Thank you for the quick reply.
I actually have taken courses on heat transfer as well as thermofluids, and have done quite a bit of calculation on this already; however with all the variables that need to be accounted for (convection coefficients for the YAG rod, surface area, flow rate, area of the rod, beginning and end temperatures of both the rod and the water) and it has just gotten a bit overwhelming. I have the Reynolds number, as well as the Nusseult number, but am not sure I'm always using the correct equations (or variables in those equations) to get it.
Ontop of this, dealing with power levels that oscillate makes it more complicated. Power can spike up to 12 kW or higher for a millisecond, then be off for another 99 milliseconds. At a constant power of 12 kW, the surface temperature is much too high.
The second equation you wrote is helpful, but I worry about it not factoring in surface area. Any suggestions?