Squealing fan belt

Hi ladies and gents we have two MX5’s my being the older the wife’s the later one ,mine is a 2004 1600 million 2 74000 mile full service history developed loud squeal AA make out changed belts and put a New alternator on everything ok then 3 hours later the same… AA back out quite stumped it appears all ok water pump no play mentioned another pulley I think crankshaft but they thiink doubtful any ideas AA said ther could have been dirt on the pulleys when new belts fitted? 

16,274,000 miles? I’m amazed. - But that’s what you apparently said! Did you mean a 1.6 litre engine?
If a belt isn’t tensioned correctly, then you will get belt ‘burn’ on the drive pulleys, this doesn’t help when it comes to getting a good drive from the engine. Yes pulleys should be cleaned IMHO, but others may say “Don’t bother”. Did the AA fit a new or reconditioned alternator, recondition’s are always open to suspicion, till proved otherwise. Belt burn on pulleys are a good lead on which belt was (is) still slipping, but old burns still left are not. Check the new belt on it’s ‘bite’ surfaces, Slip marks are usually rough to the touch. If an AA man is unsure, go to a decent garage, if you know one nearby. Cam belts don’t slip - if they did, the engine would either not start - or run roughly if it did. You may have other problems causing a squeal, bearings for one.

if it is a belt squealing it is simple to identify.

poor water on it (if accessible) and it will stop if its the belt slipping.

if it is the belt, then tension is likely the cause.

did they fit the correct belt?

I’m having the same problem with my 1999 1.8iS, which has just passed 90,000 miles. The alternator was replaced with a recon unit about 5 years ago but charges OK.
The problem is usually a “twittering” sound on startup and for the first few miles; the noise disappears if the alternator pulley is sprayed with water, and the belt seems to be centred and in good condition. Tension was checked about 2,000 miles ago.
Occasionally, usually on a damp misty morning, there’ll be a loud squeal (from the belt, not from me) which goes quiet quickly.
According to a YouTube video, the American guy giving the advice cured a similar problem by fitting a Goodyear Gatorback belt which has a special pattern on the “V” side of the belt, but these belts to fit MX-5s don’t seem to be available in the UK.
I’m thinking of replacing the existing unnamed drive belts with branded ones, and I would appreciate any recommendations or suggestions.

If the tension is OK, then the next thing to check is if the belt is bottoming in the Vee of the pulley. 

A new belt should ride the walls of the Vee on all pulleys and never bottom.  If a new belt bottoms out then either it is the wrong belt or the pulley is badly worn.

Also the Vee profile should ensure good contact of both sides of the belt on both walls of the Vee.