OK, so you use 5/16" tube. What is the wall thickness? For our instrument low pressure lines we used 0.035" wall. So 5/16" is 0.3125". Now minus 2 × 0.035" is 0.2425" ID. That is less than 1/4", not very big for cooler lines. And your mild steel may be a thicker wall tubing.
If you step up to 3/8" tube, the OD is 0.375" minus 2 × 0.035" is 0.305" ID. That is an increase in diameter of 25+%.
Now if you calculate inside area, the 5/16" is 0.046"square. The 3/8" tube is 0.073"square. That boils down to a 58.8% increase in flow area. That is substantial, and will aid keeping the trans oil cool.
Tubing is always measured by OD while hose is by ID.
Eventually for 1/2" and larger tubing the minimum wall thickness was 0.045". The pressure capability drops off rapidly as the diameter increases, which requires a heavier wall tubing. A couple of units required 1/2" tubing capable of 3,000PSI. This required the wall thickness to be 0.093".
