You're wrong.
The Qmax of 226W is not the amount of power it will use, but the maximum amount of heat it will removed from the cold side.
The 226W is removed using a current of Imax=24 amps which takes Vmax=15.2 volts to produce.
So, at 12V you're appyling 79% of Vmax and need a a supply that can deliver at least 79% of Imax, or about 19 amps plus a margin of error.