Laminar Flow Forced Convection Heat Transfer Behavior …people.tamu.edu/~jorge.alvarado/publications/Laminar Flow... · 2013-09-24 · Convection Heat Transfer Behavior of a Phase

Satyanarayana KondleDepartment of Mechanical Engineering,

Texas A&M University,

College Station, TX 77843

Jorge L. Alvarado1

Department of Engineering Technology

and Industrial Distribution,

Texas A&M University,

3367 TAMU,

College Station, TX 77843-3367

e-mail: [email protected]

Charles MarshEngineer and Research Development Center,

U.S. Army Corps of Engineers,

Champaign, IL 61822-1076

Laminar Flow ForcedConvection Heat TransferBehavior of a Phase ChangeMaterial Fluid in MicrochannelsIn this paper, a phase change material (PCM) fluid (N-eicosane) is compared with purewater as heat transfer fluid. The heat transfer behavior of PCM fluid under laminar flowconditions (Reynolds number of 700) in circular and rectangular microchannels wasstudied numerically. In the numerical study, an effective specific heat model was used totake into account the phase change process. Heat transfer results for circular and rectan-gular microchannels with PCM fluid were obtained under hydrodynamically and ther-mally fully developed conditions. A PCM fluid in microchannels with aspect ratios of 1 to2, 1 to 4, and 1 to 8 was found to enhance the thermal behavior of microchannels whichcan be beneficial in a host of cooling applications. The flow was assumed to be hydrody-namically fully developed at the inlet and thermally developing inside the microchannel.Heat transfer characteristics of PCM slurry flow in microchannels have been studiedunder three types of wall boundary conditions including constant axial heat flux with con-stant peripheral temperature (H1), constant heat flux with variable peripheral tempera-ture (H2), and constant wall temperature (T) boundary condition. The fully developedNusselt number was found to be higher for H1 than for H2 and T boundary conditions forall the geometries. Moreover, Nusselt number also increased with aspect ratio and wassensitive to the variations in effective specific heat. [DOI: 10.1115/1.4023221]

Keywords: phase change material fluid, microchannel, convective heat transfer, constantheat flux, constant wall temperature, effective specific heat

1 Introduction

PCM heat transfer fluids continue to capture the attention ofresearchers in industry and academia due to their unique thermalproperties as novel coolants for a host of heat transfer applica-tions. Such applications include electronics cooling and other rele-vant thermal management systems. Experiments have shown thatPCM fluids exhibit an enhanced heat capacity due to the phasechange material’s latent heat of fusion upon melting. However,little is known how such fluids behave in different heat exchangerconfigurations under laminar flow conditions.

Heat transfer in microchannel flow has been an important areaof research in the heat transfer community in the last decade.Modern electronics equipment requires large heat flux removalfrom small areas, and microchannels have been found to beeffective in such applications. Many geometric configurationswith different boundary conditions have been studied with singlephase fluids [1–8]. Also, PCM fluids have been studied in vari-ous configurations at the macroscale [9–15]. However, the use ofPCM fluids in microchannels has not yet been studied thoroughly[16–18].

Xing et al. [16,17] did an extensive study and evaluated the per-formance of PCM fluid in circular microchannels. They modeledthe particle flow separately from the mean flow using source termsin the momentum and energy equations. They also consideredparticle–particle interactions and particle-depletion layer effectsnear the wall. Sabbah et al. [18] performed a computational fluiddynamics (CFD) analysis of PCM fluid in microchannels withthick walls taking into account the conjugate nature of the heat

transfer process. Results from the previous efforts on PCM fluidshave been promising with the main advantage of achieving lowerwall temperature than conventional single phase fluids under thesame heat flux conditions.

Lee et al. [1] performed numerical and experimental testing offluid flow in microchannels with width between 194 lm and534 lm with water in the Reynolds number range between 300and 3500. They showed good agreement between numerical andexperimental results with an average difference of about 5%. Weiet al. [4] performed numerical and experimental study of stackedmicrochannels. They considered both counter flow and parallelflow arrangements to ensure temperature uniformity and lowertemperatures. The results showed that the counter flow arrange-ment results in more even temperature distribution, whereas theparallel flow type results in lower temperatures. Stacked micro-channels also resulted in a lower pressure drop.

