Farad to Amp-Hour Conversion?

There is no conversion between Farad to an Amp-hour rating.

Energy stored in a capacitance is calculated as 0.5CV^2 (result in joules, equivalent to watt-seconds) and in electrochemical cells estimated as the amp-hour rating multiplied by nominal cell voltage (result in watt-hours).

In practice, provisions for balancing as well as accommodations for the differing discharge profiles are required.

A further point to consider is USABLE energy. A 12V lead-acid battery’s amp-hour capacity is usually rated based on how long it will last at some defined current output before the voltage drops to around 10.5V (below which damage can occur). The energy equation given refers to completely draining the capacitor down to zero volts. Since no system can function down to zero volts, the USABLE energy is quite a bit less than that in practice.

From a size standpoint, capacitors would take up many times the volume of a battery for the same amp-hour capacity. However, in some applications, what is most important is how much current it can output over a short duration only. Under such circumstances, the capacitors can actually have an advantage. It’s all trade-offs. However, if one uses capacitors, they will really need to understand how to do cell balancing, and protect from overvoltage issues. Otherwise, they will destroy the capacitors.