Posted On: Monday - December 19th 2016 8:30AM MST
In Topics:   Global Climate Stupidity
(Continuation of Post 2.)
The best way to explain the complications of a mathematical model is by example, I think. I am going to summarize an engineer friend of mine's research project from the distant past.
Back when a whole lot of manufacturing was still going on in the US, this particular company wanting the analysis done was in the business of making personal computers. Chips were being soldered onto circuit boards (or planned to be) via a "hot-bar" process. A small heated bar (just for the purpose of picturing it, think of a roughly 1/2 in. long and 1/8 x 1/8 in bar) is placed by machinery onto a bunch of leads of a chip on one side to solder them all at the same time.
The engineering problem was to determine the electrical power input to the bar and length of time of contact with the leads that would get the joints soldered without overheating any part of the chip (maybe of the circuit board, also). Too much energy in, or at too high a rate - burned out parts. Too little energy in, or too low a rate - unreliable cold solder joints. Both situations are bad. Simple, right? No, NOT. Granted, unlike the the entire climate of the world, say, data could be obtained experimentally, though probably not easily with the small scale. Though experimental methods are a part of engineering, of course, and might still need to be done as verification with this problem, there is a reason we need to apply theory, heat transfer and thermodynamics, in this case.
The reason we need to apply theory is that we don't want to keep doing experiments every time we change any part of the process (different chips, say, or different dimensions, whatever). We want a mathematical model that works. Notice very carefully the "THAT WORKS", part, which differentiates engineering from science!). If we have a computer model, based, of course on math that describes the physics involved, we can change parameters and still get our numerical answers. In this case, we require temperatures of the chip material and the solder over time, given chip dimensions/material, size and material of the leads and solder, size, material, and power of/to the bar, and even air temperature of the work area.
This problem involves conduction, convection, and radiation heat transfer - all 3 modes. Conduction is very well defined by something called "Fourier's law of conduction". This can be modeled very well in even 3 dimensions and transiently (changes over time vs. steady-state) with "finite-element" and "finite-difference" techniques. These techniques break up some differential equations, which are very hard to solve together, into thousands or even millions of linear equations. The methods of solving linear equations simultaneously are called "matrix methods" and have been know for, say, a century or two. Nowadays, computer power is not even a factor for millions of equations, but even at the time of this example, one just needed more time - yes, you might wait on the computer to run for half an hour. Kids today would probably pull the plug to reboot the machine well before that time ;-}
Now, for the convection, the engineering analysis is much more empirical - meaning math derived not directly from theory but generalized from experiments. It is not something that gives numbers with great accuracy, and that's especially so with "free convection", meaning no forced flow of fluid, just movement of air (in this case) via buoyancy (warm air rises as it is lighter, right?). That's what this analysis involved, and much assuming and "smearing" must be done to match actual complicated geometry with the few generalized geometry set-ups that have known empirical math to describe them.
Radiation heat transfer has a simple equation for the gist of it, but the external geometry (what things the hot parts radiate to.) can make the problem complicated in a hurry.
I will get to comparing this problem to modeling the climate of the world in the next post.