Issue while charging and discharging?

LiFePO4 batteries do not like being over charged or taken to "dead"... Remember that a "12 volt" Lithium battery is 3 or 4 cells in series... So, cells can be within 10% to 100% SoC, and one cell can be over or under voltage.

Typically, a good quality Li Ion battery pack will have some sort of BMS/battery management system/ to both monitor the individual cell voltages, and can alarm or turn off the DC bus connection if one or more cells are risking damage.

BMS can have other functions such as cell balancing, over current shutoff, over charge shutoff, over/under temperature shutoff, etc... too.

Assuming the PC fan is 0.06 Watts, that works out to (that is a very small fan):

Power=Voltage*Current
Current = Power/Voltage
0.06 Watts / 12 volts = 0.005 amps
50 AH / 0.005 amps = 10,000 hours of storage (100% to 0% of battery capacity)
10,000 Hours * 1/24 hours per day * 1/30 days per month = 13.9 months of usage

In reality, probably would use 20% to 90% of battery capacity (longer life, avoid over/under voltage cells) or around 9.7 months of runtime per 50 AH bank charging...

If you are in New York, Li Ion batteries do typically have a low temperature charging limit of ~50F--So if you have hard freezes there, you might want to address that (no charging during winter, insulated/warm structure, etc.).

You could very easily recharge that battery pack 2x to 3x a year and avoid solar charging (especially during winter).

Or, many outhouses just use a vent pipe and avoid fans all together.

Just to use some round numbers... A suggested minimum solar panel size for your fan load (say 14 days of no/low sun, and worst case 1 hour of sun per day during winter):

0.06 Watts * 24 hours per day = 1.44 Watt*Hours per day
1.44 hours per day * 14 days of "no sun" * 1/0.77 solar panel+controller deratings = 26.2 WH (per 14 days)
26.2 WH per 14 days * 1/24 days * 1/1 hour of sun per day = 1.9 Watt solar panel

So your 100 Watt solar panel is "way more" than needed for your 0.06 Watt fan. And frankly charging your battery bank 2-3x per year would be more than sufficient.

If you wanted to add some "options"--Such as LED lighting and/or even some motion controlled lighting (outside path, inside reading, etc.)--You have the basics for supplying a lot more energy to those optional loads (obviously, keep the loads low--This is still a small system).

-Bill

Ooppsss. kWH... Missed that. Oh... You did write 0.06wh. So do this again with 6 Watt motor (makes more sense--A 60 Watt computer fan is pretty large). (6 Watts = 0.006 kWatt)

Power=Voltage*Current
Current = Power/Voltage
6 Watts / 12 volts = 0.5 amps
50 AH / 0.5 amps = 100 hours of storage (100% to 0% of battery capacity)
100 Hours * 1/24 hours per day = 4.17 days of usage on battery bank

6 Watts * 24 hours per day = 144 Watt*Hours per day
144 Watt*hours per day * 1/0.77 typical solar panel+controller eff * 1/1 hour of sun per per day = 187 Watt solar panel

Note: 1 hour of sun per day is pretty low--What is your (rough) location, and is this 12 months a year, or no winter usage? At 2 hours of sun per day, then a 94 Watt panel would keep up during deep winter for many areas.

-Bill

12 volts * 0.14 amps = 1.68 Watts

Power=Voltage*Current
Current = Power/Voltage
12 volts, 0.145 amps, 1.68 Watts
50 AH / 0.145 amps = 345 hours of storage (100% to 0% of battery capacity)
345 Hours * 1/24 hours per day = 14 days of usage on battery bank
Note suggest recommended battery usage of 20%-90% State of Charge for best battery life (I.e., 70% of capacity)
14 days (100% of bank capacity) * 0.70 suggested bank usage = 10 days of bank usage (longer battery life)

1.68 Watts * 24 hours per day = 40 Watt*Hours per day
40 Watt*hours per day * 1/0.77 typical solar panel+controller eff * 1/1 hour of sun per per day = 52 Watt solar panel

Still a relatively smallish fan... But certainly the "numbers" work out OK.

-Bill