model for nanofluids thermal conductivity based on

* Corresponding author: [email protected]

Model for Nanofluids Thermal Conductivity Based onModified Nanoconvective Mechanism

Yuant Tiandho1,2*, Rika Favoria Gusa1,3, Irwan Dinata1,3 and Wahri Sunanda1,3

1Research Center for Energy and Information Technology, Universitas Bangka Belitung, Bangka - Indonesia2Department of Physics, Universitas Bangka Belitung, Bangka - Indonesia3Department of Electrical Engineering, Universitas Bangka Belitung, Bangka - Indonesia

Abstract. Nuclear reactors are one of the long-term energy fulfillment solutions. Efforts to increaseoperating power density at various type of nuclear reactors are programs that are being developed toimprove the economic properties of a reactor. The use of nanofluids allows nuclear reactors to operatemore optimally through increased critical heat flux (CHF) and increased retention capability ofnuclear reactors to accidents. Thermal conductivity is a nanofluids property that intensively studiedbecause it has not been obtained an accurate model. In this paper a nanofluid thermal conductivitymodel was developed by involving all possible heat transfer mechanisms. But the modification onlyfocuses on the mechanism of nanoconvection. According to this model the thermal conductivity ofnanofluids depends on the volume fraction of nanoparticles, particle diameter, viscosity, density, andtemperature.

Keywords: Nanofluids; thermal conductivity; nuclear reactor; nanoconvective.

1. IntroductionThe urgency of reducing carbon emissions from variousenergy sources in the world is increasingly emphasized toavoid increasing the impact of climate change. Suggestionregarding the target of reducing global emissions to <1trillion tons of CO2 in 2100 become a focus on varioussectors in the world. Therefore, many countries havedeveloped various technologies of zero carbon emissionsources. One of the energy fulfillment technologies withthe lowest carbon emission level is nuclear power.Development of nuclear reactor technology is still beingdeveloped massively to satisfy its security requirementsand economic aspects [1].

Currently, efforts to increase the operating powerdensity of light water reactor (LWR) is one of theapproaches used to improve the economic properties of areactor. There are several possible attempts. Oneinteresting alternative effort besides conventional effortssuch as fuel assembly management and optimization fueldesign engineering is the seeding of cooling water withnanoparticles to produce fluids with superior heatdissipation [2]. Several tests at MIT's Nuclear ResearchReactor have shown that the use of nanofluids (i.e. fluidswith dispersed nanoparticles) can increase theproductivity of pressurized water reactor (PWR).Nanofluids can prevent the formation of a vapor layeraround the fuel rods and it can increase critical heat flux(CHF) significantly [3]. The increasing of CHF innanofluids with nanoparticles such as silicon, aluminum,and titanium oxide can reach 200% [4]. In addition, theretention of nuclear reactors increased by 40% when

accidents on the use of nanofluids was a very promisingprospect [5,6].

The ability of heat transfer is one of the mostinteresting aspects of nanofluids [7]. Some theoriesrelated to heat transfer mechanisms such as the Maxwellsuspension approach, heat propagation speed [8], andBrownian analysis on nanofluids [9] are some theoriesthat are widely used to explain the heat characteristics ofnanofluids. Nevertheless, the theory has not yet beenobtained a satisfactory description.

The nanofluids heat transfer mechanism proposed byJang and Choi [7,9] has a complex picture. In the modelare presented various possible heat transfer mechanismssuch as conduction by intermediate base fluid, inter-nanoparticles, and convection between fluid base withnanoparticles. However, in the proposed model there areseveral approaches such as the expression of the Nusseltnumber as the quadratic function of the Reynolds andPrandtl numbers which are often not appropriate for someconditions [10-12]. Furthermore, the study of theviscosity dependence of nanofluids on temperature hasnot been explicitly described in the model. Whereasviscosity is a parameter that gives a significant influenceon convection coefficient of nanofluids [13]. Therefore,in order to correct these problems, we propose amodification of the nanofluids conductivity modelespecially for the nanoconvention aspect.2. Modelling the Nanofluids ThermalConductivityIn the model constructed by Jang and Choi [7] it is statedthat basically the conductivity of nanofluids can be

, 0 (2018) https://doi.org/10.1051/e3sconf/2018E3S Web of Conferences 73 730101510ICENIS 2018

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analyzed based on four mechanisms. The first mechanismis the heat transfer mechanism generated by the collisionsbetween the fluid base molecules. The second mechanismis generated from the thermal diffusion of thenanoparticles in the fluid. The third mechanism is derivedfrom collisions between nanoparticles due to Brownianmotion. However, based on calculations, the energy fluxvalue in the third mechanism is not significant and thismechanism is often neglected. While the last mechanismis a thermal interaction of the dynamics or dancing ofnanoparticles with a fluid base molecule, also called thenanoconvection mechanism.

