Journal of Thermal Science and Technology

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1 Bulletin of the JSME Vol.13, No.1, 2018 Journal of Thermal Science and Technology Effect of geometrical parameters on turbulent flow and heat transfer behaviors in triple-start corrugated tubes Pitak PROMTHAISONG*, Withada JEDSADARATANACHAI** and Smith EIAMSA-ARD* *Department of Mechanical Engineering, Faculty of Engineering Mahanakorn University of Technology, Bangkok, Thailand **Department of Mechanical Engineering, Faculty of Engineering King Mongkut s Institute of Technology Ladkrabang, Bangkok, Thailand Received: 2 August 2017; Revised: 11 December 2017; Accepted: 20 February 2018 Abstract Computational results of flow structure, pressure loss and heat transfer characteristics in triple-start corrugated tubes are reported. The influences of the depth ratio (DR = 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14 and 0.16) and pitch ratio (PR = 0.5, 0.65, 0.75, 1.0, 1.5 and 2.0) were investigated in turbulent flow regime, Re = 5000 to 20,000. The computational results indicated that the triple-start corrugated tubes generate main swirl flow and helical swirl flow which helps to reduce the thermal boundary layer thickness and enhance the heat transfer rate. The flow and heat transfer become under fully developed periodic condition around x/d = 6.0. The friction factor monotonically increases with the rise of DR values and decrease PR values while maximum heat transfer rate is found at DR = 0.08 and PR = Nusselt numbers and friction factors of triple-start corrugated tubes in the investigated range are found to be 0.8 to 2.31 and 1.0 to times over those of the straight circular smooth tube, respectively. For the range studied, the triple-start corrugated tube with DR = 0.06 and PR = 0.75 offers the maximum thermal enhancement factor of 1.21 at Re = Keywords : Heat transfer, Heat transfer enhancement, Pressure loss, Swirl flow, Triple-start corrugated tube 1. Introduction Overall performance of a heat exchanger can be enhanced by improving heat transfer in the tube heat exchanger (Song and Nishino, 2008). The creation of swirling flow to disturb a boundary layer is one of the techniques for improving heat transfer in tube heat exchangers. Swirling and vortexing flows can be induced by using tube inserts such as; twisted tape (Song et al, 2008; Kim et al, 2009; Thianpong et al., 2012; Nanan and Eiamsa ard, 2014; Changcharoen et al., 2015; Eiamsa-ard et al., 2013; Eiamsa-ard et al., 2009), woven wire screen matrix (Zhao et al., 2011), helical screw tape (Guo et al., 2010; Sivashanmugam and Suresh, 2007) and conical spring (Karakaya and Durmuş, 2013) and modified tubes including corrugated/twisted tube and helical oval tube (Li et al., 2011, Rainieri and Pagliarini, 2002 ; Akhavan-Behabad and Esmailpour, 2014), etc. In general, modified tubes offer lower heat transfer than tube inserts. However, the major advantage of modified tube is causing low friction loss penalty which is an important key for achieving high/reasonable overall thermal performance. The geometry and parameters of the modified tubes (twisted elliptical tube, twisted tube, twisted oval tube, single-start corrugated tube, helically corrugated tube) play an important role in controlling thermal performance as reported by several researchers. Li et al. (2011) employed single-start corrugated tube with roughness heights (e) of 0.11, 0.21, 0.37, 0.68 mm, roughness widths (w) of 1.6, 1.7, 1.8 mm and roughness pitches (p) of 5 mm. They found that Nusselt number monotonically increased with increasing roughness height. However, the maximum heat transfer enhancement at the same pumping power was achieved at roughness height of Rainieri and Pagliarini (2002) carried out the experiments to compare the effect of helical and transverse corrugated tubes with e = 1.5 mm and p = 16, 32, 48, 64 mm on heat transfer enhancement. Their results showed that transverse corrugated tubes were more efficient in enhancing the rate of heat transfer. Akhavan-Behabadi and Esmailpour (2014) used the single-start corrugated Paper No

