Tube Banks

Two correlations for crossflow over tube banks (bundles). Both require the transverse pitch \(S_T\), longitudinal pitch \(S_L\), and an explicit arrangement field ("inline" or "staggered"). The arrangement field is required on the Geometry object when geometry_type = "tube_bank".

All correlations output the Nusselt number \(Nu = h D / k\), where \(D\) is the tube outer diameter.

Key

Title

Re Range

Pr Range

Year

tube_banks.zukauskas

Žukauskas Bank

20 – 2,000,000

0.7 – 500

1972

tube_banks.grimison

Grimison

2,000 – 40,000

Pr ≈ 0.7

1937

Correlation formulas

Žukauskas (1972) — tube_banks.zukauskas

\[Nu = C\,Re_D^m\,Pr^{0.36}\,\left(\frac{Pr}{Pr_w}\right)^{0.25}\,F\]

Tabular coefficients \(C\) and \(m\) depend on arrangement (inline or staggered) and Re band. For staggered arrangements at \(Re \geq 1000\), \(C\) includes a pitch-ratio factor \((S_T/S_L)^{0.2}\).

The wall-Prandtl correction \((Pr/Pr_w)^{0.25}\) is omitted (set to 1) when fluid.wall_viscosity is not provided. The row correction factor \(F = 1.0\) (full correction tables require a number-of-rows input not present in the current state model).

Grimison (1937) — tube_banks.grimison

\[Nu = C_1\,Re_D^{m_1}\,Pr^{0.36}\]

Uses simplified constants from Incropera/DeWitt Table 7.4 (historical). Valid for \(2000 \leq Re \leq 40{,}000\); developed from gas data (Pr ≈ 0.7 only). No wall-temperature correction is applied.

Notes

  • Both correlations require geometry.pitch_transverse and geometry.pitch_longitudinal to be set; the state validator enforces this for geometry_type = "tube_bank".

  • tube_banks.zukauskas is preferred over Grimison for its wider Re range, wall-Prandtl correction, and broader validation base.

  • tube_banks.grimison is retained for legacy comparison and for Re bands not covered by other tube-bank correlations.

  • The row correction factor \(F\) (which reduces \(Nu\) for banks with fewer than ~10 rows) is not currently applied. Results represent the fully-developed average over a deep bank.

References

Formulation (equations)

Žukauskas (1972): Žukauskas, A., “Heat Transfer from Tubes in Crossflow,” in Advances in Heat Transfer, vol. 8, eds. Hartnett, J.P. and Irvine, T.F., Academic Press, New York, pp. 93–160, 1972, DOI: 10.1016/S0065-2717(08)70038-8. Full volume PDF available (Internet Archive, identifier AdvancesInHeatTransfer). Updated revision: Žukauskas, A., Advances in Heat Transfer, vol. 18, pp. 87–159, 1987, DOI: 10.1016/S0065-2717(08)70166-7 (paywalled). Textbook: Incropera, F.P., DeWitt, D.P., Bergman, T.L., and Lavine, A.S., Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, 2011, Sec. 7.6, Tables 7.5–7.7, p. 446.

Grimison (1937): Grimison, E.D., “Correlation and utilization of new data on flow resistance and heat transfer for cross flow of gases over tube banks,” Transactions of the ASME, vol. 59, no. 7, pp. 583–594, Oct. 1937, DOI: 10.1115/1.4020557. (ASME Digital Collection paywall.) Experimental data from: Pierson, O.L., “Experimental investigation of the influence of tube arrangement on convection heat transfer and flow resistance in cross flow of gases over tube banks,” Transactions of the ASME, vol. 59, no. 7, pp. 563–572, 1937, DOI: 10.1115/1.4020549. Textbook: Incropera et al., 7th ed., Sec. 7.6, Table 7.4 (historical); Cengel, Y.A. and Ghajar, A.J., Heat and Mass Transfer, 5th ed., McGraw-Hill, 2015, Sec. 7-4.

Variable definitions

  • \(Nu = h D / k\) (Nusselt number, tube outer diameter \(D\)): Incropera et al., 7th ed., Sec. 7.6.

  • \(Re_D\) based on maximum free-stream velocity \(V_{max}\) in the minimum free-flow area between tubes: Žukauskas (1972), p. 93; Incropera et al., 7th ed., Sec. 7.6.

