Nanoparticles are being used in fluids to enhance the thermal conductivity, but published results are incon- sistent. A possible explanation for this inconsistency is that nanoparticles may migrate under the action of temperature gradient counteracting Brownian diffusion. This work focuses on looking at the potential for particle migration due to thermophoresis. KEY WORDS: Thermophoresis, Nanoparticle, particle migration, Thermal Diffusion, Soret Effect, Diffusion, Nanofluids,Brownian diffusion 1. INTRODUCTION The low thermal conductivity of traditional heat transfer fluid like water, oil and ethylene glycol reduces the effectiveness and the compactness of a heat exchanger. Nanometer size particles are added to the base fluid to enhance the overall thermal conductivity of the fluid. This increased heat transfer helps in lowering the maxi- mum temperature of the system as well as reducing the size of the heat exchangers.This combination of base fluid and highly conductive nanoparticles is known as nanofluids and several studies have been carried out to examine the thermal conductivity of these nanofluids. However, there are discrepancies among the results published. This is due to the lack of a formidable model that can underline the physics involved with this en- hancement [1–4]. Evans et al. [5] shows that the enhancement in the thermal conductivity of the nanofluid can be predicted by effective medium theory and there is no effect of Brownian diffusion. While Choi [2] claimed Brownian motion governs the thermal behavior of the nanofluid. Eastman [6] discussed the idea of ballistic transport to be the reason for thermal conductivity enhancement instead of diffusive transport. J. A. Eastman and Keblinski [7] stated particle size, particle agglomeration, particle-fluid interface and temperature as four major contributors to thermal transport in nanofluids. Thermophysical properties like viscosity can also play a significant role in the heat transfer enhancement [8, 9]. The enhancement in thermal conductivity is ex- perimentally observed to be temperature dependent, Gao [10] showed that thermal conductivity was higher at higher temperature for the nanofluid made from n-Octadecane and Alumina. Ho and Gao [11] reported a lower heater enhancement with the increased nanoparticle mass fraction due to reduced convection. [12–15] claim that the Brownian motion plays the dominant role in thermal conductivity enhancement, while other claims that Brownian motion only affects the nanoparticles but not the heat transfer. When particles are added to the base fluid, they move around by virtue of Brownian motion. Brownian dif- fusion is responsible for keeping the uniform concentration within the system, and continuously negates the formation of concentration gradient. But, when this nanofluid is exposed to the temperature gradient across the boundaries, there is diffusion due this temperature gradient. This diffusion is known as thermophoresis. This thermophoretic behavior tends to move the particles towards the cold end of the system while the Brow- nian diffusion tries to fight against this build up of concentration gradient. The dominant mode of diffusion will either result in uniform distribution of particle or non-uniform distribution with potential for particles to migrate towards the cold side. The analysis of the dominant mode is identified using a thermal diffusion factor. The discrepancy between the experimental results in the thermal conductivity enhancement of the base fluid ∗Corresponding Jeffrey S. Allen: jstallen@mtu.edu 1 TFEC-2024-50984 might be due to this migration of particles. 2. MODELLING AND FORMULATION OF THERMOPHORESIS Heat transfer and nanoparticle transport are thermodynamically coupled. heat flux Jq=Lqqd dx1 T +Lqµ−1 TdµT dx (1) particle flux Jp=Lµqd dx1 T +Lµµ−1 TdµT dx (2) The transport forces are thermal (T) and chemical ( µT), The L-coefficients are known as Onsager coefficients [16–19]. Mechanisms such as sedimentation or buoyancy are not considered here. This could potentially open the gate for microgravity or space applications of nanoparticle enhanced PCMs. Particle flux can be expressed in terms of gradients in temperature and concentration. Jp=−Lµq T2dT dx −Lµµ TµT cdc dx (3) The total particle flux is given by Jp=−cDTdT dx−DBdc dx=Jp,c+Jp,T (4) where DBis Brownian diffusion coefficient, DTis thermal mobility coefficient, c is concentration of particles, Jp,candJp,Tare the diffusion flux due to concentration and temperature gradients, respectively. Comparing equations 3 and 4: DT=Lµq T2c– and – DB=Lµµ TµT c(5) The relative strength of Brownian diffusion to thermal mobility is evaluated at net zero mass flux. atJp= 0 ,1 cdc dx=−DT DBdT dx(6) The Soret coefficient STis the ratio of thermal mobility coefficient to Brownian diffusion, ST=DT/DB. Using Einstein-Stokes and thermal diffusion equations [20, 21], the relative effect of the thermal mobility to the Brownian diffusion can be written in the non-dimensional form as the thermal diffusion factor, which is the Soret coefficient scaled with temperature, TST. DT=β ρnfµnf T;DB=kBT 3πµfdp(7) TST=− 1 c∂c ∂x 1 T∂T ∂x! Jp=0=3πµnfµfdpβ ρnfkBT(8) dpis particle diameter, µfis the viscosity of the base fluid, ρpandρnfare the density of particle and nanofluid, andkBis Boltzmann constant. Nanofluid is the nanoparticles dispersed in base-fluid. When TST<1, Brownian diffusion prevents the development of a nanoparticle concentration gradient . When TST>1, then a nanoparticle concentration gradient might develop due to thermophoresis. We can not assume the thermal conductivity or density of the fluid to be constant over the applicable range of temperature. So it is necessary to provide these properties as a function of temperature and it is shown in table 1. 