An Exact Analytical Solution for Convective Heat Transfer in Elliptical Pipes

Document Type : Research Article

Authors

Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

Abstract

In this paper, an analytical solution for convective heat transfer in straight pipes with
the elliptical cross section is presented. The solution is obtained for steady-state fluid flow and heat
transfer under the constant heat flux at walls using the finite series expansion method. Here, the exact
solution of Nusselt number as well as temperature distribution in terms of aspect ratio is presented as the
correlation in the Cartesian coordinate system and validated with the previous investigations. It is shown
that the minimum amount of Nusselt number, as well as the maximum absolute value of dimensionless
temperature at the center of the cross section, are related to the aspect ratio equal to 1 (circular pipe). The
solution indicated that the amount of Nusselt number is increased by changing the geometry of cross
section from circular to an elliptical shape and it finally tends to 4356/833 at large enough aspect ratios.
Our results also show that 95% of the increase in Nusselt number to the circular cylinder is related to
aspect ratio equal to 18.36. The present method of solution could be used to obtain the exact solution of
convective heat transfer in elliptical pipes for other thermal boundary conditions and fluid rheological
behaviors.

Highlights

[1] R. Shah, Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry, International Journal of Heat and Mass Transfer, 18(7) (1975) 849- 862.

[2] R.K. Shah, A.L. London, Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data, Academic press, 1978.

[3] P. Wibulswas, Laminar-flow heat-transfer in non-circular ducts, University of London, 1966.

[4] R. Lyczkowski, C. Solbrig, D. Gidaspow, Forced convection heat transfer in rectangular ducts—general case of wall resistances and peripheral conduction for ventilation cooling of nuclear waste repositories, Nuclear Engineering and Design, 67(3) (1982) 357-378.

[5] H. Zhang, M. Ebadian, A. Campo, An analytical/ numerical solution of convective heat transfer in the thermal entrance region of irregular ducts, International communications in heat and mass transfer, 18(2) (1991) 273-291.

[6] A. Barletta, E. Rossi di Schio, E. Zanchini, Combined forced and free flow in a vertical rectangular duct with prescribed wall heat flux, International journal of heat and fluid flow, 24(6) (2003) 874-887.

[7] C. Nonino, S. Del Giudice, S. Savino, Temperature dependent viscosity effects on laminar forced convection in the entrance region of straight ducts, International journal of heat and mass transfer, 49(23) (2006) 4469- 4481.

[8] T.J. Rennie, G.V. Raghavan, Thermally dependent viscosity and non-Newtonian flow in a double-pipe helical heat exchanger, Applied Thermal Engineering, 27(5) (2007) 862-868.

[9] H. Iacovides, G. Kelemenis, M. Raisee, Flow and heat transfer in straight cooling passages with inclined ribs on opposite walls: an experimental and computational study, Experimental Thermal and Fluid Science, 27(3) (2003) 283-294.

[10] A. Jaurker, J. Saini, B. Gandhi, Heat transfer and friction characteristics of rectangular solar air heater duct using rib-grooved artificial roughness, Solar Energy, 80(8) (2006) 895-907.

[11] S.W. Chang, T.L. Yang, R.F. Huang, K.C. Sung, Influence of channel-height on heat transfer in rectangular channels with skewed ribs at different bleed conditions, International Journal of Heat and Mass Transfer, 50(23) (2007) 4581-4599.

[12] S.K. Saha, Thermal and friction characteristics of laminar flow through rectangular and square ducts with transverse ribs and wire coil inserts, Experimental Thermal and Fluid Science, 34(1) (2010) 63-72.

[13] S. Ray, D. Misra, Laminar fully developed flow through square and equilateral triangular ducts with rounded corners subjected to H1 and H2 boundary conditions, International Journal of Thermal Sciences, 49(9) (2010) 1763-1775.

[14] L.-z. Zhang, Z.-y. Chen, Convective heat transfer in cross-corrugated triangular ducts under uniform heat flux boundary conditions, International Journal of Heat and Mass Transfer, 54(1) (2011) 597-605.

[15] M.M. Shahmardan, M. Norouzi, M.H. Kayhani, A.A. Delouei, An exact analytical solution for convective heat transfer in rectangular ducts, Journal of Zhejiang University SCIENCE A, 13(10) (2012) 768-781.

