Morteza Khoshvaght-Aliabadi ,Saber Deldar ,Shafiqur Rehman ,Ashkan Alimoradi

1 Department of Chemical Engineering,Shahrood Branch,Islamic Azad University,Shahrood,Iran

2 Department of Mechanical Engineering,Tarbiat Modares University,Tehran,Iran

3 Center for Engineering Research,The Research Institute,King Fahd University of Petroleum and Minerals,Dhahran 31261,Saudi Arabia

4 Department for Management of Science and Technology Development,Ton Duc Thang University,Ho Chi Minh City,Viet Nam

5 Faculty of Applied Sciences,Ton Duc Thang University,Ho Chi Minh City,Viet Nam

Keywords:Computational fluid dynamics Heat transfer Laminar flow Curved-twisted tubes Comparative study

ABSTRACT Increasing importance of heat transfer in chemical engineering science causes that investigation in the field of enhancement techniques is always one of the up-to-date topics for study.In the current comparative analysis,the thermal enhancement and friction penalty are explored numerically for curved tubes via twisted configuration.To accomplish this,three common geometries namely helical,serpentine,and Archimedes spiral,are considered at different coil-pitches and twist-pitches as well as five Reynolds numbers in the laminar flow regime.The results exhibit noticeable enhancements(up to 60%)in the thermal performance of the twisted cases as compared to the smooth cases.The highest increases are recorded for the serpentine case,followed by the helical and spiral cases.It is found that these enhancements vary via coil-pitch and twist-pitch.Increasing coil-pitch and twist-pitch augments both heat transfer coefficient and pressure drop in all curved-twisted tubes,however,the effects of twist-pitch are more pronounced.To predict Nusselt number and friction factor,new correlations are also proposed.The maximum deviations of the predicted results compared to the simulated data are within ±5%.

1.Introduction

Chemical heat exchange devices designed based on curved tubes might be a better choice in the following cases,

I.Space limitation:whenever,there is not enough space to lay the straight tubes,the curved tubes provide a greater surface area in a relatively smaller volume.

II.Low flow rates(i.e.laminar flow):the overall performance of curved tubes is higher that the straight tubes due to normal flows produced by centrifugal forces.

III.Possibility of deposition:the curved flow paths usually have a self-cleaning feature.The curved tubes cause a local increase in the velocity of flowing fluid which helps to remove the blockage and keep the surface clean.

The curved tubes are usually made in helical,serpentine,and Archimedes spiral forms,which are widely applied in chemical engineering devices,such as heat exchangers,chemical reactors,photobioreactors,etc.Several investigations have neem performed on these geometries,and comprehensive data has been reported.In this literature,different enhancement techniques in the curved tubes were applied.Some studies[1-4]focused on the use of wire turbulators in the helical tubes and noticeable findings were reported.For instance,it was found that both the pitch and the diameter affected the heat transfer of the fluid through the helical tube,so that increasing these parameters caused thermal enhancements up to 70%[1].The performance of advanced coolants such as nanofluids was investigated inside the helical tubes [5-9],and a new review article was provided by Mukesh Kumara and Chandrasekar [10].It was concluded that using nanofluids in helical tubes can be more effective in improving thermal performance than the straight tube.Kurniaet al.[11] examined different cross sectional forms for the helical tube.Their results displayed that the square form generated the maximum entropy.Recently,Wanget al.[12]examined a new cross sectional shape namely trilobal.It was found that the thermal enhancement in this geometry was in the range of 1.16-1.36 times,and the friction factor suddenly increased from 0.96 to 1.10 times.Khosravi-Bizhaemet al.[13]tested pulsating flow through the helical tube.It was exposed that the thermal enhancements were up to 39% with pressure drop augmentations between 3% and 7%.

