Laminar Forced 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 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 ϩ␯ͩ 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 and dimensionless duct length. ϭ Dh . Later it will be shown that a more appropriate length scale Even for these common channel shapes, the expressions are only should be used. valid for particular boundary conditions and flow conditions. ͓ ͔ Only a limited set of numerical data are available for the com- Stephan, see 3 , developed a correlation for the circular tube bined entrance region. In addition to numerical data for the circu- which is valid for all values of the dimensionless duct length z* lar duct and the parallel plate channel for a range of Prandtl num- Ͻ Ͻϱ and for 0.1 Pr . bers, a small set of data are available for the rectangular and ͑ ͓ ͔͒ The Stephan correlation see 3 has the following form: triangle ducts for Prϭ0.72. All of the available data and models Nu͑Pr→ϱ͒ for the combined entrance region are reviewed by Kakac and ϭ ͓ ͔ Num,T ͑ 1/6͑ ͒1/6͒ (13) Yener 10 . tanh 2.432Pr z* Finally, numerical data were obtained by Asako et al. ͓6͔ for the where polygonal ducts for a range of Prandtl numbers, while Garimella et al. ͓5͔ obtained empirical data for the rectangular channel in the 3.657 0.0499 Nu͑Pr→ϱ͒ϭ ϩ tanh͑z*͒ laminar-transition-turbulent regions for a number of aspect ratios. tanh͑2.264͑z*͒1/3ϩ1.7͑z*͒2/3͒ z* At present, no single model is available which can predict the (14) data for both the circular duct or parallel plate channel. Further, no

Journal of Heat Transfer FEBRUARY 2004, Vol. 126 Õ 55 correlating equations are available for any of the numerical data related to non-circular ducts. In the sections which follow, a new model is proposed which serves these needs.

Model Development A model which is valid for most non-circular ducts will now be developed using an approach similar to that proposed by Churchill and Ozoe ͓1,2͔. Churchill and Ozoe ͓1,2͔ developed a model for the circular duct which is valid for all Prandtl numbers over the entire range of dimensionless duct lengths, by combining a com- posite model for the Graetz problem with a composite model for laminar forced convection from a flat plate. With some additional modification, Churchill and Ozoe ͓1,2͔ were able to apply these models for all Prandtl numbers. The present approach is limited to the range 0.1ϽPrϽϱ, which is valid for most flows found in heat exchangers. Fig. 1 f ReͱA for singly connected geometries, data from Ref. The model development will be presented in several steps. †7‡ First, each characteristic region: fully developed flow (L ӷ ӷ Ӷ Lh ,Lt), Graetz flow (L Lh ,L Lt), and laminar boundary Ӷ layer flow (L Lh ,Lt) will be examined in detail to show how the mean channel spacing divided by the mean channel width. each region is dealt with in non-circular ducts. Next a model is This alternative definition was chosen since these shapes contain developed for a range of Prandtl numbers, duct lengths, thermal the parallel plate channel limit when aspect ratio becomes small. boundary condition, and Nusselt number type, i.e., local or aver- Next, Muzychka and Yovanovich ͓18͔ applied the same reason- age value. Finally, the