Effects of tube cross-sectional shapes on flow pattern,liquid film and heat transfer of n-pentane across tube bundles

Xuejing He,Zhenlin Li*,Ji Wang*,Hai Yu

College of Mechanical and Transportation Engineering,China University of Petroleum (Beijing),Beijing 102249,China

Keywords:Tube shapes Flow pattern Liquid film thickness Heat transfer Two-phase flow

ABSTRACT The heat transfer of hydrocarbon refrigerant across tube bundles have been widely used in refrigeration.Three-dimensional simulation model using volume of fluid (VOF) was presented to study the effects of tube shapes on flow pattern,film thickness and heat transfer of n-pentane across tube bundles,including circle,ellipse-shaped,egg-shaped and cam-shaped tube bundles.Simulation results agree well with experimental data in the literature.The liquid film thickness of sheet flow and heat transfer for different tube shapes were obtained numerically.The flow pattern transition occurs lower vapor quality for ellipse-shaped tube than other tube shapes.For sheet flow,the liquid film on circle tube and ellipseshaped tube is symmetrically distributed along the circumferential direction.However,the liquid film on egg-shaped tube at circumferential angles(θ)=15°-60°is thicker than θ=135°-165°.The liquid film on cam tube at θ=15°-60° is slightly thinner than θ=135°-165°.The liquid film thickness varies from thinner to thicker for ellipse-shaped,cam-shaped,egg-shape and circle within θ=15°-60°.The effect of tube shape is insignificant on thin liquid film thickness.Ellipse-shaped tube has largest heat transfer coefficient for sheet flow.In practical engineering,the tube shape could be designed as ellipse to promote heat transfer.

1.Introduction

The tube bundle heat exchanges are widely applied in industrial processes,including refrigeration [1,2],chemical industry [3],seawater desalination [4,5] and food processing [6].Especially in refrigeration,the spiral wound heat exchanger is mainly cryogenic heat exchanger in large liquefied natural gas(LNG)plants.Improving heat transfer performance can provide economic benefit.One of effective ways for improving heat transfer performance is changing the tube cross-sectional shapes[7].Designing of intensified heat exchange tube is an effective way to reduce the manufacturing cost of heat transfer equipment,which can improve the uniformity of liquid film and increase heat transfer coefficient[8].For the shell side of spiral wound heat exchanger,two-phase refrigerant flows across tube bundles.The refrigerant and the tube wall exchange heat [9],resulting in various flow patterns on the shell side.The heat transfer mechanism of two-phase flow across tube bundles is complex.Hence,it is significant to analyze the flow pattern,film thickness and heat transfer of various tube shapes,which could help the design of heat exchange tubes.

Circular tubes are widely used in heat exchangers because of easier production and higher pressure endurance [10].Therefore,many researchers have studied its heat transfer performance.Lietal.[11] investigated the influence of different Reynolds number(Re)and gas flow rate on hydrodynamic characteristics.They found liquid film thickness increases with increasing gas flow.In droplet and column flow,the gas flow rate affects the region of thinnest liquid film.Ramadan and Park[12]found that the liquid film is difficult to cover the tube wall for lowRe.The heat transfer strengthens with increasingRe.Karmakar and Acharya[13]studied effects of different surface wettabilities on heat transfer for horizontal tubes in the jet-flow.They presented local and average heat transfer coefficient for the droplet,jet and sheet flow[14].Their results showed that the flow regimes vary substantially in all directions and affect local Nusselt number distribution.Zhaoetal.[15] studied the falling film flow and heat transfer performance for different distributor height,tube pitch,liquid flow rate and tube bundle arrangement.They found the velocity fluctuation could affect flow pattern,which exists in the falling film tube bundle.Chenetal.[16]studied the film thickness distribution for droplet and sheet flow.The film thickness decreases with increasing tube spacing and tube diameter.Tube spacing mainly effected the upper part film thickness.Luetal.[17] investigated the effect of the geometrical parameters for tube bundles in spiral wound heat exchanger,including number of layers,center core diameter and space bar thickness.Lietal.[18] developed a numerical model to simulate the flow pattern and heat transfer of flow boiling across tube bundles.They analyzed flow pattern transition conditions and developed flow pattern maps.Several studies experimentally investigated the falling film flow.Austegardetal.[19] studied the flow pattern transitions for different tube spaces and diameters in shell side of spiral-wound heat exchanger tube array.They found droplet,column and sheet flow pattern.Sunetal.[20]studied the falling film flow under sloshing conditions.Houetal.[21]studied the liquid film falling around a horizontal tube for circumferential angles (θ) ranging from 15° to 165°.They found the thinner film thickness locates at θ=90°-115°.However,the fluid across circular tube bundles causes high pressure drop and vibration,even tube damage.