2 TFEC-2024-50984 Table 1 Temperature dependent material properties [22–25] Ethylene GlycolWaterAcrylic AcidFormic AcidCaprylic AcidAcetic AcidParaffin Density, [g/mL], AB−(1−T/T c)n A 0.32503 0 .34710 0.34645 0.36821 0.29231 0.35182 0.23837 B 0.25499 0.27400 0.25822 0.24296 0.26676 0.26954 0.25763 n 0.172 0.28571 0.30701 0.23663 0.28020 0.26843 0.274 Tc, K 645 647.13 615 580 692 592.71 745.26 Viscosity, [cP], log10µ=A+B/T +CT+DT2 A -16.9728 -10.2158 -15.9215 -4.2125 -10.4823 3.8937 -8.5505 B∗1033.1886 1.7925 2.4408 0.97953 2.067 .784 1.6708 C∗10−23.2537 1.7730 3.4383 0.552 1.8423 0.6665 1.5675 D∗10−5-2.448 -1.2631 -2.7677 -0.57723 -1.3722 -0.75606 -1.2341 Thermal Conductivity [W/m-K] organic compound: log10K=A+B 1−T C2/7 inorganic compound: K=A+BT+CT2 A -0.5918 0.2758 -1.6101 -0.8626 -1.6624 -1.2836 -1.5198 B - 4.61E-03 0.9742 0.3692 0.9819 0.5893 0.8067 C 645 -5.5391E-06 615 580 692 592.71 745.26 Specific Heat [J/mol-K], Cp=A+BT+CT2+DT3 A 75.878 92.053 -18.242 -16.110 70.790 -18.944 151.154 B 6.4182E-01 -3.9953E-02 1.2106 8.7229E-01 1.7647 1.0971 2.7878 C∗10−3-1.6493 -0.21103 -3.1160 -2.3665 -4.1521 -2.8921 -6.1542 D∗10−61.6937 0.53469 3.1409 -2.4454 3.9451 2.9275 5.5249 Table 2 Properties of Nanoparticles [22, 25] Thermal Density Conductivity ρp[kg/m3]kp[W/m-K] Alumina 3950 50 Copper 8960 402 Gold 19300 317.422 Silver 10500 428.227 Titania 4000 11.7 3. RESULTS AND DISCUSSIONS Five particle materials were included in this study: gold, alumina, titania, copper and silver. The density and thermal conductivity for these particles are listed in Table 2. These properties are considered constant as the variation in density and thermal conductivity at the applicable temperature is insignificant. Six types of PCMs were studied. These were modeled only in the liquid phase without solidification. The base fluids are acetic acid, acrylic acid, caprylic acid, ethylene glycol, formic acid and water. Nanoparticles of the five materials listed earlier were paired with these fluids with varying particle size and volume fraction. The potential for particles to migrate or remain uniformly distributed depends on the size of nanoparticles, volume fraction, particle type, fluid, and operating temperature range. 3 TFEC-2024-50984 3.1 Impact of Particle Size and Temperature Variations in thermal diffusion factor with temperature for copper particles of diameter 2, 5 and 20 nm at 5% volume fraction in all six fluids are shown in Fig 1. There is potential for more uniformly distributed particles at 2 nm as compared to 20 nm. This observation aligns with the results reported by L ¨usebrink and Ripoll [26]. The TSt curves shift to large values as particle size increases indicating Brownnian diffusion is dominant relative to thermophoresis. Nevertheless, caprylic acid and ethylene glycol always have some potential for particle migration. (a)2 nm diameter (b)5 nm diameter (c)10 nm diameter Fig. 1 Temperature vs Thermal Diffusion Factor with Copper Particles We can also see from these figures, that particles are expected to be more uniformly distributed at higher temperatures and thus, the migration towards cold end. This result is supported by experimental and molecular dynamic simulation results of Galliero and V olz [27], Duhr and Braun [28]. 3.2 Impact of Particle Type Nanoparticles of alumina, copper, silver, titania and gold were studied for 5 nm diameter and 2% volume fraction. Alumina (Fig 2a) and titania (Fig 2b) particles have potential for thermophoresis over the entire range of temperatures. While Brownian diffusion is dominant at temperatures greater than 300 L for gold (Fig 2c) and silver (Fig 2d) particles in water, acrylic acid, formic acid and acetic acid. The variation in thermal conductivity between different types of particles affect the migration potential. Alu- mina and titania have relatively low thermal conductivity as compared to silver, gold, and copper. While particle density also plays a role, its effect is minor due to the minimal change in nanofluid density caused by the addition of nanoparticles. The results indicate that to offset the lower thermal conductivity of alumina and 4 TFEC-2024-50984 (a)Alumina Particles (b)Titania Particles (c)Gold Particles (d)Silver Particles Fig. 2 Comparison of particle materials at 5 nm diameter. achieve the same thermal diffusion as copper, its density would need to be increased by 117 times. However, the impact of density remains insignificant because it changes proportionally with the particle fraction so the fluid density is primary in determining nanofluid density. 