Turbulent forced convection in a heat exchanger square channel with wavy-ribs vortex generator☆

Amnart Boonloi,Withada Jedsadaratanachai*

1 Department of Mechanical Engineering Technology,College of Industrial Technology,King Mongkut's University of Technology North Bangkok,Bangkok 10800,Thailand

2 Department of Mechanical Engineering,Faculty of Engineering,King Mongkut's Institute of Technology Ladkrabang,Bangkok 10520,Thailand

Keywords:Flow configuration Forced convection Heat transfer characteristic Turbulent flow Wavy-ribs

ABSTRACT Turbulent forced convective heat transfer and flow configurations in a square channel with wavy-ribs inserted diagonally are examined numerically.The influences of the 30°and 45° flow attack angles for wavy-ribs,blockage ratio,R B=b/H=0.05–0.25 with single pitch ratio,R P=P/H=1 are investigated for the Reynolds number based on the hydraulic diameter of the square channel,Re=3000–20000.The use of the wavy-ribs,which inserted diagonal in the square channel,is aimed to help to improve the thermal performance in heat exchange systems.The finite volume method and SIMPLE algorithm are applied to the present numerical simulation.The results are presented on the periodic flow and heat transfer pro files, flow configurations,heat transfer characteristics and the performance evaluations.The mathematical results reveal that the use of wavy-ribs leads to a higher heat transfer rate and friction loss over the smooth channel.The heat transfer enhancements are around 1.97–5.14 and 2.04–5.27 times over the smooth channel for 30°and 45°attack angles,respectively.However,the corresponding friction loss values for 30°and 45°are around 4.26–86.55 and 5.03–97.98 times higher than the smooth square channel,respectively.The optimum thermal enhancement factor on both cases is found at R B=0.10 and the lowest Reynolds number,Re=3000,to be about 1.47 and 1.52,respectively,for 30°and 45°wavy-ribs.

1.Introduction

The design of compact heat exchangers and the methods for improving thermal performance had been needed in many industries.The vortex generators(VGs)had been used in the heat exchanger channel to augment the heat transfer performance.The presence of vortex generators can create the jet flows which impinge on the channel wall lead to thinner boundary layer and perform better mixing of the fluid flow.Both experimental and numerical investigations on the enhancement of heat transfer and thermal performance had been published by various investigators,and the literature reviews on heat transfer augmentation with vortex generators were also available[1].

Hernon et al.[2]used hot wire and PIV technique to study VGs and concluded that the heat transfer augmentation arose from the complex unsteady flow phenomena.Henze and Wolfersdorf[3]investigated the influences of tetrahedral VG in the heat transfer test section.They claimed that the highest VG height provides the highest heat transfer enhancement.Kenan et al.[4]studied experimentally the effects of the tapes with double-sided delta-winglets in various configurations under different flow conditions.They summarized that the winglet height was the most effective factor on the pressure drop.

The investigations on effects of the spacing between the edge of VG(2–14 cm)and the flow attack angle(6°–24°)were reported by Pauley and Eaton[5].They found that the maximum augmentation is around 30%for the Stanton number(St)in the range studied.Wroblewski and Eibeck[6]established that the 12°delta-winglet vortex generators(DWVGs)enhance the heat transfer performance by around 25%at near the downwash regime.Skullong and Promvonge[7]reported that the 30°delta winglet vortex generators with Rp=1 performs the highest thermal enhancement factor.Depaiwa et al.[8]investigated the solar air heater channel with ten pairs of rectangular winglet vortex generators(RWVGs)placed on the inlet region,and the influences of the flow attack angle,α =30°–60°with pointing upstream(PU)and pointing downstream(PD)for Re=5000–23000 were reported.They concluded that the highest flow attack angle leads to the highest heat transfer rate and friction loss.Depaiwa et al.[8]also found that the PU-RWVG provides a lower heat transfer rate than PD-RWVG.Wang et al.[9]examined the longitudinal vortex generators(LVGs)in horizontal narrow rectangular channels on heat transfer and friction loss.They found that the heat transfer augmentation is around 10%–45%by using LVGs.Jian et al.[10]investigated the effect of narrow rectangular channel with four pairs of LVGs on flow structure and heat transfer characteristics.The enhancements are around 100.9%and 11.4%for heat transfer and friction factor,respectively,in the laminar regime,while around 87.1%and 100.3%,respectively,for the turbulent flow regime.Min et al.[11]studied the winglet pair vortex generators(RWPVGs)in a rectangular channel on heat transfer behavior for Re=5000–17500.Their experimental results showed that the augmentation is around 46%–55%on heat transfer.