2 tube with seven different tube inclinations ranging from -90 to +90 to study evaporation of R-134a. It was found that the one with the inclination of + 90 offered the highest heat transfer coefficient. Naphon et al. (2006) experimentally studied heat transfer and friction loss in the helical corrugated tube with rib height to diameter ratios (x/d i) = 0.12, 0.15, 0.19 and helical rib pitch to diameter (p/d i) = 0.63, 0.78 and They found that both Nusselt number and friction factor increased with increasing rib height and decreasing rib pitch. Rabienataj Darzi et al. (2012) studied the turbulent flow and heat transfer characteristics in the helically corrugated tube (single-start corrugated tube). The helically corrugated tube with e/d i = , , and p/d i =0.618, 0.862, were employed. Their experimental results showed that the heat transfer increased by 260% when compared with the smooth tube. The use of helically corrugated tube in the studied range gave thermal enhancement factors ranging from 1.15 to Yang et al. (2011) investigated the effect of twisted elliptical tube on convective heat transfer and flow characteristics compared with those of the smooth tube. The effect of aspect ratio and twist pitch were also studied. The results showed that the vortex flows induced in the transverse plan enhanced the mass transfer between tube core region and near wall zone. In turbulent regime, Nusselt number and friction factor increased to 1.4 to 2.8 and 2.4 to 3.5 times above the smooth tube, respectively. The thermal enhancement factors ranging 1.0 to 2.5 were achieved. Zhang et al. (2012) presented the effect of horizontal twisted elliptical tube on condensation heat transfer characteristics. It was found that the use of horizontal twisted elliptical tube could increase the heat transfer condensation coefficient up to 34% over the smooth tube. Tan et al. (2012, 2013a, 2013b) experimentally and numerically studied heat transfer and flow in twisted oval tubes. The results indicated that heat transfer coefficient and friction factor both decreased with increasing of twist pitch length. In addition, twisted oval tube was promising for the use at low tube side flow rate and high shell side flow rate. As the literature review shown above focused on the effect of geometries of single spirally corrugated tube on heat transfer enhancement and thermal performance characteristics. Recently, the tubes with more numbers of spirally starts (2, 3 and 6 starts) were proposed for better heat transfer rate and thermal performance. Eiamsa-ard et al. (2016) studied the effect of combined three-start spirally twisted tube with triple-channel twisted tapes with four tape width ratios (w/d = 0.1, 0.25, 0.34 and 0.5) on the flow and thermal mechanism, thermal performance characteristics. The triple-channel twisted tapes were equipped with the twisted tube in two different arrangements: belly-to-belly and belly-to-neck. The results revealed that the equipment in belly-to-neck arrangement gave higher thermal performance than the one in bellyto-belly arrangement. Jing et al. (2017) studied the effect of pitch and corrugation depth ratios of spirally corrugated tubes with six-starts on heat transfer enhancement. They found that increasing of corrugation depth strengthened secondary flow and longitudinal vortex and helped to increase heat transfer rate while increasing of pitch resulted in the opposite trend. Wang et al. (2017) investigated the effect of corrugation height and pitch ratios of helically corrugated tubes on heat transfer characteristics. They recommended that the corrugation height ratio should be less than 0.1 while the pitch ratio should be less than 2.0 to ensure that the growth rate of the heat transfer was higher than the growth rate of the pressure drop (flow friction). Haervig et al. (2017) studied the heat transfer and flow field behaviors in the sinusoidally, spirally corrugated tubes with different corrugation height ratios (0-0.16) and corrugation pitch ratios (0-2.0). Their results showed that at the low corrugation heights, only a weak secondary flow centred in the corrugated section was found while at the higher corrugations heights, the tangential velocity component increased with a small increase in heat transfer and higher pressure drop. Hyder (2017) investigated the heat transfer and pressure loss behaviors in corrugated tubes with different spiral shapes under low Reynolds numbers between 100 and They found that the heat transfer rate, friction factor and thermal performance of the corrugated tube were enhanced up to 3.7, 2.3 times and 3.4 times as compared to those of the smooth tube. The above studies indicate that helical rough surface tubes are promising in enhancing heat transfer. However, the flow structure and heat transfer characteristics in helical rough surface tubes are seldom reported. Therefore, the present work proposes the numerical study of the effect of triple-start corrugated tube heat exchangers with various depth ratios (e/d = DR = 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14 and 0.16) and pitch ratios (p/d, PR = 0.5, 0.65, 0.75, 1.0, 1.5 and 2.0) in order to achieve more information of heat transfer and flow behaviors. 2. Mathematical foundation The simulations of flow and heat transfer characteristics in the triple-start corrugated tubes are governed by the continuity equation, the Navier-Stokes equation and the energy equation. These equations are applied with three- 2

