Food Scientists: Please Explain the Final Dough Temperature (FDT) Formula This is undoubtedly a question for a food scientist. I have seen online a formula to calculate the water temperature needed to achieve a final/desired dough temperature: 

water temp = (FDT)*(mult. factor)-(room temp)-(flour temp)-(friction factor)-(preferment temp)


So for a desired dough temp of 78ºF on a straight dough (no preferment, mult. factor = 3) worked by hand (friction factor = 0) with room temperature flour (say 72º), you would need to make your water 90º, i.e.,

90 = 78*3 - 72 - 72 - 0


I've practiced this a couple of times now, and it seems to work to within a degree or so...magic! I'm baffled as to why, though. It seems quite suspect that no weights are contained in the formula. And, getting even more scientific, no specific heats. 

Can someone please explain where this useful formula comes from? I'm a physicist by trade so this is really fascinating, and I would love to know the derivation. (Math doesn't scare me.) 

For reference, here are two of the places that I found this formula:


https://www.kingarthurflour.com/blog/2018/05/29/desired-dough-temperature
https://www.theperfectloaf.com/common-bread-baking-calculators/


Thanks!
If kneading does not significantly heat up the dough, and the process is sufficiently quick then, by basic thermodynamics, the final dough temperature (FDT) is the average of the temperatures of the ingredients. (JoutlawPhysics points out in a comment below that this assumes equal masses of ingredients, which is usually only approximately true.) Hence

FDT = (water temp + room temp + flour temp + preferment temp)/4.

Here 4 is the `multiplication factor'. In general the multiplication factor is the number of temperatures involved, minus one. Rearranging the formula above we get

water temp = FDT*(mult. factor) - room temp - flour temp - preferment temp.

Your formula is the small generalization allowing for the heating effect of friction during kneading or mixing of the dough.