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Hot roof designs, aka "dense-packed" insulated sloped roofs for warm or hot humid climates:
This article describes various warm or hot or hot-humid climate hot roof designs, benefits, and risks. The text discusses solutions for un-vented cathedral ceilings and similar under-roof spaces in hot humid climates and offers advice on how to avoid condensation, leaks, attic mold, & structural damage when roof venting is not possible.
This article series about roof and ceiling ventilation describes inspection methods and clues to detect roof venting deficiencies, insulation defects, and attic condensation problems in buildings.
Hot Roofs vs Vented Roofs in Hot Humid Southern Climates - Building Heat Gain
Our photo (left) shows an extensive Penicillium sp. mold contamination in an attic where moisture condensation had gone unnoticed
The identification of this mold as Penicillium happened in the lab, of course, not based on just the visual inspection.
TenWolde and Rose pointed out in 1999 that
No scientific claims have ever been made that attic ventilation is needed for moisture control in warm, humid climates.
In these climates, the outside air is much more humid than the inside air which is cooled and dehumidified by air conditioning.
In such climates, attic venting tends to increase rather than decrease moistur4e levels in the attic .... [and] may therefore increase the danger of condensation on [the] HVAC ducts [as well]. 
Our photo (left) shows stains on the upper building walls under an inadequately-vented soffit. This New York Home suffered recurrent ice dam leaks sending moisture through the building walls, with the expected problems that result.
Joe Lstiburek's view is that particularly in the hot humid South (of the U.S. - and not the local of the buildings we are illustrating here) there is not much gained in venting a roof (2-3% reduction in heat transfer in a vented attic).
And quite properly Dr. Lstiburek continues to point out that once A/C ducts are placed into the vented attic heat transfer to the occupied space goes up to 5-7% (for tight insulated ducts) compared to routing those ducts through conditioned space of the home, and a much worse 25% heat gain to the home if the HVAC ducts are leaky.
And his argument that air movement through the attic under the roof will not flush heat being radiated towards the ceiling below is both interesting and compelling as well.
Joe also argues that the moisture from warm humid outside air run through an attic to ventilate it can move through the attic insulation where it can condense on the cool building ceilings below. 
When does warm air move "down" and when might cool air move "up" in a structure?
Those data are compelling, though I'm not entirely convinced that that nasty warm southern air my respected acquaintance describes actually moves "down" to the cool ceiling in the attic. In roofs I've worked-on and examined with air tests and smoke guns, warm air rising in the hot attic went zooming out at the ridge vent, drawing cooler (nasty moist Southern) air in at the eaves.
The incoming air followed the underside of the roof upwards on the air currents exiting at the ridge. without however, recapping construction costs, or considerations for an existing home rather than new construction.
I have found surprising down-currents in indoor air inside a few buildings where upper floor air was being cooled to a temperature and density greater than warmer air downstairs. In that circumstance cooler upstairs air flowed down stairwells to lower floors and on occasion that down-draft could de-pressurize an attic or cathedral ceiling causing warm moist outdoor to enter those areas.
Impact of venting on roof temperatures
Joe pegs the impact of venting on roof temperatures at 5%.
Really? I actually measured the temperature drop in a hot, un-vented attic in New York from over 145 degrees to 95 degrees after we cut in soffit and ridge venting
I was measuring air temperatures in the attic, not roof surface temperatures, and not radiated heat effects. That was a 35% drop in air temperature in the attic! This single "real world" case found a roof cavity temperature drop of seven times Dr. Joe's estimate.
Clearly we have seen some actual building conditions that don't behave as the scientists expected.
Watch out for real world snafus, damage, leaks in roofs. In sum, I'm left unsure about the gap between new construction designs and a perfect world where roofs never leak and the roofs I and more importantly, repair and renovation roof contractors have found when we inspected, tore apart, and repaired leaky roofs of both insulated cathedral ceiling homes, and homes with vented roof cavities.
In hot humid climates an un-vented roof is a defensible approach that may avoid both condensation and heat transfer problems.
In both warm climates and in cool northern climates an un-vented roof is a defensible approach that avoids risk of chimney effects affecting building heating and cooling costs
Moisture problems originating outside the building such as due to damage or poor workmanship involve an element of luck and appear to have been excluded by experts and studies in the field, notwithstanding the experience of repair-roofers and home inspectors that those leaks appear on a great many homes, can cause severe rot damage, costly mold damage, and that leaks eventually happen on a great many if not most homes that remain standing for 20 years or more.
Attic or roof cavity condensation due to indoor moisture sources vary widely as individual building moisture conditions vary (we start our roof inspection by looking for evidence of basement or crawl space flooding) as well as by quality of construction and the number of penetrations in the building ceilings and walls, not to mention the effects of air pressure differences within the building.
Delivering a durable hot dry roof is sound in theory but hard to deliver in practice.
Samuelson (1995) pointed out that "... to guarantee no indoor air movement into the attic, the ceiling has to be airtight and the pressure of the attic needs to be higher than that of the indoor air (i.e. pressurized attic or depressurized living space)" - not normal conditions in most climates.
An under-roof cathedral ceiling or attic ventilation system, to help the building in any way, needs to be properly designed and installed with adequate intake venting at the eaves, outlet venting at the ridge, and with careful sealing of air leaks from the occupied space into the attic or roof cavity (to avoid heat losses or increases in both moisture movement into the roof cavity and increased heating or cooling costs).
The packed un-vented hot roof system can work very well in the climates where recommended if everything is perfect in construction and if no leaks ever occur in the roof system over its life - that is, if nothing ever goes wrong. If leaks do occur, from either direction, damage is likely to be more severe than in the older vented roof cavity approach.
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