Limits of Thermal Insulations – Heat Transfer within Evacuated Porous High-Performance Insulations

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Limits of Thermal Insulations – Heat Transfer within Evacuated Porous High-Performance Insulations

https://doi.org/10.21203/rs.3.rs-2957179/v1

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published 12 Jun, 2023

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High-performance thermal insulations (HPI) have outstanding thermal properties to minimize heat transfer for a specific application. The term "high performance" is a relative rating: whether an insulating property, i.e. the thermal conductivity or thermal transmittance, is significantly lower than those of conventional insulating materials or systems depends on the use case and the related conditions. Applications fields for thermal insulations are the energy, building, industry and transport sector, but also the space and aviation industry. The total effective thermal conductivity of evacuated HPI depends on the contributions of the heat transfer via the solid skeleton of the porous insulation material and the radiative heat exchange. For a given operating temperature, the effective thermal conductivity yields a minimum depending on the density and the infrared-optically extinction properties of the insulation material. Examples for HPI with outstanding low thermal conductivity values will be presented and compared with the theoretical predictions based on a percolation model for the solid thermal conductivity of the skeleton and on a diffusion model for the radiative heat transfer.

High-Performance Insulation

Minimal Thermal Conductivity

Percolation Model

Scaling Exponent

Connectivity

High-performance thermal insulations (HPI) also known as thermal superinsulations are systems or materials with outstanding thermal properties minimizing heat transfer for a specific application. The term "superinsulation" is a relative rating: whether an insulating property, i.e. the thermal conductivity or thermal transmittance, is significantly lower than those of a conventional insulation material depends on conditions of use. The reduction of the global climate heating is one of the most urgent problems to be solved in the world and energy related CO₂-emissions contribute mainly to this effect. Thus, the application of superinsulations for the reduction of energy losses and thus the net energy demand in various applications is obvious. Applications fields for thermal insulations are energy applications, buildings [1, 2], industrial applications [3] and the transport sector [4], but also space and aviation industry. Specifically using suppression of heat transfer via the pore phase, nanoporous insulation materials, e.g. aerogel based systems or vacuum insulations, were used to provide slim, resource and energy efficient insulations for buildings or consumer products [5, 6], like refrigerators. The implementing of a hydrogen infrastructure, including the transport and storage of liquid hydrogen, needs also superior insulations, especially for cryogenic temperatures. In the industry sector, for example, heat losses in high temperature processes is an ongoing impetus for improving insulation systems. In space applications with its extreme conditions, the superior properties of superinsulation are exploited to minimize weight [7]. The transport of temperature sensitive vaccine in the frame of the Covid pandemic was done by using superinsulated transport boxes to guarantee a save transport in the latest corner of the world [8]. The mentioned examples illustrate the variety of conditions in which superinsulations have to operate, from 4 K up to 3000 K, under ambient atmospheric conditions, in vacuum or any other atmosphere. Therefore, there is not one superinsulation fitting all applications, but a whole variety of specialized approaches. One can also recognize a ‘race’ for the lowest effective thermal conductivity value, taking place in the academic and industrial research [9].

Faced with the problem to provide an excellent thermal insulation, one has to reduce the thermal conductivity to a minimum within the range of the given operation conditions. In general, heat transfer in porous materials will be determined by the heat transfer via the solid skeleton by phonons and, in the case of electrically conducting materials, by electrons and the connectivity of the solid skeleton, heat transfer via the gas within the porous structures and by radiative heat transfer [10]. It is trivial that the contribution of the gas can be limited to a negligible minimum by evacuating the porous system. Remaining is the optimization problem to balance the heat transfer via the solid and the radiative heat transfer. Reducing the mass, i.e. the effective density, of an insulation reduces the contribution of the solid conduction, however, enhances the radiative heat transfer because less material can interact with the thermal radiation. In general the conductive and radiative heat transfer is coupled and the overall heat transfer is theoretically described by coupled integro-differential equations [11]. For infrared (IR)-optically thick media, where the local radiative heat flux density depends only on the local temperature gradient, the coupling is negligible and separate expressions for conductive and radiative conductive can be derived and minimized in the sum. This paper provides some general rules to determine the minimum of thermal conductivity for a certain operating temperature and will discuss the question, whether a lower limit of thermal conductivity values for evacuated porous insulations can be given. The focus is on evacuated porous superinsulations. Other types of superinsulations, like multi-layer insulations are discussed elsewhere in detail [12].

