Heat and mass transfer empirical equations

[bill nye scienceguy!]

I guess this question applies to deriving empirical equations from any data set but in my engineering studies I've wondered this when working with heat and mass transfer.

When you do an experiment and want to identify the governing equation for the process, is it simply a case of bringing together all the variables you think appropriate and then trying to make an arrangement of them fit the curve on your graph?

It crosses my mind now as I'm working out the performance ratio of a multi-stage flash distillation plant and wondering how the hell someone derived this equation with 9 different terms, 7 different variables and 2 constants!

 

[Naty1]

When you do an experiment and want to identify the governing equation for the process, is it simply a case of bringing together all the variables you think appropriate and then trying to make an arrangement of them fit the curve on your graph?

generally, yes, that's a good description. I have no idea how the specific equation you describe was developed, but likely it did not happen all at once...likely someone took a stab at it and someone else improved it...and it was gradually developed. It's analogous to fitting a straight line to a bunch of data points via a least squares fit...maybe that's accurate enough maybe not...for Hubble, as an example, that seemed to do the trick developing H₀, the Hubble constant.

What you describe is also pretty much what Einstein did developing GR...he started from the famous insight, his equivalence principle, and so was able to derive a number of answers from the acceleration analogy to gravity...he had a number of different formulations and apparently discarded one by one them based on his understanding of how well they would match his anticipated behavior for the phenomena...In his case he did not even know how to do the math at first, his former teacher researched and found Riemannian geometry for Einstein...and Einstein leaned on other earlier formulations as well, like those of Lorentz and fitzgerald...

Another example is developing a feedback equation for some electrical system...a trick is to correlate the transfer function of the circuit with poles and zeros...one can then plot the results and get a visual representation of behavior...but before that mathematical trick was learned, it must have taken incredible trial and error and intuition to design a circuit with desired characteristics.

Today, I'm sure there must be some numerical analysis techniques that are computerized that can do a lot of what you suggest quickly and efficiently.