Why my battery pack lasts only a few months?

Using just a resistor for this, the current changes depending on the voltage difference between the source and destination.

So you'd need to size the resistor so that the largest possible difference between them limits the current to less than the max current the least-capable part was able to handle, and then live with the lower currents you'd have in every other situation.
The spec sheet says the cells are capable of 20A when new, so 4P would be 80A. Even though they are used, they probably could start the car on their own, or with the capacitor, eliminating the resistor all together.

The problem I see is that there is an unregulated charging current coming from the alternator, and that likely exceeds the max charging current of the cells. Per the spec sheet, 4P would allow a 5.44A charge rate.
 
The spec sheet says the cells are capable of 20A when new, so 4P would be 80A. Even though they are used, they probably could start the car on their own, or with the capacitor, eliminating the resistor all together.

The problem I see is that there is an unregulated charging current coming from the alternator, and that likely exceeds the max charging current of the cells. Per the spec sheet, 4P would allow a 5.44A charge rate.
In which case, the resistor would (as noted ;) ) have to be sized to allow only 5.44A at the largest possible voltage difference between source and destination, and then live with the lower current that would flow in all other situations.

The largest difference would be when hte alternator is being revved up to it's highest output voltage, and the battery is *completely empty*, at whatever voltage that is for cells at lowest possible allowed voltage by the system electronics.
 
There is enough energy stored to start the engine, even at 12v. There are 3 capacitor banks rated for 500 farad each, they're all made out of 6 2,7v caps in series for a total of 16,2v fully charged, all the caps are soldered on a balancing board and that is the pack you can find on Amazon, Aliexpress, eBay etc. So if only a resistor is needed to limit the current i'm already on the good way. The actual resistor is a 25w 0,1 ohm, which shoupd limit the current to about 10/15 amp, which is ok considering the lithium cells used.
I think you have the same cap bank as me but 3 of them, remember this means you have 250F total (83.33 each) since they are in series. I calculated from 12.8v down to 9v I can get about 3.5kJ out of mine which is enough to start an engine I think sometimes depending on conditions, if it's cold or a larger engine maybe not. But you have 3 times that much so sounds good to me to just have the cap for starting.
 
Thank you guys very much for all the info, I was focused only on the discharge rate, so it may be the charging rate that is killing my lithium packs. Have to give it a try with all the suggestions written here. What should the correct resistor be?
 
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I think you have the same cap bank as me but 3 of them, remember this means you have 250F total (83.33 each) since they are in series. I calculated from 12.8v down to 9v I can get about 3.5kJ out of mine which is enough to start an engine I think sometimes depending on conditions, if it's cold or a larger engine maybe not. But you have 3 times that much so sounds good to me to just have the cap for starting.
Yes they are in series but every capacitor is rated for 500F, so I assume the total capacity should be 500F, and I put 3 of these in parallel, so the total capacity should be 1500F. Correct me if I'm wrong. Attached there are a couple of pictures of my capacitor banks and here is the link: Amazon.it
 

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Nope that's not how series capacitors work and yes the marketing for these is deceptive. You have 250F total like I said, which is still a lot of juice, like 10.5kJ depending on starting and ending voltage probably enough to start a smaller engine even in colder conditions. You can look up capacitors in series yourself but the energy they store does of course add with more capacitors but capacitance is not the same as amp hours, the math is different since energy in a capacitor is 0.5*C*V^2, see it's that squared voltage. The killer is you can only use some of that energy since the starter will only crank down to a certain voltage, I mean after that you'll still loose all the energy in them into the starter but it will be stopped so it's wasted as heat and not starting.
 
Thank you guys very much for all the info, I was focused only on the discharge rate, so it may be the charging rate that is killing my lithium packs. Have to give it a try with all the suggestions written here. What should the correct resistor be?
I would use a cheap adjustable (voltages and current) boost converter to bring the charge voltage up to 16.4 or so, so they are closer to full charge rather than half charge. 4.1v per cell will help extend the cell life compared to fully charging to 4.2v. That might solve most of the issues.
 
Nope that's not how series capacitors work and yes the marketing for these is deceptive. You have 250F total like I said, which is still a lot of juice, like 10.5kJ depending on starting and ending voltage probably enough to start a smaller engine even in colder conditions. You can look up capacitors in series yourself but the energy they store does of course add with more capacitors but capacitance is not the same as amp hours, the math is different since energy in a capacitor is 0.5*C*V^2, see it's that squared voltage. The killer is you can only use some of that energy since the starter will only crank down to a certain voltage, I mean after that you'll still loose all the energy in them into the starter but it will be stopped so it's wasted as heat and not starting.
Can help for a better starting have only 5 caps instead of 6? 2,7x5=13,5v. As the alternator charges at about 14v can be a problem going a little overcharge, maybe up to 2,8/2,9 each capacitor? Also is strictly necessary a balancing board for the capacitors as I know they tend to balance each other?
 
I would use a cheap adjustable (voltages and current) boost converter to bring the charge voltage up to 16.4 or so, so they are closer to full charge rather than half charge. 4.1v per cell will help extend the cell life compared to fully charging to 4.2v. That might solve most of the issues.
Might give it a try but I read in the description that it can take 12A max output and 15A input, and over 7/8A it is necessary to improve heat dissipation. So that means i have to calculate the right resistor to limit the current to the lithium cells charging spec.
 
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