The paper presents some typical results of an analysis of two-dimensional, steady-state heat conduction in coolant channels subjected to the third kind of thermal boundary conditions usually encountered in a regeneratively or dump-cooled liquid rocket engine. Duct geometries of circular, rectangular, trapezoidal and isosceles triangular cross-sections with rounded corners are considered. The solution method is based on the boundary collocation technique of linear least-squares matching using Householder reflections. The results of maximum duct wall temperatures and total heat transfer to the coolant by the present analysis vis-a-vis those by one-dimensional analysis reveal that the latter predicts higher values of maximum wall temperatures in the curved region of non-circular ducts and lower values of total heat transfer to the coolant. © 1980 Springer-Verlag.