What is the relationship between fuel mixture, throttle setting and engine temperature?

I'm trying to understand the relationship between mixture, throttle, and engine temp. In this article from AOPA, it states

... a lean mixture of 16 to 1 is not going to burn as hot as a rich mixture of 8 to 1. ... an engine [runs] hotter on lean mixtures.

This seems in general contradictory but what I'm gleaning is that there are some instances in which a leaner mixture will result in a higher engine temp and other instances in which a richer mixture will. Perhaps this comes down to throttle?

Could someone clear up for me what are the "primary points" so to speak with regards to engine temp that I should be aware of?

Further, does there exist a general graph somewhere with mixture, throttle, and engine temp on the x, y, and z axes, respectively? I might expect to see local maxima at full lean, full rich, and stoichiometric and local minima at I don't know where.

This image from Lycomings leaning instructions is likely what you are looking for:

The misnomer may be that leaning always increases the temperature of the engine. If you start at full rich or something above peak lean temp, leaning the engine will cause the CHT and EGT to rise until it hits its peak. If you continue to lean beyond this the temperature will start to drop again, and rapidly, but so will the power generated and at some point you run the risk of simply cutting the engine out so generally you want to lean to "peak EGT" since thats as low as you are practically going to want to go (unless you want worse fuel economy while flying slower).

If you look at the chart you will see below peak EGT your specific fuel consumption will also start to increase again.

Broadly speaking you can also see that increasing the mixture within its general operating range has a cooling effect on the engine.

If you are flying GA planes I would say the primary points in regards to engine temp is simply:

• Keep an eye on temps and don't overheat the engine.
• Lean slowly and carefully, keeping an eye on EGT or CHT for warm up which may not be immediate while leaning

If the gauge is creeping up, richen the mixture and consider reducing your angle of climb to increase airflow over the engine.

I don’t have the graph you are looking for, but we can reason this out.

If the mixture were 100% fuel and 0% air, it would be impossible for ignition to occur, so the temperature would be at a minimum. This explains what happens when you “flood” an engine by over-priming.

If the mixture were 100% air and 0% fuel, there is nothing to ignite. This is what happens when you pull the mixture knob/lever to IDLE CUTOFF at the end of a flight to kill the engine.

Ignition can happen between these two points, and temperature must rise when that happens, so we have a curve with some sort of peak in between and sloping down toward where the reaction fails on either side.

This peak temperature happens at the stoichiometric ratio between fuel and air. Having more fuel than optimal means you are running Rich of Peak (ROP), and having more air than optimal means you are running Lean of Peak (LOP). In both cases, the excess molecules (whether they be fuel or air) will be absorbing some of the heat and carrying it with them out the exhaust ports rather than letting it do useful work.

Throttle position is irrelevant.