Lee et al. [1] did an extensive study of fluid flow and heat trans-fer through rectangular microchannels with different aspect ratios,similar to the first step performed in the current study. They con-sidered two boundary conditions (H1 and T) and came up withcorrelations for Nusselt number based on the channel aspect ratio.Their results were very helpful when taking into account H1 con-dition numerically. Their approach and results were used to vali-date the current approach for different microchannel aspect ratiosconsidered in this PCM fluid case study.

Most of the studies in the area of PCM fluids have been limitedto regular heat transfer section including circular pipes. Yama-gishi et al. [10] studied the basic characteristics of microencapsu-lated PCM slurries as heat transfer fluids. They used slurry ofwater containing C18H36 particles with a volume fraction of about0.3 in their experiments. Their research revealed that the heattransfer coefficient increases while the phase change materialundergoes phase change. Alvarado et al. [19] performed a seriesof heat transfer experiments with microencapsulated phase change

1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the

JOURNAL OF HEAT TRANSFER. Manuscript received June 1, 2011; final manuscriptreceived October 17, 2012; published online April 11, 2013. Assoc. Editor: DarrellW. Pepper.

Journal of Heat Transfer MAY 2013, Vol. 135 / 052801-1Copyright VC 2013 by ASME

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material (MPCM) slurries using a circular heat transfer sectionunder turbulent flow conditions. They measured heat transfercoefficient, latent heat of fusion, melting and freezing points, andviscosity of the MPCM slurries as heat transfer fluids. Their ex-perimental results confirmed that MPCM slurries behave as New-tonian fluids for concentrations below 18.0%.

Chen et al. [11] performed experiments using a circular tubeunder constant heat flux with slurry containing water as carrierfluid and bromohexadecane as PCM. Their experiments showedthat PCM slurries are a very promising alternative as heat transferfluids compared to water. When the PCM slurry is used, theyfound that the average wall temperature decreased by 30% andthe heat transfer enhancement was 40% compared to water. Thepressure drop of the PCM fluid was found to be 67% lower thanwater. Their results were used to validate the phase change modelused in the current study. Recently, Sabbah et al. [20] investigatedthe heat transfer characteristics of MPCM slurry flowing in aheated tube. Results indicate that MPCM can enhance the heattransfer coefficient by as much as 50% due to the latent heat offusion of the phase change material.

One of the difficulties when numerically simulating MPCMslurries is the readiness and availability of reliable, effective, andcomputationally inexpensive numerical approaches that canaccount for tens and hundreds of thousands of almost neutrallybuoyant PCM particles as small as 5 lm. Moreover, the specificgravity of PCM particles is usually in the range of 0.8–0.9, whichmakes numerical simulation even more daring and difficult. Toundertake numerical simulation of PCM fluids in heat transfersections, an effective specific heat model developed by Roy andAvanic [12] has been shown to account for the PCM melting pro-cess accurately. Their model takes into account the latent heat offusion and specific heat of the PCM particles, as well as the car-rier fluid properties in a simple mathematical model. Before theirmodel was developed, most of the researchers simulated PCMflow by using complicated source terms into the energy equation.Roy and Avanic [12] compared several specific heat models, andcame to the conclusion that accounting for the latent heat offusion of each particle was not very important as long as themelting range of the phase change material in the model was cho-sen carefully. They simulated PCM slurry flow in a circular ductwith constant wall temperature using a specific heat model, andfound that the model predicts the heat transfer behavior reason-ably well and accurately. The simple linear specific model wasused in the current study to simulate PCM slurry flow in circularchannel and microchannels with constant heat flux. They showedthat the results using the specific heat model were very close tothe experimental results.

There are several other studies on the heat transfer enhancementof fluids when PCMs are used. Chen et al. [11] experimentallyinvestigated PCM fluid flow characteristics for various concentra-tions of PCM, ranging from 5% to 27.6%. However, there are veryfew publications in which PCM flow was studied in microchan-nels. Hao and Tao [17] developed a numerical model and did anextensive study on PCM slurry flow through circular microchannelunder constant heat flux condition. Their results are significantbecause they solved the particle motion independently of the fluidmotion using separate momentum equations. They also consideredthe effects of particle-depleted layer, particle–wall interaction,interparticle interaction along with solving separate mass, momen-tum, and energy equations. Sabbah et al. [18] considered three-dimensional conjugate heat transfer through microchannels usingPCM slurry. Their results showed that lower and more uniformwall temperatures could be obtained in the case of PCM slurrycompared with water as heat transfer fluid.

