Can drying laundry inside cool down your apartment?
Evaporating water takes energy. Drying wet laundry takes this energy from the surrounding air in the form of heat. (Let's assume no direct sunlight or other weird sources.) So drying laundry inside should cool down your apartment to some degree. (At the cost of raising humidity)
The question is about the magnitude of this effect. Would drying a batch of laundry inside make any reasonable difference to inside temperature?
1 answer
Yes, it can make a noticeable difference. Having lots of wet surface area was one way rooms were cooled before air conditioning.
There is an example (at least used to be) of this at the Smithsonian Museum where water is pumped to overflow a large urn, with the outside of the urn shaped to maximize surface area of the water sheeting down its sides. The room is noticeably cooler than adjacent rooms.
One problem with your laundry method is that the amount of water is limited. Of course you also need the ambient humidity to be low, else there won't be any evaporation, and therefore cooling.

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