Method for measuring heat conductivity coefficient of pore type porous medium

Abstract

According to the method for measuring the heat conductivity coefficient of the porous medium, REV elements are selected as a porous medium heat transfer model, the pore structure of the porous medium is analyzed, and fractal characteristics of the porous medium are determined; evaluating the complexity of the pore structure of the porous medium and the distribution condition of the pore space; selecting a proper heat transfer analysis model according to the pore structure complexity and research requirements of the porous medium; the model comprises a simplified REV primitive heat transfer model and a fine REV primitive fractal heat transfer model; substituting the obtained porosity, fractal dimension and other parameters into a selected heat transfer model; and calculating the heat conductivity coefficient by applying a formula of the selected heat transfer model. According to the method, geometric limitation is exceeded through the fractal theory, a more complex pore structure can be processed, the heat conductivity coefficient of the porous medium can be predicted more accurately, and the deviation between a model and an actual structure is reduced; and a simplified rough and fine model series is provided, so that the accuracy of the heat conductivity coefficient of the pore type porous medium is ensured, and the efficiency is also improved.

Description

Technical Field

[0001] The invention relates to the field of porous medium heat transfer, and in particular to a method for measuring thermal conductivity of porous media. Background Art

[0002] Due to its unique pore structure, the measurement of thermal conductivity of porous media faces great challenges. Traditional methods such as the hot wire method and the flat plate method are difficult to accurately reflect the complexity of the pore structure, resulting in large measurement errors. These models and experimental methods can predict and calculate the thermal conductivity of porous media, but they are usually based on specific assumptions and simplifications, such as geometric models and the uniformity of pore structures. The heat transfer process in porous media is complex and involves the interaction between the solid skeleton and the pore fluid. Traditional methods often simplify the processing and are difficult to accurately reflect the actual heat transfer characteristics in porous media.

[0003] The Chinese patent with publication number CN 106248725 A discloses a method for measuring the equivalent thermal conductivity of porous media, which solves the above problems to a certain extent. However, it is based on the APDL program in ANSYS software, uses the finite element numerical simulation method to calculate the heat transfer process of the model, randomly distributes the pore structure of the porous material by volume pores, and analyzes the relationship between the equivalent thermal conductivity of the porous material model and the porosity. The true density of the porous medium is measured by a true density meter, and the thermal conductivity of the porous medium at a certain porosity is measured by the laser flash method. Combined with the finite element simulation results, the equivalent thermal conductivity of the porous medium at different temperatures is calculated. Its shortcomings are: there is a large gap between the simplified structure of the porous medium and the actual situation, and the accuracy is poor. Summary of the invention

[0004] The present invention aims to overcome the deficiencies of the prior art and provides a method for measuring thermal conductivity of porous media, which improves the accuracy and reliability of thermal conductivity measurement by accurately describing the pore structure.

[0005] The present invention provides a method for measuring thermal conductivity of porous media, which is applicable to measuring thermal conductivity of skeleton-type porous media, and comprises the following steps:

[0006] To calculate the equivalent thermal resistance of the REV primitive, the REV primitive is simplified into a cube with a side length of 1, the pore is simplified into a sphere with a radius of r, and the heat flow flows in and out from left to right in the horizontal direction; when the heat flow is only transferred in the matrix skeleton, the thermal resistance is defined as R 1 When the heat flow is transferred between the matrix skeleton and the medium pores, the thermal resistance is R 2 ;

[0007] In the skeleton-type REV heat transfer model, the matrix skeleton thermal resistance is:

[0008]

[0009] In the formula, λ 1 is the thermal conductivity of the matrix, A 1 is the area of ​​the matrix;

[0010] Take any microelement with a thickness of dx in the REV element, take the center of the REV element as the origin, and the angle between the origin and the straight line where the medium and the matrix in the microelement contact the x-axis is α, then the thermal resistance of dx can be regarded as a cross-sectional area of ​​πr 2 sin 2 The dielectric thermal resistance of α is 1-πr with respect to the cross-sectional area. 2 sin 2 Parallel connection of matrix thermal resistance of α;

[0011] In the skeleton-type REV heat transfer model, the parallel thermal resistance of the skeleton and pores is:

[0012]

[0013] In the formula, λ 2 is the thermal conductivity of the medium;

[0014] After integrating equation (1-2), we can get the thermal resistance:

[0015]

[0016] Then, the equivalent thermal conductivity of the skeleton-type REV heat transfer model is:

[0017]

[0018] For the skeleton-type simplified REV heat transfer model, the actual measured porosity Φ is obtained. The relationship between the pore characteristic length and the porosity Φ is:

[0019]

[0020] Substituting equation (1-5) into equation (1-4), we can get the equivalent thermal conductivity λ of the skeleton-type simplified REV heat transfer model: gj for:

[0021]

[0022] As a further improvement of the present invention, for the skeleton-type fine REV heat transfer model, the fractal dimension of the REV primitive is first calculated by equation (1-9), and the measure M of the REV primitive is converted into (L) The relationship with the measurement scale L is as follows:

[0023]

[0024] Where D F is the fractal dimension, M(L) It can be the mass, volume or length of an object, L is the measurement scale;

[0025] In a space with a scale of L, the spatial fractal dimension of the natural stacking of REV primitives is D Fl According to the fractal scaling relationship, the volume occupied by the pores in the primitive can be expressed as Then the porosity Φ of the unit cell is:

[0026]

[0027] Where D Fl is the fractal dimension of the pores in the primitive, D is the Euclidean dimension, l max , l min is the self-similarity upper limit and self-similarity upper limit of the primitive;

[0028] Taking the logarithm of both ends of equation (1-8), and because the dimension of Euclidean space D = 3, the fractal dimension can be expressed as

[0029]

[0030] Substituting equation (1-9) into equation (1-6), the equivalent thermal conductivity λ of the skeleton-type fine REV heat transfer model is gj for:

[0031]

[0032] The present invention provides a method for measuring thermal conductivity of porous media, which is applicable to measuring thermal conductivity of granular porous media, and comprises the following steps:

[0033] To calculate the equivalent thermal resistance of the REV primitive, the REV primitive is simplified into a sphere with a radius of r, the medium pore is simplified into a cube with a side length of 1, and the heat flow flows in and out from left to right in the horizontal direction; when the heat flow is only transferred in the medium pores, the thermal resistance is defined as R 1 When the heat flow is transferred between the matrix particles and the medium pores, the thermal resistance is R 2 ;

[0034] In the particle-type REV heat transfer model, only the medium pore thermal resistance is:

[0035]

[0036] In the formula, λ 2 is the thermal conductivity of the medium, A 2 is the area of ​​the medium.

[0037] Take any microelement with a thickness of dx in the REV element, take the center of the REV element as the origin, and the angle between the origin and the straight line where the medium and the matrix in the microelement contact the x-axis is α, then the thermal resistance of dx can be regarded as a cross-sectional area of ​​πr 2 sin 2 The matrix thermal resistance of α is 1-πr with respect to the cross-sectional area. 2 sin 2 α is the parallel connection of dielectric pore thermal resistance.

[0038] In the particle-type REV heat transfer model, for the parallel thermal resistance of the skeleton and pores:

[0039]

[0040] In the formula, λ 1 is the thermal conductivity of the matrix;

[0041] After integrating equation (1-12), we can get the thermal resistance:

[0042]

[0043] Equivalent thermal conductivity of the particle type REV heat transfer model:

[0044]

[0045] For the particle-type simplified REV heat transfer model, the actual measured porosity Φ is obtained. Since the relationship between the pore characteristic length and the porosity Φ is:

[0046]

[0047] Substituting equation (1-15) into equation (1-14), the equivalent thermal conductivity λ of the particle-type simplified REV heat transfer model is kl for:

[0048]

[0049] As a further improvement of the present invention, for the particle-type fine REV heat transfer model, the fractal dimension of the REV primitive is first calculated, and then the fractal dimension is substituted into (1-16), that is, the equivalent thermal conductivity λ of the particle-type fine REV heat transfer model is obtained. kl for:

[0050]

[0051] A method for measuring thermal conductivity of porous media of the present invention transcends geometric limitations through fractal theory and can handle more complex pore structures. The method of the present invention can more accurately predict the thermal conductivity of porous media and reduce the deviation between the model and the actual structure. The simplified coarse and fine model series provided can be flexibly selected according to the needs of actual problems and resource limitations, which not only ensures the accuracy of the thermal conductivity of porous media but also improves efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 Description and size selection for REV primitives;

[0053] Figure 2 It is a skeleton-type simplified REV heat transfer model;

[0054] Figure 3 It is a schematic diagram of the heat transfer process of the skeleton type REV model;

[0055] Figure 4 It is a skeleton-type fine REV fractal heat transfer model;

[0056] Figure 5 It is a particle-type simplified REV heat transfer model;

[0057] Figure 6 It is a schematic diagram of the heat transfer process of the particle type REV model;

[0058] Figure 7 It is a granular fine REV fractal heat transfer model. DETAILED DESCRIPTION

[0059] In order to facilitate the description of this application, this application is based on the fractal theory REV description method, and according to the internal heat transfer of porous media, the size selection and conditions of the REV primitive are given in detail, such as Figure 1 The REV scale takes a more macroscopic perspective, removing the detailed consideration of the physical and chemical properties and structural composition of porous media, and only needs to obtain the statistical parameters of porous media. This simplified processing method significantly improves the efficiency of porous media heat transfer research.

[0060] 1. Selection of REV primitive size

[0061] First, select a reasonable representative elemental volume (REV). Based on the continuity assumption of porous media, when describing and analyzing with porosity as the characteristic quantity, the selected REV element should satisfy the following requirements:

[0062] (1) The characteristic size of the REV unit is much smaller than the macroscopic size of the porous medium, ensuring that the volume of the REV unit is negligible for the overall structure;

[0063] (2) The REV element contains enough pores to represent the distribution law of the structure, and a real and effective equivalent continuous uniform model and parameters can be obtained in the REV element.

[0064] 2. REV primitive assumptions

[0065] The heat transfer process in porous media mainly includes the heat conduction process between the solid skeleton matrix and the fluid medium, the convection heat transfer of the fluid in the pores, the phase change heat transfer and the radiation heat transfer between multiple phases. According to the characteristics of porous media in actual engineering, the heat conduction process is simplified and analyzed, and the following assumptions are made:

[0066] (1) The porous medium is continuous, uniform, and isotropic within the selected REV element;

[0067] (2) There is only one medium in the internal pores of the porous medium, and there is no reaction between the matrix and the medium;

[0068] (3) Ignore the volume change of porous media caused by temperature increase, and there is no phase change;

[0069] (4) Only heat conduction occurs in porous media, ignoring convection and radiation heat transfer;

[0070] (5) Every point is in a state of thermodynamic equilibrium.

[0071] In addition, given that the heat and mass transfer characteristics of the actual pore structure in space have variability in different directions, fractal theory and methods can play a unique role in dealing with disordered and randomly distributed porous media. Therefore, for porous media, this method proposes a simplified-fine heat transfer analysis model based on REV, and introduces parameters such as porosity and fractal dimension to derive the thermal conductivity calculation formula.

[0072] According to the structural characteristics of the porous media, the porous media are divided into two units composed of base particles, namely the skeleton-type REV heat transfer model and the particle-type REV heat transfer model. The skeleton-type REV heat transfer model is further divided into the skeleton-type simplified REV heat transfer model and the skeleton-type fine REV heat transfer model. The particle-type REV heat transfer model is further divided into the particle-type simplified REV heat transfer model and the particle-type fine REV heat transfer model. The thermal conductivity of the skeleton-type simplified REV heat transfer model, the skeleton-type fine REV heat transfer model, the particle-type simplified REV heat transfer model and the particle-type fine REV heat transfer model are measured below.

[0073] Example 1

[0074] The present invention provides a method for measuring thermal conductivity of porous media, which is applicable to measuring thermal conductivity of simplified skeleton porous media, and comprises the following steps:

[0075] The skeleton-type simplified REV heat transfer model consists of a matrix skeleton and medium pores, such as Figure 2 As shown;

[0076] To calculate the equivalent thermal resistance of the REV primitive, the REV primitive is simplified into a cube with a side length of 1, the pore is simplified into a sphere with a radius of r, and the heat flow flows in and out from left to right in the horizontal direction; when the heat flow is only transferred in the matrix skeleton, the thermal resistance is defined as R 1 When the heat flow is transferred between the matrix skeleton and the medium pores, the thermal resistance is R 2 , the heat transfer process is as follows Figure 3 As shown;

[0077] In the skeleton-type REV heat transfer model, the matrix skeleton thermal resistance is:

[0078]

[0079] In the formula, λ 1 is the thermal conductivity of the matrix, A 1 is the area of ​​the matrix;

[0080] Take any microelement with a thickness of dx in the REV element, take the center of the REV element as the origin, and the angle between the origin and the straight line where the medium and the matrix in the microelement contact the x-axis is α, then the thermal resistance of dx can be regarded as a cross-sectional area of ​​πr 2 sin 2 The dielectric thermal resistance of α is 1-πr with respect to the cross-sectional area. 2 sin 2 Parallel connection of matrix thermal resistance of α;

[0081] In the skeleton-type REV heat transfer model, the parallel thermal resistance of the skeleton and pores is:

[0082]

[0083] In the formula, λ 2 is the thermal conductivity of the medium;

[0084] After integrating equation (1-2), we can get the thermal resistance:

[0085]

[0086] Then, the equivalent thermal conductivity of the skeleton-type REV heat transfer model is:

[0087]

[0088] For the skeleton-type simplified REV heat transfer model, the actual measured porosity Φ is obtained. The relationship between the pore characteristic length and the porosity Φ is:

[0089]

[0090] Substituting equation (1-5) into equation (1-4), we can get the equivalent thermal conductivity λ of the skeleton-type simplified REV heat transfer model: gj for:

[0091]

[0092] Example 2

[0093] The fine REV primitive fractal heat transfer model refers to a primitive model in which two simplified REV primitives exist at the same time but with different sizes and the primitives are arranged in a certain fractal rule. For the skeleton-type fine REV heat transfer model, such as Figure 4 As shown, it is measured based on the skeleton-type simplified REV heat transfer model obtained in Example 1. First, the fractal dimension of the REV primitive is calculated by formula (1-9), and the measure M of the REV primitive is converted into (L) The relationship with the measurement scale L is as follows:

[0094]

[0095] Where D F is the fractal dimension, M (L) It can be the mass, volume or length of an object, L is the measurement scale;

[0096] In a space with a scale of L, the spatial fractal dimension of the natural stacking of REV primitives is D Fl According to the fractal scaling relationship, the volume occupied by the pores in the primitive can be expressed as Then the porosity Φ of the unit cell is:

[0097]

[0098] Where D Fl is the fractal dimension of the pores in the primitive, D is the Euclidean dimension, l max , l min is the self-similarity upper limit and self-similarity upper limit of the primitive;

[0099] Taking the logarithm of both ends of equation (1-8), and because the dimension of Euclidean space D = 3, the fractal dimension can be expressed as

[0100]

[0101] Substituting equation (1-9) into equation (1-6), the equivalent thermal conductivity λ of the skeleton-type fine REV heat transfer model is gj for:

[0102]

[0103] Example 3

[0104] The present invention provides a method for measuring thermal conductivity of porous media, which is applicable to measuring thermal conductivity of granular porous media, and comprises the following steps:

[0105] The particle-type simplified REV heat transfer model consists of matrix particles and medium pores. Figure 5 As shown;

[0106] To calculate the equivalent thermal resistance of the REV primitive, the REV primitive is simplified into a sphere with a radius of r, the medium pore is simplified into a cube with a side length of 1, and the heat flow flows in and out from left to right in the horizontal direction; when the heat flow is only transferred in the medium pores, the thermal resistance is defined as R 1 When the heat flow is transferred between the matrix particles and the medium pores, the thermal resistance is R2, and the heat transfer process is as follows Figure 6 As shown;

[0107] In the particle-type REV heat transfer model, only the medium pore thermal resistance is:

[0108]

[0109] In the formula, λ 2 is the thermal conductivity of the medium, A 2 is the area of ​​the medium.

[0110] Take any microelement with a thickness of dx in the REV element, take the center of the REV element as the origin, and the angle between the origin and the straight line where the medium and the matrix in the microelement contact the x-axis is α, then the thermal resistance of dx can be regarded as a cross-sectional area of ​​πr 2 sin 2 The matrix thermal resistance of α is 1-πr with respect to the cross-sectional area. 2 sin 2 α is the parallel connection of dielectric pore thermal resistance.