Recently,non-circular tubes were studied numerically due to their high heat transfer performance.Qietal.[8] studied heat transfer outside elliptical tube and found that heat transfer coefficient increases 20%-22% for ellipse-shaped tubes compared with circular tubes.Leeetal.[22] studied obliquely dispensed falling film over a horizontal ellipse-shaped tube.The effects of obliquely dispensed angle and mass flow rate on heat transfer and liquid film distribution were presented.Wanetal.[23]found that increasing tube perimeter and seawater salinity decreases heat transfer.HigherRe,inlet temperature and tube ellipticity cause higher heat transfer coefficient.However,the heat transfer decreases with increasing tube ellipticity as it is larger than 3.25.

In addition,some researchers studied special shaped tubes for falling film flow,such as egg-shaped tube[9,24],drop-shaped tube[25,26],flat tube [27],corrugated tube [28],spiral grooved tube[29],cam-shaped tube [30] and enhanced tube [30,31].Zhangetal.[9] found a high,favorable pressure gradient at the front of the tube,while a low,adverse pressure gradient in the back of the tube appeared for egg-shaped tubes.The egg-shaped tube has the best overall heat transfer performance when axis ratio is 2 andRe＞11952.Hanetal.[24] studied the film thickness distribution along the axial and circumferential direction for differentRenumbers and longitudinal inter-tube spacing.Deebetal.[25] analyzed the heat transfer for a single drop-shaped tube at θ=50°.The heat transfer strengthens with increasing transverse pitch ratio across staggered drop-shaped tube bundles.The drop-shaped tube bundles has better thermal-hydraulic performance compared with circle bundles [26].Puetal.[27] studied a circular tube and three flat tubes heat transfer.Zhangetal.[28] studied liquid film thickness of corrugated tube and found the thinnest film at 90°-120°.Qietal.[29] found that the spiral-grooved tube can enhance heat transfer compared with the smooth tube.Abolfathietal.[7] studied flow performance for the tube bundles comprised by camshaped and circular tube.Luoetal.[32]compared the heat transfer characteristics for a drop-shaped tube,an oval-shaped tube and a circular tube.They found the drop-and oval-shaped tubes have higher heat transfer coefficient.

Most researchers focused on falling film flow for circle tubes.Several studies have been developed on special shaped tubes.However,further studies are still necessary.Most previous research focused on one tube shape,which,however,is a main factor affecting heat transfer.Therefore,the difference of flow pattern,film thickness and heat transfer for different tube shapes is lack of study.In practical refrigeration spiral wound heat exchanger,special flow patterns appeared except falling film flow due to the complex practical conditions,especially various vapor qualities.Heat transfer is strongly effected by flow pattern.Study on the flow pattern and flow pattern transition for various tube shapes is needed.The forces on the liquid film are different with the same circumferential angle for various tube shapes,which leads to various flow pattern,film thickness and heat transfer.

The present study investigates the influence mechanism of tube shapes on flow pattern,film thickness and heat transfer.Threedimensional model for circle,ellipse-shaped,egg-shaped and cam-shaped tube was presented.Volume of fluid (VOF) model was used.The simulation results compared with experimental data in literature [21].The flow pattern transition for different shape tubes for various vapor qualities and heat fluxes were developed.The film thickness distribution in circumferential direction for different tube shapes were analyzed.The heat transfer mechanism for different shape tubes were analyzed.The results demonstrated the heat transfer for different tube shapes in terms of liquid film thickness and flow pattern,which provides the heat exchangers with benefit for improving heat transfer in practical applications.

