Internal Forced Convection ========================== Six correlations for internal forced convection in circular tubes. All correlations output the Nusselt number :math:`Nu = h D / k`. .. list-table:: :header-rows: 1 :widths: 28 30 22 10 10 * - 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`` .. math:: Nu = \frac{(f/8)(Re - 1000)\,Pr} {1 + 12.7\sqrt{f/8}\,(Pr^{2/3} - 1)} where :math:`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`` .. math:: 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 (:math:`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`` .. math:: 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} Classic turbulent correlation. Literature uncertainty ≈ 25 %. **Sieder-Tate (1936)** — ``internal.sieder_tate`` .. math:: Nu = 0.027\,Re^{0.8}\,Pr^{1/3}\,\left(\frac{\mu}{\mu_w}\right)^{0.14} The viscosity-ratio correction :math:`(\mu/\mu_w)^{0.14}` is applied when ``fluid.wall_viscosity`` is provided; omitted otherwise. **Shah (1978)** — ``internal.shah_laminar`` .. math:: 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 :math:`Nu = 3.66` when ``developing_length`` is absent. **Churchill-Ozoe (1973)** — ``internal.churchill_ozoe`` .. math:: Nu = Nu_0 + \frac{0.0668\,Gz}{1 + 0.04\,Gz^{2/3}} where :math:`Nu_0 = 3.66` (UWT) or :math:`Nu_0 = 4.36` (UHF). Falls back to :math:`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 :math:`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 ~~~~~~~~~~~~~~~~~~~~ - :math:`Nu = h D / k` (Nusselt number for circular tube of diameter :math:`D`): Incropera et al., 7th ed., Sec. 8.1. - :math:`Re = \rho u_m D / \mu` (bulk-velocity Reynolds number): Incropera et al., 7th ed., Sec. 8.1. - :math:`Pr = \mu c_p / k` (Prandtl number): Incropera et al., 7th ed., Sec. 8.1. - :math:`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. - :math:`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. - :math:`\mu_w` (dynamic viscosity at wall temperature, Sieder-Tate only): Sieder & Tate (1936), p. 1429; Incropera et al., 7th ed., Eq. 8.61. - :math:`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. - :math:`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 :math:`\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** :math:`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** :math:`n = 0.4` **(heating) /** :math:`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. - :math:`Nu_0 = 3.66` **(UWT fully developed base)** → Incropera et al., 7th ed., Sec. 8.4, Table 8.1. - :math:`Nu_0 = 4.36` **(UHF fully developed base)** → Incropera et al., 7th ed., Sec. 8.4, Table 8.1. - **Graetz number definition** :math:`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** :math:`\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.**