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The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials
ICHMT-03261; No of Pages 4 International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3Q1
Hu Zhang, Wenzhen Fang, Zengyao Li ⁎, Wenquan Tao
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Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
5
a r t i c l e
6 7
Available online xxxx
8 9 10 11 12 13
Keywords: Transient plane source method Effective thermal conductivity Nano-porous material Contribution of gaseous heat conduction Gas pressure
a b s t r a c t
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A thermal conductivity test apparatus based on transient plane source method is built and developed to measure effective thermal conductivity of open porous materials at different gas pressures. The effective thermal conductivity of open nano-porous silica materials with porosity of 88.5% is measured under gas pressures ranging from 0.001 Pa to 1 MPa. The contribution of gaseous heat conduction to the effective thermal conductivity of materials is decomposed by subtracting the effective thermal conductivity at ultimate vacuum from that at different gas pressure. It is found that the contribution of gaseous heat conduction is much different with the gas thermal conductivity in nano-porous materials and in free space. The result is also demonstrated by theoretical analysis and numerical simulation. © 2015 Elsevier Ltd. All rights reserved.
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Aerogel, which is manufactured by applying a sol–gel process and supercritical drying technology, is a typical nano-porous material with open-cell and random structure [1]. It has merits such as high porosity, high specific area and super thermal insulation performance. Aerogels have the lowest thermal conductivity in solid or porous materials. Because aerogels are very fragile due to its extremely low density and near transparency to radiation of wavelengths of 3–8 μm, reinforced fibers and opacifiers are usually embedded and doped in the aerogels to maintain high mechanical strength and to ensure high thermal insulation performance at high temperature [2]. The composite still has high porosity, open porous random spatial structure and high thermal insulation performance. So we call aerogel and their composites nanoporous materials since their nano-scale pores with random size are the key factor of reducing its effective thermal conductivity. Owing to the outstanding thermal insulation property, they have a broad application prospect from the viewpoints of saving energy and thermal protection such as the thermal insulation materials in buildings and thermal protection system of shuttle and aircraft [3]. Therefore, great attention has been paid to analyze the heat transfer mechanism and to optimize the thermal insulation property [2,4–12]. The heat transfer paths in nano-porous materials are mainly divided into three types: solid heat conduction, gaseous heat conduction and thermal radiation. In nanoporous materials, the size effect is significant for both heat conduction via solid and gas and thermal radiation due to the sizes of solid skeleton
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1. Introduction
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The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials☆
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☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Z. Li).
and pore is close to or even smaller than the characteristic sizes of the energy carriers such as gas molecules, phonons and photons [13–15]. There are three kinds of heat transfer models to predict the effective thermal conductivity of porous materials in literatures [4,13]. The first model regarded the effective thermal conductivity as the superposition of solid (λs), gas (λg) and radiation (λr) thermal conductivities with the form of λe = λg + λs + λr [5–7,10,12–15]. Gas convection is neglected because the pore size is less than 1 mm at ambient pressure [5]. The second model treated the effective thermal conductivity as the sum of radiation and a combined solid and gas conduction (λc): λe = λc + λr [2,8,9,15,27]. The combined thermal conductivity of solid and gas, λc, can be calculated from effective structure models simplified from the practical structure or from empirical models. The third model is to conduct numerical simulation by generating the actual structure to calculate the effective thermal conductivity [4,17]. In this paper, we focus on the gas heat conduction in nano-porous materials because the gaseous heat conduction plays a significant or even dominant role on reducing heat transfer in nano-porous materials [6]. The gas thermal conductivity in nano-porous materials is lower than that in free space because the motion of gas molecules in nano-pores is suppressed by the nano-porous structure [14,15]. The gas thermal conductivity in nano-porous materials varies with gas pressure as well as the effective thermal conductivity (λe). By subtracting the effective thermal conductivity of materials at ultimate vacuum (λe,0) from that at different gas pressures, the contribution of gaseous heat conduction to the effective thermal conductivity (λg,0) could be obtained. In most of the previous works, the effective thermal conductivities of porous materials were experimentally investigated at the gas pressures of 1 bar or lower [5,6, 9,10]. However, the decomposed λg,0 was usually regarded as the gas thermal conductivity in porous materials without distinguishing their differences by applying the first heat transfer model [5,10]. Recently,
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027 0735-1933/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
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H. Zhang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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Nomenclature
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a cp, cv d D k kB l m p Ss T
T1:14 T1:15 T1:16 T1:17 T1:18 T1:19
Greek symbols γ γ = cp / cv λ thermal conductivity, W/m·K ρ density, kg/m3 П porosity, %
T1:20 T1:21 T1:22 T1:23 T1:24 T1:25 T1:26 T1:27 T1:28
Subscripts c conductive e effective e,0 effective at vacuum g gas g,0 contribution of gas heat conduction r radiative s solid
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3. Results and discussion
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3.1. Effective thermal conductivity at different gas pressures
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The effective thermal conductivity of Super-G (λe) measured at different gas pressures at 297 K is shown in Fig. 1. The results clearly show that the effective thermal conductivity varies greatly with gas pressure. The effective thermal conductivity keeps constant when the gas pressure is less than 0.01 kPa, which stands for the overall contribution of solid conduction and thermal radiation and it remains unchanged with gas pressure. However, the effective thermal conductivity increases sharply when the gas pressure is higher than 0.1 kPa due to the enhanced gaseous heat conduction.
