I came into posession of an in-line oil heater sold for heating vegetable oil. Turns out it has a negative temperature coefficient. I measured the resistance at several temperatures and did some curve-fitting just for giggles.
32 F: R = 2.5 ohm
74 F: R = 1.85 ohm
212F: R = 1 ohm First I though about quadratic regression, but you don't get asymptotes with a quadratic. The actual resistance of the material would have an asymptote for temperature increasing without limit (somthing less than
1 ohm, but obviously not a negative value). So I decided to look at a kind of hyperboloid with three coefficients: (R+a)(T+b)=c. After fussing around with the numbers some, I decided to drop the a (making the resistance asymptote zero). This only leaves two coefficients to fit three data points, but check it out. I solved for b and c using the data points at 74 and 212 degrees; plugged in the third data point, and it fit. Thought whoa, maybe I'm onto something here. The equation: R(T+88.4)=300 Looking at it for physical significance, obviously R approaches 0 as temperature increases without limit. But what about that 88.4? I decided to go ogle the internet. Found mention that the R-T characteristic is supposed to be a "negative, non-linear exponential function." So I fitted the 74 and 212 degree data points, using Rankine values now into an equation of the form R = a exp(b/T) and got R = .0929 exp(10.76/T). That predicts R = 2.39 at 32 degrees. Probably as good a fit as the hyperbolic, considering measurement error. And a better fit with physical reality, because the hyperboloid exhibits an anomaly at -88.4 F. Sort of like predicting absolute zero is -88.4 degress F instead of -459. The hyperbolic was without any real physical significance. But it points to a weakness we humans have in looking for theories to fit the facts we have. It was such a good fit, I thought it must mean something.