Numerical simulation of oscillatory flow and heat transfer inpulsating heat pipes with multi-turns using OpenFOAM

Jongwook Choia and Yuwen Zhangb

aSchool of Mechanical and Aerospace Engineering, Sunchon National University, Jeonnam, Republic of Korea;bDepartment of Mechanical and Aerospace Engineering, University of Missouri, Columbia, USA

ABSTRACTThe oscillatory flow and the heat transfer in two-dimensional pulsatingheat pipes (PHPs) with multi-turns were simulated using OpenFOAM. Thevolume of fluid method was used for the phase change, and the behaviorof the working fluid was achieved by considering the mass transfer bal-ance between the evaporation and the condensation. Ethanol was used asthe working fluid, and the liquid phase and the vapor phase were assumedto be incompressible. The result revealed that the temperature variationcurves did not converge to one pattern according to the number of gridsin the symmetric shape of the PHP because the starting time of the work-ing fluid circulation was different. In the PHP with the asymmetric shape,the circulation started earlier than in the PHP with symmetric shape. Whenthe bond number was 0, which means being in zero gravity, the workingfluid dried out in the evaporator section of the PHPs with 5 and 10 turns.However, the working fluid still remained in the PHPs with 15 and 20turns. The numerical analysis performed in this article is expected to helpto simulate the flow phenomenon in PHP.

ARTICLE HISTORYReceived 23 September 2019Accepted 9 January 2020

1. Introduction

Pulsating heat pipe (PHP) is a device capable of continuously performing heat exchange withoutthe external driving force by utilizing the sensible heat caused by the temperature difference andthe latent heat by the phase change. The device can be applied to various fields ranging from asmall electronic substrate to a solar water heating system, and it can also be used to improve theefficiency of the existing heat exchanger.

The previous studies on the PHP have focused mainly to improve the thermal performance,which can be directly affected by working fluid, input power, inclination angle, filling ratio, thenumber of turns, internal diameter, evaporator and condenser lengths, and so on. The experimen-tal studies [1–8] on working fluid have been carried out to compare the thermal performances ofthe PHPs with water, ethanol, and methanol as a working fluid. As the results of these studies,the thermal performance of PHPs with water, naturally retaining large values of specific heat andlatent heat, was generally better than PHPs with ethanol and methanol. In other experimentalconditions, however, ethanol and methanol had a better thermal performance than water [9, 10].Furthermore, some studies have pursued the experiments using nanofluid [11–15], ferrofluid[14, 16, 17], self-rewetting fluid [18, 19], and surfactant fluid [20]. Such working fluids showedeven higher thermal performance than water due to their own characteristics.

CONTACT Yuwen Zhang zhangyu@missouri.edu Department of Mechanical and Aerospace Engineering, University ofMissouri, Columbia, MO 65211, USA.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/unht.� 2020 Taylor & Francis Group, LLC

NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS2020, VOL. 77, NO. 8, 761–781https://doi.org/10.1080/10407782.2020.1717202

As for the experiments on input power, they have been conducted in all the studies mentionedearlier. Those experiments showed that as the input power increased, the thermal resistancedecreased and the thermal performance improved, which was regarded as a general trend. On theother hand, the working fluid did not circulate when the input power was too low, and the work-ing fluid dried out in the evaporator section of the PHP when the input power was too high. Inboth cases, the thermal resistance was relatively high.

In the experimental studies on effects of inclination angle [2, 14, 17, 21–24], the thermal resist-ance was the lowest in the vertical bottom heat mode. This is because the working fluid in theupper condenser section can be easily moved to the lower evaporator section by gravity. Theexperimental research on filling ratio [1, 5, 8, 9, 14, 17, 20, 25, 26] showed that the optimal ther-mal performance was achieved when the filling ratio of the PHP was generally 40%–65%. Whenthe filling ratio was too low, the dry-out phenomenon occurred in the evaporator. Whereas thefilling ratio was too high, the circulation speed of the working fluid was slow and the thermalperformance low. In the studies on the number of turns [2, 24, 27], the working fluid did not cir-culate according to the conditions at low turn numbers. However, the working fluid had relativelygood circulation at large turn numbers, making the thermal performance improved.