Charbel et al.[12]studied the effects of inclined vortex generators in circular tube with various configurations by numerical method.They presented the augmentations of the temperature gradients and vorticity near the heating wall when using inclined vortex generators,that help to increase in heat transfer rate and thermal performance.

Sarac and Bali[13]experimentally investigated the swirl flow in a horizontal pipe on heat transfer and pressure drop for Re=5000–30000.The augmentation is around 18.1%–163%of the Nusselt number depended on Reynolds number,position of VG, flow attack angle and the number of vanes.They also concluded that the increase of the flow attack angle leads to the rise of the Nusselt number.Wu and Tao[14,15]investigated the influences of RWVG placed on the lower wall of the test channel for Re=800–3000,and pointed out that the enlarge pressure loss in the test channel is because of the rise of the flow attack angle.Yang et al.[16]studied the effects of the CFU pair in a rectangular channel for VG.Biswas and Chattopadhyay[17]studied the punched DWVG in the rectangular channel on the thermal performance by numerical method.They found that the heat transfer rate increases around 45.4%over the smooth channel at an attack angle of 20°.Ahmed et al.[18]found that the delta-wing can create the counter rotating longitudinal vortices along their leading edges.Kaniewski et al.[19]found that the increase on the global mixing is more than 50%for laminar flow regime when using one pair of RWVG located on the lower wall of test channel.Hiravennavar et al.[20]studied the influences of the DWVG on both hydrodynamically developed and thermally developing laminar test channel.They found that the enhancement of heat transfer is around 33%and 67%for single winglet and winglet pair,respectively,when compared with the plain channel.Biswas et al.[21]investigated the thermal performance for the laminar regime in the rectangular channel with delta-wing and delta-winglet pair VGs.They presented that the winglet-pair gives a lower thermal performance than the delta-wing VG.Deb et al.[22]studied the heat transfer rate on both laminar and turbulent regimes in a rectangular channel with DWVGs.They found that the enhancement is about 16%at the outlet of the channel for Re=5000.Sohankar[23]investigated the effect of the angled ribs placed in the rectangular channel on heat transfer configurations,and concluded that the steady heat transfer appears at lower Reynolds number while at the higher Reynolds number,the unsteady trend is found.

Promvonge et al.[24]investigated numerically inline V-shaped discrete thin ribs with the flow attack angle of 60°located on two opposite heated walls.They found that the V-shaped discrete thin ribs can create a pair of counter-rotating vortices(P-vortex)leading to the rise of heat transfer rate.They also concluded that the rise of rib height leads to the increase in heat transfer and friction loss.

