Prediction of the overall performance of furnaces and engines in terms of the operational and intrinsic thermal efficiencies (eta and alpha(o)(o)) has been limited in the past by need to assume empirically a linear dependence of the specific exhaust enthalpy, h(g), on the useful output (heat or work), H-s. In this paper, a theoretically-based relation between h(g) and H-s is developed using an integral approach, following Hottel and Thring, by treatment of furnace enclosures as heat exchangers. The expression developed is shown to be supported by comparison with experiment, using measurements on a boiler obtained by Prengle at the University of Houston. The new, theoretically-based and experimentally-supported relation shows that the variation between h(g) and H-s is nonlinear : it also depends on anew parameter, R(o), that is the ratio, at idle, of the exhaust gas temperature above ambient to the average flame temperature above ambient. The h(g)-H-s curve is near-linear for values of R(o) near unity, and the nonlinearity increases as R(o) decreases. The near-linear case is expected to be valid for hot wall furnaces which have high exhaust temperatures, with the nonlinear case valid for systems with high heat recovery and low exhaust temperatures, as in boilers. In this study, the numerical value of R(o) obtained for the boiler is small (0.09) and the h(g)-H-s curve, as expected, is correspondingly highly nonlinear over the full potential range of output. As a general result of the analysis, the dependence of the linear or nonlinear pattern of behavior on R(o) thus creates a clear target for future test by prediction from detailed mechanistic models.