A variational method is presented for solving eigenvalue problems which arise in connection with the analysis of convective heat transfer in the thermal entrance region of ducts. Consideration is given to both situations where the temperature profile depends upon one cross-sectional co-ordinate (e.g. Circular tube) or upon two cross-sectional co-ordinates (e.g. Rectangular duct). The variational method is illustrated and verified by application to laminar heat transfer in a circular tube and a parallel-plate channel, and good agreement with existing numerical solutions is attained. Then, application is made to laminar heat transfer in a square duct as a check, an alternate computation for the square duct is made using a method indicated by Millsaps and Pohlhausen. The variational method can, in principle, also be applied to problems in turbulent heat transfer.