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Multi-Prandtl correlating equations for free convection heat transfer from a horizontal tube of elliptic cross-section. (English) Zbl 1157.80317
Summary: Steady laminar free convection from a horizontal elliptic cylinder set in unbounded space is studied numerically under the assumption of uniform surface temperature. A specifically developed computer-code based on the SIMPLE-C algorithm is used for the solution of the mass, momentum and energy transfer governing equations. Simulations are performed for ratios between the minor and major axes of the elliptic cross-section of the cylinder in the range between 0.05 and 0.98, inclination angles of the major axis of the elliptic cross-section with respect to gravity in the range between $$0^\circ$$ and $$90^\circ$$, Rayleigh numbers based on the major axis of the elliptic cross-section in the range between 10 and $$10^{7}$$, and Prandtl numbers in the range between 0.7 and 700. It is found that the heat transfer rate increases with increasing the Rayleigh and Prandtl numbers, while decreases with increasing the orientation angle of the cross-section of the cylinder, i.e., passing from the slender to the blunt configuration. In addition, a noteworthy fact is that in most cases the amount of heat exchanged at the cylinder surface has a peak at an optimum axis ratio which is practically independent of the Prandtl number, while may either increase or decrease with increasing the Rayleigh number depending on whether the orientation angle of the tube is above or below a critical value of approximately $$67.5^\circ$$. Dimensionless correlating equations are proposed both for the optimum axis ratio for maximum heat transfer and for the heat transfer rate from the cylinder surface to the undisturbed surrounding fluid reservoir.
##### MSC:
 80A20 Heat and mass transfer, heat flow (MSC2010) 76R10 Free convection
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##### References:
 [1] Langmuir, I.: Convection and conduction in gases, Phys. rev. 34, 401-422 (1912) [2] Mcadams, W. H.: Heat transmission, (1954) [3] Morgan, V. T.: The overall convective heat transfer from smooth circular cylinders, Adv. heat transfer 11, 199-264 (1975) [4] Churchill, S. W.; Chu, H. H. S.: Correlating equations for laminar and turbulent free convection from a horizontal cylinder, Int. J. Heat mass transfer 18, 1049-1053 (1975) [5] Kuehn, T. H.; Goldstein, R. J.: Correlating equations for natural convection heat transfer between horizontal circular cylinders, Int. J. Heat mass transfer 19, 1127-1134 (1976) [6] Kutateladze, S. S.: Fundamentals of heat transfer, (1963) · Zbl 0121.44502 [7] Pera, L.; Gebhart, B.: Experimental observations of wake formation over cylindrical surfaces in natural convection flows, Int. J. Heat mass transfer 15, 175-177 (1972) [8] Hesse, G.; Sparrow, E. M.: Low Rayleigh number natural convection heat transfer from high-temperature horizontal wires to gases, Int. J. Heat mass transfer 17, 796-798 (1974) [9] Fand, R. M.; Morris, E. W.; Lum, M.: Natural convection heat transfer from horizontal cylinders to air water and silicone oils for Rayleigh numbers between $$3\times 102$$ and $$2\times$$107, Int. J. Heat mass transfer 20, 1173-1184 (1977) [10] Clemes, S. B.; Hollands, K. G. T.; Brunger, A. P.: Natural convection heat transfer from long horizontal isothermal cylinders, J. heat transfer 116, 96-104 (1994) [11] R. Hermann, Heat transfer by free convection from horizontal cylinders in diatomic gases, NACA TM 1366, 1954. [12] T. Chiang, J. Kaye, On laminar free convection from a horizontal cylinder, in: Proceedings of the Fourth National Congress of Applied Mechanics, 1962, pp. 1213 – 1219. [13] Saville, D. A.; Churchill, S. W.: Laminar free convection in boundary layers near horizontal cylinders and vertical axisymmetric bodies, J. fluid mech. 29, 391-399 (1967) · Zbl 0154.23703 · doi:10.1017/S0022112067000904 [14] Elliot, L.: Free convection on a two-dimensional or axisymmetric body, Q. J. Mech. appl. Math. 23, 153-162 (1970) · Zbl 0213.54102 · doi:10.1093/qjmam/23.2.153 [15] J.H. Merkin, Free convection on an isothermal horizontal cylinder, ASME Paper