Hang out your laundry to line dry in your house rather than putting it in the dryer.
This will release a lot of moisture into the air.
Depending on how often you wash your clothes, this may be enough.
If it isn't a few damp teatowels hung on radiators (assuming they're not electric) will achieve the same effect.
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'Traditional' humidifiers look like this:
It's just a ceramic container with a hook that hangs over the radiator.
Fill it with water, and the water evaporates as the radiator heats it.
You can emulate this with a dish or bowl hanging on your radiator.
If you have the space the easiest option is to place a container on top of the radiator.
It was not uncommon to add scents to the water, like lavender; this supposedly helped combat the 'stuffiness' of the room.
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You can put many houseplants in the room, they will increase the humidity naturally.
Take some pots with plants which have big leaves.
Some examples: Fern Spathiphyllum (Peace Lily) Additional positive effect will be that they will absorb some unhealthy substances from the air indoor.
Of course they will need to be watered regularly.
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If the room is small, you can create a solution using a bucket of water and a few kg of table salt (sodium chloride or potassium chloride).
Dump a large amount of salt into the bucket, and add water until the salt will absorb no more water.
This is what's called a saturated salt solution.
The solution will attempt to maintain the humidity level of the room at around 75%.
Additionally, you can use a bucket of water, a towel and a fan to do this more easily.
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Large bowl of water in the room with you.
I do this when the kids have stuffy noses/colds.
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Use a rice cooker full of water.
Unlike most electric kettles it will continue to run even with the water boiling, but will turn off (or to a lower "keep warm" state) when the water runs out and the temperature goes above 100°C.
You will need to be careful though; it can work too well and turn your walls into slush.
Use a really small rice cooker unless it's a really large room.
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Taking a bath, or just fill your bathtub with some water will keep your room humidified.
Leave your bathroom door open after taking shower/bath to let the steam out and humidify your room.
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My grandparents would set out trays of water in front of the (floor-level) ducts.
I've noticed that a large aquarium functions as a humidifier.
When I needed relief and didn't have a humidifer yet, I tried spraying water around the bricks of the fireplace.
I've held a hot wet washcloth up to my face to breathe through.
That helps as a warm compress for sinus pain and also provides humid breathing air.
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Leave an electric kettle heating with water in it.
If you do this, however, you'll have to be vigilant for condensation on cold walls (behind dressers and bedsteads, for instance) or runoff from sweating windows causing mold growth.
The problems with your nose from dryness are nothing to what black mold will cause.
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I'm going to elaborate on Umber Ferrule's answer .
Since my estate is rather cheap, I can afford to dedicate a couple of square feet to pretty large polyethylene basin like this : Several things to consider: evaporation happens at the surface of water, so area is significant parameter here (this makes square bowl/basin much better than round one)
it works best at open space with air circulation present allowing humid and dry air to mix freely
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Brew beer.
A 5 gallon batch of beer simmers and boils for 2-3 hours (depending on the recipe and strength of your stove).
In a 2 bedroom apartment brewing 1 or 2 batches per weekend has certainly kept the air more humid.
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It's winter, it's cold, and the heat is constantly blasting throughout my house. This is making the air very dry and uncomfortable to breathe, particularly at night when I'm trying to sleep.  Are there any decent ways to humidify my bedroom without actually buying a humidifier? The only one I'm really aware of is running the shower with hot water, but I don't want to leave that much water running overnight. Plus, I don't sleep in the bathroom.
How can I humidify a room without a humidifier?
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There are a few ways to humidify your house without buying a humidifier. One trick is to hang out your laundry to dry in the house instead of using a dryer or simply handing some wet tea towels on your radiators. Alternatively, leaving a bucket of water or salt water in the room will also help humidify it and you could leave your bathroom door open after taking a bath or shower. Furthermore, you could use a rice cooker full of water, and house plants are also a great option for keeping your house humid. The options are to hang out laundry indoors, leave damp teatowels hanging on radiators, hang water containers next to radiators, get some house plants and water them regularly, place a bucket of heavily-salted water or just a bowl of ordinary water in the room (possibly with a fan blowing over it) or an open space where air cirulates, have a slow boiling rice cooker, take a bath/shower or just fill the bath and leave the door open, set out trays of water next to any floor level ducts, get an aquarium, spray water around fireplace bricks or try breathing through a warm wet washcloth, leave a kettle boiling or take up beer brewing.
Hang out your laundry indoors and maybe leave damp teatowels on a few radiators. Hang a container full of water next to the radiators - add lavender to further combat stuffiness. Get lots of house plants and water regularly. Keep a bucket full of water with a few kg of salt dissolved into it in the room, or just a bucket of water with a fan blowing over the top. Place a large bowl of water in the room. Place a rice cooker full of water in the room and switch on. Take a bath or shower or just fill your bathtub with hot water and leave the bathroom door open. Set out trays of water next to any floor level ducts, get an aquarium, spray water around fireplace bricks or try breathing through a warm wet washcloth. Leave an electric kettle boiling - but look out for possible mold growth Place a large plastic basin in an open space where there is air circulation. Take up beer brewing.