Numerical investigations of the thermal-hydraulic performance in a rib-grooved heat exchanger tube based on entropy generation analysis

Accepted Manuscript Title: Numerical investigations of the thermal-hydraulic performance in a ribgrooved heat exchanger tube based on entropy generation analysis Author: Nianben Zheng, Peng Liu, Feng Shan, Zhichun Liu, Wei Liu PII: DOI: Reference:

S1359-4311(16)30131-4 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.008 ATE 7732

To appear in:

Applied Thermal Engineering

Received date: Accepted date:

11-10-2015 5-2-2016

Please cite this article as: Nianben Zheng, Peng Liu, Feng Shan, Zhichun Liu, Wei Liu, Numerical investigations of the thermal-hydraulic performance in a rib-grooved heat exchanger tube based on entropy generation analysis, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Numerical investigations of the thermal-hydraulic performance in a rib-grooved heat exchanger tube based on entropy generation analysis Nianben Zheng*, Peng Liu, Feng Shan, Zhichun Liu, Wei Liu* School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

*Corresponding authors: Address: School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China Tel: +86-27-87542618 Fax: +86-27-87540724 E-mail: [email protected] (Wei Liu); [email protected] (Nianben Zheng)

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Highlights 

A tube with discrete inclined ribs and grooves is proposed to enhance heat transfer



Multiple longitudinal swirl flows are induced



Effects of geometric parameters on the rate of entropy generation are investigated



Nusselt number and friction factor correlations are derived

Abstract A numerical simulation has been conducted to examine the turbulent flow characteristics and heat transfer performance in a rib-grooved heat exchanger tube. Ribs and grooves characterized by a discrete and inclined distribution are alternately arranged on the inner wall surface of the test section. The main purpose of this study is to determine thermal-hydraulically superior rib-groove geometry for heat transfer enhancement. Flow structures and local heat transfer characteristics are presented and analyzed at first. Because of the disturbance of ribs and grooves, multiple longitudinal swirl flows are induced, which immensely affect the local heat transfer performance. Then, effects of geometric parameters, including the rib-groove pitch ratio (P*), the number of circumferential ribs and grooves (N), and the rib-groove inclination angle () on the heat transfer and flow performance are examined. The results show that the heat transfer ratios (Nu/Nu0) are about 1.58 to 2.46, while friction factor ratios (f/f0) are about 1.82 to 5.03. Therefore, the performance evaluation criterion (PEC) values vary from 1.19 to 1.68. In addition, entropy generation analysis indicates that the impact of the ribs and grooves is thermodynamically advantageous. The entropy generation number reaches the minimum when P* = 0.35, N = 8 and = 45°, under which condition the proposed rib grooved tube would perform the best in

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practice. Key words: Numerical simulation; Turbulent flow; Rib-grooved tube; Multiple longitudinal swirl flows; Entropy generation analysis. 1. Introduction Heat transfer enhancement techniques are usually needed in designing heat exchangers with high performance and low resistance. Artificial roughness such as ribs, grooves, protrusions and fins is a commonly used passive heat transfer enhancement technique in single-phase internal flows in channels or tubes. The presence of artificial roughness enhances the heat transfer coefficient by breaking the viscous sub-layer that leads to the reduction of thermal resistance and the increase of turbulence in a region close to artificially roughened surface [1]. Numerical and experimental investigations on the heat transfer enhancement by the application of artificial roughness have been extensively reported. In general, considerable attention has been paid to the geometric parameters, shape, arrangement, and flow conditions [2-13], which have a significant effect on both local and overall heat transfer and flow performance. The results show that ribs can effectively enhance heat transfer, and show better heat transfer performance than that of grooves due to the strong secondary flow induced by the ribs. However, the drawback associated with the application of ribs is the relatively large pressure drop. Compared to ribs, grooves show better performance of pressure drop because the wake region induced by them is fairly small, and the direct impingement of the mainstream on the grooves is also weakened. In order to achieve high heat transfer performance without much increase in the pressure

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drop, the idea of making full use of the advantages of ribs and grooves by the combined turbulence promoter in the form of rib-groove has been proposed. Zhang et al. [14] studied the heat transfer and friction in rectangular channels with ribbed or ribbed-grooved walls. Their results indicated that the rib-grooved duct showed much better heat transfer performance than that of the ribbed duct. Jaurker et al. [1] experimentally investigated the characteristics of heat transfer and friction for rib-grooved artificial roughness on a broad heated wall. They claimed that the Nusselt number could be further enhanced beyond that of ribbed duct while keeping the friction factor enhancement low. Layek et al. [15] conducted an experimental investigation on the characteristics of heat transfer and fluid flow for fully developed turbulent flow in a rectangular duct with repeated integral transverse chamfered rib-groove roughness. Later, Layek and his co-workers [16] investigated the effect of chamfering on the thermal-hydraulic performance in a solar heater. Eiamsa-ard and Promvonge [17] examined the combined effects of rib-grooved turbulators on the thermal characteristics of turbulent rib-grooved channel flows under a uniform heat flux boundary condition. Tang and Zhu [18] reported the properties of heat transfer enhancement in a rectangular channel with discontinuous crossed ribs and grooves. Their results revealed that the overall thermal-hydraulic performance for a ribbed-grooved channel was increased by 10% to 13.6% when compared to a ribbed channel. Thermal behaviors in a solar air heater channel fitted with combined wavy-rib and groove turbulators were experimentally examined by Skullong et al. [19]. Al-Shamani et al. [20] examined the characteristics of heat transfer enhancement in a channel with trapezoidal rib–groove using nanofluids. Their results showed that this trapezoidal rib-groove using nanofluids had the potential to

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dramatically increase heat transfer. Numerical simulation and optimization of turbulent nanofluids flow in a three-dimensional rectangular rib-grooved channel using the response surface methodology (RSM), and the genetic algorithm method (GA) were conducted by Yang et al. [21]. Navaei et al. [22] reported the effects of different geometrical parameters and various nanofluids on the thermal performance of rib–grooved channels. To improve the balance between heat transfer enhancement and pressure drop, there are also some researchers who have attempted to achieve the optimum flow field from the viewpoint of heat transfer optimization with different optimization objectives, including minimum power consumption [23], minimum heat consumption [24], minimum entransy dissipation [25] and minimum exergy destruction [26]. According to the results of heat transfer optimization, multiple longitudinal swirl flows are the principal characteristic in the optimized flow field, where a better balance between heat transfer enhancement and pressure drop can be obtained. Recently, our group [27] investigated the heat transfer and flow characteristics in a heat exchanger tube with double discrete inclined ribs. Visualization of the flow field showed that multiple longitudinal swirl flows were generated in the ribbed tube. Later, we [28] examined the heat transfer and flow structures in a tube with discrete inclined grooves. It is very interesting to find that heat transfer performance can be greatly enhanced by generating multiple longitudinal swirl flows with these discrete inclined grooves. These recent works indicate that the utilization of discrete inclined ribs or grooves alone can effectively enhance the heat transfer performance by generating multiple longitudinal swirl flows, which gives us a hint that it may be possible to generate multiple longitudinal swirl flows by the combination use of discrete inclined ribs and grooves to

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further enhance the heat transfer performance in the rib-grooved tube. Although the rib-grooved structure has been researched and explored by many other researchers, the idea of using the rib-grooved structure to generate the optimum flow field for heat transfer enhancement according to heat transfer optimization has been rarely reported. Therefore, we conduct a numerical simulation to examine the flow and heat transfer characteristics in the tube with discrete inclined ribs and grooves in this work. In the present study, a parameter study with respect to the rib-groove pitch ratio, the number of circumferential ribs and grooves, and the rib-groove inclination angle on the heat transfer and flow performance has been carried out to optimize the geometric configuration of ribs and grooves. In addition, entropy generation analysis has been conducted to further determine thermal-hydraulically superior rib-groove geometry for heat transfer enhancement. 2. Physical model description The schematic diagram of the rib-grooved heat exchanger tube is depicted in Fig. 1. The computational domain consists of three sections, namely, the test section, the upstream section and the downstream section. Ribs and grooves characterized by a discrete and inclined distribution are alternately arranged on the inner wall surface of the test section with a length (L2) of 0.2 m. To guarantee a nearly fully developed turbulent flow condition and eliminate the effect of backflows, the smooth upstream section and downstream section with lengths of 0.1 m (L1) and 0.1 m (L3) are designed for the test section, respectively. Figure. 2 shows the geometric configurations of the test section. The rib-grooved tube has an inner diameter (D) of 0.017 m. The length (L) and width (W) of the ribs and grooves are 0.006 m and 0.002m, respectively. The height (H) of the rib is 0.0005 m, and the depth (E)

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of the groove is 0.0005 m. In the present investigation, the main objective is to study the effects of geometric parameters on the flow and heat transfer performance in the rib-grooved tube. The parameters to be investigated are as follows: three rib-groove pitch ratios, defined as the ratio of the rib-groove pitch to the inner diameter (P* = P/D = 0.35, 0.47 and 0.59); three numbers of circumferential ribs and grooves (N = 4, 6 and 8); four rib-groove inclination angles ( = 15°, 30°, 45° and 60°). 3. Entropy generation analysis Because fluid flows in a horizontal tube and the flow is considered fully developed in the present work, effects of gravity on the flow are quite limited, and the variation in velocity is not significant. Therefore, the change of potential energy and kinetic energy is ignored. Taking the rib-grooved tube of length dz (along the flow direction) as the thermodynamic system, the first and second law of thermodynamic can be expressed as: m d h  q d z

(1)

q d z

d S gen  m d s 

(2)

Tw

where q  is the heat flux per unit length. In addition, by using the canonical relation given by Tds  dh   dp

(3)

Substituting Eq. (1) and Eq. (3) into Eq. (2) yields the entropy generation rate per length: T T 1 w d S gen  m c p  dT  dP  TT  c pT w 

   

(4)

Pressure drop in Eq. (4) is given by: f  um 2

dP  

dz

(5)

2 D1

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The bulk temperature variation of fluid along the rib-grooved tube is given by Sahin [26]: According to Sahin [29-30], the bulk temperature variation of the fluid along the rib-grooved equation  Q

tube

is

obtained

 m c p d T  h D 1  T w  T  d z

by

integrating

the

heat

balance

from the inlet of the test section assuming a

uniform heat transfer coefficient in the test section where the flow is considered fully developed. The bulk temperature variation of the fluid along the rib-grooved tube is as follows:   4h T  T w  (T w  T i ) e x p   z   u m D 1 c p 

(6)

where Ti represents the inlet temperature of the test section. Integrating Eq. (4) along the test section of the rib-grooved tube, total entropy generation can be obtained according to Sahin [29]

S gen

where  =  T w

  m c p  ln 

 Ti  T w

 1   e 4 St   1 

*

*   4 St   1 e  



 h

u

m

cp

*

Ec

    

is the dimensionless temperature difference,  * = L 2

dimensionless length of the test section, and S t



 e 4 St    f ln   1 8 St  1

Ec  um

2

c

p

Tw

(7)

D1

is the

 is the Eckert number,

 is the Stanton number.

A non-dimensional entropy generation number based on the total amount of heat transfer is defined by Sahin [30] as:

NS 

S gen

(8)

Q Tw

where Q is the total rate of heat transfer to the fluid. Based on the non-dimensional entropy generation number, we define the non-dimensional entropy generation number ratio (η) to quantize the thermodynamic impact of the ribs and

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grooves. The non-dimensional entropy generation number ratio is given by:  

N S ,e N

(9)

S ,0

where N S , e , N S , 0 are the non-dimensional entropy generation number in the enhanced tube and smooth tube, respectively. The ribs and grooves could effectively enhance the thermodynamic performance in addition to enhancing the heat transfer of the tube when the non-dimensional entropy generation number ratio is less than unity (η < 1). 4. Numerical simulation 4.1. Selection of turbulence model To simulate the present situation accurately, it is necessary to select an appropriate turbulence model. Therefore, the commonly used turbulence models in industrial applications, including the Standard k- Model, the RNG k- Model, the Realizable k-Model, the Standard k- Model and the SST k- Model are adopted to simulate the flow condition in the ribbed tube as documented in Ref. [31], which is similar to the present rib-grooved tube. In the numerical simulation, we have simplified the physical model and only the internal flow are taken into consideration. The boundary conditions between the numerical and experimental models are identical. Fig. 3 shows the comparison of the average Nusselt number between the experimental and numerical results with different turbulence models. It is clear that the results obtained from the SST k- Model are the closest to the experimental results with the maximum deviation of 6%. Note that the experimental uncertainties of Nusselt numbers for the range of Reynolds numbers investigated are below 6% [31], therefore the SST k- Model is more reliable than other

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models for the flow in the ribbed tube. The reason why the SST k- Model show the best result is that the SST k- Model is a hybrid two-equation model that combines the advantages of both k- and k- Models such that the SST k- Model functions like the k- Model close to the wall and the k- Model in the freestream. In other words, the SST k-Model is a good compromise between k- and k- Models so that it is more reliable than other turbulence models for predicting the flow and heat transfer performance of the ribbed tube. Therefore, the SST k- Model is applied to the following numerical investigations. 4.2. Governing equations and numerical methods The governing equations for the SST k- Model are expressed as follows, according to Ref. [32]. Continuity equation:  ( ui )  xi

 0

(10)

Momentum equation:    ui



t

   u iu



j



 

x j

p xi



   ui  x j  x j 



____  ' '   ui u j   

(11)

Energy equation:  xi



T  

xi

  u iT  

   T  xi  c p xi 

   

(12)

The turbulence kinetic energy equation: k t





   uik  xi



~



 Pk    k  

  k       kut    xi   xi 

(13)

The specific dissipation rate equation:    t





   u i  xi



  S

2

   

2



       t    2  1  F1     xi   xi 

1 k  w2

  xi  xi

(14)

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where the blending function F1 is defined by:    F1  ta n h   m in  m a x    

where C D k 

  1 k  10  m ax  2   2 ,1 0    xi  xi  

  k 5 0 0  4    2 k    , 2     ,   2    y y   C D k  y     4

(15)

and y is the distance to the nearest wall.

The turbulent eddy viscosity is defined as follows: t 

 1k m a x   1 , S F 2

(16)



where S is the invariant measure of the strain rate and F 2 is a second blending function defined by:   F 2  ta n h   m a x  

 2 k 5 0 0   , 2        y y   

2

   

(17)

A production limiter is used in the SST model to prevent the build-up of turbulence in stagnation regions: Pk   t

u j ui  ui    x j x j  xi 

~     Pk  m in  Pk ,1 0    k   



(18)

All constants are computed by a combination of the corresponding constants of the k– and k– models via   



 2

  1 F   2 1  F

 0 .0 9 ,  1  5 9 ,  1  3 4 0 , 

k1

 , etc. The constants for this model are as follows:

 0 .8 5 , 

1

 0 .5 ,  2  0 .4 4 ,  2  0 .0 8 2 8 , 

k 2

 1,

 0 .8 5 6

The boundary conditions considered for the rib-grooved tube are as follows. At the inlet, a uniform velocity was applied and the fluid temperature of the inlet was fixed at 293 K, while a pressure-outlet condition was used at the outlet. A turbulent intensity of 5% and the hydraulic diameter were selected as the turbulence quantities for the inlet and outlet. Non-slip velocity conditions on the walls were assumed. A constant and uniform

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temperature of 333 K was applied on all inner walls. Water was selected as the working fluid and the thermal properties were given in Table 1. All results were obtained under steady-flow conditions with inlet velocity between 0.4 m/s and 1.2 m/s. All the governing equations with the applied boundary conditions were solved on the commercial software Fluent 13.0. To couple the velocities and pressures, the SIMPLE algorithm was used. The diffusion terms were discretized using a central difference scheme, and the second-order upwind scheme was applied for the discretization of the convection terms. Enhanced wall treatment which blends the viscous sublayer formulation and the logarithmic layer formulation was used for the near-wall treatments for fluid velocity and temperature fields, and this formulations is default for SST k- Model adopted in this work. The minimum convergence criterion was 10-8 for the energy equation, while that of the rest equations was 10-6. 4.3. Parameter definition The Reynolds number (Re), average heat transfer coefficient (h), Nusselt number (Nu) and friction factor (f) are defined as follows: Re 

h 

 um D

(19)

 q

(20)

Tw  Tm

Nu 

hD

(21)

k

f 

P

 L2

/ D   um

2

(22) /2

where u m is the mean velocity in the tube, q is the average heat flux, T w is the temperature of the wall, and T m



1 2

 Ti

 To

 is the bulk temperature of the fluid.  P is the difference

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between the mass-averaged pressures of the inlet and the outlet of the test section. To evaluate the overall thermal-hydraulic performance of the rib-grooved tube at the same pump power, the extensively used performance evaluation criterion (PEC) proposed by Webb and Kim [33] is adopted. The PEC is defined as follows: PEC 

where

N u0

and

f0

Nu / Nu0



f / f0



1/ 3

(23)

are the Nusselt number and the friction factor of the plain tube,

respectively. 4.4. Grid generation and independence test The computational domain was resolved using unstructured mesh generated with the commercial software ICEM CFD 13.0, as shown in Fig. 4. The meshes consist of prisms, tetrahedrons, hex core, and pyramids. To ensure that y+ remained less than 1 and capture the flow structures near the wall, the vicinity of the tube wall was meshed into much finer cells. Prism grids, which are orthogonal to the surfaces and with faces perpendicular to the boundary layer flow direction, were extruded from the surface of the tube. A Hex core mesh, with hexahedron grids filling the majority of the volume, was generated to capture the characteristics of the fluid flow in the core region. To fill the areas between the surface of the prisms and the hex core, tetrahedron grids were used. Pyramid grids were implemented to establish a conformal connection between triangle and quadrangle faces. Grid independent tests of three mesh levels (3.4M, 5.0M and 7.6M) for the rib-grooved tube with P* = 0.47, N = 8 and  = 30° at Re = 10170 were conducted to validate the accuracy of numerical solutions. The deviation between the calculated values for the second mesh level and third mesh level is about 2.24% for Nusselt number (Nu) and 2.15% for

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friction factor (f), as shown in Fig. 5, implying that the results of the second mesh level is adequate for simulations. Therefore, the second mesh level was adopted in the subsequent simulation considering a compromise of computational time and solution precision. 5. Results and discussion 5.1. Code validation To verify the reliability of the numerical methods and procedures for the numerical simulations in this study, the numerical results were compared to the experiment results as documented in Ref. [31]. The comparison between the numerical results and experimental results are shown in Fig. 6. We can see from the figure that numerical results are similar to the experimental results for both Nusselt number and friction factor. Quantitative comparison indicates that the deviations between the results calculated by the numerical simulation and experiments are within 16%–22% for Nusselt number and 10%–24% for friction factor. The deviations can be attributed to the discrepancies between the numerical and experimental model and uncertainty in experimental measurements. Thereby, numerical methods adopted in this study for heat transfer and pressure drop predictions were judged to be reliable. 5.2. Flow structures and local heat transfer characteristics Prior to discussing the performances of heat transfer in the rib-grooved tube, it is necessary to have an understanding of the flow structures at first. Flow structures in the rib-grooved tube with P* = 0.47, N = 8 and  = 30° at Re = 10170 are shown in Figs. 7 and 8. Fig. 7 shows the local streamlines around the ribs and grooves. It is obvious to see that vortex structures are induced right behind the rib when the incoming flow passes over a rib

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(marked by dashed red lines), and fluid is swallowed inside the groove generating another vortex inside the groove as the fluid flow over a groove (marked by dashed black lines). These vortices result in the flow deviations from the mainstream. The effects of flow deviations on the mainstream are presented in the form of three-dimensional streamlines along the flow direction in the rib-grooved tube, as shown in Fig. 8. Obviously, there is a major change in the flow structures that eight longitudinal swirl flows are generated due to the disturbance of ribs and grooves. These longitudinal swirl flows are uniformly distributed near the wall, and the mainstream flow is ultimately divided into eight helical streams. This type of multiple longitudinal swirl flow pattern leads to a relatively long flow path in the tube, and fluid between the wall and core flow regions is intensively mixed, which will have a significant impact on the performance of heat transfer in the rib-grooved tube. Figure 9 shows the profile of local heat flux on the tube wall at Re = 10170. The local wall heat flux is high on the front surface of the ribs, which can be attributed to the direct flow impingement on the front surface of the ribs. The areas behind the ribs and grooves also have much higher local heat flux than those between the ribs or grooves, owing to the effects of vortices induced by the ribs and grooves. A closer examination of heat flux distributions shows that the enhanced heat transfer area behind the ribs and grooves is much larger than those on the front surface of the ribs, implying that the area behind the ribs and grooves are the main parts that contribute to the heat transfer enhancement. Besides, the local heat flux is much higher on the areas right behind the ribs and grooves than those further downstream from the ribs and grooves. Therefore, it can be concluded that heat transfer enhancement is more effective in the areas right behind the ribs and grooves, and taking full advantage of

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these areas will further enhance the heat transfer. 5.3. Effects of geometric parameters 5.3.1. Effects of rib-groove pitch ratio Figures 10 (a), (b) and (c) show the variations of Nusselt number ratio, friction factor ratio and the overall thermal performance with rib-groove pitch ratios, respectively. The number of circumferential ribs and grooves is 8 and the rib-groove inclination angle is 30°. At a given rib-groove pitch ratio, the Nusselt number ratio decreases with the increasing Reynolds number, while the friction factor ratio shows the opposite trend, implying that heat transfer enhancement is more effective at low Reynolds. The reason for this is that the turbulence intensity of the flow at low Reynolds number is low and the thermal boundary layer is much thicker, therefore the effect of the disruption of the boundary layer induced by the ribs and grooves is more remarkable at low Reynolds number. At a given Reynolds number, both Nusselt number ratio and friction factor ratio increase with the decrease of the rib-groove pitch ratio, and reach the maximum at P* = 0.35. This is because heat transfer enhancement is more effective right behind the ribs and grooves, as discussed in section 5.2, and the lower rib-groove pitch ratio helps to enlarge the enhanced heat transfer area that further enhances the heat transfer. However, the lower rib-groove pitch ratio causes more flow blockages in the tube, and thereby the flow performance becomes worse. The heat transfer enhancement is about 1.74 to 2.25 and friction factor ratio is 1.84 to 3.88 for different rib-groove pitch ratios. It is clear from Fig. 10 (c) that PEC values increase with the decreasing rib-groove pitch ratio, and vary from 1.28 to 1.63, which are all higher than 1. The maximum PEC value is obtained at P* = 0.35. Besides, PEC values decrease with the

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increase of Reynolds number. This phenomenon can be explained by the fact that PEC ( PEC



 Nu

/ Nu0

 

f / f0

 ) is determined by the Nusselt number ratio and friction factor 1/ 3

ratio together. The Nusselt number ratio decreases with the increase of Reynolds number, while the friction factor ratio increases with increasing Reynolds number. Therefore, PEC values decrease with the increase of Reynolds number. The entropy generation number and the non-dimensional entropy generation number ratio with different rib-groove pitch ratios are compared in Fig. 11. The results show that the entropy generation number decreases with the decreasing of rib-groove pitch ratio, implying that the reduction of the rib-groove pitch ratio can not only enhance the heat transfer performance, but also abate the irreversibility of the heat transfer and flow process in the rib-grooved tube. It is observed from Fig. 11 (b) that the non-dimensional entropy generation number ratios are less than unity, which shows the thermodynamic advantage of the rib-grooved tube compared to the smooth tube. Besides, the lower rib-groove pitch ratio leads to lower non-dimensional entropy generation number ratio. Therefore, the rib-grooved tube in practical applications is recommended to have a rib-groove pitch ratio of 0.35 to minimize the entropy generation. 5.3.2. Effects of the number of circumferential ribs and grooves Figures 12 (a) and (b) show the velocity vectors, and surface streamlines with different numbers of circumferential ribs and grooves in the outlet cross-section of the test section for P* = 0.35 and  = 30° at Re = 10170. It is obvious that multiple longitudinal swirl flows are induced in the rib-grooved tube, and the number of longitudinal swirl flow is proportional to the number of circumferential ribs and grooves. With the increasing of the number of

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circumferential ribs and grooves, the area affected by single longitudinal swirl flow is reduced, and the cores of longitudinal swirl flows tend to get closer to the tube wall because of the interaction of the longitudinal swirl flows. The swirling strength which represents the strength of the local swirling motion and temperature distributions in the outlet cross-section of the test section are presented in Figs. 13 (a) and (b). The swirling strength criterion which uses the imaginary part of the complex eigenvalues of the velocity gradient tensor to visualize vortices is a commonly used method for the identification of vortices. Details of the definition of the swirling strength can be found in Ref. [34]. Figure. 13 (a) shows that the swirling strength of the area where longitudinal swirl flow is generated is much higher than other area without longitudinal swirl flow, and high swirling strength area enlarges with the increasing number of circumferential ribs and grooves. As shown in Fig. 13 (b), the temperature is higher and more uniformly distributed in the cross section with more numbers of circumferential ribs and grooves. This stems from the fact the higher swirling strength leads to more intense mixing of fluid, thus enhance the heat transfer performance in the rib-grooved tube.

Figure 14 shows the entropy generation distribution in the flow and heat transfer processes with different numbers of circumferential ribs and grooves at Re = 10170. It is clear that the entropy generation near the wall is much higher than that in the core flow region, indicating the irreversibility of the heat transfer and flow process mainly occurs in the area near the wall. The reason is that the heat transfer temperature difference is much higher in the region near the wall. Moreover, it is valuable to point out that longitudinal swirl

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flows induced by ribs and grooves lead to more intense mixing of fluid near the wall and thereby decrease the heat transfer temperature difference, which leads to a reduction in entropy generation. Besides, the ability of ribs and grooves to reduce irreversibility of the heat transfer and flow process increases with the increment of the number of circumferential ribs and grooves. Figures 15 (a), (b) and (c) show the variations of Nusselt number ratio, friction factor ratio and PEC with different numbers of circumferential ribs and grooves. The results show that both Nusselt number ratio and friction factor ratio with N = 8 are considered to the highest, followed by those of N = 6, and the values for those with N = 4 are the lowest. For a given Reynolds number both Nusselt number ratio and friction ratio increase with increasing numbers of circumferential ribs and grooves. On the whole, the heat transfer in the rib-grooved tube is enhanced by a factor of 1.58 to 2.25, and friction factor is increased by a factor of 1.82 to 3.88 over those of the smooth tube. PEC values decrease with increasing Reynolds number, and increase significantly with increasing numbers of circumferential ribs and grooves, as shown in Fig. 15 (c). PEC values vary from 1.20 to 1.63, which are larger than unity, implying that the proposed rib-grooved tube is advantageous over the smooth tube. Effects of the number of circumferential ribs and grooves on the entropy generation number and the non-dimensional entropy generation number ratio are shown in Fig. 16. With a given Reynolds number, the entropy generation number decreases with the increasing of the number of circumferential ribs and grooves, which indicates that increasing the number of circumferential ribs and grooves is an effective to reduce the irreversibility in

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addition to enhancing the heat transfer performance in the rib-grooved tube. It is seen from Fig. 16 (b) that the rib-grooved tubes are thermodynamically advantageous (η < 1) for all the cases investigated. And more numbers of circumferential ribs and grooves result in lower non-dimensional entropy generation number ratios In order to minimize the entropy generation, the rib-grooved tubes are recommended to have the number of circumferential ribs and grooves of 8. 5.3.3. Effects of rib-groove inclination angle Velocity vectors and surface streamlines with different rib-groove inclination angles in the outlet cross-section of the test section for P* = 0.35 and N = 8 at Re = 10170 are presented in Fig. 17. It is apparent that eight longitudinal swirl flows are induced in the rib-grooved tubes with different rib-groove inclination angles. However, the swirling strength of the longitudinal swirl flows varies with the rib-groove inclination angles, as shown in Fig. 18 (a). From the figure, one sees that the swirling strength of the longitudinal swirl flows with  = 45° is considered the highest, followed by those with  = 30° and  = 60°, and value for that with  = 15° is the lowest. The contours of temperature in the rib-grooved tubes are in consistence with the contours of the swirling strength, as shown in Fig. 18 (b). The temperature in the rib-grooved tube with  = 45° is higher and more uniformly distributed than those with other rib-groove inclination angles. This is because the mechanism for heat transfer enhancement in the rib-grooved tube is mainly due to the turbulent mixing of fluid between the wall and core flow regions induced by the longitudinal swirl flows, and the higher swirling strength of the longitudinal swirl flows results in more intense turbulent mixing of fluid.