2.Simulation Model

2.1.Geometric model and boundary conditions

The geometric structures of cam-shaped tube,egg-shaped tube,ellipse-shaped tube and circle tube are shown in the Fig.1.Including 3 rows of tubes.Along the fluid flow direction,the first row of tube is adiabatic tube to obtain fully-developed flow,and the remaining 2 rows are heated by constant heat fluxes.The diameter of the circle tube is 12 mm,and the tube spacingPris 4 mm.Camshaped tube is generated from two semicircles and their external common tangents as shown in the Fig.1(d),while egg-shaped tube is generated from the upper semicircle and lower semi-ellipse as shown in the Fig.1(c).The shape of a single tube is obtained according to the geometric dimensions in the Fig.2,which ensures the effective heat transfer areas identically for the four kinds of tubes.The geometrical parameters are shown in Table 1.The inlet was divided into gas inlet and liquid inlet,both of which are velocity inlets as shown in Fig.1(a).Inlet gas and liquid velocity can be obtained by a constant inlet vapor quality.The relation between vapor quality and velocity are [33]:

Table 1 Geometry parameters

Fig.1.Geometry of calculation domain for different tubes: (a) circle tube,(b) ellipse tube,(c) egg-shaped tube,(d) cam-shaped tube.

Fig.2.Geometry dimension for different tubes: (a) circle tube,(b) ellipse tube,(c) egg-shaped tube,(d) cam-shaped tube.

where,xis vapor quality.Gis mass flux.ulis liquid velocity,uvis the vapor velocity.Alis liquid actual internal area.Avis vapor actual internal area.Ais actual internal area.ρlis liquid density.ρvis vapor density.

The heated tube walls have constant heat flux.The outlet was pressure outlet.At the beginning,the computational domain is set to be filled with gas phase.The simulation fluid flows in the calculation domain with vapor and liquid.The physical properties ofn-pentane are obtained by NIST REFPROP [34].

2.2.Simulation fluid

Refrigerants of spiral wound heat exchanger usually work at temperature as low as -127.3 °C and are composed from various hydrocarbon refrigerant in the process of liquefied natural gas,which belongs to non-azeotropic mixture and has complex physical properties.A model usedReandGa0.25(Galileo number) was proposed to determine the flow pattern [19].Sunetal.[20] used their model [19] and compared the physical properties of mixed refrigerant,water,ethanol andn-pentane.The value ofGa0.25ofn-pentane is similar to mixed refrigerant.Therefore,n-pentane was used as the simulation fluid.

2.3.Governing equations

The flowing domain is full ofn-pentane.Transient calculation is carried out.VOF model is used to track the gas-liquid interface.Continuous surface force (CSF)model is used to obtain the surface tension.The turbulence model of RNGk-ε model is used.The time step is 1 × 10-4s.

The continuity equation of VOF model is

where,αqis the volume fraction of theqth phase in a grid.

Momentum equation is

where,F is the surface tension,which is given by CSF model [35].

where,σ is surface tension coefficient,Kiis the curvature of the interface,n is normal unit vector,t is tangential unit vector and θ is the contact angle.

Energy equation is

where,Eis internal energy andSEis the energy source term,which is latent heat.Turbulence model of RNGk-ε is used.

2.4.Model of mass transfer

The Lee model[36]was used to simulate the phase change process.WhenTl≥Tsat,the evaporation occurs and liquid phase changes to the gas phase.The mass transfer from the liquid phase to the gas phase is:

where,β is the mass transfer time relaxation parameter and its value is 0.1.ρlis density for liquid,Tlis liquid temperature,Tsatis saturation temperature.

WhenTl≤Tsat,the condensation occurs and gas phase changes to the liquid phase.The mass transfer from the gas phase to the liquid phase is:

where,ρvis density for vapor.Tvis vapor temperature.

