Area = Pi x r 2
2 ¼ “ diameter pipe:-
1 1/8 x 1 1/8 x Pi = 3.98 in2 = 25.68cm2
2 ½ “ diameter pipe:-
1 ¼ x 1 ¼ x Pi = 4.91 in2 = 31.68cm2
Volume of a cylinder = [Pi x r 2 x height], and I litre of volume is 10cm x 10cm x 10cm.......
By using a bit of arithmetic, with a cross section of 25.68cm2, one litre will occupy:-
1000/25.68 = 39cm of exhaust pipe,
But by increasing the cross section to 31.68 cm2, one litre of exhaust gas will occupy:-
1000/31.68 = 31.5cm of exhaust pipe,
So you can get a lot more exhaust gas out of a slightly wider pipe!
Assuming that the engine is running at 3000 rpm, the volumetric displacement of the engine will be 1.6 litres (cubic capacity) for every 2 revolutions, = 2400 litres per minute (off boost).
With a turbo boost cutting in at 3000 revs, it can be twice that amount (at 1 bar boost), but lets call it 0.7 bar, so you can have a sudden(ish) jump from 2400 litres/min to 4080 litres/min, and that gas has to go somewhere, so the wider the pipe by even ¼” diameter will improve things by over 12% volumetric capacity.
I can’t be bothered to incorporate air speed losses, friction, temperature losses, direction changes and all the other variables, but they all have an effect.
Overall, think of a river, flowing down a slight gradient – where it is wider, the water flows slower than where it is narrower, even then it is slower at the sides than in the middle, due to drag on the riverbank, but back to the pipework, there is (proportionally) less circumference to area ratio on the wider pipe, so less air drag.
That said, unless you are aiming at 300bhp, with big boost, the 2 ¼“ will be enough

To put it into context, it is still a 12.6% improvement over a standard 2" system, not counting the 'squish' restriction, and other crap in the way due to the way the pipes are bent! - realistically, i would estimate the improvement in airflow to be in the region of 20-25% with a mandrel bent system with a PCE.