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Research Papers

Implicit Model Equation for Hydraulic Resistance and Heat Transfer including Wall Roughness

[+] Author and Article Information
Eckart Laurien

University of Stuttgart/Institute of Nuclear Technology and Energy Systems,
Pfaffenwaldring 31, D-70569 Stuttgart, Germany
e-mail: Laurien@ike.uni-stuttgart.de

Manuscript received May 12, 2015; final manuscript received October 13, 2015; published online February 29, 2016. Assoc. Editor: Thomas Schulenberg.

ASME J of Nuclear Rad Sci 2(2), 021016 (Feb 29, 2016) (6 pages) Paper No: NERS-15-1085; doi: 10.1115/1.4031948 History: Received May 12, 2015; Accepted October 13, 2015

Heat transfer to water at supercritical pressure within the core of a supercritical water reactor must be predicted accurately to ensure safe design of the reactor and prevent overheating of the fuel cladding. In the previous work (Laurien, 2012, “Semi-Analytic Prediction of Hydraulic Resistance and Heat Transfer for Pipe Flows of Water at Supercritical Pressure,” Proceedings of the International Conference on Advances in Nuclear Power Plants, ICAPP’12, Chicago, June 24–28), we have demonstrated that the wall shear stress and the wall temperature can be computed in a coupled way by a finite-difference method, taking the wall roughness into account. In the present paper, the classical two-layer model, consisting only of a laminar sublayer and a turbulent wall layer, is extended toward the same task. A set of implicit algebraic equations for the wall shear stress and the wall temperature is derived. It is consistent with the well-established Colebrook equation for rough pipes, which is included as a limiting case for constant properties. The accuracy of the prediction for strongly heated pipe flow is tested by comparison to experiments (Yamagata et al., 1972, “Forced Convective Heat Transfer to Supercritical Water Flowing in Tubes,” Int. J. Heat Mass Transfer, 15(12), 2575–2593) with supercritical water. The high accuracy and the generality of Laurien (2012) “Semi-Analytic Prediction of Hydraulic Resistance and Heat Transfer for Pipe Flows of Water at Supercritical Pressure,” Proceedings of the International Conference on Advances in Nuclear Power Plants, ICAPP’12, Chicago, June 24–28 are not achieved, but with the help of correction factors, the two-layer model has a potential for improved predictions of the hydraulic resistance and the heat transfer of pipe and channel flows at supercritical pressure.