model is compared with published data ing to the fully developed Nusselt number in non-circular ducts, from numerous sources. Much of these data have been collected leading to the development of a model for the classic Graetz prob- and organized by Shah and London ͓7͔. Since much of the pub- lem. lished data appears in graphical form in the original sources, tabu- Figures 3 and 4 compare the data for many duct shapes ob- lated values from ͓7͔ are used for comparison. In most cases these tained from Shah and London ͓7͔. When the results are based data were obtained from the original authors, when available. upon the square root of cross-sectional area two distinct bounds are formed for the Nusselt number. The lower bound consists of Fully Developed Flow. The fully developed flow limit for all duct shapes which have re-entrant corners, i.e., angles less than both hydrodynamic and thermal problems has been addressed by 90 deg, while the upper bound consists of all ducts with rounded ͓ ͔ ͓ ͔ Muzychka 16 and reported in Yovanovich and Muzychka 17 corners and/or right angled corners. A model has been developed and Muzychka and Yovanovich ͓18,19͔. Additional results appear in Muzychka and Yovanovich ͓20,21͔ and Yovanovich et al. ͓22͔. These references ͓17–22͔ provide models for the classic Graetz problem and the hydrodynamic entrance problem for forced flow, and for natural convection in vertical ducts of non-circular cross- section. In addition, Refs. ͓20͔, ͓21͔ applied scaling principles to all of the fundamental internal flow problems. An important result of Yovanovich and Muzychka ͓17͔ is that the characteristic length scale for non-circular ducts in laminar flows should not be the hydraulic diameter, but rather, the square root of the cross-sectional area of the duct. This conclusion was drawn from dimensional analysis performed on an arbitrarily shaped duct with validation provided by examination of the ana- lytical and numerical data from the literature. Bejan ͓23͔ also arrived at the same conclusion using his constructal theory of organization in nature. Yovanovich and Muzychka ͓17͔ showed that when the friction factor- product is based upon the square root of cross-sectional area, the vast number of data were reduced to a Fig. 2 f ReͱA for other singly connected geometries, data from single curve which was merely a function of the aspect ratio of the Ref. †7‡ duct or channel. They also showed that this curve was accurately represented by the first term of the exact series solution for the rectangular duct cross-section. This result is given by 12 f Reͱ ϭ (21) A 192⑀ ␲ ͱ⑀͑1ϩ⑀͒ͫ1Ϫ tanhͩ ͪͬ ␲5 2⑀ For most rectangular channels it is sufficient to choose 0.01 Ͻ⑀Ͻ1, since the rectangular channel approaches the parallel plate for ⑀Ͻ0.01. A plot of this model with data for many duct shapes is provided in Fig. 1. Figure 2 shows a broader comparison with several other non-circular ducts. The aspect ratio in Figs. 1 and 2 is taken to be a measure of the slenderness of the non-circular duct. In most cases, it is merely the width to length ratio of the duct. However, in cases such as the circular annulus, annular sector, and the trapezoid, it is defined as Fig. 3 Fully developed flow NuT , data from Ref. †7‡

