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Basically, since temperature as governed by the equations from the study of heat transfer, is neither uniformly distributed (a field) and changes with time (non-equilibrium), how applicable are the results you get from heat transfer back to thermodynamics? Especially considering that many of the equations are developed using energy balance from the first law which assume no work.

I know that heat transfer isn't derived from first principles and is an observational science but seeing that its results contradict the assumptions made by thermodynamic analysis, how trusted/accurate are the results, and since it's investigation have there been any reformulations of thermodynamics which account for it? Or are the violations from heat transfer simply negligible when considering them in thermodynamics?