The Pizza Lab: A Computer Simulation of a Pepperoni Slice Curling
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One of the benefits of having gone to a school well known for its concentration of nerds is that many of my close friends are, well, nerds. And there are few greater sources of excitement than when two separate spheres of nerd-dom collide in a synergistic orgy of geekitude. Those are truly the times when human knowledge seems to advance in leaps and bounds
Last week, I wrote a piece about a pet obsession of mine: What makes pepperoni slices curl? The final conclusion was that it largely has to do with temperature differentials between the top surface heating faster than the bottom, as well as meat flow patterns inside the sausage caused by the stuffing horn being slimmer than the casing.
A few hours after the post went live, I got an email from my friend Evros Loukaides, a research student at Cambridge University studying the behavior and applications of thin morphing structures. Apparently, curling pepperoni falls squarely in the line of his work:
Kenji! You are dangerously close to my research topic with your latest post. As in, I'm about to forward it to my supervisor. We study the morphing capabilities of thin structures and the resulting shapes. If you are considering doing similar work about the shapes of chinese crackers, we might wanna talk to you about a joint publication. :)
If you require any computational modelling of food structures, to make your articles geekier than they already are, I'm your man.
Require? No. Really really want? You bet! I jokingly tasked him with creating a computer model of a pepperoni slice being heated on top of a pizza. An hour later, this hit my inbox:
Challenge accepted! (sort of)
Let me start with my assumptions: I model a slice of pepperoni as a disk with a radius of 15mm and a thickness of 3mm. I start the entire model at 300°K (80°F) and apply heat as a boundary condition on the top side, until it reaches 480°K (404°F). I also apply heat on the sides and bottom but of lower magnitude.
Since the geometry is trivial, the key is knowing the properties of the material. If those are accurate, you can usually get really good approximations for reality. I'm not a material scientist—I mostly deal with the effects of geometry on structural properties. Even if I was, the mechanical properties of tissue are still only partially understood—especially if you're interested in processed meats, which contain a collection of tissues (fat, ligaments, muscle, etc.) in a casing of separate properties. So basically we'll need to make a ton of assumptions and simplifications which pretty much render the results irrelevant to reality. But hey, why not? It's all good fun.
What are the parameters we need? You already showed that conductivity of the material is significant—otherwise the directionality of the heat gradient wouldn't matter. The specific heat capacity is also relevant to this, and in turn this depends on the density of the material. The coefficient of expansion, which in this case is obviously negative is probably the controlling parameter. The Young Modulus—the stiffness of the material—will show how much it needs to move to accommodate this heat gradient. Most tissue is usually modelled as hyperelastic, but for higher temperature, this effect is reduced, and we observe almost linear behaviour. But do keep in mind that all of these parameters depend on temperature: For example intuitively you can see that dried/cooked meat is stiffer than raw meat. I'm using an elastic model here as a demonstration but of course the slice deforms plastically.
I tried to find some numbers from the literature, but they are scarce and only tangentially related to our quest. For example one reference quotes the thermal conductivity of various tissues, but for raw tissue in its natural state. Still, it gives us a rough figure.
He finished with this:
I'll spare you the details, but by pulling numbers out of thin air in a similar manner for the remaining parameters, I was able to construct something resembling an approximation for your entertainment. If you need an accurate model, I'll need a lot more experimental data and time. And pizza.
Evros, having been to Pizza Express in Cambridge, I can only say that you deserve better pizza. I'll do my best to get it to you.
About the author: J. Kenji Lopez-Alt is the Chief Creative Officer of Serious Eats where he likes to explore the science of home cooking in his weekly column The Food Lab. You can follow him at @thefoodlab on Twitter, or at The Food Lab on Facebook.