56 Õ Vol. 126, FEBRUARY 2004 Transactions of the ASME where L is an arbitrary length scale. If the following parameters are defined: y z/L w¯ L ϭ ϭ ␪ϭ Ϫ ϭ ¯y z* T To ReL L ReLPr ␯ the governing equation becomes 2␪ץ ␪ץ C*¯y ϭ (27) y 2¯ץ *zץ The governing equation may now be transformed into an ordi- nary differential equation for each wall condition using a similar- ity variable ͓22͔ ¯y ␩ϭ (28) Fig. 4 Fully developed flow NuH , data from Ref. †7‡ ͑9z*/C*͒1/3 Both the UWT and UWF conditions are examined. Solution to the ͓ ͔ which accurately predicts the data for both thermal boundary con- Leveque problem is discussed in Bird et al. 24 . The solution for ditions and both upper and lower bounds. The resulting expression local Nusselt number with the UWT condition yields: which is related to Eq. ͑21͒ is 1/3 1/3 3) ⌫͑2/3͒ f ReL f ReL ϭ ͩ ͪ Ϸ ͩ ͪ NuL 0.4273 (29) f ReͱA 2 ␲ 18z* z* Nuͱ ϭC ͩ ͪ (22) A 1 8ͱ␲⑀␥ while the solution for the local Nusselt number for the ͑UWF͒ ͑ ͒ where C1 is equal to 3.24 for the UWT boundary condition and condition yields: 3.86 for the ͑UWF͒ boundary condition. These results are the 1/3 1/3 f ReL f ReL exact solutions for fully developed flow in a circular tube when ϭ⌫͑ ͒ͩ ͪ Ϸ ͩ ͪ NuL 2/3 0.5167 (30) the characteristic length scale is the square root of cross-sectional 18z* z* area. The parameter ␥ is chosen based upon the geometry. Values for ␥ which define the upper and lower bounds in Figs. 3 and 4 are The average Nusselt number for both cases may be obtained fixed at ␥ϭ1/10 and ␥ϭϪ3/10, respectively. Almost all of the from Eq. ͑12͒, which gives available data are predicted within Ϯ10 percent by Eq. ͑22͒, with a few exceptions. 3 NuLϭ NuL (31) 2 Graetz Flow. If the velocity distribution is fully developed and the temperature distribution is allowed to develop, the classic The solution for each wall condition may now be compactly Graetz problem results. In the thermal entrance region, the results written as are weak functions of the shape and geometry of the duct. This 1/3 f ReL behavior is characterized by the following approximate analytical ϭ ͩ ͪ NuL C C (32) expression first attributed to Leveque, see ͓24͔: 2 3 z* 1/3 C* where the value of C is 1 for local conditions and 3/2 for average Nuϰͩ ͪ (23) 2 z* conditions, and C3 takes a value of 0.427 for UWT and 0.517 for UWF. where C* is the dimensionless mean velocity gradient at the duct The Leveque approximation is valid where the thermal bound- wall and z* is the dimensionless axial location. Thus, if C* is ary layer develops in the region near the wall where the velocity made a weak function of shape, then Nu will be a weaker function profile is linear. The weak effect of duct geometry in the entrance of shape due to the one third power. region is due to the presence of the friction factor-Reynolds num- In the thermal entrance region of non-circular ducts the thermal ber product, f Re, in the above expression, which is representative boundary layer is thin and it may be assumed to be developing in of the average velocity gradient at the duct wall. The typical range a region where the velocity gradient is linear. For very small dis- Ͻ Ͻ ͓ ͔ of the f Re group is 6.5 f ReD 24, Shah and London 7 . This tances from the duct inlet, the effect of curvature on the boundary h results in 1.87Ͻ( f Re )1/3Ͻ2.88, which illustrates the weak de- layer development is negligible. Thus, the we may treat the duct Dh wall as a flat plate. The governing equation for this situation is pendency of the thermal entrance region on shape and aspect ra- given by tio. Further reductions are achieved for similar shaped ducts by using the length scale LϭͱA, i.e., see Figs. 1 and 2. 2 T A model which is valid over the entire range of dimensionless ץ Tץ Cy ϭ␣ (24) -y 2 duct lengths for Pr→ϱ, was developed by Muzychka and Yoץ zץ vanovich ͓18͔ by combining Eq. ͑22͒ with Eq. ͑32͒ using the where the constant C represents the mean velocity gradient at the Churchill and Usagi ͓11͔ asymptotic correlation method. The form duct wall. For non-circular ducts, this constant is defined as: of the proposed model for an arbitrary characteristic length scale w isץ w 1ץ Cϭ ͯ ϭ Ͷ ͯ ds (25) n f Re 1/3 n 1/nץ n Pץ w w ͑ ͒ϭͩ ͭ ͩ ͪ ͮ ϩ͑ ͒n ͪ Nu z* C2C3 Nufd (33) For hydrodynamically fully developed flow, the constant C is z* related to the friction factor-Reynolds number product Now using the result for the fully developed friction factor, Eq. ͑ ͒ ,w L 21 , and the result for the fully developed flow Nusselt numberץ f ReL ϭ ͯ ϭC* (26) Eq. ͑22͒, with nϷ5 a new model ͓19͔ was proposed having the ¯n wץ 2 w form

