Convective fix for air conditioning dilemma?

Question

I live in a part of Mexico that can get really hot in the summer. I've got an un-airconditioned bedroom on the other side of a wall from a room cooled by a powerful Mirage mini-split AC.

The problem is that the only communication between the two rooms currently is the bedroom doorway which opens from a shared hallway. This is insufficient because of the dead air volume above the doorway (the ceiling is inclined similarly to that over the AC unit). Please see pictures for reference.

My idea is to bore two holes 8-10 inches in diameter through the wall as shown in the detail shot. The purpose would be to establish a convection loop, where the hot air from top of the bedroom would flow out into the area of the AC unit (and be entrained in the cool air flow from it). To compensate for the reduction in pressure, cold air would replace the ventilated hot air by flowing in through the hallway and into the bedroom doorway. I understand that this scheme, if it works, will probably put an additional load on the AC. That's acceptable.

I haven't done any math on this idea (indeed, I'm not sure what math to do), but am just guessing at the size of the holes. Intuitively, it seems feasible.

Any thoughts on whether this would be effective before I get out my chisel and hammer?

Overview shot to demonstrate problem:

Detail shot, with proposed holes:

Update:

First, many thanks to all who responded, and particularly for the exhaustive analysis provided by Olin. The hole is being cut now, and the design has changed due to some unforeseen realities in the masonry. It will now be a rectangular opening about 12" x 20", with the top corner about 6" from the ceiling and wall. I'll report back on the outcome, and whether a fan is necessary.

Thanks again!

Answer 1

Those two holes by themselves will have little effect. The pressure difference between the two ends of the holes will be so small that only a little air will flow.

If you want to go thru all this trouble, then put a hole at the bottom too, with a fan that actively blows cold air from the bottom of the living room into the bedroom. Hot air will then find its way out of the bedroom, with one of the paths being the holes at top. However, the obvious answer is to put another air conditioner in the bedroom. That will work well, be much less trouble, and probably cost less in the long run considering what the holes in the wall will do to the value of the house.

Let's see how much pressure difference there will be across the holes at top just due to passive convection. Let's say the height is 8 feet. A column of that height 1 square inch in crossection has a volume of 96 in³ = 1.57 liters.

Air is mostly N₂ and O₂, with molecular masses of 28 and 32, respectively. There is more N₂, so let's say the molecular mass of air is 29, which means the mass of one mole of air is 29 grams. At 0°C = 273°K and 1 atm, a mole occupies 22.4 l. That means the density is (29 g)/(22.4 l) = 1.295 g/l. This means the mass of our 1 square inch column is 2.033 g at 273°K.

Let's say the cold air is 68°F = 20°C = 293°K and the hot air is at 90°F = 32°C = 305°K. The cold air mass per square inch of column is:

(273°K / 293°K) 2.033 g = 1.894 g

And the same for the hot column:

(273°K / 305°K) 2.033 g = 1.819 g

The difference is therefore 0.075 g. On earth, that has a weight of 0.000168 pounds, so the air pressure difference due to 68°F and 90°F over 8 feet is only 0.00017 PSI. The flow rate that will cause thru your 8 inch holes simply won't be enough to make a meaningful difference.

Answer 2

Agree with @Olin Lathrop that the holes should be top and bottom rather than side-by-side. However, unless your bedroom door must remain shut and has a tight seal, you could probably get away with just one vent with an integral fan, like the one shown in the picture below. Put it anywhere on the wall where it's convenient, and the air leaking around the door will complete the circuit. You can always add a dedicated return vent later if you need to.

Answer 3

I did something similar in an apartment, where there were air conditioners in the bedrooms at either end of the apartment and a living room and kitchen between them. I mounted two side by side window fans in the tops of the doorways between the one bedroom and the living room and the other bedroom and the kitchen. Similar to this. I'm the only one in the family tall enough to have to duck going through the doorway, and I did not live there, so it was not a problem.

The fans were usually run to take the higher level hot air from the central rooms into the bedrooms, forcing the cool air in the bedrooms back out through the lower level. Going the other way was also possible.

For the house in question, this would include the shared hallway in the conditioned space, which may be acceptable. It would require the doors to be open, which may not be acceptable. It would force air into the hallway from the bedroom at the top of the doorway, say, and allow return at a lower level. It's not completely efficient because of short circuiting, but it should work to some level. The advantage is that it's cheap and easy. I just used a wooden bracket below the top of the doorway as a support for the fan. I did have the advantage of rectangular doorways, however.

Answer 4

Some math, if you care:

Ideally you want 12 air changes per hour for best effect. So, say your room is a standard 20' x 20' x 10' (or if you like metric, 6m x 6m x 3m). Ideally, at 12 changes per hour, you need to move 800 cfm through the holes. For (2) 8 inch holes, that works out to about 1,100 ft/min - which is in the normal range for ducting.

However much heat wants to move the air, it will. But this size hole is adequate for the application. If you get a fan (as suggested), make sure it can move at least 400 cfm through each hole, with a delta P of 0.25 inches of water minimum.