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diameter of contact area, nm isobaric/isochoric specific heat capacity, J/mol · K diameter, nm diameter of spheres on each edge, nm k = 1 − λg / λs Boltzmann constant, 1.38 × 10−23 J/K mean free path, nm mass, g gas pressure, Pa specific surface area, m2/g temperature, K
is adjusted from low to high. The effective thermal conductivity of Super-G, a commercial silica aerogel composite that was manufactured by Microtherm® with a density of 240 kg/m3, porosity of 88.5% and claimed thermal conductivity of 0.0258 W/m·K at 373 K, is measured in nitrogen atmosphere under gas pressures ranging from 0.001 Pa to 1 MPa at 297 K.
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The contribution of gaseous heat conduction to the effective thermal conductivity of Super-G (λg,0) is decomposed from the experiment results as shown in Fig. 1. The gas thermal conductivities in both free space and nano-porous materials are also theoretically depicted, as shown in Fig. 1. The gas thermal conductivity in free space is obtained from NIST database with taking the effect of actual gas into account [23]. The gas thermal conductivity in super-G used for comparison is calculated from Zeng's equation [14]. The mean free path (lm0 ) in free space:
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2. Experimental investigations
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An apparatus based on the Transient Plane Source (TPS) method [18] is established to measure the effective thermal conductivity of porous materials under the gas pressures ranging from 0.001 Pa to 1 MPa. It's the first time that the thermal conductivity has been measured at a different gas pressure using TPS method. The accuracy of the apparatus is validated by using NIST1453, an expanded polystyrene board with thermal conductivity of 0.032 W/m·K at room temperature with a deviation within 3%. We studied the theoretical accuracy of TPS method and measured thermal conductivity of nano-porous materials in our previous works [19–22]. In the experimental investigations, the nano-porous materials are initially vacuumized exhaustively for at least 3 times to remove the absorbed water vapor or impurity gas, and the gas pressure
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1 lm ¼ pffiffiffi 2 : −1 2πdg p=kB T þ 0:25Ss ρpor ∏
ð2Þ 147
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Due to the suppression of nano-porous structure, the mean free path of gas molecules in porous materials 145
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ð1Þ
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kB T lm0 ¼ pffiffiffi 2 : 2πdg p
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3.2. Contribution of gaseous heat conduction decomposed from the effective 132 thermal conductivity 133
Reichenauer et al. found that λg,0 is about seven times higher than the gas thermal conductivity in a glass sphere bed with sphere diameter of 1 mm at ambient pressure [6]. They believed that the difference is caused by the local thermal shortcuts where spheres are touching; however, no further explanation was given. Swimm et al. studied the thermal conductivity of aerogels with average pore sizes of about 600 nm and 7 μm, respectively, under the gas pressures ranging from 10 Pa to 10 MPa [7]. A significant difference between λg,0 and the gas thermal conductivity was also observed, which is believed to be caused by the coupling heat transfer effect between solid and gas. A simple unit cell of two accumulated spherical particles was proposed to explain the coupling heat transfer effect; however, only qualitative agreement was obtained. This study aims at investigating the influence of gaseous heat conduction on the effective thermal conductivity of nano-porous materials within a wide range of gas pressure. Experimental measurement, theoretical analysis and numerical simulation are conducted to clarify the relationship between the gas thermal conductivity and λg,0 in nanoporous materials.
