Reheating question: temperature gradient figures?

Are there any data about how long it takes food from the fridge to heat to a percentage of the applied temperature in the bath?

So far as I have seen, to reheat sous vice food sous vide, you set the bath temp to what you cooked at, so as to not overcook, and then use the original time, sans tenderising and pasteurisation, for a simple warming.

It occurred to me that nearly all meat is acceptable to eat at a temperature quite a bit lower than the actual cook temp: it just needs to be warm to bring out the juices and wake up the meat. It would save a bit of time (possibly quite a bit, because the temp delta speed is anti-log or like that...asymptotic) if you used the original temp to heat the food, but were happy to settle for a lower serving temp.

So it seems to me of use if you could know a time at which the food reaches some percentage of the bath temp, because it seems that the time to _reach_ bath temp is not much affected by the bath temp, just by thickness....that makes sense to me; higher "temperature pressure", faster temp climb. But I can't find anything about this, graph or figures.

So. I cook at 57 C, for however long I need to get the desired effect. I then refrigerate. Later I want to heat for eating. Say I am happy to eat at 40 C. In a 57 C bath, given a thickness, how long would it take?

Any info greatly appreciated.

Nick