The uses of vortex generators in the compact heat exchanger had been investigated.Du et al.[25]investigated the punched LVGs on the wavy fin surface of flat tube which used in the direct air-cooled condenser.They concluded that the 25°DWP gives the highest performance in comparison with other types for the inlet air flow velocity varied from 1 to 5 m·s?1.The two types of vortex generators(RWP and DWP with two different flow arrangements:CFU and CFD for plate- fin heat exchanger)were studied by Saha et al.[26].They reported that all the vortex generators change the fluid flow structure that effects for heat transfer augmentation in the test section.They also summarized that the RWP provides higher thermal performance than DWP.Khoshvaght-Aliabadi et al.[27]investigated numerically laminar forced convective and heat transfer of copper-base deionized water nano- fluid inside the vortex-generator plate- fin channels.Li et al.[28]studied on both experiments and simulations on the thermal- fluid configurations of flat- fin heat sink with a pair of vortex generators placed in cross flow channel.They claimed that the heat transfer augmentation of the heat sink by the use of vortex generators is greater,and the rise of the pressure drop is lower,at lower Reynolds numbers.Du et al.[29]presented the influence of LVGs on wavy finned flat tube for heat transfer enhancement.They found that the augmentations on the average Nusselt number and friction factor are around 21%–60%and 13%–83%for Re=1500–4500,respectively.The effects of DWVG in louvered fin heat exchanger were studied by Huisseune et al.[30],showing that the louvered fin heat exchanger with DWVG is more compact than no DWVG case at the similar heat duty and pumping power conditions.He et al.[31]investigated the influences of RWVG in fin-and-tube heat exchanger,and presented that the RWP enhances the thermal performance of the fin-and-tube heat exchangers with moderate pressure loss penalty.

Promvonge et al.[32]studied the heat transfer configurations and thermal performance evaluations for 30°inclined baffle inserted diagonally in the square channel at turbulent regime.They concluded that the 30°inclined baffle can induce the vortex flow and impinging jet flows over the test channel.The maximum heat transfer is found around 4.5 times higher than the smooth square channel with no 30°inclined baffle.

Many investigations studied on the improvement of the thermal performance with numerical and experimental methods.As the literature reviews,there are found that the V-shaped and the modified V-shaped vortex generators had been widely applied in the heat exchanger channel by placing on the channel wall.The V-shaped and the modified V-shaped vortex generators perform better thermally than the other shapes.However,there are found that the installation of the vortex generators which placed on the channel wall is not suitable for industrial system,but the inserted diagonally in the square channel as Ref.[32]is more convenient for the formation and installation in the heating system.Therefore,this work focuses on the inserts diagonally installed in the square channel and also reducing the pressure loss in the heat system by using wavy-ribs using numerical“experiments”.The main reason for using wavy-ribs(also called double V-ribs)is that they can help to decrease the pressure loss in comparison with the V-ribs,while the flow configurations and the heat transfer behavior are similar as the V-ribs.

2.Flow Descriptions

2.1.The heat exchanger square channel with wavy ribs inserted diagonally

The heat exchanger square channel with wavy ribs inserted diagonally in the test section is presented in Fig.1.The test fluid is air that enters the square channel at an inlet temperature,Tin.The hydraulic diameter,Dh=H is set to 0.05 m,b is the wavy ribs half-height,and

b/H is defined as the blockage ratio,RB.The distance between the wavy rib,P is equal to H and P/H is known as the pitch ratio,RP=1.The flow attack angles of the wavy rib are about 30°and 45°.The case studies are listed in Table 1.

2.2.Boundary condition

The velocity inlet boundary is used for the inlet while the outlet of the flow domain is set as pressure outlet boundary.The constant properties at average bulk temperature are set for the test fluid.The square channel walls,wavy rib and inserted plate are used in impermeable boundary and no-slip wall conditions The constant heat flux of all the square duct walls is maintained at 600 W·m?2while the adiabatic wall conditions is assumed for wavy ribs and inserted plate.

Fig.1.The heat exchanger square channel with wavy ribs inserted diagonally.

Table 1 Case studies

3.Mathematical

3.1.Mathematical model

The mathematical part is referred from Ref.[33].The assumptions for current computation are as follows:

·Fluid flow and heat transfer are set as steady and three-dimensional.

·Turbulent and incompressible conditions are used for fluid flow.

·Fluid properties remain constant.

·Body forces,viscous dissipation and radiation heat transfer are ignored.