Heat transfer by phonons

In the case of diffusive phononic heat transfer via the solid skeleton of a porous material Fourier’s law holds:

$${\overrightarrow{q}}_{c}= -{\lambda }_{s}\bullet \nabla T$$

with ${\overrightarrow{q}}_{c}$: heat flow density, λ_s: solid conductivity and $\nabla T$: temperature gradient. In this case the mean free path length of the phonons are small in comparison to the sample dimensions. Various theoretically models have been proposed for the description of the solid conductivity in porous material composites [13–16]. Most of these models describe the macroscopic heat flow based on the assumption of a characteristic microscopic geometry of the solid structure, e.g. within a unit cell, thus limiting the application of a specific model to other porous materials. However, in order to make general statements, simplifying models with general assumptions are beneficial. The so-called percolation model fulfills these aspects. In this model, a randomly distributed two phase medium is considered and can be simplified described by a randomly arranged three-dimensional network of thermal resistances [17]. The solid conductivity as a function of density ρ of the porous media follows a scaling rule [18–22]. Lu [20] proposed for the effective solid thermal conductivity λ_s the proportionality:

$${\lambda }_{s}\propto {(f-{f}_{c})}^{\alpha } and f=\frac{\rho }{{\rho }_{0}} ,$$

with α: scaling exponent, ρ: density of the porous media, ρ₀: bulk density of the porous media and the solid volume fraction f. f_c is the percolation threshold, which is defines the minimum solid volume fraction from which on a continuous path for heat transport between the considered macroscopic temperature gradient is present. The above mentioned two phase medium built a resistance network which consists of thermally conducting elements and non-conducting elements with zero thermal conductivity. In the vicinity of the percolation threshold Eq. (2) holds, i.e. $\left(f-{f}_{c}\right)\ll 1$. Thovert et al. provided theoretically scaling exponents for the effective thermal conductivity of two- and three-dimensional random media [23]:

α = 1.300 ± 0.002 for two-dimensional systems,

α = 1.9 ± 0.1 for three-dimensional systems. (3)

Assuming that the percolation threshold is negligible compared to the actual sold fraction and for low density materials, Eq. (2) can be simplified:

$${\lambda }_{s}\left(T\right)={\lambda }_{0}\left(T\right){f}^{\alpha }$$

with ${\lambda }_{0}$: intrinsic thermal conductivity of the solid structure from which the porous solid is built. Looking at Eq. (4), the similarity with Archie’s law is striking, which describes phenomenologically the effective electrical conductivity of a water-saturated porous non-conducting rock. Nielsen et al. found for crushed and compressed polyurethane foam with different densities a scaling exponent of α = 1.2 ± 0.3 [24]. Lu investigated different porous non-metallic powders and derived scaling exponents from 1.1 to 1.9. For highly porous monolithic opacified silica aerogels a scaling exponent of α = 1.5, for non-opacified silica aerogels α = 2.0 was reported [20, 25]. In both cases the experimental values were not corrected for the radiative contribution to the total effective thermal conductivity. Weigold et al. investigated the correlation of microstructure and thermal conductivity in polyurea aerogels and found a scaling behavior with α-values close to 1 [26]. In general, it can be simplified that in the case of fully connected structures, e.g. foams, α = 1 holds, which is already described by Gibson and Ashby [27]. In systems with a structurally fully connected skeleton, such as foams or even bonded fiber insulation, an increase in density leads to a proportional increase in sold phase thermal conductivity, since all material is always involved in heat transport. In porous structures, which also contain material areas that are not fully connected, containing e.g. dead ends, an increase in density can lead to the opening up of new heat flow paths and thus to an over proportional increase in heat conduction leading to scaling exponents α > 1.

In the further course of this work, it is assumed that Eq. (4) can be used as a good and simple approximation for the description of the effective solid conductivity.

