Hi Chet, update on the analytical solution:
I haven't been able to actually derive the analytical solution myself yet (seems more difficult than I first anticipated), however Mathematica does this and has solved the equations both numerically and analytically. I'll transfer the analytical solution functions (if correct) to Python. Here is the output for fluid temperature versus time (I have the solid plot also, its basically the same):
View attachment 293819
You can see the constant values I used, plus the T(t) and Ts(t) analytical solution functions output by Mathematica (the functions are blurry so I've typed them out at the bottom), and also the analytical solution graph from 0 to 3000s. Note it seems to curve off unexpectedly just before 3000s.
Here is the numerical solution, also done by Mathematica. The plot starts out similar to the analytical one, and continues to level off towards the inlet temperature indefinitely, which seems to make more physical sense. There is no unusual decrease in temperature behaviour happening in the numerical solution
View attachment 293820
Its late here so I'm going to check these plots against my own code first thing tomorrow.
Edit: The screenshooting doesn't seem to want to cooperate with the Mathematica page so I'm copying the analytical solution below. It doesn't seem to want to wrap at the end of the line (its two equations in the same line), so I will edit this tomorrow
$${{T(t) = 90 -4.37526 E^{(-0.0133365 t)}-5.62474 E^{(-0.000737726 t)}-4.44089*10^{-16} E^{(0.0125987 t)}]
and
Ts(t) = 90 +0.585555 E^{(-0.0133365 t)}
+8.88178*10{^-16} E^{(-0.0125987 t)}-10.5856 E^{(-0.000737726 t)}
-8.88178*10^{-16} E^{(0.0125987 t)}]}}$$