Internal Forced Convection

Six correlations for internal forced convection in circular tubes. All correlations output the Nusselt number \(Nu = h D / k\).

Key	Title	Re Range	Pr Range	Year
internal.gnielinski	Gnielinski	3,000 – 5,000,000	0.5 – 2,000	1976
internal.petukhov	Petukhov	10,000 – 5,000,000	0.5 – 2,000	1970
internal.dittus_boelter	Dittus-Boelter	10,000 – 120,000	0.6 – 160	1930
internal.sieder_tate	Sieder-Tate	10,000 – 1,000,000	—	1936
internal.shah_laminar	Shah (laminar)	Re ≤ 2,300	—	1978
internal.churchill_ozoe	Churchill-Ozoe (laminar)	Re ≤ 2,300	—	1973

Correlation formulas

Gnielinski (1976) — internal.gnielinski

\[Nu = \frac{(f/8)(Re - 1000)\,Pr} {1 + 12.7\sqrt{f/8}\,(Pr^{2/3} - 1)}\]

where \(f = (0.790 \ln Re - 1.64)^{-2}\) (Petukhov friction factor). Preferred correlation for transitional and turbulent flow. Literature uncertainty ≈ 10 %.

Petukhov (1970) — internal.petukhov

\[Nu = \frac{(f/8)\,Re\,Pr} {1.07 + 12.7\sqrt{f/8}\,(Pr^{2/3} - 1)}\]

Uses the same friction factor as Gnielinski. Valid for fully turbulent smooth tubes (\(Re \geq 10{,}000\)). Literature uncertainty: ±5–6% for Pr 0.5–200; ±10% for Pr 0.5–2,000 (primary source confirmed, Petukhov 1970 p. 523).

Dittus-Boelter (1930) — internal.dittus_boelter

\[\begin{split}Nu = 0.023\,Re^{0.8}\,Pr^n, \quad n = \begin{cases} 0.4 & T_w > T_b \\ 0.3 & T_w < T_b \end{cases}\end{split}\]

Classic turbulent correlation. Literature uncertainty ≈ 25 %.

Sieder-Tate (1936) — internal.sieder_tate

\[Nu = 0.027\,Re^{0.8}\,Pr^{1/3}\,\left(\frac{\mu}{\mu_w}\right)^{0.14}\]

The viscosity-ratio correction \((\mu/\mu_w)^{0.14}\) is applied when fluid.wall_viscosity is provided; omitted otherwise.

Shah (1978) — internal.shah_laminar

\[Nu = 3.66 + \frac{0.0668\,Gz}{1 + 0.04\,Gz^{2/3}}, \quad Gz = Re \cdot Pr \cdot \frac{D_h}{L}\]

Thermally-developing laminar flow with a constant-wall-temperature base. Falls back to \(Nu = 3.66\) when developing_length is absent.

Churchill-Ozoe (1973) — internal.churchill_ozoe

\[Nu = Nu_0 + \frac{0.0668\,Gz}{1 + 0.04\,Gz^{2/3}}\]

where \(Nu_0 = 3.66\) (UWT) or \(Nu_0 = 4.36\) (UHF). Falls back to \(Nu_0\) when developing_length is absent.

Notes

• internal.gnielinski is preferred for transitional and turbulent regimes (Re 3,000 – 5,000,000) due to its lower literature uncertainty.

• internal.petukhov is valid for smooth-tube turbulent flow (Re ≥ 10,000).

• internal.shah_laminar and internal.churchill_ozoe apply to thermally developing or fully developed laminar flow; churchill_ozoe additionally selects the correct fully-developed base according to the boundary-condition type.

• internal.sieder_tate is included for legacy compatibility; Gnielinski is preferred when wall-viscosity data is not available.

• All turbulent correlations assume a smooth, straight circular tube with constant fluid properties evaluated at bulk temperature.