Heat transfer by thermal radiation

Besides the heat transfer via the solid skeleton, radiative heat transfer is present in evacuated insulation materials. In optimized, i.e. IR-optical opaque thermal insulations considered here, it can be assumed that the radiative heat transfer is diffusive. In this case, a radiative thermal conductivity λ_r can be defined [28]:

$$\overrightarrow{{q}_{r}}=-{\lambda }_{r}\left(T\right)\bullet \nabla T= -\frac{16}{3}\bullet \frac{\sigma {\bullet n}^{2}\bullet {T}_{r}^{3}}{\rho \bullet {e}^{*}\left(T\right)}\nabla T$$

with ${\overrightarrow{q}}_{r}$: radiative heat flow density, σ: Stefan-Boltzmann constant, n: effective index of refraction, ρ: density, e^*(T): temperature-dependent effective specific extinction coefficient and T_r: mean radiative temperature. The specific extinction coefficient e*(T) is derived from the wavelength dependent specific extinction coefficient e*(Λ) and depends on the geometry and complex index of refraction of the material which interacts with the thermal radiation. For an isotropic, gray, highly porous media the radiative conductivity can be simplified:

$${\lambda }_{r}\left(T\right)= \frac{16}{3}\bullet \frac{\sigma \bullet {T}^{3}}{\rho \bullet {e}^{*}}$$

In this case, the effective index of refraction can be assumed to be close to one. For a gray media the specific extinction is independent of the wavelength and therefore constant with temperature [25].

Effective total thermal conductivity

The effective total thermal conductivity for evacuated porous thermal insulation materials with a diffusive radiative heat transfer is given by the sum of the effective solid thermal conductivity (Eq. (4)) and the radiative thermal conductivity (Eq. (6)):

$${\lambda }_{t}\left(T,\rho \right)={\lambda }_{c}\left(T,\rho \right)+{\lambda }_{r}(T,\rho )$$

$${\lambda }_{t}\left(T,\rho \right)={\lambda }_{0}\left(T\right){\left(\frac{\rho }{{\rho }_{0}}\right)}^{\alpha }+\frac{16}{3}\bullet \frac{\sigma \bullet {T}^{3}}{\rho \bullet {e}^{\text{*}}}$$

In real technical thermal insulation systems in many cases additional components are added to the primary material system, like IR-opacifier, binder material or other mechanically stabilizing additives. The effect of these materials on the solid conductivity is assumed to be neglected because of the small concentrations.

The effective total thermal conductivity Eq. (7) can now be used to calculate a minimal thermal conductivity for a given temperature. In a first step, the optimal density has to be derived. Since the solid thermal conductivity increases with density, while the radiative thermal conductivity decreases, there exit a minimum for a given temperature. The conditions for a minimum are:

$$\frac{d{\lambda }_{t}}{d\rho }=0$$

This derivation provides the optimal density for which a minimum in the effective total thermal conductivity occurs:

$${\rho }_{opt}\left(T\right)=\sqrt[\alpha +1]{\frac{B}{{\lambda }_{0}}{T}^{3}\frac{{\rho }_{0}^{\alpha }}{\alpha }}$$

with B = (16·σ)/(3·e*). The higher the temperature, the higher the optimal density since the radiative heat transfer become more and more dominant at higher temperature. Inserting Eq. (9) in Eq. (7) yields to an expression for the minimal total effective thermal conductivity λ_min:

$${\lambda }_{min}= \left(\alpha +1\right){\lambda }_{0}{\left(\frac{B\bullet {T}^{3}}{{\lambda }_{0}\bullet {\rho }_{0}\bullet \alpha }\right)}^{\frac{\alpha }{\alpha +1}}$$

By defining the radiation-conduction parameter N:

$$N\left(T\right)=\frac{B\bullet {T}^{3}}{{\rho }_{0}\bullet {\lambda }_{0}}=\frac{{\lambda }_{r}\left({T,\rho }_{0}\right)}{{\lambda }_{0}}$$

eq. (10) can be expressed as:

$$\frac{{\lambda }_{min}}{{\lambda }_{0}}= \left(\alpha +1\right){\left(\frac{N}{\alpha }\right)}^{\frac{\alpha }{\alpha +1}}$$