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Fig. 1. Effective thermal conductivity of Super-G and comparison between gas thermal conductivity and contribution of gaseous heat conduction.
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
H. Zhang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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Therefore the thermal conductivity of gas in nano-porous medium is different from those in the free space [14]: 1=2 ð2:25γ−1:25Þ0:461ðp=kB T Þ 8kB T=πmg mg cv : λg ¼ p ffiffiffi −1 2 0:25Ss ρpor ∏ þ 2ðp=kB T Þπdg
effective thermal conductivity of silica aerogels [15]. In this paper, the intersecting spherical structure (see Fig. 3) is selected as an example to calculate the λg,0 theoretically and reveal the difference between the gas thermal conductivity and λg,0. The effective thermal conductivity of the structure is derived as follows based on one-dimensional heat conduction assumption.
ð3Þ
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166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183
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Zeng et al. adopted three regular structures, intersecting square rod, intersecting cylindrical rod and intersecting sphere, to derive the
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In Eq. (4), λg at different gas pressures is calculated from Eq. (3), d and D are determined by their relationships with porosity and specific surface area, and a / d = 0.2 is selected for calculation [15]. The specific surface area of Super-G is measured as 141.4 m2/g by applying adsorption and desorption of nitrogen with PCTPro-Evo (SETARAM Inc., France). For simplicity, the bulk thermal conductivity of SiO2 (1.34 W/m · K) is used as the skeleton thermal conductivity with the size effect ignored [2,7,10,15]. Theoretically, λg,0 at different gas pressures can be calculated by subtracting the effective thermal conductivity at ultimate vacuum from λe, which is then compared with the experimental λg,0 of Super-G as shown in Fig. 3. Obviously, the theoretical λg,0 shows quantitative agreement with the experimental result at full gas pressure range except for the gas pressure close to or higher than ambient pressure. Super-G is a composite that consists of silica aerogel and doped fibers and opacifier while the theoretical calculation is conducted with an effective structure using the same porosity and specific surface area as Super-G for simplification. Therefore, the difference is mainly introduced by the simplification of the effective structure. The theoretical study in this part is primarily used for illustrating the difference between λg,0 and λg,0. The agreement between experimental result and theoretical analysis could be further improved by applying more complicated models, which is beyond the scope of this article research. It is also proved that λg,0 is different from the gas thermal conductivity in nano-porous materials and can be larger than the gas thermal conductivity in free space at a higher gas pressure. 3.4. Contribution of gaseous heat conduction decomposed from the numerical simulation
Fig. 2. Mean free paths of nitrogen molecules in free space and in Super-G.
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A random generation-growth method is employed to generate a 221 porous structure with both the solid phase and the gaseous phase 222
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3.3. Contribution of gaseous heat conduction decomposed from the effective structure
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ð4Þ
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2 ðn−1Þπλg d a D D−ka d −1 þ ln þ 1− λg Dk d kd D−kd D
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þ
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" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# a2 a2 πλg d D ndk πλs a 2 − 1− ln 1− 1− − þ 2nDk d ndk D d 4 D
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λe ¼
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As depicted in Fig. 1, λg,0 is much different from the gas thermal conductivity in nano-porous material, and it is even higher than the thermal conductivity of actual gas in free space at a higher pressure. Heat transfer in nano-porous material includes solid conduction via the solid skeleton, gaseous conduction in the pores, thermal radiation and coupled heat conduction between solid and gas. If the material is ultimately vacuumized, gaseous conduction in the pores and coupled heat conduction between solid and gas will be vanished. The experiment result proves that the variation of the effective thermal conductivity of nano-porous materials at different gas pressures is different from the variation of gas thermal conductivity at different pressures. Therefore, it is unreasonable to regard λg,0 as the thermal conductivity of gas in nano-porous material. In Fig. 1, the gas thermal conductivity in free space is almost unchanged when the gas pressure is not higher than 1 MPa while the gas thermal conductivity in nano-porous materials almost reaches zero when the pressure is less than 1 kPa and then increases rapidly with the increase of pressure. Such phenomenon is mainly attributable to the motion of gas molecules being suppressed by the nano-porous structure, and as a result, the mean free path of gas molecules in free space is different from that in nano-porous material (calculated from Eqs. (1) and (2)), as shown in Fig. 2. The gas thermal conductivity in nano-porous materials gets closer to that in free space when the pressure is higher than 1 MPa because the increment of molecules number density will shorten the mean free path, making it approach to that in free space with the increase of pressure. It is inconceivable to believe that the contribution of gas conduction to the effective thermal conductivity of nano-porous materials at different gas pressures, λg,0, can be greater than the gas thermal conductivity in free space. We studied many other nano-porous materials and the same phenomenon is observed, Super-G is selected as an illustration to clarify the difference between λg,0 and the gas thermal conductivity in nano-porous materials.