References

Formulation (equations)

Gnielinski (1976): Gnielinski, V., “New equations for heat and mass transfer in turbulent pipe and channel flow,” International Chemical Engineering, vol. 16, no. 2, pp. 359–368, 1976. English translation of: Gnielinski, V., Forschung im Ingenieurwesen, vol. 41, no. 1, pp. 8–16, 1975, DOI: 10.1007/BF02559682. Textbook presentation: Incropera, F.P., DeWitt, D.P., Bergman, T.L., and Lavine, A.S., Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, 2011, Sec. 8.5, Eq. 8.62, p. 514.

Petukhov (1970): Petukhov, B.S., “Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties,” in Advances in Heat Transfer, vol. 6, eds. Hartnett, J.P. and Irvine, T.F., Academic Press, New York, pp. 503–564, 1970, DOI: 10.1016/S0065-2717(08)70153-9. Textbook: Incropera et al., 7th ed., Sec. 8.5, Eq. 8.63.

Dittus-Boelter (1930): Dittus, F.W. and Boelter, L.M.K., “Heat Transfer in Automobile Radiators of the Tubular Type,” University of California Publications in Engineering, vol. 2, no. 13, pp. 443–461, 1930. Reprint: International Communications in Heat and Mass Transfer, vol. 12, no. 1, pp. 3–22, 1985, DOI: 10.1016/0735-1933(85)90003-X. Textbook: Incropera et al., 7th ed., Sec. 8.5, Eq. 8.60, p. 514; Cengel, Y.A. and Ghajar, A.J., Heat and Mass Transfer, 5th ed., McGraw-Hill, 2015, Sec. 8-4, Eq. 8-55.

Sieder-Tate (1936): Sieder, E.N. and Tate, G.E., “Heat Transfer and Pressure Drop of Liquids in Tubes,” Industrial & Engineering Chemistry, vol. 28, no. 12, pp. 1429–1435, 1936, DOI: 10.1021/ie50324a027. Textbook: Incropera et al., 7th ed., Sec. 8.5, Eq. 8.61; Cengel & Ghajar, 5th ed., Sec. 8-4, Eq. 8-56.

Shah (1978): Shah, R.K. and London, A.L., Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data, Supplement 1 to Advances in Heat Transfer, eds. Irvine, T.F. and Hartnett, J.P., Academic Press, New York, 1978, Ch. 5, ISBN: 978-0-12-020051-1. Textbook: Incropera et al., 7th ed., Sec. 8.4, Eq. 8.58; Cengel & Ghajar, 5th ed., Sec. 8-3.

Note on formula origin: The entry-length formula \(Nu = 3.66 + 0.0668\,Gz\,/\,(1 + 0.04\,Gz^{2/3})\) was originally proposed by Hausen (1943) and was tabulated and validated by Shah & London (1978). A full citation for Hausen (1943) is not available in the project reference set — see Missing References below.

Churchill-Ozoe (1973):

• UHF boundary condition: Churchill, S.W. and Ozoe, H., “Correlations for laminar forced convection with uniform heating in flow over a plate and in developing and fully developed flow in a tube,” Journal of Heat Transfer (ASME), vol. 95, no. 1, pp. 78–84, Feb. 1973, DOI: 10.1115/1.3450009.

• UWT boundary condition: Churchill, S.W. and Ozoe, H., “Correlations for laminar forced convection in flow over an isothermal flat plate and in developing and fully developed flow in an isothermal tube,” Journal of Heat Transfer (ASME), vol. 95, no. 3, pp. 416–419, Aug. 1973, DOI: 10.1115/1.3450078.

• Textbook: Incropera et al., 7th ed., Sec. 8.4, Eq. 8.57.

Variable definitions

• \(Nu = h D / k\) (Nusselt number for circular tube of diameter \(D\)): Incropera et al., 7th ed., Sec. 8.1.

• \(Re = \rho u_m D / \mu\) (bulk-velocity Reynolds number): Incropera et al., 7th ed., Sec. 8.1.

• \(Pr = \mu c_p / k\) (Prandtl number): Incropera et al., 7th ed., Sec. 8.1.

