This is a classical problem in literature dealing with forced steady heat convection combined with heat conduction in a duct with walls at different temperature. The first segment of the duct has "cold'' walls, while the second segment has "hot'' walls. The flow has an imposed temperature at the inlet and a known convection field (i.e., a given parabolic velocity profile). From the engineering point of view, this Worked Problem illustrates the application of conduction analysis to an important class of heat transfer problems in fluidic devices.
From a physical point of view this Worked Problem illustrates many aspects of steady convection-diffusion phenomena such as heat transfer into a channel, forced convection with an imposed velocity profile, heat conduction through walls, insulation (see VS Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966). Peclet number as a measure of axial transport velocity field (modeling the physics of the problem) and the length of the "hot'' portion of the duct are only some of the interesting parameters studied to extract average temperatures. Another aspect deals with discontinuity in boundary conditions (imposed temperature).