Front. Mater. Sci. 2012, 6(1): 79–86 DOI 10.1007/s11706-012-0156-6
RESEARCH ARTICLE
Numerical calculations of effective thermal conductivity of porous ceramics by image-based finite element method Yan‐Hao DONG, Chang‐An WANG (✉), Liang‐Fa HU , and Jun ZHOU *
State Key Lab of New Ceramics and Fine Processing, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2012
ABSTRACT: The effective thermal conductivity of heterogeneous or composite materials is an essential physical parameter of materials selection and design for specific functions in science and engineering. The effective thermal conductivity is heavily relied on the fraction and spatial distribution of each phase. In this work, imagebased finite element method (FEM) was used to calculate the effective thermal conductivity of porous ceramics with different pore structures. Compared with former theoretical models such as effective media theory (EMT) equation and parallel model, image-based FEM can be applied to a large variety of material systems with a relatively steady deviation. The deviation of image-based FEM computation mainly comes from the difference between the two dimensional (2D) image and the three dimensional (3D) structure of the real system, and an experiment was carried out to confirm this assumption. Factors influencing 2D and 3D effective thermal conductivities were studied by FEM to illustrate the accuracy and application conditions of image-based FEM. KEYWORDS: material
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thermal conductivity, image-based FEM, porous ceramic, two-phase
Introduction
The effective thermal conductivity of heterogeneous or composite materials is an essential physical parameter of materials selection and design for specific functions in science and engineering. Generally, a two-phase mixture system is the simplest in composite materials. The effective thermal conductivity is heavily relied on the fraction and spatial distribution of each phase. A significant number of models have been proposed to predict the effective thermal conductivity [1–4]. Many of these models are either purely Received November 22, 2011; accepted December 12, 2011 E-mail:
[email protected], * Current address: Materials Science and Engineering Program, Texas A&M University, 3003 TAMU, College Station, TX 77843-3003, USA E-mail:
[email protected]
empirical or theoretically derived, but limited in application to specific types of structures due to their geometrical simplification on the spatial distributions of the phases. For example, Rayleigh’s expression describes the heat flow around an obstructing cylinder [5]. In Maxwell-Eucken expression inclusions are treated as spheres [6]. Effective media theory (EMT) equation assumes the two phases are randomly distributed and thus takes the connectivity and interaction of the two phases into consideration [7–8]. These expressions are limited to specific materials because they take the ratio of the two phase contents as the only parameter to determine the effective thermal conductivity while using geometrical simplification and ignoring the influence of spatial distribution of phases in the real materials. Some other theoretical models may have wider application, but they usually include undesirable
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empirically determined parameters. Finite element method (FEM) is widely used to simulate the thermal transport process in recent years, with the development of computers and commercial software packages [9–10]. Image-based FEM has been developed as an easy way to calculate thermal properties including thermal conductivity [11–12]. However, it was only applied to a specific kind of materials, and the accuracy, application range as well as application conditions of this method were seldom mentioned. Porous ceramics are a kind of typical two-phase materials and of increasing interests due to their wide applications in ceramic filter, catalyst carrier, sensor, porous electrode, biomaterials, thermal barrier and so on [13–15]. As to applications for thermal isolation in hightemperature processes, insulated engine components, refractory and thermal insulation, low thermal conductivity is usually the goal pursued by researchers. Some models, e.g. EMT equation, are usually adopted to analyze and predict thermal conductivity of porous ceramics [16–17]. In this work, image-based FEM was used to calculate the effective thermal conductivities of porous ceramics with different pore structures. The calculating results were compared with experimental results and predictions given by EMT equation and parallel model. Origins of the deviations were analyzed and factors influencing two dimensional (2D) and three dimensional (3D) effective thermal conductivities were discussed to illustrate the accuracy and application conditions of image-based FEM.
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Fig. 1 Sample images used in image-based FEM computations: (a) original; (b) after binary.
Image-based FEM computations
Cross-section photographs of porous ceramics samples are needed in the following procedure. The effective thermal conductivity of the microstructure marked in Fig. 1(a) was computed as an example to show the procedure of the image-based FEM. First, we set a threshold value to divide the photograph into two levels of grey to obtain a binary image as shown in Fig. 1(b). The white represents the ceramic phase and the black represents the porous phase. A suitable adjustment of the threshold value between the two phases is applied to ensure the fraction of the black phase equals the porosity in the sample. The resulting image with m n pixels produces a mesh with m rows and n columns naturally. Commercial software ABAQUS and MATLAB are used for all the following image-based FEM progress. The image-based FEM computations are performed according to the following algorithm as shown in Fig. 2:
Fig. 2 Effective thermal conductivity measurement used in image-based FEM.
