# Thermal Conductivity Calculations for Nanoparticles Embedded in a Base Fluid
## 4 Discussion  
### Nanoparticles'Thermal Conductivity and Specularity Parameter, \(\beta\)  
Experimental investigations [47; 48; 83; 84] and the molecular dynamics method [85; 86] revealed that an increase in the thermal conductivity of nanofluids is related to an increase in the nanoparticle material density rather than its material. The density of a nanosized material is a size-dependent parameter. [PERSON] et al. [87] showed that the densities of Si nanoparticles with 20 nm and 5 nm diameters decrease by amounts of 6% and 18%, respectively. Experimental studies [88; 89] have also shown that the density of nanosized materials decreases as their size decreases, particularly at a diameter of less than 20 nm. Accordingly, the decreases in the density of CuO and Al\({}_{2}\)O\({}_{3}\) nanoparticles with diameters of 36 nm and 33 nm, respectively, were not significant since their size ranges were larger than 20 nm [87]. The experimental data from [PERSON] et al. [62] at room temperature is seen in Figure 1, where the enhancement of the nanofluid thermal conductivity of CuO was higher than Al\({}_{2}\)O\({}_{3}\) by 28% at a 5% volume concentration. This enhancement return to a higher density of CuO (bulk density = 6.3 gm/cm\({}^{3}\)) than Al\({}_{2}\)O\({}_{3}\) (bulk density = 3.9 gm/cm\({}^{3}\)) [47; 85; 86]. However, CuO and Al\({}_{2}\)O\({}_{3}\) cannot be compared with the other two oxides (ZnO and SiO\({}_{2}\)) because of their different diameters and the great dependence of the properties on the different preparation conditions. Comparisons should be done between nanofluids with the same base fluid, same particle size, and the same volume fraction. In this work, the thermal conductivity of CuO nanoparticles was higher than for Al\({}_{2}\)O\({}_{3}\) nanoparticles, while their bulk thermal conductivity was close to each other (as seen in Table 1). However, the values of the intrinsic thermal conductivity of nanoparticles were due to their material type. The lower value of the nanoparticles' thermal conductivities compared to that of the bulk material can be explained as follows.  
In experimental studies [90], the thermal conductivity of Si nanowires is less than the bulk value by an order of magnitude. In a molecular dynamics study [91], if the size of Si nanoparticles decreased below about 10 nm, their thermal conductivity decreased by two orders of magnitude compared to that of the bulk value. [PERSON] et al. [92; 93] showed that the thermal conductivities of GaN and Ge nanowires are size-dependent. The reduction in value is due to both phonon confinement and boundary scattering effects; however, these depend on size-dependent parameters, such as the Debye temperature, surface roughness, the Gruneisen parameter, and the group velocity of phonons [94; 95].  
In NFs, the thermal energy interaction takes place at the nanoparticles' surface [52; 63]. The specularity parameter, \(\beta\), is a function of the surface roughness, \(\mu\)[96], which reflects the probability of diffusive or specular scattering of phonons and is denoted by the following equation [55]: \(\beta=1-10\mu/D\), where \(D\) is the nanoparticles' diameter. The values of \(\beta\) are between 1 and 0. Due to the lack of experimental values for the surface roughness, \(\beta\), was treated in this work as an adjustable parameter [93; 94], where the values for all materials are listed in Table 2. These results suggested that nanoparticles' thermal conductivity through phonon scattering processes at the surface plays a significant role in transferring energy in NFs.