Journal of Heat Transfer FEBRUARY 2004, Vol. 126 Õ 57 Table 1 Coefficients for general model

Boundary Condition

ϭ ϭ 0.564 UWT C3 0.409, C1 3.24 f͑Pr͒ϭ ͓1ϩ͑1.664Pr1/6͒9/2͔2/9 ϭ ϭ 0.886 UWF C3 0.501, C1 3.86 f͑Pr͒ϭ ͓1ϩ͑1.909Pr1/6͒9/2͔2/9 Nusselt Number Type Local C ϭ1 C ϭ1 2ϭ 4ϭ Average C2 3/2 C4 2 Shape Parameter Upper Bound ␥ϭ1/10 Lower Bound ␥ϭϪ3/10

1/3 5 5 1/5 The average Nusselt number for both cases may now be ob- f ReͱA f ReͱA Nuͱ ͑z*͒ϭͫͭ C C ͩ ͪ ͮ ϩͭ C ͩ ͪͮ ͬ tained from Eq. ͑12͒, which gives: A 2 3 1 z* 8ͱ␲⑀␥ (34) NuLϭ2NuL (40) ␥ where the constants C1 , C2 , C3 , and are given in Table 1. The solution for each wall condition may now be compactly These constants define the various cases for local or average Nus- written as selt number and isothermal or isoflux boundary conditions for the C Graetz problem. The constant C3 was adjusted from that found by 4 NuLϭ f ͑Pr͒ (41) the Leveque approximation to provide better agreement with the ͱz* data. ϭ ϭ where the value of C4 1 for local conditions and C4 2 for Laminar Boundary Layer Flow. Finally, if both hydrody- average conditions, and f (Pr) are defined as namic and thermal boundary layers develop simultaneously, the results are strong functions of the fluid Prandtl number. In the 0.564 f ͑Pr͒ϭ (42) combined entrance region the behavior for very small values of z* ͓1ϩ͑1.664Pr1/6͒9/2͔2/9 may be adequately modeled by treating the duct wall as a flat for the UWT condition, and plate. The characteristics of this region are 0.886 Nuz f ͑Pr͒ϭ (43) Pr→0 ϭ0.564Pr1/2 (35) ͓1ϩ͑1.909Pr1/6͒9/2͔2/9 ͱRe z for the UWF condition. The preceding results are valid only for Nu small values of the parameter z*. z 1/3 Pr→ϱ ͱ ϭ0.339Pr (36) Rez General Model. A new model for the combined entrance re- gion may now developed by combining the solution for a flat plate for the UWT condition ͓2͔, and with the model for the Graetz flow problem developed earlier. The proposed model takes the form Nuz Pr→0 ϭ0.886Pr1/2 (37) ͱ ͑ ͒ϭ͕͑ ͑ ͖͒mϩ͕ n ϩ n ͖m/n͒1/m Rez y z y z→0 Pr y z→0 y z→ϱ (44) which is similar to that proposed by Churchill and Ozoe ͓1,2͔ for Nu z 1/3 the circular duct. This model is a composite solution of the three Pr→ϱ ͱ ϭ0.464Pr (38) Rez asymptotic solutions just discussed. This results in the following model for simultaneously devel- for the UWF condition ͓1͔. oping flow in a duct of arbitrary cross-sectional shape Composite models for each wall condition were developed by Churchill and Ozoe ͓1,2͔ using the asymptotic correlation method ͒ m 1/3 5 C4 f ͑Pr f ReͱA of Churchill and Usagi ͓11͔. The results may be developed in Nuͱ ͑z*͒ϭͫͩ ͪ ϩͩ ͭ C C ͩ ͪ ͮ A 2 3 terms of the Pr→0 behavior or the Pr→ϱ behavior. For internal ͱz* z* flow problems, the appropriate form is chosen to be in terms of 5 m/5 1/m → f ReͱA the Pr 0 characteristic which introduces the Peclet number Pe ϩͭ C ͩ ͪͮ ͪ ͬ (45) 1 ϭRe Pr: 8ͱ␲⑀␥ Nu C z ϭ o ϭ ͑ ͒ The parameter m was determined to vary between 2 and 7 for ͒1/2 1/6 n 1/n f Pr (39) all of the data examined. Values for the blending parameter were ͑RezPr CoPr ͫ1ϩͩ ͪ ͬ found to be weak functions of the duct aspect ratio and whether a Cϱ local or average Nusselt number was examined. However, the where Co and Cϱ represent the coefficients of the right hand side blending parameter was found to be most dependent upon the of Eqs. ͑35–38͒. The correlation parameter n may be found by fluid Prandtl number. solving Eq. ͑39͒ at an intermediate value of Pr where the exact A simple linear approximation was determined to provide better solution is known, i.e., Prϭ1. This leads to nϭ4.537 for the UWT accuracy than choosing a single value for all duct shapes. Due to condition and nϭ4.598 for the UWF condition. For simplicity, the variation in geometries and data, higher order approximations nϭ9/2 is chosen for both cases. offered no additional advantage. Therefore, the linear approxima-