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Fig. 3. Comparison of contribution of gaseous heat conduction and the effective structure and random structure of aerogels.
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
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taking account in scale effect/interfacial effect of solid skeleton of 271 nano-porous materials doped with fiber and opacifier. 272
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being considered as the continuous phase [17]. As shown in Fig. 3, the random structure has a porosity of 88.5% and domain size of 100 × 100 × 100. The ratio of the average pore diameter to the particle diameter is 5. Then the thermal conductivity of the random structure is calculated by applying the Lattice Boltzman method. λg,0 decomposed from the calculation is also compared with the experimental data as shown in Fig. 3. The decomposed λg,0 of the random structure also reveals the difference between λg,0 and the gas thermal conductivity.
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3.5. Influence factors of contribution of gaseous heat conduction
The authors would like to thank the support from National Natural 276 Science Foundation of China (51320105004 and 51276138). 277
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λg,0 is affected by the coupled heat conduction via solid and gas at different gas pressures when temperature is kept constant and the pores is filled with non-polar gases that are transparent to thermal radiation, then
ð5Þ
−f ðλs ; λr ; porosity; structureÞ ¼ f λs ; λg ; porosity; structure −f ðλs ; porosity; structureÞ: 237
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4. Conclusion
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In summary, the effective thermal conductivity of open nano-porous materials with high porosity is influenced by gas pressure greatly. The contribution of gaseous heat conduction to the effective thermal conductivity decomposed from the experiment result is greater than the gas thermal conductivity in nano-porous materials and even greater than that in free space at a higher gas pressures. In many previous works, the relationship between the contribution of gaseous heat conduction to the effective thermal conductivity and gas thermal conductivity is not clearly clarified and often confused with each other. In this paper, the relationship is differentiated clearly. The contribution of gaseous heat conduction to porous material depends on not only the gas thermal conductivity, but also the solid thermal conductivity, porosity and spatial structure. Regarding the contribution of gas heat conduction decomposed from experiment as the gas thermal conductivity without distinguishing their difference may result in unreliable results. Experimental investigation, theoretical calculation and numerical simulation shown in this paper are three effective ways of obtaining the effective thermal conductivity and the contribution of gas heat conduction of porous materials. However, there are still some challenges of predicting the effective thermal conductivity more accurately, such as developing multi-scale theoretical models and multi-scale simulation,
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From Eq. (5), it is very easy to understand that λg,0 is influenced by gas thermal conductivity, solid thermal conductivity, porosity and spatial structure. The increment of gas thermal conductivity within the nano-porous materials will enhance the coupled heat conduction between gas and solid greatly because the total thermal resistance of heat conduction networks decreases much more than the decrement of thermal resistance of the filling gas. That is reason why the decomposed contribution of gaseous heat conduction of nano-porous materials can be larger than the gas thermal conductivity in free space at high gas pressures. Therefore, it's not reasonable to confuse the λg,0 with the gas thermal conductivity in porous materials and such analysis can be generalized as porous materials with different scale of pore size.