3 Promthaisong, Jedsadaratanachai and Eiamsa-ard, dimensional, steady and incompressible flow while natural convection, viscous dissipation, body forces and radiation heat transfer are ignored. These equations can be written in the Cartesian tensor system as follows: Continuity equation: u i 0 xi (1) Momentum equation: p u i ui u j u 'i u 'j x j xi x j x j (2) Energy equation: ui T t T xi x j x j (4) where and t are molecular thermal diffusivity and turbulent thermal diffusivity, respectively. The diffusivities can be written as and (5) t t Pr Prt where u u j 2 k t uk u'i u'j t i x j xi 3 xk ij (6) The realizable k-ε turbulent model used in the present paper which were proposed by Shih et al. (1995), can be expressed as k ku j t x j x j k t G k G b S k k x j (8) and u j t x j x j t 2 C1 C3 Gb S C1 S C2 k k x j (9) where k 1 u j ui C1 max 0.43,, S, S 2Sij Sij, Sij 2 x x 5 j i (10) the model constants are given as follows: k 1.0, 1.2, C1 1.44, C2 1.9 The SIMPLE algorithm was used to resolve the flow and pressure equation based on finite volume approach (Patankar, 1980). The discretization scheme for resolved all of the governing equations is the QUICK method. The energy equation and other are considered to be converged when the normalized residual values are less than 10-9 and 10-5, respectively. Fig. 1. Triple start corrugated tube geometry for (a) full length and (b) periodic module Fig. 2. Computational domain for (a) full length and (b) periodic module 3 2

4 3. Triple-start corrugated tubes geometry Triple-start corrugated tube possesses three grooves in cross-section plane. The geometry and the computational domain are depicted in Figs. 1 and 2, respectively. The air (Pr = 0.707) are the working fluid at 300 K flows into the triple-start corrugated tubes in x direction. The characteristic diameter of tubes (D) was fixed at 0.05 m. The effect of the helical corrugation pitch to characteristic diameter (p/d = PR of 0.5, 0.65, 0.75, 1.0, 1.5 and 2.0) is investigated. The helical corrugation pitch (p) is defined as the 360 helical length. The study encompassed corrugation depth to characteristic diameter ratios or depth ratios (e/d or DR) of , 0.06, 0.08, 0.1, 0.12, 0.14 and Reynolds numbers spanned 5000 Re 20,000. To resolve the viscous sublayer (the first layer of the turbulent flow around near wall region in which momentum transfer and heat transfer significantly change), the near-wall modeling method with the Enhanced Wall Treatment (EWT) for the k-ε equation was used with the realizable k-ε turbulent model. The hexahedron grid with high density was generated in this region while the tetrahedral grid was generated in the core flow region. The boundary conditions applied in the present work can be summarized as follows: - For full length tubes, a uniform air mass-flow rate was applied at the inlet while pressure was used to indicate the outlet condition. For periodic modules, the inlet and outlet were set to be under periodic condition. - For tubes wall, no-slip condition and uniform heat flux of 600 W/m 2 were applied. The physical properties of fluid were assumed to be constant at mean bulk temperature. 4. Parameters The evaluation of Reynolds number (Re), friction factor (f), local Nusselt number (Nu x ), average Nusselt number (Nu) and thermal enhancement factor (TEF) was made via the following equations. u 0 D Re h (11) where D h is the hydraulic diameter for calculating the Reynolds number and can be calculated from 4A c B, A c is the cross-sectional area, B is the wetted perimeter of the cross-section while the characteristic diameter of tubes (D) was evaluated depending on the tube geometric parameters. The helical corrugation pitch to characteristic diameter (p/d = PR) and the corrugation depth to characteristic diameter ratios or depth ratios (e/d or DR). P / L Dh f (12) 1 2 u0 2 where P is the pressure drop and u 0 is mean flow velocity. hxdh Nux (13) k where h and k are the convective heat transfer coefficient and thermal conductivity, respectively. 1 Nu NuxdA A (14) Thermal performance enhancement factor (TEF) is defined as the ratio of the heat transfer coefficient of triple-start corrugated tubes, h, to that of the smooth tube, h 0, at an equal pumping power. Nu/ Nu0 TEF (15) 1 / 3 f / f 0 where Nu 0 and f 0 are the Nusselt number and friction factor of the straight circular smooth tube, respectively. 5. Numerical results and discussion 5.1 Grid independence The evaluation of grid independence was performed by using the triple-start corrugated tubes periodic module with DR = 0.08, PR = 1.0 at Re =5000 as the model. Grid independence was determined from both the accuracy of 42

