Thermal Paste heatsink compound should be removed and new put on when working with the Peltier module.

Rubbing Alcohol and Coffee Filters Rubbing Alcohol and Coffee Filters

Thermal Compound stickey

The stuff is sticky. You can use acetone, isopropyl alcohol (rubbing alcohol) or ethanol alcohol (get denatured cause if not denatured you might take a swig when you get frustrated). I used rubbing alcohol and coffee filters. Coffee filters because they do not leave fibers.
I used a 50 ml tube of Fujik BAB900 Thermal Glue, Silicone. I bought it from FCB Electronics Ann Arbor, Michigan on eBay. Not impressed with it. Top was watery and middle is dry.