(i) Create a rectangle with the length m and the width n. (ii) Define two kinds of materials with given thermal conductivities. (iii) Mesh the area with m n elements, where m and n are original length and width of the image, respectively.
Yan-Hao DONG et al. Numerical calculations of effective thermal conductivity of porous ceramics by image-based FEM
(iv) Assign material properties (thermal conductivity value) to each element, according to the serial number of nodes and elements. (v) Apply boundary conditions for thermal conductivity (a temperature gradient ΔT imposed across the upper and lower boundaries and adiabatic left and right boundaries). (vi) Calculate the vertical heat flux of each element. (vii) Calculate the effective thermal conductivity of the structure based on the average heat flux per unit area on the upper or the lower boundaries of the 2D image. The formula is as follows: X q=X Q l¼ ¼ (1) ΔT =Y ðThot – Tcold Þ=Y where l is effective thermal conductivity, Q is the average heat flux on the upper boundary of the image, q is the vertical heat flux of each element on the upper boundary, ΔT is the temperature gradient imposed across the upper and lower boundaries, and X and Y are length and height of the rectangle, respectively.
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measured using the Archimedean method. Microstructures were observed by a scanning electron microscope (SEM, JSM 6700F, JEOL, Tokyo, Japan). Thermal conductivities at room temperature were measured on 5 mm 5 mm 3 mm machined specimens, using the Thermal Transport Option (TTO) of Physical Properties Measurement System (PPMS, Model 6000, Quantum Design, US). The porosity, thermal conductivity, FEM results and EMT results for each group are shown in Table 1. With the results also displayed in Fig. 4, we can see that the general trends of the experimental thermal conductivities and FEM results are similar with relative deviations from 36.3% to 50.9%. EMT results are closer to experimental thermal conductivities than FEM results because uniformly distributed pores satisfy the assumptions of EMT equation well. However, the line of EMT results drops faster than experimental results and intersects with the line of experimental results. The deviations from the measured thermal conductivity change from a positive value to a negative one gradually, while the deviations of image-based FEM results are steadier.
3 Image-based FEM applied to different structures of materials 3.1
Porous ceramics with uniformly distributed pores
A tertiary-butyl alcohol (TBA)-based gel-casting process was used to fabricate porous yttria-stabilized zirconia (YSZ) ceramics shown in Fig. 3 [18]. Four groups of porous YSZ ceramics with different porosities and thermal conductivities were fabricated according to varied initial solid loadings. The porosities of the samples were
Fig. 4 FEM results, experimental thermal conductivity and comparison with EMT equation of porous ceramics with uniformly distributed pores.
3.2 Porous ceramics with unidirectionally aligned pore channel structure
Fig. 3 SEM image of porous ceramics with uniformly distributed pores.
A TBA-based freeze-casting process was used to fabricate YSZ ceramics with unidirectionally aligned pore channels [19]. The obtained structure was shown in Fig. 5. Three groups of the porous YSZ ceramics with different porosities and different thermal conductivities were fabricated. Samples were characterized in the same means stated above. Considering the structures in this
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Table 1 Group
Experimental thermal conductivity, FEM and EMT results of porous ceramics with uniformly distributed pores Porosity/%
Thermal conductivity/(W$m–1$K–1) Experimental results
FEM results
1
47.2
0.526
0.335
EMT results 0.689
2
59.7
0.271
0.163
0.328
3
64.7
0.216
0.105
0.212
4
72.7
0.122
0.075
0.101
Fig. 5 SEM image of porous ceramics with unidirectionally aligned pore channels.
case, EMT equation was on longer applicable so we compared FEM results with parallel model. The porosity, thermal conductivity, FEM results and parallel model results of each group are shown in Table 2. Results also shown in Fig. 6 demonstrate that the general trend of the experimental thermal conductivity (along the parallel direction of the pore channel) and FEM results are the same with relative deviations from 7.8% to 17.1%. Compared with the parallel model, FEM results are much closer to the experimental thermal conductivities and reflect the same tendency. Generally speaking, thermal conductivity decreases with the increase of porosity. However, our experimental results show the opposite relationship which cannot be reflected in many former theoretical models including parallel model. The porosity and structure both influence the thermal conductivity of porous ceramics. With a lower porosity, the size of pore channels decreases subsequently, which offers more Table 2 Group
Fig. 6 FEM results, experimental thermal conductivity and comparison with parallel model of porous ceramics with unidirectionally aligned pore channels.
thermal surface barriers and leads to a lower thermal conductivity. Image-based FEM computations can take actual changes in the structure into consideration as well as porosity and thus match the experimental results well. Image-based FEM computations were applied to porous ceramics with different pore structures. Results show that image-based FEM computations reflect the same tendency with the experiments and the deviations are insignificant. While many traditional theoretical computations such as EMT equation and parallel model simplify the structure and take porosity as the only parameter, we can get the information of the structure extracted from the image for the numerical calculation of effective thermal conductivity as well as the porosity. Thus, the use of image-based FEM is not limited to a specific category of materials. The only requirement is to divide the two phases according to grey levels or other methods.
Experimental thermal conductivity, FEM and parallel mode results of porous ceramics with unidirectionally aligned pore channels Porosity/%
Thermal conductivity/(W$m–1$K–1) Experimental results
FEM results
1
74.2
0.230
0.212
Parallel model 0.484
2
75.0
0.345
0.286
0.469
3
77.3
0.470
0.414
0.429
Yan-Hao DONG et al. Numerical calculations of effective thermal conductivity of porous ceramics by image-based FEM
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4 Analysis of deviations of the image-based FEM The relative deviations of the image-based FEM results may come from several aspects. A considerable one is derived from the difference between the 2D image and the real 3D structure. Compared with 2D cases, there are more degrees of freedom of heat flux in 3D heat transfer, resulting in the 3D thermal conductivity being larger than the 2D thermal conductivity, which was shown by results in previous section. The 2D thermal conductivity, therefore, can be considered as a lower limit of the 3D thermal conductivity [10]. In Fig. 1(b) we can see many isolated white areas i.e. ceramic phase, which are in fact continuous phase in the real structure. We may also find continuous porous phase crossing the whole area in the picture and the ceramic phase is completely separated. The heat flow is completely blocked under this circumstance, which decreases calculated thermal conductivity significantly. Completely separated ceramic phase cannot appear in 3D cases, which suggests that there must be heat bridges in real structures that can serve as extra pathways of heat flux compared with 2D cases. So the decrease in connectivity and increase in blockage are the main difference between 2D image and real 3D structure. To estimate these effects, a similar image-based FEM computation on two-dimensional materials would be the best choice since there are no change in structure concerning the 2D and 3D cases of twodimensional materials. If the calculated results of twodimensional materials are much more agreeable with experimental results, it can be certified that the difference between the 2D image and the real 3D structure is the main cause of the deviation. A sample was fabricated by the TBA-based freezecasting process described above. If we consider the thermal conductivity along the perpendicular direction of the pore channel, YSZ ceramics with unidirectionally aligned pore channels could be approximately simplified as a twodimensional material. Figure 7 is the photograph of a cross section perpendicular to the pore channels of the sample. The pore channels and the walls could be distinguished obviously. However, the phase of the walls is not dense but porous YSZ ceramics as shown in Fig. 8. Some adjustments are needed before the calculation. The overall porosity of the sample is 74.2%, while the ratio of pore channels in the image is 41.0%. Consequently, the pores in the wall take a part of 33.2% of the overall volume. Therefore the porosity of the wall is 56.3%, if we consider
Fig. 7 SEM image of a cross section perpendicular to the pore channel of porous ceramics with unidirectionally aligned pore channels.
Fig. 8 High-magnification SEM image of the channel wall.
the wall as a phase. Samples with similar porosities were fabricated and their thermal conductivities were measured to be 0.34 W$m–1$K–1. The FEM computation was then applied and the result of unidirectional ordered porous YSZ ceramics is 0.108 W$m–1$K–1, while the experimental thermal conductivity is 0.103 W$m–1$K–1. The relative deviation is only 4.85%, much less than the results stated above. Image-based FEM computation on two-dimensional materials can be very close to the experimental results. Therefore, the assumptions stated above have been proved.
5 Relationships between 2D and 3D effective thermal conductivities To further investigate the difference between 2D and 3D heat transfer and factors influencing 2D and 3D effective
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thermal conductivities, models of randomly distributed two-phase materials were studied by FEM. In the model of 2D heat transfer, a rectangle was created and meshed. The temperature gradient ΔT was imposed across the upper and lower boundaries of the rectangle with adiabatic left and right boundaries. In the model of 3D heat transfer, a cube was created and meshed. The temperature gradient ΔT was imposed across the upper and lower faces of the cube with four other adiabatic faces. Two phases with different thermal conductivity were distributed randomly according to the fractions of the two phases in the models. The rectangle in 2D model can be seen as a cross section of the cube in 3D model, which shows the same relationships between the images and the samples in all above imagebased FEM computations. Effective thermal conductivities were calculated according to the models and two-phase materials with various thermal conductivity ratio l1/l2 (l1/ l2 = 5, 10, 50, 69.2, 100, respectively) were investigated as shown in Fig. 9 (l1/l2 = 69.2 is the same value with lYSZ/ lair). The results are as follow:
system. The line firstly increases and then decreases with the increased fraction of the low-thermal-conductivity phase. All curves show a peak around the fraction of 70%. Experimental results of porous ceramics with uniformly distributed pores in Section 3.1 show a peak around 70% too, as reflected in Fig. 10 (The thermal conductivities of the samples are treated as l3D and image-based results are treated as l2D). This is also a result of the relative contribution of the extra pathways of heat flux to the effective thermal conductivity:
Fig. 10 Ratio of effective thermal conductivity of experimental results (l3D) and image-based FEM results (l2D) of porous ceramics with uniformly distributed pores with different porosity.
