- Email: [email protected]
Compressible fluid flow field synergy principle and its application to drag reduction in variable-cross-section pipeline
International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Compressible fluid flow field synergy principle and its application to drag reduction in variable-cross-section pipeline Bo Zhang ⇑, Jinsheng Lv, Jixue Zuo School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 27 February 2013 Received in revised form 12 January 2014 Accepted 6 June 2014
Keywords: Field synergy Compressible flow Drag reduction Optimisation
a b s t r a c t The compressible fluid flow field synergy principle was presented as an effective theoretical guide to reduce the drag during compressible flow. A compressible flow field synergy model was presented based on the incompressible flow field synergy principle. A variable cross-section pipeline air of compressible flow was used to verify the model. Two specific improvement schemes were studied for practical applications, which showed a 24% and 20% reduction in the resistance. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The past few decades have witnessed a growing interest in the field synergy principle. These studies have primarily focused on two aspects: heat convection and drag reduction in flow. To boost the heat transfer performance, Guo et al. [1–4] introduced the concept of field synergy and consequently garnered significant attention from other researchers. This concept focuses on the strength of the heat convection, which not only depends on the temperature gradient field, the velocity field in the fluid domain and the fluid property, but also on the included angle size between the velocity field and heat flow field. The size of the included angle negatively correlates with the degree of field synergy and the heat transfer efficiency. Field synergy theory was first applied to a laminar convection heat exchange process [1]. Subsequently, it has been numerically and experimentally validated in laminar and turbulent flow and applied to the analysis of heat exchangers [2–4]. The reduction of flow drag is a fundamental problem in hydrodynamics. So far, researchers have devised a variety of technologies to reduce flow drag. For example, objects in the external flow are designed to streamline or improve the surface roughness to delay the boundary layer separation; for internal flow, guide plates are placed in the bend to avoid secondary separation. However, most ⇑ Corresponding author. Address: 2# Linggong Road, Ganjingzi District, Dalian, Liaoning 116024, China. Tel.: +86 411 84706537; fax: +86 411 84691725. E-mail address: [email protected] (B. Zhang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.06.027 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
of the presented flow drag reduction technology is based on experience and lacks a unified theory. Chen et al. [5] began with the incompressible fluid flow drag reduction problem and explored the feasibility of solving this problem by using the field synergy theory. As a result, they presented a field synergy drag reduction model. They also deduced the incompressible flow field synergy equation according to the principle of the minimum dissipation of mechanical energy. By solving the equation, the optimal incompressible flow can be obtained to effectively guide the optimisation of pipeline design. Many researchers [6–14] have focused on studying the heat transfer field synergy presented by Guo. To quantitatively describe and compare the degree of field synergy between the velocity vector and temperature gradient vector, previous researchers have analysed and discussed the evaluation index in the heat transfer field synergy. The published literature indicates that the evaluation index of the field synergy mainly includes three aspects: (1) the included angle between the velocity vector and the temperature gradient vector; (2) the cosine of the included angle between the velocity vector and the temperature gradient vector; and (3) the field synergy number. The included angle between the velocity vector and the temperature gradient vector ranges from 0 to 90 in the heat transfer field synergy [6–10]. When the included angle ranges from 90 to 180 , the supplementary angle is examined. Zhou et al. [12] analysed and discussed the evaluation index of the heat transfer field synergy and presented five methods to calculate the included angle between the velocity vector and temperature gradient vector. They ultimately compared the five
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B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
Nomenclature B D DC F FS f L ~ n P Re S S Sij Skk t U u u V
Lagrange multiplier characteristic length, V=S, m pressure drop, defined by Eq. (7), dimensionless additional volume force, N field synergy number, defined by Eq. (8), dimensionless volume force, N length, m surface unit normal vector pressure, Pa Reynolds number, dimensionless area of the solid surface, m2 dimensionless area deformation rate tensor divergence, dimensionless time, s dimensionless velocity velocity, m/s dimensionless velocity volume, m3
calculation methods via numerical simulations. Guo proposed the field synergy number to evaluate the strength of the heat transfer field synergy. Compared with the included angle and its cosine value between velocity vector and temperature gradient vector, the field synergy number better reflects the quality of the heat transfer field synergy. However, the evaluation method has never been mentioned for compressible flow field synergy. The number of studies undertaken on heat transfer enhancement has been constantly growing since the early 1960s. In 1999, Bergles et al. mentioned nearly 4345 publications on the topic in their literature review [15]. The intensification techniques are numerous [16] and can be classified into two categories: passive and active [17]. Passive techniques enhance heat transfer by modifying the surface exchange or fluid properties, changing the surface geometry, disrupting boundary layers, or promoting liquid–vapour phase changes. Active techniques use jet, spray and electronic methods, etc. to enhance heat transfer. The relevant applications also have a promising future. The aforementioned methods all enhance the heat transfer, but their mechanisms are poorly understood. Therefore, Guo [18] provided a new concept to understand this process. This paper uses this new concept to elucidate further studies. Drag reduction in gas–liquid two-phase flow is caused by mechanical friction, which is inevitable. Drag can be reduced in pipe flow via two ways. One method consists of adding polymer to the flow [19], the other changes the boundary condition between the flow and solid or the structure of the entire flow field. This study attempts to use the second method to optimise the flow field and obtain new results. The flow drag reduction problem in compressible flow is also very important to design applications, such as the high-speed flow drag reduction problem in pneumatic conveying and aviation devices, etc. The relationship between the flow resistance and the field synergy degree in compressible flow is discussed in this paper. The evaluation method in the heat transfer field synergy is referenced to evaluate the compressible flow field synergy. According to the principle of the minimum dissipation of mechanical energy, a compressible flow field synergy equation was established that takes the equation of continuity as a constraint. Furthermore, a variable cross section of compressible pipe flow was examined as an example to verify the effectiveness of the compressible flow field synergy.
