If you used that 24V 10A transformer with a full wave rectifier (not the bridge) you would get 14.8V - 15V at 10A, depending on the actual diode losses. You won't make it to 20V, you would need a 30V secondary to do that.
I'll show you how to select some power MOSFETs to replace the Darlingtons when you are ready for it.
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