• \(f = (0.790 \ln Re - 1.64)^{-2}\) (Petukhov smooth-pipe friction factor): Petukhov (1970), pp. 503–564; Incropera et al., 7th ed., Eq. 8.21.

• \(Gz = Re \cdot Pr \cdot D / L\) (Graetz number): Shah & London (1978), Ch. 5; Incropera et al., 7th ed., Sec. 8.4, Eq. 8.58.

• \(\mu_w\) (dynamic viscosity at wall temperature, Sieder-Tate only): Sieder & Tate (1936), p. 1429; Incropera et al., 7th ed., Eq. 8.61.

• \(Nu_0 = 3.66\) (fully developed Nu for constant wall temperature): exact Graetz-problem solution, Incropera et al., 7th ed., Sec. 8.4, Table 8.1.

• \(Nu_0 = 4.36\) (fully developed Nu for uniform heat flux): exact solution, Incropera et al., 7th ed., Sec. 8.4, Table 8.1.

Range of applicability

• Gnielinski Re 3,000–5,000,000; Pr 0.5–2,000; L/D ≥ 10: Incropera et al., 7th ed., Sec. 8.5, Eq. 8.62, p. 514.

• Petukhov Re 10,000–5,000,000; Pr 0.5–2,000: Petukhov (1970), pp. 503–564; Incropera et al., 7th ed., Sec. 8.5, Eq. 8.63.

• Dittus-Boelter Re 10,000–120,000; Pr 0.6–160; L/D ≥ 10: Incropera et al., 7th ed., Sec. 8.5, Eq. 8.60, p. 514. The Re upper bound of 120,000 reflects the McAdams (1954) form of the correlation; Incropera Eq. 8.60 states Re ≥ 10,000 with no explicit upper bound. The Pr lower bound of 0.6 is from Incropera 7th ed.; some earlier sources and the original 1930 publication used Pr_min = 0.7.

• Sieder-Tate Re ≥ 10,000; Pr 0.7–16,700; L/D ≥ 10: Incropera et al., 7th ed., Sec. 8.5, Eq. 8.61. Note: the summary table on this page shows “—” for the Pr range, which omits the documented range of 0.7–16,700 from Incropera Eq. 8.61.

• Shah Re ≤ 2,300 (laminar), all Pr, with fallback to Nu = 3.66 for fully developed limit (Gz → 0): Incropera et al., 7th ed., Sec. 8.4, Eq. 8.58.

• Churchill-Ozoe Re ≤ 2,300 (laminar): Incropera et al., 7th ed., Sec. 8.4, Eq. 8.57.

Assumptions

• Smooth, straight circular tube (Gnielinski, Petukhov, Dittus-Boelter, Sieder-Tate): Gnielinski (1976); Petukhov (1970); Incropera et al., 7th ed., Sec. 8.5.

• Fluid properties at bulk mean temperature (Gnielinski, Petukhov, Dittus-Boelter, Shah, Churchill-Ozoe): Incropera et al., 7th ed., Sec. 8.5 (turbulent) and Sec. 8.4 (laminar).

• Wall viscosity \(\mu_w\) evaluated at wall temperature (Sieder-Tate only): Sieder & Tate (1936); Incropera et al., 7th ed., Eq. 8.61.

• Thermally developing, hydrodynamically fully developed flow (Shah, Churchill-Ozoe): Shah & London (1978), Ch. 5; Churchill & Ozoe (1973).

• Constant wall temperature (UWT) boundary condition for Shah: Shah & London (1978), Ch. 5.

• Boundary-condition type selectable between UWT and UHF for Churchill-Ozoe: Churchill & Ozoe (1973), DOI: 10.1115/1.3450009 (UHF) and DOI: 10.1115/1.3450078 (UWT).

