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Numerical investigation of heat transfer enhancement inside heat exchanger tubes fitted with perforated hollow cylinders
International Journal of Thermal Sciences 147 (2020) 106153
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Numerical investigation of heat transfer enhancement inside heat exchanger tubes fitted with perforated hollow cylinders M.E. Nakhchi, J.A. Esfahani * Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Perforated hollow cylinder Turbulent flow Thermal performance Vortex generation Heat transfer enhancement
This paper investigates the flow structure and thermal performance characteristics of the fluid flow through a heat exchanger tube fitted with perforated hollow cylinders (PHCs) under turbulent flow regime. The effects of the perforated index (0.08 < PI < 0.24), diameter ratio of the hollow cylinder (0.5 < d/D < 0.9) and Reynolds number ð6000 < Re < 16000Þ on the thermal performance and heat transfer enhancement are investigated. The vortex flow generated near the holes leads to better fluid mixing between the tube wall and the core regions and this recirculating flow enhances the heat transfer rate in comparison with a plain tube. The numerical results indicate that the flow resistance can be reduced up to 86.2% with increasing the perforated index from 0.08 to 0.24. The maximum thermal performance value of 1.456 could be achieved for the case of d/D ¼ 0.74 and PI ¼ 24% at Re ¼ 6000. The results show that the fluid mixing between the core region and the tube walls for the case of 0.7 is higher than the other cases. Therefore, the heat transfer rate is expected to be better for the case of PI ¼ 0.08 and d/D ¼ 0.7 due to the thermal boundary layer destruction caused by the PHCs.
1. Introduction Heat transfer augmentation is essential in the design of the heat exchangers and other thermal systems such as solar collectors [1], microchannel heat sinks [2], melting process [3], chemical industry, etc. Reducing the size of heat exchangers and improving the performance play an important role in the design of heat exchangers [4]. Several passive techniques such as using grooved geometries [5], wavy surfaces [6–9], twisted tapes [10–12], conical inserts [13], louvered strip inserts [14] V-shaped ribs [15], transverse ribs [16,17] and other types of vortex generators [18–20] are employed in the past years to improve the thermal performance of heat transfer devises. Several numerical and experimental studies have investigated the effects of different vortex generators and turbulators on the thermal performance enhancement of different types of heat exchangers. Chang and Huang [21] conducted experimental work to investigate the effect of spiky ribbed twisted-tapes with and without edge notches on heat transfer enhancement of heat exchanger tubes. They observed that this modified twisted tape could significantly improve the thermal perfor mance in comparison with conventional twisted tapes. Eiamsa-Ard and Promvonge [22] experimentally investigated the Nusselt number and friction factor of water fluid flow through tubes enhanced with
clockwise and counter-clockwise twisted tapes. It was concluded that the Nusselt numbers in the heat exchanger tube equipped with the counter-clockwise twisted tapes are higher than those with the con ventional twisted tapes up to 41.9%. Promvonge and Eiamsa-Ard [23] studied the effect of combined use of conical rings and twisted tapes on heat transfer enhancement inside circular tubes and reported that the maximum performance factor of 1.96 is obtained for using the conical-ring and the twisted-tape with twist ratio of 3.75. Different types of perforated turbulators are developed in the past years to generate vortex flows and improve the thermal performance of different types of heat exchangers. Bhuiya et al. [24] experimentally investigated the heat transfer enhancement and friction loss of air fluid flow through tubes fitted with perforated twisted tapes. They observed that the Nusselt number and thermal performance factor could be enhanced up to 340% and 59%, respectively in comparison with con ventional simple circular heat exchanger tubes. Kongkaitpaiboon et al. [25] developed a new type of perforated conical rings (PCRs) to enhance the fluid mixing between the core regions and the tube wall. It was concluded that 137% heat transfer enhancement could be achieved over that in a plain tube by using PCRs with pitch ratio of 4. Chamoli et al. [26] numerically investigated the heat transfer, friction factor and the thermal performance factor in a heat exchanger tube fitted with
* Corresponding author. E-mail addresses: [email protected] (M.E. Nakhchi), [email protected] (J.A. Esfahani). https://doi.org/10.1016/j.ijthermalsci.2019.106153 Received 23 May 2019; Received in revised form 10 September 2019; Accepted 21 October 2019 Available online 1 November 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Fig. 1. Schematic of the circular channel equipped with PHCs. Table 2 Grid independency for Re ¼ 16000, PI ¼ 24% and d/D ¼ 0.9.
Table 1 The geometrical parameters of the heat exchanger tube equipped with PHCs. Parameter
Value
Tube diameter (D) Tube length (L) Perforated index ðPIÞ
66 mm 1350 mm 8%, 16%, 24%
Perforated cylinder diameter ðdÞ
33, 46.2, 59.4 mm
Diameter ratio (d/D) Perforated cylinder length ðlÞ
0.5, 0.7, 0.9 135 mm
Perforated cylinder thickness ðtÞ
1 mm
diameter of hole ðdh Þ
5 mm
Pitch ratio Re ¼ ρuD=μ
Grid number
Nu
� � iþ1 �Nu Nui �� � � Nuiþ1 �
f
� � �f iþ1 f i � � � � iþ1 � � � f
467721 893448 1253583 1605402
71.66 75.85 77.10 77.12
– 0.055 0.016 0.0003
0.0510 0.0540 0.0572 0.0571
– 0.055 0.059 0.002
Table 3 Maximum values of yþ for Re ¼ 16000.
2 6000–16000
Case
Tube wall
PHC wall
d/D ¼ 0.5, PI ¼ 8% d/D ¼ 0.7, PI ¼ 8% d/D ¼ 0.9, PI ¼ 8% d/D ¼ 0.5, PI ¼ 16% d/D ¼ 0.7, PI ¼ 16% d/D ¼ 0.9, PI ¼ 16% d/D ¼ 0.5, PI ¼ 24% d/D ¼ 0.7, PI ¼ 24% d/D ¼ 0.9, PI ¼ 24%
1.241 1.354 1.862 1.382 1.508 2.115 1.498 1.973 2.341
1.352 1.446 1.903 1.426 1.634 2.163 1.605 2.225 2.583
perforated vortex generators. It was observed that the thermal perfor mance of 1.65 at Re ¼ 3000 could be reached by using turbulators with perforated index of 16%. Skullong et al. [18] presented a numerical analysis to investigate the effect of staggered-winglet perforated tapes on the thermal performance of turbulent fluid flow through heat exchanger tubes. They concluded that using perforated tape with a blockage ratio of 0.15, pitch ratio of 1.0 at Re ¼ 4180 could enhance the thermal performance factor up to 1.71. Sheikholeslami and Ganji [27] conducted a numerical investigation on perforated turbulators with different pitch ratios for thermal performance enhancement through heat exchanger tubes. It was observed that the thermal performance factor of 1.59 could be achieved by using perforated turbulators with a pitch ratio of 1.07. Nakhchi and Esfahani [28] numerically investigated the effects of perforated conical rings with different geometries on the thermal performance enhancement of heat exchangers. Their work revealed that the maximum thermal performance factor of 1.241 could be obtained by using 10 PCRs with d/D ¼ 0.1 at Re ¼ 4000. They also concluded that the Nusselt number of PCRs reduces up to 35.48% with
Fig. 2. The computational domain with detailed views of the tube inlet and the PHCs. 2
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Table 4 Validation of the numerical simulations with experimental results of Singh et al. [29] for the Nusselt number and friction factor for two different PHC geometries. Re
d/D ¼ 0.5 and PI ¼ 8%
d/D ¼ 0.5 and PI ¼ 24%
Nu 6000 11000 16000
f
Nu
f
Present study
Exp. [29]
Present study
Exp. [29]
Present study
Exp. [29]
Present study
Exp. [29]
46.22 74.51 96.23
46.67 74.82 96.88
0.140 0.120 0.113
0.142 0.123 0.112
36.05 57.94 74.65
36.30 58.13 75.03
0.058 0.051 0.047
0.060 0.052 0.048
increasing the number of PCR holes from 4 to 10. Singh et al. [29] conducted an experimental study to investigate the effect of perforated hollow cylinders on the thermal performance factor and heat transfer augmentation of the air fluid flow through circular tubes. Their results show that the thermal performance factor was in the range of 1.21–1.47 and it is maximum at perforated index ðPI ¼ Aholes =Atot Þ of 0.08 and diameter ratio of 0.65. Based on the above literature review, the perforated hollow cylinders can significantly improve the thermal performance of thermal systems. Although the experimental investigation of Singh et al. [29] investigated the effect of design parameters on the Nusselt number and friction factor of the fluid flow through tubes fitted with PHCs, a numerical study is necessary for better understanding the physical effects of different geometrical parameters on the turbulent flow characteristics in the presence of the PHCs. Also, the effect of the diameter ratio needs to be further investigated to determine in which conditions the maximum thermal performance factor could be obtained. Accordingly, the main goals in the current investigation are to cover this topic and develop correlations to predict the thermal performance and heat transfer enhancement of the system as functions of the design parameters. For this purpose, different perforated hollow cylinders with variable diam eter ratios and perforated index are mounted at the center of a heat exchanger tube.
length (L ¼ 1350 mm), the thickness of the hollow cylinders (t ¼ 1 mm) and holes diameter ðdh ¼ 5mmÞ are kept constant in the present study. The water inlet temperature (Tin) is assumed to be 300 K and the tube wall temperature (Tw) is kept uniform at 350 K. To reduce the inlet ef fects, an extra length ðLin ¼ 1000mmÞ is added to the tube inlet. The gravitational effects are assumed to be negligible and the Reynolds number is varied from 6000 to 16000. Pressure outlet boundary con dition is selected at the outlet of the tube. The PHCs are assumed to be adiabatic and no-slip boundary conditions are imposed on the walls. The hollow cylinders inside the heat exchanger tube are usually fixed with a thin connecting rod. However, the effects of this thin rod are neglected in the present numerical simulations to make better discussions about the physical effects of the PHC on turbulent kinetic energy, velocity vectors and other CFD results. It also should be pointed out that the effects of this thin rod on the thermal characteristics are negligible. The boundary values of the turbulence parameters near the wall are described by the enhanced wall function method. The turbulence in tensity is the tube inlet is kept constant at 10%. 3. Physical and numerical models 3.1. Governing equations The fluid flow is steady, incompressible and three-dimensional. Gravitational effects are neglected. Based on the Reynolds average Navier Stokes (RANS) model, the continuity, momentum, and energy equations are as follows [5,11,30]:
2. Physical description Fig. 1 shows a schematic of the heat exchanger tube fitted with perforated hollow cylinders and the coordinate system. Water with constant physical properties is selected as the working fluid. Table 1 shows the geometrical parameters used in the present numerical study. The perforated index varies from 8% to 24%, while the diameter ratio (d/D) is ranged from 0.5 to 0.9. The tube diameter (D ¼ 66 mm), tube
∂ ðρui Þ ¼ 0 ∂xi
Fig. 3. Turbulent kinetic energy contours in the tube fitted with PHCs at Re ¼ 6000. 3
(1)
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Fig. 4. Turbulent dissipation rate contours in the tube fitted with PHCs at Re ¼ 6000.
� �
� ∂ u ρu ¼ ∂xj j i
��
∂p ∂ ∂u μ i þ ∂x ∂xj ∂xj
∂ ∂ ðρui TÞ ¼ ∂xj ∂xi
��
μ
Pr
�
þ
μt ∂T Prt ∂xi
finite volume method with SIMPLE algorithm. The momentum, energy, turbulent kinetic energy, and dissipation rate are discretized by using second-order upwind scheme. The residuals of continuity, momentum, k and ε equations for the convergence of the numerical computations are set to be less than 10 5 and 10 8 for the energy equation. A grid independence test is performed (Table 2) to check the accu racy of the computational grid for calculating the Nusselt number and friction factor through heat exchanger tube fitted with PHCs with PI ¼ 24% and d/D ¼ 0.9 at Re ¼ 16000. It can be seen that the relative difference between the mesh numbers of 1253583 and 1605402 is about 0.03% for the Nusselt number and 0.2% for the friction factor. There fore, the grid number of 1253583 is chosen for further numerical simulations. The maximum values of yþ on the tube wall and the walls of the PHCs with different geometries at Re ¼ 16000 are shown in Table 3. It can be observed that the maximum yþ values are less than 3 for all of the test cases. This means that the viscous sub-layer effects are captured correctly in the numerical simulations of turbulent fluid flow through tube equipped with PHCs. It can be seen the yþ values near the tube wall are generally smaller than that on the PHC wall. This is mainly due to boundary layer mesh (inflation) used near the tube walls.
(2)
ρu’i u’j �
(3)
where ρ, p, u and μ, are fluid density, pressure, turbulent fluctuations, pressure and dynamic viscosity of the water, respectively. The Reynolds 0
stress term ðρu’i u’j Þ can be formulated as: � � ∂u ∂u 2 2 ∂uk ρu’i u’j ¼ μt i þ j ρkδij μ δij 3 3 t ∂xk ∂xj ∂xi
(4)
The equations of turbulent kinetic energy (k) and dissipation rate (ε) are: �� � � ∂ ∂ μt ∂k þ Gk ρε ðρui kÞ ¼ þμ (5) ∂xi ∂xj σ k ∂xj
∂ ∂ ðρui εÞ ¼ ∂xi ∂xj
��
�
�
μt ∂ε ε þμ þ ½C1ε Gk ∂xj σε k
ρC2ε ε�
(6)
In the above equations, σ k ¼ 1, σ ε ¼ 1:3, C1ε ¼ 1:42 and C2ε ¼ 1:68 are constants of the (RNG) k ε turbulent model.
3.3. Code validation
3.2. Computational domain
In order to show the accuracy of the numerical analysis, comparison between the results of the present study and the experimental results of Singh et al. [29] for the case of uniform wall heat flux at three different Reynolds numbers is made in Table 4. Four different turbulence models were employed for the numerical simulations (RNG k-e, standard k-e, k-w and SST k-w model). It was observed that the numerical results of the RNG k-e model are in excellent agreement with experimental data.
The computational domain with detailed views of the mesh at the inlet of the heat exchanger tube and the PHCs are shown in Fig. 2. The smaller mesh is used near the tube walls to capture the turbulence flow characteristics and resolve the viscous sub-layer effects near the tube walls. ANSYS Fluent 18.1 is selected for the numerical analysis by using 4
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Re ¼
Nu ¼
uD
ν hD q} D ¼ k ðTw Tb Þ k
(7) (8)
where h is the local heat transfer coefficient, k the thermal conductivity of water u the inlet velocity, ν the kinematic viscosity of water, Tw the wall temperature and Tb the bulk temperature of the fluid flow. The friction factor (f) and thermal performance factor ðηÞ are defined as follows: f¼
η¼
2D ΔP L ρu2 ðNu=Nus Þ ðf =fs Þ1=3
(9) (10)
ΔP is the total pressure difference and L is length of the pressure section. Nus and fs are defined as the Nusselt number and friction loss of the fluid flow through a smooth tube which are obtained from the nu merical simulations. The heat transfer (Nu) and friction factor results (f) are validated with the experimental correlations of Dittus-Boelter and Gnielinski [33] for Nusselt number and the correlations of Blasius and Petukhov [33] for the friction factor (f). It was observed that the nu merical results are in excellent agreement with experimental data. 4. Results and discussion Fig. 3 shows the turbulent kinetic energy contours of water fluid flow through heat exchanger tube equipped with PHCs with different diam eter ratios and perforated index values. It can be observed that the Fig. 5. The y-velocity contours and velocity vectors in the tube fitted with PHCs at Re ¼ 6000.
