Without knowing anything about the engine, what would you think the noise it?
It does sound like a diesel, and after rev-ing engine there is a noise that sounds like a chain. However, after acceleration pedal has been let go off, and not while it is pressed down.

 

That sounds like lifter noise to me. The grinding noise may be caused by the VVT solenoid, camshaft bearing or timing belt tensioner and the timing belt itself.
The lifters seem noisy on all 4 cylinders. So I would look for something that affects the camshafts.

 

Two easy things to try, remove the two drive belts and eliminate the aux drive items, then unplug the VVT solenoid and see if there is any change. Also how much oil is in it? Condition?

You may want to check the oil pressure & engine compression too.

 

Oil has less than 500 miles on it.

In original post I asked about the noise without any prior knowledge about the engine. Well, here is some additional info.

• It all started with engine knocking because of bad bearing. The bearing might have failed because there was not enough oil. This happened after a long trip, after which we found out there was very little oil. It didn't leak anywhere, just burned...
• Rod bearings and crankshaft were replaced, engine filled with oil. It worked, but timing didn't feel right.
• After re-adjusting the timing, something happened that caused two exhaust valves to stuck open during first start up, inevitably smashing piston, spark plug, and eventually destroying the head.
• Engine went on a stand again to have head along with other parts replaced and rebuilt.

Now about the head. We found a 2.0 engine with low miles for low price. From 2.0 only head block with valves was used. The model numbers matched, so we figured there would be no issues. Timing shafts, chains, etc from 2.0 engine are different, therefore we didn't use it.

Lastly, things we are considering now that engine is in the car, with a noise... When two exhaust valves were stuck open and smashed into piston, multiple things could have been altered.

• The chain could have been stretched due to strain. Less likely.
• Chain tensioner could have got damaged. Very possible.
• Timing shafts might have shifted a bit on a new head, therefore there is slack in a chain. Highly unlikely because heads blocks themselves are exactly same.
• VVT system was damaged and does function properly. Possible. Have to try unplugging the oil control valve, or stuffing something into VVT to make it not move.
• Spacing between valve lifters and timing shaft is different. Less likely to be the cause since it was checked for spacing and clearance according to factory specs. It checkouts well.

As for now, the plan is to replace chain tensioner from salvaged engine which is exactly same, but hopefully not damaged. Along with that, a new chain will be installed, in case chain did stretch.

Any thoughts?