5 computational results and convergent time of the systems. The systems with different grid numbers of 50520, , , and were comparatively tested. The numerical results (Fig. 3) show that the use of the grid number of 502,962 provided comparable Nusselt number and friction factor results to those obtained by using the grid number of 650,756. However, the convergent time for the simulation using the grid number at 502,962 is considerable shorter. Therefore, grid number of 502,962 was adopted for the simulations of all periodic modules of the triple-start corrugated tubes. 5.2 Validation test for smooth and twisted tubes Fig. 3. Results of the grid dependence test. The validation test was performed by comparing the numerical results of the straight circular smooth tube obtained from the Realizable k-ε, the standard k-ε and, Renormalized Group (RNG) k-ε turbulent models with those achieved from the standard correlations (Incropera et al., 1996) and experimental results from Harleß et al. (2016) as shown in Fig. 4(ab). Evidently, the results computed using the Realizable k-ε turbulent model show the minimum deviations with the standard correlations of 10.1% and 6.3% for the average Nusselt number and friction factor, respectively, and with the experimental results of 7.2% and 9.1%. Therefore, the Realizable k-ε turbulent model was selected for the computations of the triple-start corrugated tubes. The comparison between the numerical result with different turbulent modelling (realizable k-ɛ, standard k-ε and, Renormalized Group (RNG) k-ε turbulent models) and the data from experimental by Rabientaj Darzi et al. (2012) and Harleß et al. (2016) are presented in Fig. 4(c). The boundary conditions of the test tube for the numerical method is defined as similar as the experimental condition. The results of the comparison are presented in term of Nusselt number, Nu and friction factor, f. The results indicated that the realizable k-ɛ model shows sufficiently accurate prediction as compared to those by Rabientaj Darzi et al. (2012) with deviations within 9.45% for Nusselt number (Nu) and 3.17% for friction factor (f). (a) (b) (c) Fig. 4. Comparison of the turbulent model with standard correlations for (a) Nusselt number of straight circular smooth tube, (b) friction factor of straight circular smooth tube and (c) for twisted tubes. 5.3 Concept of fully developed periodic flow and heat transfer The triple-start corrugated tube with DR = 0.06 and PR = 1.0 at Re = 5000 is the selected case for the demonstration of velocity profiles and the heat transfer profiles as presented in Figs. 5 to 8. 52

6 Promthaisong, Jedsadaratanachai and Eiamsa-ard, Fig. 5. Velocity profiles along the triple-start corrugated tube length at DR = 0.06, PR = 1.0 and Re = x/d = 0.5 x/d = 1.5 x/d = 2.5 x/d = 3.5 x/d = 4.5 x/d = 5.5 x/d = 6.5 x/d = 7.5 (a) x/d = 8.5 x/d = 9.5 (b) Fig. 6. TKE contours and flow structures in transverse plane for (a) full length and (b) periodic module at DR = 0.06, PR = 1.0 and Re =