Basically in thermal conduction analysis it alwaysinvolves the transfer of energy from particles that havehigher thermal energy to particles that have lower thermalenergy due to the interaction between particles. In gasesand liquids, the thermal energy can be transferred throughthe interaction of collisions between particles. Based onthe kinetic theory, the net energy flux passing through thez-plane is given by [7],

ˆV

dTJ z z l C lC Cdzυ υ υρ ρ (1)

where Jυ , C , ˆVC , l , T, and υρ are energy flux,

molecular average velocity, heat capacity per unitvolume, mean free path, temperature and energy densityrespectively. The spatial averages of these are [14]. Thusthe thermal conductivity is given by,

1 ˆ3 Vk lCC (2)

In the first heat transfer mechanism because itinvolves only collisions between the fluid base moleculesand if there is a fraction of nanoparticles in the fluid, f,then the net flux is,

1 ,1 ˆ 1 13 BF V BF BF BF

dT dTJ l C C f k fdz dzυ (3)

where k is the thermal conductivity and a parameter withsubscript BF related to physical parameter for base fluid.Furthermore it can easily defined the net flux due to thesecond mechanism as,

2 ,1 ˆ3 p V P P p

dT dTJ l C C f k fdz dzυ (4)

where a parameter with subscript P related to physicalparameter for nanoparticle. However in the secondmechanism there is an obstacle that interrupt the heattransfer process called the Kapitza resistance factor thusthe nanoparticle conductivity of the second mechanismcan be expressed as,

P Pok kβ (5)

where kPo is the conductivity of the particle before itinvolves the Kapitza resistance. The third mechanisminvolving collisions between nanoparticles due toBrownian motion and it produces energy flux as,

3 ,1 ˆ3 P V P T

dTJ C C fdzυ (6)

where P and VC are the mean collision length ofparticles and the translational velocity of particles.Through the kinetic theory the translational velocity can

be expressed as 3

18 bT

P

k TC

dρπwhere KB is Boltzman

constant. From the formula it can be found that the energyflux range is at the order of 10-5 and the value is muchsmaller than other mechanisms [15]. Therefore in thispaper the third mechanism is not taken into account. In thelast mechanism, i.e. the nanoconvective mechanism, theenergy flux in the heat transfer process can be defined as,

4 ~P BFP BF T T

T

T T dTJ h T T f hf hfdzυ δ δ

δ(7)

where h and Tδ are convection heat transfer coefficientand thickness of the boundary layer respectively. Thuswithout involving the third mechanism, the effectivethermal conductivity (keff) of the nanofluids will lead to,

1eff BF P Tk k f k f hfβ δ (8)In nanoscale, thermal boundary layer will reduce toPrandtl number relation,

~PrTδδ (9)

Where δ is the hydrodynamic boundary layer. In Jang andChoi's model [7] it is postulated that nanolayers of orderedliquid molecules act as hydrodynamic boundary layersand it expressed as,

~ BFAdδ (10)where A is a proportional constant and in literature thevalue is 618 10A [7,9]. In the heat transfer to a surfacein the fluid, the Nusselt number (Nu) describes the ratioof convection heat transfer to conduction over the surface.Hence the Nusselt number is often used as a bridge toexplain the convection coefficient in conduction heattransfer frames. In general the Nusselt number is definedas,

hDNuk

(11)

In the nanoscale case, the characteristic length ofnanofluids, D, can be defined as the nanoparticleequivalent diameter and k is kBF [9]. However in amacroscopic view with the interaction between nanofluidsand fluid flow geometry the characteristic length can beexpressed as the hydraulic diameter of channel while theparameter k in eq. (11) is expressed as keff [16,17]. Thusfor this article the expression of the convective coefficientcan be written as

BF

p

k Nuhd

(12)

Heat transport in nanofluids systems has the samecharacteristics as heat transfer in fluidized beads. Brodkeyet al. [17] have reported that fluidized beds (containmicron-sized particles) have Nusselt numbers that are 20-100 times larger than single particles. Therefore, to satisfythese conditions, based on the relationship betweenNusselt numbers for particles on heat transfer fluidizedbeds, the expression h must be modified [19]. In general,the Nusselt number is a function that contains [20,21],

