# Thermal Conductivity Calculations for Nanoparticles Embedded in a Base Fluid
## 3 Results  
According to the experimental data obtained at room temperature by [PERSON] et al. [62], calculations of the thermal conductivity ratio (the ratio between the effective thermal conductivity of the NFs and the thermal conductivity of the base fluid) for copper oxide (CuO) and alumina (Al\({}_{2}\)O\({}_{3}\)) nanoparticles of diameters 36 nm and 33 nm, respectively, that were dispersed in deionized water were performed using Equation (1). The results are delineated (dotted lines) in Figure 1. The thermal conductivities of CuO and Al\({}_{2}\)O\({}_{3}\) nanoparticles found using the trial and error method were equal to 7.8 W/m\(\cdot\)K and 4 W/m\(\cdot\)K, respectively. According to [PERSON] and [PERSON]'s model (Equation (6)), these results (7.8 W/m\(\cdot\)K and 4 W/m\(\cdot\)K) give a good thermal conductivity ratio compared to that of the experimental data shown in Figure 1. The data of the base fluid, nanoparticles, and the parameter values are shown in Table 1. However, the thermal conductivities of the two oxide nanoparticles were calculated based on Equation (4) and Tables 1 and 2, which give 7.76 W/m\(\cdot\)K and 4.04 W/m\(\cdot\)K, respectively, and were in good agreement with the results obtained using Equation (1).  
In Figure 1, for comparative purposes, [PERSON] and [PERSON]'s [63] calculations were considered for the Al\({}_{2}\)O\({}_{3}\) nanoparticles represented by [PERSON] et al.'s [62] data. [PERSON] and [PERSON] [63] used [PERSON] et al.'s [64] model, in which contributions from both the interfacial layer and aggregation of the nanoparticles were taken into account. The results underpredicted the reported data. [PERSON] and [PERSON] [63] attributed this underprediction of their calculations to another contributing parameter that may be required to predict the thermal conductivity ratio of NFs. A detailed exposition that discusses the results and parameters is provided in the next section.  
The same calculation procedure was performed for 20 nm ZnO nanoparticles that were dispersed in EG with [PERSON] et al.'s [52] experimental data by using Equations (1) and (6) and Table 1. The results are reported in Figure 2. The obtained thermal conductivity value 1.6 W/m\(\cdot\)K of ZnO nanoparticles obtained from Equation (1) was close to that of the previously reported values of 1.4 W/m\(\cdot\)K for polycrystalline ZnO thin films [65], and 1.16 W/m\(\cdot\)K for 20 nm ZnO nanoparticles [66]. However, according to the theoretical model of [PERSON] et al. [67], calculations of the thermal conductivity ratio due to aggregations of nanoparticles were performed by [PERSON] et al. [52] for their experimental data (solid blue-line), as seen in Figure 2. Our calculation was more precise than [PERSON] et al.'s [52] calculation.  
Figure 1: (Color online) Calculated thermal conductivity ratio (K\({}_{\rm eff}\)/K\({}_{\rm m}\)) versus the particle volume fraction for Al\({}_{2}\)O\({}_{3}\) and CuO nanoparticles in a water base fluid using the [PERSON]’s model (Equation (1)) and [PERSON]’s model (Equation (6)). Expt. data are from [62]. The solid black-line is from [PERSON] and [PERSON]’s calculations [63].  
Figure 2: (Color online) Calculated thermal conductivity ratio (K\({}_{\rm eff}\)/K\({}_{\rm m}\)) versus the nanoparticle volume fraction for a ZnO-ethylene glycol (EG) nanofluid, using the [PERSON]’s (Equation (1)) and [PERSON]’s models (Equation (6)). Experimental data are from [52]. The solid blue-line represents [PERSON] et al.’s calculations [52].