Graetz problem with axial conduction

Consider the Graetz problem with axial conduction.[8] [7] The governing equation is ... 

M. L. Michelsen, and J. Villadsen, The Graetz Problem with Axial Heat Conduction, Int. J. Heat Mass Transfer, (17) 1391-1402,1974. 

Example 7.4 Modified Graetz problem with coupled heat and mass flows The Graetz problem originally addressed heat transfer to a pure fluid without the axial conduction with various boundary conditions. However, later the Graetz problem was transformed to describe various heat and mass transfer problems, where mostly heat and mass flows are uncoupled. In drying processes, however, some researchers have considered the thermal diffusion flow of moisture caused by a temperature gradient. 

Solve the following Graetz problem using the convection and conduction module in FEMLAB. When there is no axial conduction, you can absorb the Peclet number into the distance, z, and solve the problem once for all Pe. Solve with a radius of 1.0... 

In Fig. 3, the variation of local Nusselt number along the constant wall temperature tube is presented as a function of Peclet number, representing axial conduction in the fluid. For Pe = 50, which represents a case with negligible axial conduction, the solution of the classical Graetz problem, Nu = 3.66, is reached [44], while for Pe = 1, Nu = 4.03 [45] is obtained as the fully developed values of Nu. The temperature gradient at the wall decreases at low Pe values, thus the local and fully developed Nu values increase with decreasing Pe. 

Heat transfer on Walls With Uniform Wall Temperature. Heat transfer in a duct with uniform wall temperature is known as the Graetz or Graetz-Nusselt problem. In this case, a fluid with a fully developed velocity distribution (Eq. 5.15) and a uniform temperature flows into the entrance, and the fluid axial conduction, viscous dissipation flow work, and energy resources are negligible. Graetz [14] and Nusselt [15] solved this problem as follows ... 

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