  • \(Pr\) evaluated at fluid inlet/outlet arithmetic mean temperature (Žukauskas bank): Incropera et al., 7th ed., Sec. 7.6, p. 446.

  • \(Pr_w\) evaluated at tube wall surface temperature \(T_s\) (Žukauskas bank): Žukauskas (1972); Incropera et al., 7th ed., Table 7.5, p. 446.

  • \(C\), \(m\) (tabular coefficients, both correlations): Incropera et al., 7th ed., Tables 7.5–7.6 (Žukauskas bank) and Table 7.4 historical (Grimison); Žukauskas (1972).

  • \(F\) (row correction factor for \(N_L < 20\)): Žukauskas (1972); Incropera et al., 7th ed., Table 7.7, p. 446. Set to 1.0 in this implementation (see Notes).

  • \(S_T\), \(S_L\) (transverse and longitudinal pitch): Incropera et al., 7th ed., Sec. 7.6.

Range of applicability

  • Žukauskas bank Re 20–2,000,000; Pr 0.7–500; \(N_L \geq 20\) for \(F = 1\). The lower bound Re = 20 is confirmed directly from the primary source: Žukauskas (1972), p. 147 explicitly states Eqs. (38)–(44) describe “49 banks…at Re from 20 to 2×106 and Pr from 0.7 to 500.” Three-source discrepancy: the correlation YAML previously recorded \(Re_{min} = 10\); the Incropera text records \(Re_D = 16\); the primary source p. 147 states Re = 20. The primary source is taken as authoritative and the YAML has been corrected accordingly.

  • Grimison Re 2,000–40,000; Pr ≈ 0.7 (gases only). The valid range is sourced from Incropera et al., 7th ed., Sec. 7.6, Table 7.4 and Cengel & Ghajar, 5th ed., Sec. 7-4. The primary source (Grimison 1937, Trans. ASME 59:583–594, DOI: 10.1115/1.4020557) is paywalled and unverifiable; the range Re 2,000–40,000 is a textbook-attributed bound. UNCERTAIN — primary source not accessible.

Assumptions

  • \(Re_D\) based on maximum velocity \(V_{max}\) between tubes (Žukauskas bank): Žukauskas (1972), p. 93; Incropera et al., 7th ed., Sec. 7.6.

  • Fluid properties at arithmetic mean of inlet and outlet temperatures; \(Pr_w\) at wall temperature (Žukauskas bank): Incropera et al., 7th ed., Sec. 7.6, p. 446.

  • \(N_L \geq 20\) rows for full validity (\(F = 1.0\)); row correction factor from Table 7.7 required for shallower banks: Žukauskas (1972); Incropera et al., 7th ed., Table 7.7.

  • For Re 100–1,000 (Žukauskas bank) treat as single isolated cylinder (no bank coefficients tabulated): Incropera et al., 7th ed., Table 7.5.

  • \(S_T / S_L \geq 0.7\) required for aligned (in-line) arrangement: Incropera et al., 7th ed., Table 7.5, note.

  • No wall Prandtl correction applied (Grimison): Grimison (1937) original data was for gases with Pr ≈ 0.7 only; the \(Pr^{0.36}\) factor in the textbook form is a later generalization — see Flag below.

  • Inline vs. staggered arrangement supplied explicitly via the arrangement field on Geometry; this is a required input for geometry_type = "tube_bank" (state validator enforces it). Not inferred from pitch ratios.

Uncertainty / error bounds

  • Žukauskas bank ±25%: textbook-attributed figure (Incropera 7th ed., Sec. 7.6; Cengel & Ghajar, 5th ed., Sec. 7-4). The primary source (Žukauskas 1972, “Heat Transfer from Tubes in Crossflow,” Advances in Heat Transfer, Vol. 8, pp. 93–160) was reviewed in full during the 2026-03-28 traceability audit. No explicit ±% uncertainty statement was found for the tube-bank correlation in that chapter. UNCERTAIN — primary source reviewed; no explicit bound found.

  • Grimison ±25%: textbook-attributed. Primary source (Grimison 1937) is paywalled (ASME Digital Collection, DOI: 10.1115/1.4020557) and unverifiable. UNCERTAIN — primary source not accessible.

