In progress. Math has not been checked!

A friend of mine and I have been talking about orchids lately. The dendrobium papilo comes from a relatively cool growing area in the Phillipines. What that means is it probably is happy at typical household temperatures during Canadian winters, slightly warmer during the day, and a cool drop at night. What it is not so happy about is the humidity. Some quick googling suggests that 70-80% RH would be appropriate for this plant in its natural habitat. Naturally, this is impractical to maintain in a home and good practice would be to support the plants needs by keeping it regularly watered and preventing the growing media from growing out. But what if we could bump up the humidity even a little bit? Surely 50% RH is better than 30%? My friend wondered if putting a pebble tray beneath the plant might do any good.

Here’s where the sketchy science comes in. At 21C, 30% RH gives us an absolute humidity of 5.5 g/m3 (https://www.ready.noaa.gov/READYmoistcal.php). To get it up to 50% RH, we would need 9.1 g/m3, with the inrease required being 3.6 g/m3, that doesn’t seem too much. Let’s pretend we’re in the bathroom in the average US home of 40 square feet, with a height of 10 ft. That gives us 11.3 m3 and would require about 40.7 g of water added into the air. This is readily done by taking a shower. However, hopefully shower users have noticed their bathrooms drying out over time. The turnover rate of air in a house starts around once every 3 hours. For convenience, let’s consider that 1/3 of the air is replaced every hour. At the start of the hour, we’ve added the 40.7 g. We lose a third of this if the air (TWC: 102.8 g) is well-mixed (-34.3 g) and it is replaced with the 30% RH air (+20.7). To maintain the 50% RH, we need to make up for the difference and add 13.6 g. Since we’re no longer runing the shower, how long does it take to add 13.6 g into the air passively via evaporation?

Let’s use an airflow rate of 20 m/s (72 km/h) if the pebble tray is sitting right in front of the vent. For a 3 inch saucer, that’s about182 cm2 of surface area. Let’s say 300 cm2 (0.03 m2) if you add lots of porous pebbles. Under the most efficient evaporation rate (when the RH is 30%), the evaporation rate is 131.7 g/hour, or 131.7 mL/hour. That’s pretty darned good! All you need to do now is add 3 kg of water a day. The tray is probably at most 3 cm deep by the way, so it can hold just under 550 cm3 of water, or 550 mL. So you only need to top off every 3-4 hours really.

The problem is, none of what I’ve posed above is realistic. First, the air being exchanged in is probably close to 0. Currently where I am, it’s 80% RH outside. From -10 C outside to 21 C indoors, we would have just under 10% RH air coming in. So instead of adding 20.7 g, into the air with the exchange, we’re adding about 6.4 g of water. Which means we need to make up for an additional 14.3 g, double what we had earlier for a total of 27.9g. Our plants usually don’t sit right by the vent, and I had used the *main* duct velocity. According to this reddit post (https://www.reddit.com/r/AskEngineers/comments/t5yfj0/how_to_calculate_air_velocity_after_it_exits/), the plants need at least 0.5 m/s for good airflow (which I pointed out might be a concern with resting your plant right on top of a tray if it covers up the entire surface). Let’s say the airflow is 5 m/s right out of the vent, which is about the upper realistic value for a low pressure duct. It is likely much lower elsewhere, but let’s call it 5 m/s anyhow. Now we’re at an evaporation rate of 390.0 g/hour (https://www.omnicalculator.com/physics/evaporation-rate#google_vignette). At 0.5 m/s, this is reduced to 112.1 g/hour. Still, about 3 times the replacement rate that we needed. Right now we’re assuming that the cooler, higher humidity air from above the tray is mixing into the overall bathroom and the airflow is directly passing over the surface.

At this point, the bathroom is looking like a pretty good spot. There are some unrealistic assumptions I’ve made, like the hourly time step, but overall it might be possible to maintain 50% RH in the bathroom!

Now let’s think about the living space. Let us increase the size of the space and call it 50 m3. The water content at 50%RH in this room is 455 g, and the water loss from the 3x turn over is -151.7 g + 29.3 g = 122.4 g. The same saucer is placed somewhere with good airflow, and evaporates at 112.1 g/hour. This is now less than our target 50%RH. If you run this forward by 24 hours with continual water top-ups, you end up with 455 g – (122.4 g – 112.1 g) x24 = 455 g – 247.2 g = 207.8 g of water. That’s 4.15 g/m3 and about 23%RH (note that evaporation will of course increase once you hit below by previous threshold of 30% RH).

Here’s the real practical test. Spill about 100 g of water in a relatively small room that still has airflow. Measures the relative humidity every few minutes in the room. How long does it actually take to evaporate? If you start at 30% RH, what do you end up with after a few minutes? (For reference, I add about 1.5 kg of water into my ~ 20-30 m3 room each day just topping up plants and fish tanks, my room sits between 45 – 50%RH. The water loss is mostly via transpiration through plants, and dehydration of the extremely high surface area of spaghnum moss topping most my plants, not the fish tank! The surface area of the tank is a bit under a square meter, but it’s a shallow tank where the humidity is partially trapped by a plexiglass pane so it does not mix readily into the rest of the room.)