Heat Mass Transfer (2007) 43:603–611 DOI 10.1007/s00231-006-0133-7

ORIGINAL

Numerical investigation of heat transfer and fluid flow characteristics inside a wavy channel Gong-Nan Xie Æ Qiu-Wang Wang Æ Min Zeng Æ Lai-Qin Luo

Received: 8 January 2005 / Accepted: 4 May 2006 / Published online: 7 July 2006  Springer-Verlag 2006

Abstract The periodically fully developed laminar heat transfer and fluid flow characteristics inside a twodimensional wavy channel in a compact heat exchanger have been numerically investigated. Calculations were performed for Prandtl number 0.7, and Reynolds number ranging from 100 to 1,100 on non-orthogonal non-staggered grid systems, based on SIMPLER algorithm in the curvilinear body-fitted coordinates. Effects of wavy heights, lengths, wavy pitches and channel widths on fluid flow and heat transfer were studied. The results show that overall Nusselt numbers and friction factors increase with the increase of Reynolds numbers. According to the local Nusselt number distribution along channel wall, the heat transfer may be greatly enhanced due to the wavy characteristics. In the geometries parameters considered, friction factors and overall Nusselt number always increase with the increase of wavy heights or channel widths, and with the decrease of wavy lengths or wavy pitches. Especially the overall Nusselt number significantly increase with the increase of wavy heights or channel widths, where the flow may become into transition regime with a penalty of strongly increasing in pressure drop. Keywords Heat transfer Æ Flow friction Æ Wavy channel Æ SIMPLER algorithm List of symbols cp Specific heat, J/(kg K) D Hydraulic diameter (=2H), mm G.-N. Xie Æ Q.-W. Wang (&) Æ M. Zeng Æ L.-Q. Luo State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China e-mail: [email protected]

f H J L Lp Nu Num Nu(n) p P, Q q(n) Re SL S/ Sp T ui xi

Friction factor Channel width, mm Jacobean of coordinate transformation Wavy pitch, mm Wavy length, mm Overall Nusselt number Average Nusselt number Local Nusselt number Pressure, Pa Control functions in elliptic equation systems Local heat flux, W/m2 Reynolds number Expansion length of wall, mm Source term Wavy height, mm Temperature, K Velocity components in Cartesian coordinates, m/s Cartesian coordinates

Greek symbols a, b, c Metric coefficients k Thermal conductivity, W/(m K) l Fluid dynamics viscosity, Pa s n, g Coordinates in transformed plane q Fluid density, kg/m3 / General dependent variable G General diffusion coefficient

Subscripts b Bulk in Inlet

123

604

m w n, g xx, yy

Heat Mass Transfer (2007) 43:603–611

Mean Wall Partial derivatives with respect to n and g Second order partial derivatives with respect to xx and yy

1 Introduction Compact heat exchangers are characterized by a large heat transfer surface area per unit volume of the exchanger, resulting in reduced space, weight, support structure and footprint energy requirements, and cost, as well as improved process design, plant layout, and processing conditions. When gas is used as working medium, an enhancement technique may be used to improve intensity of heat transfer. One way to enhance gas-side heat transfer is to modify the fluid flow channel. Originally, plain fin had been used until 1960s when corrugated or herringbone wavy fins were introduced to enhance heat transfer by lengthening the flow path and cause better mixing in the fin flow channels.Currently, the trend is to develop various slit fin designs, such as offset strip or louvered fins, where the principle of enhancing heat transfer is to destroy boundary layer growth and improve mixing in the flow channels. Although slit fins generally have higher heat transfer coefficients than wavy fins, wavy fins have some advantages in these designs such as low pressure drop and high reliability. That is why wavy fin is dominant fin used in various compact heat exchangers. Therefore, different modified wavy-fin channels have been presented for enhancing heat transfer during these years. The flow and heat transfer in wavy channels have been extensively investigated in recent years. Xin and Tao [1], Yang and Asako [2], Wang and Vanka [3], Niceno and Nobile [4], Comini and Nonino [5], Niu and Zhang [6], as well as references [7–10] numerically studied laminar flow and heat transfer characteristics in different wavy channels. Experimental studies on the turbulent flow and heat transfer in wavy channels were reported in [11–13]. Patel and colleagues [14], Masoud and Thomas [15], Ergin and Ota [16, 17] numerically investigated the turbulent flow in wavy channels. Three dimensionality flow in wavy-walled channels was studied in [18, 19]. Cross wavy ducts in primary surface heat exchanger were investigated by Utriainen and Sunden [20, 21]. Recent progress in studying various fin surface in compact heat exchanger were summarized in [22].