No. 76-HT-16, 1976. [16] Muntasser, M. A.; Mulligan, J. C.: A local non-similarity analysis of free convection from a horizontal cylindrical surface, J. heat transfer 100, 165-167 (1978) [17] Kuehn, T. H.; Goldstein, R. J.: Numerical solution to the Navier – Stokes equations for laminar natural convection about a horizontal isothermal circular cylinder, Int. J. Heat mass transfer 23, 971-979 (1980) · Zbl 0431.76026 · doi:10.1016/0017-9310(80)90071-X [18] Farouk, B.; Guceri, S. I.: Natural convection from a horizontal cylinder-laminar regime, J. heat transfer 103, 522-527 (1981) [19] Badr, H. M.: Heat transfer in transient buoyancy driven flow adjacent to a horizontal rod, Int. J. Heat mass transfer 30, 1997-2012 (1987) [20] Wang, P.; Kahawita, R.; Nguyen, T. H.: Numerical computation of the natural convection flow about a horizontal cylinder using splines, Numer. heat transfer 17, 191-215 (1990) · Zbl 0703.76070 · doi:10.1080/10407789008944739 [21] Saitoh, T.; Sajiki, T.; Maruhara, K.: Bench mark solutions to natural convection heat transfer problem around a horizontal circular cylinder, Int. J. Heat mass transfer 36, 1251-1259 (1993) · Zbl 0801.76085 · doi:10.1016/S0017-9310(05)80094-8 [22] Lin, F. N.; Chao, B. T.: Laminar free convection over two-dimensional and axisymmetric bodies of arbitrary contour, J. heat transfer 96, 435-442 (1974) [23] Raithby, G. D.; Hollands, K. G. T.: Laminar and turbulent free convection from elliptic cylinders with a vertical plate and horizontal circular cylinder as special cases, J. heat transfer 98, 72-80 (1976) [24] Hassani, A. V.: Natural convection heat transfer from cylinders of arbitrary cross section, J. heat transfer 114, 768-773 (1992) [25] Merkin, J. H.: Free convection boundary layers on cylinders of elliptic cross section, J. heat transfer 99, 453-457 (1977) [26] Cheng, C. -Y.: The effect of temperature-dependent viscosity on the natural convection heat transfer from a horizontal isothermal cylinder of elliptic cross section, Int. commun. Heat mass transfer 33, 1021-1028 (2006) [27] Huang, S. Y.; Mayinger, F.: Warmeubergang bei freier konvektion um elliptische rohre, Warme stoffubertrag. 18, 175-183 (1984) [28] Badr, H. M.; Shamsher, K.: Free convection from an elliptic cylinder with major axis vertical, Int. J. Heat mass transfer 36, 3593-3602 (1993) · Zbl 0782.76081 · doi:10.1016/0017-9310(93)90177-8 [29] Chen, Y. -M.; Wang, K. -C.: Numerical and experimental studies on natural convection from a horizontal elliptic cylinder, J. chin. Inst. chem. Eng. 27, 353-362 (1996) [30] Badr, H. M.: Laminar natural convection from an elliptic tube with different orientations, J. heat transfer 119, 709-718 (1997) [31] Mahfouz, F. M.; Kocabiyik, S.: Transient numerical simulation of buoyancy driven flow adjacent to an elliptic tube, Int. J. Heat fluid flow 24, 864-873 (2003) [32] Elsayed, A. O.; Ibrahim, E. Z.; Elsayed, S. A.: Free convection from a constant heat flux elliptic tube, Energy convers. Manage. 44, 2445-2453 (2003) [33] Van Doormaal, J. P.; Raithby, G. D.: Enhancements of the simple method for predicting incompressible fluid flows, Numer. heat transfer 11, 147-163 (1984) · Zbl 0553.76005 · doi:10.1080/01495728408961817 [34] Patankar, S. V.; Spalding, D. B.: A calculation procedure for heat mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat mass transfer 15, 1787-1797 (1972) · Zbl 0246.76080 · doi:10.1016/0017-9310(72)90054-3 [35] Leonard, B. P.: A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. meth. Appl. mech. Eng. 19, 59-78 (1979) · Zbl 0423.76070 · doi:10.1016/0045-7825(79)90034-3 [36] Patankar, S. V.: Numerical heat transfer and fluid flow, (1980) · Zbl 0521.76003 [37] Patankar, S. V.: Recent developments in computational heat transfer, J. heat transfer 110, 1037-1045 (1988) [38] Ramanujan, S.: Modular equations and approximations to $$\pi$$, Quart. J. Pure appl. Math. 45, 350-372 (1913 – 1914) · JFM 45.1249.01 · www.imsc.res.in [39] Cianfrini, C.; Corcione, M.; Habib, E.: Free convection heat transfer from a horizontal cylinder affected by a downstream parallel cylinder of different diameter, Int. J. Therm. sci. 45, 923-931 (2006)
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