Laminar Forced Convection Heat Transfer in the Combined Entry Y
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Laminar Forced Convection Heat Transfer in the Combined Entry Y. S. Muzychka Asst. Prof. Region of Non-Circular Ducts Faculty of Engineering and Applied Science, Memorial University of Newfoundland, A new model for predicting Nusselt numbers in the combined entrance region of non- St. John’s, NF, Canada, A1B 3X5 circular ducts and channels is developed. This model predicts both local and average Nusselt numbers and is valid for both isothermal and isoflux boundary conditions. The M. M. Yovanovich model is developed using the asymptotic results for convection from a flat plate, thermally Distinguished Professor Emeritus, developing flows in non-circular ducts, and fully developed flow in non-circular ducts. Fellow ASME Through the use of a novel characteristic length scale, the square root of cross-sectional Department of Mechanical Engineering, area, the effect of duct shape on Nusselt number is minimized. Comparisons are made University of Waterloo, with several existing models for the circular tube and parallel plate channel and with Waterloo, ON, Canada, N2L 3G1 numerical data for several non-circular ducts. Agreement between the proposed model and numerical data is within Ϯ15 percent or better for most duct shapes. ͓DOI: 10.1115/1.1643752͔ Keywords: Forced Convection, Heat Transfer, Heat Exchangers, Internal, Laminar, Modeling Introduction Combined Entry, 0ËPrËؕ. In cartesian coordinates the governing equations for an incompressible and constant property Heat transfer in the combined entry region of non-circular ducts fluid in the combined thermal entrance region are ͓7͔: is of particular interest in the design of compact heat exchangers. wץ vץ uץ In these applications passages are generally short and usually composed of cross-sections such as triangular or rectangular ge- ϩ ϩ ϭ0 (1) zץ yץ xץ ometries in addition to the circular tube or parallel plate channel. 2wץ 2wץ w 1 dpץ wץ wץ Also, due to the wide range of applications, fluid Prandtl numbers usually vary between 0.1ϽPrϽ1000, which covers a wide range ϩ ϩ ϭϪ ϩͩ ϩ ͪ (2) 2 ץ 2 ץ ץ w ץ v ץ u of fluids encompassing gases and highly viscous liquids such as x y z dz x y automotive oils. 2Tץ 2Tץ Tץ Tץ Tץ A review of the literature reveals that the only models available u ϩv ϩw ϭ␣ͩ ϩ ͪ (3) y 2ץ x2ץ zץ yץ xץ for predicting heat transfer in the combined entry region are those of Churchill and Ozoe ͓1,2͔ for the circular duct and Baehr and The pressure gradient may be written Stephan ͓3͔ and Stephan ͓4͔ for the circular duct and parallel plate channel. Recently, Garimella et al. ͓5͔ developed empirical ex- 1 dp dwc pressions for the rectangular channel, while numerical data for Ϫ ϭw (4) dz c dz polygonal ducts were obtained by Asako et al. ͓6͔. Additional data ϭ for the rectangular, circular, triangular, and parallel plate channel where wc wc(z) is the velocity of the inviscid core. The above ͓ ͔ ͓ ͔ ϭ are available in Shah and London 7 , Kakac et al. 8 , Rohsenow equations are subject to the no slip condition wwall 0, the bound- et al. ͓9͔, and Kakac and Yener ͓10͔. A complete review of design edness condition w(x,y,z)Þϱ along the duct axis, and the initial correlations is presented later. In addition, data are reported for the condition w(x,y,0)ϭw¯ . In cartesian coordinates, one additional circular annulus in ͓7͔. The models reported in this paper are equation is required to relate the two components of transverse applicable to the annulus, but only for the special cases where velocity. To date, very few solutions to this set of equations, Eqs. both surfaces of the annulus are at the same wall temperature or ͑1–4͒, have been obtained. wall flux condition. The present work will develop a new model using the Churchill Fully Developed Hydrodynamic Flow, Pr\ؕ. In cartesian and Usagi ͓11͔ asymptotic correlation method. In this method, the coordinates the governing equation for fully developed laminar special asymptotic solutions of the combined entry problem are flow in a constant cross-sectional area duct is used to develop a more general model for predicting heat transfer 2w 1 dpץ 2wץ coefficients in non-circular ducts. ϩ ϭ (5) y 2 dzץ x2ץ Governing Equations which represents a balance between the pressure and viscous In order to fully appreciate the complexity of the combined forces. entry problem, the