More recently, Ravi et al. [9] numerically studied the perform-ance of PCM slurry flow in finned tubes. They found that thespecific heat model gave good results comparable with experi-mental results available in the literature. They presented theresults using PCM fluid for different fin heights, and differentboundary conditions (H1 and T). The study showed that PCM

fluid significantly enhances heat transfer in finned tubes. Dammeland Stephan [21] studied the heat transfer behavior of MPCMslurries in a rectangular minichannels both experimentally andnumerically. The study reveals that the effects of viscosity andthermal conductivity should also be considered in any thermalanalysis of MPCM slurries.

A thorough literature review suggests that not much researchhas been done with PCM fluids in microchannels. However, at themacro scale, a good amount of data is available in the literature onPCM slurry performance as heat transfer fluid. Hence, this studyis very important and complements the available literature onPCM fluid flow particularly in microchannels.

2 Specific Heat Model

Simulating the complete phase change process of phase changematerials as particles is difficult with the currently available multi-phase commercial software and models. Moreover, simulationschemes that account particle behavior besides the melting pro-cess are computationally expensive. Two types of effective mod-eling approaches were found in the literature. One uses a heatsource term in the energy equation, while the other considers aspecific heat model. While both methods are found to be effective,the use of the specific heat model is simpler and easier to imple-ment into a computer code. The latter approach was used in thecurrent simulation which was already found effective for similarapplications [12].

The model assumes that the phase change of PCM can beapproximated accurately by changes in the specific heat of thebulk fluid within the melting temperature range of the phasechange material. Ravi et al. [9] found that this model gives resultsthat are comparable with experimental data. The effective specificheat model is simple and straight forward to implement in com-mercial software like FLUENT. The following equations were usedto account for the effective specific heat:

For T< T1 or T> T2 [9]

Cp;b ¼ c � Cp;p þ 1� cð Þ � Cp;f (1)

For T1< T< T2 [9]

Cp;b ¼ 1� cð Þ � Cp;f þc � Lh

T2 � T1ð Þ (2)

where T1 and T2 are the temperatures at which the phase changeprocess starts and ends, respectively. Lh and c are the latent heatof fusion of the phase change material and phase change materialconcentration, respectively. The model was incorporated intoFLUENT as a user defined function and allowed for variation in spe-cific heat as a function of temperature at the local level. Eventhough Eqs. (1) and (2) are basically step functions, they havebeen proven to be effective when taking into account the heattransfer behavior of PCM fluid [9].

3 Geometry and Mesh

Once the specific heat model was incorporated in the numericalscheme, PCM fluid through rectangular microchannels with aspectratios of 1:2, 1:4, and 1:8 were simulated. The height of the micro-channel was taken as 150 lm and the width was varied to get thedesired aspect ratio. Except for the H1 condition (constant axialheat flux with constant peripheral temperature), the wall thicknessof the microchannel was neglected and considered to have zerothickness.

Taking advantage of symmetry, only a quarter of the flow do-main was simulated numerically as seen in Fig. 1(a). Two adja-cent sides were specified as no-slip walls, and the other two werespecified as symmetric, having no heat flux or flow perpendicularto them as shown in Fig. 1(a).

052801-2 / Vol. 135, MAY 2013 Transactions of the ASME


GAMBIT (2.3) was used to create the geometry and mesh the do-main. The geometry was meshed using hexahedral elements. Themesh consisted of finer elements near the wall to capture the largetemperature gradients normal to flow direction. The mesh near theentrance region was also made finer to account for the entry lengtheffects. Figure 1(b) shows the typical grid near the inlet of themicrochannel. Three different grid resolutions, 10� 20� 100,15� 30� 1000, and 15� 30� 1500 were used first to make surea grid-independent solution could be found. The results for thesegrids are presented in Sec. 6.

