing four 18650 cells will be 2.4142*18.6 mm = 44.90 mm.
The minimum diameter of a tube enclosing three 18650 cells will be 2.1547*18.6 mm = 40.08 mm.
These numbers are theoretical minimums and do not include wall thickness of the container tube or internal cell holder structure. I measured the outer body diameter of a four-cell NiteCore TM-15 at 50.0 mm, and the body diameter of a three-cell Thrunite TN31 is given as 49.1 mm.
Conclusion: A three-cell configuration wastes more internal space than a four-cell configuration, and resulting diameter of a four-cell flashlight will be only very slightly greater than that of a three-cell flashlight.

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#### leon2245

##### Flashlight Enthusiast
Interesting, well done, thanks for sharing!

#### monkeyboy

##### Flashlight Enthusiast
Thanks for posting this. I don't think that this is such a small difference though. The 11% difference in diameter between 3 and 4 cells makes a big difference on how comfortable the light is to hold for extended periods of time IMO. I don't like 1 x 4 18650 cell format lights, as it's just beyond the comfort threshold for me. I'd much rather have a 2 x 2 format battery tube, even though there is much more wasted space.
e.g. I have a custom dual bored 2.5D maglite that takes 4 x 18650. Incidentally, the packing efficiency of a 2 cell setup is exactly 50%.

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#### Conte

##### Enlightened
Hmm, I never actually thought about it before, but now that I see it on paper it makes sense and doesn't surprise me at all .

Now I'm wondering how 5 would work out, or more, hah.

#### watt4

##### Enlightened
I had a flashback to classroom discussions of crystal structures.

#### ven

##### Flashaholic
Thank you for posting,very interesting:thumbsup: I always thought there was not much in it regarding light diameter in general between 3 and 4 cell lights. For me i prefer 4 as longer run times and less ask per cell over 3(depending on set up maybe).

Here are 2 carriers from fenix,tk75vn(4 cell) and tk51(3 cell)

4 cell on left and 3 cell on right

Thrunite tn series have the 3 cell carriers,wonder why it is chosen over 4 cells,not sure on weight savings but i am sure it would not be much in the type of light, but a loss of run time/efficiency ........

#### leon2245

##### Flashlight Enthusiast
I like the idea of the ones without removable battery carriers at all, for maximum packing efficiency. & maybe battery change time too?

#### ven

##### Flashaholic
Yes Leon agree ,I like the way the ea4 and 8 has been machined out of one piece. I see advantages of both in a way .If you had a spare carrier for the fenix you could swap them over(if did not want size with ext for example). I find the fenix carriers tight regarding my protected cells ,not over easy removing them.
Some carriers like the tn35vn and built in mm15vn are far easier swapping cells .

Still easier though to slide out of body.

Sent from my iPhone whilst passing the time in kids play centre.........:laughing: