Numerical Solution to the Problem of Thermal Conductivity in a Porous Plate with a Topology of Triply Periodic Minimal Surfaces

Introduction. Porous materials are widely used in various industries, such as metallurgy [1], microelectronics [2], and construction [3]. Their unique properties, due to the presence of cavities and voids filled with gases that conduct heat poorly, make them ideal for use in the production of thermal insulation [4]. The urgency of this problem is related to the need to improve energy efficiency in various areas, which is an important aspect in the context of sustainable development. In light of this, in [4], the authors developed an energy- and resource-saving technology for producing an effective porous heat and sound insulation structural material, which emphasizes the importance of research in this area.

A comparative analysis of the literature shows that the problem raised by the authors has been sufficiently studied, but there are still a number of aspects that require more in-depth study. Paper [5] presents modern porous thermal insulation materials, including the use of silicate raw materials in their production. In [6], various types of porous material matrices are studied, and it is emphasized that the geometric size of the sample (specifically, its thickness, optimal for sound absorption — about 20 mm) significantly affects their characteristics. In addition, it should be noted that porous materials have a lower mass compared to homogeneous media, which makes them mostly attractive for use under conditions of mass and size restrictions, such as the aviation [7] and space industries [8]. In the context of the petrochemical industry, studies conducted by Mazitov A.A. [9] demonstrated the capabilities of a new web application for mathematical modeling of non-stationary oil flow in a porous medium, which emphasizes the importance of porous materials in this area as well. Paper [10] clearly demonstrated the relevance of the analysis of the mechanical properties of porous plastics, since an applied methodology for calculating the vibrations of piezoceramic porous plates was developed using the finite element method.

Despite their advantages, such as low weight, high strength characteristics and predictable thermal properties, porous materials have disadvantages related to the stochastic distribution of pores and the anisotropy of their properties, which complicates the mathematical and physical analysis of transport processes. In this regard, the analysis of a material with a porous structure created on the basis of triply periodic minimal surfaces (TPMS) is of considerable interest, since it demonstrates predictable properties depending on the geometric characteristics of the elementary cells.

Thermal properties of materials based on TPMP are actively studied in scientific circles [11]. To model the processes occurring in porous media, both numerical [12] and analytical methods are used. However, one of the main problems remains the search for a qualitative connection between the thermophysical characteristics [13] and the geometry of porous media [14]. In this context, the study of the temperature field in porous bodies, taking into account their geometric characteristics and cell porosity, as well as the application of modern numerical methods and algorithms, is an area with high potential and considerable topicality. The development of numerical algorithms capable of describing the behavior of temperature by coordinate and time, as well as heat flow depending on changes in the porosity coefficient, allows for a more accurate and effective approach to the design of heat exchangers and the development of thermal protection of equipment.

Thus, the objectives of this research were to study the process of thermal conductivity in a porous plate with a given porosity made of PETG plastic, and to search for dependences to determine the coefficient of effective thermal conductivity of such materials based on Fischer-Koch S cells. To achieve this goal, a problem of thermal conductivity in a porous plate was formulated and solved, and an analysis of the obtained solutions was performed. The results of this study can be useful for practitioners in the design of heat exchangers and in other areas of application.

Research Methods. In the study of thermal conductivity in porous materials, a combination of two approaches was used: representative volume elementary (RVE) and computational homogenization. This methodology was implemented on Fisher-Koch S TPMP cells to calculate the effective thermal conductivity [15]. The effective thermal conductivity coefficient of the material was determined from formula [16]:

where λ_s — thermal conductivity coefficient of the frame material; φ — porosity [17].

The density of the porous material was determined from formula [18]:

where ρ_s — density value of the material from which the cell is made.

The geometric characteristics of the cellular structure are the determining factors on which the degree of porosity of the material depends. They were calculated using the mathematical expression:

where V_ТПМП — cell volume; V — cube volume.

Numerical Solution. The solution to problem (1) – (4) was obtained by the finite difference method [19]. According to this method, a space-time grid with coordinate steps Δξ and time steps ΔFo is introduced. In this case:

(5)

where I, K — number of steps along coordinate ξ and time Fo, respectively.

On grid (5), the grid functions are introduced [20]. Using the explicit approximation scheme of differential operators, problem (1) – (4) takes the following form [21]:

2.5 million grid cells were used in the calculations.

Two key limitations were established during the calculations. First of all, the PETG material was characterized by constant and predetermined thermal parameters. It was also assumed that heat exchange occurred exclusively on surfaces with specified boundary conditions, while the remaining cell faces were thermally insulated.

In the course of the study, it was found that using a grid containing 2.5 to 3.0 million cells leads to an optimal solution to the problem. Increasing the number of grid elements does not significantly affect the accuracy of the solution, but drastically increases the time costs and complexity of the process.

To confirm the correctness of the results obtained in the MathCAD environment, the authors conducted additional modeling in the ANSYS system. The dynamics of temperature changes is graphically presented in Figure 2.

Research Results. Figure 3 shows the calculation results obtained by the finite element method (FEM) and the finite difference method (FDM). The analysis of the results allows us to conclude that the dimensionless temperatures obtained by the two methods are in satisfactory agreement. By analyzing the computational experiments, it is established that the Fisher-Koch S-type TPMP cell retains its structure (there are no internal intersections, complete filling of pores, etc.) in the wall thickness range 0 < δ < 0.002 m. The time evolution of isotherms and the dynamics of their speed mode are shown in Figures 4, 5. A notable feature is that in the process of time development, each isothermal line is formed on the surface with a characteristic initial speed inherent in it. It is noteworthy that an increase in the porosity of the material causes an increase in the heat flow over the entire time interval of the research, which is clearly demonstrated in Figure 5. The dependence of the heat flow density on time with varying porosity is shown in Figure 6.

Discussion and Conclusion. In the course of the study, a mathematical model of heat transfer for a flat porous plate based on the Fisher-Koch prefractal S type TPMP was developed. The accuracy of the solution was confirmed by comparison with alternative calculation methods, such as finite-difference and finite-element analysis. The maximum deviation of the results according to the Chebyshev norm did not exceed 3%.

One of the significant advantages of the proposed approach is the ability to calculate the temperature distribution and heat losses in porous media without using complex computer programs and expensive computing equipment. The expressions obtained for the temperature function significantly simplify the process of engineering calculations of heat transfer in such materials.

The numerical modeling results, consistent with fundamental studies on thermal conductivity [20], methods for specifying boundary and initial conditions [15], as well as with various approaches to numerical calculations [11], open up new prospects for further scientific research. In particular, the developed approach can be applied to the analysis of one-dimensional heat transfer in porous media, including the calculation of temperature distribution, heat fluxes, and effective thermal conductivity coefficients of materials based on TPMP.