Based on above assumptions.The equations can be written as follows:

Continuity equation:

Momentum equation:

where ρ,ui,p,μ and u′are the fluid density,mean component of velocity in the direction xi,pressure,dynamic viscosity and fluctuating component of velocity,respectively.

Energy equation:

where Γ and Γtare the molecular thermal diffusivity and turbulent thermal diffusivity,respectively and are given by

The coordinate system is indicated in Fig.1,and the x-axis is in the plane of installed ribs,y-axis is perpendicular to this plane,and the origin is at the beginning of the test section.

The Reynolds-averaged approach to turbulent modeling requires that the Reynolds stresses,in Eq.(2),be modeled.Eq.(5)shows that the Boussinesq hypothesis relates the Reynolds stresses to the mean velocity gradients:

where the turbulent kinetic energy,k,is defined byand δijis a Kronecker delta.An advantage of the Boussinesq approach is the relatively low computational cost associated with the computation of the turbulent viscosity,μtgiven is μt= ρCμk2/ε.The RNG k–ε model is an example of the two-equation models that use the Boussinesq hypothesis.The RNG k–ε model is derived from the instantaneous Navier–Stokes equation using the “renormalization group”(RNG)method.The steady state transport equations are expressed as

Theαkandαεdata are the inverse effective Prandtl number for k and ε,respectively.C1εand C2εare the constants.The effective viscosity μeffis written as follows:

where Cμis constant and equal to 0.0845.

The governing equations were discretized by the QUICK scheme,decoupling with the SIMPLE algorithm and solved using a finite volume approach[34].When the normalized residual values were less than 10?5for all variables,but less than 10?9for the energy equation,the solutions were accepted as convergent.

The Reynolds number,friction factor,local Nusselt number,average Nusselt number and thermal performance enhancement factor are presented in Eqs.(10)through(14),respectively:

where Nu0and f0are the Nusselt number and friction factor for the smooth channel,respectively.

3.2.Grid independency test

The variations in Nu and f values for the diagonal wavy rib at RB=0.20 are less than 0.1%when rising the number of cells from 360000 to 424700,as depicted in Table 2 and Fig.2.Therefore,the grid system of 360000 cells is used for the current computational model.

Table 2 Grid system test

Fig.2.Grid independent test.

4.Results and Discussion

4.1.Validation on smooth square channel

Fig.3 shows the verifications of the smooth square channel for Nu0and f0by comparing the values form empirical correlations[34]with similar conditions.There are found that the results shown agree well within±3%on both Nu0and f0.

Fig.3.Verification with correlation for the Nusselt number and friction factor of the smooth square channel.

Fig.4.Verification of the present simulation with previous exp.result for(a)Nusselt number and(b)friction factor at R B=0.2,R P=1 and α =30°[32].

Fig.5.Flow configurations for(a)α =30°and(b)α=45°at Re=6000 and R B=0.20.

Due to the experimental investigation for 30°and 45°wavy ribs have not been reported,therefore,the present model(replaced the wavy ribs with inclined ribs under similar model structures)is compared with very close configuration as Ref.[32].Our numerical results on both heat transfer and friction loss are found in excellent agreement within±4%for 30°inclined baffle with RP=1 and RB=0.2 as depicted in Fig.4(a)and(b),respectively.

4.2.Fully developed periodic flow and heat transfer pro files

The flow and heat transfer pro files of the wavy ribs are presented in terms of the u/u0and Nu/Nu0at various x/D values as Figs.5 and 6,respectively.

Fig.5(a)and(b)present the flow pro files of the wavy ribs for 30°and 45°,respectively,at Re=6000 and RB=0.20.There are found that the flow structure can divide into two parts:developing and fully developed periodic pro files.The periodic pro files mean that the flow pattern repeats periodically,while for the fully developed periodic profiles,the velocity patterns are similar in both configuration and value.The periodic pro files and the fully developed periodic flow pro files are found at the 2nd module and around 6th–8th modules,respectively,on both 30°and 45°elements.