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In this paper, the flow and heat transfer characteristics inside a wavy channel in a compact heat exchanger were investigated numerically. The analyses were performed for two-dimension periodically fully developed laminar flow. The wall temperature is kept constant. The current work presents numerical computation aimed at studying the effects of various parameters such as the flow Reynolds number, wavy height and wavy length. The mathematical model is based on finite volume method using nonorthogonal body-fitted coordinates and a non-staggered grid. The computations were performed for the flow Reynolds number ranging from 100 to 1,100, and Prandtl number 0.7. Furthermore, the pressure drop and heat transfer were compared to those of parallel-plate channel.

2 Physical model and mathematical formulation The problem to be analyzed is schematically shown in Fig. 1. The engineering background of this study is heat transfer and fluid flow in wavy channel used in compact heat exchangers, such as the internally finned tubes with insert blocked core-tubes of inter-cooler of compressors, as shown in Fig. 1a, where the side walls are formed by various wavy shapes, as expanded wavy fin shown in Fig. 1b. The computational domain is chosen within one cycle as shade region shown in Fig. 1b. The typical values for the wavy channel geometrical parameters are as follows: the width of the channel is H=2 mm, wavy height is Sp=1 mm, wavy length is Lp=5 mm, and wavy pitch is L=13 mm. The numerical computation is based on the following assumptions: (1) the fluid properties are constant; (2) the flow and heat transfer are in steady state, laminar and periodically fully developed; (3) the body force and dissipation terms are neglected. The equations for continuity, momentum and energy in physical domain may be expressed in tensor form as follows. Continuity equation @ui ¼ 0; @xi

ð1Þ

Momentum equations   @ l @ @uk 1 @p ðui uk Þ ¼ ;  @xi q @xi @xi q @xk

ð2Þ

Energy equation   @ k @ @T ðui T Þ ¼ : @xi qcp @xi @xi

ð3Þ

Heat Mass Transfer (2007) 43:603–611

605

Fig. 1 Schematic diagram of wavy channel

wavy fin

block tube

L A

Fluid flow

Sp B

C

D

H

y E

x O

Lp

ui ðx; yÞjABCD ¼ ui ðx; yÞjEFGH ¼ 0;

ð4Þ

Tðx; yÞjABCD ¼ Tðx; yÞjEFGH ¼ Tw ;

ui ðx; yÞjAE ¼ ui ðx; yÞjDH ;

gxx þ gyy ¼ Qðn; gÞ;

ð5Þ

Tðx; yÞjAE ¼ Tðx; yÞjDH :

Attention is turned to coordinate transformation and grid generation. The continuity equation and the conservation form of transport equation for a general dependent variable in curvilinear coordinate systems (n, g) can be written as follows: @U @V þ ¼ 0; @n @g

ð6Þ

   @ @ @ 1 ðU/Þ þ ðV/Þ ¼ C/ a/n  b/g @n @g @n J    @ 1 b/n þ c/g þ S/ ; þ C/ @g J ð7Þ where G/ denotes general diffusion coefficient, for u, v it is l/q, while for T, it is k/q cp. Here l, q, k and cp represent dynamics viscosity, fluid density, thermal conductivity and specific heat respectively. S/ is the source term in computational space. The contra-variant velocities U, V and geometric parameters are given as: U ¼ uyg  vxg ; a¼

x2g

þ

y2g ;

V ¼ vxn  uyn ; b ¼ xn xg þ yn yg ;

The conservation equations are solved on a curvilinear non-orthogonal grid. The non-orthogonal grid is generated by elliptic equation systems, by solving the Poisson equations: nxx þ nyy ¼ Pðn; gÞ;

where Tw represents the wall temperature. At the inlet and outlet boundaries:

H

(b)

(a)

Boundary conditions At the top and bottom walls:

G

F

ð9Þ

where P, Q are control functions. The detail of the grid generation methods can be found in [23]. A typical grid system generated by this method is shown in Fig. 2. It is found that although the non-orthogonal grid generation method is used, the grid lines keep good orthogonality in regular region.

3 Numerical procedure The discretization of the transport equations in the computational domain was performed on a non-staggered grid by using the finite volume method with SIMPLER (Semi-Implicit Method for Pressure Linked Equations Revised) algorithm to deal with the linkage between pressure and velocities [24]. The second-order upwind difference (SUD) scheme and central-order difference (CD) scheme were used in discretizing convection terms and diffusion terms, respectively.

J ¼ xn yg  xg yn ; c ¼ x2n þ y2n : ð8Þ

Fig. 2 Typical grid used in computation

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606

Heat Mass Transfer (2007) 43:603–611

Because the grid is non-orthogonal, the pressure equation and pressure correction equation contain cross-derivatives, which leads to a nine-point formulation in two-dimensional computation. In this study, the cross-derivatives were treated explicitly, where the pressure value and pressure correction value were coupling with interface contra-variant velocities, more details can be found in [25]. Therefore a five-point solver was used to solve the algebraic equations in twodimension computation. The solutions of the implementation of the periodical boundary conditions in the computational domain were listed in [26]. In this computation, the method of mutual replacements of field variables at one end with those at the other end is used. Local Nusselt number was determined by NuðnÞ ¼

qðnÞ D ; TW  TðnÞ k

ð10Þ

while the local heat flux at the wall surface was computed by Fourier’s law of heat conduction. The value of local heat flux was determined by qðnÞ ¼ 

pffiffiffi kTg c : J

ð11Þ

ð16Þ

Re ¼ quin D=l:

The friction factor of one cycle was determined by f ¼

pm ðnAE Þ  pm ðnDH Þ D : SL qu2in =2

ð17Þ

The iteration convergence is that the maximum relative residual of control-volume (SMAX) in the continuity equation is less then a pre-specified value, ð18Þ

SMAX=flowin

where flowin is the inlet mass flow rate. A preliminary computation was performed on four grid systems (60 · 12, 80 · 20,100 · 30, 120 · 40) to assure the grid independence of the numerical solution for the case Re=120. The test results are shown in Table 1. From the Table, it can be seen that, deviation between the result by grid 120 · 40 and that by 80 · 20 is less than 3%, therefore, the gird 80 · 20 was chosen for all the computations.

4 Results and discussion 4.1 Test problem

The mean Nusselt number at wall surface was determined by R NuðnÞds R Num ¼ : ð12Þ ds The average heat flux of wall surface was determined by R nD qm ¼

nA

Rn pffiffiffi pffiffiffi qðnAD Þ cdn þ nEH qðnEH Þ cdn : R n pffiffiffi R nD pffiffiffi cdn þ nEH cdn nA

ð13Þ

The log-mean temperature difference was determined by Dtm ¼

Tb ðnDH Þ  Tb ðnAE Þ Tb ðnAE Þ ln TTWWT b ðnDH Þ

:

ð14Þ

The code developed according to the solution methodology in the preceding cross section was first applied to compute flow in parallel-plate channel where the wavy height is set to zero. The computed velocity profile of fully developed region was compared with the corresponding analytical solution. The excellent agreement is exhibited in Fig. 3. The Nusselt number and friction factor were 7.548 and 95.9, respectively. 4.2 Flow field The typical predicted flow field is shown in Fig. 4. The configuration of the domain is under original geometrical parameters. The characteristics of flow field shown in Fig. 4 may be summarized as follows. Firstly, no recirculation zone appears at low Reynolds number (about less than 300), that is, heat transfer in the wavy

Therefore, the overall Nusselt number within one cycle was computed by Table 1 Grid independence study (Re=120)

Nu ¼

qm D : Dtm k

The Reynolds number was defined as

123

ð15Þ

Variable

60 · 12

80 · 20

100 · 30

120 · 40

f Nu

0.9733 7.5287

0.9851 7.5886

0.9973 7.6487

1.0408 7.7685

Heat Mass Transfer (2007) 43:603–611

607

1.0

Re=120

u/umax

0.8

0.6

0.4 The present work Exact solution

0.2

0.0 0.0

0.2

0.4

0.6

0.8

y/D Fig. 3 Velocity profile of parallel-plate channel (Re=120)

1.0

channel is not enhanced, and therefore its characteristic is closing to that of parallel-plate channel. Secondly, there are two little recirculation flows within one cycle flow field. One is located in the top-left region below top wavy, the other one is located in bottomright region behind bottom wavy at Re=386. This is quite reasonable as these two regions are leeward surfaces. Thirdly, with Reynolds number increasing, the sizes of the two flow recirculation zones increase. Furthermore, little recirculation flow appeared near the exit of channel. This variation will surely affect both the pressure drop and the overall Nusselt number. Lastly, the size of exit recirculation flow zone will increase when Reynolds number is larger than 1,000. 4.3 Local Nusselt number and pressure The local Nusselt number and pressure distribution along wall surfaces are shown in Fig. 5. It can be seen in Fig. 5a, that there exist a wavy through Nusselt number distribution in the top wavy region, corre-

2.0

(a) C

Re=120

1.6

Nu(ξ)/Num

(a) Re=120, uin=0.45m/s

B

1.2

0.8

(b) Re=386, uin=1.45m/s

Bottom Top

F parallel-plate channel G

0.4 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14

x 0.5

(b)

0.0

F

-0.5

p/(ρuin2/2)

-1.0

(c) Re=676, uin=2.53m/s

-1.5

par al l e l-pla te c G han n

-2.0 B

-2.5

el

bottom Top

-3.0 -3.5 -4.0

C

-4.5

(d) Re=963, uin=3.61m/s Fig. 4 Flow fields at different Reynolds numbers

-1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14

x

Fig. 5 Local Nu and pressure along wall surfaces

123

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Heat Mass Transfer (2007) 43:603–611

sponding to local maximum values at points B and C. While there exist a wavy apex in the bottom wavy region, corresponding to local values at points F and G. This may be due to the relatively strong heat transfer at bottom-left wavy that locating windward surface, and the low intensity of heat transfer bottom-right wavy that locating leeward surface. The heat transfer at top wall surface is almost contra-symmetrical to that at bottom wall surface. It is can be seen in Fig. 5b, that there exist a wavy pressure distribution at the top wall surface, and a wavy peak at bottom wall surface. This may be the pressure value at leeward surface as existing low velocity potential is much higher than that at windward surface as existing high velocity potential. Obviously, it is found that the pressure drop of the wavy channel is higher than that of parallel-plate channel. The local Nusselt number at two wall surfaces is presented in Fig. 6. It can be seen that, at the bottom wall surface, with Reynolds number increasing, the local maximum value of Nusselt number increase, whose location move ahead, and the minimum values at points F and G decrease, Nusselt number near inlet

and exit slightly increase. While at the top wall surface, the local minimum value of Nusselt number decrease, whose location move ahead, and the maximum values at points B and C increase, Nusselt number near inlet and outlet slightly decrease, especially, there occur fluctuation at high Reynolds numbers. 4.4 Friction factor and overall Nusselt number The friction factors and overall Nusselt numbers versus Reynolds number under different wavy height are shown in Fig. 7. It can be seen from Fig. 7a, that in the low Reynolds number region (approximately less than 300), the variation of f with Reynolds number has similar trend as that of fully developed laminar flow in a parallel-plate channel, that is, f decreases linearly with increase in Reynolds numbers. This is the flow region where pressure drop is mainly caused by the surface friction effect. In the high Reynolds number region, with further increasing in Reynolds numbers,