Therefore RNG k-e model is employed for the numerical simulations. It can be observed that the numerical results are in good agreement with experimental data. The results show that both of the Nusselt number and friction factor decreases with increasing the perforated index (PI) from 8% to 24%. As the number of holes increases, the amount of fluid mixing between the PHCs and the tube wall decreases which reduces the ther mal boundary layer disruption. The results show that the friction factor decreases with increasing the Reynolds number, which is similar to friction loss trend on the Moody chart for turbulent flows. 3.4. Data reduction The Reynolds number (Re) and Nusselt number (Nu), can be defined as [31,32]:
Fig. 7. The axial velocity contours in the tube fitted with PHCs at Re ¼ 6000.
Fig. 6. Contours of the turbulent kinetic energy inside a circular tube equipped with PHC with φ ¼ 8% and d=D ¼ 0:7 at Re ¼ 6000. 5
International Journal of Thermal Sciences 147 (2020) 106153
M.E. Nakhchi and J.A. Esfahani
highest turbulent dissipation rate appears near the PHC holes. Thus, the best fluid mixing condition can occur in this region. It is clear that the turbulent dissipation rate for the case of d/D ¼ 0.9 is very small near the tube walls. The physical reason of this is that the fluid flow through the narrow gap between the cylinder and the tube is much less than the flow rate in the core region. As a result, there is limited possibility of mixing the flow between these two regions. Velocity vectors can also explain the physical reason. It can be seen that for the cases of d/D ¼ 0.9, the fluid flow is only moves toward the center of the tube, resulting in a reduction in the mixing and heat transfer rate. The results show that the fluid mixing between the core region and the tube walls for the cases of (a) and (b) is higher than the other cases. It can also be seen that for the case (a), the fluid flow doesn’t have an effective interaction with the tube wall and as a results, it is not able to effectively destruct the thermal boundary near the wall. Therefore, the heat transfer rate is expected to be better for the case of PI ¼ 0.08 and d/D ¼ 0.7. Fig. 5 shows the y velocity contours and velocity vectors for different PHC geometries. The velocity vectors show that the fluid mixing between the core regions and tube wall near the holes is considerably higher than the other parts. When the fluid flow passes through the hole, a part of the fluid is directed upwards and the other part moves downwards. As a result, the flow mixing is improved and the hot fluid near the wall is moved to the central part of the pipe. The red contours show the upward flow direction, while the blue velocity contours show the fluid flow in the downward direction. It can be observed that the y-velocity contours near the PHC with d/D ¼ 0.7 is higher than the other cases. Fig. 6 shows a detailed view of the turbulent kinetic energy contours near the PHC and the tube walls for the case of φ ¼ 8% and d=D ¼ 0:7. It can be seen that the flow disturbance near the holes increases in the axial direction. This means that better fluid mixing between the tube wall and the core regions results in thermal boundary layer disruption near the tube walls. It also can be observed that strong vortex flow occurs near the outlet of the PHC. This strong turbulence intensification can considerably in crease the Nusselt number and friction factor of the water fluid flow through the heat exchanger tube. Fig. 7 shows the axial velocity contours of turbulent fluid flow (Re ¼ 6000) through heat exchanger tube equipped with PHCs with different diameter ratios. It can be observed that for the case of d/D ¼ 0.5, the axial velocity near the tube wall is strong, while the axial velocity through the hollow cylinder is low. The results indicate that for the case of d/D ¼ 0.7, fluid velocity through the hollow cylinders and near the walls is more uniform than the other cases. This means that better fluid mixing makes the flow more uniform and as a result, the heat transfer would be enhanced. It also can be observed that for the case of d/D ¼ 0.9, the axial flow velocity is small and negligible in the region between the PHC and the tube wall. This reduces the fluid mixing and thermal boundary layer disruption near the walls. One of the objects in the present study is to drive correlations for the Nusselt number, friction factor and thermal performance factor as func tions of the design parameters (Re, PI, d/D). Response surface methodol ogy is employed as a powerful method to obtain second-order polynomial correlations for the flow characteristics through a heat exchanger tube equipped with PHCs. MINITAB software with RSM face-centered design is employed for the analysis. Based on the consents of this method, 30 nu merical data are required as inputs. Regression analysis of the numerical results is performed. The following polynomial correlations for Nu, f, and η are generated in Eqns. (11)–(13), respectively. These correlations are valid in the range of (0.08 < PI < 0.24, 0.5 < d/D < 0.9, 6000 < Re < 16000).
Fig. 8. The effect of design parameters on averaged Nusselt number inside tube fitted with PHCs.
Fig. 9. The effect of design parameters on friction factor inside tube fitted with PHCs.
turbulent kinetic energy for the case of PI ¼ 8%, d= D ¼ 0:7 is the highest among the tested geometries. It can be observed that for PI ¼ 8% the turbulent kinetic energy enhances with increasing the diameter ratio from 0.5 to 0.7 and then it decreases as the diameter ratio increases from 0.7 to 0.9. Physically speaking, for d/D ¼ 0.5 the hollow cylinder is too small and it is not able to increase the flow disturbance between the tube walls and the core regions. With increasing the diameter ratio to 0.7, turbulent kinetic energy (TKE) and flow perturbation significantly in crease due to better fluid mixing between the tube wall and the central regions. It also can be observed that the holes have a considerable effect on TKE enhancement for all of the test cases. In other words, the flow disturbance near the perforated walls is the main reason for thermal boundary layer disruption near the tube walls which improves the heat transfer rate in comparison with a tube without any inserts. The results show that with increasing the PHC diameter ratio to 0.9, the gap be tween the cylinder and the tube wall becomes very small. As a result, the fluid flow mixing reduces near the holes of the hollow cylinder. The results show that the turbulent kinetic energy production near the PHC with PI ¼ 24% considerably reduces in comparison with other geome tries. This is mainly due to the smaller mass flow rate in the presence of higher number of holes. Fig. 4 depicts the turbulent dissipation rate contours in cross section plane of Z ¼ 0.32 for different d/D and PI values at Re ¼ 6000. The
Nu ¼
60:88 þ 7:34. � 10 3 Re 3
� 2:41 �.PI þ 204:34 � d D
þ0:25 � 10 Re � d D þ 46:87PI � d D
6
8:12 � 10 3 Re � PI 2
0:061 � 10 Re
85:22PI 2
151:14ðd=DÞ2
6
(11)
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Fig. 10. Surface contours of Nusselt number, friction factor and η as functions of the design parameters.