4 Phase Change Material Fluid Properties

In the simulation study, water was taken as the carrier fluid witha PCM volume concentration of 15%. N-eicosane was selected asphase change material with properties listed in Table 1. Water prop-erties were assumed to be constant with temperature. The bulk fluid(waterþPCM) properties were calculated using the equationsgiven below. The latent heat of PCM particles was assumed to be230 kJ/kg [15], and the volumetric concentration in the bulk fluidwas 15%. The exact melting point of N-eicosane is 310 K. For thecurrent work, the melting range of PCM (N-eicosane) was assumedto be 309–310 K for stability and convergence of the numericalsolution.

Viscosity [12]:

lb

lf

¼ 1� c� 1:16c2� ��2:5

(3)

Density [12]:

qb ¼ c � qp þ 1� cð Þ � qf (4)

Thermal conductivity [12]:

kb ¼ kf �2þ kp

kfþ 2c

kp

kf� 1

� �

2þ kp

kf� c

kp

kf� 1

� � (5)

It is important to highlight that the PCM fluid was modeled as aliquid all the time, with the properties of the PCM particlesaccounted for using Eqs. (1)–(5). This simplified the simulationprocess considerably since the fluid behavior of the PCM particlesdid not have to be taken into account. This assumption holds wellfor PCM particles as small as 5 lm [11,12].

5 Simulation Procedure

As a part of the simulation procedure, the flow at the inlet ofthe microchannel was hydrodynamically developed. All the simu-lations were performed assuming a Reynolds number of 700unless otherwise specified. First, a simple case with no heat trans-fer was run, and the fully developed velocity profile was extractedat the outlet. The extracted flow velocity profile then was appliedas inlet boundary condition (i.e., hydrodynamically developed ve-locity profile). Inlet temperature of the fluid was assumed to beconstant (298 K).

H2 (constant heat flux with variable peripheral temperature)[22] and T (constant wall temperature) boundary conditions werespecified at the wall. However, taking into account H1 boundarycondition (constant axial heat flux with constant peripheral tem-perature) was more difficult to impose as this boundary conditionis usually not directly available in most of the commercial CFDcodes. Garimella and Lee [8] explained how this condition can betaken into account in FLUENT using a thin wall conduction modelalong with the shell conduction option applied at the wall

Fig. 1 (a) Diagram showing the geometry cross section and the portion modeled inFLUENT. (b) Mesh near the inlet to the microchannel.

Table 1 Carrier fluid and PCM physical properties

Fluid k (W/m K) Viscosity (cP) Density (kg/m3) Cp (kJ/kg K)

Water 0.613 0.855 997 4.179PCM 0.15 Not applicable 946.4 1.973Bulk 0.525 1.387 989.4 3.862

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boundary. The “thin wall” model (with shell conduction enabled)lets the heat applied at the outer boundary of the wall spreadthrough the wall material in the direction normal to the flow. Thisredistribution of heat flux facilitates achieving uniform tempera-ture along the periphery of the channel and hence the H1 bound-ary condition can be satisfied. Along with this boundarycondition, a large value of thermal conductivity for the wall mate-rial (Al) was chosen to ensure the H1 condition. This kind ofboundary condition, though not straight forward, was found to bein best agreement with experimental results under constant heatflux boundary conditions as observed by Garimella and Lee [8].

5.1 Governing Equations and Solution Procedure. Thegoverning equations for flow and heat transfer were solved usingthe commercial CFD solver FLUENT (6.3.26). FLUENT uses the finitevolume method to discretize the domain to solve convective heattransfer problems. The governing equations, continuity, momen-tum, and energy equations [23] solved by FLUENT are shownbelow, respectively.

r � ~vð Þ ¼ 0 (6)

Table 2 Friction factors for various geometries

Geometry f*Re—current simulation f*Re—literature [24]

Circular 16 161:2 15.5 15.61:4 18.1 18.31:8 20.6 20.7

Table 3 Nusselt number with T boundary condition for singlephase fluid (water)

Geometry

Nu—simulationunder T boundary

conditionNu—literature

[24]

Circular 3.67 3.661:2 3.388 3.391:4 4.45 4.441:8 5.62 5.95

Table 4 Nusselt number with H1 boundary condition for singlephase fluid (water)

Geometry

Nu—simulationunder H1 boundary


[24]

Circular 4.40 4.361:2 4.1 4.111:4 5.29 5.351:8 6.41 6.6

Table 5 Nusselt number with H2 boundary condition for singlephase fluid (water)

Geometry

Nu—simulationunder H2 boundary


[25]

1:2 3.02 3.071:4 2.94 3.031:8 2.94 3.03

Fig. 2 (a) Grid independence test for 1:2 geometry under H2boundary condition using water. (b) Grid independence testfor 1:4 geometry under H1 boundary condition using PCMfluid.