The radiation-conduction parameter N is the ratio of the radiative to conductive heat flux within the non-porous, purely absorbing medium. In the case of a medium, where thermal radiation is also reduced by scattering effects, scattering occurs on the interfaces of solid and pore volumes and internal grain boundaries, for example. If the macroscopic density of the skeleton increases close to the bulk density, the interface is more and more reduced. Additionally, dependent scattering occurs if the distances between the scattering centres become smaller than the wavelength of the scattered radiation. Both effects reduce the extinction coefficient by scattering within the material. Assuming a constant bulk thermal conductivity λ₀, N increases strongly nonlinear with temperature. This assumption is justified because just a limiting value is sought and rather than the exact value of the thermal conductivity. The resulting ratio of the minimal total effective thermal conductivity to the solid thermal conductivity of the non-porous material follows a root function (see Fig. 1).

In Fig. 1 can be also seen that the higher the scaling exponent, the lower the minimal total thermal conductivity for a given temperature and density. This again, follows the fact that for values of α > 1, a specific amount of mass present is not involved in the solid heat transfer. The functional dependence of the effective total thermal conductivity as a function of temperature and density is depicted in Fig. 2 for the arbitrarily set solid thermal conductivity of 1.2 Wm^-1K^-1 and extinction coefficient of 40 m²kg^-1, which holds for a glassy material, and α = 2. For a given temperature, a minimum regarding the thermal conductivity can be observed shifting from low densities at low temperatures to increasing density values at higher temperature (cf. Eq. (9)). At higher temperatures, radiation becomes more and more dominant and more material per volume lead to a higher extinction of radiation.

With the minimum effective total thermal conductivity for a given temperature, the lower limits of the thermal conductivity values can be derived section by section, assuming that for a given temperature section the effective thermal conductivity of the solid is constant. This sequestering is useful, because for lower temperatures organic materials such as plastics may be considered, which can have much lower values of total thermal conductivity than inorganic materials, e.g. amorphous silica. At higher temperatures, temperature stable ceramic materials have to be considered, which have principally higher values of the bulk thermal conductivity because of their crystalline character in comparison to amorphous materials. At the highest temperature carbon based insulation materials are used. Here also, graphite or carbonaceous microcrystallites structures result in in higher values of the solid skeleton.

The experimentally determined values of the effective thermal conductivity of different selected superinsulation materials in wide temperature range will be used to validate the theoretically considerations. A brief description of the insulation materials is presented in Table 1. The selection of published thermal conductivity data was guided by the need to consider materials with particularly low thermal conductivity values, as far as this was known to the authors. Furthermore, the measuring devices and measurement uncertainties are described in the literature.

Table 1

Compilation of evacuated thermal superinsulation materials with outstanding low thermal conductivity values.
Insulation Material	Description	Temperature Range / K	Lit.
Polyamide fibers insulation	trilobal polyimide fiber (radius ∼ 3.7µm)	118–423	[29]
Opacified Silica Aerogel Powder	90 wt.% silica aerogel powder, 10 wt. % carbon black	100–523	[30]
Hollow Microspheres	hollow glass bubbles, type K, black sleeve	204, mean temperature	[31]
Opacified Silica Aerogel	Monolithic porous silica aerogels with infrared opacifier	280–535	[20]
‘Microtherm®’ Powder Board	Commercial compressed silica powder board with infrared opacifier	293–523	[30]
Opacified Silica Powder	Fumed silica with infrared opacifier	448–663	[32]
Furnace Black ‘Printex 95 ®’	powder, particle size 0.015 µm, manufacturer: Orion Engineered Carbons	296–1306	[33]
Carbon Aerogel	monolithic porous carbons with density 225 kgm^− 3	573–1775	[34]