� 3 3 0:006 . � 10 Re 0:77 � PI þ 0:62d D þ 0:007 � 10 Re � PI 2 6 2 2 0:43ðd=DÞ þ0:13PI � d D þ 0:00014 � 10 Re þ 0:56PI
f ¼ 0:02
(12) �
3 3 η ¼ 0:84 0:007 . � 10 Re þ 0:28 � PI þ 1:65 � d D þ 0:0012 � 10 Re � PI
0:03PI � d D
0:32 � PI 2
1:11 � ðd=DÞ
2
(13) The deviation between the data predicted by the correlations of Nu, f and η and numerical results are in �3.4% for Nu, �6.9% for f and �7.4 for η, respectively. The variations of the average Nusselt number with respect to the Reynolds number is presented in Fig. 8. It can be observed that the Nusselt number increases with the Reynolds number. This is mainly due to stronger temperature gradients near the tube wall in the presence of water fluid flows with higher velocities. The results indicate that the Nusselt number enhances up to 25.9% with decreasing the PI from 0.24 to 0.08 at Re ¼ 16000 and d/D ¼ 0.7. As discussed earlier, fluid mixing through tubes equipped with PHCs with smaller number of holes is stronger which results in better fluid mixing and thermal boundary layer disruption near the tube walls. It can be observed that
Fig. 11. Effect of diameter ratio on η for PI ¼ 0.24.
7
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
the heat transfer rate for the fluid flow through tube fitted with PHCs with d/D ¼ 0.7 is higher than the other diameter ratios, which is in agreement with previous physical discussions of Fig. 4. The results show that the heat transfer rate enhances up to 222.7% in comparison with the plain tube. The maximum average Nusselt number of 104.21 could be achieved by mounting PHCs with PI ¼ 0.08 and d/D ¼ 0.7 through heat exchanger tubes at Re ¼ 16000. Fig. 9 shows the effect of different design parameters on the friction factor variations of the fluid flow inside the tube fitted with PHCs. The results show that the average friction factor reduces with increasing the Reynolds number. It also can be observed that the friction factor in creases up to 86.2% with decreasing the number of holes (PI) from 0.24 to 0.0.08 at Re ¼ 11000 and d/D ¼ 0.7. When the PHCs with smaller number of holes are mounted in the circular tube, viscous dissipation rate of the water flow near the tube walls enhances and therefore, the pressure drop enhances. The other main physical reason for friction factor increment is vortex flow between the tube wall and core region near the holes. It also can be observed that the friction factor of the fluid flow through tube fitted with PHCs with b/D ¼ 0.70 is higher than that for the other diameter ratios. The friction factor increases up to 421.2% in comparison with the plain tube. Surface contours of the Nusselt number, friction factor and thermal performance factor ðηÞ of turbulent fluid flow through heat exchanger tube fitted with PHCs are shown in Fig. 10. Based on the definition of the thermal performance, η > 1 is favorable and indicates that the PHCs have sufficient influence on the heat transfer enhancement. The results indicate that thermal performance decreases with increasing the Rey nolds number. It also should be pointed out that the thermal perfor mance increases up to 5.24% with increasing the perforated index (PI) from 8% to 24%. Based on the results of Fig. 10, it was observed that the maximum thermal performance factor is occurred in the range of ð0:6 < d =D < 0:8Þ. In order to better investigate the effect of the diam eter ratio on the thermal performance of the heat exchanger, the η variations with respect to d/D are shown in Fig. 11. It is seen that the η parameter increases with increasing d/D from 0.6 to 0.74 and then de creases. As a result, at d/D ¼ 0.74, the thermal performance is maximum by considering the heat transfer rate and the pressure drop. The results show that the thermal performance factor decreases with increasing the Reynolds number from 6000 to 16000. The maximum thermal perfor mance value of 1.456 could be achieved for the case of d/D ¼ 0.74 and PI ¼ 24% at Re ¼ 6000.
rate of the water flow near the tube walls enhances and conse quently, the pressure drop enhances. � The maximum thermal performance value of 1.456 could be ach ieved for the case of d/D ¼ 0.74 and PI ¼ 24% at Re ¼ 6000. Acknowledgement This research was supported by the Office of the Vice Chancellor for Research, Ferdowsi University of Mashhad, under Grant No. 49607. References [1] A. Saravanan, S. Jaisankar, Heat transfer augmentation techniques in forced flow V-trough solar collector equipped with V-cut and square cut twisted tape, Int. J. Therm. Sci. 140 (2019) 59–70. [2] L. Lin, J. Zhao, G. Lu, X.-D. Wang, W.-M. Yan, Heat transfer enhancement in microchannel heat sink by wavy channel with changing wavelength/amplitude, Int. J. Therm. Sci. 118 (2017) 423–434. [3] M. Jourabian, M. Farhadi, A.R. Darzi, Constrained ice melting around one cylinder in horizontal cavity accelerated using three heat transfer enhancement techniques, Int. J. Therm. Sci. 125 (2018) 231–247. [4] E.-J. 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Fan, J. Deng, A. Nakayama, W. Liu, Parametric study on turbulent heat transfer and flow characteristics in a circular tube fitted with louvered strip inserts, Int. J. Heat Mass Transf. 55 (2012) 5205–5213. [15] S. Li, G. Xie, B. Sunden, Numerical investigation of fluid flow structure and heat transfer in a passage with continuous and truncated V-shaped ribs, Int. J. Numer. Methods Heat Fluid Flow 25 (2015) 171–189. [16] N. Massarotti, P. Nithiarasu, O. Manca, S. Nardini, D. Ricci, Numerical investigation of air forced convection in channels with differently shaped transverse ribs, Int. J. Numer. Methods Heat Fluid Flow 21 (2011) 618–639. [17] O. Manca, S. Nardini, D. Ricci, Numerical analysis of water forced convection in channels with differently shaped transverse ribs, J. Appl. Math. (2011) 2011. [18] S. Skullong, P. Promvonge, C. Thianpong, M. Pimsarn, Heat transfer and turbulent flow friction in a round tube with staggered-winglet perforated-tapes, Int. J. Heat Mass Transf. 95 (2016) 230–242. [19] X. Zhang, Z. Liu, W. Liu, Numerical studies on heat transfer and friction factor characteristics of a tube fitted with helical screw-tape without core-rod inserts, Int. J. Heat Mass Transf. 60 (2013) 490–498. [20] M.-A. Moon, M.-J. Park, K.-Y. Kim, Evaluation of heat transfer performances of various rib shapes, Int. J. Heat Mass Transf. 71 (2014) 275–284. [21] S.W. Chang, B.J. Huang, Thermal performances of tubular flows enhanced by ribbed spiky twist tapes with and without edge notches, Int. J. Heat Mass Transf. 73 (2014) 645–663. [22] S. Eiamsa-Ard, P. Promvonge, Performance assessment in a heat exchanger tube with alternate clockwise and counter-clockwise twisted-tape inserts, Int. J. Heat Mass Transf. 53 (2010) 1364–1372. [23] P. Promvonge, S. Eiamsa-ard, Heat transfer behaviors in a tube with combined conical-ring and twisted-tape insert, Int. Commun. Heat Mass Transf. 34 (2007) 849–859. [24] M. Bhuiya, M. Chowdhury, M. Saha, M. Islam, Heat transfer and friction factor characteristics in turbulent flow through a tube fitted with perforated twisted tape inserts, Int. Commun. Heat Mass Transf. 46 (2013) 49–57.
5. Conclusion In the present numerical study, turbulent flow characteristics through heat exchanger tube equipped with perforated hollow cylinders are investigated, to investigate the effects of diameter ratio (d/D) and perforated index (PI). The Reynolds number varies from 6000 to 16000. (RNG) k–ε turbulent model is employed in the three-dimensional sim ulations. The main findings are as follows: � The turbulent kinetic energy enhances with increasing the diameter ratio from 0.5 to 0.7 and then it decreases as the diameter ratio in creases from 0.7 to 0.9. Physically speaking, for d/D ¼ 0.5 the hollow cylinder is too small and it is not able to increase the flow disturbance between the tube walls and the core regions. With increasing the diameter ratio to 0.7, turbulent kinetic energy (TKE) and flow perturbation significantly increase due to better fluid mixing be tween the tube wall and the central regions. � Correlations are developed to predict the thermal performance, friction factor and Nusselt number of the fluid flow through a tube fitted with PHCs as functions of the design parameters. � The friction factor reduces up to 86.2% with increasing the perfo rated index (PI) from 0.08 to 0.24. When the PHCs with smaller number of holes are mounted in the circular tube, viscous dissipation 8
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
[25] V. Kongkaitpaiboon, K. Nanan, S. Eiamsa-Ard, Experimental investigation of heat transfer and turbulent flow friction in a tube fitted with perforated conical-rings, Int. Commun. Heat Mass Transf. 37 (2010) 560–567. [26] S. Chamoli, R. Lu, P. Yu, Thermal characteristic of a turbulent flow through a circular tube fitted with perforated vortex generator inserts, Appl. Therm. Eng. 121 (2017) 1117–1134. [27] M. Sheikholeslami, D. Ganji, Heat transfer improvement in a double pipe heat exchanger by means of perforated turbulators, Energy Convers. Manag. 127 (2016) 112–123. [28] M. Nakhchi, J. Esfahani, Numerical investigation of different geometrical parameters of perforated conical rings on flow structure and heat transfer in heat exchangers, Appl. Therm. Eng 156 (2019) 494–505.
[29] S.K. Singh, M. Kumar, A. Kumar, A. Gautam, S. Chamoli, Thermal and friction characteristics of a circular tube fitted with perforated hollow circular cylinder inserts, Appl. Therm. Eng. 130 (2018) 230–241. [30] M. Nakhchi, J. Esfahani, Entropy generation of turbulent Cu–water nanofluid flow in a heat exchanger tube fitted with perforated conical rings, J. Therm. Anal. Calorim 138 (2019) 1423–1436. [31] M. Nakhchi, J.Esfahani, Numerical investigation of turbulent Cu-water nanofluid in heat exchanger tube equipped with perforated conical rings, Adv. Powder Technol. 30(7) 1338-1347. [32] H. Mohammed, A.K. Abbas, J. Sheriff, Influence of geometrical parameters and forced convective heat transfer in transversely corrugated circular tubes, Int. Commun. Heat Mass Transf. 44 (2013) 116–126. [33] F. Incropera, D. DeWitt, Introduction to Heat Transfer, 1985.
9
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Numerical investigation of heat transfer enhancement inside heat exchanger tubes fitted with perforated hollow cylinders M.E. Nakhchi, J.A. Esfahani * Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Perforated hollow cylinder Turbulent flow Thermal performance Vortex generation Heat transfer enhancement
This paper investigates the flow structure and thermal performance characteristics of the fluid flow through a heat exchanger tube fitted with perforated hollow cylinders (PHCs) under turbulent flow regime. The effects of the perforated index (0.08 < PI < 0.24), diameter ratio of the hollow cylinder (0.5 < d/D < 0.9) and Reynolds number ð6000 < Re < 16000Þ on the thermal performance and heat transfer enhancement are investigated. The vortex flow generated near the holes leads to better fluid mixing between the tube wall and the core regions and this recirculating flow enhances the heat transfer rate in comparison with a plain tube. The numerical results indicate that the flow resistance can be reduced up to 86.2% with increasing the perforated index from 0.08 to 0.24. The maximum thermal performance value of 1.456 could be achieved for the case of d/D ¼ 0.74 and PI ¼ 24% at Re ¼ 6000. The results show that the fluid mixing between the core region and the tube walls for the case of 0.7 is higher than the other cases. Therefore, the heat transfer rate is expected to be better for the case of PI ¼ 0.08 and d/D ¼ 0.7 due to the thermal boundary layer destruction caused by the PHCs.
1. Introduction Heat transfer augmentation is essential in the design of the heat exchangers and other thermal systems such as solar collectors [1], microchannel heat sinks [2], melting process [3], chemical industry, etc. Reducing the size of heat exchangers and improving the performance play an important role in the design of heat exchangers [4]. Several passive techniques such as using grooved geometries [5], wavy surfaces [6–9], twisted tapes [10–12], conical inserts [13], louvered strip inserts [14] V-shaped ribs [15], transverse ribs [16,17] and other types of vortex generators [18–20] are employed in the past years to improve the thermal performance of heat transfer devises. Several numerical and experimental studies have investigated the effects of different vortex generators and turbulators on the thermal performance enhancement of different types of heat exchangers. Chang and Huang [21] conducted experimental work to investigate the effect of spiky ribbed twisted-tapes with and without edge notches on heat transfer enhancement of heat exchanger tubes. They observed that this modified twisted tape could significantly improve the thermal perfor mance in comparison with conventional twisted tapes. Eiamsa-Ard and Promvonge [22] experimentally investigated the Nusselt number and friction factor of water fluid flow through tubes enhanced with
clockwise and counter-clockwise twisted tapes. It was concluded that the Nusselt numbers in the heat exchanger tube equipped with the counter-clockwise twisted tapes are higher than those with the con ventional twisted tapes up to 41.9%. Promvonge and Eiamsa-Ard [23] studied the effect of combined use of conical rings and twisted tapes on heat transfer enhancement inside circular tubes and reported that the maximum performance factor of 1.96 is obtained for using the conical-ring and the twisted-tape with twist ratio of 3.75. Different types of perforated turbulators are developed in the past years to generate vortex flows and improve the thermal performance of different types of heat exchangers. Bhuiya et al. [24] experimentally investigated the heat transfer enhancement and friction loss of air fluid flow through tubes fitted with perforated twisted tapes. They observed that the Nusselt number and thermal performance factor could be enhanced up to 340% and 59%, respectively in comparison with con ventional simple circular heat exchanger tubes. Kongkaitpaiboon et al. [25] developed a new type of perforated conical rings (PCRs) to enhance the fluid mixing between the core regions and the tube wall. It was concluded that 137% heat transfer enhancement could be achieved over that in a plain tube by using PCRs with pitch ratio of 4. Chamoli et al. [26] numerically investigated the heat transfer, friction factor and the thermal performance factor in a heat exchanger tube fitted with
* Corresponding author. E-mail addresses: [email protected] (M.E. Nakhchi), [email protected] (J.A. Esfahani). https://doi.org/10.1016/j.ijthermalsci.2019.106153 Received 23 May 2019; Received in revised form 10 September 2019; Accepted 21 October 2019 Available online 1 November 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Fig. 1. Schematic of the circular channel equipped with PHCs. Table 2 Grid independency for Re ¼ 16000, PI ¼ 24% and d/D ¼ 0.9.
Table 1 The geometrical parameters of the heat exchanger tube equipped with PHCs. Parameter
Value
Tube diameter (D) Tube length (L) Perforated index ðPIÞ
66 mm 1350 mm 8%, 16%, 24%
Perforated cylinder diameter ðdÞ
33, 46.2, 59.4 mm
Diameter ratio (d/D) Perforated cylinder length ðlÞ
0.5, 0.7, 0.9 135 mm
Perforated cylinder thickness ðtÞ
1 mm
diameter of hole ðdh Þ
5 mm
Pitch ratio Re ¼ ρuD=μ
Grid number
Nu
� � iþ1 �Nu Nui �� � � Nuiþ1 �
f
� � �f iþ1 f i � � � � iþ1 � � � f
467721 893448 1253583 1605402
71.66 75.85 77.10 77.12
– 0.055 0.016 0.0003
0.0510 0.0540 0.0572 0.0571
– 0.055 0.059 0.002
Table 3 Maximum values of yþ for Re ¼ 16000.
2 6000–16000
Case
Tube wall
PHC wall
d/D ¼ 0.5, PI ¼ 8% d/D ¼ 0.7, PI ¼ 8% d/D ¼ 0.9, PI ¼ 8% d/D ¼ 0.5, PI ¼ 16% d/D ¼ 0.7, PI ¼ 16% d/D ¼ 0.9, PI ¼ 16% d/D ¼ 0.5, PI ¼ 24% d/D ¼ 0.7, PI ¼ 24% d/D ¼ 0.9, PI ¼ 24%
1.241 1.354 1.862 1.382 1.508 2.115 1.498 1.973 2.341
1.352 1.446 1.903 1.426 1.634 2.163 1.605 2.225 2.583
perforated vortex generators. It was observed that the thermal perfor mance of 1.65 at Re ¼ 3000 could be reached by using turbulators with perforated index of 16%. Skullong et al. [18] presented a numerical analysis to investigate the effect of staggered-winglet perforated tapes on the thermal performance of turbulent fluid flow through heat exchanger tubes. They concluded that using perforated tape with a blockage ratio of 0.15, pitch ratio of 1.0 at Re ¼ 4180 could enhance the thermal performance factor up to 1.71. Sheikholeslami and Ganji [27] conducted a numerical investigation on perforated turbulators with different pitch ratios for thermal performance enhancement through heat exchanger tubes. It was observed that the thermal performance factor of 1.59 could be achieved by using perforated turbulators with a pitch ratio of 1.07. Nakhchi and Esfahani [28] numerically investigated the effects of perforated conical rings with different geometries on the thermal performance enhancement of heat exchangers. Their work revealed that the maximum thermal performance factor of 1.241 could be obtained by using 10 PCRs with d/D ¼ 0.1 at Re ¼ 4000. They also concluded that the Nusselt number of PCRs reduces up to 35.48% with
Fig. 2. The computational domain with detailed views of the tube inlet and the PHCs. 2
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Table 4 Validation of the numerical simulations with experimental results of Singh et al. [29] for the Nusselt number and friction factor for two different PHC geometries. Re
d/D ¼ 0.5 and PI ¼ 8%
d/D ¼ 0.5 and PI ¼ 24%
Nu 6000 11000 16000
f
Nu
f
Present study
Exp. [29]
Present study
Exp. [29]
Present study
Exp. [29]
Present study
Exp. [29]
46.22 74.51 96.23
46.67 74.82 96.88
0.140 0.120 0.113
0.142 0.123 0.112
36.05 57.94 74.65
36.30 58.13 75.03
0.058 0.051 0.047
0.060 0.052 0.048
increasing the number of PCR holes from 4 to 10. Singh et al. [29] conducted an experimental study to investigate the effect of perforated hollow cylinders on the thermal performance factor and heat transfer augmentation of the air fluid flow through circular tubes. Their results show that the thermal performance factor was in the range of 1.21–1.47 and it is maximum at perforated index ðPI ¼ Aholes =Atot Þ of 0.08 and diameter ratio of 0.65. Based on the above literature review, the perforated hollow cylinders can significantly improve the thermal performance of thermal systems. Although the experimental investigation of Singh et al. [29] investigated the effect of design parameters on the Nusselt number and friction factor of the fluid flow through tubes fitted with PHCs, a numerical study is necessary for better understanding the physical effects of different geometrical parameters on the turbulent flow characteristics in the presence of the PHCs. Also, the effect of the diameter ratio needs to be further investigated to determine in which conditions the maximum thermal performance factor could be obtained. Accordingly, the main goals in the current investigation are to cover this topic and develop correlations to predict the thermal performance and heat transfer enhancement of the system as functions of the design parameters. For this purpose, different perforated hollow cylinders with variable diam eter ratios and perforated index are mounted at the center of a heat exchanger tube.
length (L ¼ 1350 mm), the thickness of the hollow cylinders (t ¼ 1 mm) and holes diameter ðdh ¼ 5mmÞ are kept constant in the present study. The water inlet temperature (Tin) is assumed to be 300 K and the tube wall temperature (Tw) is kept uniform at 350 K. To reduce the inlet ef fects, an extra length ðLin ¼ 1000mmÞ is added to the tube inlet. The gravitational effects are assumed to be negligible and the Reynolds number is varied from 6000 to 16000. Pressure outlet boundary con dition is selected at the outlet of the tube. The PHCs are assumed to be adiabatic and no-slip boundary conditions are imposed on the walls. The hollow cylinders inside the heat exchanger tube are usually fixed with a thin connecting rod. However, the effects of this thin rod are neglected in the present numerical simulations to make better discussions about the physical effects of the PHC on turbulent kinetic energy, velocity vectors and other CFD results. It also should be pointed out that the effects of this thin rod on the thermal characteristics are negligible. The boundary values of the turbulence parameters near the wall are described by the enhanced wall function method. The turbulence in tensity is the tube inlet is kept constant at 10%. 3. Physical and numerical models 3.1. Governing equations The fluid flow is steady, incompressible and three-dimensional. Gravitational effects are neglected. Based on the Reynolds average Navier Stokes (RANS) model, the continuity, momentum, and energy equations are as follows [5,11,30]:
2. Physical description Fig. 1 shows a schematic of the heat exchanger tube fitted with perforated hollow cylinders and the coordinate system. Water with constant physical properties is selected as the working fluid. Table 1 shows the geometrical parameters used in the present numerical study. The perforated index varies from 8% to 24%, while the diameter ratio (d/D) is ranged from 0.5 to 0.9. The tube diameter (D ¼ 66 mm), tube
∂ ðρui Þ ¼ 0 ∂xi
Fig. 3. Turbulent kinetic energy contours in the tube fitted with PHCs at Re ¼ 6000. 3
(1)
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Fig. 4. Turbulent dissipation rate contours in the tube fitted with PHCs at Re ¼ 6000.
� �
� ∂ u ρu ¼ ∂xj j i
��
∂p ∂ ∂u μ i þ ∂x ∂xj ∂xj
∂ ∂ ðρui TÞ ¼ ∂xj ∂xi
��
μ
Pr
�
þ
μt ∂T Prt ∂xi
finite volume method with SIMPLE algorithm. The momentum, energy, turbulent kinetic energy, and dissipation rate are discretized by using second-order upwind scheme. The residuals of continuity, momentum, k and ε equations for the convergence of the numerical computations are set to be less than 10 5 and 10 8 for the energy equation. A grid independence test is performed (Table 2) to check the accu racy of the computational grid for calculating the Nusselt number and friction factor through heat exchanger tube fitted with PHCs with PI ¼ 24% and d/D ¼ 0.9 at Re ¼ 16000. It can be seen that the relative difference between the mesh numbers of 1253583 and 1605402 is about 0.03% for the Nusselt number and 0.2% for the friction factor. There fore, the grid number of 1253583 is chosen for further numerical simulations. The maximum values of yþ on the tube wall and the walls of the PHCs with different geometries at Re ¼ 16000 are shown in Table 3. It can be observed that the maximum yþ values are less than 3 for all of the test cases. This means that the viscous sub-layer effects are captured correctly in the numerical simulations of turbulent fluid flow through tube equipped with PHCs. It can be seen the yþ values near the tube wall are generally smaller than that on the PHC wall. This is mainly due to boundary layer mesh (inflation) used near the tube walls.