Fig. 3 (a) Nusselt number variation for various aspect ratioswith H1 boundary condition using water. (b) Nusselt numbervariation for various aspect ratios with H2 boundary conditionusing water.



r � q~v~vð Þ ¼ �rpþr � s� �

(7)

r � ~v qEþ pð Þð Þ ¼ kr � rT þr � s �~v� �

(8)

The default under-relaxation factors of 0.3 for continuity, 0.7for momentum (0.7), and 1.0 for energy were used to get theproper numerical solution. The residual limit for convergence wasset as 10�6 for all the equations. Initially, each solution wasobtained with first-order discretization. Then, the first-order con-verged solution was taken as an initial guess to get a convergedsolution with second-order accuracy.

The Nusselt number and other parameters were based on thefollowing equations [24]:

Nu ¼ hDh

kb

(9)

where Dh is hydraulic diameter. The local heat transfer coefficientwas found using the following equation:

h ¼ q00

Tw � Tbð Þ (10)

where q00, Tw, and Tb are the average wall heat flux along the pe-riphery, average wall temperature, and the mass weighted averageof fluid temperature on the cross section at a particular axial loca-tion, respectively.

Fig. 4 Nusselt number variation for a circular channel underH2 boundary condition. Experimental results by Chen et al.[11].

Fig. 5 (a) Centerline temperature variation for 1:2 geometryunder H1 boundary condition. (b) Centerline temperature varia-tion for 1:4 geometry under H1 boundary condition. (c) Center-line temperature variation for 1:8 geometry under H1 boundarycondition.

Fig. 6 (a) Centerline temperature variation for 1:2 geometryunder H2 boundary condition. (b) Centerline temperature varia-tion for 1:4 geometry under H2 boundary condition. (c) Center-line temperature variation for 1:8 geometry under H2 boundarycondition.



The mass weighted average of fluid temperature (Tb) was foundas follows:

Tb ¼Pn

i¼1 Tiqi ~vi � ~Ai

�� Pn

i¼1 qi ~vi � ~Ai

�� (11)

Nondimensional distance was calculated using

zþ ¼ z

r � Re � Pr(12)

where

r ¼ Dh

2(13)

with Re, Pr, and Z representing Reynolds number, Prandtl num-ber, and the nondimensional axial distance, respectively. Equa-tions for Prandtl number, Reynolds number, and friction factor(based on pressure drop) are provided below

Pr ¼ Cp;b � lb

kb

(14)

Re ¼ qVDh

l(15)

f ¼ Dh � Dp

2LqV2(16)

6 Results and Discussion

First, a single phase fluid (water) was considered to ensureaccurate numerical results. The fully developed Nusselt Numberand pressure drop were compared with those available in theliterature for validation of the different boundary conditions.Table 2 depicts the simulated and published pressure droplosses (i.e., f-Re) for different microchannel configurations.Tables 3–5 show Nusselt number values under T, H1, and H2boundary conditions, respectively, for different microchannelconfigurations.

It can be clearly seen that the results match very well with theexpected literature values. It should be noted that there is no differ-ence between H1 and H2 boundary conditions for a circular micro-channel or duct due to perfect symmetry. Also, it can be observed(Table 5) that the Nusselt number for H2 boundary conditionremains constant for various aspect ratios. This is because the 1:2geometry or aspect ratio is the limiting case for the H2 boundarycondition [25–27], which can be found using Eq. (17) given below.From the equation, it can be found that the Nusselt number variessignificantly for aspect ratios from 0.5 to 1.0, but for aspect ratioslower than 0.5, its value remains between 2.8 and 3.4.

Fig. 7 (a) Centerline temperature variation for 1:2 geometryunder T boundary condition. (b) Centerline temperature varia-tion for 1:4 geometry under T boundary condition. (c) Center-line temperature variation for 1:8 geometry under T boundarycondition.

Fig. 8 (a) Average fluid temperature variation for 1:2 geometryunder H1 boundary condition using PCM fluid. (b) Averagefluid temperature variation for 1:4 geometry under H1 boundarycondition using PCM fluid. (c) Average fluid temperature varia-tion for 1:8 geometry under H1 boundary condition using PCMfluid.