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References
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λg;0 ¼ λe −λe ;0 ¼ f λs ; λg ; λr ; porosity; structure
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Acknowledgments
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H. Zhang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
3Q1
Hu Zhang, Wenzhen Fang, Zengyao Li ⁎, Wenquan Tao
4
Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
5
a r t i c l e
6 7
Available online xxxx
8 9 10 11 12 13
Keywords: Transient plane source method Effective thermal conductivity Nano-porous material Contribution of gaseous heat conduction Gas pressure
a b s t r a c t
D
P
A thermal conductivity test apparatus based on transient plane source method is built and developed to measure effective thermal conductivity of open porous materials at different gas pressures. The effective thermal conductivity of open nano-porous silica materials with porosity of 88.5% is measured under gas pressures ranging from 0.001 Pa to 1 MPa. The contribution of gaseous heat conduction to the effective thermal conductivity of materials is decomposed by subtracting the effective thermal conductivity at ultimate vacuum from that at different gas pressure. It is found that the contribution of gaseous heat conduction is much different with the gas thermal conductivity in nano-porous materials and in free space. The result is also demonstrated by theoretical analysis and numerical simulation. © 2015 Elsevier Ltd. All rights reserved.
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Aerogel, which is manufactured by applying a sol–gel process and supercritical drying technology, is a typical nano-porous material with open-cell and random structure [1]. It has merits such as high porosity, high specific area and super thermal insulation performance. Aerogels have the lowest thermal conductivity in solid or porous materials. Because aerogels are very fragile due to its extremely low density and near transparency to radiation of wavelengths of 3–8 μm, reinforced fibers and opacifiers are usually embedded and doped in the aerogels to maintain high mechanical strength and to ensure high thermal insulation performance at high temperature [2]. The composite still has high porosity, open porous random spatial structure and high thermal insulation performance. So we call aerogel and their composites nanoporous materials since their nano-scale pores with random size are the key factor of reducing its effective thermal conductivity. Owing to the outstanding thermal insulation property, they have a broad application prospect from the viewpoints of saving energy and thermal protection such as the thermal insulation materials in buildings and thermal protection system of shuttle and aircraft [3]. Therefore, great attention has been paid to analyze the heat transfer mechanism and to optimize the thermal insulation property [2,4–12]. The heat transfer paths in nano-porous materials are mainly divided into three types: solid heat conduction, gaseous heat conduction and thermal radiation. In nanoporous materials, the size effect is significant for both heat conduction via solid and gas and thermal radiation due to the sizes of solid skeleton
N C O
28
1. Introduction
U
25 27
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R O
i n f o
F
2
The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials☆
1
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Z. Li).
and pore is close to or even smaller than the characteristic sizes of the energy carriers such as gas molecules, phonons and photons [13–15]. There are three kinds of heat transfer models to predict the effective thermal conductivity of porous materials in literatures [4,13]. The first model regarded the effective thermal conductivity as the superposition of solid (λs), gas (λg) and radiation (λr) thermal conductivities with the form of λe = λg + λs + λr [5–7,10,12–15]. Gas convection is neglected because the pore size is less than 1 mm at ambient pressure [5]. The second model treated the effective thermal conductivity as the sum of radiation and a combined solid and gas conduction (λc): λe = λc + λr [2,8,9,15,27]. The combined thermal conductivity of solid and gas, λc, can be calculated from effective structure models simplified from the practical structure or from empirical models. The third model is to conduct numerical simulation by generating the actual structure to calculate the effective thermal conductivity [4,17]. In this paper, we focus on the gas heat conduction in nano-porous materials because the gaseous heat conduction plays a significant or even dominant role on reducing heat transfer in nano-porous materials [6]. The gas thermal conductivity in nano-porous materials is lower than that in free space because the motion of gas molecules in nano-pores is suppressed by the nano-porous structure [14,15]. The gas thermal conductivity in nano-porous materials varies with gas pressure as well as the effective thermal conductivity (λe). By subtracting the effective thermal conductivity of materials at ultimate vacuum (λe,0) from that at different gas pressures, the contribution of gaseous heat conduction to the effective thermal conductivity (λg,0) could be obtained. In most of the previous works, the effective thermal conductivities of porous materials were experimentally investigated at the gas pressures of 1 bar or lower [5,6, 9,10]. However, the decomposed λg,0 was usually regarded as the gas thermal conductivity in porous materials without distinguishing their differences by applying the first heat transfer model [5,10]. Recently,
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027 0735-1933/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
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H. Zhang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
T1:1
Nomenclature
T1:2 T1:3 T1:4 T1:5 T1:6 T1:7 T1:8 T1:9 T1:10 T1:11 T1:12 T1:13
a cp, cv d D k kB l m p Ss T
T1:14 T1:15 T1:16 T1:17 T1:18 T1:19
Greek symbols γ γ = cp / cv λ thermal conductivity, W/m·K ρ density, kg/m3 П porosity, %
T1:20 T1:21 T1:22 T1:23 T1:24 T1:25 T1:26 T1:27 T1:28
Subscripts c conductive e effective e,0 effective at vacuum g gas g,0 contribution of gas heat conduction r radiative s solid
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3. Results and discussion
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3.1. Effective thermal conductivity at different gas pressures
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The effective thermal conductivity of Super-G (λe) measured at different gas pressures at 297 K is shown in Fig. 1. The results clearly show that the effective thermal conductivity varies greatly with gas pressure. The effective thermal conductivity keeps constant when the gas pressure is less than 0.01 kPa, which stands for the overall contribution of solid conduction and thermal radiation and it remains unchanged with gas pressure. However, the effective thermal conductivity increases sharply when the gas pressure is higher than 0.1 kPa due to the enhanced gaseous heat conduction.