Considering evaporation and condensation,the equation of mass transfer source term is:

The setting parameters of the CFD solver are listed in Table 2.

Table 2 Setting parameters of the CFD solver

2.5.Grid

Fig.3 shows the tetrahedral unstructured grid generated for the 4 kinds of tubes.Different numbers of grids were generated for the 4 kinds of tubes to verify the meshes independence.When the grid number exceeds 191,156,the heat transfer coefficient of circle tube is basically stable as shown in Fig.4.In the same way,the grid independence for other three tubes were presented.The heat transfer coefficient is stable as the number of grid for the ellipseshaped tube,egg-shaped tube and cam-shaped tube are higher than 183,791,206,951 and 209,879,respectively.

Fig.3.Meshing of calculation area for different tubes: (a) circle tube,(b) ellipse tube,(c) egg-shaped tube,(d) cam-shaped tube.

Fig.4.Grid independence validation.

2.6.Model validation

The numerical method is validated using the same experimental condition given by Houetal.[21].The predicted results and experimental data were compared in Fig.5.The average deviation between predicted results and experimental data is 8.59%.

Fig.5.Comparisons of simulation results and experimental data: (a) prediction δ for various circumferential angles and (b) prediction deviation of δ.

3.Results and Discussion

3.1.Flow patterns for different tube shapes

The flow patterns for different tube shapes are simulated with different inlet vapor qualities and heat fluxes.Table 3 shows the simulation conditions.Physical properties ofn-pentane are shown in Table 4.Fig.6 shows the flow patterns of different tube shapes.Three kinds of flow patterns appeared in each tube shape,including sheet flow,semi-annular flow and droplet flow.

Table 3 Simulation conditions

Table 4 Physical properties of the fluid

Fig.6.Flow patterns for different tubes: (a) circle tube,(b) ellipse tube,(c) egg-shaped tube,(d) cam-shaped tube.

The flow pattern is mainly effected by vapor quality.When vapor quality is 0.2,the velocity of gas and liquid is low and the liquid mass flow is large,showing a sheet flow with the tube wall completely covered,as shown in Fig.6(A).As the vapor quality increases,the gas phase velocity and the shear stress increases,leading to the tearing of the liquid film,showing a semi-annular flow as shown in Fig.6(B).The gas mass flow increases when the vapor quality is 0.8.The gas phase outside the tube dominates the flow.The liquid on the tube wall is swept to form droplets by gas flow.The droplet flow is shown in Fig.6(C).

The flow pattern transition ofn-pentane across tube bundles for various tube shapes was simulated for various vapor qualities and heat fluxes as shown in Fig.7.In the flow pattern map,different flow patterns,including sheet flow,semi-annular flow and droplet flow were presented for various simulation conditions.For circle tube,egg-shaped tube and cam-shaped tube,the transition line of sheet flow to semi-annular flow is for vapor quality of 0.4.The transition line of semi-annular flow to droplet flow is for vapor quality of 0.6-0.7.Therefore,the transition condition of flow pattern is not significantly affected by shape of circle tube,eggshaped tube and cam-shaped tube.However,for the ellipseshaped tube,the transition line of sheet flow to semi-annular flow is for vapor quality of 0.3.The transition line of semi-annular flow to droplet flow is for vapor quality of 0.6.The transition vapor quality of elliptic tube is lower than other tube shapes.The possible reason is that the upper part of the circle tube,egg-shaped tube and cam-shaped tube is semicircle.The upper part of the ellipseshaped tube is semi-ellipse,whose tangential component of the gravity is larger than the semicircle,resulting in larger liquid flow velocity than semicircle and the liquid film easier to crack.Therefore,the transition line of ellipse-shaped tube has lower vapor quality than other tube shapes.

Fig.7.Flow pattern map for different tubes.