Lp=5mm,H=2mm,L=13mm

1 20

(a)

Nu

16

Bottom transition

12

f

Re=120 Re=386 Re=675 Re=963

Sp/mm 5/3 5/4 1 5/6

8

(a)

4

0.1 100

F

1000

G

0

0

1

2

3

4

5

6

7

8

9

Re 10 11 12 13

14

x 25

Lp=5,H=2mm,L=13mm

(b)

C

12

Top Re=120 Re=386 Re=675 Re=963

B

15

Nu

SP /mm

20

Nu

fR e= 96

5/3 5/4 1 5/6

10

8

10

Nu=7.54

5

0

(b) 6 100 0

1

2

3

4

5

6

7

8

9

10 11 12 13

x

Fig. 6 Local Nusselt numbers at different Reynolds numbers

123

1000

Re

Fig. 7 Variation of friction factor and overall Nusselt number under different wavy heights

Heat Mass Transfer (2007) 43:603–611

Fig. 9. It can be seen that, f and Nu increase with the decrease of wavy pitch. Furthermore, the enhancement intensity for decrease wavy pitch of heat transfer is similar to that for decreasing wavy lengths. The effects of different channel widths on f and Nu are shown in Fig. 10. It can be seen that f and Nu decrease with the decrease of channel widths. From Fig. 10a, at small channel width, when Reynolds number exceeds a certain value, the friction factor ceases to change with Reynolds number where the flow may become into transition flow. One can find the critical Reynolds numbers Recr at H=1.5, 2 and 3 are about 840, 1,080 and 1,200, respectively. It is found that the Recr decrease with the decrease of channel widths. That is, at small channel width, the flow may quickly become into transition flow, or turbulent flow at low Reynolds number.

1

f

Sp=1mm,H=2mm,L=13mm

Lp/mm 6 5 4.5 4

fR e= 96

(a) 100

1000

Re 8.5 Sp=1,H=2mm,L=13mm

Lp/mm 6 5 4.5 4

8

Nu

the friction factor increases, that is, the flow resistance increases due to those recirculation vortices. Furthermore, the higher the wavy height, the larger the value of f for each case with different heights. This becomes obvious at high Reynolds numbers. However, at high wavy height, when the Reynolds number exceeds a certain value the flow may become into transition regime, the per-cycle pressure drop ceases to change with Reynolds number. One can find the critical Reynolds number Recr for Sp=5/4 and 5/3 are about 800 and 650, respectively. It is found that as wavy height increasing, Recr decreases. That is, at high wavy height, the flow may quickly become into transition flow, even turbulent flow at low Reynolds numbers. From Fig. 7b, it can be seen that in low Reynolds number region, the overall Nusselt number almost remains constant, the increase in Reynolds number leads to an appreciable increase in overall Nusselt number. Furthermore, with the wavy height increasing, the overall Nusselt number increase, especially there is a significant increase in Nu compared to the value (7.541) for parallel-plate channel at higher wavy heights, because the fluid may have better mixing due to the size increase of recirculation zones. Thus, the wavy channel can be an effective heat transfer enhancement technique at high Reynolds numbers and high wavy heights. It should be noted that despite the increasing in overall heat transfer, the penalty in pressure drop is usually large. For the case of Re=1,000, Sp=1 mm, the overall Nusselt number is approximately 1.1 times than that of parallel-plate channel, while the pressure drop is about 2 times than that of parallel-plate channel. For the case of Re=1,000, Sp=5/3, the overall Nusselt number is approximately two times than that of parallel-plate channel, while the pressure drop is about eight times than that of parallelplate channel. In Fig. 7, we also notice a strong dependence of Nu and f on Reynolds numbers. With the decrease of Re, both Nu and f approa1ch the corresponding values for the parallel-plate channel case. The results presented in [1, 3, 5] showed similar trends. The effects of different wavy lengths on friction factor and overall Nusselt number are presented in Fig. 8. As wavy lengths decrease, f and Nu increase. When Reynolds number decrease, both Nu and f approach their asymptotic values, that is, Nusselt number is close to 7.54 and fRe is close to 96. It is found that the effect of decreasing wavy lengths on heat transfer enhancement is much slighter than that of increasing wavy heights. For the case of Re=1000, the enhancement intensity of Lp=4 mm is about 5% higher than that of Lp=6 mm, while pressure drop of Lp=4 mm is about 40 percent higher than that of Lp=6 mm. The effects of different wavy pitch on f and Nu are shown in

