Dittus-Boelter, Sieder-Tate, Gnielinski: Choosing a Turbulent Heat Transfer Correlation

Why Correlations, and What They Correlate

Turbulent heat transfer inside a tube cannot be calculated from first principles any more than turbulent friction can. Instead, experiments are condensed into correlations for the Nusselt number, Nu = hD/k, the heat transfer coefficient made dimensionless by the tube diameter and fluid conductivity. Dimensional analysis says that, for fully developed forced convection, Nu should depend on the Reynolds number, which describes the flow, and the Prandtl number, which describes how momentum diffusion compares with heat diffusion in the fluid.

For laminar flow, the answer is exact and has no Reynolds dependence at all: in fully developed flow, Nu = 3.66 for a constant wall temperature and 4.36 for a constant wall heat flux. Turbulent flow needs empirical correlations, and three of them account for most use.

Dittus-Boelter: Simple, Famous, and Approximate

The familiar form is Nu = 0.023·Re^0.8·Pr^n, with n = 0.4 when the fluid is being heated and 0.3 when it is being cooled. It is usually quoted as valid for Re above about 10,000, Pr between about 0.6 and 160, and tubes longer than about ten diameters. Its simplicity makes it the first correlation students learn and the one most exam problems assume.

Its history is more tangled than its fame suggests. Winterton's 1998 note traced the equation and found that the version everyone quotes was popularised through McAdams' textbook, with coefficients that differ slightly from those in the 1930 Dittus and Boelter paper on automobile radiators. That does not make the correlation wrong, but its accuracy is modest. Incropera's widely used text cautions that errors can reach about 25%, especially when fluid properties vary strongly across the tube.

Sieder-Tate for Viscous Liquids, Gnielinski for Accuracy

For viscous liquids like oils, viscosity can differ greatly between the bulk fluid and the hot or cold wall, which changes the velocity profile near the wall. Sieder and Tate's 1936 correlation, Nu = 0.027·Re^0.8·Pr^(1/3)·(μ/μ_w)^0.14, adds a viscosity-ratio correction for exactly this, with μ_w evaluated at the wall temperature. It is quoted for Prandtl numbers from about 0.7 up to roughly 16,700, which covers heavy oils that Dittus-Boelter cannot.

When accuracy matters, Gnielinski's correlation is the better choice: Nu = (f/8)(Re − 1000)Pr/[1 + 12.7√(f/8)(Pr^(2/3) − 1)], valid for about 3000 < Re < 5 × 10⁶ and 0.5 < Pr < 2000. It builds on Petukhov's analysis, uses the Darcy friction factor f, often from Petukhov's smooth-tube formula f = (0.790 ln Re − 1.64)⁻², and is typically within about 10% of experimental data. Crucially, it extends down into the transition region, where Dittus-Boelter should not be used at all; Gnielinski's 2013 paper revisited how to bridge that region smoothly.

The Same Ideas Carry Over to Mass Transfer

Heat, mass, and momentum are all carried across the boundary layer by the same turbulent eddies, so their transfer coefficients are closely linked. Chilton and Colburn made this explicit in 1934 with the j-factor analogy: j_H = St·Pr^(2/3) = f/8 = j_D, where St is the Stanton number, f the Darcy friction factor, and j_D the equivalent mass transfer factor built from the Sherwood and Schmidt numbers.

That analogy is why a heat transfer correlation can often be turned into a mass transfer correlation by swapping Nu for the Sherwood number and Pr for the Schmidt number. It is also why it pays to know where each correlation came from. The choice of correlation, and checking that the Reynolds and Prandtl numbers fall inside its stated range, matters as much in an absorber as in a heat exchanger.

References

1. Dittus, F. W., & Boelter, L. M. K. (1930). Heat transfer in automobile radiators of the tubular type. University of California Publications in Engineering, 2(13), 443–461.
2. Winterton, R. H. S. (1998). Where did the Dittus and Boelter equation come from? International Journal of Heat and Mass Transfer, 41(4–5), 809–810. doi:10.1016/S0017-9310(97)00177-4
3. Sieder, E. N., & Tate, G. E. (1936). Heat transfer and pressure drop of liquids in tubes. Industrial & Engineering Chemistry, 28(12), 1429–1435. doi:10.1021/ie50324a027
4. Petukhov, B. S. (1970). Heat transfer and friction in turbulent pipe flow with variable physical properties. Advances in Heat Transfer, 6, 503–564. doi:10.1016/S0065-2717(08)70153-9
5. Gnielinski, V. (2013). On heat transfer in tubes. International Journal of Heat and Mass Transfer, 63, 134–140. doi:10.1016/j.ijheatmasstransfer.2013.04.015
6. Chilton, T. H., & Colburn, A. P. (1934). Mass transfer (absorption) coefficients: Prediction from data on heat transfer and fluid friction. Industrial & Engineering Chemistry, 26(11), 1183–1187. doi:10.1021/ie50299a012
7. Bergman, T. L., Lavine, A. S., Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer (7th ed.), Chapter 8. Hoboken, NJ: Wiley.