Our first given condition is that the digit at the unit place is double the digit in the tens place.

∴ Let the tens digit be y.
The digit in the units place is 2y.
Number = 10y + 2y = 12y

Now the second condition is that the number exceeds the sum of its digits by 18.

∴ By given condition,

(y + 2y) + 18 = (10y + 2y)
∴ 3y + 18 = 12y
12y - 3y = 18
9y =18

∴ y = 2

Hence, the digit in the tens place is 2.

So, digit in the units place is 4.

∴ Our number is 24.