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Figure 19 shows the local entropy generation in the outlet cross-section of the test section with different rib-groove inclination angles. Similar to the entropy generation distribution in Fig. 14, the higher entropy generation is mainly located in the region near tube wall. Due to the disturbance of ribs and grooves, the heat transfer temperature difference is decreased and thereby the entropy generation is reduced. From Fig. 19, we can also see that the entropy generation, especially in the near-wall region, decreases with the increasing rib-groove inclination angle up to = 45°, and beyond which it tends to increase, implying that the rib-grooved tube with = 45° could enhance heat transfer rate more effectively. Variations of Nusselt number ratio, friction factor ratio and PEC with different rib-groove inclination angles are shown in Fig. 20. With a given Reynolds number, Nusselt number ratio increases with the increasing rib-groove inclination angle and reaches the maximum at the rib-groove inclination angle of 45°, then it starts to decrease with further increase of rib-groove inclination angle. Nusselt number ratios vary from 1.63 to 2.46, and the maximum Nusselt number ratio is obtained at the rib-groove inclination angle of 45° with Re = 10170. Friction factor ratio is increased by a factor of 1.88 to 5.03 on the whole. Friction factor ratio shows the similar variation trend to Nusselt number ratio, but the decreasing trend is less apparent than that of the Nusselt number ratio when the rib-groove inclination angle is more than 45°. PEC values, firstly, increase with the increasing rib-groove inclination then tends to decrease with further increasing the rib-groove inclination angle. PEC values vary from 1.19 to 1.68; the maximum value is obtained with  = 45 and Re = 6780. Effects of rib-groove inclination angle on the entropy generation number and the

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non-dimensional entropy generation number ratio are further investigated in the present work. Figures. 21 (a) and (b) show the variations of the entropy generation number and the non-dimensional entropy generation number ratio with different rib-groove inclination angles in the rib-grooved tubes, respectively. With a given Reynolds number, the entropy generation number decreases with the increasing rib-groove inclination angle up to = 45°, and beyond which it tends to increase. The entropy generation number reaches the minimum at = 45° and Re = 6780. It is seen from Fig. 21 (b) that the rib-grooved tubes are thermodynamically advantageous (η < 1) for all the cases investigated, and the rib-grooved tube with = 45° has the minimum non-dimensional entropy generation number ratio for a given Reynolds number. Therefore, the rib-grooved tube is recommended to have the rib-groove inclination angle of 45° in practical applications. In addition, a closer examination of Fig. 20 (a) and Fig. 21 (b) shows that red curve for Re = 10170 behaves different from others. The reason for this is that the Nusselt number ratio and the non-dimensional entropy generation number ratio are quite sensitive to both Reynolds number and the rib-groove inclination angle, especially at  = 45° and Re=10170, and the combined effects of Reynolds number and the rib-groove inclination angle result in the phenomena. 5.4. Correlations for Nusselt number (Nu) and friction factor (f) The results discussed above imply that the heat transfer and flow performance in the rib-grooved tube are strongly affected by geometric parameters and Reynolds numbers. Therefore, the Nusselt number and friction factor correlations for the rib-grooved tubes were derived to predict the heat transfer and pressure drop with respect to and geometric

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parameters, including three rib-groove pitch ratios (P* = P/D = 0.35, 0.47 and 0.59), three numbers of circumferential ribs and grooves (N = 4, 6 and 8) and four rib-groove inclination angles ( = 15°, 30°, 45° and 60°), and five Reynolds numbers in the range of 6780–20340. Derived Nusselt number and friction factor correlations are given as follows. N u  0 .0 0 6 9 4 R e f  0 .0 0 1 5 6 R e

0 .8 9 8 3 1

0 .1 0 0 1 5

 P *

 P *

 0 .3 7 7 5 7

 0 .7 3 6 3 7

N

N

0 .3 6 8 1 5





0 .5 9 2 5 1

0 .1 7 6 2 1

0 .3 4 5 4 8

(24) (25)

The coefficient of determination R2 is 0.97 for Nusselt number correlation and 0.95 for friction factor correlation. Fig. 22 (a) and (b) show the comparison of Nusselt number and friction factor between correlations and numerical data. As shown in Fig. 22, Nusselt numbers and friction factors predicted from Eqs. (24) and (25) agree well with the numerical results within the data range of ± 10% for Nusselt number and ± 20% for friction factor. 6. Conclusions A numerical study has been conducted to examine the impact of geometric parameters on the thermal-hydraulic performance in a rib-grooved heat exchanger tube. Moreover, entropy generation analysis was applied to determine thermal-hydraulically superior rib-groove geometry for heat transfer enhancement. The following results are obtained: 1. Multiple longitudinal swirl flows are generated in the rib-grooved tube due to the disturbances of ribs and grooves. This type of flow pattern leads to a relatively long flow path in the tube, and fluid between the wall and core flow regions is intensively mixed, which has a significant impact on the performance of heat transfer in the rib-grooved tube. 2. For the range of Reynolds numbers investigated, in the rib-grooved heat exchanger tube, the heat transfer ratios are about 1.58 to 2.46, while friction factor ratios are approximately

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1.82 to 5.03. PEC values vary from 1.19 to 1.68 for various combinations of the geometric parameters. 3. For all the cases studied, non-dimensional entropy generation number ratios are below unity indicating the thermodynamic advantage of the proposed rib-grooved tube. The entropy generation number decreases with the decreasing of rib-groove pitch ratio and the increasing of the number of circumferential ribs and grooves. The entropy generation number reaches the minimum when P* = 0.35, N = 8 and = 45°. Therefore, the proposed rib-grooved tube is recommended to have this combination of the geometric parameters in practical applications. Conflict of interest None Acknowledgments The work was supported by the National Key Basic Research Program of China (973 Program) (2013CB228302) and the National Natural Science Foundation of China (51376069). Nomenclature cp

specific heat (J kg-1 K-1)

D

inner diameter of the tube (m)

Ec

Eckert number

E

groove depth (m)

f

friction factor

H

rib height (m)

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h

average heat transfer coefficient (W m-2 K-1)

k

kinetic energy (m2 s-2)

L1

length of the upstream section (m)

L2

length of the test section (m)

L3

length of the downstream section (m)

L

length of rib or groove (m)

m

mass flow rate (kg s-1)

N

number of circumferential ribs and grooves

Nu

Nusselt number

Ns

entropy generation number

p

pressure (Pa)

p

pressure drop (Pa)

