Broan has made dozens of 110 CUFT exhaust fans it cannot tell what type if fan yours might be. Low end units have a small fan and better ones use a drum fan. Drum fan models are very quit compared to a fart fan. I looked up the installation manual on two fans and they no longer recommend any maximum length of exhaust duct. They only had static air pressure that determines volume of air flow.
About three things that might be happening. I'm assuming the staining is around the exhaust fan?
1) The 24' run is too long causing too much back pressure therefore limits the amount of air flow.
2) There is a damper at the duct connector. Be sure that it is not stuck closed. It it is made of metal you'll usually hear it click around during very windy days. Check the outside vent hood. If it has a damper cover, be sure that is now stuck closed. It could be possible a bird had built a nest at the exhaust hood even with a flapper. This happened my my parents house.
3) When you say kids, are these teenagers that never seem to get out of the shower? We all have at least one in the household. With restricted or limited air flow all that humidity is collecting at the ceiling and it doesn't have to be cold for condensation, it's called dew point. Same as the mirrors fogging over. Perhaps the fan is not running long enough. Install a timer switch for the fan. Some are up to 60 minutes, other 30 minutes. For a bathroom of this size 30 minutes should be enough to run after the shower has ended. For larger bathrooms up to 60 minutes is what we use for our master bath.
Decora Countdown Timer, 120 VAC, 60 Hz, 1800-Watt Incandescent, 600-Watt CFL, 5 Amp LED, 20 Amp Resistive/Inductive, 1HP, Single pole, 3-way or more locations (with VP0SR). Has 4 preset buttons for 10-minute,
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