Fig. 9 FEM results of 2D and 3D effective thermal conductivity of randomly distributed two-phase system.
Firstly, the ratio of l3D/l2D varies with different l1/l2 under the same relevant amount of the two phases. As discussed above, decreased connectivity and increased blockage leads to different l3D and l2D and 3D structure offers more pathways of heat flux than 2D. So with a larger difference between the two phases, i.e. a larger ratio of l1/ l2, the contribution of the extra pathways between 2D and 3D structures to the effective thermal conductivity is more obvious and correspondently the line is higher as reflected in the picture. Secondly, the ratio of l3D/l2D varies with the volume fractions of the two phases within the same material
a) When the fraction of the low-thermal-conductivity phase is small, there are fewer obstacles for heat transfer and it has plenty of the pathways of heat flux in 2D to achieve a relatively high thermal conductivity. The extra pathways of heat flux contribute little to the change in effective thermal conductivity. b) When the fraction of the low-thermal-conductivity phase is very high, although the transfer in 2D is obstructed, the transfer in 3D is also obstructed and there are not enough extra pathways of heat flux to provide significant influence on the heat transfer. The difference between 2D and 3D effective thermal conductivities is also negligible. c) Concerning above processes a) and b), when the amounts of the two phases are appropriate (approximately 70% in this work), the extra pathways in 3D structures may cause a noticeable difference between 2D and 3D heat transfer and the ratio of l3D/l2D reaches a maximum. The curves are results of the contribution of the extra pathways of heat flux in 3D structure. With the fraction of the low-thermal-conductivity phase of approximately 70%,
Yan-Hao DONG et al. Numerical calculations of effective thermal conductivity of porous ceramics by image-based FEM
the influence of the extra pathways reaches a peak. Therefore, image-based FEM can be more accurate with either lower or higher fractions. Grandjean et al. [11] also found similar phenomenon. According to their research, up to 20% of pore volume fraction, image-based simulations are in close agreement with analytical expressions. It is also noticed that the peak value (l3D/l2D = 2.06) in Fig. 10 is smaller than that of random distribution model result (l3D/l2D = 3.80) in Fig. 9. This is because in real materials phases are continuous/partly continuous and the effect of extra pathway is weakened compared to completely random distribution. To conclude, the difference between 2D and 3D effective thermal conductivities in real materials should be smaller than numerical modeling results shown in Fig. 9. Moreover, the porosities of our samples were just around 70% while the deviations were steady and not very large, which further demonstrates the accuracy of image-based FEM results. In all the above FEM calculations of this paper, porous phase is just treated as a low thermal conductivity phase, no specific illustrations whether it is solid or fluid. Therefore, although the focus of this paper is on the effective thermal conductivity of porous ceramics, the analysis procedure can be applied to any two-phase materials and other mechanical and thermal properties can also be calculated in similar ways.
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The difference of thermal conductivities between the two components obviously influences the ratio of 2D and 3D effective thermal conductivities. The ratio of l3D/l2D varies with different l1/l2 under the same relevant amount of the two phases and different fractions of the two phases within the same material system. In particular, the difference is most significant when the fraction of the low-thermalconductivity phase reaches about 70% and the deviations of image-based FEM results can be smaller with either lower or higher fractions. Our results show the accuracy of image-based FEM as well. Acknowledgements The authors would like to thank the financial support from the National Natural Science Foundation of China (Grant Nos. 90816019 and 51172119).
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Conclusions
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Through adjusting the grayscale of the cross-section photographs of porous ceramics, the distributions and volume fractions of ceramic phase and porous phase can be distinguished. Based on this image treatment, FEM was successfully used to calculate the effective thermal conductivity of porous ceramics with different pore structures and results agree with experiments well. Compared with former models such as EMT equation and parallel model, image-based FEM can be applied to a large variety of material systems and has a relatively steady deviation. The deviation of image-based FEM computation mainly comes from the difference between 2D and 3D heat transfer, i.e., the difference between the 2D image and the 3D structure of the real system. The decrease in connectivity and the increase in blockage account for the difference between 2D and 3D effective thermal conductivities. An approximately 2D material was fabricated to confirm this assumption.
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