123 124
R O
O
F
diameter of contact area, nm isobaric/isochoric specific heat capacity, J/mol · K diameter, nm diameter of spheres on each edge, nm k = 1 − λg / λs Boltzmann constant, 1.38 × 10−23 J/K mean free path, nm mass, g gas pressure, Pa specific surface area, m2/g temperature, K
is adjusted from low to high. The effective thermal conductivity of Super-G, a commercial silica aerogel composite that was manufactured by Microtherm® with a density of 240 kg/m3, porosity of 88.5% and claimed thermal conductivity of 0.0258 W/m·K at 373 K, is measured in nitrogen atmosphere under gas pressures ranging from 0.001 Pa to 1 MPa at 297 K.
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E
D
The contribution of gaseous heat conduction to the effective thermal conductivity of Super-G (λg,0) is decomposed from the experiment results as shown in Fig. 1. The gas thermal conductivities in both free space and nano-porous materials are also theoretically depicted, as shown in Fig. 1. The gas thermal conductivity in free space is obtained from NIST database with taking the effect of actual gas into account [23]. The gas thermal conductivity in super-G used for comparison is calculated from Zeng's equation [14]. The mean free path (lm0 ) in free space:
T
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2. Experimental investigations
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An apparatus based on the Transient Plane Source (TPS) method [18] is established to measure the effective thermal conductivity of porous materials under the gas pressures ranging from 0.001 Pa to 1 MPa. It's the first time that the thermal conductivity has been measured at a different gas pressure using TPS method. The accuracy of the apparatus is validated by using NIST1453, an expanded polystyrene board with thermal conductivity of 0.032 W/m·K at room temperature with a deviation within 3%. We studied the theoretical accuracy of TPS method and measured thermal conductivity of nano-porous materials in our previous works [19–22]. In the experimental investigations, the nano-porous materials are initially vacuumized exhaustively for at least 3 times to remove the absorbed water vapor or impurity gas, and the gas pressure
96 97 98 99
104 105 106 107 108 109 110 111 112 113 114
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94 95
129 130 131
134 135 136 137 138 139 140 141 142
144
1 lm ¼ pffiffiffi 2 : −1 2πdg p=kB T þ 0:25Ss ρpor ∏
ð2Þ 147
O
92 93
127 128
Due to the suppression of nano-porous structure, the mean free path of gas molecules in porous materials 145
C
90 91
125 126
ð1Þ
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88 89
kB T lm0 ¼ pffiffiffi 2 : 2πdg p
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86 87
C
100 101
84 85
119 120
3.2. Contribution of gaseous heat conduction decomposed from the effective 132 thermal conductivity 133
Reichenauer et al. found that λg,0 is about seven times higher than the gas thermal conductivity in a glass sphere bed with sphere diameter of 1 mm at ambient pressure [6]. They believed that the difference is caused by the local thermal shortcuts where spheres are touching; however, no further explanation was given. Swimm et al. studied the thermal conductivity of aerogels with average pore sizes of about 600 nm and 7 μm, respectively, under the gas pressures ranging from 10 Pa to 10 MPa [7]. A significant difference between λg,0 and the gas thermal conductivity was also observed, which is believed to be caused by the coupling heat transfer effect between solid and gas. A simple unit cell of two accumulated spherical particles was proposed to explain the coupling heat transfer effect; however, only qualitative agreement was obtained. This study aims at investigating the influence of gaseous heat conduction on the effective thermal conductivity of nano-porous materials within a wide range of gas pressure. Experimental measurement, theoretical analysis and numerical simulation are conducted to clarify the relationship between the gas thermal conductivity and λg,0 in nanoporous materials.