Fig.6.Heat transfer characteristics for α =30°and 45°at Re=6000 and R B=0.20.

Fig.7.Streamlines in transverse planes for α =45°at Re=6000 and R B=0.20 at the fully developed periodic flow region.

The heat transfer characteristics provide similar results as flow configurations which produce periodic and fully developed periodic profiles around the 2nd module and 3rd–4th modules,respectively,as displayed in Fig.6.

4.3.Flow visualizations and heat transfer characteristics

The flow configurations are shown in forms of streamlines in transverse planes and the longitudinal vortex flows over wavy ribs as Figs.7 and 8,respectively.In general,the streamlines in transverse planes for both 30°and 45°show similar pattern:four main vortex flows,counter rotating flow with common- flow-up when considering at the lower part of the plane and the small vortices at the corner of the planes.The main flow structure is sketched in Fig.9.The flow profiles repeat itself from one to another cell due to the periodic installation of wavy ribs as shown in Fig.8.The impinging jet flows over the other ribs are found to be a similar pattern as the 2nd rib(orange streamlines)due to the periodic flow configurations which occurred by array of wavy rib vortex generators as described in Fig.5.The longitudinal vortex flows are a key to augmenting heat transfer rate over the heat exchanger channel.

Fig.10(a)and(b)presents the local Nusselt number contours over the square channel walls in the x–z and x–y planes,respectively,for α =45°,Re=6000 and RB=0.2.It is found that the use of the wavy ribs offers higher heat transfer rate than the smooth square channel.The longitudinal vortex flows and the impinging jet of the flows on the square channel wall,that leads to the enhancement of the heat transfer over the smooth square channel.As seen,the peak of heat transfer regimes is found,except for the corner of the square channel.

The wavy ribs give a better local Nusselt number distributions than the inclined rib[32]when considering at the contours of Nusselt number over the channel surfaces.In addition,the wavy ribs inserted diagonally in the square channel give more comfort to install than the vortex generators that were placed on the channel walls and help more in convenient maintenance.

4.4.Performance evaluation

The variations of Nu/Nu0with Reynolds number are presented in Fig.11(a)while the variations of Nu/Nu0with RBvalues are displayed in Fig.11(b).In general,the uses of wavy ribs lead to the higher heat transfer rate over the smooth square channel on both 30°and 45°.The increasing Reynolds number performs reducing Nu/Nu0,but the increasing RBresults in the rise of the Nu/Nu0.The wavy ribs with 45°provide a higher heat transfer rate than the 30°for all Reynolds numbers and RB.The augmentation of the Nusselt number is around 1.97–5.14 and 2.04–5.27 times higher than the smooth square channel for wavy ribs with 30°and 45°,respectively.Although the inclined ribs provide slightly higher heat transfer rate than the wavy ribs[32],the wavy ribs give greater Nusselt number distributions on the square channel walls than the inclined ribs.

Fig.8.Impinging flows in front of first rib and second rib for wavy ribs at α =45°,Re=6000 and R B=0.2.

Fig.9.Diagram of the secondary flow structure for wavy ribs.

Fig.12(a)and(b)show the variations of the f/f0with Reynolds number and with the RB,respectively.The rise of the Reynolds number leads to slightly increase in the f/f0.The trends of the f/f0are found to increase when the RBvalue performs higher.The 45°wavy ribs give higher f/f0than the 30°wavy ribs.The maximum friction factor value is found at RB=0.25 for 45°wavy ribs around 97.98 times higher than the smooth square channel.In range studied,the augmentation of the f/f0is around 4.26–97.98 depending on the Re,RBand flow attack angle.Although,the use of the wavy ribs gives a very large pressure drop compared with inclined ribs[32],the durability of the wavy ribs seems to be higher than the inclined ribs when using in the heating system and in high viscosity fluids.

