Hi Stephen,
If I'm understanding your question correctly, you want to know if you can use a hairdryer to completely dry the prints between new layers (?) The first exposure is always on dry paper, of course
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I always use a hairdryer to dry the gum/pigment before exposure. In my climate, it's the only way I'd ever get a coating to dry. Works great.
But, after each exposure's final developer/wash tray, I hang my prints to dry. I have a small, free-standing electric oil radiator heater going in my darkroom most of the time. If I close the darkroom door with the heater going on low, and the ventilation fan on, the wet prints dry pretty fast. One thing to remember and avoid is a significantly different drying regime from one time to the next. Different drying speeds and temperatures will affect the shrinkage of the paper --more or less/depending, but there's a possibility the the differences will be enough to screw up your registrations.
One additional thought: a drying cabinet can be as simple as a cheap food dehydrator unit with screen trays. They seem to be everywhere in stores this time of year. I have one for drying other kinds of prints, but I've standardized on the 'hang in a warm room' method for my gum prints.
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