# How to Derive Ceiling Temp from Roof Temp and R-value

My Zone 2A ceiling has exposed rafters, so my only insulation is exterior polyiso. I’m trying to decide how thick to make it — and my only concern is performance (mainly thermal comfort, as I’m already close to net zero), *not* construction difficulty (I’m the builder).

GBA Super Advisor Dana posted this in reply to my general post: “Even at a relatively modest peak temp of 120F on the metal roof the average temp through the [6″] foam layer would be around 100F.”

Can anyone explain how to perform this calculation? E.g., what would the ceiling temp be if the foam layer were 7″ or 9″ etc… I’d truly appreciate this as it’s the last design decision I have after a year of planning ðŸ™‚ I think this’d be very helpful info for anyone building in hot climates, in which rooftop SHG is a huge part of cooling load. Thanks!

### GBA Detail Library

A collection of one thousand construction details organized by climate and house part

## Replies

You need at least code R value. With that much insulation and with AC adjusted to what feels good, you won't notice any difference in comfort by adding more insulation. But if you want to dive into it, radiant temperature and variation has some effect on comfort. Windows are the big issue. So is proper mixing of the AC air with room air (which reduces stratification).

Hi Jon, thanks for the response. Yes, radiant temperature from my (large) ceiling is my concern. Everything (including windows) is already set in stone -- thanks in large part to the GBA community :) -- so this foam thickness thing is my last question.

I will go well above code on this if it's worth it, so that's what I'm trying to determine. I put the more specific details of the build here as far as target R-values and such -- https://www.greenbuildingadvisor.com/question/roof-insulation-for-28-vaulted-octagon-zone-2a

I do not think you can predict the ceiling temp without knowing the volume and speed of the air moving across the ceiling will be.

My guess is if you build an R38 ceiling you will be very comfortable.

If you want to understand how many BTUs you will use I say build a BEopt model of your home.

https://www.nrel.gov/buildings/beopt.html

Walta

I read Dana's response differently. What I get is he's saying that if the ceiling is at 80F and the roof is at 120F, the average temperature is the average of those two numbers, or 100F. He's not calculating those values, just suggesting them as plausible numbers for the sake of example.

That's right. I made no attempt to calculate the peak temp of the roof (instead relying on old marketing fluff from high SRI roofing vendors), and a WAG at what the ceiling temp MIGHT be for a roughly code-min insulated roof in a room maintained at 70-75F near the floor.

Low SRI roofs get a LOT hotter than 120F, even when outdoor temps are < 100F, and even high SRI roofs are going to hit temps a lot hotter than 120F when it's 110F outside. For a quickie view of some work done by the some folks at the Lawrence Berkeley Nat'l Labs, see Figure 2 in this document:

https://heatisland.lbl.gov/sites/all/files/Cool-roof-Q+A.pdf

For a white metal high SRI roof on a reasonably steep pitch (not flat or low-angle) it looks like the roof temp will run about 25F above ambient. So in a location with a 1% outside design temp of ~95F (like say, Houston TX) the roof temp would hit 120F or higher for ~87 hours in a typical year. That's a reasonable baseline for 120F in the example.

Clearly with a roughly conical shaped ceiling the temp at the very top is likely to be much higher than at the edge where it meets the wall unless there is something like a ceiling fan to break up stratification. Even on the exterior of a sloped roof the lower edge of the roof will be cooler than the temp at the top due to convection on the exterior. But with most roofs (including high SRI roofs) that temperature delta would is moderated by radiational cooling back at the sky. The indoor temperature delta from low edge to top is probably going to be higher than on the exterior. With a conical shaped ceiling I would expect the stratification delta-T from the pointy-tip to the floor to be somewhat higher than with a simple gabled ceiling, since the volume of stagnant air 2 feet below the very top is smaller. But it probably doesn't affect the distributed average ceiling temp by very much.

Just now I took a quick set of measurements of the roof temp, outdoor temp floor temp and indoor ridge temp in an addition on my home with a 14' (floor to ceiling ridge) peak ceiling, insulated with crummy R38s in 16" on center 2x12 rafters.

Outdoors:

--South facing roof temp: 155F (way hotter than an high SRI roof)

---North facing wall temp (outdoor temp) 80F

That's a 75F delta roof to ambient, pretty typical for dark shingled roof.