7 5.3.1 The velocity profile in fully developed periodic region Figure 5 displays the velocity profiles in the triple-start corrugated tube at various locations (y/d = 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42 and z/d = 0.0). In general, the velocity profiles at different y/d values are in similar patterns. The velocity profiles are divided into 2 parts; (1) the hydrodynamic entrance region at the entry regime and (2) the fully developed periodic region which commences at around x/d 6.0. In the figure, it is obvious that the velocities in the core region (small y/d) are higher than those near wall region (large y/d). The turbulent kinetic energy contours and the swirling flows in y-z plane for the full length and periodic modules are displayed in Fig. 6(a-b). For the full-length module, the turbulent kinetic energy in the hydrodynamic entrance region is extremely high which is significantly different from those in the fully developed periodic region. The turbulent kinetic energy fields in the fully developed periodic region are similar to that of the periodic module. As shown, the turbulent kinetic energies near wall region are high except around the ridge while those in the core region are low. This indicates that swirling flows are generated by the corrugated tube wall. The creation of the swirling flow leads to fluid mixing and thermal boundary layer destruction which helps to improve the heat transfer rate and subsequently enhance thermal performance especially near the tube wall The heat transfer profile in fully developed periodic region Figure 7 shows the local Nu/Nu 0 profiles on tube wall of the triple-start corrugated tube. In the figure, the local Nu/Nu 0 profiles become fully developed periodic at around x/d 6 as confirmed by the local Nusselt number contour for the full length shown in Fig. 8(a). In addition, the local Nusselt number contour for the periodic module is in similar pattern with that of the full length in the fully developed periodic region as depicted in Fig. 8(b). Fig. 7. Nu/Nu0 profiles for the corrugated tube at DR = 0.06, PR = 1.0 and Re = (a) (b) Fig. 8. Local Nux contours for (a) full length and (b) periodic module at DR = 0.06, PR = 1.0 and Re =

8 Promthaisong, Jedsadaratanachai and Eiamsa-ard, straight circular smooth tube DR = 0.02 DR = 0.04 DR = 0.06 DR = 0.08 DR = 0.1 DR = 0.12 DR = 0.14 DR = 0.16 Fig. 9. Flow structures in triple-start corrugated tubes at PR = 1.0 and Re = Effect of depth and pitch ratio The effect of depth ratio (DR = e/d = , 0.06, 0.08, 0.1, 0.12, 0.14 and 0.16) and pitch ratio (p/d = PR = 0.5, 0.65, 0.75, 1.0, 1.5 and 2.0) of triple-start corrugated tubes on flow and heat transfer characteristics are reported in the forms of the plots of iso-surface flow, turbulent kinetic energy fields, temperature fields and local Nusselt numbers on tube walls as shown in Figs. 9 to 16. The plots of flow structure in Figs. 9 and 10 reveal that the swirl flows in the triplestart corrugated tubes consist of a main swirl flow and a helical swirl flow. The helical swirl flows are induced when DR 0.04 and PR At DR = 0.02 to 0.04, DR = 0.10 to 0.16 and PR = 0.5, the helical swirl flow moves over the groove surface while at DR = 0.08 the helical swirl flow transfers the fluid to thoroughly contact with the groove surface. 8 2

9 Promthaisong, Jedsadaratanachai and Eiamsa-ard, PR = 0.5 PR = 0.65 PR = 0.75 PR = 1.0 PR = 1.50 PR = 2.0 (a) (b) Fig. 10. Flow structures in triple-start corrugated tubes at DR = 0.08 and Re = 5000 for (a) side view and (b) isometric view. straight smooth tube DR = 0.08 DR = 0.02 DR = 0.10 DR = 0.04 DR = 0.06 DR = 0.12 Fig. 11. TKE contour in transverse plane in triple-start corrugated tube at PR = 1.0 and Re =

10 Promthaisong, Jedsadaratanachai and Eiamsa-ard, straight smooth tube PR = 0.5 PR = 0.65 PR = 0.75 PR = 1.0 PR = 1.5 PR = 2.0 Fig. 12. TKE contour in transverse plane in triple-start corrugated tube at DR = 0.08 and Re = straight smooth tube DR = 0.08 DR = 0.02 DR = 0.1 DR = 0.04 DR = 0.06 DR = 0.12 Fig. 13. Temperature contour in transverse plane in triple-start corrugated tube at PR = 1.0 and Re = straight smooth tube PR = 0.5 PR = 0.65 PR = 0.75 PR = 1.0 PR = 1.5 PR = 2.0 Fig. 14. Temperature contour in transverse plane in triple-start corrugated tube at DR = 0.08 and Re =