Uncertainty / error bounds

• Gnielinski ±10%: UNCERTAIN regarding original paper. The 1976 original paper (Gnielinski 1976, Int. Chem. Eng. 16(2):359–368) is not available in any accessible digital form and could not be verified. The available 2013 update (Gnielinski, IJHMT 63:134–140, reviewed during 2026-03-28 audit) does not restate a ±10% bound explicitly. The ±10% is sourced from Cengel & Ghajar, 5th ed., Sec. 8-5.

• Petukhov: The primary source (Petukhov 1970, p. 523) states the accuracy in two parts: ±5–6% for Re 10,000–5,000,000 and Pr 0.5–200; ±10% for the wider range Pr 0.5–2,000. These are the only directly verified uncertainty bounds in the catalog. Source: Petukhov (1970), Adv. Heat Transfer 6:503–564, p. 523, Eq. (50).

• Dittus-Boelter ±25%: textbook-attributed (Incropera et al. Eq. 8.60; Cengel & Ghajar Eq. 8-55). The original 1930 publication (UC Pubs. Eng. 2(13):443–461) is unavailable; the 1985 reprint (ICHMT 12:3–22, DOI: 10.1016/0735-1933(85)90003-X) is paywalled. Cannot verify against primary source. UNCERTAIN.

• Sieder-Tate ±20%: textbook-attributed. The original paper (Sieder & Tate 1936, Ind. Eng. Chem. 28:1429–1435, Table II, p. 1433, reviewed during 2026-03-28 audit) shows deviations ranging from ±4% to ±63% by fluid type and Re range. No single ±20% bound is stated anywhere in the paper. The ±20% is a textbook-consensus estimate. UNCERTAIN.

• Shah ±10%: textbook-consensus estimate. The original conference paper (Shah 1978, Proc. National HMT Conference, IIT Bombay) is not available in any accessible digital form. Cannot verify. UNCERTAIN.

• Churchill-Ozoe ±10%: textbook-consensus estimate. The original papers (Churchill & Ozoe 1973, J. Heat Transfer 95:78–84 and 95:416–419) are paywalled (ASME). Cannot verify. UNCERTAIN.

Missing references

• Hausen (1943): the entry-length formula used by both Shah (1978) and Churchill-Ozoe was originally proposed by Hausen. Full citation: Hausen, H., “Darstellung des Wärmeüberganges in Rohren durch verallgemeinerte Potenzbeziehungen,” Z. VDI Beih. Verfahrenstechnik, vol. 4, pp. 91–98, 1943. This source is not available in the project reference set; the formula is accessed through Shah & London (1978) and Incropera et al., 7th ed., Sec. 8.4, Eq. 8.58.

Traceability mapping

• Gnielinski equation form → Gnielinski (1976), Int. Chem. Eng. 16(2):359–368; Incropera et al., 7th ed., Eq. 8.62, p. 514.

• Petukhov equation (denominator constant 1.07) → Petukhov (1970), Adv. Heat Transfer 6:503–564, DOI: 10.1016/S0065-2717(08)70153-9; Incropera et al., 7th ed., Eq. 8.63.

• Friction factor \(f = (0.790 \ln Re - 1.64)^{-2}\) → Petukhov (1970), ibid.; Incropera et al., 7th ed., Eq. 8.21.

• Dittus-Boelter coefficients (0.023, exponent 0.8) → Dittus & Boelter (1930), UC Pubs. Eng. 2(13):443–461; 1985 reprint DOI: 10.1016/0735-1933(85)90003-X; Incropera et al., 7th ed., Eq. 8.60, p. 514.

• Dittus-Boelter exponent \(n = 0.4\) (heating) / \(0.3\) (cooling) → Dittus & Boelter (1930); Incropera et al., 7th ed., Eq. 8.60, p. 514.

• Sieder-Tate coefficients (0.027, 0.8, 1/3) and viscosity exponent (0.14) → Sieder & Tate (1936), Ind. Eng. Chem. 28(12):1429–1435, DOI: 10.1021/ie50324a027; Incropera et al., 7th ed., Eq. 8.61.

