Open-cell under the roof deck: the moisture math for an Erie winter

The question

A rafter bay sprayed full with open-cell foam and one sprayed with 3 inches of closed-cell have about the same R-value, around R-21. Both stop air. In a Western PA winter, how much water vapor from the house reaches the roof sheathing through each one?

Inputs

Input	Value	Where it comes from
Indoor air	70°F, 35% RH	A typical, comfortable winter setting
Indoor dew point	41.1°F	Psychrometric formulas (ASHRAE Fundamentals)
Outdoor monthly means	Dec 34°F, Jan 28°F, Feb 29°F, Mar 36°F	Erie, 1991–2020 NOAA normals
Sheathing temperature	Outdoor mean plus about 2°F	Steady-state heat flow through R-21 of foam and the roofing above
Open-cell permeance	11 to 17 perms	Open-cell data sheets at 3½ to 5½ in. (one lists 16.9 perms at 3.5 in.; both makers under Sources)
Closed-cell permeance, 3 in.	0.5 to 0.8 perm	About 1 perm at 1¼ to 1⅞ in. on data sheets, falling roughly in proportion with thickness

November is left out: its 44°F mean keeps the sheathing above the indoor dew point.

Method

Water vapor diffuses from high vapor pressure to low. The vapor pressure of the indoor air is fixed by its temperature and humidity. At the cold sheathing, the most the air can hold is the saturation pressure at the sheathing's temperature. Vapor flow equals the foam's permeance times the pressure difference. This is a standard first-order estimate (the "Glaser" method) on monthly averages.

Results

Month	Sheathing	Pressure difference (in. Hg)	Through open-cell (gr/h·ft²)	Through 3 in. closed-cell (gr/h·ft²)
December	35.9°F	0.048	0.53 to 0.81	0.02 to 0.04
January	30.2°F	0.093	1.02 to 1.57	0.05 to 0.07
February	31.2°F	0.085	0.94 to 1.45	0.04 to 0.07
March	37.8°F	0.032	0.35 to 0.54	0.02 to 0.03

Added up over December to March, that is about 0.29 to 0.45 pound of water per square foot of roof through open-cell, or roughly 9 to 14 pounds for every 4 × 8 sheet of sheathing. Through 3 inches of closed-cell it is about 0.01 to 0.02 pound per square foot, under a pound per sheet. The ratio is roughly 14 to 34 times.

In both assemblies the 41°F plane sits inside the foam in January: about 1.6 inches below the deck in open-cell, about 0.8 inch in closed-cell. The foam faces stay near 69°F, so nothing condenses on the room side of either one.

What the numbers do and don't say

They overstate the total. Sun on the roof warms the sheathing during the day and pushes some moisture back into the foam, and the wood and foam store and release moisture in ways a steady-state estimate ignores. Real roofs are likely to pick up less than the pounds above, and some open-cell roofs in zone 5 do fine.
The ratio is the useful part. Whatever the real total, open-cell delivers one to two orders of magnitude more vapor to the deck than closed-cell. That is the gap a Class II vapor retarder on the open-cell closes, and why IRC R806.5 item 4 requires one in zones 5 to 8.
Humidity moves it a lot. At 30% RH indoors the January pressure difference drops by about 40%; at 40% RH it rises by about 40%. A house that runs humid in winter is the worst case for an uncoated open-cell roof.
Where to look. Home Innovation Research Labs treats OSB as dry below 16% moisture content and 20% as the decay-safety threshold. The ridge and the north slope are the places to measure, in February or March.

The practical reading is the one in the open-cell roof deck article: under a roof deck in Western PA, use closed-cell against the deck, or put a rated vapor retarder over open-cell, and check an existing open-cell roof with a moisture meter in late winter.

Assumptions: one-dimensional steady state on monthly means; air-tight foam with no gaps; R-21 of foam (about 5½ in. of open-cell at R-3.8 per inch or 3 in. of closed-cell at R-7 per inch); roofing, sheathing and outside air film about R-1.2; inside air film R-0.68. A worked estimate, not a lab test or a field measurement.