11 straight smooth tube DR = 0.02 DR = 0.04 DR = 0.06 DR = 0.08 DR = 0.1 DR = 0.12 Fig. 15. Nux contour on triple-start corrugated tube wall at PR = 1.0 and Re = PR = 0.5 PR = 0.65 PR = 0.75 PR = 1.0 PR = 1.5 PR = 2.0 Fig. 16. Nux contour on triple-start corrugated tube wall at DR = 0.08 and Re = Fig. 17. Relationship between Nu/Nu0 and Re. The intensity of rotational and swirl flows is enhanced with an increase of the DR from 0.06 to 0.08, and it increases with a decrease in the PR except PR = 0.5. In addition, the helical pitch length of the main swirl flow decreases with the rise of DR and PR value. Figures 11 and 12 reveal that turbulent kinetic energy increases when DR increases from 0.02 to 0.12 and PR increases from 0.5 to This can be attributed to the stronger helical swirl flow. It is also observed that at low DRs (0.02 to 0.04), only a weak TKE centered in the corrugated section is detected while the tangential velocity component and TKE are low resulting in low heat transfer rate. At low PRs (0.65 to 1.0) and high DRs (0.08 to 0.12), TKE is strengthened near the groove corrugated wall. 11 2

12 Figures 13 and 14 show the plot of temperature field for the straight smooth tube and the triple-start corrugated tubes with DR = 0.02 to 0.12, PR =0.5 to 2.0 and Re = In general, the use of triple-start corrugated tubes leads to a major change of the temperature field when compared to that of the smooth tube. For the triple-start corrugated tubes, fluid mixing is promoted when DR increases from 0.02 to 0.08 and PR increases from 0.65 to 1.0, indicated by the thinning thermal boundary layer as swirl flow intensity increases. This can be explained that the DR = 0.02 to 0.08 and PR = 0.65 to 1.0, are capable of induction of more consistent flow turbulence and swirl with higher frequency, resulting in more efficient disruption of the viscous boundary layer. However, when DR increases from 0.08 to 0.12 and PR increases from 0.65 to 1.0, the opposite trend is found notified by the thicker thermal boundary due to the swirl flow moves over the tube surface resulting in diminished heat transfer between tube wall and the fluid. The results suggest that the heat transfer optimum condition corresponding to maximum heat transfer rate is found at DR = 0.08 and PR = 0.75 while the thermal enhancement optimum corresponding to the maximum thermal enhancement is found at DR = 0.06 and PR = The heat transfer characteristic is presented in form of local Nusselt number contours on tube wall as shown in Figs. 15 and 16. Evidently, the triple-start corrugated tubes yield superior heat transfer to the smooth tube. Heat transfer increases when DR increases from 0.02 to 0.08 and PR increases from 0.65 to 1.0. The opposite trend is found when DR increases from 0.08 to 0.12 and PR increases from 1.5 to 2.0 due to the reasons mentioned above. Among the studied cases, the triple-start corrugated tube with DR = 0.08 and PR = 0.75 yields the highest heat transfer rate. 5.5 Performance evaluation Fig. 18. Relationship between Nu/Nu0 and DR at different PRs. The relative between Nusselt number ratio (Nu/Nu 0 ), friction factor ratio (f/f 0 ), thermal enhancement factor ( TEF) and depth/pitch ratio (DR/PR) at different Reynold numbers (Re) are displayed in Figs. 17 to 22. In general, Nusselt number ( Nu) of triple- start corrugated tubes is considerable higher than that in the straight smooth tube ( Nu 0 ). This is responsible by the swirl induction by the corrugated wall. In general, Nusselt number ratio ( Nu/ Nu 0 ) decreases as Reynolds number increases owing to the increase of turbulence intensity and decrease of thermal thickness at high flow rate. The variation of the pressure drop in terms of friction factor ratio (f/f 0 ) as a function of Reynolds number is in Fig. 19 indicates that the presence of triple- start corrugated tubes causes significantly higher friction factor than that in the straight smooth tube. Friction factor ratio (f/f 0 ) tends to decrease as Reynolds number increase. Variation of thermal enhancement factor (TEF) with Reynolds number (Re) for the triple-start corrugated tubes with various depth/pitch ratio (DR/PR) is demonstrated in Figs. 21 and 22, respectively. In general, TEF of the triple-start corrugated tubes increases when DR increases from to but decreases when DR increases beyond and the maximum of TEF is found around PR = 0.65 to 1.0. This accords with the results of heat transfer (Nusselt number). In addition, f/f 0 increases with the rise of DR value (Fig. 20). This makes the thermal enhancement for DR > 0.08 even worse. Although, the tube with 12 2