V x; y; z
dimensionless volume Cartesian coordinates, m
Greeks b dij k
field synergy angle, degree second order unit tensor, dimensionless constant dynamic viscosity, kg/(m s) kinematic viscosity, m2/s Lagrange function, W density, kg/m3 viscous dissipation function, W
l m P
q u
Subscripts i; j i direction, j direction x; y; z Cartesian coordinates in inlet m mean f boundary X domain
2. Compressible flow field synergy equation The following model was derived by taking the incompressible field synergy drag reduction model as reference and basing the model on the compressible Navier–Stokes equation. For compressible flow, the Navier–Stokes equation can be written as follows:
@ui @ui 1 @ þ uj ¼ fi @t @xj q @xj
2 P þ lSkk dij 2lSij 3
ð1Þ
The Navier–Stokes equation for steady-state compressible fluid flow without volume force can be written as follows:
Z
quj
X
@ui xj dV ¼ @
Z X
@P dV @xi
Z X
@ 2 lSkk dij 2lSij dV @xj 3
ð2Þ
Defining D ¼ V=S as characteristic length and introducing the following dimensionless variables:
ui ¼
ui ; uin
uj ¼
uj ; uin
rui ¼
rui ; uin =D
dV dV ¼ ; V
dS dS ¼ S
ð3Þ
yields the following form of Eq. (2):
D
q
u2in
Z X
Z @P l 2 xi dV ¼ 2 Skk dij 2Sij ~ ndS @ quin 1 3 Z @ ui dV þ Duj @xj X
The following has been well established:
@ ui @xj
ð4Þ
i ¼ @u @xj ¼ ru ui . Thus, u in
@ ui @ui rui rui ¼ @xj ¼ ¼ @xj uin uin D
in
ð5Þ
When combined with Eqs. (5) and (4) can be transformed into the following:
D qu2in
Z X
Z @P l 2 dV ¼ 2 Skk dij 2Sij ~ ndS @xi quin 1 3 Z þ Urui dV
ð6Þ
X
The term on the left side of Eq. (6) is the dimensionless pressure drop in the xi direction:
B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
DC i ¼
Z
D
qu2in
X
@P dV @xi
ð7Þ
The second term on the right is the integration of the dot product between the dimensionless velocity and the dimensionless velocity gradient vectors, which can be written as follows:
FSm ¼
Z
Urui dV ¼
X
Z
jrui j cos b dV U m
ð8Þ
X
When defining bm as the compressible flow field synergy angle, the field synergy angle and its cosine value reflect the degree of synergy between the velocity and the velocity gradient vectors, which is called the degree of compressible flow field synergy. While the field synergy angle is small, its cosine value is large, and the degree of the field synergy between the velocity and the velocity gradient vectors is quite high; conversely, the degree of the field synergy is low. According to the definition of Reynolds number
Re ¼
uin L
m
¼
uin V
l=q S
¼
uin qD
l
D uin
Z 2 ndS þ Re FSm Skk dij 2Sij ~ 1 3
ð10Þ
ZZZ
½u þ Br qudV
@ @B @q @q @q @ux @uy @uz ðBr quÞ ¼ ux þ uy þ uz þq þq þq @ux @ux @x @y @z @x @y @z @q @ þB þq @x @x ð18Þ divðquÞ ¼ ux
@q @q @q @ux @uy @uz þ uy þ uz þq þq þq ¼0 @x @y @z @x @y @z
@ @q @ @q @B @ ðBr quÞ ¼ B ¼B þq þq ¼ ðqBÞ @ux @x @x @x @x @x
@ux @uy @uz þ þ @x @y @z
ð21Þ
When combined with Eqs. (17) and (21), the change in P as a function of the velocity vector for u can be written as follows:
1 3
1 2
l r2 u þ rðr uÞ þ rðqBÞ ¼ 0
ð11Þ
Fig. 1. Internal structure diagram of variable cross-section pipeline.
ð12Þ
The change in u as a function of the velocity vector for ux can be obtained as follows:
ð14Þ
Fig. 2. Grid of the pipeline before optimisation.
When k ¼ 23 l is constant, the compressible fluid in the flow process meets the following criterion:
@ux @uy @uz þ þ – 0; @x @y @z
@k ¼0 @x
ð15Þ
When combined with Eqs. (15) and (14) can be transformed into the following:
@u 1 @ ¼ l 2r2 ux þ 2 ðr uÞ 3 @x @ux
ð16Þ
Thus, the change in u as a function of the velocity vector for u can be expressed as follows:
ð22Þ
Furthermore, the compressible fluid Navier–Stokes equation is defined as follows:
ð13Þ
@u @ @ ¼ l 2r2 ux þ 2 ðdiv uÞ þ 2kdiv u @x @x @ux
ð20Þ
Eq. (20) can be transformed into a three-dimensional form:
where div u can be written as follows:
divu ¼
ð19Þ
When combined with Eqs. (19) and (18) can be transformed into the following:
where u is the viscous dissipation function, which can be expressed as follows:
" 2 2 2 2 @ux @uy @uz @ux @uy þ2 þ2 þ þ @x @y @z @y @x 2 2 # @ux @uz @uy @uz 2 þ kðdivuÞ þ þ þ þ @z @x @z @y
ð17Þ
Furthermore,
X
u¼l 2
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@ ðBr quÞ ¼ rðqBÞ @u
Eq. (10) indicates that the dimensionless pressure drop for a compressible fluid depends on the degree of synergy between the velocity field and velocity gradient field. This paper defines Eq. (10) as the compressible flow field synergy model for drag reduction. A Lagrange function for the viscous dissipation is established based on continuity equation:
P¼
@u 1 ¼ l 2r2 u þ 2 rðr uÞ 3 @u
ð9Þ
Eq. (6) can be transformed into the following:
Re DC i ¼
Fig. 3. Grid of the pipeline after adding a wedge-shaped face.
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B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
2 F rP r lr u þ r ð2lSÞ ¼ qu ru 3
ð23Þ
When contrasting Eqs. (22) and (23), the following equation can be obtained:
rðqBÞ ¼ rP
ð24Þ 2
F ¼ qu ru r ð2lSÞ 2lr u
ð25Þ
Combining Eqs. (23) and (25) yields the following:
qu ru r ð2lSÞ 2lr2 u rP r
2 lr u 3
þ r ð2lSÞ ¼ qu ru
ð26Þ
Eq. (26) denotes the compressible flow field synergy equation. In contrast with the incompressible flow field synergy, the additional volume force is different because the size of the additional volume force, F, is not equal to the inertial force, in the fluid flow process. This paper proposes that solving Eq. (26) can yield the optimal velocity field based on a given set of boundary conditions. 3. Application of field synergy to reduce drag in a variable crosssection pipeline
Fig. 4. Grid of the pipeline after narrowing the pipeline outlet width.
Fig. 5. Air streamline field near the top right corner in the pipeline before optimisation.
Fig. 6. Air streamline field near the top right corner after optimisation.
A variable cross-section pipeline was studied to verify the model presented above. The configuration and pipeline dimensions are presented in Fig. 1. The unit is mm. The wall was assumed to be adiabatic and its thickness was neglected. The inlet pressure and
Fig. 7. The shape and the position of the wedge-shaped face.
Fig. 8. Air streamline field near the top right corner after adding wedge-shaped face to the pipeline.
B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
temperature were 2 105 Pa and 300 K, respectively. The outlet pressure was 1 105 Pa. The air dynamic viscosity was 1:7894 105 kg/(ms). The Mach number was 0.35; thus, the flow pattern was a subsonic compressible flow.
Fig. 9. Air streamline field near the top right corner after narrowing the pipeline outlet width.
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The FLUENT 6.3 software was employed to solve the governing equation in the flow process to obtain the heat transfer field and the velocity field of the pipeline. The pipeline was modelled as a two-dimensional body according to Fig. 1, and the grid had a quadrilateral structure. As shown in Figs. 2–4, the grid initially consisted of 12,180 elements and was later adapted to 10,962 elements. The inlet boundary condition was a pressure-inlet and the outlet boundary condition was a pressure-outlet. The solving method was couple implicit. The realisable k-e turbulence model was selected. The pressure and velocity were coupled by the SIMPLE algorithm, and the diffusion and convection terms were discretised using a first order upwind scheme. The additional volume force term was added by the user-defined function (UDF) in FLUENT. After calculating the flow in the variable cross-section pipeline, the distribution of the streamline field could be obtained. Backflow appeared at the top right and bottom right corners of the streamline field. The backflow was caused by a combination of the inertial and viscous forces. Because the pipeline was a symmetric structure along the axis, the air streamline field near the top right corner was examined in this study. As shown in Fig. 5, a series of backflow streamlines appeared that flowed opposite to the main flow direction in the top right corner. In addition to the friction between fluid and wall, the backflow streamline can impede fluid flow. The viscous dissipation was 5840 W. A new streamline field was obtained by solving compressible flow field synergy equation. As shown in Fig. 6, the backflow streamline completely disappeared, and the viscous dissipation was only 646 W. Thus, the resistance was reduced.
Fig. 10. Local field synergy angles in the four cases.
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B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
Table 1 Various values of the parameters in the four cases.
Number of grid Mass flow (kg/s) Viscous dissipation (W) Arithmetic average angle ( ) Volume weighted average angle ( ) Vector die average angle ( ) Die dot product average angle ( ) Integral median average angle ( )
Before optimisation
After optimisation
After adding wedge-shaped face
After narrowing outlet width
12,180 5.0 5840 85.0 87.9 87.6 86.7 88.2
12,180 5.3 646 95.6 100.0 94.4 94.4 94.0
10,962 4.9 4444 84.1 86.5 87.1 86.5 87.6
11,312 5.0 4647 87.9 87.7 87.7 86.8 88.4
Since the inlet pressure and outlet pressure were constant, the reduction of drag mainly depended on two factors: changing the pipeline geometry structure and changing the inlet turbulence parameters. The first approach mainly includes changing the outlet wall shape (such as the exit angle, variable angle transition, etc.) and adding an air spoiler to the flow field. This study adopted a wedge-shaped face at the exit or decreased the width of the exit to reduce drag. The geometric dimensions of the wedge-shaped face were defined by the air streamline field in the pipeline after optimisation. As shown in Fig. 7, the height of the wedge-shaped face was 0.6 mm, and its edges were in line with the direction of the streamline. Fig. 8 shows the distribution of the streamline field in the upper right corner of the pipeline with the wedge-shaped face. As shown in Fig. 8, the backflow streamline weakened significantly, most of the opposing streamline disappears, and the streamline appeared as only a small offset to the main streamline direction. The viscous dissipation was 4444 W, and the resistance was reduced by 24%. As show in Fig. 9, narrowing the pipeline outlet width along the inner edge of the backflow streamline resulted in a viscous dissipation of 4647 W, which constituted a resistance reduction 20%. The angles of field synergy before optimisation, after optimisation, after adding wedge-shaped face and after narrowing outlet width were shown in Fig. 10a–d, respectively. The region the local filed synergy angle less than 80 was defined as A. It could be seen that: Ab Ac < Ad < Aa . Table 1 shows the various parameters in the four cases. According to the five methods of heat transfer field synergy evaluated by Zhou et al. [12], five average field synergy angles were examined in the compressible flow drag reduction model. As shown in Table 1, the mass flow rate did not significantly change among the cases. Thus, it was considered constant in this study. The five average field synergy angles after optimisation are greater than those before optimisation. Based on the fluid flow field synergy principle, the field synergy angle negatively correlates with the degree of field synergy and the drag reduction. However, the field synergy angle did not significantly change in response to the optimisation after adding the wedge-shaped face or narrowing the pipeline outlet width. However, the viscous dissipation reduced to 4444 W and 4647 W in the two cases. The above analysis indicates that solving the compressible field synergy equation can optimise the entire flow field; changing the specific structure can only alter the local field synergy angles, but the entire flow field synergy optimisation has not been achieved.
4. Concluding remarks According to the streamline filed of a variable cross-section pipeline, the structure has deficiencies that clearly require improvement. The optimal streamline field was obtained by solving compressible flow field synergy equation, which is the ultimate goal of structural optimisation design. Although the optimal streamline field was obtained, the specific structure of this streamline field was difficult to determine. In this
study, we attempted to modify the partial structure by adding a wedge-shaped face built along the optimal streamline or narrowing the pipeline outlet width along the inner edge of the backflow streamline. (a) Compared with the streamline filed of the original pipeline, the backflow was clearly reduced. (b) The resistance of the improved pipeline reduced by 24% and 20% for the two modification schemes. Conflicts of interest The authors declare there is no conflicts of interest regarding the publication of this paper. Acknowledgement This work was supported by the National Nature Science Foundation of China (No. 51276025). References [1] Z.Y. Guo, W.Q. Tao, R.K. Shah, The field synergy (coordination) principle and its applications in enhancing single phase convective heat transfer, Int. J. Heat Mass Transfer 48 (2005) 1797–1807. [2] W.Q. Tao, Z.Y. Guo, B.X. Wang, Field synergy principle for enhancing convective heat transfer-its extension and numerical verifications, Int. J. Heat Mass Transfer 45 (2002) 3849–3856. [3] J.F. Guo, M.T. Xu, L. Cheng, Numerical investigations of circular tube fitted with helical screw-tape inserts from the viewpoint of field synergy principle, Chem. Eng. Process.: Process Intensif. 49 (2010) 410–417. [4] X.W. Li, J.A. Meng, Z.Y. Guo, Turbulent flow and heat transfer in discrete double inclined ribs tube, Int. J. Heat Mass Transfer 52 (2009) 962–970. [5] Q. Chen, J.X. Ren, Z.Y. Guo, Fluid flow field synergy principle and its application to drag reduction, Chin. Sci. Bull. 53 (2008) 1768–1772. [6] Y.L. He, P.B. Huang, Z.G. Qu, W.Q. Tao, Numerical verification of field synergy principle in oscillating flow of gap regenerator, J. Eng. Thermophys. 24 (2003) 649–651 (in Chinese). [7] Q. Chen, J.X. Ren, Z.Y. Guo, Field synergy analysis and optimization of decontamination ventilation designs, Int. J. Heat Mass Transfer 51 (2008) 873–881. [8] W.Q. Tao, Y.L. He, Q.W. Wang, Z.G. Qu, F.Q. Song, A unified analysis on enhancing single phase convective heat transfer with field synergy principle, Int. J. Heat Mass Transfer 45 (2002) 4871–4879. [9] J.F. Guo, M.T. Xu, L. Cheng, The application of field synergy number in shelland-tube heat exchanger optimization design, Appl. Energy 86 (2009) 2079– 2087. [10] W. Liu, Z.C. Wei, T.Z. Ming, Z.Y. Guo, Physical quantity synergy in laminar flow field and its application in heat transfer enhancement, Int. J. Heat Mass Transfer 52 (2009) 4669–4672. [11] Q. Chen, M.R. Wang, Z.Y. Guo, Field synergy principle for energy conservation analysis and application, Adv. Mech. Eng. (2010). [12] J.J. Zhou, W.Q. Tao, D.B. Wang, Qualitative analysis and quantitative discussion of index for field synergy principle, J. Zhenzhou Univ. 27 (2006) 45–47 (in Chinese). [13] Y.X. Li, J.H. Wu, L. Zhang, L.P. Kou, Comparison of fluid flow and heat transfer behavior in outer and inner half coil jackets and field synergy analysis, Appl. Thermal Eng. 31 (2011) 3083–3087. [14] C. Habchi, T. Lemenand, D.D. Valle, L. Pacheco, O.L. Corre, H. Peerhossaini, Entropy production and field synergy principle in turbulent vortical flows, Int. J. Thermal Sci. 50 (2011) 2365–2376. [15] A. Bergles, Enhanced heat transfer: endless frontier, or mature and routine, Enhanced Heat Transfer 6 (1999) 79–88.
B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101 [16] R. Webb, Principles of Enhanced Heat Transfer, Wiley Interscience, 1994. [17] L. Léal, M. Miscevic, P. Lavieille, M. Amokrane, F. Pigache, F. Topin, B. Nogarède, L. Tadrist, An overview of heat transfer enhancement methods and new perspectives: focus on active methods using electroactive materials, Int. J. Heat Mass Transfer 61 (2013) 505–524.
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[18] Z.Y. Guo, D.Y. Li, B.X. Wang, A novel concept for convective heat transfer enhancement, Int. J. Heat Mass Transfer 41 (1998) 2221–2225. [19] R. Benzi, A short review on drag reduction by polymers in wall bounded turbulence, Phys. D: Nonlinear Phenom. 239 (2010) 1338–1345.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Compressible fluid flow field synergy principle and its application to drag reduction in variable-cross-section pipeline Bo Zhang ⇑, Jinsheng Lv, Jixue Zuo School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 27 February 2013 Received in revised form 12 January 2014 Accepted 6 June 2014
Keywords: Field synergy Compressible flow Drag reduction Optimisation
a b s t r a c t The compressible fluid flow field synergy principle was presented as an effective theoretical guide to reduce the drag during compressible flow. A compressible flow field synergy model was presented based on the incompressible flow field synergy principle. A variable cross-section pipeline air of compressible flow was used to verify the model. Two specific improvement schemes were studied for practical applications, which showed a 24% and 20% reduction in the resistance. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The past few decades have witnessed a growing interest in the field synergy principle. These studies have primarily focused on two aspects: heat convection and drag reduction in flow. To boost the heat transfer performance, Guo et al. [1–4] introduced the concept of field synergy and consequently garnered significant attention from other researchers. This concept focuses on the strength of the heat convection, which not only depends on the temperature gradient field, the velocity field in the fluid domain and the fluid property, but also on the included angle size between the velocity field and heat flow field. The size of the included angle negatively correlates with the degree of field synergy and the heat transfer efficiency. Field synergy theory was first applied to a laminar convection heat exchange process [1]. Subsequently, it has been numerically and experimentally validated in laminar and turbulent flow and applied to the analysis of heat exchangers [2–4]. The reduction of flow drag is a fundamental problem in hydrodynamics. So far, researchers have devised a variety of technologies to reduce flow drag. For example, objects in the external flow are designed to streamline or improve the surface roughness to delay the boundary layer separation; for internal flow, guide plates are placed in the bend to avoid secondary separation. However, most ⇑ Corresponding author. Address: 2# Linggong Road, Ganjingzi District, Dalian, Liaoning 116024, China. Tel.: +86 411 84706537; fax: +86 411 84691725. E-mail address: [email protected] (B. Zhang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.06.027 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
of the presented flow drag reduction technology is based on experience and lacks a unified theory. Chen et al. [5] began with the incompressible fluid flow drag reduction problem and explored the feasibility of solving this problem by using the field synergy theory. As a result, they presented a field synergy drag reduction model. They also deduced the incompressible flow field synergy equation according to the principle of the minimum dissipation of mechanical energy. By solving the equation, the optimal incompressible flow can be obtained to effectively guide the optimisation of pipeline design. Many researchers [6–14] have focused on studying the heat transfer field synergy presented by Guo. To quantitatively describe and compare the degree of field synergy between the velocity vector and temperature gradient vector, previous researchers have analysed and discussed the evaluation index in the heat transfer field synergy. The published literature indicates that the evaluation index of the field synergy mainly includes three aspects: (1) the included angle between the velocity vector and the temperature gradient vector; (2) the cosine of the included angle between the velocity vector and the temperature gradient vector; and (3) the field synergy number. The included angle between the velocity vector and the temperature gradient vector ranges from 0 to 90 in the heat transfer field synergy [6–10]. When the included angle ranges from 90 to 180 , the supplementary angle is examined. Zhou et al. [12] analysed and discussed the evaluation index of the heat transfer field synergy and presented five methods to calculate the included angle between the velocity vector and temperature gradient vector. They ultimately compared the five
1096
B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
Nomenclature B D DC F FS f L ~ n P Re S S Sij Skk t U u u V
Lagrange multiplier characteristic length, V=S, m pressure drop, defined by Eq. (7), dimensionless additional volume force, N field synergy number, defined by Eq. (8), dimensionless volume force, N length, m surface unit normal vector pressure, Pa Reynolds number, dimensionless area of the solid surface, m2 dimensionless area deformation rate tensor divergence, dimensionless time, s dimensionless velocity velocity, m/s dimensionless velocity volume, m3
calculation methods via numerical simulations. Guo proposed the field synergy number to evaluate the strength of the heat transfer field synergy. Compared with the included angle and its cosine value between velocity vector and temperature gradient vector, the field synergy number better reflects the quality of the heat transfer field synergy. However, the evaluation method has never been mentioned for compressible flow field synergy. The number of studies undertaken on heat transfer enhancement has been constantly growing since the early 1960s. In 1999, Bergles et al. mentioned nearly 4345 publications on the topic in their literature review [15]. The intensification techniques are numerous [16] and can be classified into two categories: passive and active [17]. Passive techniques enhance heat transfer by modifying the surface exchange or fluid properties, changing the surface geometry, disrupting boundary layers, or promoting liquid–vapour phase changes. Active techniques use jet, spray and electronic methods, etc. to enhance heat transfer. The relevant applications also have a promising future. The aforementioned methods all enhance the heat transfer, but their mechanisms are poorly understood. Therefore, Guo [18] provided a new concept to understand this process. This paper uses this new concept to elucidate further studies. Drag reduction in gas–liquid two-phase flow is caused by mechanical friction, which is inevitable. Drag can be reduced in pipe flow via two ways. One method consists of adding polymer to the flow [19], the other changes the boundary condition between the flow and solid or the structure of the entire flow field. This study attempts to use the second method to optimise the flow field and obtain new results. The flow drag reduction problem in compressible flow is also very important to design applications, such as the high-speed flow drag reduction problem in pneumatic conveying and aviation devices, etc. The relationship between the flow resistance and the field synergy degree in compressible flow is discussed in this paper. The evaluation method in the heat transfer field synergy is referenced to evaluate the compressible flow field synergy. According to the principle of the minimum dissipation of mechanical energy, a compressible flow field synergy equation was established that takes the equation of continuity as a constraint. Furthermore, a variable cross section of compressible pipe flow was examined as an example to verify the effectiveness of the compressible flow field synergy.
V x; y; z
dimensionless volume Cartesian coordinates, m
Greeks b dij k
field synergy angle, degree second order unit tensor, dimensionless constant dynamic viscosity, kg/(m s) kinematic viscosity, m2/s Lagrange function, W density, kg/m3 viscous dissipation function, W
l m P
q u
Subscripts i; j i direction, j direction x; y; z Cartesian coordinates in inlet m mean f boundary X domain
2. Compressible flow field synergy equation The following model was derived by taking the incompressible field synergy drag reduction model as reference and basing the model on the compressible Navier–Stokes equation. For compressible flow, the Navier–Stokes equation can be written as follows:
@ui @ui 1 @ þ uj ¼ fi @t @xj q @xj
2 P þ lSkk dij 2lSij 3
ð1Þ
The Navier–Stokes equation for steady-state compressible fluid flow without volume force can be written as follows:
Z
quj
X
@ui xj dV ¼ @
Z X
@P dV @xi
Z X
@ 2 lSkk dij 2lSij dV @xj 3
ð2Þ
Defining D ¼ V=S as characteristic length and introducing the following dimensionless variables:
ui ¼
ui ; uin
uj ¼
uj ; uin
rui ¼
rui ; uin =D
dV dV ¼ ; V
dS dS ¼ S
ð3Þ
yields the following form of Eq. (2):
D
q
u2in
Z X
Z @P l 2 xi dV ¼ 2 Skk dij 2Sij ~ ndS @ quin 1 3 Z @ ui dV þ Duj @xj X
The following has been well established:
@ ui @xj
ð4Þ
i ¼ @u @xj ¼ ru ui . Thus, u in
@ ui @ui rui rui ¼ @xj ¼ ¼ @xj uin uin D
in
ð5Þ
When combined with Eqs. (5) and (4) can be transformed into the following:
D qu2in
Z X
Z @P l 2 dV ¼ 2 Skk dij 2Sij ~ ndS @xi quin 1 3 Z þ Urui dV
ð6Þ
X
The term on the left side of Eq. (6) is the dimensionless pressure drop in the xi direction:
B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
DC i ¼
Z
D
qu2in
X
@P dV @xi
ð7Þ
The second term on the right is the integration of the dot product between the dimensionless velocity and the dimensionless velocity gradient vectors, which can be written as follows:
FSm ¼
Z
Urui dV ¼
X
Z
jrui j cos b dV U m
ð8Þ
X
When defining bm as the compressible flow field synergy angle, the field synergy angle and its cosine value reflect the degree of synergy between the velocity and the velocity gradient vectors, which is called the degree of compressible flow field synergy. While the field synergy angle is small, its cosine value is large, and the degree of the field synergy between the velocity and the velocity gradient vectors is quite high; conversely, the degree of the field synergy is low. According to the definition of Reynolds number
Re ¼
uin L
m
¼
uin V
l=q S
¼
uin qD
l
D uin
Z 2 ndS þ Re FSm Skk dij 2Sij ~ 1 3
ð10Þ
ZZZ
½u þ Br qudV
@ @B @q @q @q @ux @uy @uz ðBr quÞ ¼ ux þ uy þ uz þq þq þq @ux @ux @x @y @z @x @y @z @q @ þB þq @x @x ð18Þ divðquÞ ¼ ux
@q @q @q @ux @uy @uz þ uy þ uz þq þq þq ¼0 @x @y @z @x @y @z
@ @q @ @q @B @ ðBr quÞ ¼ B ¼B þq þq ¼ ðqBÞ @ux @x @x @x @x @x
@ux @uy @uz þ þ @x @y @z
ð21Þ
When combined with Eqs. (17) and (21), the change in P as a function of the velocity vector for u can be written as follows:
1 3
1 2
l r2 u þ rðr uÞ þ rðqBÞ ¼ 0
ð11Þ
Fig. 1. Internal structure diagram of variable cross-section pipeline.
ð12Þ
The change in u as a function of the velocity vector for ux can be obtained as follows:
ð14Þ
Fig. 2. Grid of the pipeline before optimisation.
When k ¼ 23 l is constant, the compressible fluid in the flow process meets the following criterion:
@ux @uy @uz þ þ – 0; @x @y @z
@k ¼0 @x
ð15Þ
When combined with Eqs. (15) and (14) can be transformed into the following:
@u 1 @ ¼ l 2r2 ux þ 2 ðr uÞ 3 @x @ux
ð16Þ
Thus, the change in u as a function of the velocity vector for u can be expressed as follows:
ð22Þ
Furthermore, the compressible fluid Navier–Stokes equation is defined as follows:
ð13Þ
@u @ @ ¼ l 2r2 ux þ 2 ðdiv uÞ þ 2kdiv u @x @x @ux
ð20Þ
Eq. (20) can be transformed into a three-dimensional form:
where div u can be written as follows:
divu ¼
ð19Þ
When combined with Eqs. (19) and (18) can be transformed into the following:
where u is the viscous dissipation function, which can be expressed as follows:
" 2 2 2 2 @ux @uy @uz @ux @uy þ2 þ2 þ þ @x @y @z @y @x 2 2 # @ux @uz @uy @uz 2 þ kðdivuÞ þ þ þ þ @z @x @z @y
ð17Þ
Furthermore,
X
u¼l 2
1097
@ ðBr quÞ ¼ rðqBÞ @u
Eq. (10) indicates that the dimensionless pressure drop for a compressible fluid depends on the degree of synergy between the velocity field and velocity gradient field. This paper defines Eq. (10) as the compressible flow field synergy model for drag reduction. A Lagrange function for the viscous dissipation is established based on continuity equation:
P¼
@u 1 ¼ l 2r2 u þ 2 rðr uÞ 3 @u
ð9Þ
Eq. (6) can be transformed into the following:
Re DC i ¼
Fig. 3. Grid of the pipeline after adding a wedge-shaped face.
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B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
2 F rP r lr u þ r ð2lSÞ ¼ qu ru 3
ð23Þ
When contrasting Eqs. (22) and (23), the following equation can be obtained:
rðqBÞ ¼ rP
ð24Þ 2
F ¼ qu ru r ð2lSÞ 2lr u
ð25Þ
Combining Eqs. (23) and (25) yields the following:
qu ru r ð2lSÞ 2lr2 u rP r
2 lr u 3
þ r ð2lSÞ ¼ qu ru
ð26Þ
Eq. (26) denotes the compressible flow field synergy equation. In contrast with the incompressible flow field synergy, the additional volume force is different because the size of the additional volume force, F, is not equal to the inertial force, in the fluid flow process. This paper proposes that solving Eq. (26) can yield the optimal velocity field based on a given set of boundary conditions. 3. Application of field synergy to reduce drag in a variable crosssection pipeline
Fig. 4. Grid of the pipeline after narrowing the pipeline outlet width.
Fig. 5. Air streamline field near the top right corner in the pipeline before optimisation.
Fig. 6. Air streamline field near the top right corner after optimisation.
A variable cross-section pipeline was studied to verify the model presented above. The configuration and pipeline dimensions are presented in Fig. 1. The unit is mm. The wall was assumed to be adiabatic and its thickness was neglected. The inlet pressure and
Fig. 7. The shape and the position of the wedge-shaped face.
Fig. 8. Air streamline field near the top right corner after adding wedge-shaped face to the pipeline.
B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
temperature were 2 105 Pa and 300 K, respectively. The outlet pressure was 1 105 Pa. The air dynamic viscosity was 1:7894 105 kg/(ms). The Mach number was 0.35; thus, the flow pattern was a subsonic compressible flow.
Fig. 9. Air streamline field near the top right corner after narrowing the pipeline outlet width.
1099
The FLUENT 6.3 software was employed to solve the governing equation in the flow process to obtain the heat transfer field and the velocity field of the pipeline. The pipeline was modelled as a two-dimensional body according to Fig. 1, and the grid had a quadrilateral structure. As shown in Figs. 2–4, the grid initially consisted of 12,180 elements and was later adapted to 10,962 elements. The inlet boundary condition was a pressure-inlet and the outlet boundary condition was a pressure-outlet. The solving method was couple implicit. The realisable k-e turbulence model was selected. The pressure and velocity were coupled by the SIMPLE algorithm, and the diffusion and convection terms were discretised using a first order upwind scheme. The additional volume force term was added by the user-defined function (UDF) in FLUENT. After calculating the flow in the variable cross-section pipeline, the distribution of the streamline field could be obtained. Backflow appeared at the top right and bottom right corners of the streamline field. The backflow was caused by a combination of the inertial and viscous forces. Because the pipeline was a symmetric structure along the axis, the air streamline field near the top right corner was examined in this study. As shown in Fig. 5, a series of backflow streamlines appeared that flowed opposite to the main flow direction in the top right corner. In addition to the friction between fluid and wall, the backflow streamline can impede fluid flow. The viscous dissipation was 5840 W. A new streamline field was obtained by solving compressible flow field synergy equation. As shown in Fig. 6, the backflow streamline completely disappeared, and the viscous dissipation was only 646 W. Thus, the resistance was reduced.
Fig. 10. Local field synergy angles in the four cases.
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B. Zhang et al. / International Journal of Heat and Mass Transfer 77 (2014) 1095–1101
Table 1 Various values of the parameters in the four cases.
