UPS battery calculator

Determine the required energy needed to run the UPS for the given load, time, and inverter efficiency.
$\mathrm{needed\; energy}=\frac{\mathrm{output\; power}\times \mathrm{run\; time}}{\mathrm{inverter\; efficiency}}=\frac{50W\times 2h}{0.80}=117.65\mathrm{Wh}$

Convert energy requirement into battery capacity, accounting for depth of discharge (DoD).
$\mathrm{needed\; battery\; capacity}=\frac{\mathrm{needed\; energy}}{\mathrm{battery\; voltage}\times \mathrm{DoD}}=\frac{117.65\mathrm{Wh}}{12V\times 0.2}=49.02\mathrm{Ah}$

Use Peukert's Law to calculate battery C20 rating.
$\mathrm{rated\; battery\; capacity}=\frac{\mathrm{effective\; capacity}}{{\left(\frac{\mathrm{rated\; time}}{\mathrm{actual\; time}}\right)}^{(1-\mathrm{Peukert\; coefficient})}}=\frac{49.02\mathrm{Ah}}{{\left(\frac{20h}{2h}\right)}^{(1-1.25)}}=87.17\mathrm{Ah}$

Check against maximum discharge rate.
$\mathrm{discharge\; rate}=\frac{\mathrm{output\; power}}{\mathrm{system\; voltage}\times \mathrm{inverter\; efficiency}\times \mathrm{rated\; battery\; capacity}}=\frac{50W}{12V\times 0.80\times 87.17\mathrm{Ah}}=0.06C$

You need a battery of at least $87\mathrm{Ah}$.

Determine ideal charging current.
${\mathrm{charging\; current}}_{\mathrm{ideal}}=\mathrm{battery\; capacity}\times \mathrm{ideal\; charging\; rate}=100\mathrm{Ah}\times 0.1C=10.00A$

Check charging time against requirements.
${\mathrm{charging\; time}}_{\mathrm{ideal}}=\frac{\mathrm{DoD}}{\mathrm{ideal\; charging\; rate}\times \mathrm{CEF}}=\frac{0.2}{0.1C\times 0.85}=2.35h$

Minimum and maximum charging current and time are calculated in the same way.
${\mathrm{charging\; current}}_{\min }=100\mathrm{Ah}\times 0.07C=7.00A$

${\mathrm{charging\; time}}_{\min }=\frac{0.2}{0.07C\times 0.85}=3.36h$

${\mathrm{charging\; current}}_{\max }=100\mathrm{Ah}\times 0.15C=15.00A$

${\mathrm{charging\; time}}_{\max }=\frac{0.2}{0.15C\times 0.85}=1.57h$

You need a charger that can supply between $7A$ and $15A$, idealy $10A$.