13 DR = 0.08 yields the highest heat transfer rate, the tube with DR = 0.06 yield the highest thermal performance factor due to the best trade-off between Nu/Nu 0 and f/f 0 (Fig. 22). In range studied, the Nu/Nu 0, f/f 0 and TEF obtained by the use of the triple-start corrugated tubes are found to be around 0.84 to 2.2, 1.0 to and 0.72 to 1.2, respectively. The highest TEF of 1.23 is achieved by using the triple-start corrugated tube with DR = 0.06 and PR = 0.75 at Re = Fig. 19. Relationship between f/f0 and Re. Fig. 20. Relationship between f/f0 and Re at different PRs 13 2

14 Fig. 21. Relationship between TEF and Re. Fig. 22. Relationship between TEF and Re for the corrugated tubes with different PRs. 14 2

15 (a) Nu/Nu 0 (b) f/f 0 (c) TEF Fig. 23. Comparison between the present work and the previously published works. 5.6 Comparison with previously published works The numerical results of the triple-start corrugated tube with DR = 0.06 and PR = 1.0 are selected to compare with those appeared in the previously published works including the corrugated tube with x/d i = 0.12 and p/d i =1.05, two-start corrugated tube with DR = 0.06 and PR=0.25, the micro-fin tube, the dimpled tube, which were presented by Naphon et al. (2006); Promthaisong et al. (2016), Eiamsa-ard and Wongcharee (2012) and Thianpong et al. (2009), respectively, as seen in Fig. 23. Manifestly, the present triple-start corrugated tubes offer higher TEF than the two-start corrugated tube, corrugated tube and micro-fin tube at Reynolds number between 5000 and 10,000 and higher heat transfer and friction factor than the corrugated tube (Promthaisong et al., 2006), micro-fin tube (Eiamsa-ard and Wongcharee, 2012) and dimpled tube (Thainpong et al., 2009). 6. Conclusions The heat transfer, friction loss and thermal enhancement in the triple-start corrugated tubes for Re = 5000 to 20,000 were numerically studied. The effect of the depth ratio (e/d, DR =0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14 and 0.16) and pitch ratios (p/d, PR =0.5, 0.65, 0.75, 1.0, 1.5 and 2.0) on flow and thermal characteristics were also investigated. The major findings of the present work can be concluded as follows: - Fully developed periodic flow and heat transfer commences at around x/d of Triple-start corrugated tubes induce swirl flow which helps to reduce the boundary layer thickness, improve the fluid mixing between the core and near wall and eventually enhance heat transfer on tube wall. - The Nusselt numbers and friction factors of triple-start corrugated tubes in the investigated range are found to be 0.8 to 2.31 and 1.0 to times over those of the smooth tube, respectively. - For the range studied, the triple-start corrugated tube with DR = 0.06 and PR = 0.75 offers the maximum thermal enhancement factor (TEF) of 1.21 at the lowest Reynolds number of Re =

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18 k turbulent kinetic energy, /m 2 s -2 k a thermal conductivity of air, /W m 1 K 1 Nu Nusselt number PR corrugation pitch ratio p corrugation pitch, /m P static pressure, /Pa Pr Prandtl number Re Reynolds number S magnitude of mean strain rate, /s -1 S ij mean strain rate tensor, /s -1 S k user-defined source term for k, /kg m 1 s 3 S ɛ user-defined source term for ɛ, /kg m 1 s 4 T temperature, /K TEF thermal performance enhancement factor u 0 mean or uniform velocity in smooth tube, /m s 1 fluctuation velocity in x i -direction, /m s 1 u j velocity component in x j -direction, /m s 1 fluctuation velocity in x j -direction, /m s 1 x i, x j coordinate direction, /m x x-position, /m y y-position, /m z z-position, /m Greek letters μ dynamic viscosity, /kg s 1 m 1 Γ thermal diffusivity, /kg s 1 m 1 Γ t turbulent thermal diffusivity, /kg s 1 m 1 σ k turbulent Prandtl numbers for k σ ε turbulent Prandtl numbers for ε δ ij Kronecker delta, /m ratio of the turbulent to mean strain kinematic viscosity, /m 2 s -1 ε dissipation rate, /m 2 s -3 ρ density, /kg m 3 θ twisted angle, /degree Subscripts 0 straight circular smooth tube pp pumping power a air Superscripts average 18 2