Hey Dan, not sure what I can do with this information, but I love that you did it, and it will make a cool math lesson.

Meanwhile, it made me really want some cheese.

“I have no idea why it works, though.”

If that’s not a hook into mathematical inquiry, then I don’t know what is. The fact that it’s genuine (I think) makes it even that much better, rendering useless even the must stubborn pseudocontext detectors.

There’s a heat transfer problem going on here. You’re basically raising the energy of the cheese until it undergoes a phase change (not sure if it’s precisely a phase change because cheese isn’t a pure substance, but we’ll run with it). Energy is transferred to and from the cheese by three mechanisms:

1: Energy from the microwave emission is absorbed uniformly throughout the volume of the cheese block.
2: Heat is conducted between the bottom face of the cheese block and the plate. Note that the plate also absorbs some of the microwave energy and will likely warm up.
3: Heat is also transferred by convection from the cheese block to the air inside the microwave. The air doesn’t absorb much of the microwave energy, so it stays relatively cool until the cheese and plate warm it up.

If the cheese and plate are at roughly the same temperature, then there’s not much conductive heat transfer (#2). I’m guessing that the dominant effects are the energy absorbed by the volume of the cheese block (#1) and the energy lost by cheese block through convection to the air in the microwave (#3).

As you move to bigger and bigger blocks of cheese, I think the air in the microwave has time to warm up, so it doesn’t affect things as much. I bet if you put an upside down tupperware over the cheese block so that it is in contact with a much smaller volume of air that it would melt faster.

@Steven: While the plate will warm up slightly, remember that microwaves work by exciting water molecules, of which I would guess that there are relatively few in ceramic as compared to cheese. I’m guessing that #2 is going to be a relatively large effect as well, and probably even larger than the convection in a relatively small microwave.

There’s also going to be a radiative component, but that one I am okay ignoring.

Microwaves don’t heat the cheese uniformly. The center of the block is not exposed to as much microwave radiation, since the cheese around it has absorbed most of it. Thus the radius of the block (distance from center to surface) is important. If I’m reading the numbers right, this seems to be the strongest effect.

The total energy absorbed by the cheese is proportional to the time, so if the block were being heated uniformly, only the volume would matter.

I suspect that determining when the cheese has melted is also subject to surface tension effects, with smaller blocks holding their shape better even when they are nearly liquid.
Melting ice, rather than cheese, might make for more obvious determination of when everything is melted, since ice does not form a skin the way melted cheese sometimes does.

This is one of my favorite videos you’ve done yet. The mathematician in me likes the naturalness of the question and the scaling, but the kid in me likes the cheese and the microwaving.

Seriously, I often feel this relief as a pure mathematician that I don’t have to worry about the problems you present: they’re applied, and I work in a realm where beautiful simplifications make my life easier, and my work more aesthetically pleasing and natural. Still, actually figuring real world stuff out holds tremendous appeal for me.

Incidentally, have you seen Conrad Wolram’s Computer Based Math stuff? He has a nice TED talk on the importance of doing what you’re doing, essentially.

I would recommend combining these results with the toaster regression via the Sandwich Theorem.

Time vs. nachos eaten maybe?

Good stuff. Keep up the good work.

Cheese is conductive.

So the microwaves do not penetrate all that far into the surface of the cheese, so the microwave energy absorbed is going to be largely a function of the exposed surface area of the cheese.

The energy required to melt the cheese is going to scale linearly with the volume.

So the time it takes to melt should be roughly V/ESA.

As for the little guy, that’s interesting, but microwave wavelengths are about 12 cm, so from a node to a high point is about 6 cm. I imagine the little guy took longer because it happened to be near a node in the microwave.

You actually should be able to verify this pretty easily. If you make the little 1cm^3 blocks, and distribute a handful of them on a nonrotating plate (I don’t know if you can turn off the rotation, or remove the rotating platter and place another one in there), you ought to notice that there will be a large variation in the melting times for the little guys.

In fact, I’ve seen a demo where you take a plate full of minimarshmellows, and then by looking at the distance separating the ones that grow the largest, looking on the back of the microwave for the frequency, and knowing about the wave nature of light, you can estimate the speed of light to within 10%

In short, the small one melts slowly because our assumption of uniform micowave absorption breaks down on small length scales

Also took a look at the ratio of volume to exposed surface area versus time, and I still go with volume as the better indicator.

http://goo.gl/O3FLf

What else can we put in the microwave?

Different kinds of cheese?

From what I know of microwaves interacting with metals (i.e. conductive things and electromagnetic radiation at GHz frequencies), “The Virtuosi” has a good starting hypothesis.

Said another way, the electronic properties of the cheese will limit the heat flux into the bulk material. The thermal conductivity of the material limits the diffusion of heat into the center. We could probably rig a small 3D thermal conductivity model (assuming no melting) to examine how the heat flux and volume are related in a (highly) idealized way.

The cavity in the oven can, indeed, have hot spots. Think of a string vibrating and you’ll have the mental picture of “nodes” within the microwave cavity.

Nice experiment.

If you do the many small cheese experiment, I’d like to see that data, too.

Dan, Are you aware of Moody’s Mega Math Challenge? I thought of your students and your teaching methods when I heard about it. The registration deadline is Feb 25th. More info at: http://m3challenge.siam.org

Does anyone know if there is an equivalent contest to Mood’s Math Challenge in Canada.

