Enhancement of Turbulent Convective Heat Transfer Using a Microparticle Multiphase Flow

energies

Article Enhancement of Turbulent Convective Heat Transfer using a Microparticle Multiphase Flow

Tao Wang 1,2,*, Zengliang Gao 1,* and Weiya Jin 1 1 Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou 310032, China; [email protected] 2 Institute of Mechanical Engineering, Quzhou University, Quzhou 324000, China * Correspondence: [email protected] (T.W.); [email protected] (Z.G.)

Received: 12 February 2020; Accepted: 9 March 2020; Published: 10 March 2020

Abstract: The turbulent heat transfer enhancement of microfluid as a heat transfer medium in a tube was investigated. Within the Reynolds number ranging from 7000 to 23,000, heat transfer, friction loss and thermal performance characteristics of graphite, Al2O3 and CuO microfluid with the particle volume fraction of 0.25%–1.0% and particle size of 5 µm have been respectively tested. The results showed that the thermal performance of microfluids was better than water. In addition, the graphite microfluid had the best turbulent convective heat transfer effect among several microfluids. To further investigate the effect of graphite particle size on thermal performance, the heat transfer characteristics of the graphite microfluid with the size of 1 µm was also tested. The results showed that the thermal performance of the particle size of 1 µm was better than that of 5 µm. Within the investigated range, the maximum value of the thermal performance of graphite microfluid was found at a 1.0% volume fraction, a Reynolds number around 7500 and a size of 1 µm. In addition, the simulation results showed that the increase of equivalent thermal conductivity of the microfluid and the turbulent kinetic energy near the tube wall, by adding the microparticles, caused the enhancement of heat transfer; therefore, the microfluid can be potentially used to enhance turbulent convective heat transfer.

Keywords: heat transfer enhancement; multiphase ﬂow; microﬂuid; Eulerian–Eulerian model

1. Introduction The viscous layer is dominated by heat conduction, which hinders the heat transfer in a tube. It has been found that nanofluid can enhance heat transfer. Typical nanoparticles of metals, such as Al and Cu, and metal oxides such as TiO2, Fe3O4, Al2O3 and CuO, have been used for nanofluid [1–8]. Further, nanofluids have also been used with other enhancing devices for the heat transfer process, such as wire coils, twisted tape and roughened heat exchanger surfaces [9–13]. Recently, the combination of nanofluid, inserts and roughened heat exchanger surfaces has been reported [14,15]. Although nanofluids can improve heat transfer inside a tube at a low particle volume fraction, they have some disadvantages. It is difficult to prepare nanoparticles and nanofluid suspensions, and the smaller the particles, the more serious the particle aggregation. Further, some poisonous particles are small and easily diffused into the air and inhaled by the human body [16]. In contrast, the particle size of the microparticles is large, which can alleviate such problems. Chang et al. [17] studied the characteristics of heat transfer of Al-water microfluid with swirl in a circular tube. Their results showed that the Nusselt number can be enhanced by increasing the swirl intensity or Reynolds numbers. Ghorbani et al. [18] investigated the enhancement of the laminar characteristics of convective heat transfer using microparticle suspensions with a Reynolds number ranging from 6 to 24. Their results indicated a significant enhancement of the convective heat transfer coefficient by adding the microparticles. Wang et al. [19] employed a numerical model to study the convective heat transfer of

Energies 2020, 13, 1282; doi:10.3390/en13051282 www.mdpi.com/journal/energies Energies 2020, 13, 1282 2 of 16 a Cu–water microfluid. Their simulation results showed the thermal performance can be improved from 1.1 to 2.3. This paper aims to study the turbulent convective heat transfer of Graphite, Al2O3 and CuO microfluids with a volume fraction of 0.25%, 0.5%, 0.75% and 1.0%, as well as a particle size of 5 µm and Reynolds numbers ranging from 7000 to 23,000. To determine the best heat transfer effect for the microfluid, the influence of particle size on thermal performance was also tested.

2. Experiments

2.1. Microﬂuid Preparation Table1 lists the properties of the purchased microparticles. Firstly, the microparticles were weighed by an electronic scale with an accuracy of 0.001 g, according to the volume fraction of the microparticles. Secondly, graphite, Al2O3 and CuO microparticles were added into the water, measured according to their volume fraction, and stirred evenly by a bar. Finally, the microparticle suspensions were oscillated for at least 45 min by using an ultrasonic vibrator (35 kHz, 120 W) to reduce microparticle aggregation. The ﬁlter papers of 10 µm were used to remove the big aggregated microparticles. The microparticles with a volume fraction of 0.25%, 0.5%, 0.75% and 1.0% were prepared.

Table 1. Parameters of the microparticles.

Parameter Graphite Al2O3 CuO Average grain diameter/µm 5 5 5 Purity/% 99.9 99.9 99.9 Density/kg m 3 2250 3970 6500 · − Speciﬁc heat/J kg 1 C 1 708 765 540 · − ◦ − Thermal 127 25 18 conductivity/W m 1 K 1 · − · −

For the sedimentation analysis of the microfluids, the microfluid was stirred evenly and placed into a graduated cylinder of the specified scale (100 ml). The heights of the clear liquid layer and the sedimentation layer were recorded at different periods during the standing process by visualization. The velocity of sedimentation was calculated from the relationship between the height of the clear liquid layer and time. Table2 shows the velocity of the sedimentation calculated by the experimental data matched with that calculated by the Stokes formula [20] as shown in Equation (1), within an error of 10%.

Table 2. The velocity of sedimentation.

Experimental Velocity Calculated Velocity by the Material Property Error/% /mm s 1 Stokes Formula [17] /mm s 1 · − · − Graphite 0.021 0.0198 6 Al2O3 0.050 0.0469 7 CuO 0.091 0.0866 5

Stokes formula: 2 (ρp ρw)dp g V = − (1) 18µw

2.2. Experimental Device An experimental device was set up to study the turbulent convective heat transfer of the microfluid, which is shown in Figure1. This device consists of two loops, i.e., a hot water loop and a cold fluid loop. A counter-flow heat exchanger was used to link the two loops. There were 4 Pt100 thermal resistance Energies 2020,, 13,, 1282x FOR PEER REVIEW 3 of 16 thermal resistance apparatus to measure the inlet and outlet temperatures with two for the cold water flowapparatus and two to measurefor hot water the inlet flow. and The outlet Pt100 temperatures and the flowmeters with two were for thesepara coldtely water connected flow and to ADAM two for 4015hot water and ADAM flow. The 4117. Pt100 Hot and water, the pumped flowmeters from were the separately hot water connectedtank, was heated to ADAM by a 4015 3 kW and heater ADAM rod at4117. around Hot water,320 K, pumpedthen the fromhot water the hot was water cooled tank, down was by heated flowing by a through 3 kW heater the shell rod at and around eventually 320 K, heatedthen the back hot to water the hot was water cooled tank. down The by cold flowing microflu throughid from the the shell cool and water eventually tank at heatedaround back 300 K to was the heatedhot water by tank.flowing The through cold microfluid the tube; from it then the flowed cool water back tankto the at cool around water 300 tank K was after heated being by cooled flowing in thethrough cooler. the Therefore, tube; it then the flowed inlet backtemperatures to the cool of water the tankhot and after cold being water cooled flows in the were cooler. fixed Therefore, in the experiments.the inlet temperatures The inner of tube, the hot with and an cold inner water diamet flowser of were 12 mm fixed and in themade experiments. of copper, Thehad innera thermal tube, with an inner diameter of 12 mm and made of copper, had a thermal conductivity of 376 W m 1 K 1, conductivity of 376 W·m−1·K−1, while the stainless steel shell had a shell-inner diameter of· 38− ·mm.− whileBoth of the the stainless tube and steel shell shell wall had thicknesses a shell-inner were diameter 2 mm. ofThe 38 one-meter-long mm. Both of the test tube section and shellwas wallwell insulatedthicknesses with were wrapping 2 mm. The glass one-meter-long wool inside test and section aluminum was well foil insulated tape outside. with wrappingVolume flow glass rates, wool pressuresinside and and aluminum temperatures foil tape were outside. measured Volume by flowturbine rates, flowmeters, pressures andpressure temperatures transducer were and measured thermal resistanceby turbine (Pt100), flowmeters, respectively. pressure Al transducerl test data andwere thermal recorded resistance through (Pt100),an ADAM respectively. 4561, which All converted test data wereRS485 recorded signals to through USB for an use ADAM on a computer. 4561, which converted RS485 signals to USB for use on a computer.

Figure 1. Experimental process.

For accurate results, the device’s uncertainty waswas evaluated [21]. [21]. Table Table 33 showsshows thethe uncertaintiesuncertainties of the experimental instruments’instruments’ performance,performance, dependingdepending on on the the accuracy accuracy of of each each instrument. instrument. Table Table4 4lists lists the the uncertainties uncertainties of theof the experimental experimental variables. variables. All theAll rangesthe ranges of experimental of experimental uncertainties uncertainties were wereless than less 15%;than the15%; instruments the instruments and variables and variable thuss showedthus showed strong strong reliability reliability and accuracy. and accuracy.

Table 3. Uncertainty of the experimental instruments.

ApparatusApparatus Name Name Uncertainty Uncertainty /%/% Pt100Pt100 0.5 0.5 pressurepressure transducer transducer 0.5 0.5 turbineturbine ﬂowmeters flowmeters 0.5 0.5

Table 4. Uncertainty of the experimental parameters.

Experimental Parameters Uncertainty /% f 1.42 Nu 12

Energies 2020, 13, 1282 4 of 16

Table 4. Uncertainty of the experimental parameters.

Experimental Parameters Uncertainty /% f 1.42 Nu 12 Re 0.55

2.3. Data Processing

2.3.1. The Thermophysical Properties of Microfluids The physical properties used for the microfluid were calculated. The density of the microfluid was evaluated by the following; ρ = (1 φ)ρw + φρmp (2) m f − The specific heat of the microfluid was calculated by

φρmpCp,mp + (1 φ)ρwCp,w Cp,m f = − (3) ρm f

The thermal conductivity was evaluated by the Maxwell model [22], as follows;

km f kmp + 2kw + 2φ(kmp kw) = − (4) kw kmp + 2kw φ(kmp kw) − − The viscosity of the microﬂuids was evaluated by Einstein’s formula [23]

µm f = µw(1 + ηφ) (5)

2.3.2. Heat Transfer and Friction Factor The heat ﬂow was calculated as follows;

. . Q = Cp,cρcVc(Tc,out T ) (6) c − c,in . . Q = C ρ V T T (7) h p,h h h h,out − h,in It was found from the experimental results that the heat provided by the hot fluid was 5% to 8% higher than that absorbed by the cold fluid due to the heat loss caused by the convection and radiation of the test part and the surrounding environment. Considering the heat loss, the average heat transfer rate is calculated as follows; . . . Qave = Qc + Qh /2 (8) . . where Qc is the heat flux absorbed by the cold fluid, and Qh is the heat flux provided by the hot fluid. The overall heat transfer coefficient (K) can be calculated as . K = Qave/Ai∆TLMTD (9) where Ai is the heat transfer inner-surface area of the tube, and ∆TLMTD is the logarithmic mean temperature difference. The inside surface area of the heat transfer of the tube is calculated as follows;

Ai = πdiL (10) where di is the inside diameter of the tube, and L is the length of the tube. Energies 2020, 13, 1282 5 of 16

The heat transfer coeﬃcient (hi) was evaluated as below;

1/K = do/hidi + doln(do/di)/2k + 1/ho + R f (11) where do is the outside diameter of the tube, k is the length of the tube, hi is the heat transfer coefficient on the inside surface of the tube, ho is the heat transfer coefficient on the outside surface of the tube, k is the thermal conductivity of the material of the tube, and Rf is the thermal resistance of dirt and scale. The last three elements of Equation (11) are constant, and hi can be expressed as Equation (12), based on the Wilson plot method [24]; n hi = cRe (12) where c and n are the constant. The heat transfer coefficient can be re-expressed as

h = d /(1/K b)do (13) i i − where b is the constant. The average Nusselt number is evaluated by

Nu = hidi/k (14) where k is the heat transfer coeﬃcient of water or microﬂuids. The tube Reynolds number was evaluated as follows;

Re = ρVdi/µ (15) where µ is the viscosity of water or microﬂuids, ρ is the density of water or microﬂuids, and V is the mean velocity of tube inlet. The friction factor (f ) is expressed by

2 f = 2∆Pdi/ρV L (16) where ∆p is pressure loss between tube inlet and outlet. The thermal performance factor can be calculated as

1/3 η = (Num f /Nuw)/( fm f / fw) (17) where Numf and fmf are the Nusselt number and friction loss of the microﬂuids, and Nuw and fw are the Nusselt number and friction loss of the water, respectively.

2.4. Experimental Results and Discussion

2.4.1. Heat Transfer and Flow Characteristics The Nusselt number, friction factor and thermal performance at different Reynolds numbers for the graphite, A12O3 and CuO with different volume fractions were calculated with experimentally measured temperatures, pressure loss values and flow rates. The Nusselt number results for three kinds of microfluids are shown in Figure2; the Nusselt number does not only increase with Reynolds number increasing but also increases with the volume fraction of microparticle increasing. In addition, the Nusselt number of the graphite microfluid was higher than that of any other microfluid under the same conditions and was about 1.3–1.4 times, 1.5–1.7 times, 1.7–2.0 times and 2.0–2.3 times the Nusselt number of the water at a volume fraction of 0.25%, 0.5%, 0.75% and 1.0%, respectively. The reason is that the equivalent thermal conductivity of the microfluid increased according to Equation (4), due to the higher thermal conductivity of adding graphite microparticle about 127 W m 1 K 1, which is more · − · − Energies 2020, 13, 1282 6 of 16

than 200 times that of water and 5 times the thermal conductivity of A12O3, and 7 times the thermal conductivity of CuO. The A12O3 microfluid was about 1.1–1.2 times, 1.2–1.4 times, 1.3–1.5 times and 1.4–1.6 times the Nusselt number of the water at volume fraction of 0.25%, 0.5%, 0.75% and 1.0%, respectively, and the CuO microfluid was about 1.1–1.2 times, 1.2–1.3 times, 1.3–1.4 times and 1.3–1.5 times the Nusselt number of the water at a volume fraction of 0.25%, 0.5%, 0.75% and 1.0%, respectively. The Nusselt numbers of the A12O3 and CuO microfluids were similar because the thermal conductivity of A12O3 and CuO were almost the same, and more than about 29 times the thermal conductivity of water. Therefore, the heat transfer performance could be enhanced by adding Energiesmicroparticles 2020,, 13,, xx withFORFOR PEERPEER a higher REVIEWREVIEW thermal conductivity than water. Further, graphite has good chemical6 of 16 stability, electrical conductivity and thermal conductivity, so the microfluid mixed by this particle had chemical stability, electrical conductivity and thermal conductivity, so the microfluid mixed by this better heat transfer performance than A12O3 and CuO. particle had better heat transfer performance than A12O3 andand CuO.CuO.

(a) (b) (c)

FigureFigure 2.2. Variations ofof NuNu withwithwith ReRe:(:: ((aa)) graphite;graphite; ((bb)) A1A1222O33;;;( ((cc)) CuO. CuO.

The results of the friction factor with the Reynol Reynoldsds numbers are shown in Figure3 3.. InIn thethe testedtested range, thethe frictionfriction factors factors of of the the three three kinds kinds of microfluids of microfluids were were all higher all higher than that than of that water. of Thewater. friction The frictionfactor of factor the graphite of the microfluidgraphite microfluid was 0.006–0.003, was 0.006–0.003, 0.008–0.005, 0.008–0.005, 0.01–0.006 and0.01–0.006 0.012–0.007 and higher0.012–0.007 than that of water, the friction factor of the A1 O microfluid2 3 was 0.007–0.004, 0.01–0.006, 0.01–0.007 and higher than that of water, the friction factor2 of3 the A12O3 microfluidmicrofluid waswas 0.007–0.004,0.007–0.004, 0.01–0.006,0.01–0.006, 0.01–0.01– 0.0070.0144–0.008 and 0.0144–0.008 higher than higher that of than water, that and of water, the friction and the factor friction of the factor CuO of microfluid the CuO was microfluid 0.009–0.008, was 0.009–0.008,0.012–0.01, 0.014–0.01 0.012–0.01, and 0.014–0.01 0.016–0.011 and higher0.016–0.011 than thathigh ofer waterthan that at a of volume water fractionat a volume of 0.25%, fraction 0.5%, of 0.25%,0.75% and0.5%, 1.0%, 0.75% respectively. and 1.0%, The respectively. reason was thatThe thereason equivalent was that viscosity the equivalent of the microfluid viscosity increased of the microfluiddue to the additionincreased of due microparticles to the addition according of micropar to Equationticles according (5). The to higher Equation the volume(5). The fractionhigher the of volumethe microparticles, fraction of thethe greatermicroparticles, the equivalent the greater viscosity the equivalent of the microfluid. viscosity The of frictionthe microfluid. factor of The the frictiongraphite factor microfluid of the wasgraphite lower microfluid than that ofwas any lowe otherr than microfluid that of any under other the microfluid same conditions under duethe same to its conditionslubrication. due Generally, to its lubrication. the friction Generally, factor decreases the friction with Reynolds factor decreases number increasingwith Reynolds but increases number withincreasing volume but fraction increases of microparticlewith volume fraction increasing. of microparticle increasing.

(a) (b) (c)

Figure 3. Variations of f with Re: (a) graphite; (b) A12O3; (c) CuO. FigureFigure 3.3. Variations ofof ff with Re:(: (a) graphite; (b) A12OO33; ;((cc)) CuO. CuO.

2.4.2. Thermal Performance The results of the thermal performance factor with different Reynolds numbers are shown in Figure 4; the thermal performance for all tested microfluids was greater than 1, which was better than water. Therefore, in the experimental range, it can be concluded that the positive effects of the enhanced heat transfer outweighed the negative effects of the increased friction losses. The higher the volume fraction of the microparticles, the greater the thermal performance of the microfluid. The thermal performance of the graphite microfluidmicrofluid waswas higherhigher thanthan thatthat ofof anyany otherother microfluidmicrofluid underunder the same conditions, and the thermal performance factor was about 1.2–1.4, 1.5–1.7, 1.7–1.9 and 1.9– 2.2, at a volume fraction of 0.25%, 0.5%, 0.75% and 1.0%, respectively. The thermal performance factor of the graphite microfluid was the highest, which showed the comprehensive characteristics of the heat transfer performance. Further, thethe frictionfriction losseslosses ofof thethe graphigraphite microfluid were the best among the tested microfluids. The maximum thermal performance of 2.23 was found with the graphite

Energies 2020, 13, 1282 7 of 16

2.4.2. Thermal Performance The results of the thermal performance factor with different Reynolds numbers are shown in Figure4; the thermal performance for all tested microfluids was greater than 1, which was better than water. Therefore, in the experimental range, it can be concluded that the positive effects of the enhanced heat transfer outweighed the negative effects of the increased friction losses. The higher the volume fraction of the microparticles, the greater the thermal performance of the microfluid. The thermal performance of the graphite microfluid was higher than that of any other microfluid under the same conditions, and the thermal performance factor was about 1.2–1.4, 1.5–1.7, 1.7–1.9 and 1.9–2.2, at a volume fraction of 0.25%, 0.5%, 0.75% and 1.0%, respectively. The thermal performance factor of the graphite microfluid was the highest, which showed the comprehensive characteristics of the heat transfer performance. Further, the friction losses of the graphite microfluid were the best among the Energies 2020, 13, x FOR PEER REVIEW 7 of 16 tested microfluids. The maximum thermal performance of 2.23 was found with the graphite microfluid microfluidat a 1.0% volume at a 1.0% fraction volume and fraction a Reynolds and a numberReynolds around number 7500. around As 7500. shown As in shown Figure in4, Figure thermal 4, thermalperformance performance decreases decreases with an increasewith an inincrease the Reynolds in the Reynolds number and number increases and withincreases an increasing with an increasingvolume fraction volume of microparticles.fraction of microparticles.

(a) (b) (c)

2 3 Figure 4. Variations ofof ηη withwith ReRe:(: (a) graphite; (b)) A12O3;;( (cc)) CuO. CuO.

2.4.3. Regression Regression Equation According toto the the obtained obtained experimental experimental data data of the of graphite the graphite microfluid microfluid and the and correlative the correlative formula, formula,comparisons comparisons between thebetween experimental the experimental value and value the predicted and the value predicted for the value Nusselt for numberthe Nusselt and numberthe friction and factor the friction are illustrated factor are in Figuresillustrated5 and in6 Figures. The nonlinear 5 and 6. curveThe nonlinear fit equation curve coe fitffi cients,equation as coefficients,shown in Equations as shown (18) in andEquations (19), were (18) calculatedand (19), we withre calculated the least square with the method least square solved method by the iteration solved byalgorithm the iteration of Levenberg algorithm Marquardt. of Levenberg It can Marquardt. be observed Itthat can thebe observed predicted that results the are predicted in good results agreement are inwith good the agreement experimental with data the withinexperimental a deviation data wi ofthin10% a deviation for the Nusselt of ± 10% number for the and Nusselt5% number for the ± ± andfriction ± 5% factor. for the friction factor. Nu = 0.0002Re0.7454Pr3.4007(1 + φ)67.6574 (18)

0.3679 17.6099 f = 1.0299Re− (1 + φ) (19)

Figure 5. Comparison of Nu between the experiments and the predictions from the regression equation.

Energies 2020, 13, x FOR PEER REVIEW 7 of 16

microfluid at a 1.0% volume fraction and a Reynolds number around 7500. As shown in Figure 4, thermal performance decreases with an increase in the Reynolds number and increases with an increasing volume fraction of microparticles.

(a) (b) (c)

Figure 4. Variations of η with Re: (a) graphite; (b) A12O3; (c) CuO.

2.4.3. Regression Equation According to the obtained experimental data of the graphite microfluid and the correlative formula, comparisons between the experimental value and the predicted value for the Nusselt number and the friction factor are illustrated in Figures 5 and 6. The nonlinear curve fit equation coefficients, as shown in Equations (18) and (19), were calculated with the least square method solved by the iteration algorithm of Levenberg Marquardt. It can be observed that the predicted results are Energies 2020in good, 13, 1282 agreement with the experimental data within a deviation of ± 10% for the Nusselt number 8 of 16 and ± 5% for the friction factor.

Energies 2020, 13, x FOR PEER REVIEW 8 of 16

Figure 5. Comparison of Nu between the experiments and the predictions from the regression FigureEnergies 5. 2020Comparison, 13, x FOR PEER of Nu REVIEWbetween the experiments and the predictions from the regression equation.8 of 16 equation.

Figure 6. Comparison of f between the experiments and the predictions from the regression equation.

Nu=+0.0002 Re0.7454 Pr 3.4007 (1φ ) 67.6574 (18)

−0.3679 17.6099 fRe=+1.0299 (1φ ) (19) Figure 6.FigureComparison 6. Comparison of f ofbetween f between the the experiments experiments and and the the predictions predictions from from the regression the regression equation. equation. 2.4.4. Graphite Graphite Particle Particle Size Size and and Regression Regression Equation Nu=+0.0002 Re0.7454 Pr 3.4007 (1φ ) 67.6574 (18) The above results show the enhanced heat transfer of the graphite microfluid. microfluid. The The effect effect of the − particle size on thermal performance fRe=+ is1.0299 discussed0.3679 (1 in thisφ ) 17.6099 section. The results for particle (19) sizes of 1 μµm and 5 μµm are shown in Figures7 7 and and8 .8. Compared Compared with with a a graphite graphite particle particle size size of of 5 5 μµm, the Nusselt number2.4.4. and Graphite frictionfriction Particle factorfactor forSizefor a aand graphite graphite Regression particle particle Equation size size of of 1 µ1m μm were were more more than than those those for for a size a size of 5 ofµm. 5 The thermal performance of a graphite microfluid with a particle size of 1 µm was higher than that for μm. The thermalThe above performance results show of the a graphiteenhanced heatmicrofluid transfer ofwith the agraphite particle microfluid. size of 1 The μm effect was of higher the than thata particle for particlea particle size ofsize 5 sizeonµm. thermal of 5 μ performancem. is discussed in this section. The results for particle sizes of 1 μm and 5 μm are shown in Figures 7 and 8. Compared with a graphite particle size of 5 μm, the Nusselt number and friction factor for a graphite particle size of 1 μm were more than those for a size of 5 μm. The thermal performance of a graphite microfluid with a particle size of 1 μm was higher than that for a particle size of 5 μm.

(a) (b)

Figure 7. Relation diagram diagram of of(a )NuNu with Re::( (aa)) a a graphite graphite particle particle size size (of ofb) 1 1 μµm; ( b) a graphite particle size of 5 Figure μµm. 7. Relation diagram of Nu with Re: (a) a graphite particle size of 1 μm; (b) a graphite particle size of 5 μm.

Energies 2020,, 13,, 1282x FOR PEER REVIEW 9 of 16 EnergiesEnergies 2020 2020, 13, 13, x, xFOR FOR PEER PEER REVIEW REVIEW 9 9of of 16 16

(a) (b) (a()a ) (b(b) ) Figure 8. Relation diagram of f with Re of microparticle size: (a) a graphite particle size of 1 μm; (FigureFigureb)Figure a graphite 8.8. 8. Relation Relation particle diagram diagram size of ofof of f f 5withf withwithμm Re. ReRe of of microparticle microparticle size: size: (a () (a aa)) agraphite a graphite graphite particle particle size sizesize of of of 1 1μµ m;μmm; ; ((bb)()b a a) graphite agraphite graphite particle particle particle size size size of 5ofµ ofm. 5 5μ μmm. . Based on the obtained experimental data on particle size, comparisons between the experimental valueBasedBased andBased predicted onon on thethe the obtainedobtained obtained value experimentalforexperimental experimental different particle datadata data onon on sizes particlepart part icleforicle size,size,the size, Nusselt comparisonscomparisons comparisons number betweenbetween between and friction thethe the experimentalexperimental experimental factor are valueshownvaluevalue and andin and Figures predictedpredicted predicted 9 and valuevalue value 10. forforThe for didifferent regressiondifferentﬀerent particleparticle particle equation sizessizes sizes can forfor for be thethe the expressed NusseltNusselt Nusselt number numberas number in Equations andand and frictionfriction friction (20) factorfactorand factor (21), areare are whereshownshownshown D inin refin Figures Figures= Figures 2.5 μ9 m, 9and 9and andand 10 10. .D10. The mpThe The= regression theregression regression diameter equation equation equation of the can particlescan can be be be expressed expressed expressedin micrometers. as as as in in in Equations Equations Equations The deviation (20) (20) (20) and and and of (21), (21),the (21), µ wherecorrelationwherewhere DD refDrefref =data= =2.52.5 2.5 fromμ m,μm,m, andtheand and experimental DD mpDmpmp == =the the diameterdiameter diameter data is ofof± of 10% thethe the particlesparticlesand particles ± 5% inin forin micrometers.micrometers. micrometers.the Nusselt ThenumberThe The deviationdeviation deviation and friction ofof of thethe the factor,correlationcorrelation respectively. datadata fromfrom thethe experimentalexperimental datadata isis ± 10%10% andand ± 5%5% forfor thethe NusseltNusselt numbernumber andand frictionfriction correlation data from the experimental data is± ± 10% and± ± 5% for the Nusselt number and friction factor,factor,factor, respectively. respectively. respectively.

Figure 9. Comparison of Nu between the experiments and the predictions from the regression equation.FigureFigureFigure 9.9. Comparison9. Comparison Comparison of Nuof of Nubetween Nu between between the experimentsthe the experiments experiments and the and and predictions the the predictions predictions from the from regressionfrom the the regression equation.regression equation.equation.

Figure 10. Comparison of ff between the experiments andand thethe predictionspredictions from from the the regression regression equation. equation.FigureFigure 10. 10. Comparison Comparison of of f betweenf between the the experiments experiments and and the the pr predictionsedictions from from the the regression regression equation.equation.

Energies 2020, 13, 1282 10 of 16

! 0.0523 Dmp − Nu = 0.0094Re0.7891Pr1.0986(1 + φ)63.9181 (20) Dre f ! 0.0558 0.3389 22.3896 Dmp − f = 0.7891Re− (1 + φ) (21) Dre f

3. Numerical Study

3.1. Simulation Model For studying the heat transfer mechanism in detail, the heat transfer process of the microﬂuid ﬂow through the tube were simulated by an Eulerian–Eulerian model using the CFD software. The details of the continuity equations given by Equation (22), the momentum conservation equation given by Equations (23) and (24), and the energy equation given by Equation (25) are shown, respectively, as follows; ∂ (α ρ ) + (α ρ V ) = 0 (22) ∂t i i ∇· i i i ∂ α ρ V + α ρ V V = α p + τ + α ρ g + K (Vs V ) (23) ∂t f f f ∇· f f f f − f ∇ ∇· f f f s f − f ∂ (αsρsVs) + (αsρsVsVs) = αs p ps + τs + αsρs g + K (V Vs) (24) ∂t ∇· − ∇ − ∇ ∇· s f f − ∂ (α ρ h ) + (α ρ v h ) = α k T h (T T ) (25) ∂t i i i ∇· i i i i ∇· i i,e f f ∇ i − ij i − j

In the Equations, αi is the volume fraction of phase i, ρi is the density of phase i, νi is the velocity of phase i, hi is the speciﬁc enthalpy of phase i, g is the gravity acceleration and i represents either the ﬂuid (f ) or solid (s) phase;

 h i  18 0.687 αsρ f Vs V f 2.65  1 + 0.15(α f Res) − α f − α f > 0.8  Res ds Ks f = (26)  150αs(1 α )µ αsρ Vs V  − f f f − f  2 + 1.75 α f 0.8 α f ds ds ≤ h i 2 τ = α µ V + ( V )T + α λ µ ( V )I (27) i i i ∇ i ∇ i i i − 3 i ∇· i where Ksf is the interphase momentum exchange coefficient and defined by the Gidaspow model [25], which is a combination of the Wen and Yu models [26] and Ergun model [27], τi is the phase i stress–strain tensor shown in Equation (27), ds is the particle diameter of solid phase and µi is the shear viscosity of phase i and λi is the bulk viscosity of phase i. Equation (28) is the transport equation for the closure of the solid phase momentum equation " # 3 ∂ (αsρsΘs) + (αsρsVsΘs) = ( psI + τs) : Vs + (κ Θs) γ + ϕ (28) 2 ∂t ∇· − ∇ ∇ Θs ∇ − Θs f s where (-psI + τs): νs is energy generated by the solid stress tensor, k Θs is the energy diffusion, ∇ Θs∇ Θs is the granular temperature of the solid phase, γΘs is the collisional dissipation of energy due to collisions between particles and ϕfs is the energy exchange between the solid phase and the fluid or solid phase. The two-fluid model (TFM) has been successfully applied in the literature to predict the heat transfer coefficient [28,29]. The details of the computational conditions are shown in Table5. Two-equation turbulence models (K–ε) are widely used to simulate the turbulent flow and heat transfer due to their robustness, economy and reasonable accuracy [30]. Bayat et al [31] found that the simulation results of applying the Realizable K–ε model agreed well with the Gnielinski [32] experiment results and the test Energies 2020, 13, x FOR PEER REVIEW 11 of 16

The two-fluid model (TFM) has been successfully applied in the literature to predict the heat transfer coefficient [28,29]. The details of the computational conditions are shown in Table 5. Two- equation turbulence models (K−ε) are widely used to simulate the turbulent flow and heat transfer due to their robustness, economy and reasonable accuracy [30]. Bayat et al [31] found that the simulation results of applying the Realizable K−ε model agreed well with the Gnielinski [32] experiment results and the test results in this paper also agree well with the simulation results by using the Realizable K−ε turbulence model. A near-wall modeling method (enhanced wall treatment), combined with a two-layer model with enhanced wall functions, can be used with coarse meshes as wall-function meshes, as well as fine meshes as low-Reynolds number meshes, typically with the first near-wall node placed at y+ ≈ 1. Grid convergence was checked on the grid size of 0.5–2 mm and the growth ratio of the boundary layer of 1.1, 1.3 and 1.5 under the condition of Y+ < 1. Through some trial simulations of the grid models, when the grid size was 1 mm and the growth ratio was 1:1, the grid size decreased and the changes of simulation results were less than 1%. The computational domain is shown in Figure 11; the thickness of the tube was 2 mm, and the details of the computational boundary conditions are shown in Table 6. SIMPLE was used to solve the pressure– velocity coupling algorithm. After solving the initial model, the velocity profiles at the outlet Energiesboundary2020 ,were13, 1282 imposed on the inlet boundary, and the last solution was recalculated for precision.11 of 16 The iterative calculations were performed until the normalized residual fell below a convergence criterion (less than 10−6 for the energy residual and less than 10−4 for all other variables), and the results in this paper also agree well with the simulation results by using the Realizable K ε turbulence variation of the Nusselt number, pressure loss and outlet temperature profiles were kept− steady. model. A near-wall modeling method (enhanced wall treatment), combined with a two-layer model with enhanced wall functions,Table can be5. Details used with of the coarse computational meshes asconditions. wall-function meshes, as well as fine meshes as low-Reynolds number meshes, typically with the first near-wall node placed at y+ 1. Grid ≈ convergence was checkedFluid on the grid size of 0.5–2Microfluid mm and the growth ratioWater of the boundary layer of 1.1, 1.3 and 1.5 underFluid the flow condition of Y+ < Eulerian1. Through multiphase some trial simulations single-phase of model the grid models, when the grid sizeTurbulence was 1 mm modeling and the growth Realizable ratio was 1:1,k–ε themodel grid size Same decreased as microfluid and the changes of simulationNear-wall results were modeling less than method 1%. The computational Enhanced wall domain treatment is shown Same in as Figure microfluid 11; the thickness of the tube wasMicrofluid 2 mm, and concentrations the details of the 0.25%, computational 0.5%, 0.75%, boundary 1.0% conditions 0% are shown in Table6. SIMPLE was used to solvey+ the pressure–velocity coupling < 1 algorithm. After solving < 1 the initial model, the velocity profiles at the outlet boundary were imposed on the inlet boundary, and the last solution was recalculated for precision. TheTable iterative 6. Details calculations of the boundary were conditions. performed until the normalized residual 6 4 fell below a convergenceBoundary criterion (less Conditions than 10− for the energy Setting residual and less than 10− for all other variables), and the variation oftube the Nusselt wall number, pressure loss No-slip and outlet temperature profiles were kept steady. shell wall No-Slip tube inlet velocity Table 5. Details of the computational conditions. tube outlet pressure Fluidshell inlet Microfluid velocity Water Fluid flowshell outlet Eulerian multiphase pressure single-phase model Turbulence modelingtube inlet temperature Realizable k–ε model 300 K Same as microfluid Near-wall modelingshell method inlet temperature Enhanced wall treatment 320 K Same as microfluid Microfluid concentrationstwo sides of the tube 0.25%, wall 0.5%, coupled0.75%, 1.0% thermal conditions 0% y+ <1 <1 shell wall adiabatic

Figure 11.11. The computational domain.

Table 6. Details of the boundary conditions.

Boundary Conditions Setting tube wall No-slip shell wall No-Slip tube inlet velocity tube outlet pressure shell inlet velocity shell outlet pressure tube inlet temperature 300 K shell inlet temperature 320 K two sides of the tube wall coupled thermal conditions shell wall adiabatic

3.2. Simulation Results and Discussion The results of the experiment and simulation of the water and the graphite microﬂuid at a volume fraction of 1.0% are shown in Figure 12. The results of the simulation matched those of the experiments Energies 2020, 13, x FOR PEER REVIEW 12 of 16

3.2. Simulation Results and Discussion Energies 2020, 13, 1282 12 of 16 The results of the experiment and simulation of the water and the graphite microfluid at a volume fraction of 1.0% are shown in Figure 12. The results of the simulation matched those of the experimentswithin an error within of 12%. an error The simulationof 12%. The results simulation for the results turbulent for kineticthe turbulent energy, kinetic the interphase energy, heatthe interphasetransfer coe heatfficient transfer and coefficient the temperature and the of temperature graphite particle of graphite sizes of particle 1 µm andsizes 5 ofµ m1 μ inm the and middle 5 μm incross-section the middle ofcross-section the tube at of a volumethe tube fraction at a volume of 1.0% fraction witha of flow 1.0% of with Re = a23,000 flow of are Re demonstrated = 23,000 are demonstratedin Figures 13– in15 Figures. As seen 13–15. in Figure As seen 13 in, the Figure water-phase 13, the water-phase and the solid-phase and the solid-phase (graphite) (graphite) turbulent turbulentkinetic energy kinetic for energy the 1% for graphite the 1% microfluid graphite inmicrof the middleluid in cross-section the middle ofcross-section the tube is slightlyof the tube higher is slightlythan that higher of the than base that fluid of (water) the base with fluid a radius(water of) with 0–0.0055 a radius m. Moreover,of 0–0.0055 the m. solid-phase Moreover, the (graphite) solid- phaseturbulent (graphite) kinetic turbulent energy nearkinetic the energy tube near wall the with tube a radiuswall with of 0.0055–0.006a radius of 0.0055–0.006 m gradually m gradually increases increasesby about by 17 about times, 17 and times, 10 times and 10 the times turbulent the turbulent kinetic kinetic energy energy of the waterof the water at particle at particle sizes ofsizes 1 µ ofm 1and μm 5 andµm, 5 respectively. μm, respectively. Near theNear tube the wall, tube thewall, addition the additi of microparticleson of microparticles increases increases the turbulent the turbulent kinetic kineticenergy energy due to thedue irregular to the irregular movement movement of microparticle of microparticle suspensions, suspensions, which does which not onlydoes e ffnotectively only effectivelyenhance the enhance disturbance the dist ofurbance the boundary of the boundary layer but destroyslayer but the dest effroysect ofthe the effect conductive of the conductive boundary boundarylayer. The layer. enhancement The enhancement of the particle of the size particle of 1 µ msize is of higher 1 μm than is higher that atthan 5 µ mthat due at to5 μ them due number to the of numbermicroparticles of microparticles in the same volumein the same fraction. volume In Figure fraction. 14, theIn interphaseFigure 14, heatthe interphase transfer coe heatfficient transfer of the coefficientgraphite microfluid of the graphite for a particle microfluid size offor 1 aµ particlem is about size 5 timesof 1 μ thatm is ofabout a particle 5 times size that of 5ofµ am. particle The reason size offor 5 thisμm. is The that reason a smaller for particlethis is that size a indicates smaller particle a larger size contact indicates surface a arealarger for contact the heat surface transfer. area In for the thesame heat volume, transfer. the In number the same of volume, particles the with number a size of of particles 1 µm is 125with times a size that of 1 of μm 5 µism 125 in times size, and that the of 5surface μm in areasize, is and 5 times the larger.surface As area shown is 5 intimes Figure larger. 15, theAs temperatureshown in Figure in the 15, middle the temperature cross-section in of the the middletube is highercross-section than that of the of thetube water, is higher because than the that increase of the water, of the equivalentbecause the thermal increase conductivity of the equivalent of the thermalmicrofluid conductivity and the turbulent of the microfluid kinetic energy and the improves turbulent the kinetic heat transfer energy by improves adding the the microparticles heat transfer bywith adding good the thermal microparticles conductivity. with The good greater thermal the conductivity. volume fraction The of greater the microparticle, the volume thefraction greater of the the microparticle,temperature change. the greater the temperature change.