Missing references

None identified beyond those flagged under individual entries below.

Traceability mapping

  • Žukauskas bank equation form \(C\,Re_D^m\,Pr^{0.36}\,(Pr/Pr_w)^{0.25}\,F\) → Žukauskas (1972), Adv. Heat Transfer 8:93–160; Incropera et al., 7th ed., Sec. 7.6, Tables 7.5–7.7, p. 446.

  • Žukauskas bank C, m tabular coefficients (inline and staggered, Re bands) → Incropera et al., 7th ed., Tables 7.5–7.6, p. 446, sourced from Žukauskas (1972). Primary source Eqs. (38)–(44) verified against Žukauskas (1972), “Heat Transfer from Tubes in Crossflow,” Advances in Heat Transfer, Vol. 8 (pp. 142–147). Verified coefficients: Aligned Re 20–100: C=0.80, m=0.40; Staggered Re 20–100: C=0.90, m=0.40; Aligned Re 1,000–200,000: C=0.27, m=0.63; Staggered Re 1,000–200,000: C=0.35×(\(S_T/S_L\))0.2, m=0.60 (\(S_T/S_L < 2\)); C=0.40, m=0.60 (\(S_T/S_L \geq 2\)); Aligned Re 200,000–2,000,000: C=0.021, m=0.84; Staggered Re 200,000–2,000,000: C=0.022, m=0.84.

  • Žukauskas bank Pr exponent 0.36 → Žukauskas (1972); Incropera et al., 7th ed., Tables 7.5–7.6.

  • Žukauskas bank \((Pr/Pr_w)^{0.25}\) correction → Žukauskas (1972); Incropera et al., 7th ed., Sec. 7.6.

  • Žukauskas bank staggered pitch factor \((S_T/S_L)^{0.2}\) for Re ≥ 1,000 → Incropera et al., 7th ed., Table 7.6; Žukauskas (1972).

  • Row correction factor \(F = 1.0\) (for \(N_L \geq 20\) ) → Žukauskas (1972); Incropera et al., 7th ed., Table 7.7, p. 446.

  • Žukauskas bank Re 20–2,000,000; Pr 0.7–500 → Žukauskas (1972), p. 147 (primary source, confirmed); Incropera et al., 7th ed., Sec. 7.6, p. 446 (documents Re ≥ 16 — slightly inconsistent with primary source).

  • Žukauskas bank \(N_L \geq 20\) for full validity → Žukauskas (1972); Incropera et al., 7th ed., Sec. 7.6, Table 7.7.

  • Grimison equation form \(C_1\,Re_D^{m_1}\,Pr^{0.36}\) → Incropera et al., 7th ed., Sec. 7.6 (historical table). Flag: the Pr:sup:`0.36` factor is not present in the original Grimison (1937) paper, which was developed for gases (Pr ≈ 0.7) only. This generalization to other fluids via Pr:sup:`0.36` originates in the textbook form (Incropera et al.) and is not traceable to Grimison (1937) directly.

  • Grimison C1, m1 constants → Incropera et al., 7th ed., Sec. 7.6, Table 7.4 (historical); Cengel & Ghajar, 5th ed., Sec. 7-4; ultimately Grimison (1937), Trans. ASME 59:583–594, DOI: 10.1115/1.4020557.

  • Grimison Re 2,000–40,000 → Incropera et al., 7th ed., Sec. 7.6, Table 7.4; Cengel & Ghajar, 5th ed., Sec. 7-4. Primary source (Grimison 1937, Trans. ASME 59:583–594) is paywalled; range is textbook-attributed only. UNCERTAIN.

  • Grimison — no wall Prandtl correction → Grimison (1937) original data for gases; Incropera et al., 7th ed., Sec. 7.6.

  • Arrangement (inline / staggered) as explicit required input → project implementation decision (see dev-log 2026-03-27). The arrangement field on Geometry is required when geometry_type = "tube_bank"; it is not inferred from pitch ratios. Inline/staggered terminology follows Incropera et al., 7th ed., Sec. 7.6.

  • Žukauskas bank uncertainty ±25% → UNCERTAIN (specific primary-source page not identified).

  • Grimison uncertainty ±25% → UNCERTAIN (specific primary-source page not identified).