[0111] In the particle-type REV heat transfer model, for the parallel thermal resistance of the skeleton and pores:

[0112]

[0113] In the formula, λ 1 is the thermal conductivity of the matrix;

[0114] After integrating equation (1-12), we can get the thermal resistance:

[0115]

[0116] Equivalent thermal conductivity of the particle type REV heat transfer model:

[0117]

[0118] For the particle-type simplified REV heat transfer model, the actual measured porosity Φ is obtained. Since the relationship between the pore characteristic length and the porosity Φ is:

[0119]

[0120] Substituting equation (1-15) into equation (1-14), the equivalent thermal conductivity λ of the particle-type simplified REV heat transfer model is kl for:

[0121]

[0122] Example 4

[0123] The fine REV primitive fractal heat transfer model refers to a primitive model in which two simplified REV primitives exist at the same time but with different sizes and the primitives are arranged in a certain fractal rule. For the particle-type fine REV heat transfer model, such as Figure 7 As shown, based on the simplified REV heat transfer model of the particle type obtained in Example 3, after implementing (1-7) and (1-8) of 2 and calculating the fractal dimension (1-9) of the REV primitive, equation (1-9) is substituted into (1-16), and the equivalent thermal conductivity λ of the particle type fine REV heat transfer model is obtained. kl for:

[0124]

[0125] As can be seen from Examples 1-4, the method for determining thermal conductivity of porous media is mainly divided into three steps: Step 1 is to analyze the pore structure characteristics, including parameters such as porosity and fractal dimension; Step 2 is to select a thermal conductivity model; Step 3 is to substitute the parameters obtained in Step 1 to calculate the thermal conductivity; specifically, as follows:

[0126] (1) Pore structure analysis:

[0127] Analyze the pore structure of porous media and determine its fractal characteristics, including parameters such as porosity and fractal dimension; evaluate the complexity of the pore structure of porous media and the distribution of pore space.

[0128] (2) Model selection:

[0129] According to the complexity of the pore structure of the porous medium and the research requirements, a suitable heat transfer analysis model is selected; the model includes a simplified REV primitive heat transfer model and a refined REV primitive fractal heat transfer model.

[0130] (3) Parameter input:

[0131] The obtained parameters such as porosity and fractal dimension are introduced into the selected heat transfer model; the thermal conductivity is calculated using the formula of the selected heat transfer model.

[0132] For the simplified REV primitive heat transfer model, the thermal conductivity of the primitive model is calculated using formulas (1-6) and (1-16); for the refined REV primitive fractal heat transfer model, the thermal conductivity of the primitive model is calculated using formulas (1-10) and (1-17).

[0133] The effect of the present invention is further described below in conjunction with specific porous media. Based on the simplified and refined heat transfer model of porous media established above, calcium silicate is taken as the research object, the thermal conductivity of the porous media with different porosities is calculated, and the relationship between the porosity and thermal conductivity of the porous media is analyzed.

[0134] Take the macroscopic length L 0 Taking the calcium silicate sample with a diameter of =1.140mm as an example, the maximum pore radius is 48.5μm, the minimum pore radius is 1.43μm, and the air thermal conductivity is 0.0244W / (m·K) under standard conditions; the thermal conductivity of the solid skeleton of calcium silicate is 0.0550W / (m·K).

[0135] The actual effective thermal conductivity of calcium silicate measured is shown in Table 1:

[0136] Table 1 Effective thermal conductivity of calcium silicate with different porosity

[0137]

[0138] The measured values ​​of thermal conductivity of calcium silicate porous media with different porosities are compared with the calculated results of the proposed fractal primitive heat transfer model.

[0139] Table 2 Relative error between REV fractal element thermal conductivity and actual thermal conductivity

[0140]

[0141] From the analysis of Table 2, it can be seen that when the porosity is 30-90%, the relative error between the thermal conductivity calculated by the REV fractal element model and the thermal conductivity measured experimentally is within 7%. The thermal conductivity measured by this method has high accuracy.