FigureFigure 12. 12. ComparisonComparison of of the the simulation simulation and and experiment. experiment.

Energies 2020, 13, x FOR PEER REVIEW 13 of 16 Energies 2020, 13, 1282 13 of 16 EnergiesEnergies 2020 2020, 13, 13, x, xFOR FOR PEER PEER REVIEW REVIEW 1313 of of 16 16

(a) (b)

(a) (b) Figure 13. The turbulent(a) kinetic energy in the middle cross-section of the(b) tube at Re = 23,000: (a) a FigureFiguregraphite 13. 13. The TheparticleThe turbulent turbulent turbulent size kinetic ofkinetic kinetic1 μ menergy energy with energy ain involume the inthe themiddle middle fraction middle cross-section cross-section cross-sectionof 1.0% ;of (ofb the) thea of graphitetube tube the at tube at Re Reparticle at= = 23,000: Re23,000:= size23,000: ( a(a) of)a a 5 graphite(graphiteaμ)m a graphite with particle particle a volume particle size size fraction size of of 1 of1 μ μ 1mm µof withm with 1.0% with a a volume. avolume volume fraction fraction fraction of ofof 1.0% 1.0%;1.0%; ;( b(b) )a a graphite graphite particle particle particle size size size of of 5 ofµ 5m 5 μwithμmm with with a volume a a volume volume fraction fraction fraction of 1.0%. of of 1.0% 1.0%. .

Figure 14. The interphase heat transfer coefficient in the middle cross-section of the tube at Re = FigureFigure23,000. 14. 14. The The interphase interphase heat heatheat transfer transfertransfer coefficient coecoefficientﬃcient in in the the middle middle cross-sectioncross-section cross-sectionof of of the the the tube tube tube at at atRe Re Re= = 23,000.= 23,000.23,000.

(a) (b)

(a) (b) FigureFigure 15. 15.The The temperature temperature(a) in in the the middle middle cross-section cross-section of of the the tube tube at Reat =Re23,000: =(b 23,000:) (a) ( aa) graphite a graphite particle FiguresizeFigureparticle of 15. 115.µ sizeThem; The (ofbtemperature temperature) 1 a μ graphitem; (b) a in particlegraphitein the the middle middle size particle of cross-section cross-section 5 µ sizem. of 5 μ of m.of the the tube tube at at Re Re = = 23,000: 23,000: ( a(a) )a a graphite graphite particleparticle size size of of 1 1 μ μm;m; ( b(b) )a a graphite graphite particle particle size size of of 5 5 μ μm.m.

Energies 2020, 13, 1282 14 of 16

In general, the turbulent convective heat transfer enhancement of microﬂuid ﬂow is mainly due to two aspects, i.e., the increase in turbulent kinetic energy due to the irregular movement of the microparticle suspensions and the increased equivalent thermal conductivity and interphase heat transfer by adding microparticles with higher thermal conductivity, as illustrated in Equation (3).

4. Conclusions The results of this study indicate the feasibility of convection heat transfer enhancement by using microparticle multiphase flow for turbulent flow in a tube, based on the Nusselt number, friction factor and thermal performance characteristics of graphite, Al2O3 and CuO microfluid, with a particle volume fraction of 0.25%–1.0% and a Reynolds number ranging from 7000 to 23,000. The conclusions can be drawn as follows:

(1) A variety of microfluids (Graphite, A12O3 and CuO) were studied in a vertical tube. The results show that the heat transfer performance of the various microfluids is better than that of water. The graphite microfluid provides the best thermal performance compared to the other two microparticle suspensions. All the microfluids enhanced heat transfer but increased flow resistance. (2) Both graphite microfluids (those with 5 µm and 1 µm graphite particles) further show that the microfluid with 1 µm particles experiences higher heat transfer enhancement than that with 5 µm particles. It is concluded that the smaller the size of particles, the stronger the heat transfer effect and the more significant the friction loss. (3) The regression equations obtained in the present paper provide the correlative coefficient of the Nusselt number and the friction factor. The predicted results are in good agreement with the experimental results within a deviation of 10% and 5% for the Nusselt number and friction ± ± factor, respectively. (4) The simulation results using the Eulerian–Eulerian granular flow model and considering the microparticle suspension as a continuum show that an increase in the equivalent thermal conductivity of the microfluid and an increase in the turbulent kinetic energy near the wall are two key factors that enhance turbulent convective heat transfer.

Author Contributions: Conceptualization, T.W. and Z.G.; methodology, T.W. and Z.G.; validation, T.W., Z.G. and W.J.; investigation, T.W.; data curation, T.W.; writing—original draft preparation, T.W.; writing—review and editing, T.W., Z.G. and W.J.; supervision, Z.G. and W.J.; funding acquisition, T.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Program of Science and Technology Development Projects of Zhejiang Province, grant number 2015C31133. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

A heat transfer surface area (m2) cp specific heat capacity of fluid (J/kg K) · d diameter of the tube (m) f friction factor h heat transfer coefficient (W/m2 K) · k thermal conductivity (W/m K) · L length of the test section (m) m mass flow rate (kg/s) Nu Nusselt number ∆P pressure drop (Pa) Q heat transfer rate (W) Energies 2020, 13, 1282 15 of 16

Re Reynolds number T temperature (K) V mean axial flow velocity (m/s) K overall heat transfer coefficient (W/m2 K) · φ volume fraction of microparticle (%) ρ density (kg/m3) µ fluid dynamic viscosity (Pa s) · η thermal performance factor Greek symbols c cold fluid h hot fluid i inner o outer w water mf microfluid mp microparticle in inlet out outlet