58 Õ Vol. 126, FEBRUARY 2004 Transactions of the ASME tion which predicts the blending parameter within 30 percent was Table 5 Comparison of model and data for rectangular duct Ä Õ found to be satisfactory. Variations in the blending parameter of †7‡:Pr 0.72, „min max—% diff… this order will lead to small errors in the model predictions, ⑀ϭb/a Nu Nu Nu whereas variations on the order of 100 percent or more, i.e., m,T z,H m,H choosing a fixed value, produce significantly larger errors. The 1 Ϫ20.7/1.21 Ϫ0.18/13.14 Ϫ3.75/14.8 resulting fit for the blending parameter m is 1/2 Ϫ18.5/7.48 Ϫ0.11/13.13 Ϫ2.42/14.6 1/3 Ϫ16.5/7.43 Ϫ0.05/13.4 Ϫ0.42/13.16 mϭ2.27ϩ1.65Pr1/3 (46) 1/4 Ϫ14.6/6.73 Ϫ0.04/12.04 Ϫ0.33/13.5 The above model is valid for 0.1ϽPrϽϱ which is typical for most low Reynolds number flow heat exchanger applications. Table 6 Comparison of model and data for equilateral triangu- lar ducts 7 minÕmax—% diff Finally, the following model was developed by Muzychka and † ‡„ … Yovanovich ͓18,20͔ for the apparent friction factor in the entrance Pr Nuz,T Num,T Nuz,H Num,H region. It is given here for completeness as follows: 0.72 Ϫ14.17/Ϫ1.85 Ϫ11.1/12.27 Ϫ1.50/4.19 Ϫ8.92/4.84 2 2 1/2 12 3.44 ϱ Ϫ12.11/Ϫ6.81 Ϫ7.24/Ϫ2.09 f Reͱ ϭ ϩͩ ͪ app A ͫͩ 192⑀ ␲ ͪ ͱ ϩ ͬ z ͱ⑀ ͑1ϩ⑀͒ͫ1Ϫ tanhͩ ͪͬ ␲5 2⑀ (47) It predicts most of the non-circular friction data within Ϯ10 percent.

Comparisons of Model With Data Comparisons with the available data from ͓7͔ are provided in Tables 2–6 and Figs. 5–11. Good agreement is obtained with the data for the circular duct and parallel plate channel. Note that comparison of the model for the parallel plate channel was ob- tained by considering a rectangular duct having an aspect ratio of ⑀ϭ0.01. This represents a reasonable approximation for this sys- tem. The data are also compared with the models of Churchill and Ozoe ͓1,2͔ and Stephan ͓3,4͔ for the circular duct and parallel plate channel in Figs. 5–8. Fig. 5 Simultaneously developing flow in a circular tube The model is compared with the data by determining the per- NuH,z , data from Ref. 7 cent difference between the data and the model predictions. Maxi- † ‡ mum and minimum values of the percent difference are given in Tables 2–6 for each set of data. The numerical data for the the UWT circular duct fall short of the model predictions at low Pr numbers. However, all of the models are in excellent agreement with the integral formulation of

Table 2 Comparison of model and data for circular duct 7 Õ † ‡ „min max—% diff…

Pr Nuz,T Num,T Nuz,H 0.7 Ϫ6.59/Ϫ0.07 Ϫ14.69/3.87 Ϫ0.08/3.98 2.0 Ϫ8.43/Ϫ0.07 Ϫ19.07/1.34 Ϫ0.07/5.19 5.0 Ϫ8.65/Ϫ0.07 Ϫ24.41/Ϫ0.60 Ϫ0.20/5.40 ϱ Ϫ1.24/8.57 Ϫ1.60/5.27

Fig. 6 Simultaneously developing flow in a circular tube Nu , data from Ref. 7 Table 3 Comparison of model and data for parallel plate chan- T,z † ‡ Õ nel †7‡„min max—% diff…

Pr Num,T Nuz,H 0.1 Ϫ13.33/6.32 N/A 0.7 Ϫ5.17/7.43 Ϫ1.59/5.32 2.0 Ϫ0.24/7.43 Ϫ1.79/4.49 5.0 Ϫ0.02/13.75 Ϫ0.05/6.01 ϱ 1.60/10.0 Ϫ9.93/7.03

Table 4 Comparison of model and data for square duct 7 Õ † ‡ „min max—% diff…

Pr Nuz,H Num,H 0.1 Ϫ0.45/11.32 Ϫ8.14/11.42 1.0 Ϫ2.19/2.56 Ϫ6.72/4.82 10 Ϫ1.96/1.52 Ϫ1.15/5.07 ϱ Ϫ2.76/1.73 Fig. 7 Simultaneously developing flow in a circular tube NuT,m , data from Ref. †7‡

Journal of Heat Transfer FEBRUARY 2004, Vol. 126 Õ 59 Fig. 8 Simultaneously developing flow in channel NuT,m , data Fig. 11 Simultaneously developing flow in square duct NuH,m , from Ref. †7‡ data from Ref. †7‡

verse velocities with the data of Wibulswas ͓25͔ for Prϭ0.72 Kreith ͓14͔. Good agreement is also obtained for the case of the shows that the discrepancy is likely due to the data and not the square duct for all Prandtl numbers. Comparisons of the model model. with data for the rectangular duct at various aspect ratios, see The accuracy for each case may be improved considerably by Table 5, and the equilateral triangular duct, see Table 6, show using the optimal value of the parameter m. However, this intro- that larger discrepancies arise. Also included in Tables 2–6 are the duces an additional parameter into the model which is deemed results of Muzychka and Yovanovich ͓19͔ for the case of unnecessary for purposes of heat exchanger design. The proposed Pr→ϱ. model predicts most of the available data for the combined entry The data used for comparison in Tables 5 and 6 were obtained problem to within Ϯ15 percent and may be used to predict the by Wibulswas ͓25͔. In this work, the effects of transverse veloci- heat transfer characteristics for other non-circular ducts for which ties in both the momentum and energy equations were ignored. there are presently no data. Comparison of the data for the square duct at Prϭ1.0 obtained by The present model also agrees well with the published models Chandrupatla and Sastri ͓26͔ which includes the effects of trans- of Churchill and Ozoe ͓1,2͔, Eqs. ͑19,20͒, and the models of Stephan ͓3,4͔, Eqs. ͑13,16͒, for the tube and channel. It is evident from Figs. 5–8 that the present model provides equal or better accuracy to the existing expressions.

Summary and Conclusions A general model for predicting the heat transfer co-efficient in the combined entry region of non-circular ducts was developed. This model is valid for 0.1ϽPrϽϱ,0Ͻz*Ͻϱ, both uniform wall temperature ͑UWT͒ and uniform wall flux ͑UWF͒ conditions, and for local and mean Nusselt numbers. Model predictions agree with numerical data to within Ϯ15 percent for most non-circular ducts and channels. The model was developed by combining the asymptotic results of laminar boundary layer flow and Graetz flow for the thermal entrance region. In addition, by means of a novel characteristic length, the square root of cross-sectional area, re- sults for many non-circular ducts of similar aspect ratio collapse onto a single curve. Fig. 9 Simultaneously developing flow in channel NuH,z , data from Ref. 7 † ‡ Acknowledgments The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada ͑NSERC͒ under a post-doctoral research fellowship for the first author and research grants for both authors.

Nomenclature A ϭ flow area, m2 a ϭ major axis of ellipse or rectangle, m b ϭ minor axis of ellipse or rectangle, m C ϭ constant ϭ ϭ Ci constants, i 1..5 D ϭ diameter of circular duct, m ϭ ϵ Dh hydraulic diameter of plain channel, 4A/P f ϭ friction factor ϵ¯␶/(1/2␳w¯ 2) Gz ϭ ϵ␲ * Fig. 10 Simultaneously developing flow in square duct NuH,z , Graetz number, /4z h ϭ 2 data from Ref. †7‡ heat transfer coefficient, W/m K