It is obvious, looking at Table 1, that in these optimized thermal insulations in most cases the amount of thermal radiation were reduced by the use of infrared opacifier or by the intrinsic extinction property of the base material, even at moderate temperatures. Data for lower temperatures as 100 K were not considered because a strong temperature dependence for the solid thermal conductivity can be expected in this range. For amorphous materials, the solid conductivity decreases strongly with decreasing temperature. For crystalline materials with falling temperatures, the solid thermal conductivity raises to a maximum before it decreases which reflects the increasing mean free path of phonons with decreasing temperatures and the limiting of it by the finite size of material structures, e.g. grain or specimen boundaries. Figure 3 shows the thermal conductivity values of selected superinsulation materials (cf. Table 1) as a function of temperature. Additionally plotted are the lower boundary function for the thermal conductivity according to Eq. (12). Three different parameter set have be used to cover the whole temperature range. In all cases the scaling exponent α was set to α = 2, which is close to the theoretically expected value for a three-dimensional random media. The solid lines represent the minimal effective total thermal conductivity assuming that the optimal effective density as function of temperature is given. Since the experimental data sets have a constant density for each material investigated, the experimental derived thermal conductivity data have to be only at one temperature close to the lower limit. However, this is usually not necessarily the case for the highest temperatures, as in many practical cases the thermal insulation is optimized for this.

In Fig. 4, where the same data of Fig. 3 are plotted on a double-logarithm scale. The gray shaded area more or less represents the limit below which thermal conductivity values is no longer achievable at a given temperature, as far as the assumptions made here are valid.

The objective of this work was to provide general rules to determine the minimum of the effective total thermal conductivity of evacuated insulations for a certain operating temperature. Based on a general approach for the descriptions of the solid conductivity, which is based on the percolation theory, and for the diffusive radiative heat transfer, a model is presented for the calculation of a minimum effective total thermal conductivity for a given temperature. Eq. (12) with the scaling exponent α = 2 describes the general slope a = 2 for the lower limit value curve, describing the lowest thermal conductivity values for a given temperature (cf. Figure 3 and Fig. 4). The presented selected experimental literature data are close to the proposed limit curves. It is clear that, due to the necessary simplicity of the model and the wide variety of thermal conductivity data used, a qualitative rather than quantitative agreement can be expected and the chosen double-logarithm plot makes larger deviations look smaller. It can be concluded that the gray shaded area, shown in Fig. 4, more or less represents the lower limit of the effective total thermal conductivity of evacuated insulations at a given temperature, as far as the assumptions made here are valid. Reported values far below this limit should be considered with care and discussed in detail, concerning in respect to the measurement conditions, used thermal conductivity equipment and uncertainty budget. Generally, it can be stated, that it is advantageous to design insulation materials with a widespread solid skeleton where not all branches are connected, so there are also so-called dead ends [35]. From the model point of view, this means systems with a high scaling exponent have to be preferred. In addition, the bulk thermal conductivity of the solid backbone material should be low. Depending on the application, it is important to ensure that the thermal insulation has sufficient mechanical stability. This applies less to loose powder fills and fiber fleeces, but does apply to self-supporting insulation boards. In some cases, this pushes up the limit of what can be achieved in terms of low thermal conductivity values.

