very complex, including radiation convection fouling factors and loss of energy to exhaust gases, if heated by gas, or view factors if heated by electric.
As dooron said, it is very complex, & involves uncertainties in conduction, convection and radiation heat transfer modes. For this reason, boiler efficiencies are usually done by 'black box' or heat balance principle: Efficiency = (Steam heat content - feedwater heat content)/Caloric value of fuel; the inefficiency is (mostly) the flue gas heat. Normally, an exhaust temperature of 300 F or above is necessary for a good heat transfer rate, good exhaust draft and to avoid condensation (corrosion). The necessary exhaust temp. of course increases going from low pressure HW (hot water) to low pressure steam to medium pressure HW to high pressure HW to high pressure steam to superheated steam, although heat recovery systems are more frequently used for the latter systems.
For more on evaluating boiler efficiency, see
The theoretical approach is daunting. The most recent , theoretical modeling was done by W. Fiveland ( B+W) using a Galerkin method , but the inaccuracies are tied into the lack of knowledge of the radiation physical properties of the entrained flyash.
A more practical approach to obtain the average furnace heat absorption and funace exit gas temperature is what I call the "Russian Normative Method". A simplified summary of this is presented in the text by S. Kakac "Boilers, Evaporators, and Condensers", in the chapter written by the chinese engineer.
After obtaining the average exit gas temperature and furnace heat absorption rate, it is neccesary to estimate the variation as a function of height, and also to impose limits on the expected variation between average and peak values of absorption rate ( such as when the slag falls of a section of furnace wall).