[16] M. Shahmardan, M. Sedaghat, M. Norouzi, An analytical solution for fully developed forced convection in triangular ducts, Heat Transfer—Asian Research, 44(6) (2015) 489-498.

[17] M. Sayed-Ahmed, K.M. Kishk, Heat transfer for Herschel–Bulkley fluids in the entrance region of a rectangular duct, International Communications in Heat and Mass Transfer, 35(8) (2008) 1007-1016.

[18] M. Norouzi, M. Kayhani, M. Nobari, Mixed and forced convection of viscoelastic materials in straight duct with rectangular cross section, World Applied Sciences Journal, 7(3) (2009) 285-296.

[19] H. Claiborne, HEAT TRANSFER IN NONCIRCULAR DUCTS PART I, Oak Ridge National Lab., 1951.

[20] L. Tao, On some laminar forced-convection problems, Journal of Heat Transfer, 83(4) (1961) 466-472.

[21] C.-Y. Cheng, The effect of temperature-dependent viscosity on the natural convection heat transfer from a horizontal isothermal cylinder of elliptic cross section, International communications in heat and mass transfer, 33(8) (2006) 1021-1028.

[22] V. Sakalis, P. Hatzikonstantinou, N. Kafousias, Thermally developing flow in elliptic ducts with axially variable wall temperature distribution, International journal of heat and mass transfer, 45(1) (2002) 25-35.

[23] K. Velusamy, V.K. Garg, G. Vaidyanathan, Fully developed flow and heat transfer in semi-elliptical ducts, International journal of heat and fluid flow, 16(2) (1995) 145-152.

[24] K. Velusamy, V.K. Garg, Laminar mixed convection in vertical elliptic ducts, International journal of heat and mass transfer, 39(4) (1996) 745-752.

[25] V. Javeri, Analysis of laminar thermal entrance region of elliptical and rectangular channels with Kantorowich method, Heat and Mass Transfer, 9(2) (1976) 85-98.

[26] R. Abdel-Wahed, A. Attia, M. Hifni, Experiments on laminar flow and heat transfer in an elliptical duct, International journal of heat and mass transfer, 27(12) (1984) 2397-2413.

[27] http://en.wikipedia.org/wiki/Ellipse.

[28] W. Kays, M. Crawford, B. Weigand, Convective Heat and Mass Transfer-Four edition, in, Mc Graw-Hill Publishing Co. Ltd, 2005.

[29] T. Papanastasiou, G. Georgiou, A.N. Alexandrou, Viscous fluid flow, CRC Press, 1999.

