Humidity example problem #2

Tair = 90 F	r = ?
RH = 50%	Td = ?

The problem is worked out in detail below




First you fill in the air temperature and the RH data that you are given.

(A) since you know the air's temperature you can look up the saturation mixing ratio (30 g/kg). 

(B)  Then you might be able to figure out the mixing ratio in your head.  Air that could hold up to 30 g/kg of water vapor is filled to 50% of its capacity.  Half of 30 is 15, that is the mixing ratio.  Or you can substitute into the relative humidity formula and solve for the mixing ratio.  The details of that calculation are shown above at B.


Finally you imagine cooling the air (I added more intermediate temperatures in the table above than we did in class).  Notice how the saturation mixing ratio decreases, the mixing ratio stays constant, and the relative humidity increases as the air is cooled.   In this example the RH reached 100% when the air had cooled to 70 F.  That is the dew point temperature.

What does the difference Tair - Td tell you about the relative humidity?

We can use results from humidity problems #1 and #2 to learn and understand a useful rule. 




In the first example the difference between the air and dew point temperatures was large (45 F) and the RH was low (20%).

In the 2nd problem the difference between the air and dew point temperatures was smaller (20 F) and the RH was higher (50%). 

The easiest way to remember this rule might be to remember the case where there is no difference between the air and dew point temperatures. 
The RH then would be 100%.

Humidity example problem #3

Tair = ?	r = 10.5 g/kg
RH = 50%	Td = ?

We skipped this problem in class.  But I've included all the details below just in case you want to try to solve the problem on your own.

You're given the the mixing ratio = 10.5 g/kg and a relative humidity of 50%.    You need to figure out the air temperature and the dew point temperature.  Here's the play by play solution to the question:


(1) The air contains 10.5 g/kg of water vapor.  This is 50% (half) of what the air could potentially hold.  So the air's capacity, the saturation mixing ratio must be 21 g/kg (you can either do this in your head or use the RH equation following the steps shown above). 

(2) Once you know the saturation mixing ratio you can look up the air temperature in a table (80 F air has a saturation mixing ratio of 21 g/kg)

(3) Then you imagine cooling the air until the RH becomes 100%.  This occurs at 60 F.  The dew point is 60 F

Humidity example problem #4

Tair = 90 F	r = ?
RH = ?	Td = 50 F

One of the dew point's jobs is the same as the mixing ratio - it gives you an idea of the actual amount of water vapor in the air.  This problem will show that if you know the dew point, you can quickly figure out the mixing ratio and vice versa.  Knowing the dew point is equivalent to knowing the mixing ratio.


We enter the two temperatures given on a chart and look up the saturation mixing ratio for each.


We ignore the fact that we don't know the mixing ratio.  We do know that if we cool the 90 F air to 50 F the RH will become 100%.  So on the 50 F row, we can set the mixing ratio equal to the value of the saturation mixing ratio at 50 F, 7.5 g/kg.  The two have to be equal in order for the RH to be 100%.


Remember back to the three earlier examples.  When we cooled air to the the dew point, the mixing ratio didn't change.  So the mixing ratio must have been 7.5 all along.   Once we know the mixing ratio in the 90 F air it is a simple matter to calculate the relative humidity, 25%.

Drying moist air
The figure below is on p. 87 in the photocopied ClassNotes.  It explains how you can dry moist air. 





At Point 1 we start with some 90 F air with a relative humidity of 25%, fairly dry air.   These are the same numbers that we had in Example Problem #4.  We imagine cooling this air to the dew point temperature, 50 F.  While doing that the mixing ratio, r, would stay constant.  Relative humidity would increase and eventually reach 100%.  A cloud would form (Pt. 2 in the figure above). 

Then we continue to cool the air below the dew point, to 30 F.  Air that is cooled below the dew point finds itself with more water vapor than it can contain.  The excess moisture must condense (we will assume it falls out of the air as rain or snow).  Mixing ratio will decrease, the relative humidity will remain 100%.  When air reaches 30 F it contains 3 g/kg, less than half the moisture that it originally did (7.5 g/kg). 

The air is being warmed back up to 90 F along Path 4.  As it warms the mixing ratio remains constant.  Cooling moist air raises the RH.  Warming moist air, as is being down here, lowers the RH.  Once back at the starting temperature, Point 5, the air now has a RH of only 10%.

Drying moist air is basically wringing moisture from a wet sponge.


You start to squeeze the sponge and it gets smaller.  That's like cooling the air and reducing the saturation mixing ratio, the air's capacity for water vapor.  At first squeezing the sponge doesn't cause anything to happen (that's like cooling the air, the mixing ratio stays constant as long as the air doesn't lose any water vapor).  Eventually water will start to drop from the sponge (with air this is what happens when you reach the dew point and continue to cool the air below the dew point).  Then you let go of the sponge and let it expand back to its original shape and size (the air warms back to its original temperature).  The sponge (and the air) will be drier than when you started.

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Dry air indoors in the winter
The air indoors in the winter is often quite dry.


In the winter, cold air is brought inside your house or apartment and warmed.  Imagine foggy 30 F air (with a RH of 100% this is a best case scenario, the cold air outdoors usually has a relative humidity less than 100% and is drier). Bringing the air inside and warming it will cause the RH to drop from 100% to 20%..  This can cause chapped skin, can irritate nasal passages, and causes cat's fur to become charged with static electricity. 




We can see the effects of a rain shadow illustrated well in the state of Oregon.  The figure above at left shows the topography (here's the source of that map).  Winds generally blow from west to east across the state. 

Coming off the Pacific Ocean the winds first encounter a coastal range of mountains.  On the precipitation map above at right (source) you see a lot of greens and blue on the western sides of the coastal range.  These colors indicate yearly rainfall totals that range from about 50 to more than 180 inches of rain per year.  Temperate rainforests are found in some of these coastal locations.  The line separating the green and yellow on the left side of the precipitation map is the summit, the ridgeline, of the coastal mountain range.

That's the Willamette River valley, I think, in between the coastal range and the Cascades.  This valley is somewhat drier than the coast because air moving off the Pacific has lost some of its moisture moving over the coastal range. 

What moisture does remain in the air is removed as the winds move up and over the taller Cascades.  The boundary between yellow/green and the red is the ridgeline of the Cascade Mountains.  Yearly rainfall is generally less than 20 inches per year on the eastern side, the rain shadow side, of the Cascades.    That's not too much more than Tucson which averages about 12 inches of rain a year.



Death valley is found on the downwind side of the Sierra Nevada mountains (source of left image).  The Chihuahuan desert and the Sonoran desert are found downwind of the Sierra Madre mountains in Mexico (source of the right image).  Mexico might be a little harder to figure out because moist air can move into the interior of the country from the east and west.  But there are mountains along both coasts, so some of that moisture will be removed before arriving in the center of the county.

Most of the year,  the air that arrives in Arizona comes from the west, from the Pacific Ocean (this changes in the summer).  It usually isn't very moist by the time it reaches Arizona because it has traveled up and over the Sierra Nevada mountains in California and the Sierra Madre mountains further south in Mexico.  The air loses much of its moisture on the western slopes of those mountains.   Beginning in early July in southern Arizona we start to get air coming from the south or southeast.  This air can be much moister and leads to development of our summer thunderstorms.

Just as some of the world's driest regions are found on the downwind side (the rain shadow side) of mountain ranges, some of the wettest locations on earth are on the upwind sides of mountains.  There seems to be some debate whether Mt. Wai'ale'ale in Hawaii or Cherrapunji India gets the most rain per year.  Both get between 450 and 500 inches of rain per year.