609

7.5

Nu=7.54

(b) 7 100

1000

Re

Fig. 8 Variation of friction factor and overall Nusselt number under different wavy lengths

123

610

Heat Mass Transfer (2007) 43:603–611 1

1

Lp=5mm,Sp=1mm,H=2mm

L=13 mm, Lp=5 mm, Sp=1 mm

H=2 mm

f

f

H=3 mm L/mm 11 13 15

fR e= 96

H=1.5 mm

transition

(a) 100

(a)

1000

100

Re

1000

Re

8.5 Lp=5mm,Sp=1mm,H=2mm

L=13mm, Lp=5mm,Sp=1mm L

8

H=3 mm

Nu

Nu

11 13 15

7.5

8

H=2 mm

Nu=7.54

H=1.5 mm

(b)

(b)

7 100

7 100

1000

Re

Fig. 9 Variation of friction factor and overall Nusselt number under different wavy pitches

Fig. 10 Variation of friction factor and overall Nusselt number under different channel widths

5 Conclusions An investigation on fluid flow and heat transfer characteristics of periodically fully developed flow inside a wavy channel in a compact heat exchanger has been performed numerically on a non-orthogonal non-staggered grid, based on SIMPLER algorithm. The following conclusions can be drawn. 1.

2.

Intensity of heat transfer for the windward surfaces is relatively strong to that for the leeward surfaces. In the low Reynolds number region, the characteristics of fluid flow and heat transfer within one cycle are close to those for parallelplate channel. Intensity of heat transfer may be improved with the increase of wavy height and channel width, or the decrease of wavy length and wavy pitch in the high Reynolds number region. Increasing the wavy height has the most significant effect on heat

123

1000

Re

3.

transfer enhancement. This may be attributed to enlargement of the recirculation flow zones. At higher wavy height or smaller channel width, the pressure drop ceases to change with change of Reynolds number, where the flow may quickly become into transition flow, even turbulent flow in low Reynolds numbers region.

Acknowledgments This work was supported by Higher Academy Young Teacher Foundation Project of Fok Ying-Tung Education Foundation (Grant No. 91056) and National Natural Science Foundation of China (Grant No.50323001).

References 1. Xin RC, Tao WQ (1988) Numerical prediction of laminar flow and heat transfer in wavy channels of uniform crosssectional area. Numer Heat Transf 14:465–481