Re, Pr, , , ,BF pNu Nu K K f particle shape (13)


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where Re is the Reynolds number. Currently there aremany Nusselt number expressions that have been used asin Table 1. Altough almost all Nusselt numberexpressions contain parameters of Reynolds and Prandtlnumbers and almost all of them satisfy the expression ofthe Nusselt number function as eq. (13) but theseexpressions contain significantly different numbers ofReynolds and Prandtl numbers. Therefore declaring thekeff formula as a function of the Reynolds number andPrandtl number explicitly through the Nusselt numberexpression reduction is quite complicated. There are manypossible combinations allowed. Except by the expressionof the Nusselt number proposed by Zerradi [12] whichstates the Reynolds number and Prandtl number as thetermophysical coefficient and it depends on theexperimental data fitting.

Table 1. Definition of Nusselt number in the literatures

Ref. Nusselt Number[22] 2 0.5RePrNu[9] 2 2Re PrNu

[11] 0.5 0.50.021Re PrNu

[10]

0.0010.6886

0.9238 0.4

0.0059 1 7.6286 RePr

Re Pr

pdNu f

D

[12]Pr Re Rep q q

m mNu f fα γ χ δ where theGreek symbols correspond to thermophysicalcoefficient

One possible alternative to solve the differenceproblem in determining the Nusselt number is still todeclare the keff to be explicitly dependent on the Nusseltnumber. The nanofluids conductivity remains presentedin eq. (15) and does not need to expand it to the expressionof the Reynolds number and the Prandtl number. Whilethe experimental value of Nusselt number of a fluid canbe determined through the instrument system [23].However various studies have reported that the Nusseltnumber continues to change with changing rheologicalfactors. Thus it will create another problem. Therefore, toavoid the inconsistency of the formula with experimentaldata it is worthy to declare the Nusselt number as afunction of the termophysical coefficient as in Zerradimodel [12].

Using the fluidized beds analogy to the generalrelation of the convection coefficient for Brownianmotion-induced Stokes flow of multiple nanoparticles hasa modified form as,

Re Prm nBFm

p

kh C fd

(14)

Through the substitution of eq. (11), (9), and (10) into eq.(8) it will lead us to,

1 Re PrPr

eff m nP BF

BF BF p

k k Ddf f fk k d

β (15)

In Fig. 1 a comparison between models in eq. (15) withthe experiments by Mintsa et. al [24] is shown. Wecompared the model on two types of nanofluids i.e. CuO

and Al2O3 which were dispersed in water. The materialproperties of the fluid and nanoparticles used we refer toJang & Choi experiments [7]. It appears the model weproposed is quite appropriate with the experimental databecause the variance is small. From the results it can bestated that the volume fraction of the dispersednanoparticles in the fluid directly proportional with theconductivity. For information, through fitting the data canbe obtained the parameters of thermophysical parameters:m and n. For CuO the value m = 1.257 and n = 1, whilefor Al2O3 the value m = 1.576 and n = 1. Based on theresults it appears that the value of the thermophysicalparameters obtained through this model is lower than themodel proposed by Jang and Choi [7] but higher than thatthe model proposed by Pak and Cho [11]. In addition, it isclear that the thermophysical parameters for differentnanofluids may have different values. Thus declaringthose thermophysical parameters in a constant willdecrease the accuracy of the proposed model.

(a)

(b)Figure 1. Comparison of the model in eq. (15) with theexperiment [24] for keff/kBF in nanofluids (water based): (a) CuOwith d = 29 nm and (b) Al2O3 with d = 36 nm.

Other interesting results that can be found in CuO andAl2O3 nanofluids the both Prandtl numbers are almostsame. Physically, the Prandtl number represents themomentum diffusivity ratio to the thermal diffusivity thatis,

Pr pckμ

(16)

where cp and μ are heat type and dynamic viscosityrespectively. In nanofluids due to occurs interactionbetween nanoparticles, fluid base, or both, the physicalparameter of the above equations can be viewed aseffective conditions. The effective specific heat ofnanofluids can be expressed as [25],


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3

, ,p,

1 BF p BF p pp eff

eff

f c f cc

ρ ρρ

(17)

While the effective density of nanofluids can bedetermined from the equation,

1eff BF pf fρ ρ ρ (18)

The effective viscosity of nanofluids as a function ofvolume fraction nanoparticles is defined as,

2

72p B p

eff BF

V dF

ρμ μ

δ(19)

While VB is Brownian velocity, 181 bB

p p p

k TV

d dπρ, δ is

the distance between particles,6 pdπδφ

, and F is

correction factor correspond to the volume fraction andthe average diameter of nanoparticles. Looking at theexpression of the above number, it is safe to say thePrandtl number reduce to unity. Because in the case ofCuO and Al2O3 an increase in specific heat value is alsofollowed by an increase in its conductivity. It has alsofrequently used Prandtl numbers of unity on a variety ofnanofluids parameter analysis although in extremeconditions the Prandtl number can reach 2000 when itsReynolds number revolves around the order of 106 [26].

