Model for Nanofluids Thermal Conductivity Based on Modified Nanoconvective Mechanism

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


Introduction
The urgency of reducing carbon emissions from various energy sources in the world is increasingly emphasized to avoid increasing the impact of climate change. Suggestion regarding the target of reducing global emissions to <1 trillion tons of CO2 in 2100 become a focus on various sectors in the world. Therefore, many countries have developed various technologies of zero carbon emission sources. One of the energy fulfillment technologies with the lowest carbon emission level is nuclear power. Development of nuclear reactor technology is still being developed massively to satisfy its security requirements and economic aspects [1].
Currently, efforts to increase the operating power density of light water reactor (LWR) is one of the approaches used to improve the economic properties of a reactor. There are several possible attempts. One interesting alternative effort besides conventional efforts such as fuel assembly management and optimization fuel design engineering is the seeding of cooling water with nanoparticles to produce fluids with superior heat dissipation [2]. Several tests at MIT's Nuclear Research Reactor have shown that the use of nanofluids (i.e. fluids with dispersed nanoparticles) can increase the productivity of pressurized water reactor (PWR). Nanofluids can prevent the formation of a vapor layer around the fuel rods and it can increase critical heat flux (CHF) significantly [3]. The increasing of CHF in nanofluids with nanoparticles such as silicon, aluminum, and titanium oxide can reach 200% [4]. In addition, the retention of nuclear reactors increased by 40% when accidents on the use of nanofluids was a very promising prospect [5,6].
The ability of heat transfer is one of the most interesting aspects of nanofluids [7]. Some theories related to heat transfer mechanisms such as the Maxwell suspension approach, heat propagation speed [8], and Brownian analysis on nanofluids [9] are some theories that are widely used to explain the heat characteristics of nanofluids. Nevertheless, the theory has not yet been obtained a satisfactory description.
The nanofluids heat transfer mechanism proposed by Jang and Choi [7,9] has a complex picture. In the model are presented various possible heat transfer mechanisms such as conduction by intermediate base fluid, internanoparticles, and convection between fluid base with nanoparticles. However, in the proposed model there are several approaches such as the expression of the Nusselt number as the quadratic function of the Reynolds and Prandtl numbers which are often not appropriate for some conditions [10][11][12]. Furthermore, the study of the viscosity dependence of nanofluids on temperature has not been explicitly described in the model. Whereas viscosity is a parameter that gives a significant influence on convection coefficient of nanofluids [13]. Therefore, in order to correct these problems, we propose a modification of the nanofluids conductivity model especially for the nanoconvention aspect.

Modelling the Nanofluids Thermal Conductivity
In the model constructed by Jang and Choi [7] it is stated that basically the conductivity of nanofluids can be analyzed based on four mechanisms. The first mechanism is the heat transfer mechanism generated by the collisions between the fluid base molecules. The second mechanism is generated from the thermal diffusion of the nanoparticles in the fluid. The third mechanism is derived from collisions between nanoparticles due to Brownian motion. However, based on calculations, the energy flux value in the third mechanism is not significant and this mechanism is often neglected. While the last mechanism is a thermal interaction of the dynamics or dancing of nanoparticles with a fluid base molecule, also called the nanoconvection mechanism.
Basically in thermal conduction analysis it always involves the transfer of energy from particles that have higher thermal energy to particles that have lower thermal energy due to the interaction between particles. In gases and liquids, the thermal energy can be transferred through the interaction of collisions between particles. Based on the kinetic theory, the net energy flux passing through the z-plane is given by [7], where J υ , C ,ˆV C , l , T, and υ ρ are energy flux, molecular average velocity, heat capacity per unit volume, mean free path, temperature and energy density respectively. The spatial averages of these are [14]. Thus the thermal conductivity is given by, 13 In the first heat transfer mechanism because it involves only collisions between the fluid base molecules and if there is a fraction of nanoparticles in the fluid, f, then the net flux is, 1 , where k is the thermal conductivity and a parameter with subscript BF related to physical parameter for base fluid. Furthermore it can easily defined the net flux due to the second mechanism as, 2 , where a parameter with subscript P related to physical parameter for nanoparticle. However in the second mechanism there is an obstacle that interrupt the heat transfer process called the Kapitza resistance factor thus the nanoparticle conductivity of the second mechanism can be expressed as, where kPo is the conductivity of the particle before it involves the Kapitza resistance. The third mechanism involving collisions between nanoparticles due to Brownian motion and it produces energy flux as, 3 , 13 where P and V C are the mean collision length of particles and the translational velocity of particles. Through the kinetic theory the translational velocity can be expressed as 3 18 where KB is Boltzman constant. From the formula it can be found that the energy flux range is at the order of 10 -5 and the value is much smaller than other mechanisms [15]. Therefore in this paper the third mechanism is not taken into account. In the last mechanism, i.e. the nanoconvective mechanism, the energy flux in the heat transfer process can be defined as, where h and T δ are convection heat transfer coefficient and thickness of the boundary layer respectively. Thus without involving the third mechanism, the effective thermal conductivity (keff) of the nanofluids will lead to, 1 In nanoscale, thermal boundary layer will reduce to Prandtl number relation,~P r T δ δ (9) Where δ is the hydrodynamic boundary layer. In Jang and Choi's model [7] it is postulated that nanolayers of ordered liquid molecules act as hydrodynamic boundary layers and it expressed as,~B F Ad δ (10) where A is a proportional constant and in literature the value is 6 18 10 A [7,9]. In the heat transfer to a surface in the fluid, the Nusselt number (Nu) describes the ratio of convection heat transfer to conduction over the surface. Hence the Nusselt number is often used as a bridge to explain the convection coefficient in conduction heat transfer frames. In general the Nusselt number is defined as, hD Nu k (11) In the nanoscale case, the characteristic length of nanofluids, D, can be defined as the nanoparticle equivalent diameter and k is kBF [9]. However in a macroscopic view with the interaction between nanofluids and fluid flow geometry the characteristic length can be expressed as the hydraulic diameter of channel while the parameter k in eq. (11) is expressed as keff [16,17]. Thus for this article the expression of the convective coefficient can be written as Heat transport in nanofluids systems has the same characteristics as heat transfer in fluidized beads. Brodkey et al. [17] have reported that fluidized beds (contain micron-sized particles) have Nusselt numbers that are 20-100 times larger than single particles. Therefore, to satisfy these conditions, based on the relationship between Nusselt numbers for particles on heat transfer fluidized beds, the expression h must be modified [19]. In general, the Nusselt number is a function that contains [20,21], Re, Pr, , , ,

BF p
Nu Nu K K f particle shape (13)  where Re is the Reynolds number. Currently there are many Nusselt number expressions that have been used as in Table 1. Altough almost all Nusselt number expressions contain parameters of Reynolds and Prandtl numbers and almost all of them satisfy the expression of the Nusselt number function as eq. (13) but these expressions contain significantly different numbers of Reynolds and Prandtl numbers. Therefore declaring the keff formula as a function of the Reynolds number and Prandtl number explicitly through the Nusselt number expression reduction is quite complicated. There are many possible combinations allowed. Except by the expression of the Nusselt number proposed by Zerradi [12] which states the Reynolds number and Prandtl number as the termophysical coefficient and it depends on the experimental data fitting. 2 0.5RePr Nu [9] 2 2 Re Pr Nu [11] 0.5 0.5 0.021Re Pr Nu [10]  Greek symbols correspond to thermophysical coefficient One possible alternative to solve the difference problem in determining the Nusselt number is still to declare the keff to be explicitly dependent on the Nusselt number. The nanofluids conductivity remains presented in eq. (15) and does not need to expand it to the expression of the Reynolds number and the Prandtl number. While the experimental value of Nusselt number of a fluid can be determined through the instrument system [23]. However various studies have reported that the Nusselt number continues to change with changing rheological factors. Thus it will create another problem. Therefore, to avoid the inconsistency of the formula with experimental data it is worthy to declare the Nusselt number as a function of the termophysical coefficient as in Zerradi model [12].
Using the fluidized beds analogy to the general relation of the convection coefficient for Brownian motion-induced Stokes flow of multiple nanoparticles has a modified form as, Re Pr m n BF m p k h C f d (14) Through the substitution of eq. (11), (9), and (10) into eq. (8) it will lead us to,

R eP r Pr
In Fig. 1 a comparison between models in eq. (15) with the experiments by Mintsa et. al [24] is shown. We compared the model on two types of nanofluids i.e. CuO and Al2O3 which were dispersed in water. The material properties of the fluid and nanoparticles used we refer to Jang & Choi experiments [7]. It appears the model we proposed is quite appropriate with the experimental data because the variance is small. From the results it can be stated that the volume fraction of the dispersed nanoparticles in the fluid directly proportional with the conductivity. For information, through fitting the data can be obtained the parameters of thermophysical parameters: m and n. For CuO the value m = 1.257 and n = 1, while for Al2O3 the value m = 1.576 and n = 1. Based on the results it appears that the value of the thermophysical parameters obtained through this model is lower than the model proposed by Jang and Choi [7] but higher than that the model proposed by Pak and Cho [11]. In addition, it is clear that the thermophysical parameters for different nanofluids may have different values. Thus declaring those thermophysical parameters in a constant will decrease the accuracy of the proposed model. Other interesting results that can be found in CuO and Al2O3 nanofluids the both Prandtl numbers are almost same. Physically, the Prandtl number represents the momentum diffusivity ratio to the thermal diffusivity that is, Pr p c k μ (16) where cp and μ are heat type and dynamic viscosity respectively. In nanofluids due to occurs interaction between nanoparticles, fluid base, or both, the physical parameter of the above equations can be viewed as effective conditions. The effective specific heat of nanofluids can be expressed as [25],  Figure 3. Comparison of conductivity model in eq. (26) with the experimental data [29] for Al2O3 nanofluids (d = 38.4 nm, f = 1%).

Conclusions
Through modification of the nanoconvection formula in the nanofluids heat transfer mechanism proposed by Jang and Choi we propose a model of thermal conductivity of nanofluids. The Reynolds number and Prandtl number are expressed as functions of the thermophysical parameters. The model is used to formulate the conductivity relationship with the volume fraction of suspended nanoparticles in nanofluids. This model allows different materials to have different thermophysical parameters. In this article we also propose a model of viscosity as a function of temperature because it can provide very significant results in nanoconvection mechanisms. The substitution of the viscosity model into the nanofluids conductivity formulation lead us to the conductivity equation as a temperature function. The accuracy of our models can be seen through comparison between the models with the experimental data.