*4.2. Dependence of Conductivity on the Fractal Dimension, the Number of Particles in the Aggregate, and the Polydispersity of the Particles*

In an earlier work by the authors [38], the effect of aggregation on the thermal conductivity of monodispersed particles was studied by varying the fractal dimension and the number of particles in the aggregate. A considerable increase in the thermal conductivity has been shown, even for aggregates consisting of a small number of particles. However, in the present work, a different behaviour is noticed upon the introduction of polydispersity in the particle size. The results are presented next.

**Figure 3.** Effective thermal conductivity as a function of the fractal dimension of aggregates consisting of polydispersed particles by the present method (open, blue symbols) and by simulations from the DLA method (filled, red symbols).

Figure 4 shows the dimensionless effective thermal conductivity of a nanofluid as a function of the fractal dimension, for two volume fractions, namely *f<sup>p</sup>* = 0.03 (Figure 4a,b) and for *f<sup>p</sup>* = 0.1 (Figure 4c). Two different values of the number of particles per aggregate are also investigated, namely *N* = 42 (Figure 4a,c) and *N* = 9 (Figure 4b,c). To enable a comparison, the mean radius of the particles for the polydispersed cases was set equal to the ones in corresponding cases of monodispersed particles. The standard deviation of the particle size varies from *σ* = 0.1*r*<sup>0</sup> to *σ* = 0.5*r*<sup>0</sup> in Figure 4a,b, whereas in Figure 4c it is kept at *σ* = 0.5*r*0. Figure 4 also shows the thermal conductivity predictions of two analytical models, namely, the two-step Maxwell (Equations (7) and (8)) and the single-step Maxwell model (Equation (7)). Every simulation point is the average of 10 realizations with the same parameters. The conductivity ratio of the nanoparticles and the base fluid is chosen to be 130 (*kr p* = 130), which is a representative value for several practical nanofluids, such as water–Fe, engine oil–Al2O3, and water–CuO.

The effective conductivity decreased as the fractal dimension increased in all cases studied here. The level of reduction was affected by the polydispersity degree, the volume fraction, and the number of particles per aggregate. For cohesive aggregates, corresponding to relatively high fractal dimension (*d<sup>f</sup>* = 2.5), polydispersity did not affect the effective conductivity, whereas for smaller values of the fractal dimension and high polydispersity level, a notable variation was observed. More specifically, the thermal conductivity decreased with the increase of the deviation of the particle radius compared with monodisperse particle cases. However, a small deviation (*σ* = 0.1*r*0, *σ* = 0.2*r*0) affected the thermal conductivity only slightly (Figure 4a,b). A 10% reduction was observed for polydispersity degree *σ* = 0.5*r*<sup>0</sup> and volume fraction *f<sup>p</sup>* = 0.1 (Figure 4c), whereas the reduction was only about 5% for volume fraction *f<sup>p</sup>* = 0.03 (Figure 4a,b).

**Figure 4.** Effective thermal conductivity as a function of fractal dimension for: (**a**) volume fraction *f<sup>p</sup>* = 0.03 and number of particles per aggregate *N* = 42; (**b**) *f<sup>p</sup>* = 0.03, *N* = 9; (**c**) *f<sup>p</sup>* = 0.1, *N* = 9 (solid lines, solid symbols), *N* = 42 (dashed lines, open symbols). Black lines: monodispersed cases. Blue lines: two-step Maxwell model. Green line: Maxwell model. Symbols: polydispersed cases.

The previous results are in contrast to the increase of the projected area of an aggregate as the polydispersity level is increased [32]. Particle size distribution affects heat transfer in two distinct ways. Large particles accelerate heat transfer, while small particles hinder it. According to the present results, the smaller particles act as regulators of heat transport; therefore, the effective conductivity is reduced. Experimental works have observed that polydispersity of the nanoparticles has a significant impact on the thermal properties of the nanofluid [56]. Specifically, the largest enhancement has been found for highly monodisperse particles [57].

The single-step Maxwell relation (Equation (7)) remains insensitive to the fractal dimension. However, the two-step Maxwell model, although affected by the fractal dimension and the number of particles in the aggregate, cannot predict the effective conductivity of such systems. It is worth noting that aggregation increased the effective conductivity of the nanofluid significantly in all cases studied. For highly polydispersed particles, when organized into aggregates consisting of, say, *N* = 9 particles per aggregate, a 10% volume fraction (shown in Figure 4c) resulted in a 70% increase of the thermal conductivity. The higher the particles per aggregate, the higher the conductivity increase.