Claims

1. A method for measuring thermal conductivity of porous media, characterized in that: It is applicable to the determination of thermal conductivity of skeleton-type porous media, and includes the following steps: To calculate the equivalent thermal resistance of the REV primitive, the REV primitive is simplified into a cube with a side length of 1, the pores are simplified into a sphere with a radius of r, and the heat flow flows in and out from left to right in the horizontal direction; when the heat flow is only transferred in the matrix skeleton, the thermal resistance is defined as R1, and when the heat flow is transferred in the matrix skeleton and the medium pores, the thermal resistance is defined as R2; In the skeleton-type REV heat transfer model, the matrix skeleton thermal resistance is: Where λ1 is the thermal conductivity of the matrix, and A1 is the area of ​​the matrix; Take any microelement with a thickness of dx in the REV element, take the center of the REV element as the origin, and the angle between the origin and the straight line where the medium and the matrix in the microelement contact the x-axis is α, then the thermal resistance of dx can be regarded as a cross-sectional area of ​​πr 2 sin 2 The dielectric thermal resistance of α is 1-πr with respect to the cross-sectional area. 2 sin 2 Parallel connection of matrix thermal resistance of α; In the skeleton-type REV heat transfer model, the parallel thermal resistance of the skeleton and pores is: Where λ2 is the thermal conductivity of the medium; After integrating equation (1-2), we can get the thermal resistance: Then, the equivalent thermal conductivity of the skeleton-type REV heat transfer model is: For the skeleton-type simplified REV heat transfer model, the actual measured porosity Φ is obtained. Since the relationship between the pore characteristic length and the porosity Φ is: Substituting equation (1-5) into equation (1-4), we can get the equivalent thermal conductivity λ of the skeleton-type simplified REV heat transfer model: gj for:

2. A method for measuring thermal conductivity of porous media according to claim 1, characterized in that: For the skeleton-type fine REV heat transfer model, the fractal dimension of the REV primitive is first calculated by equation (1-9), and the measure M of the REV primitive is (L) The relationship with the measurement scale L is as follows: Where D F is the fractal dimension, M (L) It can be the mass, volume or length of an object, L is the measurement scale; In a space with a scale of L, the spatial fractal dimension of the natural stacking of REV primitives is D Fl According to the fractal scaling relationship, the volume occupied by the pores in the primitive can be expressed as Then the porosity Φ of the unit cell is: Where D Fl is the fractal dimension of the pores in the primitive, D is the Euclidean dimension, l max , l min is the self-similarity upper limit and self-similarity upper limit of the primitive; Taking the logarithm of both ends of equation (1-8), and because the dimension of Euclidean space D = 3, the fractal dimension can be expressed as Substituting equation (1-9) into equation (1-6), the equivalent thermal conductivity λ of the skeleton-type fine REV heat transfer model is gj for:

3. A method for measuring thermal conductivity of porous media, characterized in that: It is applicable to the determination of thermal conductivity of granular porous media, including the following steps: To calculate the equivalent thermal resistance of the REV primitive, the REV primitive is simplified into a sphere with a radius of r, the medium pore is simplified into a cube with a side length of 1, and the heat flow flows in and out from left to right in the horizontal direction; when the heat flow is only transferred in the medium pores, the thermal resistance is defined as R1, and when the heat flow is transferred between the matrix particles and the medium pores, the thermal resistance is defined as R2; In the particle-type REV heat transfer model, only the medium pore thermal resistance is: In the formula, λ2 is the thermal conductivity of the medium, and A2 is the area of ​​the medium; Take any microelement with a thickness of dx in the REV element, take the center of the REV element as the origin, and the angle between the origin and the straight line where the medium and the matrix in the microelement contact the x-axis is α, then the thermal resistance of dx can be regarded as a cross-sectional area of ​​πr 2 sin 2 The matrix thermal resistance of α is 1-πr with respect to the cross-sectional area. 2 sin 2 Parallel connection of dielectric pore thermal resistance of α; In the particle-type REV heat transfer model, for the parallel thermal resistance of the skeleton and pores: Where λ1 is the thermal conductivity of the matrix; After integrating equation (1-12), we can get the thermal resistance: Equivalent thermal conductivity of the particle type REV heat transfer model: For the particle-type simplified REV heat transfer model, the actual measured porosity Φ is obtained. Since the relationship between the pore characteristic length and the porosity Φ is: Substituting equation (1-15) into equation (1-14), the equivalent thermal conductivity λ of the particle-type simplified REV heat transfer model is kl for:

4. A method for measuring thermal conductivity of porous media according to claim 1, characterized in that: For the particle-type fine REV heat transfer model, the fractal dimension of the REV primitive is first calculated, and then the fractal dimension is substituted into (1-16), which is the equivalent thermal conductivity λ of the particle-type fine REV heat transfer model. kl for:

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