156 CHAPTER 5  Forced convection

Usually ReD = 2300 is considered to be the limit for transition to turbulence in internal flows. In view of this, for a smooth, circular duct, fD = 64/ReD needs to be used for ReD < 2300, except for situations where the boundary layer is “engineered” to be turbulent below this Reynolds number. Further, since x/D =10 for turbulent flows, unless the tube length is very short, fully developed flow relations like the ones given above should hold. The friction factor is a strong function of the surface roughness, and the Moody’s chart gives a graphical depiction of the variation of friction factor for laminar and turbulent flows. For the latter, the relative roughness is a key parameter. One can intuit that for the same Reynolds number, for a turbulent pipe flow, the higher the roughness, the higher the friction factor.

Noncircular ducts

The results we derived above for circular ducts are valid for noncircular ducts too. However, we need to use an effective diameter. This is known as the hydraulic diameter and is given by

where Ac is the cross-sectional area of the duct, and p is the wetted perimeter. For a circular duct of diameter d, Ac = π d2 /4 and p = π /d and so Dh = d.

Thermal considerations

Entry length: Using scale analysis, we can show that the dimensionless thermal entry length is given by xth /D ≈ 0.05 ReD Pr for laminar flow and approximately 10 for turbulent flow. The concept of thermal entry length is quite tricky from a conceptual viewpoint and needs an elaboration, as follows.

The mean temperature

For a duct flow, there is an absence of free stream velocity. Similarly, there is an absence of the free stream temperature. Just as we used mean velocity, here we define a mean temperature. The rate at which energy transport, Et , occurs may be obtained by integrating the product of mass flux (ρu) and the internal energy per unit mass (cvT ) over the cross-section.