60 Õ Vol. 126, FEBRUARY 2004 Transactions of the ASME k ϭ thermal conductivity, W/mK ͓5͔ Garimella, S., Dowling, W. J., Van derVeen, M., Killion, J., 2000, ‘‘Heat L ϭ length of channel, m Transfer Coefficients for Simultaneously Developing Flow in Rectangular ϭ Tubes,’’ Proceedings of the 2000 International Mechanical Engineering Con- Lh hydrodynamic entry length, m gress and Exposition, Vol. 2, pp. 3–11. ϭ Lt thermal entry length, m ͓6͔ Asako, Y., Nakamura, H., and Faghri, M., 1988, ‘‘Developing Laminar Flow L* ϭ dimensionless thermal length, ϵL/L ReLPr and Heat Transfer in the Entrance Region of Regular Polygonal Ducts,’’ Int. J. L ϭ characteristic length scale, m Heat Mass Transfer, 31, pp. 2590–2593. ϭ ͓7͔ Shah, R. K., and London, A. L., 1978, Laminar Flow Forced Convection in m correlation parameter Ducts, Academic Press, New York, NY. m˙ ϭ mass flow rate, kg/s ͓8͔ Kakac, S., Shah, R. K., and Aung, W., 1987, Handbook of Single Phase Con- n ϭ inward directed normal vective Heat Transfer, Wiley, New York. ͓ ͔ NuL ϭ Nusselt number, ϵhL/k 9 Rohsenow, W. M., Hartnett, J. P., and Cho, Y. I., eds., 1988, Handbook of Heat ϭ Transfer, McGraw-Hill, New York. P perimeter, m ͓10͔ Kakac, S., and Yener, Y., 1983, ‘‘Laminar Forced Convection in the Combined p ϭ pressure, Pa Entrance Region of Ducts,’’ in Low Reynolds Number Heat Exchangers,S. Pr ϭ Prandtl number, ϵ␯/␣ Kakac, R. K. Shah and A. E. Bergles, eds., Hemisphere Publishing, Washing- q ϭ heat flux, W/m2 ton, pp. 165–204. ͓ ͔ Q ϭ heat transfer rate, W 11 Churchill, S. W., and Usagi, R., 1972, ‘‘A General Expression for the Corre- ϭ lation of Rates of Transfer and Other Phenomena,’’ American Institute of r radius, m Chemical Engineers, 18, pp. 1121–1128. ReL ϭ Reynolds number, ϵw¯ L/␯ ͓12͔ Kays, W. M., and Crawford, M. E., 1993, Convective Heat and Mass Transfer, s ϭ arc length, m McGraw-Hill, New York, NY. T ϭ temperature, K ͓13͔ Kays, W. M., 1955, ‘‘Numerical Solutions for Laminar Flow Heat Transfer in ϭ Circular Tubes,’’ Trans. ASME, 77, pp. 1265–1274. Tm bulk temperature, K ͓ ͔ nd ϭ 14 Kreith, F., 1965, Principles of Heat Transfer,2 ed., International Textbook Tw wall temperature, K Co., Scranton, PA. u, v, w ϭ velocity components, m/s ͓15͔ Sparrow, E. M., 1955, ‘‘Analysis of Laminar Forced Convection Heat Transfer w ϭ axial velocity, m/s in Entrance Region of Flat Rectangular Ducts,’’ NACA Technical Note 3331. w¯ ϭ average velocity, m/s ͓16͔ Muzychka, Y. S., 1999, ‘‘Analytical and Experimental Study of Fluid Friction ϭ and Heat Transfer in Low Reynolds Number Flow Heat Exchangers,’’ Ph.D. x, y, z cartesian coordinates, m thesis, Uinversity of Waterloo, Waterloo, ON. z ϭ axial coordinate, m ͓17͔ Yovanovich, M. M., and Muzychka, Y. S., 1997, ‘‘Solutions of Poisson Equa- ϩ z ϭ dimensionless position for hydrodynamically de- tion within Singly and Doubly Connected Domains,’’ AIAA Paper 97-3880, veloping flows, ϵz/L ReL presented at the National Heat Transfer Conference, Baltimore MD. ͓ ͔ z* ϭ dimensionless position for thermally developing 18 Muzychka, Y. S., and Yovanovich, M. M., 1998, ‘‘Modeling Fricton Factors in ϵ L Non-Circular Ducts for Developing Laminar Flow,’’ AIAA Paper 98-2492, flows, z/ ReLPr presented at the 2nd Theoretical Fluid Mechanics Meeting, Albuquerque, NM. Greek Symbols ͓19͔ Muzychka, Y. S., and Yovanovich, M. M., 1998, ‘‘Modeling Nusselt Numbers for Thermally Developing Laminar Flow in Non-Circular Ducts,’’ AIAA Paper ␣ ϭ thermal diffusivity, m2/s 98-2586, presented at the 7th AIAA/ASME Joint Thermophysics and Heat ⑀ ϭ aspect ratio, ϵb/a Transfer Conference, Albuquerque, NM. ͓ ͔ ␥ ϭ shape parameter 20 Muzychka, Y. S., and Yovanovich, M. M., 2002, ‘‘Laminar Flow Friction and ⌫ ϭ Heat Transfer in Non-Circular Ducts and Channels: Part I—Hydrodynamic (•) Gamma function Problem,’’ Compact Heat Exchangers: A Festschrift on the 60th Birthday of ␮ ϭ dynamic viscosity, Ns/m2 Ramesh K. Shah, Grenoble, France, August 24, 2002, G. P. Celata, B. Thonon, ␯ ϭ 2 A. Bontemps, and S. Kandlikar, eds., pp. 123–130. kinematic viscosity, m /s ͓ ͔ ␳ ϭ 3 21 Muzychka, Y. S., and Yovanovich, M. M., 2002, ‘‘Laminar Flow Friction and fluid density, kg/m Heat Transfer in Non-Circular Ducts and Channels: Part II—Thermal Prob- 2 ␶ ϭ wall shear stress, N/m lem,’’ Compact Heat Exchangers: A Festschrift on the 60th Birthday of ␪ ϭ Ϫ Ramesh K. Shah, Grenoble, France, August 24, 2002, G. P. Celata, B. Thonon, temperature excess, T Tb , K A. Bontemps, and S. Kandlikar, eds., pp. 131–139. ͓22͔ Yovanovich, M. M., Teertstra, P. M., and Muzychka, Y. S., 2001, ‘‘Natural References Convection Inside Vertical Isothermal Ducts of Constant Arbitrary Cross- th ͓1͔ Churchill, S. W., and Ozoe, H., 1973, ‘‘Correlations for Laminar Forced Con- Section,’’ AIAA Paper 01-0368, presented at the 39 Aerospace Sciences vection with Uniform Heating in Flow Over a Plate and in Developing and Meeting and Exhibit, Reno, NV, January 8–11. Fully Developed Flow in a Tube,’’ ASME J. Heat Transfer, 95, pp. 78–84. ͓23͔ Bejan, A., 2000, Shape and Structure, From Engineering to Nature, Cambridge ͓2͔ Churchill, S. W., and Ozoe, H., 1973, ‘‘Correlations for Laminar Forced Con- University Press, Cambridge, UK. vection in Flow Over an Isothermal Flat Plate and in Developing and Fully ͓24͔ Bird, R. B, Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena, Developed Flow in an Isothermal Tube,’’ ASME J. Heat Transfer, 95, pp. Wiley, New York, NY. 416–419. ͓25͔ Wibulswas, P., 1966, ‘‘Laminar Flow Heat Transfer in Noncircular Ducts,’’ ͓3͔ Baehr, H., and Stephan, K., 1998, Heat Transfer, Springer-Verlag. Ph.D. thesis, London University. ͓4͔ Stephan, K., 1959, ‘‘Warmeubergang und Druckabfall bei Nicht Ausgebildeter ͓26͔ Chandrupatla, A. R., and Sastri, V. M. K., 1977, ‘‘Laminar Forced Convection Laminar Stromung in Rohren und in Ebenen Spalten,’’ Chem-Ing-Tech, 31, pp. Heat Transfer of a Non-Newtonian Fluid in a Square Duct,’’ Int. J. Heat Mass 773–778. Transfer, 20, pp. 1315–1324.

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