83
117 118
Fig. 1. Effective thermal conductivity of Super-G and comparison between gas thermal conductivity and contribution of gaseous heat conduction.
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
H. Zhang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
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Therefore the thermal conductivity of gas in nano-porous medium is different from those in the free space [14]: 1=2 ð2:25γ−1:25Þ0:461ðp=kB T Þ 8kB T=πmg mg cv : λg ¼ p ffiffiffi −1 2 0:25Ss ρpor ∏ þ 2ðp=kB T Þπdg
effective thermal conductivity of silica aerogels [15]. In this paper, the intersecting spherical structure (see Fig. 3) is selected as an example to calculate the λg,0 theoretically and reveal the difference between the gas thermal conductivity and λg,0. The effective thermal conductivity of the structure is derived as follows based on one-dimensional heat conduction assumption.
ð3Þ
150
166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183
185 186
Zeng et al. adopted three regular structures, intersecting square rod, intersecting cylindrical rod and intersecting sphere, to derive the
F
O
In Eq. (4), λg at different gas pressures is calculated from Eq. (3), d and D are determined by their relationships with porosity and specific surface area, and a / d = 0.2 is selected for calculation [15]. The specific surface area of Super-G is measured as 141.4 m2/g by applying adsorption and desorption of nitrogen with PCTPro-Evo (SETARAM Inc., France). For simplicity, the bulk thermal conductivity of SiO2 (1.34 W/m · K) is used as the skeleton thermal conductivity with the size effect ignored [2,7,10,15]. Theoretically, λg,0 at different gas pressures can be calculated by subtracting the effective thermal conductivity at ultimate vacuum from λe, which is then compared with the experimental λg,0 of Super-G as shown in Fig. 3. Obviously, the theoretical λg,0 shows quantitative agreement with the experimental result at full gas pressure range except for the gas pressure close to or higher than ambient pressure. Super-G is a composite that consists of silica aerogel and doped fibers and opacifier while the theoretical calculation is conducted with an effective structure using the same porosity and specific surface area as Super-G for simplification. Therefore, the difference is mainly introduced by the simplification of the effective structure. The theoretical study in this part is primarily used for illustrating the difference between λg,0 and λg,0. The agreement between experimental result and theoretical analysis could be further improved by applying more complicated models, which is beyond the scope of this article research. It is also proved that λg,0 is different from the gas thermal conductivity in nano-porous materials and can be larger than the gas thermal conductivity in free space at a higher gas pressure. 3.4. Contribution of gaseous heat conduction decomposed from the numerical simulation
Fig. 2. Mean free paths of nitrogen molecules in free space and in Super-G.
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A random generation-growth method is employed to generate a 221 porous structure with both the solid phase and the gaseous phase 222
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N C O
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3.3. Contribution of gaseous heat conduction decomposed from the effective structure
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ð4Þ
R O
164 165
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2 ðn−1Þπλg d a D D−ka d −1 þ ln þ 1− λg Dk d kd D−kd D
P
162 163
þ
D
160 161
187 188
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# a2 a2 πλg d D ndk πλs a 2 − 1− ln 1− 1− − þ 2nDk d ndk D d 4 D
E
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λe ¼
T
156 157
C
154 155
E
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As depicted in Fig. 1, λg,0 is much different from the gas thermal conductivity in nano-porous material, and it is even higher than the thermal conductivity of actual gas in free space at a higher pressure. Heat transfer in nano-porous material includes solid conduction via the solid skeleton, gaseous conduction in the pores, thermal radiation and coupled heat conduction between solid and gas. If the material is ultimately vacuumized, gaseous conduction in the pores and coupled heat conduction between solid and gas will be vanished. The experiment result proves that the variation of the effective thermal conductivity of nano-porous materials at different gas pressures is different from the variation of gas thermal conductivity at different pressures. Therefore, it is unreasonable to regard λg,0 as the thermal conductivity of gas in nano-porous material. In Fig. 1, the gas thermal conductivity in free space is almost unchanged when the gas pressure is not higher than 1 MPa while the gas thermal conductivity in nano-porous materials almost reaches zero when the pressure is less than 1 kPa and then increases rapidly with the increase of pressure. Such phenomenon is mainly attributable to the motion of gas molecules being suppressed by the nano-porous structure, and as a result, the mean free path of gas molecules in free space is different from that in nano-porous material (calculated from Eqs. (1) and (2)), as shown in Fig. 2. The gas thermal conductivity in nano-porous materials gets closer to that in free space when the pressure is higher than 1 MPa because the increment of molecules number density will shorten the mean free path, making it approach to that in free space with the increase of pressure. It is inconceivable to believe that the contribution of gas conduction to the effective thermal conductivity of nano-porous materials at different gas pressures, λg,0, can be greater than the gas thermal conductivity in free space. We studied many other nano-porous materials and the same phenomenon is observed, Super-G is selected as an illustration to clarify the difference between λg,0 and the gas thermal conductivity in nano-porous materials.
