I was thinking something similar to LittleInch's comment above. You could develop a pretty sophisticated model that might approximate it relatively well. But to what purpose? I would personally rather take the conservative approach of assuming the steam temperature is the maximum the trunnion would experience and design based off of that. Unless the trunnion is made from some exotic material or extremely large/long (ie. prohibitively expensive), I would think that would be the better option.
The Trunnion is very long. If I use the same temperature as the process pipe we will have expansion problems plus we have to be concerned for the building steel.
There was something similar here about temperature of a dead leg I think. I'll see if I can find it, but from what I recall this also was a hybrid set of heat energy vs convection etc.
But the basis is the same - energy in and along the trunnion ( soild or a hollow pipe?) miuns heat losses from convection ( well known fomulae) and the repeat in small sections. You should get a parabolic temp curve vs distance from the pipe.
But a drawing, sketch or photo would help a lot.
So deciding on what your parameters are for how this pipe trunnion cools is a key part of the thinking, e.g. natural convection or forced / wind and then work from there.
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To use that graph you need to then split the trunnion into short lend as the temperature will then vary along its length as it's only heated from one end.
Remember - More details = better answers
Also: If you get a response it's polite to respond to it.
What is the problem you suspect that you want to solve? Generally, if you want to keep something from getting hot, when it is connected to something hot, you apply some cooling between it and the heat source. Steel is not a great heat conductor,so cooling fins often solve any problem.
Trunnions have been around for a long time and are frequently the cause of steam piping system failure and accidents. Failure by long term fatigue is a concern
Trunnions on piping elbows are a "not recommended" design and IMHO, only tolerable on low temperature (less than 300F) systems.
Elbow Trunnions are a challenge for the piping fabricator and the welders !!!
Trunnions complicate the stress analysis of the piping system needlessly.
EPRI, recognizing the implicit long term fatigue threat of elbow trunnions, issued this report/ guideline for those troublesome supports.
Common problem in our designs. We usually model the trunnion at the full or near steam temperature. Expansion of the trunnion may lift off adjacent supports so the stress model and solution should be conservative. A very long hot trunnion may clause more problems than it is worth.
From experience long uninsulated trunnions outside on 150 -200C steam lines do not get near the temperature of the steam line. But they will be hotter than support steel.
I really don't understand the issue here though. Whatever the expansion of the trunnion it will be dwarfed by the expansion of the pipe. You need to allow things to move in a controlled manner.
A drawing, photo or diagram would help a LOT.
Remember - More details = better answers
Also: If you get a response it's polite to respond to it.
Lisega had several papers on the pipe supports and temperature distribution. I cannot reach them at the moment but, from the memory, every mm distance from the heat source on the pipe cause around 2 deg C reduction of temperature on the pipe supports of steam piping.
You probably contact Lisega for the related paper.