• Shah equation coefficients (0.0668, 0.04) → Shah & London (1978), Ch. 5; Incropera et al., 7th ed., Eq. 8.58. Original proposal by Hausen (1943) — MISSING REFERENCE (see above).

• Churchill-Ozoe equation coefficients (0.0668, 0.04) → Churchill & Ozoe (1973), J. Heat Transfer 95(1):78–84, DOI: 10.1115/1.3450009; companion: 95(3):416–419, DOI: 10.1115/1.3450078; Incropera et al., 7th ed., Eq. 8.57.

• \(Nu_0 = 3.66\) (UWT fully developed base) → Incropera et al., 7th ed., Sec. 8.4, Table 8.1.

• \(Nu_0 = 4.36\) (UHF fully developed base) → Incropera et al., 7th ed., Sec. 8.4, Table 8.1.

• Graetz number definition \(Gz = Re \cdot Pr \cdot D / L\) → Shah & London (1978), Ch. 5; Incropera et al., 7th ed., Sec. 8.4, Eq. 8.58.

• Gnielinski Re 3,000–5,000,000; Pr 0.5–2,000; L/D ≥ 10 → Incropera et al., 7th ed., Sec. 8.5, p. 514.

• Petukhov Re 10,000–5,000,000; Pr 0.5–2,000 → Petukhov (1970), p. 503; Incropera et al., 7th ed., Sec. 8.5, Eq. 8.63.

• Dittus-Boelter Re 10,000–120,000; Pr 0.6–160; L/D ≥ 10 → Incropera et al., 7th ed., Sec. 8.5, Eq. 8.60, p. 514.

• Sieder-Tate Re ≥ 10,000; Pr 0.7–16,700; L/D ≥ 10 → Incropera et al., 7th ed., Sec. 8.5, Eq. 8.61.

• Shah Re ≤ 2,300 → Incropera et al., 7th ed., Sec. 8.4, Eq. 8.58.

• Churchill-Ozoe Re ≤ 2,300 → Incropera et al., 7th ed., Sec. 8.4, Eq. 8.57.

• Smooth-tube assumption (all turbulent correlations) → Petukhov (1970); Gnielinski (1976); Incropera et al., 7th ed., Sec. 8.5.

• Properties at bulk mean temperature (turbulent) → Incropera et al., 7th ed., Sec. 8.5.

• Wall viscosity \(\mu_w\) at wall temperature (Sieder-Tate) → Sieder & Tate (1936), p. 1429; Incropera et al., 7th ed., Eq. 8.61.

• UWT/UHF boundary condition switching (Churchill-Ozoe) → Churchill & Ozoe (1973), DOI: 10.1115/1.3450009 and DOI: 10.1115/1.3450078.

• Gnielinski uncertainty ±10% → Cengel & Ghajar, 5th ed., Sec. 8-5. UNCERTAIN regarding primary source: Gnielinski (1976) is inaccessible; the available 2013 update (IJHMT 63:134–140, reviewed 2026-03-28) does not restate this value.

• Petukhov uncertainty ±5–6% (Pr 0.5–200) / ±10% (Pr 0.5–2,000) → Petukhov (1970), Adv. Heat Transfer 6:503–564, p. 523, Eq. (50). Primary source confirmed. This is the only uncertainty bound in the catalog verified directly from an accessible primary source.

• Dittus-Boelter uncertainty ±25% → textbook attribution only (Incropera Eq. 8.60; Cengel Eq. 8-55). Primary source not accessible. UNCERTAIN.

• Sieder-Tate uncertainty ±20% → textbook-consensus estimate. Primary source (Sieder & Tate 1936, Table II, p. 1433, reviewed 2026-03-28) shows deviations ±4–63% by fluid/regime; no single ±20% stated. UNCERTAIN.

• Shah uncertainty ±10% → textbook-consensus. Primary source (Shah 1978 conference paper) not accessible digitally. UNCERTAIN.

• Churchill-Ozoe uncertainty ±10% → textbook-consensus. Primary sources paywalled (ASME). UNCERTAIN.