The experimental researches on internal diameter [2, 3, 18, 25, 28] showed that the larger thediameter was, the higher the thermal performance gained because the gravity effect was strongerthan the capillary effect. In the study of the lengths of evaporator and condenser [29], the greaterthe ratio of evaporator length to condenser length was, the quicker the circulation of the workingfluid began, leading the thermal performance to be high. As to the studies on the shape of thePHP, dual diameter [6, 23, 30, 31], cavity [32], microgrooves [33], and interconnecting channels[11] were applied to the PHP, resulting in higher thermal performance than the conventionalPHPs without applying such shapes.

Most of the studies mentioned earlier have been experimentally conducted on the PHP with4–5 turns in the evaporator. The effect of turns was investigated in the PHPs with 10 or moreturns in the work by Charoensawan et al. [2], Jun and Kim [24], and Mameli et al. [27]. In thework by Charoensawan et al. [2], the experiment on the copper pipe of 2-mm diameter with the

Nomenclature

Bo bond numberCp specific heat capacity [J/kg K]D diameter of pulsating heat [m]g gravitational acceleration [m/s2]h height [m]J mass transfer [kg/m3 s]k thermal conductivity [W/m K]L specific latent heat [J/kg]max maximum valueprgh total pressure excluding hydrostatic pres-

sure [Pa]q heat flux [W/m2]R thermal resistance [K m2/W]T temperature [K]t time [sec]tot the total number of gridsU velocity [m/s]

Greek symbolsa volume fractionb mass transfer time relaxation parameter

[1/s]

j curvature [1/m]l dynamic viscosity [kg/m s]q density [kg/m3]_q total derivative of density [kg/m3 s]r surface tension [N/m]

Subscriptsboil boilingc condensation; condensere evaporation; evaporatori grid numberl liquid phaselv from liquid phase to vapor phasev vapor phasevl from vapor phase to liquid phase

Superscriptsn previous levelnþ 1 current level

762 J. CHOI AND Y. ZHANG

filling ratio of 50% was performed with varying the number of turns in the evaporator to 5, 7,11, 16, and 23. The number of turns played a significant role in the thermal performance of thePHP. The number of critical turns was suggested to be 16. In the work by Jun and Kim [24], thethermal performance was experimentally examined on the Closed Loop Micro PHP (CLMPHP)and the Closed-End Micro PHP (CEMPHP) with the channel width of 1mm and the height of0.5mm; the numbers of turns were 5, 10, 15, and 20 in these PHPs. The performances of theCLMPHP with 20 turns and the CEMPHP with 10 or more turns were not dependent on theinclination angle. In the work by Mameli et al. [27], the numerical simulation was carried outusing a one-dimensional model with 2, 5, and 20 turns in the evaporator. The result indicatedthat it did not work for the PHP with two turns in the horizontal mode but worked for the PHPwith five turns.

The studies on the number of turns have mostly focused on the operation and the thermalperformance of the PHP. However, the visualization of the phase change process and the workingfluid flow in the PHP with multi-turns has not been done much. In this article, the numericalanalysis was carried out using OpenFOAM (version 5.0), an open-source code, for the visualiza-tion mentioned earlier. The numerical analyses of the heat pipes [34–40] have mainly adoptedthe commercial software FLUENT that the source code was not provided to the users. In thisstudy, the contents of the numerical analysis process were presented in more detail, such as thegoverning equations used for the PHP, the mass transfer parameters required for the phasechange, and the simulation conditions. The grid independence test was conducted on the PHPwith five turns and also the thermal resistance of the PHP was obtained. In addition, the behav-iors of the working fluid were compared between the symmetric-shaped PHP and in the asym-metric-shaped PHP. In order to investigate the effect of gravity, the visualization of the workingfluid flow was presented according to the bond number and the number of turns.

2. Numerical analysis

2.1. Governing equations

For the numerical analysis of the phase change and working fluid flow in the PHP, the liquidphase and the vapor phase of the working fluid are assumed to be two-dimensional incompress-ible fluids [39]. The OpenFOAM offers the phase change solver only with the incompressible flu-ids. The volume of fluid (VOF) method [41] is used in the phase change, and the heatconduction in the wall is not considered. The volume fraction (al) for the liquid phase can beobtained from Eq. (1) [42], and the volume fraction (av) for the vapor phase from av ¼ 1� al:

@al@t

þr � al~Ul

� �¼ alav

_qvqv

� _qlql

� �þ al r � ~Uð Þ þ Jvl � Jlvð Þ 1

ql� al

1ql

� 1qv

� �� �(1)

where Jvl represents the mass transfer from the vapor phase to the liquid phase in the condensationprocess (T < Tboil) and Jlv is the mass transfer from the liquid phase to the vapor phase in theevaporation process (T > Tboil). These values can be acquired from Eqs. (2) and (3) based on theboiling temperature of the working fluid [43]

Jvl ¼ bcavqvTboil � TTboil

�������� (2)

Jlv ¼ bealqlTboil � TTboil

��������: (3)

Here bc is the mass transfer time relaxation parameter for the condensation and be is the masstransfer relaxation parameter for evaporation. Generally, the value of 0.1 is used for both bc andbe [43].