Nusselt number for H2 boundary condition can be found as fol-lows [25]:

NuH2 ¼ 8:235 � 1� 10:6044aþ 61:1755a2�

�155:1803a3 þ 176:9203a4 � 72:9236a5Þ (17)

where a is the ratio of the cross section dimensions (height/width).

Three different grids of varying resolution were considered forPCM fluid simulation including 10� 20� 1000, 15� 30� 1000,and 15� 30� 1500. The overall solution varied within 2% forthese three grids. Figures 2(a) and 2(b) show that the solutions forthe three grids yield the same solution (all the curves overlap eachother), so a 10� 20� 1000 grid was found to be sufficient to pro-vide grid-independent results for single phase fluid and PCM fluid,respectively. Figure 2(b) shows the results of using three differentgrids when simulating heat transfer in a microchannel with PCMfluid flowing through it.

Once the formulated numerical scheme was validated usingknown results, the effect of using microchannels with different as-pect ratio was evaluated using water only as the heat transfer fluid.Figure 3(a) shows how Nusselt varies with aspect ratio and axialdistance in the case of a single phase fluid (water) under H1boundary condition. Figure 3(b) shows similar results when H2boundary condition is imposed. These results are consistent with

previous data [24] and show the effect of aspect ratio on the com-pressed temperature gradient of the fluid which gives rise tohigher Nusselt number values specifically under H1 boundaryconditions. Under H2 boundary conditions, all the three casesreached the same value indicating that the temperature gradientwithin the fluid is sensitive to variations of peripheral surface tem-perature at each axial location. This also suggests that the thermalconductivity of the surface can play a critical role in convectiveheat transfer enhancement. Also, the nondimensional distancerequired to obtain fully developed flow is greater in the case of H2boundary condition when compared to H1 boundary condition.

6.1 Results for PCM Fluid Flow. PCM fluid flow in circularmicrochannel was simulated under H2 boundary condition firstand the results were compared with the experimental findings byChen et al. [11]. Figure 4 shows the simulation results with exper-imental data. It is important to note that in case of a circularmicrochannel, there is no difference between H1 and H2 boundaryconditions since the surface temperature does not vary along theperiphery (at a fixed axial distance) due to perfect symmetry.

Chen et al. [11] calculated the bulk temperature of the PCMfluid across a cross section using a simple energy balance equation,and assuming that all the PCM melting process starts at a particularaxial distance (independent of radial distance) and ends at a partic-ular axial location. In the current work, this assumption was not

Fig. 9 (a) Fluid temperature variation for 1:2 geometry underH2 boundary condition using PCM fluid. (b) Fluid temperaturevariation for 1:4 geometry under H2 boundary condition usingPCM fluid. (c) Fluid temperature variation for 1:8 geometryunder H2 boundary condition using PCM fluid.

Fig. 10 (a) Fluid temperature variation for 1:2 geometry underT boundary condition using PCM fluid. (b) Fluid temperaturevariation for 1:4 geometry under T boundary condition usingPCM fluid. (c) Fluid temperature variation for 1:8 geometryunder T boundary condition using PCM fluid.



necessary as the exact temperatures at each location on the crosssection were found during the simulation process. Figure 4 showsthat the numerical results can predict variations in Nusselt numberwith axial distance relatively well. The next step was to assess theeffect of boundary conditions (i.e., H1, H2, and T) on the fluidtemperature in the axial direction.

Fluid temperatures along the centerline in the flow direction areshown in Figs. 5(a)–5(c) for H1 boundary condition. The resultsclearly show the phase change process affects fluid temperaturewhen compared to a single (bulk) fluid. During the melting pro-cess, the latent heat of fusion of PCM becomes available allowingfor a significant reduction in the axial temperature gradient. Bylooking at Figs. 5(a)–5(c), it is clear that aspect ratio has a signifi-cant effect when the melting process starts under H1 boundaryconditions. On the other hand, Figs. 6(a)–6(c) show that H2boundary conditions have a significant effect on the meltinglength or axial distance required for complete melting.