P

groove pitch (m)

P*

rib-groove pitch ratio

Pr

Prandtl number

PEC

performance evaluation criterion

Q

total rate of heat transfer (W)

q

heat flux per unit length (W m-3)

q

average heat flux (W m-2)

Re

Reynolds number

St

Stanton number

S gen

entropy generation rate (W K-1)

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T

temperature (K)

um

mean velocity (m s-1)

W

width of rib or groove (m)

Greek symbols

 

rib-groove inclination angle (o)





expansion coefficient (K-1)





dissipation (m2 s-3)

η



the non-dimensional entropy generation number ratio 

thermal conductivity (W m-1 K-1)



dimensionless length of the test section



dynamic viscosity (kg m-1 s-1)

t

turbulent eddy viscosity (kg m-1 s-1)





density (kg m-3)





dimensionless temperature difference

 

specific dissipation rate (s-1)

Subscripts e

enhanced tube

m

mean

i

inlet of the test section

o

outlet of the test section

0

smooth tube

w

wall

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f

fluid

References [1] 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 (2006) 895-907. [2] X.-Y. Tang, D.-S. Zhu, Flow structure and heat transfer in a narrow rectangular channel with different discrete rib arrays, Chemical Engineering and Processing: Process Intensification, 69 (2013) 1-14. [3] M.S. Lee, S.S. Jeong, S.W. Ahn, J.C. Han, Effects of angled ribs on turbulent heat transfer and friction factors in a rectangular divergent channel, International Journal of Thermal Sciences, 84 (2014) 1-8. [4] G. Xie, J. Liu, P. M. Ligrani, B. Sunden, Flow structure and heat transfer in a square passage with offset mid-truncated ribs, International Journal of Heat and Mass Transfer, 71 (2014) 44-56. [5] Z. Shen, Y. Xie, D. Zhang, Numerical predictions on fluid flow and heat transfer in U-shaped channel with the combination of ribs, dimples and protrusions under rotational effects, International Journal of Heat and Mass Transfer, 80 (2015) 494-512. [6] H.T. Wang, W.B. Lee, J. Chan, S. To, Numerical and experimental analysis of heat transfer in turbulent flow channels with two-dimensional ribs, Applied Thermal Engineering, 75 (2015) 623-634. [7] J. Liu, G. Xie, T.W. Simon, Turbulent flow and heat transfer enhancement in rectangular channels with novel cylindrical grooves, International Journal of Heat and Mass Transfer, 81 (2015) 563-577.

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[8] X.-Y. Tang, G. Jiang, G. Cao, Parameters study and analysis of turbulent flow and heat transfer enhancement in narrow channel with discrete grooved structures, Chemical Engineering Research and Design, 93 (2015) 232-250. [9] J. Lu, X. Sheng, J. Ding, J. Yang, Transition and turbulent convective heat transfer of molten salt in spirally grooved tube, Experimental Thermal and Fluid Science, 47 (2013) 180-185. [10] G. Xie, S. Zheng, B. Sundén, Heat Transfer and Flow Characteristics in Rib-/Deflector-Roughened Cooling Channels with Various Configuration Parameters, Numerical Heat Transfer, Part A: Applications, 67 (2015) 140-169. [11] G. Xie, Y. Song, M. Asadi, G. Lorenzini, Optimization of Pin-Fins for a Heat Exchanger by Entropy Generation Minimization and Constructal Law, Journal of Heat Transfer, 137 (2015) 061901. [12] Y. Song, M. Asadi, G. Xie, L. Rocha, Constructal wavy-fin channels of a compact heat exchanger with heat transfer rate maximization and pressure losses minimization, Applied Thermal Engineering, 75 (2015) 24-32. [13] J. Liu, Y. Song, G. Xie, B. Sunden, Numerical modeling flow and heat transfer in dimpled cooling channels with secondary hemispherical protrusions, Energy, 79 (2015) 1-19. [14] Y. Zhang, W. Gu, J. Han, Heat transfer and friction in rectangular channels with ribbed or ribbed-grooved walls, Journal of Heat Transfer, 116 (1994) 58-65. [15] A. Layek, J. Saini, S. Solanki, Heat transfer and friction characteristics for artificially roughened ducts with compound turbulators, International Journal of Heat and Mass

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Transfer, 50 (2007) 4845-4854. [16] A. Layek, J. Saini, S. Solanki, Effect of chamfering on heat transfer and friction characteristics of solar air heater having absorber plate roughened with compound turbulators, Renewable Energy, 34 (2009) 1292-1298. [17] S. Eiamsa-Ard, P. Promvonge, Thermal characteristics of turbulent rib-grooved channel flows, International communications in heat and mass transfer, 36 (2009) 705-711. [18] T. Xinyi, Z. Dongsheng, Experimental and numerical study on heat transfer enhancement of a rectangular channel with discontinuous crossed ribs and grooves, Chinese Journal of Chemical Engineering, 20 (2012) 220-230. [19] S. Skullong, S. Kwankaomeng, C. Thianpong, P. Promvonge, Thermal performance of turbulent flow in a solar air heater channel with rib-groove turbulators, International communications in heat and mass transfer, 50 (2014) 34-43. [20] A.N. Al-Shamani, K. Sopian, H. Mohammed, S. Mat, M.H. Ruslan, A.M. Abed, Enhancement heat transfer characteristics in the channel with Trapezoidal rib–groove using nanofluids, Case Studies in Thermal Engineering, 5 (2015) 48-58. [21] Y.-T. Yang, H.-W. Tang, B.-Y. Zeng, C.-H. Wu, Numerical simulation and optimization of turbulent nanofluids in a three-dimensional rectangular rib-grooved channel, International communications in heat and mass transfer, 66 (2015) 71-79. [22] A. Navaei, H. Mohammed, K. Munisamy, H. Yarmand, S. Gharehkhani, Heat transfer enhancement of turbulent nanofluid flow over various types of internally corrugated channels, Powder Technology, 286 (2015) 332-341. [23] H. Jia, W. Liu, Z. Liu, Enhancing convective heat transfer based on minimum power