3.3 Impact of Volume Fraction The effect of volume fraction is not insignificant. Figure 3 shows the change in thermal diffusion factor for a range of volume fractions up to 5%. Only copper particles at 360 k are shown. Results are similar for other particle types and temperatures. 3.4 Comparison with Reported Published Data Figure 4 illustrates the trends in TSt for a few select particle-fluid pairs. All combination except for acetic acid with 2.5 nm copper particles have thermal diffusion factors greater than one, indicating a potential for non- uniform particle distributions due to thermophoresis. Figure 4 also includes selected test conditions (square symbols) from studies on thermal conductivity enhancement of phase change materials [3, 29–32]. Data from Iyahraja1 and Rajadurai [29] and Kang et al. [31] (black and blue symbols, respectively) align with the pre- dicted trend lines that indicate an increase in the size of silver nanoparticles dispersed in water increases the potential for thermophoretic migration. Also, when particles of lower thermal conductivity like alumina are used, as shown by red square [30], there is a shift towards higher thermophoretic potential. The fluid viscosity has the most prominent impact on particle migration as can be seen from the dashed green trend line, with test conditions from Eastman et al. [3] (green square). The thermophoretic potential is 1000 times greater than Brownian diffusion even though the particle is only 4 nm radius. The most commonly used PCM, paraffin, 5 TFEC-2024-50984 Fig. 3 TSt dependency on volume fraction for 5 nm copper particles at 360 K. Fig. 4 TSt dependency on particle radius, particle-fluid pair, volume fraction, and temperature. also shows that the dispersion of copper nanoparticles may result in thermophoretic migration with a particle radius of 10 nm [32]. The experimental data shown in Figure 4 all have potential for thermophoretic migration except for the sce- nario of dispersed copper nanoparticles of 5 nm diameter in acetic acid shown by the magenta square. This combination of particle size, material, and base fluid at 320 K is dominated by Brownian diffusion and ought to remain uniformly distributed. Choosing the appropriate combination of PCM and nanoparticle might avoid the development of concentration gradients. Though such concentration gradients have not been reported, most thermal conductivity studies are not cycled in the same manner as PCM energy storage systems. PCM thermal storage systems have much longer operational times and particle concentrations become locked be- tween cycles due to solidification. 4. SUMMARY The analysis reported herein has accounted for the experimentally verified empirical relations and considers the nanoparticle and fluid as a continuum. Values of thermal diffusion coefficient DTare determined from previously validated equations 7 and thermophoretic force FTthat match well with experimental values of Semenov and Schimpf [33], Ramachandran et al. [34]. The work presented here shows that depending on the range of temperature, particle size, type of fluid and type of particle, the dispersed nanoparticles have the 6 TFEC-2024-50984 potential for migration. Nanoparticles will likely be more uniformly distributed at large temperature gradients or high viscosity base fluids. This work looks at the nanoparticles dispersed in PCM in a single phase liquid model. This will be extended to include the phase transition effect going forward. During the phase transition, the PCM will have certain interfacial velocity which can be compared to the thermophoretic velocity. This could reveal if the particles will lag, or always lead the interface leading to the sticking condition at that interface. ACKNOWLEDGMENTS Support for this work was provided by the Department of Mechanical Engineering - Engineering Mechanics at Michigan Tech and through the John F. and Joan M. Calder Professorship in Mechanical Engineering. NOMENCLATURE β non-dimensional nanofluid thermal conductivity (-) c concentration kg/m3 DB Brownian diffusion coefficient m2/s dp particle diameter m DT thermodiffusion coefficient m2/s-K f subscript for base fluid - FT thermophoretic force N Jp mass diffusive flux kg/m2/s Jp,c mass diffusive flux due to concentration gradient kg/m2/s Jp,T mass diffusive flux due to temperature gradient kg/m2/s kB Boltzmann Constant m2kg/s2-K kf base fluid thermal conductivity W/m-K knf nanofluid thermal conductivity W/m-K kp particle thermal conductivity W/m-K Lµµ Onsager coefficient - µnf viscosity of nanofluid N-s/m2 µf viscosity of base fluid N-s/m2 nf subscript for nanofluid - p subscript for particle - φ volume fraction of particles - φe equivalent volume fraction of particles - rp radius of particle nm ρf density of base fluid kg/m3 ρnf density of nano fluid kg/m3 ρp density of nanoparticle kg/m3 ST Soret Coefficient K−1 T temperature K Tm melting temperature K VT drift velocity m/s 7 TFEC-2024-50984