I always mount my baryta prints using wet adhesive tape. This ensures the photographic paper lies flat, just as it came out of the box. On the subject of affordable photographic paper: Foma baryta paper is a very good choice. So far, I’ve only compared it with older Ilford paper, and in that case I actually preferred the Foma slightly more.
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On rinsing: the paper must not be left ‘motionless’ in standing water. The water should be changed gradually and the paper should always be moving slightly. I would rinse for about 45 minutes. If you want to build your own tray for this, make sure that the fresh water flows in at the top and the drain is at the very bottom. The chemicals collect at the bottom. The soaking time can be shortened by using a “soaking accelerator bath” after a short soaking period of about 10 minutes. In the simplest case, this consists of a soda solution (a small heap in the palm of your hand to 1 litre) and the print is soaked in this for about 3 minutes. Afterwards, it goes back into the washing, which will now work more quickly. Shorter rinsing times can also be achieved by preventing the fixer from penetrating too deeply into the paper fold beforehand, using two more concentrated fixers with a very short fixing time, as described here:
http://www.fotografie-in-schwarz-weiss.de/sw-fotografie/kolumne/100-wollstein02.html
We have plenty of free water at uni. My baryta prints are happy to float around in the tub for up to five hours.
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You could also pre-rinse the prints at uni and then rinse and dry them at home. But it’s certainly advisable to use the free water at uni for this. You could also let your baryta prints dry partially there first, then soak them again at home and dry them properly on a glass pane using wet adhesive tape, for example. Here’s a very good link on this, in case you don’t already know it:
http://www.fotografische.de/fotolabor/baryt-trocknen
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Incidentally, the PE drying machine in your lab won’t be suitable for baryta paper, as it’s much thicker. It contains far more moisture than PE paper. Furthermore, less moisture will penetrate the PE paper itself, as it is, so to speak, ‘sealed’.
There is (or rather: was) also paper-weight, very thin baryta paper, which was much easier to dry. But I assume you mean ‘proper’, card-weight baryta paper.
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Best regards,
Thomas