Correlation of pore size distribution with thermal conductivity of precipitated silica and experimental determination of the coupling effect

TL;DR: In this article, the pore size distributions of the samples were measured by mercury intrusion porosimetry and used to calculate the gas thermal conductivity as a function of the residual pressure.

About: This article is published in Applied Thermal Engineering.The article was published on 2019-03-05 and is currently open access. It has received 20 citations till now. The article focuses on the topics: Vacuum insulated panel & Thermal insulation.

Summary

1. Introduction

• Thermal insulation materials are essential for sustainable energy management of processes and devices.
• This leads to a phenomenon where the gaseous thermal conductivity inside the pores decreases.
• This special property makes such insulation ⁎.
• Paul-Wittsack-Str. 10, 68163 Mannheim, Germany, also known as Corresponding author at.
• Thus, precipitated silica can be attractive for some applications, such as thermal transport boxes, where long lifetimes are unnecessary.

2. Measurements and analysis

• To characterize and compare properties of different precipitated silica samples, five different types have been examined.
• In the subsequent sections, the following shortcuts are used: GT, MX, CP, KS, SM.
• These samples were selected to get a good overview of the company's product portfolio.
• All samples were prepared with external pressure loads of 5 bar and 30 bar with a hydraulic press.

2.1. Mercury intrusion porosimetry (MIP)

• Mercury porosimetry analysis is the progressive intrusion of mercury into a porous structure under strictly controlled pressure [8].
• The main advantage is that it allows pore size analysis to be undertaken over a wide range of mesopore–macropore widths (routinely, from ca. 0.003 to ca. 400 μm) [9].
• To calculate the minimum pore size x as a function of the particular mercury pressure level P, the surface tension of mercury and the contact angle between mercury and silica are required.
• Nomenclature Symbols Q heat flux [W] x̄ average pore size [m].
• D fractal dimension [–] d diameter [m] E transport extinction coefficient [m−1].

2.2. Test apparatus for thermal conductivity at various pressure levels

• To measure thermal conductivity of silica samples at various pressure levels, a test bench, which is able to operate under vacuum, has been developed.
• A view into the opened tank is shown on Fig. 2.
• Two samples of equivalent thickness, between 5 and 15mm, are placed between the heating plate and the two cooling plates.
• When testing loose samples, the powder samples can be pressed directly into the mold within the chamber.
• For all control loops, 12 PT100 temperature sensors T1.1 to T4.2 are positioned at several locations around the test stand.

2.2.1. Uncertainty analysis of thermal conductivity measurement

• Calculating the propagation of uncertainty using Table 1, together with the uncertainty for the thickness and area of the VIP, 1.5% and 1% respectively a total uncertainty of 2.07% for the thermal conductivity measurement is obtained.
• For this purpose a standard polystyrene board from NIST (National Institute of Standards and Technology) is used.
• The measured value of the newly developed guarded hot plate device is 0.0341 WmK .
• Thus, it matches perfectly within the uncertainty range of the test material.
• Two capacitive pressure sensors are used, a “MKS Baratron Type 122A” and a “Siemens SITRANS P200” sensor to measure at low and high pressures respectively.

3. Thermal transport theory and analytical model

• A fourth mechanism which depends on the microscopic shape of the material and also on gas pressure is the coupling effect between solid and gaseous thermal conductivity c [12].
• C is negligible for most foams [13] but plays a decisive role for silica based core materials, which are investigated in this work.
• All heat transfer mechanisms are temperature dependent, however, in this work constant and steady state temperature are assumed.

3.1. Solid thermal conductivity

• Heat flow over the solid phase of a bulk powder s depends on various material properties.
• Of course, the thermal conductivity SM of the solid material itself plays an important role, but also the porosity and the thermal contact resistance between the particles are influential.
• The contact resistance is not easy to capture and depends on the Poisson's ratio , the elastic modulus Y, the radius of the particles and the effecting pressure load F.
• These influences are summarized in Eq. (4), which is used by many authors to calculate solid thermal conductivity [4,15].

3.3. Gaseous thermal conductivity

• Considering the Knudsen effect, thermal conductivity of gases surrounded by a solid material is a function of geometric size and gas pressure or rather the mean free path of gas molecules.
• The Knudsen Number Kn is used to quantify the relation between the mean free path of molecules L and the geometric size of the confined space, for example the pore size x.
• The accommodation coefficient is a measure for the quality of energy exchange between the gas molecules and the solid surface.

3.4. Coupling effect

• Using Eq. (3) it is assumed, that all thermal resistances are arranged in a simple parallel configuration.
• Basically this assumption is correct, but as already mentioned for most materials a coupling of the different heat transfer mechanisms occurs; for example a coupling between solid thermal conductivity and radiation [20] or, and this is the focus of the present study, a coupling between solid and gaseous thermal conductivity.
• The effect is mainly caused by intervening gas molecules in the contact area of two particles.
• Some models to describe the coupling between these two mechanisms have been proposed, in most of the cases for silica aerogels [21–23].
• In the simplest case, the phenomenon can be described as a series connection of thermal resistances.

