Currently my clothes dryer vents into my garage. This needs to change, but I am not sure how to go about doing it. I can't vent directly to an outside wall, so need to figure out how to get to the outside from the inside of the garage.

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I am not sure what you have, I'm guessing the vent is on the back wall of the garage, if so! then maybe you could put on a 90' and run it to the left or right all the way out the outside wall, Build a sturdy shelf or ledge over the venting (stove pipe),

My dryer vent runs from the first floor, down to the basement and over to the right to the outside vent opening. The vent pipe has been leaking water at the first and second elbow.

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The reason you have water in the vent is because the vent is blocked. The water is coming from the clothes you have dried in the dryer and it has no where to go.

You need to dismantle the vent and give it a good cleaning or replace it

I bought this town house a year ago. Last week my washer broke down. To fix this i had to move the laundry center. While moving it I just noticed that the dryer's vent is inside the house. Is this normal? No wonder while i did my laundry my house

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It is a FIRE hazard, which is even more serious. Don't know about costs, but it needs to be done soon. The hot, humid air will also allow mold and mildew to develop and mold IS a health hazard. Sorry you had to discover this, but better this way than