The Virtuosi’s argument that melting time should be proportional to volume / exposed_surface_area rings true to me. The only difficulty is explaining why the relationship appears to be exponential, not linear!

From a theoretical point of view, melting cheese is hard to model, because there’s so much going on. If you want a nice theoretical model, you might want to look at simpler things, like time-lapse videos of melting ice cubes.

I would guess that if you have an ice block shaped like a box, the length, width, and height of the block will all decrease at the same constant rate as the block melts—until the corners of the block get rounded enough that it’s no longer even approximately box-shaped. I’ve never tried it out, though, and I was never good at thermodynamics…

It was mentioned here that the magnitude of the electric field inside the oven may not be homogeneous, i. e. ‘hot spots’ occur where there are antinodes in the electric field.

Furthermore, Rob suggested that the radiation would mainly absorbed close to the surface of the cheese.

I agree with those theories, but that’s not what I see in the movie. For each cube, most of the melting process takes place at the bottom, most clearly seen at 00:33 (3 cm cube).

So my model was plotting melting time vs. HEIGHT^2 x LENGTH x WIDTH which results in an affine function.

Nice experiment!

Maybe this is a good opportunity to play with this software:

“Eureqa is a software tool for detecting equations and hidden mathematical relationships in your data.”

http://ccsl.mae.cornell.edu/eureqa

Keep it up!

Hi Dan,

I fit your data by hand to an exponential with the following equation:

melting time = 4.55 exp(3.93*volume/surface area) – 1.15

As a metric, I minimized the sum of the squares of the differences between my model and the actual data for each point.

The reason I think the exponential is a reasonable model is that it is a solution to the heat equation, which is more or less what we’re dealing with here. Temperature should rise as a function of volume (microwave heats polar molecules in the cheese), and temperature should decrease as a function of the difference in temperature between the cheese and the environment, subject to the amount of surface area available for radiation from cheese to environment. (You’ve got a partial differential equation in space and time.)

As a sidebar, I teach biology in addition to math, and this problem is similar to the one cells in your body face. In the cell case, surface-to-volume ratio is the main factor which limits cell size. The idea is this:

1) cells need to transport food and oxygen into the cell to make energy

2) cells need to transport waste (such as heat and carbon dioxide) out of the cell to avoid poisoning themselves

3) the rate of transport processes is proportional to the surface area of the cell

4) the rate of waste generation and energy useage is proportional to the volume of the cell

End result: if a cell gets too big, it generates waste and uses energy so fast that it can’t transport food and waste fast enough to survive. In practice, that means that most cells are less than 1mm in diameter. We as large animals require a circulatory system to get around this difficulty.

If you’re ok with it, I would love to develop this idea into an inquiry-based lab for my students. Please let me know if you have suggestions.

Having thought hard about the all-important issue of cubeyness (link below), I am delighted to see it in application here. My students’ cubeyness measure (sum of the reciprocals of the dimensions) needs to be adapted for this problem. I will get to work on the exposed-cubeyness-measure and its correlation to cheese melting time.

On why it works, it seems the exposed surface area is how the microwaves get into the cheese. I imagine heat loss to be a minimal concern over the time scales involved here, and in comparison to the bombardment of microwaves.

And the exposed surface area idea is lovely. Kids often wonder about orientation of prisms. Abstractly, orientation doesn’t matter-a triangular prism is still a triangular prism no matter what face it is resting on (or whether it is resting on a face at all). But in the applied problem, this matters very much. Did you try different orientations of your cheese? The melting time of your cheese ought to vary greatly depending on how it stands. And what is your cheese budget, anyway? Can you afford several dozen more trials so we can examine variation in your data?

More on cubeyness:
http://wp.me/pAG7Q-3D

More on middle schoolers and prisms:
http://wp.me/pAG7Q-55

Dave, if you’re going to have a huge y-intercept, I’d like to see some data points closer to 0.

Without a physical explanation and very few data points, fitting an arbitrary curve with high R^2 values is pretty meaningless.

I can see volume being an important parameter, linearly related to time (total energy needed to melt that much cheese).

I can also see “thickness” as a important parameter (as energy is preferentially absorbed nearer the surface).

I have trouble with the y-intercept, though. Why should melting microscopic amounts of cheese take 13 seconds?

I just told my wife about this experiment. Between this and the toast, she suggested that we all make grilled cheese and get over it.

Actually, I’m not opposed to models that have no physical explanation. I use them all the time in my research. The key point is that using max R^2 to pick a model out of many hypotheses makes it very likely that you will pick a bogus model, particularly if there are few data points. (With 5 data points having different x values, I can always find a 4th-degree polynomial that fits, but it may have no predictive value for new points.)

When choosing models, one either needs to have a convincing mechanism OR one must test the model on data not used to choose the model. (Best is to have both.)

In my field, cross-validation is the standard, since we often have lots of data from past experiments, but collecting new data is slow and expensive. In cross validation, you do all your model building and model selection from part of the data, then check to see how well the model fits on the remaining data.

You have to be very careful when doing this, since you can only use the verification set once. If you go back and change your model, based on a failure on the verification set, then you’re using all the data for training, and have no independent verification.

If the model-building and selection process is automatic, you can do random splits of the data, and see the distribution of how well the data fits. This can give you a good idea whether you have enough data to fit the sort of model you are using.