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3
Fig. 3. Comparison of contribution of gaseous heat conduction and the effective structure and random structure of aerogels.
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027
4
taking account in scale effect/interfacial effect of solid skeleton of 271 nano-porous materials doped with fiber and opacifier. 272
230
being considered as the continuous phase [17]. As shown in Fig. 3, the random structure has a porosity of 88.5% and domain size of 100 × 100 × 100. The ratio of the average pore diameter to the particle diameter is 5. Then the thermal conductivity of the random structure is calculated by applying the Lattice Boltzman method. λg,0 decomposed from the calculation is also compared with the experimental data as shown in Fig. 3. The decomposed λg,0 of the random structure also reveals the difference between λg,0 and the gas thermal conductivity.
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3.5. Influence factors of contribution of gaseous heat conduction
The authors would like to thank the support from National Natural 276 Science Foundation of China (51320105004 and 51276138). 277
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λg,0 is affected by the coupled heat conduction via solid and gas at different gas pressures when temperature is kept constant and the pores is filled with non-polar gases that are transparent to thermal radiation, then
ð5Þ
−f ðλs ; λr ; porosity; structureÞ ¼ f λs ; λg ; porosity; structure −f ðλs ; porosity; structureÞ: 237
249
4. Conclusion
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In summary, the effective thermal conductivity of open nano-porous materials with high porosity is influenced by gas pressure greatly. The contribution of gaseous heat conduction to the effective thermal conductivity decomposed from the experiment result is greater than the gas thermal conductivity in nano-porous materials and even greater than that in free space at a higher gas pressures. In many previous works, the relationship between the contribution of gaseous heat conduction to the effective thermal conductivity and gas thermal conductivity is not clearly clarified and often confused with each other. In this paper, the relationship is differentiated clearly. The contribution of gaseous heat conduction to porous material depends on not only the gas thermal conductivity, but also the solid thermal conductivity, porosity and spatial structure. Regarding the contribution of gas heat conduction decomposed from experiment as the gas thermal conductivity without distinguishing their difference may result in unreliable results. Experimental investigation, theoretical calculation and numerical simulation shown in this paper are three effective ways of obtaining the effective thermal conductivity and the contribution of gas heat conduction of porous materials. However, there are still some challenges of predicting the effective thermal conductivity more accurately, such as developing multi-scale theoretical models and multi-scale simulation,
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254 255
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246 247
C
244 245
N
242 243
U
240 241
278
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From Eq. (5), it is very easy to understand that λg,0 is influenced by gas thermal conductivity, solid thermal conductivity, porosity and spatial structure. The increment of gas thermal conductivity within the nano-porous materials will enhance the coupled heat conduction between gas and solid greatly because the total thermal resistance of heat conduction networks decreases much more than the decrement of thermal resistance of the filling gas. That is reason why the decomposed contribution of gaseous heat conduction of nano-porous materials can be larger than the gas thermal conductivity in free space at high gas pressures. Therefore, it's not reasonable to confuse the λg,0 with the gas thermal conductivity in porous materials and such analysis can be generalized as porous materials with different scale of pore size.
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References
F
λg;0 ¼ λe −λe ;0 ¼ f λs ; λg ; λr ; porosity; structure
275 Q3
Acknowledgments
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[16]
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5. Uncited reference
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H. Zhang et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx
Please cite this article as: H. Zhang, et al., The influence of gaseous heat conduction to the effective thermal conductivity of nano-porous materials, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.027