H.P. Ebert, In Nearly Zero Energy Building Refurbishment, ed. by F.P. Torgal, M. Mistretta, A. Kaklauskas, C.G. Granqvist, L.F. Cabeza (Springer 2013)
A. Berge, P. Johansson, Report in Building Physics (Chalmers University of Technology, Gothenburg, Sweden, 2012)
A. Hunt, (Lawrence Berkeley National Laboratory, 2006)
J. Wernery, S. Brunner, B. Weber, C. Knuth, M. Koebel, Appl. Sci. 11(7), 19 (2021). https://doi.org/10.3390/app11072939
I. Smirnova, P. Gurikov, J. Supercrit. Fluids. 134, 228 (2018). https://doi.org/10.1016/j.supflu.2017.12.037
U. Berardi, In Nanotechnology in Eco-efficient Construction (Second Edition), ed. by F. Pacheco-Torgal, M.V. Diamanti, A. Nazari, C.G. Granqvist, A. Pruna, S. Amirkhanian (Woodhead Publishing, 2019), p. 395–416
Aerogels handbook, (Springer, News York, 2011)
G. Gärtner, C. Geier, H.P. Ebert, H. Singh, Pharmaceutical Manufacturing and Packing Sourcer, (2021)
B. Merillas, J.P. Vareda, J. Martín-de León, M. Rodríguez-Pérez, L. Durães, Polymers. 14(13) (2022). 10.3390/polym14132556
D. Buettner, A. Kreh, J. Fricke, H. Reiss, High Temp. - High Pressures. 21(1), 39 (1989)
H.P. Ebert, J. Fricke, High Temp. - High Pressures. 30(6), 655 (1998)
J.E. Fesmire, S.D. Augustynowicz, B.E. Scholtens, In Crygenic Engineering Conference – CEC ed. by W.I. J.G. (American Institute of Physics, 2008), p. 1359–1366
E.U. Schlünder, In 7th International Heat Transfer Conference (Hemisphere Publishing Co., Munich, Germany, 1982), p. 195–212
A.D. Brailsford, K.G. Major, Br. J. Appl. Phys. 15, 313 (1964)
H.W. Godbee, W.T. Ziegler, J. Appl. Phys. 37(1), 56 (1966)
E. Tsotsas, H. Martin, Chem. Eng. Process. 22(1), 19 (1987)
J. Bernasconi, H.J. Wiesmann, Phys. Rev. B 13 (3), (1976)
S.R. Broadbent, J.M. Hammersley, Mathematical Proceedings of the Cambridge Philosophical Society 53 (3), 629 (1957) doi:10.1017/S0305004100032680
P.G. de Gennes, Le J. de Phys. - Lettres. 37, L (1976)
X. Lu, Physics Department (University, Würzburg, Germany, 1991)
C. Scholz, F. Wirner, M.A. Klatt, D. Hirneise, G.E. Schröder-Turk, K. Mecke, C. Bechinger, Phys. Rev. E. 92(4), 043023 (2015). 10.1103/PhysRevE.92.043023
T. S., Random Heterogeneous Materials - Microstructure and Macroscopic Properties. (Springer New York, NY, 2001), p XXI, 703
J.F. Thovert, F. Wary, P.M. Adler, J. Appl. Phys. 68(8), 3872 (1990). 10.1063/1.346274
L.V. Nielsen, H.-P. Ebert, F. Hemberger, J.P.D. Fricke, A. Biedermann, M. Reichelt, U.D. Rotermund, High. Temperatures-high Pressures. 32, 701 (2000)
H.-P. Ebert, In Aerogels Handbook, ed. by M.A. Aegerter, N. Leventis, M.M. Koebel (Springer Science + Business Media, New York, 2011), p. 537–564
L. Weigold, D.P. Mohite, S. Mahadik-Khanolkar, N. Leventis, G. Reichenauer, J. Non Crystalline Solids. 368, 105 (2013). 10.1016/j.jnoncrysol.2013.02.029
L.J. Gibson, A. M.F., Cellurar Solids. 2nd edition edn. (Cambridge University Press, Oxford, 1997)
R. Siegel, J.R. Howell, Thermal Radiation heat transfer (McGraw-Hill Kogakushka, Ltd., Tokyo, 1972)
M. Geisler, J. Wachtel, F. Hemberger, T. Schultz, S. Vidi, H.-P. Ebert, Int. J. Thermophys. 29, 9 (2008)
R. Caps, J. Fricke, Int. J. Thermophys. 21(2), 7 (2000)
J.E. Fesmire, Phys. Procedia. 67, 8 (2015)
H. Reiss, In AIAA 16th Thermophysics Conference, (American Institute for Aeronautics and Astronautics, Inc., Palo Alto, Cal., 1981)
H.P. Ebert, B. Laudensack, F. Hemberger, O. Nilsson, J. Fricke, High Temp. - High Pressures. 30(3), 261 (1998)
M. Wiener, G. Reichenauer, S. Braxmeier, F. Hemberger, H.P. Ebert, Int. J. Thermophys. 30(4), 1372 (2009)
K. Swimm, S. Vidi, G. Reichenauer, H.P. Ebert, Int. J. Thermophys. 43(1), 8 (2021). 10.1007/s10765-021-02936-4

No competing interests reported.

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You are reading this latest preprint version

Limits of Thermal Insulations – Heat Transfer within Evacuated Porous High-Performance Insulations

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Journal Publication

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Abstract

Figures

1. Introduction

2. Heat transfer mechanisms in evacuated porous insulations

3. Minimal effective thermal conductivity

4. Comparison of experimental date with theoretical predictions

5. Conclusions

References

Additional Declarations

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