### Engineering Motivation

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.

### Physical Principles

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).