(2)
ρu’i u’j �
(3)
where ρ, p, u and μ, are fluid density, pressure, turbulent fluctuations, pressure and dynamic viscosity of the water, respectively. The Reynolds 0
stress term ðρu’i u’j Þ can be formulated as: � � ∂u ∂u 2 2 ∂uk ρu’i u’j ¼ μt i þ j ρkδij μ δij 3 3 t ∂xk ∂xj ∂xi
(4)
The equations of turbulent kinetic energy (k) and dissipation rate (ε) are: �� � � ∂ ∂ μt ∂k þ Gk ρε ðρui kÞ ¼ þμ (5) ∂xi ∂xj σ k ∂xj
∂ ∂ ðρui εÞ ¼ ∂xi ∂xj
��
�
�
μt ∂ε ε þμ þ ½C1ε Gk ∂xj σε k
ρC2ε ε�
(6)
In the above equations, σ k ¼ 1, σ ε ¼ 1:3, C1ε ¼ 1:42 and C2ε ¼ 1:68 are constants of the (RNG) k ε turbulent model.
3.3. Code validation
3.2. Computational domain
In order to show the accuracy of the numerical analysis, comparison between the results of the present study and the experimental results of Singh et al. [29] for the case of uniform wall heat flux at three different Reynolds numbers is made in Table 4. Four different turbulence models were employed for the numerical simulations (RNG k-e, standard k-e, k-w and SST k-w model). It was observed that the numerical results of the RNG k-e model are in excellent agreement with experimental data.
The computational domain with detailed views of the mesh at the inlet of the heat exchanger tube and the PHCs are shown in Fig. 2. The smaller mesh is used near the tube walls to capture the turbulence flow characteristics and resolve the viscous sub-layer effects near the tube walls. ANSYS Fluent 18.1 is selected for the numerical analysis by using 4
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Re ¼
Nu ¼
uD
ν hD q} D ¼ k ðTw Tb Þ k
(7) (8)
where h is the local heat transfer coefficient, k the thermal conductivity of water u the inlet velocity, ν the kinematic viscosity of water, Tw the wall temperature and Tb the bulk temperature of the fluid flow. The friction factor (f) and thermal performance factor ðηÞ are defined as follows: f¼
η¼
2D ΔP L ρu2 ðNu=Nus Þ ðf =fs Þ1=3
(9) (10)
ΔP is the total pressure difference and L is length of the pressure section. Nus and fs are defined as the Nusselt number and friction loss of the fluid flow through a smooth tube which are obtained from the nu merical simulations. The heat transfer (Nu) and friction factor results (f) are validated with the experimental correlations of Dittus-Boelter and Gnielinski [33] for Nusselt number and the correlations of Blasius and Petukhov [33] for the friction factor (f). It was observed that the nu merical results are in excellent agreement with experimental data. 4. Results and discussion Fig. 3 shows the turbulent kinetic energy contours of water fluid flow through heat exchanger tube equipped with PHCs with different diam eter ratios and perforated index values. It can be observed that the Fig. 5. The y-velocity contours and velocity vectors in the tube fitted with PHCs at Re ¼ 6000.
Therefore RNG k-e model is employed for the numerical simulations. It can be observed that the numerical results are in good agreement with experimental data. The results show that both of the Nusselt number and friction factor decreases with increasing the perforated index (PI) from 8% to 24%. As the number of holes increases, the amount of fluid mixing between the PHCs and the tube wall decreases which reduces the ther mal boundary layer disruption. The results show that the friction factor decreases with increasing the Reynolds number, which is similar to friction loss trend on the Moody chart for turbulent flows. 3.4. Data reduction The Reynolds number (Re) and Nusselt number (Nu), can be defined as [31,32]:
Fig. 7. The axial velocity contours in the tube fitted with PHCs at Re ¼ 6000.
Fig. 6. Contours of the turbulent kinetic energy inside a circular tube equipped with PHC with φ ¼ 8% and d=D ¼ 0:7 at Re ¼ 6000. 5
International Journal of Thermal Sciences 147 (2020) 106153
M.E. Nakhchi and J.A. Esfahani
highest turbulent dissipation rate appears near the PHC holes. Thus, the best fluid mixing condition can occur in this region. It is clear that the turbulent dissipation rate for the case of d/D ¼ 0.9 is very small near the tube walls. The physical reason of this is that the fluid flow through the narrow gap between the cylinder and the tube is much less than the flow rate in the core region. As a result, there is limited possibility of mixing the flow between these two regions. Velocity vectors can also explain the physical reason. It can be seen that for the cases of d/D ¼ 0.9, the fluid flow is only moves toward the center of the tube, resulting in a reduction in the mixing and heat transfer rate. The results show that the fluid mixing between the core region and the tube walls for the cases of (a) and (b) is higher than the other cases. It can also be seen that for the case (a), the fluid flow doesn’t have an effective interaction with the tube wall and as a results, it is not able to effectively destruct the thermal boundary near the wall. Therefore, the heat transfer rate is expected to be better for the case of PI ¼ 0.08 and d/D ¼ 0.7. Fig. 5 shows the y velocity contours and velocity vectors for different PHC geometries. The velocity vectors show that the fluid mixing between the core regions and tube wall near the holes is considerably higher than the other parts. When the fluid flow passes through the hole, a part of the fluid is directed upwards and the other part moves downwards. As a result, the flow mixing is improved and the hot fluid near the wall is moved to the central part of the pipe. The red contours show the upward flow direction, while the blue velocity contours show the fluid flow in the downward direction. It can be observed that the y-velocity contours near the PHC with d/D ¼ 0.7 is higher than the other cases. Fig. 6 shows a detailed view of the turbulent kinetic energy contours near the PHC and the tube walls for the case of φ ¼ 8% and d=D ¼ 0:7. It can be seen that the flow disturbance near the holes increases in the axial direction. This means that better fluid mixing between the tube wall and the core regions results in thermal boundary layer disruption near the tube walls. It also can be observed that strong vortex flow occurs near the outlet of the PHC. This strong turbulence intensification can considerably in crease the Nusselt number and friction factor of the water fluid flow through the heat exchanger tube. Fig. 7 shows the axial velocity contours of turbulent fluid flow (Re ¼ 6000) through heat exchanger tube equipped with PHCs with different diameter ratios. It can be observed that for the case of d/D ¼ 0.5, the axial velocity near the tube wall is strong, while the axial velocity through the hollow cylinder is low. The results indicate that for the case of d/D ¼ 0.7, fluid velocity through the hollow cylinders and near the walls is more uniform than the other cases. This means that better fluid mixing makes the flow more uniform and as a result, the heat transfer would be enhanced. It also can be observed that for the case of d/D ¼ 0.9, the axial flow velocity is small and negligible in the region between the PHC and the tube wall. This reduces the fluid mixing and thermal boundary layer disruption near the walls. One of the objects in the present study is to drive correlations for the Nusselt number, friction factor and thermal performance factor as func tions of the design parameters (Re, PI, d/D). Response surface methodol ogy is employed as a powerful method to obtain second-order polynomial correlations for the flow characteristics through a heat exchanger tube equipped with PHCs. MINITAB software with RSM face-centered design is employed for the analysis. Based on the consents of this method, 30 nu merical data are required as inputs. Regression analysis of the numerical results is performed. The following polynomial correlations for Nu, f, and η are generated in Eqns. (11)–(13), respectively. These correlations are valid in the range of (0.08 < PI < 0.24, 0.5 < d/D < 0.9, 6000 < Re < 16000).
Fig. 8. The effect of design parameters on averaged Nusselt number inside tube fitted with PHCs.