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consumption principle, Chemical Engineering Science, 69 (2012) 225-230. [24] W. Liu, H. Jia, Z.C. Liu, H.S. Fang, K. Yang, The approach of minimum heat consumption and its applications in convective heat transfer optimization, International Journal of Heat and Mass Transfer, 57 (2013) 389-396. [25] H. Jia, Z.C. Liu, W. Liu, A. Nakayama, Convective heat transfer optimization based on minimum entransy dissipation in the circular tube, International Journal of Heat and Mass Transfer, 73 (2014) 124-129. [26] J. Wang, W. Liu, Z. Liu, The application of exergy destruction minimization in convective heat transfer optimization, Applied Thermal Engineering, 88 (2015) 384-390. [27] N. Zheng, W. Liu, Z. Liu, P. Liu, F. Shan, A numerical study on heat transfer enhancement and the flow structure in a heat exchanger tube with discrete double inclined ribs, Applied Thermal Engineering, 90 (2015) 232-241. [28] N. Zheng, P. Liu, F. Shan, Z. Liu, W. Liu, Heat transfer enhancement in a novel internally grooved tube by generating longitudinal swirl flows with multi-vortexes, Applied Thermal Engineering, 95 (2016) 421-432. [29] A. Şahin, A second law comparison for optimum shape of duct subjected to constant wall temperature and laminar flow, Heat and mass transfer, 33 (1998) 425-430. [30] A.Z. Şahin, Entropy generation in turbulent liquid flow through a smooth duct subjected to constant wall temperature, International Journal of Heat and Mass Transfer, 43 (2000) 1469-1478. [31] J. Meng, Enhanced heat transfer technology of longitudinal vortices based on field-coordination principle and its application, Power Engineering and Engineering

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Thermophysics, Tsinghua University, Beijing, (2003). [32] F. Menter, M. Kuntz, R. Langtry, Ten years of industrial experience with the SST turbulence model, Turbulence, heat and mass transfer, 4 (2003). [33] R.L. Webb, N.-H. Kim, Principles of enhanced heat transfer, Taylor Francis: New York, NY, USA, (1994). [34] J. Zhou, R.J. Adrian, S. Balachandar, T. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow, Journal of Fluid Mechanics, 387 (1999) 353-396.

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Figure captions Fig. 1. Schematic diagram of the computational domain with three sections.

Fig. 2. Configurations of the test section: (a) three-dimensional plot; (b) two-dimensional plot.

(a)

(b)

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Fig. 3. Comparison of the average Nusselt number between the experimental [31] and numerical results with different turbulence models.

Fig. 4. Grids generated for the computational domain.

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Fig. 5. Nusselt number and friction factor calculated by different mesh levels for the rib-grooved tube with P* = 0.47, N = 8 and  = 30° at Re = 10170.

Fig. 6. Comparison between numerical results and experimental results for the enhanced tube.

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Fig. 7. Local streamlines around ribs and grooves at Re = 10170.

Fig. 8. Three-dimensional streamlines along the flow direction in the rib-grooved tube at Re = 10170.

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Fig. 9. Profile of local heat flux on the tube wall at Re = 10170.

Fig. 10. Effects of rib-groove pitch ratio on the heat transfer and flow performance: (a) Nusselt number ratio; (b) friction factor ratio; (c) PEC.

(a)

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(b)

(c)

Fig. 11. Effects of rib-groove pitch ratio on the entropy generation number and the non-dimensional entropy generation number ratio: (a) entropy generation number; (b) the non-dimensional entropy generation number ratio.

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(a)

(b)

Fig. 12. Velocity vectors and surface streamlines with different numbers of circumferential ribs and grooves in the outlet cross-section of the test section at Re = 10170: (a) velocity vectors; (b) surface streamlines.

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Fig. 13. Contours of swirling strength and temperature in the outlet cross-section of the test section with different numbers of circumferential ribs and grooves at Re = 10170: (a) swirling strength; (b) temperature.

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Fig. 14. Distribution of the local entropy generation in the outlet cross-section of the test section with different numbers of circumferential ribs and grooves at Re = 10170.

Fig. 15. Effects of the number of circumferential ribs and grooves on the performances of heat transfer and flow: (a) Nusselt number ratio; (b) friction factor ratio; (c) PEC.

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(a)

(b)

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(c)

Fig. 16. Effects of the number of the circumferential ribs and grooves on the entropy generation number and the non-dimensional entropy generation number ratio: (a) entropy generation number; (b) the non-dimensional entropy generation number ratio.

(a)

(b)

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Fig. 17. Velocity vectors and surface streamlines with different rib-groove inclination angles in the outlet cross-section of the test section at Re = 10170: (a) velocity vectors; (b) surface streamlines.

Fig. 18. Contours of swirling strength and temperature in the outlet cross-section of the test section with different rib-groove inclination angles at Re = 10170: (a) swirling strength; (b) temperature.

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Fig. 19. Distribution of the local entropy generation in the outlet cross-section of the test section with different rib-groove inclination angles at Re = 10170.

Fig. 20. Effects of rib-groove inclination angle on the heat transfer and flow performance: (a) Nusselt number ratio; (b) friction factor ratio; (c) PEC.

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(a)

(b)

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(c)

Fig. 21. Effects of rib-groove inclination angle on the entropy generation number and the non-dimensional entropy generation number ratio: (a) entropy generation number; (b) the non-dimensional entropy generation number ratio.

(a)

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(b)

Fig. 22. Comparison of Nusselt number and friction factor between correlations and numerical data: (a) Nusselt number; (b) friction factor.

(a)

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(b)

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Table 1. Thermal properties of water. Parameter specific heat cp (J kg-1 K-1) dynamic viscosity  (kg m-1 s-1) density  (kg m-3) thermal conductivity (W m-1 K-1)

Value 4182 0.001003 998.2 0.6

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