It's a hipped roof- I didn't take the temps of the east, west & north pitches, but used only the ceiling temps on the southern ceiling pitch, near where the pitches converge.

Indoors:

--Floor temp: 74F

--Wall temp 5' off the floor: 75F

--Ridge temp: 79F

That's about a 4-5F delta from stratification. It's only 10:15 AM, bright & sunny, so I expect the indoor stratification and average temps to rise a bit before ~2-3PM when the trees on the SW slope of the property begin to shade the roof. But I don't expect the stratification delta to hit 10F.

So in a 70-75F room the 80F ceiling temp WAG in this configuration might be about right too, but it might be even less than that with a high SRI roof and continuous polyiso.

The mean temp through the material in the previous post was calculated exactly as it would be in an ASTM C518 test plate: (Hot side temp + cold side temp) /2 The purpose for calculating the mean temp was to be able to estimate the as-used (as opposed to labeled) R-value of the polyiso under peak conditions.

Bottom line- there are no simple mathematical models that would come up with an accurate ceiling temp. Even without factoring in the thermal mass time lags (the lignin in the 2x lumber wood ceiling provides enough thermal mass to matter) there are too many variables to reduce it to a set of simple math that would fit into a forum post.

Thanks for the responses all! Dana, that makes sense and is fair enough and appreciated, I should've thought it through more. But I am still curious as to the basis of your 2016 post below -- I'm not trying to be annoying here, it's just that nothing is stopping me from going from 6" of polyiso (R-30 min at 100F*) to 8" (R-40) or even 12" (R-60). This is a build I'm doing for family, so I don't want to skimp... but I also don't want to, say, go from 10" to 20" to save 1 degree, per your post here:

Reply #9: â€œOnce youâ€™re at R50 taking it to R100 would make less than a 1F difference in peak & average ceiling temp, and isnâ€™t worth doing for cooling comfort or cooling energy use. (It's sometimes worth going higher than R50 from a heating energy use point of view though, but not in cooling dominated climates.)" (https://www.greenbuildingadvisor.com/question/are-there-any-advantages-to-using-a-radiant-barrier-in-an-attic)

*Based on your other reply, in which you said "around 100F... even brand new roofing polyiso performs closer to R5.0"

If you are worried about ceiling temp (I wouldn't be), add the ceiling fan. But OK, evidently you want to better understand the numbers.

You can estimate with ceiling deltaT = 75 x (.61/50) where 75 is the indoor to exterior roof delta-T, .61 is the approximate R value of the interior side air film and 50 is the total roof R value. As Dana expands on, stratification is the much larger issue.

Summary (same as #1):

more than code min roof insulation won't have a noticeable effect on comfort.As far as ceiling temps, my title may be misleading... just trying to come up with a narrow technical way to answer my question, when I don't really have that expertise. Apologies there.

My concern is simply: given that the rest of my assembly is set, how much polyiso should I put on the roof, in order to minimize the Summertime heat transmitted to the interior, in order to minimize my AC usage? And Dana's "R50 vs. R100" claim indicates that 20" isn't much improvement over 10". But what about 6" vs. 10", or X" vs. Y"?

The difference in heat gain/loss is easy to calculate (roughly). If you double the R value, you cut heat transfer in half. So, going from R1 to R2 provides tremendous bang for the buck. Going from R20 to R40 also cuts heat transfer in half, but it costs 10x as much as going from R1 to R2 and the total heat transfer wasn't much at R20.

Calculating the value of extra insulation can be done, and that's what the various energy models are designed to do, based on your heating and cooling degree days and local energy prices. But none of this addresses your "comfort" criteria. Once you build a relatively air-tight enclosure, "comfort" is going to be driven primarily by mean radiant temperature of the ceiling, and that's really your original question. Unfortunately, the actual temperature of the ceiling is going to be a function of roof R-value, but it will also be very affected by interior conditions: do the fans and/or HVAC system reduce stratification and how effectively, does chilled air from the HVAC directly impinge on the ceiling, even how efficiently radiant heat transfers from the ceiling to the rest of the room. Probably the best you can do is to shoot for at least current R-value for your climate. More insulation will save more energy and make the interior more comfortable. You just can't really define exactly how much more comfortable. Suffice it to say that the 2021 IECC insulation levels will generally result in well-performing and comfortable houses.

