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Water at 300 K in a 25 mm tube at 1 m/s: Reynolds and Prandtl numbers, the turbulent correlations side by side and the heat rate to a 50 °C wall

Tool	Convection Correlations
Source	Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer, 6th ed. John Wiley and Sons, Hoboken, NJ. Chapter 8 (Dittus–Boelter, Sieder–Tate, Gnielinski and Petukhov correlations, Section 8.5) and Table A.6; Gnielinski, V. (1976). "New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow." International Chemical Engineering, Vol. 16, No. 2, pp. 359–368; Lemmon, E. W., McLinden, M. O., and Friend, D. G. (2023). "Thermophysical Properties of Fluid Systems." NIST Chemistry WebBook, SRD 69 (saturated-water table).
Level	Undergraduate

Problem

Water at 300 K (26.85 °C) flows at 1 m/s through a 25 mm tube 2 m long whose wall is held at 50 °C. Find the Reynolds and Prandtl numbers, the Nusselt number and convection coefficient by the Dittus–Boelter, Sieder–Tate, Gnielinski and Petukhov correlations with their ranges of validity, and the heat flux and heat rate for the entered temperature difference.

Inputs to Enter

Fluid water (NIST WebBook table), fluid temperature 26.85 °C, surface temperature 50 °C
Tube, internal flow: D 25 mm, L 2 m, V 1 m/s, uniform wall temperature

Open the Convection Correlations tool (loaded with this example)

Answers

Water at 26.85 °C: ρ 996.5 kg/m³, μ 0.8538 cP, k 0.6094 W/(m·K), c_p 4.181 kJ/(kg·K), Pr = μ c_p/k = 5.857 (Table A.6 at 300 K: 997, 0.855, 0.613, 5.83); μ_s(50 °C) 0.5465 cP, μ/μ_s 1.562
Re = ρ V D/μ = 996.5 × 1 × 0.025 / 8.538 × 10⁻⁴ = 29,180, turbulent; entry lengths about 10 D = 0.25 m
Petukhov f = (0.790 ln Re − 1.64)⁻² = 0.0238; Gnielinski Nu = (f/8)(Re − 1000) Pr/[1 + 12.7 (f/8)^½ (Pr^(2/3) − 1)] = 191.9 (recommended), h = Nu k/D = 4,679 W/(m²·K)
Dittus–Boelter Nu = 0.023 Re^0.8 Pr^0.4 = 174.1 (h 4,245 W/(m²·K), 9 % below Gnielinski); Sieder–Tate 193.4; Petukhov 193.5; spread of the four 11 %
q″ = h (T_s − T_f) = 4,679 × 23.15 = 108.3 kW/m²; q = q″ π D L = 17.0 kW over 0.1571 m²
USCS: h = 824.0 Btu/(h·ft²·°F), q = 58,060 Btu/h

Notes

The tool's default inputs are this case; press Submit. The case is the tool's own default (the brief's water-at-300-K tube) checked in test_convection_correlations.php (234 checks) against an independent numpy re-implementation of every correlation (References/Convection/checks/convection_check.py); the same test reproduces Incropera's Examples 7.4 (cylinder in air: Churchill–Bernstein 40.6, Hilpert 37.3, Zukauskas 50.5) and 9.2 (vertical door: Ra 1.813 × 10⁹, Nu 147, 1,060 W), Ostrach's Table I of NACA Report 1111 through LeFevre's g(Pr) and a scipy solution of his similarity equations, the Blasius–Pohlhausen plate solution and the Graetz eigenvalue 3.6568.