This is one of the most common problems I see in kitchens and bathrooms, especially those that have been custom finished by spraying, rather than factory finished or brush finished. The edges of cabinet doors and drawers under sinks don’t get enough finish to protect them well against splashed water. The water breaks through the thin finish and causes it to peel. Repairs to the finish seldom work well, so the doors and drawers need to be stripped and refinished.
So it’s important to get enough finish on the edges to begin with.
The most common method of spraying doors and drawer fronts is to lay them flat and spray from above. Typically, the edges are sprayed at a 45° angle. This doesn’t get enough finish on the edges.
The first pass on the edge should be made all the way around at a 90° angle to the flat, horizontal surface. Then another pass at 45° should be made to hit the routed or shaped edges well. Finally, apply full passes perpendicular to the top surface. Do this with each coat.
To avoid refinishing the doors and drawer fronts, you can sometimes be successful applying a stain to the damaged areas (if the they have been stained originally), then brush on a finish. The finish can be anything, but alkyd or polyurethane varnish will hold up best and be easiest to brush because they dry slowly. To avoid the odor, you could use a water-based finish instead.
Be sure the surfaces are clean. The best way to clean them is to wipe over them with soap and water. Then wipe again with just water on a cloth to remove the soap. Any soap will do. Dishwater detergent will probably be the most available near a sink.
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