We all know that headers allow for better breathing. One down side to them is that they increase under hood temp due to heat escaping through a thin wall design. Manifolds on the other hand are thicker and offer more limited breathing but have a decreased under hood temp. My question is with the manifolds, where does the heat go? Would It in fact raise overall engine temp as a result of it not being released under the hood? Thank you in advance for your time.
I'm not directly addressing your question because I believe it's a very complicated answer. Instead, I've tried to offer a little insight to my thought process on the matter. But, I'm not 100% I clearly understand your question though

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Manifold:
-- Less efficient breathing means more burnt (hot) air/fuel mixture remains in the cylinder. This, in turn, reduces the amount of fresh (cooler) air/fuel mixture into the cylinder and, subsequently, more heat goes through the cylinder walls/cylinder heads and into the anti-freeze. In other words, more heat is actually going into the engine's mass (during this time period in the engine's cycle).
-- The decreased fresh air/fuel charge means the cylinder has less energy (heat) at the end of the power stroke. Therefore, less 'new' heat is produced.
-- On the other hand, the heat retention properties of the exhaust manifold force more heat out of the engine bay and out the exhaust.
Header:
-- Increased cylinder evacuation means a larger charge of fresh (cooler) air/fuel. The larger mass of cooler air will help cool the engine. BUT, the larger mass of cooler air also increased the heat energy inside the cylinder because more air/fuel is burning (vs. the manifold). So, does that, in turn, transfer more or less heat into the engine???
-- The increased flow characteristics of the headers allows the burnt air/fuel charge to flow through the headers more easily. After that hot air/fuel flows through the headers, it goes through the rest of the exhaust.
-- Since the headers allow more heat into the engine bay, most engines probably see an increase in intake air temperatures (IAT). Thereby, increasing the heat transferred into the engine.
So, to find an answer to your question, you would need to run a detailed heat transfer analysis on each of the above scenarios (and probably several more I didn't think to mention). If you done that, you'd figure out which variable was the 'strongest'. I don't know that answer though

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Have a good day!
Michael