First, I assume you are talking about a desktop or desk side computer, and not a laptop, correct?
Second, which fan is loud? There are probably at least two or three depending on the size of your computer. There is one on the cpu, one in the back of the case, and one on the power supply.
The cpu fan is typically variable in speed, driven by the load on the cpu and is usually very quiet. The other two fans are probably fixed speed unless the computer is newer or higher end old. It's possible the new hard drive is making the cpu fan work harder,, but I doubt it.
And yes, fans do go bad and get very noisy as the bearings wear out.
Changing the hard drive should have no bearing on fan speed unless it draws a lot more power than it's predecessor for some reason (higher rpm, higher capacity, etc). It's possible the power supply fan is working harder because of increased current draw, but that again is unlikely unless you really amped up the drive.
It's more likely that either a fan is going bad or there is more heat and the fans are just working harder.
So my money is on the fans themselves. Open the case and listen to see if you can find one making noise. If it gets quiet with the cover off, just leave it off.
Does it make less noise if you turn it off overnight and star it up cold?
You can buy an external enclosure for a hard drive and hook it up with a usb cable. I'm thinking $25-$50 but I haven't looked recently.
But that won't cur down on noise unless it is a very hot drive, and then you will have the fan in the enclosure to deal with.
Short ending. I doubt it is an air flow layout issue. The fans make the noise, not the air flow. They are either working harder or they are failing.