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On the other hand, if the same value (bc ¼ be ¼ 0:1Þ is used for the mass transfer parameterin the evaporation process and condensation process, the mass transfer amounts for the conden-sation and the evaporation are not balanced in the phase change process. This is because Eqs. (2)and (3) are calculated on the basis of the mass while the VOF method is designed on the basis ofthe volume. For example, when ethanol is used as the working fluid, the density ratio of theliquid phase to the vapor phase is about 550 times or more, and when the working fluid is water,the density ratio comes as about 2000 times or more. Thus, as time passes, the mass transferamount (Jlv) from the liquid phase to the vapor phase becomes greater than the one (Jvl) fromthe vapor phase to the liquid phase. This also leads to an unexpected dry-out phenomenon in thePHP. To prevent this phenomenon, the condensation mass transfer parameter (bc) is calculatedwith the evaporation mass transfer parameter (be) of 0.1 as shown in Eq. (4) [44]

bnþ1c ¼ bnc � bnc

Ptoti¼1Jvl, i �

Ptoti¼1Jlv, i

maxPtot

i¼1Jvl, i,Ptot

i¼1Jlv, i

264

375: (4)

The condensation mass transfer parameter is obtained at each time level. The difference betweenthe evaporation amount and the condensation amount is calculated in all the cells of the compu-tational domain and then corrected to correspond to the evaporation amount. The mass transferamount between the evaporation and the condensation is balanced in the phase change process.

The momentum equation is described in Eq. (5).

@ q~U� �@t

þr � q~U~U� �

� ~U@q@t

� �� ~U r � q~U

� �� �¼ �rprgh �~g �~hrqþr � l r~U þr~U

T h i

þ rjraþ Jvl � Jlvð Þ � Jvl Jlvð Þh i

~U :

(5)

For the incompressible fluids, the sum of the first and third terms on the left side represents thelocal acceleration and the sum of the second and fourth terms represents the convective acceler-ation. The first and second terms on the right side mean the total pressure and the hydrostaticpressure, respectively. The third term on the right side is the viscous force, and the fourth term isthe force due to the surface tension. The last term on the right side represents the force causedby the phase change, which is the driving force of moving the working fluid. When the workingfluid evaporates from the liquid phase to the vapor phase, the expanding force is generated inproportion to the evaporation mass transfer amount. On the other hand, the shrinking force isgenerated in proportion to the condensation mass transfer amount when the workingfluid condenses.

The energy equation to obtain the temperature is shown in Eq. (6)

@ qCpT� �@t

þr � q~UCpT

� T@ qCpð Þ

@t

� �� T r � q~UCp

h i¼ r � krTð Þ þ Jvl � Jlvð ÞL: (6)

The sum of the first and third terms on the left side represents the energy change with time, andthe sum of the second and fourth terms stands for the energy change due to the convection. Thefirst term on the right side means the energy change caused by heat conduction. The secondterm on the right side refers to the energy change due to the latent heat, which has a negativevalue in the evaporation and the positive value in the condensation.

Density, dynamic viscosity, specific heat capacity, and thermal conductivity used in the govern-ing equations are as follows in Eqs. (7)–(10), respectively. The specific heat capacity is obtainedon the basis of the mass fraction while the other properties are based on the volume fraction [43]

q ¼ avqv þ alql (7)

764 J. CHOI AND Y. ZHANG

l ¼ avlv þ alll (8)

Cp ¼avqvCpv þ alqlCpl

avqv þ alql(9)

k ¼ avkv þ alkl (10)

2.2. OpenFOAM solver

Unlike commercial programs, a solver for the numerical analysis should be selected first inOpenFOAM. There are 20 solvers associated with the multiphase flow in OpenFOAM [45]. Amongthese solvers, the interPhaseChangeFoam solver that is capable of analyzing the phase change is chosen.This solver can implement the phase change process with the VOF method for two incompressible andisothermal immiscible fluids and is appropriate to solve complex problems like cavitation. For thenumerical analysis of the PHP, the original solver is modified as follows. The condensation mass trans-fer parameter in Eq. (4) is applied to Eq. (1) to obtain the volume fractions of the liquid phase and thevapor phase. The last term on the right hand side of Eq. (5) is added to the original pressure equationto consider the force due to the phase change. The source code for Eq. (6) is generated to obtain thetemperature because the selected solver does not provide the energy equation.

The term Jvl � Jlvð Þ~U in the last terms on the right hand side of Eq. (5) affects the motion of theworking fluid, which results in a decrease or increase of the local pressure in the place where the con-

densation and the evaporation take place. The second term, Jvl Jlvð Þ~U in the last terms on the righthand side of Eq. (5) refers to a decrease or increase of the entire average pressure in the PHP andmaintains the reference pressure at a constant value in an enclosed space. This term is necessarybecause two incompressible fluids are not allowed to compress and expand due to the differencebetween the condensation mass transfer amount and the evaporation mass transfer amount.

The reference pressure and position are required when performing the fluid flow analysis on theenclosed space with the incompressible fluid. In the case of an open space, on the other hand, the ref-erence pressure and position are not necessary because the atmospheric condition is usually given tothe boundary condition. Since the PHP corresponds to the former, it is necessary to initialize the ref-erence pressure. The absolute value of the reference pressure has no effect on the phase change or theworking fluid flow but is just used to determine the relative pressure in the PHP.

2.3. Boundary conditions

The interPhaseChangeFoam solver of the OpenFOAM offers only the constant temperatureboundary condition and the temperature gradient boundary condition. The temperature gradientboundary condition can be used as the heat flux boundary condition in a single-phase fluid. Inthe PHP with a two-phase fluid, however, the OpenFOAM source code has to be modified toapply the heat flux boundary condition to it.

The difference between the heat flux boundary condition and the temperature gradient bound-ary condition is shown in Figure 1. In the case of the heat flux boundary condition, the wall tem-perature is given by considering the thermal conductivities of both the liquid and vapor phases asin Eq. (10). In the case of the temperature gradient boundary condition, however, only the ther-mal conductivity of the liquid phase is considered.

In the numerical analysis process, the heat flux boundary condition on the evaporator sectionis set to the wall temperature. The wall temperature depends on the temperature and the volumefraction of the working fluid inside the PHP. If the temperature of the working fluid inside thePHP is relatively high, the wall temperature on the evaporator section will also be on the increasealong with it to keep the heat flux constant.

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3. Grid independence

3.1. Grid generation

The grid independence test is performed in order to ensure the accuracy of the results for theanalysis on the thermal fluid flow including the phase change of the two-dimensional PHP. Thegrids are generated for the PHP with five turns as shown in Figure 2, and the program for thegrid generation is created using the Cþþ language. The lengths of the evaporator section, theadiabatic section, and the condenser section of the PHP are 30mm, 60mm, and 60mm, respect-ively. The condenser section is given the longer length than the evaporator section because thecooling region is larger than the heat source region in the actual practice. The inner diameterand the radius of curvature are 2mm and 3mm, respectively. The distance between the pipes is6mm. The grids are generated with the equal interval considering the visualization of the phasechange using the VOF method. The numbers of the grids used for the grid independence test are7605, 12,240, 19,150, 26,094, 35,595, 48,800, 61,056, and 72,710, respectively.

3.2. Simulation conditions

The initial and boundary conditions for the numerical analysis on the PHP are shown in Table 1,and the properties of the ethanol used as the working fluid are displayed in Table 2. For the ini-tial conditions, the temperatures for the vapor phase and the liquid phase are set to 356.39K and346.39K, respectively. These values are based on ±5K at 351.39K, which is the boiling point ofthe ethanol in which the phase change occurs. The filling ratio of the working fluid is set to 50%,which is the ratio of the liquid phase volume to the total volume of the PHP. The vapor phaseoccupies the remaining 50% of the total volume.

For the boundary conditions, the temperature on the condenser section is set to 293.15K,which is a general room temperature. The temperature gradient of 0 in the inward direction ofthe wall is given to the adiabatic section. The heat flux on the evaporator section is set to1000W/m2. The time step and the maximum Courant number are set to 1� 10�5 s and 0.5,respectively.

