## Wednesday, December 26, 2012

### Degree of Saturation versus Relative Humidity

These two quantities are similar and as I worked through ASHRAE Handbook Fundamentals (2009) IP, chapter 1, I was initially baffled as to what the difference was.  First, the definitions based on ASHRAE Fundamentals:

Relative humidity:  The mole fraction of water vapour in a sample divided by the mole fraction of water vapour in saturated air at the same temperature and pressure.
Degree of saturation:  The humidity ratio of water vapour in a sample divided by the humidity ratio of water vapour in saturated air at the same temperature and pressure.
Humidity ratio:  The humidity ratio is the mass of water vapour divided by the mass of dry air in a sample. This value can also be expressed in terms of molar fractions.

Discussion of Differences

In relative humidity, imagine taking two samples, one of the air which you want to know the relative humidity of and one of saturated air at the same temperature and pressure. For example, we might take samples of 1 mole each. In the non-saturated air, we will have less water vapour than in the saturated air. On the other hand, we will have more dry air in the non-saturated sample than in the saturated sample. The same general relationship will be true of mass, but the masses and the mole fractions will relate differently due to differing molecular masses. Here's the bottom line: we compare the samples based on moles (which amounts to the number of molecules) not on a given mass or volume. When you're dealing with gases, moles is the way to go.  In other words, our non-saturated and saturated samples relate by the equation:
where the items on the left hand side are the mole fractions of water vapour and dry air in the non-saturated sample and the items on the right hand side are for saturated air.

Understanding that we need to compare these quantities with a fixed number of total moles as in the previous equation is what makes ASHRAE's equation (14) work out.  From the last given equation we have
Therefore,
which verifies equation (14) as given in ASHRAE Fundamentals (2009).

It is worth noting that we only had to worry about this molar equality in developing this equation. When it comes to computing the degree of saturation from the relative humidity and vice versa, we don't have to think about that. The formula will do that for us. Also, we can still compute the degree of saturation directly from values off of a psychrometric chart (for example) by reading across to the right to get $$W$$and finding $$W_S$$ by finding where the dry bulb temperature intersects the saturation curve and reading across to the right.  The definition of degree of saturation doesn't require us to account for the number of moles, only the relative humidity (and yes, if you're using charts you can read it off there as well).  If you are familiar with reading psychrometric charts, you will know that reading W values is generally much easier than reading RH values.

So if you're getting different values for these quantities in your work, it's not (necessarily) a mistake or a misreading of the chart—they really are different.