Fig. 9. The effect of design parameters on friction factor inside tube fitted with PHCs.
turbulent kinetic energy for the case of PI ¼ 8%, d= D ¼ 0:7 is the highest among the tested geometries. It can be observed that for PI ¼ 8% the turbulent kinetic energy enhances with increasing the diameter ratio from 0.5 to 0.7 and then it decreases as the diameter ratio increases from 0.7 to 0.9. Physically speaking, for d/D ¼ 0.5 the hollow cylinder is too small and it is not able to increase the flow disturbance between the tube walls and the core regions. With increasing the diameter ratio to 0.7, turbulent kinetic energy (TKE) and flow perturbation significantly in crease due to better fluid mixing between the tube wall and the central regions. It also can be observed that the holes have a considerable effect on TKE enhancement for all of the test cases. In other words, the flow disturbance near the perforated walls is the main reason for thermal boundary layer disruption near the tube walls which improves the heat transfer rate in comparison with a tube without any inserts. The results show that with increasing the PHC diameter ratio to 0.9, the gap be tween the cylinder and the tube wall becomes very small. As a result, the fluid flow mixing reduces near the holes of the hollow cylinder. The results show that the turbulent kinetic energy production near the PHC with PI ¼ 24% considerably reduces in comparison with other geome tries. This is mainly due to the smaller mass flow rate in the presence of higher number of holes. Fig. 4 depicts the turbulent dissipation rate contours in cross section plane of Z ¼ 0.32 for different d/D and PI values at Re ¼ 6000. The
Nu ¼
60:88 þ 7:34. � 10 3 Re 3
� 2:41 �.PI þ 204:34 � d D
þ0:25 � 10 Re � d D þ 46:87PI � d D
6
8:12 � 10 3 Re � PI 2
0:061 � 10 Re
85:22PI 2
151:14ðd=DÞ2
6
(11)
M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
Fig. 10. Surface contours of Nusselt number, friction factor and η as functions of the design parameters.
� 3 3 0:006 . � 10 Re 0:77 � PI þ 0:62d D þ 0:007 � 10 Re � PI 2 6 2 2 0:43ðd=DÞ þ0:13PI � d D þ 0:00014 � 10 Re þ 0:56PI
f ¼ 0:02
(12) �
3 3 η ¼ 0:84 0:007 . � 10 Re þ 0:28 � PI þ 1:65 � d D þ 0:0012 � 10 Re � PI
0:03PI � d D
0:32 � PI 2
1:11 � ðd=DÞ
2
(13) The deviation between the data predicted by the correlations of Nu, f and η and numerical results are in �3.4% for Nu, �6.9% for f and �7.4 for η, respectively. The variations of the average Nusselt number with respect to the Reynolds number is presented in Fig. 8. It can be observed that the Nusselt number increases with the Reynolds number. This is mainly due to stronger temperature gradients near the tube wall in the presence of water fluid flows with higher velocities. The results indicate that the Nusselt number enhances up to 25.9% with decreasing the PI from 0.24 to 0.08 at Re ¼ 16000 and d/D ¼ 0.7. As discussed earlier, fluid mixing through tubes equipped with PHCs with smaller number of holes is stronger which results in better fluid mixing and thermal boundary layer disruption near the tube walls. It can be observed that
Fig. 11. Effect of diameter ratio on η for PI ¼ 0.24.
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M.E. Nakhchi and J.A. Esfahani
International Journal of Thermal Sciences 147 (2020) 106153
the heat transfer rate for the fluid flow through tube fitted with PHCs with d/D ¼ 0.7 is higher than the other diameter ratios, which is in agreement with previous physical discussions of Fig. 4. The results show that the heat transfer rate enhances up to 222.7% in comparison with the plain tube. The maximum average Nusselt number of 104.21 could be achieved by mounting PHCs with PI ¼ 0.08 and d/D ¼ 0.7 through heat exchanger tubes at Re ¼ 16000. Fig. 9 shows the effect of different design parameters on the friction factor variations of the fluid flow inside the tube fitted with PHCs. The results show that the average friction factor reduces with increasing the Reynolds number. It also can be observed that the friction factor in creases up to 86.2% with decreasing the number of holes (PI) from 0.24 to 0.0.08 at Re ¼ 11000 and d/D ¼ 0.7. When the PHCs with smaller number of holes are mounted in the circular tube, viscous dissipation rate of the water flow near the tube walls enhances and therefore, the pressure drop enhances. The other main physical reason for friction factor increment is vortex flow between the tube wall and core region near the holes. It also can be observed that the friction factor of the fluid flow through tube fitted with PHCs with b/D ¼ 0.70 is higher than that for the other diameter ratios. The friction factor increases up to 421.2% in comparison with the plain tube. Surface contours of the Nusselt number, friction factor and thermal performance factor ðηÞ of turbulent fluid flow through heat exchanger tube fitted with PHCs are shown in Fig. 10. Based on the definition of the thermal performance, η > 1 is favorable and indicates that the PHCs have sufficient influence on the heat transfer enhancement. The results indicate that thermal performance decreases with increasing the Rey nolds number. It also should be pointed out that the thermal perfor mance increases up to 5.24% with increasing the perforated index (PI) from 8% to 24%. Based on the results of Fig. 10, it was observed that the maximum thermal performance factor is occurred in the range of ð0:6 < d =D < 0:8Þ. In order to better investigate the effect of the diam eter ratio on the thermal performance of the heat exchanger, the η variations with respect to d/D are shown in Fig. 11. It is seen that the η parameter increases with increasing d/D from 0.6 to 0.74 and then de creases. As a result, at d/D ¼ 0.74, the thermal performance is maximum by considering the heat transfer rate and the pressure drop. The results show that the thermal performance factor decreases with increasing the Reynolds number from 6000 to 16000. The maximum thermal perfor mance value of 1.456 could be achieved for the case of d/D ¼ 0.74 and PI ¼ 24% at Re ¼ 6000.
rate of the water flow near the tube walls enhances and conse quently, the pressure drop enhances. � The maximum thermal performance value of 1.456 could be ach ieved for the case of d/D ¼ 0.74 and PI ¼ 24% at Re ¼ 6000. Acknowledgement This research was supported by the Office of the Vice Chancellor for Research, Ferdowsi University of Mashhad, under Grant No. 49607. References [1] A. Saravanan, S. Jaisankar, Heat transfer augmentation techniques in forced flow V-trough solar collector equipped with V-cut and square cut twisted tape, Int. J. Therm. Sci. 140 (2019) 59–70. [2] L. Lin, J. Zhao, G. Lu, X.-D. Wang, W.-M. Yan, Heat transfer enhancement in microchannel heat sink by wavy channel with changing wavelength/amplitude, Int. J. Therm. Sci. 118 (2017) 423–434. [3] M. Jourabian, M. Farhadi, A.R. Darzi, Constrained ice melting around one cylinder in horizontal cavity accelerated using three heat transfer enhancement techniques, Int. J. Therm. Sci. 125 (2018) 231–247. [4] E.-J. 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5. Conclusion In the present numerical study, turbulent flow characteristics through heat exchanger tube equipped with perforated hollow cylinders are investigated, to investigate the effects of diameter ratio (d/D) and perforated index (PI). The Reynolds number varies from 6000 to 16000. (RNG) k–ε turbulent model is employed in the three-dimensional sim ulations. The main findings are as follows: � The turbulent kinetic energy enhances with increasing the diameter ratio from 0.5 to 0.7 and then it decreases as the diameter ratio in creases from 0.7 to 0.9. Physically speaking, for d/D ¼ 0.5 the hollow cylinder is too small and it is not able to increase the flow disturbance between the tube walls and the core regions. With increasing the diameter ratio to 0.7, turbulent kinetic energy (TKE) and flow perturbation significantly increase due to better fluid mixing be tween the tube wall and the central regions. � Correlations are developed to predict the thermal performance, friction factor and Nusselt number of the fluid flow through a tube fitted with PHCs as functions of the design parameters. � The friction factor reduces up to 86.2% with increasing the perfo rated index (PI) from 0.08 to 0.24. When the PHCs with smaller number of holes are mounted in the circular tube, viscous dissipation 8
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