References

1. Hajmohammadi, M.R.; Maleki, H.; Lorenzini, G.; Nourazar, S.S. Effects of Cu and Ag nano-particles on flow and heat transfer from permeable surfaces. Adv. Powder Technol. 2015, 26, 193–199. [CrossRef] 2. Zhu, H.; Zhang, C.; Liu, S.; Tang, Y.; Yin, Y. Effects of nanoparticles clustering and alignment on thermal

conductivities of Fe3O4 aqueous nanoﬂuids. Appl. Phys. Lett. 2006, 89, 1–3. [CrossRef] 3. Vajjha, R.S.; Das, D.K.; Namburu, P.K. Numerical study of ﬂuid dynamic and heat transfer performance of

Al2O3 and CuO nanoﬂuids in the ﬂat tubes of a radiator. Int. J. Heat Fluid Flow 2010, 31, 613–621. [CrossRef] 4. Demir, H.; Dalkilic, A.S.; Kürekci, N.A. Numerical investigation on the single phase forced convection heat

transfer characteristics of TiO2 nanofluids in a double-tube counter flow heat exchanger. Int. Commun. Heat Mass Transf. 2011, 38, 218–228. [CrossRef] 5. Utomo, A.T.; Haghighi, E.B. The effect of nanoparticles on laminar heat transfer in a horizontal tube. Int. J. Heat Mass Transf. 2014, 69, 77–91. [CrossRef]

6. Takabi, B.; Shokouhmand, H. Effects of Al2O3-Cu/water hybrid nanofluid on heat transfer and flow characteristics in turbulent regime. Int. J. Mod. Phys. C. 2015, 26, 15–47. [CrossRef]

7. Takabi, B.; Gheitaghy, A.M.; Tazraei, P. Hybrid water-based suspension of Al2O3 and Cu nanoparticles on laminar convection eﬀectiveness. J. Thermophys. Heat Transf. 2016, 30, 523–532. [CrossRef] 8. Azmi, W.H.; Abdul, H.K.; Mamat, R.; Sharma, K.V.; Mohamad, M.S. Eﬀects of working temperature on

thermo-physical properties and forced convection heat transfer of TiO2 nanofluids in water-Ethylene glycol mixture. Appl. Therm. Eng. 2016, 106, 1190–1199. [CrossRef] 9. Sun, D.; Yang, D. Experimental study on the heat transfer characteristics of nanorefrigerants in an internal thread copper tube. Int. J. Heat Mass Transf. 2013, 64, 559–566. [CrossRef] 10. Yang, S.; Zhang, L.; Xu, H. Experimental study on convective heat transfer and flow resistance characteristics of water flow in twisted elliptical tubes. Appl. Therm. Eng. 2011, 31, 2981–2991. [CrossRef] 11. Tang, X.; Dai, X.; Zhu, D. Experimental and numerical investigation of convective heat transfer and fluid flow in twisted spiral tube. Int. J. Heat Mass Transf. 2015, 90, 523–541. [CrossRef] 12. Bhadouriya, R.; Agrawal, A.; Prabhu, S.V. Experimental and numerical study of fluid flow and heat transfer in a twisted square duct. Int. J. Heat Mass Transf. 2015, 82, 143–158. [CrossRef] 13. Pal, S.; Saha, S.K. Laminar fluid flow and heat transfer through a circular tube having spiral ribs and twisted tapes. Exp. Therm. Fluid Sci. 2015, 60, 173–181. [CrossRef] 14. Wongcharee, K.; Eiamsa-ard, S. Heat transfer enhancement by using CuO/water nanofluid in corrugated tube equipped with twisted tape. Int. Commun. Heat Mass Transf. 2012, 39, 251–257. [CrossRef] Energies 2020, 13, 1282 16 of 16

15. Sun, B.; Yang, A.; Yang, D. Experimental study on the heat transfer and flow characteristics of nanofluids in the built-in twisted belt external thread tubes. Int. J. Heat Mass Transf. 2017, 95, 712–722. [CrossRef] 16. Buzea, C.; Pacheco, I.I.; Robbie, K. Nanomaterials and nanoparticles: sources and toxicity. Biointerphases 2007, 2, MR17–MR71. [CrossRef] 17. Tae-Hyun, C.; Kwon-Soo, L.; Chang-Hoan, L. An experimental study on velocity and temperature of Al/water microfluid in a circular tube with swirl. J. Vis. 2014, 17, 59–68. 18. Zhu, J.Y.; Tang, S.; Pyshar, Y.; Thomas, B.; Khoshmanesh, K.; Ghorbani, K. Enhancement of laminar convective heat transfer using microparticle suspensions. Heat Mass Transf. 2017, 53, 169–176. [CrossRef] 19. Wang, T.; Jin, W.; Gao, Z.; Xiao, J.; Tao, W.; Tang, J. Turbulent flow and enhanced heat transfer induced by Cu-water microfluid in tube. J. Vib. Shock 2018, 37, 108–114. [CrossRef] 20. Stokes, G.G. On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. Trans. Camb. Philo. Soc. 1851, 9, 89–106. 21. Moffat, R.J. Describing the uncertainties in experimental results. Exp. Fluid Therm. 1988, 1, 3–17. [CrossRef] 22. Maxwell, J.C. Treatise on Electricity and Magnetism; Dover: New York, NY, USA, 1954. 23. Einstein, A. Berichtigung, zu, meiner, Arbeit. eine neue Bestimmung der Molekul-dimensionen. Ann. Phys. 1911, 34, 591. [CrossRef] 24. Wilson, E.E. A basis of rational design of heat transfer apparatus. ASME J. Heat Transf. 1915, 37, 47–70. 25. Gidaspow, D. Multiphase Flow and Fluidization; Academic Press: Boston, MA, USA, 1994. 26. Wen, C.Y.; Yu, Y.H. Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 1966, 62, 100–111. 27. Ergun, S. Fluid Flow through Packed Columns. Chem. Eng. Prog. 1952, 48, 89–94. 28. Schmidt, A.; Renz, U. Numerical Prediction of Heat Transfer in Fluidized Beds by a Kinetic Theory of Granular Flows. Int. J. Therm. 2000, 39, 871–885. [CrossRef] 29. Martinek, J.; Ma, Z. Granular Flow and Heat-Transfer Study in a Near-Blackbody Enclosed Particle Receiver. J. Sol. Energy Eng. 2015, 137, 1–9. [CrossRef] 30. White, F.M. Viscous Fluid Flow; McGraw-Hill: New York, NY, USA, 2005. 31. Bayat, J.; Nikseresht, A.H. Thermal performance and pressure drop analysis of nanofluids in turbulent forced convective flows. Int. J. Therm. Sci. 2012, 60, 236–243. [CrossRef] 32. Gnielinski, V. New equations for heat and mass-transfer in turbulent pipe and channel flow. Int. Chem. Eng. 1976, 16, 359–368.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).