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How does keeping the heat in the engine improve the torque of the engine per given amount of fuel consumed
Energy cannot be created or destroyed, only transformed. When fuel is burned, the chemical energy it contains is transformed into thermal energy. When it's burned inside an engine with the piston near TDC, the expansion of the burned fuel pushes the piston, transforming the energy from thermal into kinetic (it's actually a little more complicated; depending on load, timing, and speed there's a mix of potential and kinetic energy, but for our purposes at the moment we can think of it all as kinetic). The transformation from thermal to kinetic isn't perfect, because some of the heat is absorbed by engine parts that cannot move and therfore cannot convert thermal energy into kinetic. The cylinder walls and cylinder head, for example. Heat that escapes into the engine block and cylinder head is eventually carried off by the cooling system and shed into the atmosphere, having done no useful work for us. Likewise, there's heat left over in the burnt fuel/air mix and it is ejected via the exhaust system, also having done no work for us unless we've got a turbocharger in the line.
Remember, as the heat of a gas increases, so does its (unconstrained) volume or its (constrained) pressure. Anything that increases combustion chamber pressure — increased compression, forcing more fuel/air mix into the cylinder, advancing the ignition timing, colder=denser air charge, etc. — means more heat when combustion occurs, which means more push on the piston. It also means that for any given percentage of heat lost without transformation into kinetic energy, there'll be a higher absolute amount of such heat loss. And of course one must consider the limits (raise cylinder pressure too high, and you'll get detonation).
But it's not just a simple matter of crankin' up the heat in the intake tract. Consider EGR, which admits very hot exhaust into the intake tract. Its effect is to
lower peak combustion chamber temperatures; the exhaust is mostly inert/noncombustible, and so despite the higher intake charge temperature, there's less fuel in the charge, therefore a "smaller bang", less heat = less pressure in the combustion chamber, and therefore less production of NOx and less tendency to ping. Ping abatement is good, NOx abatement is good, but the reduction in engine performance is...not so good. If you will think about this example for a moment, you will probably get some insight into why the fuel-heating idea is being looked upon dimly.
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If the combustion chamber has a higher temperature coolant around it, is more of the heat transfered into the downward push of the piston?
Yep. The higher the temperature differential, the greater the heat loss. What's more, the cooler the coolant surrounding the combustion chamber, the greater the quenchout effect (wherein fuel on/near the surfaces does not burn, or does not burn completely). This explains the lower emissions with a hotter thermostat, and the upshot is pretty obvious: Burn more of the fuel, exhaust is cleaner, and bigger push on the piston.
But, there are conflicting forces at work. Increasing the engine temperature means more complete fuel burn, but also tends to mean a hotter, less-dense intake air charge, which means less cylinder pressure, which means a "smaller bang" and less piston pushing power. So you want to run the engine hot, the intake air and fuel cold, and watch out for pinging. Take a look at how most all vehicles are configured from the factory nowtimes: Hot thermostat, cold air intake, and undercar/underhood fuel line, filter, rail, & injectors constructed, positioned, & mounted so as to minimise fuel heat absorption. It's also worth a moment's consideration that the biggest (almost said "hottest") thing these days in propane motor fuel is
liquid propane injection. Colder, denser fuel = bigger bang = more cylinder push.
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And, does it seem logical that running a hot engine is more efficent for cruise, and then go with meth/water injection for when we need to cool down the combustion chamber for big power?
Water/methanol injection, depending on proportion and control, can cool down the combustion chamber and/or increase the cylinder pressure.
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I understand that at that particular moment we are not too concerned about economy
Well...efficiency, yeah, we are. It just refers to how much of the fuel's chemical energy we can transform into kinetic energy. Higher efficiency is better than lower efficiency. We can choose to use the kinetic energy in such a way as to maximise fuel economy or in such a way as to maximize acceleration and speed, or something in between those two extremes.
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And while we are at it, how does EGT, Exhaust Gas Temperature relate to this? Is higher better, or is lower better.
Generally lower, because you want to extract as much heat energy from the exhaust
inside the engine as possible, because that's where it can do work for us. Heat thrown out the tailpipe is lost and gone forever without doing any work for us. Note that cam and ignition timing affects EGT: If the spark occurs late or the exhaust valve opens early, the fuel/air mix will still be burning (or at least will still be hotter than with an earlier spark or a later valve opening) when it is ejected into the exhaust system. Retarded timing is a well-known cause of engine overheating; the still-burning fuel/air mix acts like a torch on the cylinder head via the exhaust ports, which heats up the coolant in a big hurry. If the timing is
very retarded, and you open the hood at night, you might get to see the exhaust manifold glowing.