Entrance region length problem

The above boundary conditions are however insufficient for the unique determination of the two-dimensional flow field U(x,z), V(x,z), an additional information about p(x,z) is necessary. For the original problem with smooth walls, the problem closure was suggested by the transition to a steady-state flow solution at a certain sufficient distance Lx called the entrance region length [566, 605], To this end, the pressure gradient can be taken constant = -/ , and the steady-state solution gives the relation/ = A = ... 

Problem (3.29) with the formulated boundary conditions possesses a unique solution over the length of the entrance flow region 0 < x < Lx, but the value Lx, the unknown length of the entrance region, is to be chosen at a distance, where no further transformation of the flow field takes place. Lx can be easily adjusted in the course of the numerical performance. 

For many calculations of heat and mass transfer, we neglect the entrance region and assume fuUy developed flow. At what path length L, in the flow field, will the flow in problem 5.23 part a above become fully developed What is the effect of the entrance length on pressure drop in the flow channel - will it be greater or less than we expect if we neglect the entrance effects ... 

The knowledge of this entrance length is important for the design of inlet sections or to define the validity of simplified models derived on the basis of fully developed laminar flow conditions. In practice, an inlet presection in the channel (with an uncoated/inert wall) can be used to allow for flow development before the fluid reaches the catalytically active region. A similarity between the entrance length of velocity and concentration/temperature profiles can be found, particularly when the wall temperature can be assumed to be uniform or severe external mass transfer control [42]. In Lopes et al. [43], the thickness of this region is discussed for the mass transfer problem with a finite wall reaction. 

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