3.2.The film thickness distribution in circumferential direction for different tube shapes

The thickness distribution of liquid film in sheet flow and the circumferential direction of θ=15°-165° were presented.The vapor qualities of circle,egg-shaped and cam-shaped tubes were 0.1-0.4.The vapor qualities of ellipse-shaped tubes were 0.1-0.3,as shown in Fig.8.The liquid flows upward along the circumferential direction.The liquid film gets thinner at θ=15°-60°and keeps steady at θ=60°-35°,while the liquid film becomes thicker at θ=135°-165°.The liquid film is thinnest at around θ=90°.The upstream liquid flows to the top of the tube due to gravity.The covering area of the liquid film increases with increasing θ.The liquid film spreads on the tube surface due to surface tension.Therefore,the liquid film becomes thinner.Gravity and velocity are in the same direction at θ=90°,where the liquid film is the thinnest.In the lower part of the tube,the liquid velocity increases,resulting in the liquid accumulates at θ=135°-165° and thicker liquid film.

Fig.8.The film thickness distribution in circumferential direction: (a) circle tube,(b) ellipse tube,(c) egg-shaped tube,(d) cam-shaped tube.

The liquid film of circular and elliptical tubes is symmetrically distributed along the circumferential direction.The liquid film of egg-shaped tube at θ=15°-60° is thicker than θ=135°-165°.The liquid film of the cam tube at θ=15°-60° is slightly thinner than θ=135°-165°.The thickness of the liquid film is effected by tube shapes.Circle tube and ellipse-shaped tube are symmetrical structure along the circumferential direction.Therefore,the liquid film is symmetrical in the upper and lower part.The lower part of egg-shaped tube is semi-ellipse.The height of the lower semiellipse is larger than the upper semicircle.The gravity tangential component along the tube wall is larger.The liquid film flows quicker,which promotes convective heat transfer,resulting in thinner liquid film.The lower cam-shaped tube is the external common tangent of two circles and semicircle with a radius of 2 mm.The tangential component of gravity is larger along the common tangent part,resulting in quicker liquid velocity and thinner liquid film.The liquid film thickens when the liquid flows to the lower semicircular tube wall.

The effect of vapor quality on liquid film thickness is nearly independent of tube shapes as shown in Fig.8.In sheet flow,the liquid film thickness gets thinner with increasing vapor quality.The liquid mass flow decreases with increasing vapor quality.The increasing gas velocity leads to increasing shear stress and thinner liquid film.

3.3.The film thickness distribution for different tube shapes

Fig.9 shows the distribution of liquid film along the circumferential direction for different tube shapes.At θ=15°-60°,the liquid film thickness for ellipse-shaped tube,cam-shaped,egg-shaped and circle tube ranks from thinner to thicker.At θ=60°-135°,the effect of tube shape on liquid film thickness is not significantly.At θ=135°-165°,the thickness of circular tube is the thickest,followed by cam-shaped tube.However,the thickness of ellipseshaped tube and egg-shaped tube have no significant difference.For the fluid flow with the same vapor quality,smaller radius of curvature at the top of the tube shows larger upstream liquid covering area.The radius of the circle tube is the largest,resulting in smaller liquid covering area.The liquid is easy to accumulate in the upper tube due to viscous force.Therefore,at θ=15°-60°,the liquid film is thicker than other parts.The upper egg-shaped tube and cam-shaped tube are semicircles with smaller curvature than the circle tube,whose liquid covering area is larger and the liquid film is thinner for the same vapor quality compared with circle tube.The upper ellipse-shaped tube is semi-ellipse,whose gravity component along the velocity direction is larger than the circular structure at θ=15°-60°.The upstream liquid is dragged down,resulting the thinner liquid film.In the thin liquid film region,θ=60°-135°,the gravity dominants liquid film flow.The liquid film velocity reaches the highest in the thin liquid film region,where the effect of tube shape on liquid film thickness is negligible.The circle tube curvature radius is the largest in the lower tube resulting in the thickest liquid film at θ=135°-165°compared with other tube shapes.The liquid film thickness in the lower part is thinner for egg-shaped tube and ellipse-shaped tube because their lower part is semi-ellipse.

Fig.9.The film thickness distribution for different tube shape: (a) x=0.1,(b) x=0.2,(c) x=0.3,(d) x=0.4.

