Noise is not related to:
load on engine
Occurs only when accelerating uphill or on flat surface
Noise stops immediately when the accelerator is released
No noise if the car is accelerating down hill - Live in the mountains
You might look at the engine pulleys to see if one is rubbing on a plastic splash shield. If you see signs of that, suspect a collapsed engine mount that is allowing one end of the engine to sag a little.
There is a tool you might be able to borrow or rent from an auto parts store that borrows them called the "Chassis Ear". It is a set of six microphones, a switch box, and headphones. You clip the microphones to suspect points, then drive around while listening with the headphones. You can move the microphones around to zero in on the source of the noise. Be aware that many mechanics have never seen or even heard of this tool. Suspension and alignment mechanics use it to find rattles, squeaks, and other noises.