3.3. Temperature variations with the number of grids

The average temperature variation of the entire surface of the evaporator section with the numberof grids is demonstrated in Figure 3. The reason to choose the evaporator temperature is that it

Figure 1. Boundary conditions for heat flux and temperature gradient.

766 J. CHOI AND Y. ZHANG

Figure 2. Geometry of pulsating heat pipe.

Table 1. Simulation conditions.

Regions Values

Initial conditions Vapor phase Volume fraction: 0Velocity: 0 (m/s)Pressure: 101,325 (Pa)Temperature: 356.39 (K)

Liquid phase Volume fraction: 1Velocity: 0 (m/s)Pressure: 101,325 (Pa)Temperature: 346.39 (K)

Filling ratio for liquid volume 50(%)Boundary conditions Condenser section Velocity: no slip

Temperature: 293.15 (K)Adiabatic section Velocity: no slip

Heat flux: 0 (W/m2)Evaporator section Velocity: no slip

Heat flux: 1000 (W/m2)

Table 2. Properties of ethanol.

Liquid phase Vapor phase

Density (kg/m3) 783.9 1.409Kinematic viscosity (m2/s) 1.368� 10–6 8.414� 10–6

Specific heat capacity (J/kg K) 2570 1600Thermal conductivity (W/m K) 0.167 0.0258Surface tension (N/m) 0.02197Specific latent heat (J/kg) 919,000Contact angle (degree) 33.3Boiling temperature (K) 351.39

NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 767

is a significant factor in obtaining the thermal resistance that indicates the thermal performanceof the PHP.

In the average temperature variation on the evaporator section with time, the temperature sud-denly increases and then decreases between about 15 s and 25 s. This is because the working fluidin the liquid phase rises up in temperature due to heating the evaporator, and the working fluidcooled in the condenser moves down to the evaporator because of the pressure difference. This

Figure 3. Temperature variations on evaporator section with time (0 s–30 s).

Figure 4. Distributions of volume fraction with time.

768 J. CHOI AND Y. ZHANG

phenomenon can be seen evidently in Figures 4 and 5. The number of grids used in these resultsis 61,056. Figure 4 shows the distribution of the volume fraction at 17.5 s, 18.3 s, and 18.8 s. Theworking fluid circulates counterclockwise and the vapor phase of the lower evaporator moves upto the upper condenser. The temperature distribution under the same conditions is seen inFigure 5. As the working fluid circulates, the temperature of the evaporator decreases.

On the other hand, the average temperature variation on the evaporator section does not convergeto one pattern with the number of grids as shown in Figure 3. This is because it is influenced by thestarting time of the circulation of the working fluid through the phase change. For example, whenthe number of grids is 72,710, the peak temperature rises abnormally higher than in other numbersof grids. The volume fraction distribution with time explains this phenomenon as shown in Figure 6when the numbers of grids are 61,056 and 72,710. In Figure 6a, b, the working fluid in the evaporatormoves upwards, showing a symmetrical flow structure regardless of the number of grids. When thenumber of grids is 61,056, the working fluid starts to rotate counterclockwise as the circle in the dot-ted line in Figure 6c. However, when the number of grids is 72,710, the working fluid is still on holdand the vapor phase with the high temperature is distributed more widely in the lower evaporator asshown in Figure 6d. For the symmetrical shape of the PHP, the dense grids turn out to affect thestarting time of the circulation of the working fluid while maintaining the pressure balance both onthe left and on the right. Table 3 presents the circulation directions of the working fluid according tothe number of grids in the PHP. When the number of grids is small, the circulation occurs in thesame direction. When the number of grids is large, the working fluid circulates in the opposite direc-tion. When the number of grids is as small as 7605 and 12,240, it is difficult to observe the thin filmof the working fluid condensed on the wall.

Since the results are not clearly converged in the grid independence test, analyzing the tem-perature variation needs to be conducted right before the phase change occurs. Therefore, thetemperature curves between 5 s and 10 s, which are marked as the dotted lined circles in Figure 3,are enlarged and depicted in Figure 7. The increase and decrease of the temperature in the curvesare attributed to the latent heat in the phase change process. In order to make clear indication ofthe convergence according to the number of grids, the average temperature variations on the

Figure 5. Temperature distributions with time.

NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 769

evaporator section at 5 s and 10 s are featured in Figure 8. In the case of 5 s, which is not influ-enced by the latent heat, the temperature quickly converges to 350.9 K with over 35,595 gridnumbers. In comparison with the case of 5 s, the case of 10 s has a slight difference due to thelatent heat, but the temperature converges to 352.2 K with over 61,056 grid numbers. These con-verged temperatures are obtained by rounding off the second decimal point. Based on the resultsso far, the optimal number of grids in consideration of the phase change process and the workingfluid flow is determined as 61,056. These grids are at intervals of about 0.225mm in the trans-verse and longitudinal directions, respectively. These intervals are applied to the followingsimulations.

Figure 6. Distributions of volume fraction with time for the number of grids of 61,056 (left side) and 72,710 (right side).

Table 3. Circulation directions of working fluid with the number of grids.

The number of grids Circulation directionFilm condensed

on wall

7605 Counterclockwise No12,240 Counterclockwise No19,150 Counterclockwise Yes26,094 Clockwise Yes35,595 Counterclockwise Yes48,800 Clockwise Yes61,056 Counterclockwise Yes72,710 Clockwise Yes

770 J. CHOI AND Y. ZHANG

4. Thermal resistance

For the thermal resistance, the average temperature variations are obtained over the entire surfaceof the evaporator with the heat flux boundary conditions as shown in Figure 9. Here, the heatflux values are set to 500, 750, 1000, 1250, and 1500W/m2, respectively. These values are obtainedwhen the input powers are 0.9828, 1.4742, 1.9656, 2.4570, and 2.9484W, respectively, at the sur-face area of the evaporator of 1.9656� 10�3 m2. As the heat flux increases, the circulation of the

Figure 7. Temperature variations on evaporator section with time (5 s–10 s).

Figure 8. Temperature variations on evaporator section with the number of grids.

NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 771

working fluid occurs due to the phase change at a relatively early stage. As a result, the tempera-ture plunges down immediately after the temperature increases in the solid line. After the work-ing fluids circulate once within the region of the solid lines, the temperatures are on the excessiverise within the region of the dashed lines. The reasons for this phenomenon are that the secondcirculation does not occur for a considerable time and that the symmetrical flow balance is notbroken in the PHP and that the vapor phase of the working fluids continues to be generated inthe evaporator.

Figure 9. Temperature variations on evaporator section with heat flux.

Figure 10. Variations of thermal resistances with heat flux.

772 J. CHOI AND Y. ZHANG

On the other hand, the thermal resistances with the heat flux are obtained using Eq. (11) asobserved in Figure 10

R ¼ Te � Tc

q: (11)

Here, the thermal resistance is defined as dividing the temperature difference between theevaporator section and the condenser section by the heat flux. Since the temperature on the con-denser section is constant at 293.15K, the temperature on the evaporator section determines thethermal resistance under the heat flux boundary condition.

As shown in Figure 10, the thermal resistance decreases as the heat flux increases, which isconsistent with the results of the previous researches [1–20]. Here, the temperatures on the evap-orator section indicate the average values during a single circulation of the working fluid, that is,the average values for the solid lines excluding the dotted lines in Figure 9.

Though the simulation conditions of this study and the experimental conditions of the workby Shafii et al. [5] are not the same, the result in this study shows the similarity to that of thestudy by Shafii et al. [5]. Since the input powers in the study by Shafii et al. [5] are relativelylarger, the results of the two thermal resistances cannot be comparable in the same axes, which isthe reason why the independent blue axes are used in Figure 10. When the heat flux is at lowvalue, the thermal resistance has a high value as in the red axes and when the heat flux is at highvalue, the thermal resistance has a low value as in the blue axes.

Figure 11. Distributions of volume fraction with time for symmetric and asymmetric PHPs.

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5. Working fluid flows with different geometry and bond numbers

5.1. Working fluid flow in asymmetric PHP

In the PHP with a symmetric structure, as the circulation of the working fluid is delayed at 72,710 inthe number of grids, the temperature on the evaporator section rises abnormally as appeared in

Figure 12. Temperature variations with time for symmetric and asymmetric PHPs.

Figure 13. Distributions of volume fraction at Bo ¼ 0.0.

774 J. CHOI AND Y. ZHANG

Figure 3. In order to investigate the working fluid flow in the PHP according to the symmetric shapeand the asymmetric shape, the results with time are compared in the same conditions such as the sur-face areas, the boundary conditions, and the number of grids as can be seen in Figure 11. At 12 s, the

Figure 14. Distributions of volume fraction at Bo ¼ 1.0.

Figure 15. Distributions of volume fraction at Bo ¼ 2.0.