Keywords

References

[1] R. Shah, Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry, International Journal of Heat and Mass Transfer, 18(7) (1975) 849- 862.
[2] R.K. Shah, A.L. London, Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data, Academic press, 1978.
[3] P. Wibulswas, Laminar-flow heat-transfer in non-circular ducts, University of London, 1966.
[4] R. Lyczkowski, C. Solbrig, D. Gidaspow, Forced convection heat transfer in rectangular ducts—general case of wall resistances and peripheral conduction for ventilation cooling of nuclear waste repositories, Nuclear Engineering and Design, 67(3) (1982) 357-378.
[5] H. Zhang, M. Ebadian, A. Campo, An analytical/ numerical solution of convective heat transfer in the thermal entrance region of irregular ducts, International communications in heat and mass transfer, 18(2) (1991) 273-291.
[6] A. Barletta, E. Rossi di Schio, E. Zanchini, Combined forced and free flow in a vertical rectangular duct with prescribed wall heat flux, International journal of heat and fluid flow, 24(6) (2003) 874-887.
[7] C. Nonino, S. Del Giudice, S. Savino, Temperature dependent viscosity effects on laminar forced convection in the entrance region of straight ducts, International journal of heat and mass transfer, 49(23) (2006) 4469- 4481.
[8] T.J. Rennie, G.V. Raghavan, Thermally dependent viscosity and non-Newtonian flow in a double-pipe helical heat exchanger, Applied Thermal Engineering, 27(5) (2007) 862-868.
[9] H. Iacovides, G. Kelemenis, M. Raisee, Flow and heat transfer in straight cooling passages with inclined ribs on opposite walls: an experimental and computational study, Experimental Thermal and Fluid Science, 27(3) (2003) 283-294.
[10] A. Jaurker, J. Saini, B. Gandhi, Heat transfer and friction characteristics of rectangular solar air heater duct using rib-grooved artificial roughness, Solar Energy, 80(8) (2006) 895-907.
[11] S.W. Chang, T.L. Yang, R.F. Huang, K.C. Sung, Influence of channel-height on heat transfer in rectangular channels with skewed ribs at different bleed conditions, International Journal of Heat and Mass Transfer, 50(23) (2007) 4581-4599.
[12] S.K. Saha, Thermal and friction characteristics of laminar flow through rectangular and square ducts with transverse ribs and wire coil inserts, Experimental Thermal and Fluid Science, 34(1) (2010) 63-72.
[13] S. Ray, D. Misra, Laminar fully developed flow through square and equilateral triangular ducts with rounded corners subjected to H1 and H2 boundary conditions, International Journal of Thermal Sciences, 49(9) (2010) 1763-1775.
[14] L.-z. Zhang, Z.-y. Chen, Convective heat transfer in cross-corrugated triangular ducts under uniform heat flux boundary conditions, International Journal of Heat and Mass Transfer, 54(1) (2011) 597-605.
[15] M.M. Shahmardan, M. Norouzi, M.H. Kayhani, A.A. Delouei, An exact analytical solution for convective heat transfer in rectangular ducts, Journal of Zhejiang University SCIENCE A, 13(10) (2012) 768-781.
[16] M. Shahmardan, M. Sedaghat, M. Norouzi, An analytical solution for fully developed forced convection in triangular ducts, Heat Transfer—Asian Research, 44(6) (2015) 489-498.
[17] M. Sayed-Ahmed, K.M. Kishk, Heat transfer for Herschel–Bulkley fluids in the entrance region of a rectangular duct, International Communications in Heat and Mass Transfer, 35(8) (2008) 1007-1016.
[18] M. Norouzi, M. Kayhani, M. Nobari, Mixed and forced convection of viscoelastic materials in straight duct with rectangular cross section, World Applied Sciences Journal, 7(3) (2009) 285-296.
[19] H. Claiborne, HEAT TRANSFER IN NONCIRCULAR DUCTS PART I, Oak Ridge National Lab., 1951.
[20] L. Tao, On some laminar forced-convection problems, Journal of Heat Transfer, 83(4) (1961) 466-472.
[21] C.-Y. Cheng, The effect of temperature-dependent viscosity on the natural convection heat transfer from a horizontal isothermal cylinder of elliptic cross section, International communications in heat and mass transfer, 33(8) (2006) 1021-1028.
[22] V. Sakalis, P. Hatzikonstantinou, N. Kafousias, Thermally developing flow in elliptic ducts with axially variable wall temperature distribution, International journal of heat and mass transfer, 45(1) (2002) 25-35.
[23] K. Velusamy, V.K. Garg, G. Vaidyanathan, Fully developed flow and heat transfer in semi-elliptical ducts, International journal of heat and fluid flow, 16(2) (1995) 145-152.
[24] K. Velusamy, V.K. Garg, Laminar mixed convection in vertical elliptic ducts, International journal of heat and mass transfer, 39(4) (1996) 745-752.
[25] V. Javeri, Analysis of laminar thermal entrance region of elliptical and rectangular channels with Kantorowich method, Heat and Mass Transfer, 9(2) (1976) 85-98.
[26] R. Abdel-Wahed, A. Attia, M. Hifni, Experiments on laminar flow and heat transfer in an elliptical duct, International journal of heat and mass transfer, 27(12) (1984) 2397-2413.
[27] http://en.wikipedia.org/wiki/Ellipse.
[28] W. Kays, M. Crawford, B. Weigand, Convective Heat and Mass Transfer-Four edition, in, Mc Graw-Hill Publishing Co. Ltd, 2005.
[29] T. Papanastasiou, G. Georgiou, A.N. Alexandrou, Viscous fluid flow, CRC Press, 1999.
AUT Journal of Mechanical Engineering
Volume 1, Issue 2
December 2017
Pages 131-138

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