**5. Conclusions**

The effect of particle size polydispersity and the sintering level on the thermal conductivity of aggregated nanoparticles was studied in the present paper. It was shown that both parameters examined here have the potential to change the heat performance of nanofluids drastically.

A method for reconstructing aggregates with the desired polydispersity degree was developed, satisfying simultaneously the requirements for certain morphological characteristics of the aggregate, namely, the fractal dimension and the number of particles in the aggregate. Particle sintering in aggregates was simulated for monodispersed cases and encoded as an overlapping mechanism in two steps: a penetration step and a growth step. In order to ensure mass conservation, the progression of each step was controlled through the minimisation of the error in the volume fraction of the sintered aggregate compared with the volume fraction of the initial aggregate. A meshless method with local refinement was used for the solution of the heat transfer equation and was found to be stable for the complex systems that were studied here. This is of key importance in the present problem as it allows using relatively large working domains that contain overlapping particles or particles at point contact with others and being able to extract statistically meaningful conclusions.

The effective thermal conductivity was calculated for aggregates that resulted from the present method of aggregation of polydispersed particles, then compared with the thermal conductivity of aggregates that were constructed with the Diffusion Limited Aggregation (DLA) method. The dependence of the effective thermal conductivity on the fractal dimension was found to be in good agreement with that in DLA method aggregates, which indicates that the proposed method produces aggregates that are thermally equivalent to those resulting from methods that describe the physical process of particle aggregation. Consequently, one can employ the present method for the investigation of the behaviour of nanofluids in heat transport problems, taking advantage of the increased simplicity of the aggregation algorithm and its rapid convergence to the final configuration.

The variation of the effective thermal conductivity was investigated over a wide range of fractal dimension values, number of particles per aggregate, and standard deviation of the particle size. Compared to fully dispersed particles, aggregation was shown to increase the thermal conductivity in all cases studied here. Small radius deviation does not substantially change the thermal conductance compared to monodispersed cases; however, a further increase of polydispersity leads to a clear reduction of the effective conductivity. On the contrary, strong polydispersity leads to an increase in the projected area, which implies an increase of heat transfer. A possible explanation for our result could be the existence of small particles within the aggregate that hinder heat transfer. This result is qualitatively confirmed by experimental measurements [56,57] according to which nanofluids consisting of particles with low polydispersity levels have higher heat performance compared to particles with high polydispersity.

The two-step Maxwell model predicts a monotonic decrease of the effective conductivity with increasing fractal dimension; however, large deviations from the numerical results were found for most of the cases examined here.

The effect of sintering of the aggregates was investigated and quantified as a function of the overlapping coefficient. Sintered aggregates have a lower effective size than the original aggregates, so a reduction in the effective conductivity should be expected. At the same time, sintering increases the heat conduction by forming larger heat pathways. This interplay yields a maximum in the thermal conductivity as a function of the degree of coalescence. The precise value of the overlapping coefficient that provides the highest conductivity increase depends on the morphological properties and the volume fraction of the initial aggregates. The present study indicates that the conditions of the production and dispersion of nanoparticles have a major impact on the thermal properties of the nanofluids. This is a possible explanation for the large deviations that have been observed between experimental works. Nanofluids with monodispersed particles, which are organized into aggregates with small overlapping, offer the highest heat transfer coefficient over the range of parameter values that were examined here. The results and conclusions of this work are also relevant to nanocomposite materials that contain polydispersed particle inclusions, which are organized in aggregates either at simple contact or in sintered form.

**Author Contributions:** Conceptualization, N.P.K. and V.N.B.; funding acquisition, V.N.B.; investigation, N.P.K., E.D.S. and V.N.B.; methodology, N.P.K.; project administration, V.N.B.; software, N.P.K.; supervision, V.N.B.; validation, N.P.K.; visualization, N.P.K.; writing—original draft, N.P.K.; writing—review and editing, E.D.S. and V.N.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the project "Materials and Processes for Energy and Environment Applications-AENAO" (MIS 5002556) which is implemented under the "Action for the Strategic Development on the Research and Technological Sector", funded by the Operational Programme "Competitiveness, Entrepreneurship and Innovation" (NSRF 2014-2020) and co-financed by Greece and the European Union (European Regional Development Fund).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Computational infrastructure was provided by the Institute of Chemical Engineering Sciences, Foundation for Research and Technology, Hellas (FORTH/ICE-HT).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