governing equations for each of the three fun- If the velocity field develops quickly, then the energy equation damental forced covection problems are reviewed. These are as in cartesian coordinates for thermally developing laminar flow in follows: combined entry or simultaneously developing flow, the ducts of constant cross-sectional area is given by thermal entrance problem or Graetz flow, and thermally fully de- ץ 2ץ 2ץ veloped flow. T T w T ϩ ϭ (6) zץ ␣ y 2ץ x2ץ Contributed by the Heat Transfer Division for publication in the JOURNAL OF ϭ HEAT TRANSFER. Manuscript received by the Heat Transfer Division March 14, where w w(x,y) is the fully developed velocity profile. This is 2003; revision received October 2, 2003. Associate Editor: N. K. Anand. the classic Graetz problem for a non-circular duct. 54 Õ Vol. 126, FEBRUARY 2004 Copyright © 2004 by ASME Transactions of the ASME When the flow becomes thermally fully developed the energy A solution for the circular tube based upon the Karman- equation may be written in terms of the mixing cup temperature Pohlhausen integral method was first developed by Kays ͓13͔ and ͓ ͔ ͓ ͔ Tm(z), Kays and Crawford 12 later corrected by Kreith 14 . It is given by the following expres- :*sion which is only valid for small values of the parameter z 2ץ 2ץ T T w dTm ϩ ϭ (7) y 2 ␣ dz 1 1ץ x2ץ ϭ ͩ ͪ Num,T ln 1/2 Ϫ1/6 (15) for the uniform wall flux ͑UWF͒ condition, and 4z* 1Ϫ2.65͑z*͒ Pr Ϫ 2ץ 2ץ T T w Tw T dTm which is valid for PrϾ2 and z*Ͻ0.001. ϩ ϭ ͩ ͪ (8) ͓ ͔ y 2 ␣ T ϪT dz For the case of a parallel plate channel, Stephan 4 correlatedץ x2ץ w m numerical results in the following way: for the uniform wall temperature ͑UWT͒ condition, where 0.024͑z*͒Ϫ1.14 1 ϭ ϩ ͑ ͒ϭ ͵͵ Num,T 7.55 ϩ 0.17͑ ͒Ϫ0.64 (16) Tm z wTdA (9) 1 0.0358Pr z* w¯ A A which is valid for 0.1ϽPrϽ1000. Shah and Bhatti, see Ref. ͓8͔, Later, a model is developed which utilizes a number of limiting obtained the following expression for the local Nusselt number approximate solutions to these equations. These are as follows: from the correlation developed by Stephan ͓4͔ fully developed flow, thermally developing flow, and laminar boundary layer flow. 0.024͑z*͒Ϫ1.14͑0.0179Pr0.17͑z*͒Ϫ0.64Ϫ0.14͒ A dimensionless heat transfer coefficient or Nusselt number Nu ϭ7.55ϩ z,T ͑ ϩ 0.17͑ *͒Ϫ0.64͒2 may be defined as 1 0.0358Pr z (17) ͑ ͒ L L ¯qw z h Sparrow ͓15͔ using the Karman-Pohlhausen integral method, NuLϭ ϭ (10) k͑¯T ͑z͒ϪT ͑z͒͒ k obtained the following approximate analytical expression for the w m Nusselt number ¯ where Tm(z) is the bulk fluid temperature, Tw(z) is the average wall temperature, and ¯qw(z) is the average wall heat flux at any 0.664 ϭ Nu ϭ ͑1ϩ6.27͑Prz*͒4/9͒1/2 (18) point along the duct. For UWT, Tw Constant and for UWF, qw m,T ͱ 1/6 ϭConstant. z*Pr ͑ ͒ In terms of the solutions to Eqs. 1–4 , the Nusselt number, which is valid for PrϾ2 and z*Ͻ0.001. NuL , may be defined as follows: Churchill and Ozoe ͓1,2͔ developed Eqs. ͑19,20͒, for the local T Nusselt number for the UWF and UWT conditions, where Gzץ 1 Ͷ Ϫ ͯ ds ϭ n /(4 z*) is the Graetz number, in the combined entrance regionץ P w of a circular duct. These models were developed using the NuLϭL (11) 1 1 asymptotic correlation method of Churchill and Usagi ͓11͔ and are Ͷ T dsϪ ͵͵wTdA P w w¯ A valid for all Prandtl numbers 0ϽPrϽϱ, but only for the circular A duct. n represents the temperature gradient at the duct wallץ/Tץ where ϩ with respect to an inward directed normal, ds is the differential of Nuz,T 1.7 arc length, L is an arbitrary characteristic length scale to be de- 5.357͓1ϩ͑Gz/97͒8/9͔3/8 termined later, A is the cross-sectional area and P is the wetted 4/3 3/8 perimeter of the duct. Traditionally, Lϭ4 A/P, the hydraulic di- Gz/71 ϭͫ1ϩͩ ͪ ͬ ameter of the duct. Finally, the flow length averaged Nusselt num- ͓1ϩ͑Pr/0.0468͒2/3͔1/2͓1ϩ͑Gz/97͒8/9͔3/4 ber is related to the local Nusselt number through (19) 1 z ϭ ͵ ͑ ͒ ϩ Nu Nu z dz (12) Nuz,H 1 z 0 5.364͓1ϩ͑Gz/55͒10/9͔3/10 Literature Review Gz/28.8 5/3 3/10 ϭͫ1ϩͩ ͪ ͬ A review of the literature reveals that very little work has been ͓1ϩ͑Pr/0.0207͒2/3͔1/2͓1ϩ͑Gz/55͒10/9͔3/5 done in the area of modeling heat transfer in the combined en- trance region of non-circular ducts. Only the circular duct and (20) parallel plate channel have models or correlations which cover a In all of the above models, the characteristic length is L wide range of Prandtl number and dimensionless duct length.