Similar enhancement techniques were proposed and studied for two other curved tubes,i.e.serpentine and spiral.Nanofluids as working fluid were examined in both the serpentine tube[14-16] and the spiral tube [17-20].Overall,better hydrothermal performances were reported for the systems working with nanofluids as compared with their base fluid.Modifications in the geometry of these types of curved tube were also proposed.Non-uniform straight lengths were studied by Khoshvaght-Aliabadi and Alizadeh [21] for the serpentine tube in the presence of Cu/water nanofluid.It was found that short straight lengths at upstream can intensify the thermal performance in the serpentine tube.Rahimiet al.[22] tested the performance of the serpentine tube equipped with the classic as well as three modified twisted tape inserts.Maximum heat transfer enhancement of 31% and 22% were observed for a modified insert namely jagged.In the other study[23],they evaluated the performance of jagged twisted tape as compared with butterfly configuration.Khodabandehet al.[24] studied different cross sectional shapes (rectangle,elliptic,trapezoid,and circle) for the spiral tube utilized in solar ponds.It was explored that at low flow rates,the thermal performance of the spiral tube was not dependent to the cross sectional shape,but the effects of cross sectional shape increased as the flow rate was increased.

Scrutiny of the previous literature shows that the transport characteristics of curved tubes with twisted structure have not been examined sufficiently due to existence of special complexity in these tubes.However,the curved-twisted tubes can be a good replacement for the curved-smooth tubes due to not only better thermal performance,but also lower fouling creation as a critical problem in heat exchangers.Hence,it seems that sufficient understanding of the curved-twisted tubes is essential for designers and engineers.The current study provides details of quantitative and qualitative information about thermal and hydraulic behaviors of fluid inside different curved-twisted tubes with the square crosssection in laminar regime.The effects of design factors and Reynolds number are investigated for each case,and the obtained results are compared with the smooth model.It is hoped that this analysis arouses interest among designers and engineers,whom work on the heat transfer enhancement techniques.

Fig.1. (a) Well-known curved tubes and (b) Smooth and twisted configurations of curved tubes.

2.Physical Models and Modeling

In the current work,a comparative study is carried out on two configurations of curved tubes,namely smooth and twisted.It is performed for three well-known curved tubes of helical,serpentine,and Archimedes spiral (spiral),which are shown in Fig.1(a).The curved tube are recognized with two general geometrical parameters (tube-diameter and tube-length) and some specific design factors.In this study,all curved tubes have the identical cross-sectional area and the same length.As defined in Fig.1(a),the design factors of the helical and spiral tubes are coil-diameter and coil-pitch,and those of the serpentine tube are straight-length and coil-pitch.To investigate the effects of design factors,the coil-diameter or straight-length is kept constant,and the coil-pitch and twist-pitch are changed.A parts of the considered geometries at the middle levels of specific design factors is shown in Fig.1(b),and details are tabulated in Table 1.Totally,24 models are tested for five different Reynolds numbers (Re) of water flow as working fluid at laminar regime.

The computational domain of each physical model is divided into three sections,i.e.entrance,test,and exit.Both the entrance and the exit sections are straight with smooth configuration and adiabatic boundary condition,but the test section refers to the heat transfer zone with the constant temperature and curved structure.The inlet velocity of the fluid,which is estimated based onRe,is introduced at the upstream of the entrance section with the constant temperature of 298.15 K.The outlet boundary condition is employed on the other side of the computational domain,i.e.at the downstream of the exit section.Also,the no-slip velocity coupled with the conjugate heat transfer condition is adopted for the wetted boundaries.

Fig.2. Pattern of generated grids for helical-smooth tube.

Fig.3. Comparison of present results with predicted data by previous correlations.

Fig.4. (a)Streamlines and temperature/velocity distributions and(b)Velocity vectors of smooth and twisted curved tubes at coil-pitch of 25 mm,number of twist-pitch of 4,and Reynolds number of 900.

Table 1 Details of considered geometries.

Water as a Newtonian and incompressible fluid is considered to flow continuously through the physical models at the steady-state laminar condition without radiation mechanisms,hence,the following equations are applied to model the fluid flow and heat transfer,

Continuity equation

Momentum equation

Fig.5. Heat transfer coefficient and pressure drop of smooth and twisted curved tubes at coil-pitch of 25 mm,number of twist-pitch of 4,and Reynolds number of 900.

Energy equation

The Richardson number,which represents the importance of natural (or free) convection relative to the forced convection,is much less than unity in this study,meaning that the buoyancy effect could be neglected here.However,to enhance the precision of the results,the temperature dependency of the density and other properties is also considered by using third-order polynomial equations as follows,

Fig.6. Effects of coil-pitch on heat transfer coefficient ratio and pressure drop ratio of curved tubes at number of twist-pitch of 4 and Reynolds number of 900.

