Temperature gradients in boiled/baked eggs?

Presumably the question is referring to temperature gradients within the egg, i.e. the difference in temperature between the outside and inside. That is actually pretty relevant cooking-wise, because it tells you how the yolk and white will be cooked.

tl;dr I'd expect to see a steeper gradient from boiling than from the convection oven, but the actual results from your lab will let you tell much more confidently.

This depends on how fast the heat transfer is, which is made most obvious by the temperature the shell is held at. You can also see it from total cooking time. If one method takes longer, then it's transferring heat more slowly, and thus you can expect a flatter temperature gradient inside the egg (there's time for the heat to propagate inward), while the faster one will have a steeper gradient (the outside heats much faster than the heat propagates in).

With boiling, there's really only one possibility: the water holds the shell at very close to 100C. You have very efficient heat transfer, the water is well-mixed, and it has a very high heat capacity, so it can easily do this. Hard-boiling eggs generally takes less than ten minutes.

With an oven, it's less clear. Without any convection, the shell will actually remain well below the oven's air temperature for quite a long time, because the air can't transfer heat rapidly enough. This is likely true even with a realistic convection oven: if you put the egg in, wait a minute, and then check the shell temperature, I highly doubt it will be at the oven air temperature. Baking without convection in the 350F ballpark generally takes around half an hour, and I believe convection only reduces that to 20-25 minutes. So I'm pretty sure the oven has less efficient heat transfer, longer cooking time, and flatter gradient, but if you had a powerful, hot enough oven with really good convection, that could flip around.

So overall, it's hard to say for sure just from your description. If you actually did an experiment, though, you can tell from the results. Egg whites are soft but solid at 155F, and totally set at 180F, while egg yolks are firm at 158F and dry/crumbly at 180F. A steeper temperature gradient within the egg means that the yolk will be less cooked compared to the white, while a flatter temperature gradient means that they'll be cooked more similarly.

If you noticed differences in cooking time, or differences in yolk vs white doneness, there you are.

All this said, dry vs wet heat does have other significant effects. Dry heat will dry out the membrane and even some of the white underneath. Given enough time, the white will shrink away from the shell and end up noticeably drier and tougher, and it can even brown. So the differences you notice in the resulting egg may not be only about temperature gradient.

A temperature gradient is the variation of temperature along some distance. Maybe I took one too many semesters of thermodynamics in college, but on first blush, this is a tricky question. There are 3 big pieces to the puzzle to consider. Heat transfer between the egg and the “medium” and the gradients across the different regions.

I can think of two gradients that are worth considering. First is the gradient across the radius of the egg. Since the egg is the same between condition A (water) and B (oven) we don’t really have to think about this because any gradient difference are solely a function of the differences outside the egg. The second is the gradient throughout the “heat transfer medium” (boiling water, or oven air). Since the water is in a rolling boil, it is well mixed and there is no temperature gradient. If the oven is a convection oven, we can also say there is no gradient, but if it is a standard oven, there will be a gradient from the surface of the egg, out into the air.

The next thing to consider is the heat transfer between the medium and the surface of the egg. Since air is less dense than water, there are fewer atoms bouncing around in a given volume. This means there are fewer collisions between the medium and the egg shell...meaning fewer opportunities to transport energy from the medium to the eggshell.

Once heat does enter the eggshell, it will conduct through the egg the same in both cases.

What does this all mean? Since the heat transfer in air is less efficient than in water, the air has to be much hotter than water in order to heat the middle to a certain temperature in the same amount of time. However, since the air is hotter, the outside shell will be much higher than 100 degrees when the middle is done, meaning there is a larger temperature gradient.

The tricky part is, without knowing what the oven temperature is I can’t say for sure that the gradient is higher.

I will answer from a chemical engineering perspective.

The oven is also a good deal of radiant heat transfer.

In a moving liquid yes it is convection but you can model it as conduction.

Once you get a surface temperature on the egg how that got there makes not difference. The inside of the egg is not aware of how the shell was heated and thermal gradient will be the same. Believe it or not the slope of the thermal gradient remains very constant as the egg is cooked.

If you start with the egg in cold water versus waiting for the water to boil is a big difference.

Start with a cold oven would be a very bad plan. A hot oven will take a little bit of time to get the skin warm.

The water in the water is the main difference. It will hydrate between the shell and membrane for a more even heat transfer. Oven will dry out the egg and create pockets.