When T boundary condition is imposed, the temperature rise isno longer linear as seen in Figs. 7(a)–7(c). This is because underconstant temperature boundary condition, the wall heat fluxdecreases along the flow direction and eventually becomes zero asthe fluid temperature equals that of wall (350 K). The temperaturevariation is clearly asymptotic as shown in Figs. 7(a)–7(c) for allaspect ratios. However, higher aspect ratio leads to a faster rise inaxial temperature as depicted in the figures. Figures 8(a)–8(c)

show the average fluid temperature variation along the length ofthe microchannel for PCM fluid under H1 boundary condition.The average fluid temperature at a specific cross section is themass weighted average of temperature as given by Eq. (11).

It can be seen from the figures that there is significant reductionin the fluid temperature when using PCM fluid compared to thebulk fluid (water). Moreover, higher aspect ratio leads to a fastermelting process. Similar behavior for H2 boundary condition is

Fig. 11 (a) Nusselt number variation for 1:2 geometry under H1boundary condition using PCM fluid. (b) Nusselt number varia-tion for 1:4 geometry under H1 boundary condition using PCMfluid. (c) Nusselt number variation for 1:8 geometry under H1boundary condition using PCM fluid.

Fig. 12 (a) Nusselt number variation for 1:2 geometry under H2boundary condition using PCM fluid. (b) Nusselt number varia-tion for 1:4 geometry under H2 boundary condition using PCMfluid. (c) Nusselt number variation for 1:8 geometry under H2boundary condition using PCM fluid.

Fig. 13 Temperature variation along the periphery of the wallfor 1:4 geometry under H2 boundary condition using PCM fluid



shown in Figs. 9(a)–9(c). However, H2 boundary condition has asignificant effect on the length of the melting process in terms ofaxial distance (Zþ). In the case of H2 boundary condition, the pe-ripheral surface temperature is not constant which induces a dif-ferent temperature gradient within the fluid. Similar to theobservations of centerline temperature, the average fluid tempera-ture variation for the T boundary condition is different from thatof H1 and H2 boundary conditions. The temperature variation isasymptotic as shown in Figs. 10(a)–10(c).

Figures 11(a)–11(c) show Nusselt number for rectangularmicrochannels of three different aspect ratios using PCM fluidflow under H1 boundary condition. The dashed lines show the

Nusselt number for the bulk (single phase) fluid (without phasechange, constant specific heat), and the solid lines show the Nus-selt number of the PCM fluid. The results show that higher aspectratio leads to higher Nusselt number. It also accelerates the melt-ing process significantly. During the phase change process, theNusselt number increases drastically and then becomes smallerthan for a single fluid until the melting process is complete. Thevariation in Nusselt number during the phase change process canbe attributed to a significant variation in temperature gradientwithin the fluid which is intensified by increases in aspect ratio.

Figures 12(a)–12(c) show Nusselt number for rectangularmicrochannels of three different aspect ratios using PCM fluidflow under H2 boundary condition. The dashed lines show theNusselt number for bulk fluid (without phase change) and thesolid lines show the Nusselt number of the PCM fluid. In thiscase, a higher aspect ratio leads to greater Nusselt number (likeunder H1 boundary conditions), but a delay on the onset of melt-ing in terms of axial distance, unlike the behavior seen under H1boundary conditions. As discussed above, H2 boundary conditionhas a unique effect on the temperature gradient within the fluidthat leads to a different convective heat transfer behavior.

This can be understood by knowing how the wall temperaturealong the periphery varies under H2 boundary condition. Figure13 shows variation of wall temperature under H2 boundary condi-tion for 1:4 aspect ratio when using PCM fluid. Figure 13 showstemperature variation along only one quarter of the whole sectionof the wall since symmetry was used for simulation purposes.Position 1 in the figure shows temperature at the center of the lon-ger edge, position 2 in the figure shows temperature at the cornerof the wall, and position 3 shows temperature at the center of theshorter edge. Variation in surface temperature is expected to havea direct effect on the temperature gradient of the fluid. The sameeffect is not possible under H1 boundary condition since the pe-ripheral temperature in constant at each axial location.

