- In internal fluid flows, a boundary layer develops from the surface where viscous effects become important, eventually merging at the centerline and resulting in a fully developed flow where the velocity profile no longer changes with increasing distance down the pipe. - The Reynolds number is used to characterize internal flows, with laminar flow occurring below a critical Reynolds number of around 2300 and fully turbulent flow requiring a Reynolds number over 10,000. - For laminar flow, the hydrodynamic entry length can be determined from an expression involving Reynolds number, while the entry length for turbulent flow is approximately independent of Reynolds number.

3. - We know that when the fluid makes contact with the surface, viscous effects become
important, and a boundary layer develops with increasing x. This development occurs at the
expense of a shrinking inviscid flow region and concludes with boundary layer merger at the
centerline.
- the velocity profile no longer changes with increasing x. The flow is then said to be fully
developed.
The Reynolds number for flow in a circular tube is defined as
Where um is the mean fluid velocity over the tube cross section and D is the tube diameter.
, the critical Reynolds number corresponding to the onset of turbulence is
although much larger Reynolds numbers (ReD 10,000) are needed to achieve fully turbulent conditions.

4. - For laminar flow (ReD 2300), the hydrodynamic entry length may be obtained from an expression of the
- general expression for the entry length in turbulent flow, we know that it is approximately independent of Reynolds number
and that

6. Because the velocity varies over the cross section and there is no well-defined free
stream, it is necessary to work with a mean velocity when dealing with internal
flows.

8. by which fluid temperatures exceed the entrance temperature increases with increasing x.

20. L> Xth
L<Xh
L> Xth
L > Xh
At L is larger then Xh & Xth