Temperature is a physical parameter that can modifythe viscosity of a fluid. This aspect is important becausethe viscosity also contributes significantly to the Reynoldsnumber. Through the comparison between the Reynoldsmodel and the Vogel model, in this paper the viscosityvalue as a function of temperature is expressed as,

0 1 expT T aμ μ μ (20)

where, μ0, μ1, and a are parameter whose value dependson the type of nanoparticles and fluids. Fig. 2 shows thecomparation between the model we proposed with theviscosity value as a function of temperature for water-based nanofluids: CuO and Al2O3 [13]. Through thecomparison it appears that the viscosity model as afunction of the temperature is quite in accordance with theexperimental results. In addition, the accuracy of thismodel is higher when compared with the various modelsthat have been studied by Masoumi et. al [13].

According to MacLaurin series the viscosityexpression in eq. (2) can written as,

21 01 TT O T

aμ μ μ (21)

Basically eq. (21) can be expressed as a generalexpression of the relationship between viscosity andtemperature as reported by Nguyen et. al. In themeasurement Nguyen obtained a viscosity relationship asa function of temperature for nanofluids withconcentrations of 1% and 4% respectively as [27],

1.125 0.0007eff BF Tμ μ (22)22.1275 0.0215 0.0002eff BF T Tμ μ (23)

The definition of Reynolds numbers in nanofluids canbe expressed as [8],

Re RM pGC dν

(24)

where CRM, v, and G are velocity of random motion ofnanoparticles, kinematic viscosity, and correction factor.The definition of velocity of random motion can beobtained through the relation between macroscopicdiffusion coefficient D0 with the mean free path of basefluid, lBF as,

0RM

BF

DC

l(25)

and the nanoparticles diffusion coefficient given byEinstein as, 0 3b pD k T dπμ [27].

(a)

(b)Figure 2. Comparison of viscosity model in eq. (20) withexperiment result [13] for water-based nanofluids: (a) Al2O3 (d= 36 nm, f = 4.5%) and (b) CuO (d = 29 nm, f= 4.5%).

Through the relationship between kinematic anddynamic viscosity for its effective value based on eq. (21)it lead us to the relationship of conductivity of nanofluidsas temperature function. Assuming the value of Prandtlnumber is one the expression of nanofluids conductivityis,

1m

eff p RM p effBF

BF BF p BF

k k GC dDd ff fk k d T

ρβ

μ λ γ(26)

where λ and γ are viscosity parameter of nanofluids. Thecomparison between our proposed model withexperimental data i.e. water-based 1% Al2O3 nanofluidsfraction is shown in Fig. 3. It appears that the model has asmall variance. Compared to other models such as theMaxwell model and the Jang and Choi model [9] ourmodel has a higher level of accuracy.


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Figure 3. Comparison of conductivity model in eq. (26) with theexperimental data [29] for Al2O3 nanofluids (d = 38.4 nm, f =1%).

3. ConclusionsThrough modification of the nanoconvection formula inthe nanofluids heat transfer mechanism proposed by Jangand Choi we propose a model of thermal conductivity ofnanofluids. The Reynolds number and Prandtl number areexpressed as functions of the thermophysical parameters.The model is used to formulate the conductivityrelationship with the volume fraction of suspendednanoparticles in nanofluids. This model allows differentmaterials to have different thermophysical parameters. Inthis article we also propose a model of viscosity as afunction of temperature because it can provide verysignificant results in nanoconvection mechanisms. Thesubstitution of the viscosity model into the nanofluidsconductivity formulation lead us to the conductivityequation as a temperature function. The accuracy of ourmodels can be seen through comparison between themodels with the experimental data.

The authors are grateful for the financial support provided byDirectorate for Research and Community Service, Ministry ofResearch, Technology and Higher Education of the Republic ofIndonesia (RISTEKDIKTI) (contract no120.E/UN50.3.1/PM/2018).

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model for nanofluids thermal conductivity based on

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