On mine, the clutch fan frequently makes a lot of noise on a cold start. This is normal. But on mine, it is not a whine. It is more like a lot of lower frequency fan type noise.
I suspect that one of your idler pulleys is going bad. It will whine like that for a few thousand miles, maybe even 10K miles, and then it may lock up or come right off. When it is doing this it is causing excessive wear on the belt. However, maybe the belt will fail first.
Also, if it is a squeal/squeak more than a whine, the belt itself could be bad as the back of the belt slips over a cold pulley. The belt back could be glazed. But then you got to ask yourself how long is that pulley's bearing actually going to last, too. I'd be looking at the belt and pulleys, unless you clearly just have a lot of fan noise.
For testing, you could add a small bit of belt dressing to the back of the belt and see if the noise disappears. Then try a dab of dressing on the ID of the belt. If used sparingly and on the belt one side at a time, often you can tell which pulley bearing is suspect. If the noise noise is still there after testing both sides of the belt, you may have two bad pulley bearings.
Using the belt dressing momentarily takes the belt surface out of the loop for testing. However, it is not a good long term fix.
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