Solutions for the arithmetic average of the temperatures over the cross-section of a circular tube at each axial position are presented for elliptic heat transfer problems with a Neumann wall boundary condition and a finite heating section. Using the Green's function method, solutions in the form of simple expressions are derived. Although these solutions are only approximations to the available exact solutions, they do not require the solution of an eigenvalue problem and are reasonably accurate under conditions for which the temperature field is approximately radially uniform. This new approach and the straightforward way of determining how the average temperature field depends on the Peclet number and on the length of the heating section allow for computation of temperature fields for lower values of the Peclet number and for higher values of heating length than have been reported using analytical solutions. Peclet number bounds are also given for the importance of axial conduction.