Fig.7. Effects of number of twist-pitch on heat transfer coefficient ratio and pressure drop ratio of curved tubes at coil-pitch of (a) 25 mm and (b) 50 mm and Reynolds number of 900.

The developed model consisting of boundary conditions together with the governing equations and thermo-physical properties are solved based on a commercial software(Fluent 6.3).The finite volume technique is adopted as solution procedure.The pressure based solver with implicit formulation is used.The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm is employed to separate the pressure equation,while the second-order upwind method is utilized for the momentum and energy equations.According to the iterative technique,the solutions are continued until all variables reach steady regimes,and the residuals drop under 10-5for all variables.

To follow the effects of the water velocity,Reis used,

The hydraulic diameter is estimated as follows,

The relation between the dimensional and the dimensionless heat transfer coefficients is as follows,

where the dimensional coefficient is estimated from the heat transfer rate and logarithmic mean temperature difference (LMTD),

Also,the hydraulic performance is evaluated by using the friction factor and pressure drop,

Fig.2 shows the pattern of the produced grids on a model.On can see that the grids are properly intensified near the walls,where the velocity and temperature variations are noticeable.Also,they are increased in both the cross-section (a×a) and the length (l)to detect the optimum numbers,which lead to accurate numerical results with reasonable computational times.For instance,for the helical-smooth tube,four sets of grids,including coarse (40 × 40 and 500),intermediate (50 × 50 and 600),fine (60 × 60 and 700),and very fine (70 × 70 and 800),are tested.From Fig.2,as the number of grids is increased from (60 × 60 and 700) to(70 × 70 and 800),the deviation of Nusselt number (Nu) and friction factor (f) is within 0.5%.

Fig.8. Effects of number of twist-pitch on velocity and temperature distributions of serpentine case at coil-pitch of 25 mm and Reynolds number of 900.

3.Comparing Numerical Results with Available Correlations

Before presenting the results,the reliability and accuracy of the current modeling are checked by comparing with some available correlations for the helical tubes,which were collected by Gouet al.[25].The employed correlations for the comparison are those suggested by Xin and Ebadian[26],Dravidet al.[27],and Kalb and Seader[28].The noticeable is that these equations can well predict the experimental data [25].

In these equations,the Dean number (De) and Prandtl number(Pr) are estimated as follows,

Since these correlations are valid for the original helical tubes,the smooth-helical tube is considered for the comparison,and the results are presented in Fig.3.Overall,there is acceptable agreements between our results and those of the correlations.Nevertheless,Eqs.(16)and(18)can correlate our results satisfactorily.The mean deviations of our results from the data predicted by Eqs.(16),(17),and (18) are about 6.1%,22.1%,and 5.8%,respectively.

4.Results and Discussion

4.1.Comparison between smooth and twisted curved tubes

Fig.4(a) shows the streamline and velocity/temperature distributions of the fluid inside the smooth and twisted configurations of curved tubes.It can be seen that the streamlines in the twisted models differ from the smooth models.As the flow path is twisted,the streamlines become chaotic.It affects both the velocity and the temperature distributions of fluid and results to the variations of thermal-hydraulic characteristics in the curved tubes.However,the highest variations are found for the serpentine case,followed by the helical and spiral cases.For quantitative evaluation,the heat transfer coefficient(h)and pressure drop(Δp)of the twisted models are compared with the smooth models in Fig.5.It can be seen that twisting the serpentine tube leads to 22.1% increase inhand 16.2% augmentation in Δp,while these variations for the helical and spiral tubes are 13.7%-13.1% and 8.4%-10.7%,respectively.It is interesting to state that the robust axial flow (or highest velocity)is found at the core of the serpentine case,while in the helical and spiral cases,it happen in the corners.Moreover,the position and size of produced vortices are unique to each case.As shown in Fig.4(b),twisting the curved tubes changes the location and strength of these swirl flows.More disrupted velocity vectors are seen for the spiral case leading to better fluid mixing.Fig.5 illustrates thathin this case is greater than that in the helical and serpentine cases.However,with a closer scrutiny one can see that the helical case results in higher values of Δpamong the studied cases.

