| GMW Water: 18g/mol |
.018g/mmol |
| LHV Water: 540 cal/g |
|
| 1 BTU: 262 cal |
|
| 1 kWhr: 3412 BTU |
893944 cal |
|
Transpiration Rate (lowest for Maple): 2mmol/m2-sec
= 0.036g/m2-sec x 540 cal/g
|
19.44 cal/m2-sec
|
|
Transpiration cooling per m2 per hour
= 19.44 x 3600 cal
|
69984 cal |
| Transpiration cooling per m2 per day |
559872 cal |
| Energy per m2 per day |
0.626 kWhr/m2-day |
| Cooling |
267 BTU/m2-hr |
| Alternative formulae |
|
| 1 cal: 4.184 joule |
|
| 1 Watt: 1 joule/s |
|
| Therefore: |
|
19.44 cal/m2-sec
= 19.44 x 4.184 joule/m2-sec |
81 W/m2 |
|
Broad assumptions:
- Canopy cover is equal to total leaf area.
- Transpiration Rate varies from 0 at sunrise, peaking shortly after local apparent noon, and falling to 0 at sunset. We estimate that 8 hours of average transpiration approximates 12 hours of varying transpiration.
- Sufficient water to support this level of transpiration is available.
- Water requirement/m2 of canopy / day would be 1.296 litres per m2 per day.
- Some quoted rates (day) for Norway Maple are in excess of 200 litres per tree per day for mature (but not senescent) trees.
- Kirnak, Short and Hansen (Studies on the relationship among moisture tension, microclimate and transpiration rate of container grown Acer rubrum – Jnl. Appl. Hort. 4(2): 65-69) quote 80g/tree-hour for 1.5 meter whip stock in suitable media. (30cm diameter container). Our Transpiration Rate of 0.036g/m2s translates to ~130g/m2-hour. This would seem to be consistent.
|
| Solar Array of Kyocer KC-187G panels (rated output 187W max) |
| 1400 panels at ~ 1m2 per panel |
1400 m2 |
| Energy Output |
235 kW (specified) |
| Energy Output / 8 hour day |
1880 kW-hr/day |
| Energy per m2 per day |
0.709 kW-hr/m2-day |
| Alternative formulae: |
|
| Energy output/m2 = 235000/1400 Watts/m2 |
~ 168 W/m2 |
|
Broad Assumptions:
-
Energy output (187W panel / 235kW array) is as quoted by Kyocera (the solar panel manufacturer), under Test Conditions. We have assumed that output would follow a curve similar to solar flux density (0 at sunrise, peak at local apparent noon, falling to 0 at sunset). Again, to avoid messy calculus we have assumed quoted output for a maximum of 8 hours per day.
-
Array size of 1400 panels is quoted from Kyocera Solar Grove Parking Lot.
|
A discussion arose several summers back, on the Canadian Urban Forest discussion group, regarding an array of solar panels, which were being installed over a parking lot in California. Quite impressively, these were expected to produce about 89 Watts per square meter of covered surface. The energy thus produced was intended to reduce the facility’s reliance on the local power grid.
In our industry, it is long standing wisdom that “an average sized tree has a cooling capacity equal to a window air conditioner.” This has been accepted as fact; indeed, we have all heard it many times quoted in advertising literature. On the strength of this folk wisdom, it was suggested in the discussion group, “… would not the preservation or planting of trees not have reduced the facilities power needs by as much or even more?” At this point, we begin to understand why lawyers never ask a question to which they do not already know the answer. The reply was simply, “Has any research been done to back this up?” In other words, we say that a tree can cool like an air conditioner, and it seems plausible, but has anyone actually “done the math”.
How much cooling effect does a tree produce. Simple question, no immediately available answersanywhere that I looked. Therefore, we must go back to first principles and derive our answer! I should point out that this is not a scholarly article, so we are not going to arrive at anything but a very loose approximation, but it should be enough for a comparison to the stated 89 Watts/m². It all starts with evapotranspiration.
Kirnak, Short and Hansen measured a transpiration rate for whip-stock Red Maple at about 2mmol per square meter per second. That’s a lot of help, if we know what a mmol is!
A millimole (mmol) is one-thousandth of a mole; and a mole is among other things equal to one gram-molecular weight of a pure substance, in this case water. The molecular weight of water is 18 (2 Hydrogen at 1 each and an Oxygen at 16). That means that a mole of water weighs 18grams; so, a mmol weighs in at a paltry 0.018g (or 18 mg). Not a lot, but it’s happening every second, of every hour, all the while the sun is shining!
So, for easy figuring, let’s assume that each of those Red Maple whips had a leaf surface area of about 1 square meter – the area might have been less, but one would seriously doubt that it would be more.
Now let’s think about heat. How much heat does one gram of water give up (or take away with it) when it turns from liquid to vapor. This value is known as the Latent Heat of Vaporization, which for water is 540 calories (heat not food) per gram. Combining this with our transpiration rate, we calculate that our tree is pumping 19.44 calories per square meter per second – every second, of every hour, all the while the sun is shining!
Unfortunately, air conditioners are rated in either BTU/hour and electricity is measured in Watts. Both can be derived from calories using the following conversion factors: 1BTU = 262 calories, 1 calorie is 4.184 joules, and 1 Watt is 1 joule per second.
Our tree is losing heat at a rate of 19.44 calories per meter per second, which may not seem like much, but that’s 69984 calories per square meter per hour! That’s 267 BTUs per square meter per hour. Once again not much, but most trees have a fair bit more than 1 square meter of canopy, let alone leaf surface! Looking at this in Watts, we can easily calculate that 19.44 cal/m²-s is about 81 Watts per square meter.
As to the window air conditioner model; it looks like we would need only about 40 square meters of leaf area to approximate a 10,000 BTU machine. That is not a lot of canopy – less than 4m diameter even if leaf area was the same as canopy surface area.
To reiterate, we have made quite a few broad assumptions – mostly about canopy cover and transpiration rate. We have also broadly assumed that the tree in question is located “just right” relative to our house and the path of the sun – and prevailing wind. The tree might very well be cooling its little heart out, but it might not be cooling our house. Then again, cooling is cooling, and even if it’s not cooling us, it’s cooling someone.