The design temperature of a refractory lined vessel is addressed in VIII-1 UG-20(a):
If necessary, the metal temperature shall be determined by computation or by measurement from equipment in service under equivalent operating conditions
You should know the various heat transfer variables and be able to calculate the wall temperature (hopefully conservatively!) using some reasonably simple textbook equations. Alternatively, find another vessel operating with similar fluids at similar temperatures with the same refractory and take a measurement on a hot summer day.
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