# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 1 Introduction  
The study of heat transport can provide engineering solutions for a plethora of important problems in many scientific and technological disciplines [1; 2]. The complexity of the description of heat transport is increased when the system under study is a heterogeneous material, consisting of more than one phases or materials in discrete forms. Many natural substances, such as stones, soil, and sand (aquifers, oil reservoirs), zeolites, biological tissues (bones, wood), as well as man-made materials, such as cement, ceramics, foams, and colloidal solutions, are considered heterogeneous materials [3]. The characteristic length of heat transport ranges from a few tens of nanometers, for colloidal solutions and nanofibers, to the field scale, for geological applications. A very useful parameter for characterizing heat conduction in a material is the effective thermal conductivity (\(k_{eff}\)) [3; 4; 5]. For isotropic materials, the effective thermal conductivity is the ratio of the magnitude of the heat flux to that of the temperature gradient in the material. In most occasions, the conductivity and the volume fraction of each component are not sufficient to determine theeffective thermal conductivity of the medium. Parameters such as the composition, the shape, and the layout of the various components also play an important role. Thus, the thermal conductivity of a heterogeneous material is related to the thermal conductivity of individual phases (air, water, solid particles), the volume fraction, and the microstructure geometry [3; 4; 5]. [PERSON] introduced the idea of increasing the thermal conductivity of a fluid by mixing suspended metallic or non-metallic solid particles [1; 3; 6]. He was also the first to calculate the effective conductivity of a set of spheres in a dilute solution. Since then, many models were proposed to describe the thermal conductivity of composite materials, the most popular being [PERSON]'s expressions [3]. However, these models are unable to include the particularities of each structure. Ultrafine particles tend to form aggregates, as a result of their random movement and the action of Van der Waals interparticle forces. Given the increased connectivity of the particles between themselves, as well as between them and the walls, aggregated systems have different thermal properties from fully dispersed particles. Nanofluids and aerogels are typical systems that are comprised of fractal-like aggregates [7]. For the calculation of the equivalent properties in such systems, numerical simulations of the underlying internal transport phenomena should be performed. Conventional numerical techniques, such as finite differences (FD) and finite elements (FE), have been used extensively with success in a wide range of problems [8; 9]. The FE method, in particular, has enjoyed extensive use and found vast commercialization, due to its stability and applicability. However, the above methods present certain inherent problems associated with the requirement for a well-organized and connected mesh in the boundaries and the main domain of the problem. Creating this well-coherent mesh can be particularly difficult and usually requires timely user involvement, especially in complex geometries with steep gradients of the field variables, where substantial mesh refinement is usually needed. Because of this difficulty, new numerical methods have appeared recently, commonly called meshfree methods [10; 11; 12; 13]. These methods overcome many of the problems associated with well-connected mesh creation through its complete elimination in the solution process. This makes them more flexible in difficult geometries, as the grid can be easily refined in strenuous areas, such as the interface between the materials. The main advantage over traditional mesh-based numerical methods is related to the approximation of the field variables. Local-type interpolation is used without the necessity of a connected mesh in the domain.