In cold weathers it is suggested to put a humidifier since the air gets too dry. I wonder how the humidity affects how much time is needed to get the air at a temperature of 20 Celsius degrees? I mean suppose you have a cold room and you want to heat the air, Will the process be slower or faster in relation to variations in humidity?
Since the water tends to keep its temperature I feel the process will be slower with higher humidity, but I am not sure.
The heat capacity of humid air is approximately given by:
$$ C_p = 1.005 + 1.82H $$
where 1.005 kJ/kg°C is the heat capacity of dry air, 1.82 kJ/kg°C the heat capacity of water vapor, and H is the absolute humidity in kg water vapor per kg dry air in the mixture. So the specific heat capacity of humid air is greater than dry air and humid air will take more energy to heat by a given amount.
But the difference is quite small. I think 100% RH at 25C is only about 2% water, and if you need to heat the room the temperature, and therefore the water content, is presumably even lower. Taking the 2% water content only increases the specific heat by about 3.6%.
Air has a specific heat capacity of slightly more than 1kJ/kgk at room temperature
So it takes a 1kW heater 1 second to heat 1kg (roughly 1 m^3) of air 1 deg C
Fairly humid air (say 60% RH at 20C) will contain around 10g/m^3 of water vapour with a specific heat capacity of 1.8 kJ/kgk - so it takes almost twice as much energy (per unit mass) to heat the water in the air than the dry air itself.
But, only 10g in every 1kg of air is water vapour ( ie 1%) so you only have to do twice as much work to that 1%.
In other words - no you won't see any measurable difference.
The information from Martin & John is great however there are other factors that make a considerable difference. Heating air be it 'dry' or humid doesn't make a lot of difference, however the materials that forms the room and contents does. The air will stratisfy as the heat will expand the position of the molecules. Humid air is less dense than dry air therefore rise slightly more easily.
The air in the room will transfer heat energy into any material that it makes contact with until the air temperature and the material are at the same temperature.
If the material is dense the amount of heat required to raise the temperature 1 Kelvin will be greater than for a less dense material. (Specific heat capacity). A porous material will contain pockets of air. Air as stated by Martin & John has a shc of about 1000J/kgK. Water has a shc of 4200J/kgK therefore requires about 4 times more heat energy to raise the temperature.
If the materials in the room and the materials that form the room (fabric) are dry they will heat up more quickly on the surface than damp materials. Therefore although humid air as gas will not make much difference from a heating point of view the materials that the gas is in contact with will have a significant influence. It is more expensive to heat a damp room than a dry one.
There are other factors such as the permiability and resitivity of the materials, the size, shape and position of furniture and walls etc. so it does become complicated. Although from a math's point of view it is interesting to produce loads of data it is not an exact science. All heat calculations are comparative only as it is impossible to achieve all the measured units.
In this problem you need to use the heat capacity in units of volume (or moles) not mass. As a gas, the mass/volume of the components of dry air (predominantly N2, CO2 and O2) is different from that of humid air.
