12V LED strips in series with a 24V power supply?
Well, yes and no. On the one hand, connecting 2 in series ought to work. On the other hand, you will need 24 volts at 36 watts. 

Since each string takes 12 volts at 18 watts, its current must be 1.5 amps. Connecting them in series will still take 1.5 amps, but the total power will be 1.5 x 24, or 36.