4.2.Influences of coil-pitch

Fig.9. Effects of Reynolds number on velocity and temperature distributions of curved tubes (a) Helical,(b) Serpentine,and (c) Spiral.

Fig.9 (continued)

The coil-pitch defined in Fig.1(b)is a special design factor in the curved tubes,so the effects of this parameter are investigated for all twisted cases and the results are discussed as compared with smooth configuration.Fig.6 discloses that the effects of the coilpitch on the thermal results (htwisted/hsmooth) of the curved tubes are more evident that the pressure drop results (Δptwisted/Δpsmooth).Likewise,the highest impacts belong to the serpentine case.For instance,as the coil-pitch is increased two-fold (from 25 mm to 50 mm),hratio of the serpentine case increases 9.6%,whereas for the same condition,hratio of the helical and spiral cases decreases 1.5% and 1.1%,respectively.These variations can be discussed by using Fig.1(b).As displayed in Fig.1,increasing the coil-pitch in the serpentine case increases the curvature length,but increasing the coil-pitch in other two cases decreases the curvature values.

4.3.Influences of number of twist-pitch

The other effective design factor in the twisted configurations is the number of 360°twist-pitch which is defined in Fig.1(b).In this analysis,three different levels of this parameter are considered as 2,4,and 8.Fig.7(a and b)displays the influences of the number of twist-pitch onhand Δpratios of different models.It can be seen that bothhand Δpratios augment with the number of twistpitch in all cases.In order to discuss the possible mechanisms,the velocity and temperature contours of the fluid through the serpentine case are selected and presented in Fig.8.The corresponding contours are displayed on a stream-wise and three normal planes.It is found that as the fluid flows through the twisted models,its velocity changes and has an important role.At the same operating condition,the twisted-serpentine tubes have greater velocities than the smooth-serpentine tube,and the difference enhances as the number of twist-pitch is enlarged.This intensifies the fluid mixing and heat exchange between the walls fluid and the core fluid,leading to more uniform temperature distributions.The mentioned mechanisms happen for the two other cases,i.e.helical and spiral.It is interesting to note that the influences of this parameter on the heat transfer and flow characteristics of the serpentine case are higher than those of the other cases.For instance,for the serpentine case at the coil-pitch of 25 mm,as the number of twist-pitch is enlarged four times (from 2 to 8),hratio enhances about 26.7%,but this enhancement for the helical and spiral cases are about 20.4%and 23.5%,respectively.A similar discussion can be made for Δpratio.Evidently,Δpaugments due to the larger flow blockage and disturbance.

Fig.10. Effects of Reynolds number on heat transfer coefficient and pressure drop of curved tubes (a) Helical,(b) Serpentine,and (c) Spiral.

4.4.Influences of Reynolds number

Fig.9(a-c)clarifies the influences ofReon velocity and temperature contours,which are presented for a normal plane.For all cases (helical,serpentine,and spiral),at the minimumRe(Re=100),the effect of design factors,i.e.coil-pitch and twistpitch number,is not noticeable,however,it becomes significant with theRe.Actually,an adequate flow velocity is required to produce efficient vortices through the curved tubes,which is obtained at higher values ofRe.As previously discussed,these swirl flows are formed due to the centrifugal forces,which their size and shape depends on the flow velocity.WhenReis high enough,the vortices cover most of the cross-sectional area,leading to a better mixing and more uniform temperature distribution.

The quantitative results of theReeffects ofhand Δpof the models are summarized in Fig.10(a)-(c).It is found thathaugments with theRe,however,the influences ofReon the heat transfer of the twisted cases are higher than the smooth one.For instance,at the coil-pitch of 25 mm,with increasing theRefrom 100 to 900,hof the helical-smooth tube increases about 196.1%,while thehof the helical-twisted tube with the twist-pitch number of 2 increases about 203.2%.Another noticeable point is that the effect of theRefurther enhances with the twist-pitch number.As an example,for the same model,these enhancements for the twist-pitch numbers of 4 and 8 are 224.2% and 261.3%,respectively.A same trend is also detected for the other models.The maximum variations are detected for the spiral-twisted tube with coilpitch of 50 mm,which are 218.2%,229.5%,253.7%,and 322.1%.It indicates that the effects ofReon the spiral case are higher than the others.It is clear that the Δpalso increases with theRe.Also,the twisted models have greater values as compared to the smooth models because of the larger surface area of contact with water and higher dynamical friction loss produced by twisted walls.