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phase change occurs in the lower evaporator section, and the working fluid moves slightly up to theupper condenser section for both geometries. After 15 s, the results show that the circulation doesnot occur in the symmetric structure, whereas the circulation proceeds early in the asymmetric struc-ture. The temperature variations with time for both cases are shown in Figure 12. Unlike the sym-metric structure, the temperature does not increase abnormally in the asymmetric structure. In theactual practice, PHP is mostly installed asymmetrically. In numerical analysis, however, when theperfect symmetric shape is used in the simulation, such abnormal phenomenon takes place.Although the circulation direction is not constant according to the simulation conditions in the sym-metric structure, the circulation direction is always clockwise in the asymmetric structure.

5.2. Effects of bond number on working fluid flow

To investigate the effect of gravity on PHP, the numerical simulation is performed for differentbond numbers. The definition of the bond number [1] is given in Eq. (12), which can bedescribed as the ratio of gravity to surface tension

Bo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2gðql � qvÞ

r

r: (12)

Figure 16. Distributions of volume fraction in PHP with 10 turns.

776 J. CHOI AND Y. ZHANG

The bond numbers are set to 0.0, 1.0, and 2.0, respectively, and their results are shown in Figures13–15. Figure 13 demonstrates that when the bond number is 0.0, there is no gravity effect, sothe working fluid in the upper condenser section does not move to the lower evaporator section.This is the same as the phenomenon that occurs in a horizontal mode. The PHP cannot carryout the heat transfer because the working fluid does not circulate. Unlike the bond number of0.0, Figure 14 shows that when the bond number is 1.0, the working fluid moves to the lowerevaporator section. Figure 15 depicts that when the bond number is 2.0, the upper part of thebubble is asymmetrically shaped against the lower part of the bubble at the top of the condensersection because the influence of gravity is greater than that of surface tension. Furthermore, it isobserved that the film of the working fluid flows down from the wall surface of the condensersection due to the influence of gravity. For reference, the bond number of 1.18 is used inthis study.

5.3. Effects of the number of turns on the working fluid flow

In the above simulations, when the bond number is 0.0 for the PHP with five turns, the dry-out phenomenon occurs, that is, all the working fluid evaporates in the evaporator section.Under the same conditions, the behaviors of the working fluids are investigated in the PHPwith 10, 15, and 20 turns, respectively. Figure 16 shows the case with 10 turns, where theworking fluid in the lower evaporator section almost evaporates with time. It indicates the factthat the behavior in the PHP with 10 turns is similar to that with five turns. Figures 17 and18 show the numerical results for the PHPs with 15 and 20 turns, respectively. Unlike the caseswith 5 and 10, the working fluid does not evaporate entirely, remaining in the evaporatorsection. These results are close to the critical number of turns of 16 presented in the work byCharoensawan et al. [2].

Figure 17. Distributions of volume fraction in PHP with 15 turns.

NUMERICAL HEAT TRANSFER, PART A: APPLICATIONS 777

Figure 18. Distributions of volume fraction in PHP with 20 turns.

778 J. CHOI AND Y. ZHANG

6. Conclusion

To visualize the behavior of the working fluid in the two-dimensional PHPs with multi-turns, thenumerical simulation was carried out using OpenFOAM. The liquid phase and the vapor phaseof the working fluid were assumed to be incompressible. In the phase change process, the masstransfer parameter for the condensation was determined by the mass balance between the evapor-ation and the condensation.

The heat flux boundary condition for the two-phase fluid was added to the original solver.The average temperature on the evaporator section and the thermal resistance were obtained withvarious heat fluxes. The thermal resistance decreased as the heat flux increased.

The circulation of the working fluid in the PHP with the asymmetry structure started earlierthan that in the PHP with the symmetry structure. When the numbers of turns were 5 and 10with the bond number of 0.0, the dry-out phenomenon occurs, and the working fluid evaporatedin the lower evaporator section. However, when the numbers of turns are 15 and 20 with thesame bond number, the dry-out phenomenon did not appear, and the working fluid remained.

It is expected that other researchers will be able to use the numerical analysis process in devel-oping the advanced numerical code for PHP. Also, it is expectable for them to be able to opti-mize the PHP configurations by comparing the relative values of the thermal resistance.

ORCID

Jongwook Choi http://orcid.org/0000-0002-2335-5758Yuwen Zhang http://orcid.org/0000-0001-8915-1769

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