Figures 14(a)–14(c) show the Nusselt number for rectangularmicrochannels of three different aspect ratios using PCM fluidunder T boundary condition. The solid lines show the Nusseltnumber for bulk fluid (without phase change, constant specificheat) and the dashed lines show the Nusselt number of the PCMfluid. Figures 14(a)–14(c) show the variation in Nusselt numberalong the flow direction under T boundary condition. In this case,the phase change process and the thermal boundary layer develop-ment take place simultaneously. It is difficult to separate the phasechange process from the boundary layer development for such aboundary condition. It can be observed from the plots that theNusselt number increases due to the phase change process in allcases.

For all the H1 and H2 boundary condition cases, the Nusseltnumber of the PCM fluid was found to reach a maximum first andthen decreased until it reached a minimum. This behavior wasalso observed in PCM flow through finned tubes by Ravi et al. [9].This can be explained by understanding the PCM melting processunder both boundary conditions. The melting process of PCMstarts at the wall and then slowly spreads through the cross sectionas shown in Fig. 15. At the beginning, only the region near thewall experiences phase change and hence the temperature near the

Fig. 14 (a) Nusselt number variation for 1:2 geometry under Tboundary condition using PCM fluid. (b) Nusselt number varia-tion for 1:4 geometry under T boundary condition using PCMfluid. (c) Nusselt number variation for 1:8 geometry under Tboundary condition using PCM fluid.

Fig. 15 Phase change process in a circular channel [25]



wall remains constant, whereas the bulk fluid temperature keepsincreasing. Due to this, the difference between the wall tempera-ture and the bulk temperature decreases, increasing the heat trans-fer coefficient and hence increasing the Nusselt number.

At the end of phase change process, all the phase change occursonly at the center of the cross section. Hence, the bulk temperatureremains constant (or slowly increases), whereas the wall tempera-ture keeps increasing. Due to this, the difference between wall andbulk temperature increases, decreasing the heat transfer coeffi-cient and thus decreasing the Nusselt number significantly.

7 Conclusions

A numerical study has been undertaken to understand the effectof microchannel aspect ratio and heat transfer fluid characteristics(single phase or PCM) on convective heat transfer. The followingspecific conclusions can be drawn from the study:

• The fully developed flow Nusselt number is higher for H1boundary condition than for H2 or T boundary conditions forall aspect ratios considered in the study. This is attributed tothe effect of peripheral surface temperature on the tempera-ture gradient within the fluid.

• The Nusselt number was found to increase when using PCMfluid compared to a single phase fluid (water).

• The specific heat model predicts the heat transfer behavioraccurately when compared with experimental results.

• The trend in Nusselt number during the phase change processwas found to be same for both H1 and H2 boundaryconditions.

• For H1 and H2 boundary conditions, Nusselt number wasfound to increase during the beginning of phase change pro-cess and dropped (lower than for the bulk) considerably asthe phase change process was near completion.

• The axial distance required to obtain fully developed flow ismuch higher for H2 boundary condition compared to H1boundary condition when taking into account the phasechange process of the PCM fluid. This is attributed to theinduced temperature gradient within the fluid under H1boundary condition.

• When using PCM fluid, the difference between the peakvalue and fully developed values of Nusselt number is higherfor H2 boundary condition than for H1 boundary condition.

The numerical simulations demonstrate that PCM fluid canindeed enhance the heat transfer performance of microchannelswith different aspect ratios. PCM fluid tends to perform betterunder constant heat flux condition than under constant wall tem-perature. Moreover, it is clear that the type of boundary conditionhas a significant effect on convective heat transfer behavior thanthe microchannel’s aspect ratio. This suggests that microchannelmaterial selection is important since low and thermal conductivitymaterials can be used to impose either a H1 or H2 boundary con-ditions under constant heat flux.

Acknowledgment

The authors would like to thank Gurunarayana Ravi for his val-uable advice and guidance throughout the study.

Nomenclature

c ¼ volumetric concentration of PCMCp ¼ specific heat of bulk fluid (slurry)

f ¼ friction factorh ¼ convective heat transfer coefficientk ¼ thermal conductivityL ¼ distance along the axial direction

Lh ¼ latent heat of PCM particlesNu ¼ Nusselt number

p ¼ pressure

q00 ¼ heat fluxT ¼ temperature~v ¼ velocityl ¼ viscosityq ¼ density

Zþ ¼ nondimensional axial distance

Subscripts

f ¼ property of the carrier fluidp ¼ property of the PCM particle (N-eicosane)b ¼ property of the bulk fluid (PCMþwater)

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