Number of grid Mass flow (kg/s) Viscous dissipation (W) Arithmetic average angle ( ) Volume weighted average angle ( ) Vector die average angle ( ) Die dot product average angle ( ) Integral median average angle ( )
Before optimisation
After optimisation
After adding wedge-shaped face
After narrowing outlet width
12,180 5.0 5840 85.0 87.9 87.6 86.7 88.2
12,180 5.3 646 95.6 100.0 94.4 94.4 94.0
10,962 4.9 4444 84.1 86.5 87.1 86.5 87.6
11,312 5.0 4647 87.9 87.7 87.7 86.8 88.4
Since the inlet pressure and outlet pressure were constant, the reduction of drag mainly depended on two factors: changing the pipeline geometry structure and changing the inlet turbulence parameters. The first approach mainly includes changing the outlet wall shape (such as the exit angle, variable angle transition, etc.) and adding an air spoiler to the flow field. This study adopted a wedge-shaped face at the exit or decreased the width of the exit to reduce drag. The geometric dimensions of the wedge-shaped face were defined by the air streamline field in the pipeline after optimisation. As shown in Fig. 7, the height of the wedge-shaped face was 0.6 mm, and its edges were in line with the direction of the streamline. Fig. 8 shows the distribution of the streamline field in the upper right corner of the pipeline with the wedge-shaped face. As shown in Fig. 8, the backflow streamline weakened significantly, most of the opposing streamline disappears, and the streamline appeared as only a small offset to the main streamline direction. The viscous dissipation was 4444 W, and the resistance was reduced by 24%. As show in Fig. 9, narrowing the pipeline outlet width along the inner edge of the backflow streamline resulted in a viscous dissipation of 4647 W, which constituted a resistance reduction 20%. The angles of field synergy before optimisation, after optimisation, after adding wedge-shaped face and after narrowing outlet width were shown in Fig. 10a–d, respectively. The region the local filed synergy angle less than 80 was defined as A. It could be seen that: Ab Ac < Ad < Aa . Table 1 shows the various parameters in the four cases. According to the five methods of heat transfer field synergy evaluated by Zhou et al. [12], five average field synergy angles were examined in the compressible flow drag reduction model. As shown in Table 1, the mass flow rate did not significantly change among the cases. Thus, it was considered constant in this study. The five average field synergy angles after optimisation are greater than those before optimisation. Based on the fluid flow field synergy principle, the field synergy angle negatively correlates with the degree of field synergy and the drag reduction. However, the field synergy angle did not significantly change in response to the optimisation after adding the wedge-shaped face or narrowing the pipeline outlet width. However, the viscous dissipation reduced to 4444 W and 4647 W in the two cases. The above analysis indicates that solving the compressible field synergy equation can optimise the entire flow field; changing the specific structure can only alter the local field synergy angles, but the entire flow field synergy optimisation has not been achieved.
4. Concluding remarks According to the streamline filed of a variable cross-section pipeline, the structure has deficiencies that clearly require improvement. The optimal streamline field was obtained by solving compressible flow field synergy equation, which is the ultimate goal of structural optimisation design. Although the optimal streamline field was obtained, the specific structure of this streamline field was difficult to determine. In this
study, we attempted to modify the partial structure by adding a wedge-shaped face built along the optimal streamline or narrowing the pipeline outlet width along the inner edge of the backflow streamline. (a) Compared with the streamline filed of the original pipeline, the backflow was clearly reduced. (b) The resistance of the improved pipeline reduced by 24% and 20% for the two modification schemes. Conflicts of interest The authors declare there is no conflicts of interest regarding the publication of this paper. Acknowledgement This work was supported by the National Nature Science Foundation of China (No. 51276025). References [1] Z.Y. Guo, W.Q. Tao, R.K. Shah, The field synergy (coordination) principle and its applications in enhancing single phase convective heat transfer, Int. J. Heat Mass Transfer 48 (2005) 1797–1807. [2] W.Q. Tao, Z.Y. Guo, B.X. Wang, Field synergy principle for enhancing convective heat transfer-its extension and numerical verifications, Int. J. Heat Mass Transfer 45 (2002) 3849–3856. [3] J.F. Guo, M.T. Xu, L. Cheng, Numerical investigations of circular tube fitted with helical screw-tape inserts from the viewpoint of field synergy principle, Chem. Eng. Process.: Process Intensif. 49 (2010) 410–417. [4] X.W. Li, J.A. Meng, Z.Y. Guo, Turbulent flow and heat transfer in discrete double inclined ribs tube, Int. J. Heat Mass Transfer 52 (2009) 962–970. [5] Q. Chen, J.X. Ren, Z.Y. Guo, Fluid flow field synergy principle and its application to drag reduction, Chin. Sci. Bull. 53 (2008) 1768–1772. [6] Y.L. He, P.B. Huang, Z.G. Qu, W.Q. Tao, Numerical verification of field synergy principle in oscillating flow of gap regenerator, J. Eng. Thermophys. 24 (2003) 649–651 (in Chinese). [7] Q. Chen, J.X. Ren, Z.Y. Guo, Field synergy analysis and optimization of decontamination ventilation designs, Int. J. Heat Mass Transfer 51 (2008) 873–881. [8] W.Q. Tao, Y.L. He, Q.W. Wang, Z.G. Qu, F.Q. Song, A unified analysis on enhancing single phase convective heat transfer with field synergy principle, Int. J. Heat Mass Transfer 45 (2002) 4871–4879. [9] J.F. Guo, M.T. Xu, L. Cheng, The application of field synergy number in shelland-tube heat exchanger optimization design, Appl. Energy 86 (2009) 2079– 2087. [10] W. Liu, Z.C. Wei, T.Z. Ming, Z.Y. Guo, Physical quantity synergy in laminar flow field and its application in heat transfer enhancement, Int. J. Heat Mass Transfer 52 (2009) 4669–4672. [11] Q. Chen, M.R. Wang, Z.Y. Guo, Field synergy principle for energy conservation analysis and application, Adv. Mech. Eng. (2010). [12] J.J. Zhou, W.Q. Tao, D.B. Wang, Qualitative analysis and quantitative discussion of index for field synergy principle, J. Zhenzhou Univ. 27 (2006) 45–47 (in Chinese). [13] Y.X. Li, J.H. Wu, L. Zhang, L.P. Kou, Comparison of fluid flow and heat transfer behavior in outer and inner half coil jackets and field synergy analysis, Appl. Thermal Eng. 31 (2011) 3083–3087. [14] C. Habchi, T. Lemenand, D.D. Valle, L. Pacheco, O.L. Corre, H. Peerhossaini, Entropy production and field synergy principle in turbulent vortical flows, Int. J. Thermal Sci. 50 (2011) 2365–2376. [15] A. Bergles, Enhanced heat transfer: endless frontier, or mature and routine, Enhanced Heat Transfer 6 (1999) 79–88.
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