4.5.Overall performance

Before utilizing the twisted-curved tubes in engineering applications,the overall performance of these models should be evaluated compared to the smooth cases.The performance index of Webb [29] is adopted here as performance evaluation criterion,

This index clarifies that for a givenRe,the effective parameters areNuandfof different models.

Fig.11(a-c) provides the obtained values for each curvedtwisted tube.An overall survey on the plots discloses that the performance of all curved-twisted tubes improves asRegoes up.Yanget al.[30],which conducted experimentations on twisted-tubes,also reported that the performance index increases up to a certainRe(around 2300)then decreases.However,the effects ofRein the serpentine case are more noticeable than the others.For instance,asReis increased from 100 to 900,the performance index of the serpentine case enhances by about 50.4%,while this enhancement for the helical and spiral cases is about 19.6% and 24.1%,respectively.Note that,at the minimumRe(Re=100),almost all twisted models show weaker performance as compared to the corresponding smooth model,and values lower than 1 are recorded.However,at higher values ofRe,the twisted models are promising and reveals better overall performances.

Fig.11. Performance index versus Reynolds number of curved tubes.

Furthermore,the overall performance improves at larger coilpitch and the twist-pitch.It means that by increasing these parameters,the thermal enhancements overcome the friction augmentations in the curved-twisted tubes.Also,the serpentine case owns the best overall performance,while the helical and spiral cases have the comparable performances.The maximum value of about 1.45 is found for the serpentine-twisted tube at coil-pitch of 50 and twist-pitch number of 8.

4.6.Correlations

In this study,new equations are formulated for predictingNuandfof all curved-twisted tubes with respect toRe,Pr,and specific design factor(coil-pitch and twist-pitch).The correlations have the following forms and the corresponding constants are listed in Table 2.

Table 2 Constants of Nusselt number and fiction factor correlations.

The precision of the these equations is assessed by the average absolute error (ABE),

It is found that the correlation results are in a good compromise with the simulated data,with an average deviation of 5.28%.

5.Conclusions

The steady-state laminar flow of water through different curved tubes (helical,serpentine,and spiral) with both smooth and twisted structures are developed and simulatedviafinite volume method.Distributions of velocity and temperature as well ashand Δpare investigated at different coil-pitches and twistpitches.The most important conclusions that might be drawn from the current analysis include the following items.Bothhand Δpof the curved-twisted tubes are higher than those of the curvedsmooth tubes.At coil-pitch of 25 mm,number of twist-pitch of 4,andReof 900,the deviations ofhand Δpbetween twisted and smooth models are 13.7% and 13.1% for the helical case,22.1%and 16.2%for the serpentine case,and 8.4%and 10.7%for the spiral case.It can be seen that the serpentine case is more sensitive to the twisting structure as compared to the helical and spiral cases.Also,increasing coil-pitch enhances the heat transfer performance of the serpentine-twisted tube,but it has an inverse effect on the helicaltwisted and spiral-twisted tubes.However,increasing number of twist-pitch enhances the thermal performance of all cases.Likewise,the influences ofReon the heat transfer of the twisted tubes are higher than the smooth case.Finally,correlations with a prediction average accuracy of 5.28% are proposed to predictNuandfof curved-twisted tubes at the studied ranges.

Nomenclature

Acfrontal surface area,m2

Attotal surface area,m2

cpspecific heat capacity,J·kg-1·K-1

Dcoil-dimeter,m

DeDean number

Dhhydraulic diameter,m

ffriction factor

hheat transfer coefficient,W·m-2·K-1

lchannel length,m

kthermal conductivity,W·m-1·K-1

mmass flow rate,kg·s-1

NuNusselt number

PrPrandtl number

ppressure,Pa

Δppressure drop,Pa

Qheat transfer rate,W

ReReynolds number

Ttemperature,K

uininlet velocity,m·s-1

Vvelocity vector,m·s-1

η performance index

μ dynamic viscosity,kg·m-1·s-1

ρ density,kg·m-3

Subscripts

b bulk

in inlet

out outlet

w wall

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors are grateful for the support by the Islamic Azad University of Shahrood Branch.