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Edit to add: In the summer, another thing to consider is that internal insulation covers up your thermal mass, so overheating is more likely to be a problem
How does that work? Presumably overheating is due to too much heat moving from outside to inside. That process would be slowed by internal insulation. It might stop temperatures equilibrating overnight (assuming that the outside temperature drops below that of your house), but I'd suspect it's more important not to allow your house to get too hot during the daytime in the first place.
ffm-
The ways that homes overheat in summer:
- Heat conducts through the roof/walls from outside to inside, due to poor insulation and made worse by dark colours and direct sun on the outside surfaces.
- Hot air gets inside through uncontrolled gaps/leaks or open doors/windows.
- Solar gain (heat from sunlight) comes through windows/openings to the interior and heats up the inside floor/walls/air.
- Internal heat gains from people, cooking, lighting, washing, computers, fridges etc
These can be combatted by a combination of insulation, keeping windows closed, external shading and being conscious of how much heat you're producing inside (difficult to avoid, most things we do generate heat).
Fortunately even in the summer, the UK is relatively cool at night. One of the most effective ways of preventing overheating is to make use of thermal mass to 'store coolth' or buffer heat gains. When outside temps cool down in the evening, open things up and get the heat out of the house, cool down the fabric. Materials like stone, concrete, brick and to some extent plaster, if they start the day cool, will then absorb the heat energy during the day keeping the ambient temp down and giving you more of that radiant thermal comfort by keeping the surface materials cooler for longer. Water also has high thermal mass but it's not that practical to have lots of it sitting around inside.
- Heat conducts through the roof/walls from outside to inside, due to poor insulation and made worse by dark colours and direct sun on the outside surfaces.
hoefla
There's a couple of things - one is efficiency of running the home eg damp walls conduct more heat to the outside, the other is how temperature, humidity and damp (condensation) are connected. These affect ambient temps and humidity, which are really important. Thermal comfort (how you feel) is also influenced by other things - a really significant one is radiant heat - if the surfaces in your line of sight (your body's thermal line of sight, really) are cold, you will feel colder. If they are warm you will feel warmer. Internal insulation provides a warm surface. It also improves overall thermal efficiency (less heat lost to the outside), so whatever heat you are inputting leads to warmer temps. The big thing to watch out for is interstitial condensation between the warm insulation and the cold wall. The existing cavity will still be doing a job to prevent damp from the outside getting to the inner leaf, and it will still be doing a job of reducing heat loss but much less significant compared with the new insulation.
Edit to add: In the summer, another thing to consider is that internal insulation covers up your thermal mass, so overheating is more likely to be a problem - not because of the insulation but because you lose that exposed thermal mass which acts to buffer heat gains.