3.5. Calculation of thermal conductivity of various silica samples via pore size distribution

• To predict the thermal conductivity of the investigated precipitated silica samples, pore size distribution measured with MIP of the compacted powder have been used.
• Following, for every pore size x and gas pressure p the Knudsen number Kn can be calculated [4].

4.1. Results of thermal conductivity measurements

• Each silica sample was measured twice with applied pressure loads of 5 bar and twice with 30 bar, over a range of 14 pressure levels between 0.05mbar and atmospheric pressure.
• The results for all the samples are shown in Fig. 4 (dash and dash-dot lines).
• As expected, thermal conductivity approaches a constant value at very low internal pressures and increases between 1mbar and 10mbar.
• A characteristic behavior for nanoporous silica samples is a slope 0ddp > for p 10 mbaratm 3= , which is evidence for the presence of pores smaller than the mean free path of air molecules at atmospheric pressure (68 nm), following the Knudsen-theory.

4.2. Mercury intrusion porosimetry

• To measure the pore size distribution of the investigated silica samples, each sample was prepared with different pressure loads (5 bar and 30 bar), heated at 350 C° and 50 mbar for 16 h and subsequently analyzed with mercury intrusion porosimetry.
• Fig. 3 shows the pore size distribution of the sample “GT” pressed with 5 bar and 30 bar respectively.
• It could be shown for all samples, that mechanical treating of the powder had negligible influence on pores smaller than about 200 nm.
• The measured pore size distribution can be used to calculate gaseous thermal conductivity, as it is necessary to know the volumetric percentage of every pore size.

4.3. Calculation of thermal conductivity without coupling effect

• Thermal conductivity curve progressions calculated with p p( ) ( )sr g= + and Eq. (10) are shown in Fig. 4 (solid and dotted lines), purposely leaving out the coupling effect.
• It can also be seen that the thermal conductivity of the 30 bar samples increases less strongly with an increase of the internal pressure than that of the 5 bar samples.
• On the other hand, taking a closer view at low pressure levels it is noticeable that the calculated curves rise faster than the experimental ones, although no coupling effect is considered, which leads to an intersection of both curves.
• This means, the probability for a gas molecule to “see” in the direction of heat flowQ, a smaller distance than the measured pore size is very high.
• Therefore, the following correction of the pore size dis- tribution for spherical pores is assumed.

4.4. Determination of the coupling effect

• The previous results visualize the important role of the coupling effect.
• Thus, the coupling effect factor f is determined by fitting the calculated curves to the experimental data using the method of least squares.
• Also listed in Table 3 are the porosity and the surface fractal dimension D of the pore-surface, both determined by MIP.
• The coupling effect basically depends on the relationship between gaseous and solid thermal conductivity, which is influenced by the geometry of the solid material.
• Therefore, the results can be used to compare the surface structures of the materials.

5. Conclusion

• The developed guarded hot plate apparatus delivers reliable results.
• This gap can now be closed with the newly gained knowledge.
• The comparison of the tested materials shows that the pore size distribution is a decisive influencing factor on the thermal properties of a silica product, nevertheless the coupling effect must not be neglected.
• This should be possible by reducing the basic particle size and therefore increase the number of particle-particle resistances.
• Furthermore, investigating the effect of alternative pore gases could be helpful in order to validate the presented model and also as another strategy to decrease the overall thermal conductivity.

Figures

Table 2 Average pore sizes of silica samples in µm.

Fig. 4. Results of the thermal conductivity measurements of five samples of precipitated silica, pre-pressed with 5 and 30 bar compared to the corresponding calculated curves without consideration of the coupling effect.

Fig. 5. Illustration of the available distances for a gas molecule inside a spherical pore.

Table 3 Coupling effect factor f, porosity measured with MIP and fractal dimension D for the five samples prepared with 5 bar and 30 bar external pressure load.

Fig. 6. Results of the thermal conductivity measurements of five samples of precipitated silica, pre-pressed with 5 and 30 bar compared to the corresponding calculated curves with adapted linear fitting parameter f.

Fig. 1. Illustration of the test stand for measuring thermal conductivity at various pressure levels.

Fig. 2. View into the opened vacuum chamber. Cooling plate 1 can be seen outside the sample mold. Both samples, the heating plate and the cooling plate 2 are already in position.

Fig. 7. Linear relationship of the porosity measured with MIP and the coupling effect factor f.

Fig. 3. Pore size distribution of “GT” prepared with two different pressure loads (5 bar and 30 bar), measured with mercury intrusion porosimetry, is the porosity of the sample.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Correlation of pore size distribution with thermal conductivity of precipitated silica and experimental determination of the coupling effect" ?

To extend the application range of vacuum insulation, the authors try to replace the high-priced core material fumed silica with the cheaper precipitated silica. For this purpose, a correction factor for the measured pore size distribution is introduced. A model is presented to predict the thermal conductivity curve, even of unknown silica samples, solely using mercury intrusion porosimetry data. 

Q2. What are the future works mentioned in the paper "Correlation of pore size distribution with thermal conductivity of precipitated silica and experimental determination of the coupling effect" ?

In future work, attempts could be made to reduce the overall thermal conductivity by reducing the solid thermal conductivity.