Thanks for responding Peter, have learned a lot from your posts this past year. You phrased my question better than I did ("Once you build...").

"More insulation will save more energy and make the interior more comfortable. You just can't really define exactly how much more comfortable. "... "... and the total heat transfer wasn't much at R20."

What I'm curious about is this "wasn't much at R20." What was it? Say a 100F day, high-SRI roof at ~120F. I understand that going from R20 to R40 halves heat gain. But to make my decision, I need to know where I'm at with R20. Whoever Dana was talking to must have had only 2 degrees of heat gain at R50 (hence Dana's claim that "once youâ€™re at R50 taking it to R100 would make less than a 1F difference in peak & average ceiling temp"). I'd be happy to get there (though I understand this is also a function of the indoor temp, etc... by the way, I don't plan to use a ceiling fan, and my minisplit will be ~9' H, whereas the ~conical ceiling will be 17' at the peak).

"Probably the best you can do is to shoot for at least current R-value for your climate."

Again, I will do *more* than that. The question is how much more. That said, thanks for alerting me to IECC 2021... it calls for R-49 attic insulation in Zone 2 (vs. the old R-38, or BSC's 40). That alone drives my point that these recommendations are minimums. I know that there's no guaranteed "right" answer... but it's also not like the "right" answer was 38 last year and 49 this year... hence my paranoia today! :)

It sounds like you are shifting from a concern about comfort to perhaps "what's the $/year difference when adding more insulation?". Or perhaps "what's the CO2e/year difference?".

Here's a simple calculation that might be what you are looking for:

Very roughly, the thermal resistance between a surface and the space it faces is R-1. That's affected by lots of factors, of the orientation of the surface, the temperature difference, etc., but for small temperatures differences, it's dominated by radiation, so we won't be too terribly far off ignoring the other factors.

If we assume a roof temperature of 125 and the indoors at 75, that's a 50 F difference. Lets call the polyiso R-6/inch. At 6" thick, that's R-36, and the total including the surface R-1, is R-37. Of the total 50 F difference, 1/37th is between the ceiling surface and the space. So the celing is 1.35 F hotter: 76.35 F. If we went to 8 inches thick, we'd have a total of R-49, and the ceiling would be 1.02 F hotter: 76.02 F.

That's not an exact prediction of the ceiling temperature, which will depend on stratification, etc., but it gives you an idea of the effect on comfort, which is not very much once you have pretty good insulation.

I've been reading all you guys' posts here for a year so really appreciate the opportunity to be in conversation :). You're all celebrities to us lurkers!

RE: JonR: No, my concern is definitely comfort -- with PV, I will be net-zero even with poor insulation. That said, of course I don't want to spend money on, say, doubling polyiso thickness only to reduce my heat gain by 1F.

RE: Charlie: That makes sense. I understand that the amount of heat that flows from the exterior to the interior must be determined relative to given ext/int temps. E.g. Your math shows that, at 75F int / 125F ext, R-49 keeps the ceiling at 76F... and R-25 keeps it at 77F. Could that 1 degree difference possibly make any difference, comfort or energy use or otherwise?

If not, then why go over R-25? Besides "Code says so" -- again, I'm more than happy to *exceed* code R-values if sensible (even though there are no building codes where I live). Is there some other consideration that I'm ignorantly overlooking? I assume so...

It will affect energy use. The affect on comfort will be minor, but real. I notice, for example, the difference between keeping my thermostat at 69 F vs. 70 F. The ceiling is just one surface, but it's a big one.

"The ceiling is just one surface, but it's a big one." Yes, in my case, I have ~1000 SF of ceiling for ~650 SF of living space.

"I notice [a 1 degree difference]." I guess we could call that one vote for "a 1 degree difference is noticeable. Less than that may or may not be." I've admittedly never lived somewhere with a thermal envelope good enough to allow me to test / see if I can subjectively detect such a small difference. That said, as big as the ceiling is, you're talking about ambient air temperature... I'd think that a 1 degree difference there would be far more noticeable than for even, say, all the walls...

Energy use calcs need the math in this ballpark. We've left the comfort math ballpark, unless you're considering going *below* code min.

It's tough to do math on comfort anyways. Too many variables and we're definitely not playing with the big ones at this point.