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References

Schulenberg, T., Starflinger, J., Marault, P., Bittermann, D., Maraczy, C., Laurien, E., Lycklama a Niejeholt, J. A., Anglart, H., Andreani, M., Ruzikowa, M., and Toivonen, A., 2011, “European Supercritical Water Cooled Reactor,” Nucl. Eng. Des., 241(9), pp. 3505–3513. 0029-5493 10.1016/j.nucengdes.2010.09.039
Pioro, I. L., Khartabil, H. F., and Duffey, R. B., 2004, “Heat Transfer to Supercritical Fluids flowing in Channels—Empirical Correlations (Survey),” Nucl. Eng. Des., 230(1–3), pp. 69–91. 0029-5493 10.1016/j.nucengdes.2003.10.010
Lycklama a Niejeholt, J. A., Visser, D. C., Laurien, E., Anglart, H., and Chandra, L., 2011, “Development of a Heat Transfer Correlation for the HPLWR Fuel Assembly by Means of CFD Analyses,” 5th International Symposium on Supercritical Water-Cooled Reactors (ISSCWR-4), Vancouver, CA, Mar. 13–16, T. Schulenberg and J. Starflinger, Heidelberg, Germany.
Löwenberg, M. F., Laurien, E., Class, A., and Schulenberg, T., 2008, “Supercritical Water Heat Transfer in Vertical Tubes: A Look-Up Table,” Prog. Nucl. Energy, 50(2–6), pp. 532–538. 0149-1970 10.1016/j.pnucene.2007.11.037
Zahlan, H., Groeneveld, D. C., and Tavoularis, S., 2011, “Derivation of a Look-Up Table for Trans-Critical Heat Transfer for Water-Cooled Tubes,” Proceedings of the 14th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-14), Sept. 25–30, American Nuclear Society, Toronto, Canada.
Laurien, E., 2012, “Semi-Analytic Prediction of Hydraulic Resistance and Heat Transfer for Pipe Flows of Water at Supercritical Pressure,” Proceedings of the International Conference on Advances in Nuclear Power Plants, ICAPP’12, Chicago, June 24–28, American Nuclear Society, Chicago, IL.
Laurien, E., 2014, “Prediction of Hydraulic Resistance and Heat Transfer of Super-Critical Water Pipe Flows with Wall Roughness,” Workshop on Heat Transfer at Supercritical Pressure in Nuclear Reactors and Solar Energy Systems, Manchester, UK, June 30–July 1, D. Jackson, Manchester, GB.
He, S., Kim, W. S., and Bae, J. H., 2008, “Assessment of Performance of Turbulence Models in Predicting Supercritical Pressure Heat Transfer in Vertical Tube,” Int. J. Heat Mass Transfer, 51(19–20), pp. 4659–4675. 0017-9310 10.1016/j.ijheatmasstransfer.2007.12.028
Zhu, Y., 2010, “Numerical Investigation of the Flow and Heat Transfer within the Core Cooling Channel of a Supercritical Water Reactor,” Dissertation, University of Stuttgart, IKE-8-122.
Kiss, A., Laurien, E., Aszodi, A., and Zhu, Y., 2010, “Numerical Simulation on a HPLWR Fuel Assembly Flow with One Revolution of Wrapped Wire Spacers,” Kerntechnik, 75(4), pp. 148–157. 0932-3902 10.3139/124.110080
Kays, W., Crawford, M., and Weigand, B., 2005, Convective Heat and Mass Transfer (Int. ed.), McGraw-Hill, New York.
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Petukhov, B. S., 1970, “Heat Transfer and Friction in Turbulent Pipe Flow with Variable Properties,” Adv. Heat Transfer, 6, pp. 503–564. 0065-2717 10.1016/S0065-2717(08)70153-9
Colebrook, C. F., 1939, “Turbulent Flow in Pipes with particular Reference to the Transition Region between Smooth and Rough Pipe,” J. Inst. Civ. Eng., 11(4), pp. 133–156. 10.1680/ijoti.1939.13150
Moody, L. F., and Princeton, L. J., 1944, “Friction Factors for Pipe Flows,” Trans. ASME, pp. 671–684. 0097-6822
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Kader, B. A., 1981, “Temperature and Concentration Profiles in Fully Turbulent Boundary Layers,” Int. J. Heat and Mass Transfer, 24(9), pp. 1541–1544. 0017-9310 10.1016/0017-9310(81)90220-9
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Avci, A., and Karagoz, I., 2009, “A Novel Explicit Equation for Friction Factor for Smooth and Rough Pipes,” J. Fluids Eng., 131(6), pp. 061203-1–061203-4. 0098-2202 10.1115/1.3129132
Yamagata, K., Nishikawa, K., Hasegawa, S., Fujii, I., and Yoshida, S., 1972, “Forced Convective Heat Transfer to Supercritical Water Flowing in Tubes,” Int. J. Heat Mass Transfer, 15(12), pp. 2575–2593. 0017-9310 10.1016/0017-9310(72)90148-2

Figures

Grahic Jump Location
Fig. 1

Nondimensional velocity profile in wall units for a smooth wall and with wall roughness ϵ in wall units, constant properties

Grahic Jump Location
Fig. 2

Full line: curve-fit of the iterative solution (symbols) of Eq. (15) versus the nondimensional roughness ϵ+

Grahic Jump Location
Fig. 3

Comparison of the present work Nu(Re) from Eq. (25), smooth, with Gnielinski’s correlation

Grahic Jump Location
Fig. 4

Nu(Re) of the present work (Eq. 25) for Pr=1, smooth and rough wall

Grahic Jump Location
Fig. 5

Wall shear stress using Eq. (36) for the experimental parameters of Ref. [20]. Dot-dashed line: constant properties (no heating); full lines: present work; dashed lines: results from Ref. [6] for the same qw

Grahic Jump Location
Fig. 6

Symbols: Tw of the experiments [20]; large dashes: bulk temperature; small dashes: Tcs, full line Tw of the present work, qw=233  kW/m2

Grahic Jump Location
Fig. 7

Symbols: Tw of the experiments [20]; large dashes: bulk temperature; small dashes: Tcs, full line Tw of the present work, qw=465  kW/m2

Grahic Jump Location
Fig. 8

Symbols: Tw of the experiments [20]; large dashes: bulk temperature; small dashes: Tcs, full line Tw of the present work, qw=698  kW/m2

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