Developing Tube Flow (Combined Entry Length)
Calculates the mean heat transfer coefficient along the length of a tube from an abrupt square entry, where both the velocity and temperature profiles are still developing.
Film Temperature
Tf=2Tw+TbReynolds Number
ReD=μρumDLaminar Developing Flow (Re < 2 300)
Define the entry-length parameter:
ξ=(L/DRePr)1/3- If ξ ≥ 2 (thermally developing): Num=1.86ξ(μ/μw≈1)
- If ξ < 2 (effectively fully developed): Num=3.66
The viscosity ratio μ/μ_w is assumed unity (appropriate for gases and near-ambient liquids).
Turbulent Developing Flow (Re ≥ 2 300)
Gnielinski fully-developed Nu:
f=(0.79lnRe−1.64)−2 Nudev=1+12.7f/8(Pr2/3−1)(f/8)(Re−1000)PrEntry-length correction (Incropera eq. 8.57):
Num=Nudev(1+(L/D)0.6762.4254)The entry factor is validated for Pr ≈ 0.7 (air).
Heat Transfer Coefficient and Heat Rate
h=DNum⋅k,Q=h⋅(πDL)⋅(Tw−Tb)Validity
- Laminar: 0.48 < Pr < 16 700
- Turbulent: 0.5 < Pr < 2 000, Re < 5 × 10⁶
- Entry correction for turbulent regime validated for Pr ≈ 0.7 (air); apply with caution for other fluids
References
- Gnielinski, V., Int. Chem. Eng., 16, 359, 1976.
- Incropera, F.P. and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 3rd ed., John Wiley & Sons, eq. 8.57, 8.63a & 8.63b, 1990.
- Rohsenow, W.R., Hartnett, J.P. and Cho, Y.I., Handbook of Heat Transfer, 3rd ed., McGraw-Hill, 1998, p. 5.29.