3.4.The heat transfer coefficient for different tube shape

Fig.10 shows the heat transfer coefficient for different tube shapes.The heat transfer coefficients are given by:

where,qis heat flux,TwandTfare the wall average temperature of heated tubes and the average fluid temperature,respectively.

At sheet flow,the heat transfer coefficients ranging from highest to lowest are for ellipse-shaped tube,cam-shaped tube,eggshaped tube and circle tube.At the semi-annular flow and droplet flow,the heat transfer coefficient is not significantly affected by tube shape.The main reasons effecting heat transfer are liquid film heat conduction and gas flow convection.At sheet flow,the tube surface is covered by liquid film.Liquid film thermal conduction dominants heat transfer.The liquid film thickness in the top part of tube thickness ranking from thinner to thicker are for ellipseshaped tube,cam shaped tube,egg-shape tube and circle tube as shown in Fig.9.The thinner liquid film leads to the lower thermal resistance and higher heat transfer coefficient.At thin liquid film,the change of liquid film thickness is little,resulting in little variation of heat transfer coefficient.At the bottom of the tube,the circle tube has the thickest liquid film,while the egg-shaped tube and the ellipse tube have the thinner liquid film.Therefore,the ellipse tube has the highest heat transfer coefficient and circle tube has the lowest at sheet flow.However,the liquid cannot cover the tube for semi-annular flow.Droplets hang on the tube surface in droplet flow.The convective heat transfer dominants heat transfer,resulting in little difference for heat transfer coefficient with the same vapor quality.

4.Conclusions

This paper studies the effects of tube shape on flow pattern and heat transfer ofn-pentane across tube bundles with various vapor qualities,heat fluxes.The VOF model was used to track the gas-liquid interface.The simulation model agrees well with the experimental data.The liquid film thickness and heat transfer for different tube shapes were investigated and analyzed.The main conclusions are as follows:

(1) Sheet flow,semi-annular flow and droplet flow appeared in each tube shape.The ellipse tube flow pattern transition occurs for low vapor quality compared with other tube shapes.

(2) In sheet flow,the liquid film of circle and ellipse-shaped tubes are symmetrically distributed along the circumferential direction.The liquid film of egg-shaped tube at θ=15°-60° is thicker than θ=135°-165°.The liquid film of the cam-shaped tube at θ=15°-60° is slightly thinner than θ=135°-165°.

(3) At θ=15°-60°,the liquid film thickness ranking from thinnest to thickest are for ellipse-shaped tube,cam-shaped tube,egg-shape tube and circle tube.At θ=60°-135°,the effect of tube shape on liquid film thickness is insignificant.At θ=135°-165°,the liquid film on circle tube is the thickest,followed by cam-shaped tube.

(4) In sheet flow,the heat transfer is affected by tube shape.Ellipse-shaped tube has largest heat transfer coefficient.Circle tube has lowest heat transfer coefficient.In the semiannular flow and droplet flow,the effect of tube shape on heat transfer is negligible.In practical application,the tube shape could be designed as ellipse to promote heat transfer.

Data Availability

No data was used for the research described in the article.

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

This study is supported by National Natural Science Foundation of China(52006242),National Natural Science Foundation of China(52192623),Science Foundation of China University of Petroleum,Beijing (ZX20200126) and Science and technology program for strategic cooperation of CNPC-China University of Petroleum(ZLZX2020-05).

Nomenclature

Eenergy,J

Fsurface tension force,N

gacceleration due to gravity,m.s-2

hheat transfer coefficient,kW.m-2.s-1

Kicurvature of the interface

n normal unit vectors

Prlongitude pitch,mm

Sαqsource term,VOF equation,kg.m-3

SEsource term,energy equation,W.m-3

Ttemperature,K

Tsatsaturation temperature,K

ttime,s

uvelocity,m.s-1

α volume fraction

β time relaxation parameters

θ contact angle

μ dynamic viscosity,Pa.s

ρ density,kg.m-3

σ surface tension coefficient,N.m-1

Subscripts

lv liquid to vapor

lliquid

qtheqth fluid

sat saturation

vlvapor to liquid

v vapor