Heat Mass Transfer (2007) 43:603–611 2. Yang LC, Asako Y, Yamaguchi Y, Faghri M (1997) Numerical prediction of Transitional characteristics of flow and heat transfer in a corrugated duct. ASME J Heat Transf 119:62–69 3. Wang G, Vanka SP (1995) Convective heat transfer in wavy passages. In J Heat Mass Transf 38:3219–3230 4. Niceno B, Nobile E (2001) Numerical analysis of fluid flow and heat transfer in periodic wavy channels. Int J Heat Fluid Flow 22:156–167 5. Comini G, Nonino C, Savino S (2003) Effect of aspect ratio on convection enhancement in wavy channel. Numer Heat Transf Part A 44:21–27 6. Niu JL, Zhang LZ (2002) Heat transfer and friction coefficients in corrugated ducts confined by sinusoidal and arc curves. Int J Heat Mass Transf 45:571–578 7. Metwally HM, Manglik RM (2004) Enhanced heat transfer due to curvature-induced lateral vortices in laminar flow in sinusoidal corrugated-plate channels. Int J Heat Mass Transf 47:2283–2292 8. Asako Y, Nakamura H, Faghri M (1988) Heat transfer and pressure difference in a corrugated duct with rounded corners. Int J Heat Mass Transf 31:1237–1245 9. Xie GN, Wang QW, Luo LQ (2006) A numerical study of heat transfer and fluid flow characteristics in a periodically wavy channel. Chin J Comput Phys 23:93–97 (in Chinese) 10. Tanda G, Vittori G (1996) Fluid flow and heat transfer in a two-dimensional wavy channel. Heat Mass Transf 33:411– 418 11. Zeng M, Wang QW, Qu ZG, Bai W, Tao WQ (2002) Experimental study on the pressure drop and heat transfer characteristics in corrugated tubes (in Chinese). J Xi’an Jiao Tong Univ 36:237–240 12. Rush TA, Newell TA (1999) An experimental study of flow and heat transfer in sinusoidal wavy passages. Int J Heat Mass Transf 42:1541–1553 13. Ress G, Beer H (1997) Heat transfer and fluid flow in pipe with sinusoidal wavy surface experimental investigation. Int J Heat Mass Transf 40:1071–1081

611 14. Patel VC, Chon JT (1991) Turbulent flow in a channel with wavy wall. J Fluids Eng 113:579–586 15. Masoud R, Thomas BG (2001) Predicting turbulent convective heat transfer in fully developed duct flows. Int J Heat Fluid Flow 22:381–392 16. Ergin S, Ota M, Yamaguchi H (2001) Numerical study of periodic turbulent flow through a corrugated duct. Numer Heat Transf Part A 40:139–151 17. Erigin S, Ota M, Yamaguchi H, Sakamoto M (1997) Analysis of periodically fully developed turbulent flow in a corrugated duct using various turbulence model and comparison with experiment. Proc of Int Conf on Fluid Eng 3:1572–1532 18. Sawyers DR, Sen M, Chang H-C (1998) Heat transfer enhancement in three-dimensional corrugated channel flow. Int J Heat Mass Transf 41:3559–3573 19. Rokni M, Sunden B (1998) 3D numerical investigation of turbulent forced convection in wavy ducts with trapezoidal cross-section. Int J Numer Meth Heat Fluid Flow 8:118–141 20. Utriainen E, Sunden B (2000) Numerical analysis of a primary surface trapezoidal cross wavy duct. Int J Numer Meth Heat Fluid Flow 10:634–648 21. Utriainen E, Sunden B (2002) A numerical investigation of primary surface rounded cross wavy duct. Heat and Mass transfer 38:537–542 22. Shah RK, Heikal MR, Thonon B, Tochon P (2001) Progress in the numerical analysis of compact heat exchanger surfaces. In: Hartnelt JP (eds) Advances in heat transfer, vol. 34. Academic, San Diego, pp 363–443 23. Tao WQ (2001) Numerical heat transfer (in Chinese). Xi’an Jiaotong University press, Xi’an. pp 442–452 24. Ferziger JH, Peric M (2002) Computational methods for fluid dynamics. Springer, Berlin Heidelberg New York, pp 217–259 25. Xie GN, Wang QW, Zhang DS, Tao WQ (2006) A pressurevelocity coupling algorithm on non-orthogonal non-staggered grid (in Chinese). Chin J Comput Mech 23:93–96 26. Tao WQ (2000) Recent advances in computational heat transfer (in Chinese). Science press, Bejing, pp 252–256

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