Fig.13(a)and(b)illustrates the thermal enhancement factor,η,with the Reynolds number and RB,respectively.The thermal enhancement factor is found to be optimum at RB=0.10 and Re=3000 on both 30°and 45°wavy ribs around 1.47 and 1.52,respectively.The increase in the Reynolds number leads to the decrease in the η for all cases.The RB=0.25 that performs the lowest η due to the increasing rate of the friction factor at this point is much enlarged.η varied from 0.90 to 1.52 for 30°and 45°wavy ribs for Re=3000–20000 and RB=0.05–0.25.

5.Conclusions

3D numerical investigations on the turbulent forced convection,heat transfer and thermal performance in the heat exchanger square channel with wavy ribs inserted diagonally are presented.The influences of RB=0.05–0.25,α =30°and 45°for Reynolds number based on the hydraulic diameter of the square channel,Re=3000–20000 are investigated.The conclusions of the finding are as follows:

–The flow and heat transfer profiles can divide into two parts;periodic and fully developed periodic pro files.For flow structure,the periodic and the fully developed periodic pro files are found at 2nd rib and 6th–8th ribs,respectively.For heat transfer behavior,the periodic profiles appeared at 2nd ribs and then develop to the fully developed periodic pro files around 3rd–4th ribs.

Fig.10.Nu x contours on the square channel walls(a)xz plane and(b)xy plane of α =45°,Re=6000 and R B=0.2.

Fig.11.The variations of the Nu/Nu0 with(a)Reynolds number and(b)R B.

– The enhancement of the Nusselt number is around 1.97–5.14 and 2.04–5.27 times over the smooth square channel for 30°and 45°,respectively,for wavy ribs which RB=0.05–0.25 and Re=3000–20000.

– In range studied,the f/f0is around 4.26–86.55 and 5.03–97.98 for wavy ribs with α =30°and 45°,respectively.

–The optimum point of the thermal enhancement factor is found around 1.47 and 1.52 at RB=0.1 and Re=3000 for 30°and 45°wavy ribs,respectively.

– The 45°wavy-ribs give higher heat transfer rate than the 30°wavyribs,while the 30°wavy-ribs can help to save friction loss in the system.

–Although,the wavy-ribs provide higher friction loss than the inclinedribs[32]with nearly the same values on both heat transfer rate and thermal enhancement factor,the wavy-ribs give more stable configuration in comparison with inclined-ribs that help save the maintenance cost when installing wavy-ribs in the heating system.

Nomenclature

b rib height,m

To which nettle-plant? asked she; I don t talk to nettle-plants. If thou didst not do it, then thou art not the true bride, said he. So she bethought herself, and said,

D hydraulic diameter of square channel,D=H

f friction factor

H square channel height,m

h convective heat transfer coefficient,W·m?2·K?1

k thermal conductivity,W·m?1·K?1

Nu Nusselt number,(=hD/k)

Pr Prandtl number(Pr=0.707)

p static pressure,Pa

Re Reynolds number,(=ρūD/μ)

RBflow blockage ratio,(=b/H)

RPpitch or spacing ratio,(=P/D)

T temperature,K

uivelocity in xi-direction,m·s?1

ū mean velocity in channel,m·s?1

α winglet inclination angle or angle of attack,(°)

Γ thermal diffusivity,(=k/ρcp)

η thermal enhancement factor,(=(Nu/Nu0)/(f/f0)1/3)

μ dynamic viscosity,kg·s?1·m?1

ρ density,kg·m?3

Fig.12.The variations of the f/f0 with(a)Reynolds number and(b)R B.

Subscripts

in inlet

pp pumping power

w wall

0 smooth channel

Fig.13.The variations of the η with(a)Reynolds number and(b)R B.

Acknowledgements

The funding of this work is supported by College of Industrial Technology,King Mongkut's University of North Bangkok,Thailand.The authors would like to thank Assoc.Prof.Dr.Pongjet Promvonge,KMITL for suggestions.