Conjugate heat transfer through nano scale porous media to optimize vacuum insulation panels with lattice Boltzmann methods

TL;DR: A new holistic approach provides a distinct advantage over similar porous media approaches by providing direct control and tuning of particle packing characteristics such as aggregate size, shape and pore size distributions and studying their influence directly on conduction and radiation independently.

Abstract: Due to reduced thermal conductivity, vacuum insulation panels (VIPs) provide significant thermal insulation performance. Our novel vacuum panels operate at reduced pressure and are filled with a powder of precipitated silicic acid to further hinder convection and provide static stability against atmospheric pressure. To obtain an in depth understanding of heat transfer mechanisms, their interactions and their dependencies inside VIPs, detailed microscale simulations are conducted. Particle characteristics for silica are used with a discrete element method (DEM) simulation, using open source software Yade-DEM, to generate a periodic compressed packing of precipitated silicic acid particles. This aggregate packing is then imported into OpenLB (openlb.net) as a fully resolved geometry, and used to study the effects on heat transfer at the microscale. A three dimensional Lattice Boltzmann method (LBM) for conjugated heat transfer is implemented with open source software OpenLB, which is extended to include radiative heat transport. The infrared intensity distribution is solved and coupled with the temperature through the emissivity, absorption and scattering of the studied media using the radiative transfer equation by means of LBM. This new holistic approach provides a distinct advantage over similar porous media approaches by providing direct control and tuning of particle packing characteristics such as aggregate size, shape and pore size distributions and studying their influence directly on conduction and radiation independently. Our aim is to generate one holistic tool which can be used to generate silica geometry and then simulate automatically the thermal conductivity through the generated geometry.

Summary

1. Introduction

• Energy efficiency is currently a very active topic of discussion and concern [1].
• By increasing the internal pressure of the VIP while maintaining a low effective thermal conductivity, not only can the lifetime of the panel be extended, the increased thermal conductivity of a damaged panel as well as the cost of production would decrease.
• The geometry is then used to simulate effects of diffusion and conduction.
• While this method allows direct control over pore size and overall porosity of the generated geometry, the method is implemented only in two dimensions.
• This work begins by presenting the different thermal transfer LBM components simulated in VIPs, including the effects of low pressure and small pore size.

2. Methods

• The authors holistic approach is composed of two parts, the first generates the silica particle geometry with YADE, the second simulates the effective thermal conductivity through the geometry usingOpenLB.
• The authors follow these two sections by describing the physical measurements of thermal conductivity through VIPs which are used to compare against the simulation results.

2.1. Geometry generation approach

• Yet Another Dynamic Engine (YADE DEM) is used to generate the geometry and compress the resulting aggregates [22].
• This geometry is passed automatically to OpenLB to simulate the effective thermal conductivity as described in Section 2.2.
• These aggregates are saved as individual text files defining particle locations and radii.
• The functionmakeClumpCloud creates a loose periodic packing of aggregates chosen equally from the input set to fill the volume.
• The end result is a set of similarly packed geometries of slightly varying pore and particle sizes.

2.2.1. LBM for energy equation

• ∆x and ∆t denote the grid spacing and time step width, respectively, and τ denotes the relaxation time.
• F eqi = wiT (x, t), (4) wherein wi are the discrete lattice direction’s weights and T (x, t) is the local temperature.
• This diffusion equation for the energy distribution is solved for the solid and fluidmaterial on a single lattice with respect to different temperature conductivities through space variant relaxation times τ.

2.2.2. LBM for radiation transfer equation

• Mink et al. [26] have shown to solve the P1-approximation of the radiative transfer equation (RTE) by means of LBM by introducing amesoscopic sink term.
• The spatial light intensity is given by the distribution function’s zeroth moment φ(x) = ∑ i gi. (11) The boundary radiation intensities are chosen following the Stephan Boltzmann law [27] according to the local lattice temperatures by φ = ( T(x,t) Thot )4 . (12).

2.2.3. Effective lambda

• It should be noted that heat conductivity due to the coupling term is negligible due to the relatively large mean free path of the molecules [7].
• The coupling term arises from the increased heat transport bridging between neighboring particles or fibers on the micro-scale and increases as the heat conductivity of the gas and solid increases.
• The effective heat conductivity λeff though the resolved packing is calculated by λeff = qeffL ∆T (14) where L is the cell length,∆T is the temperature difference between the upper (Γt ) and lower boundary (Γb) and the effective heat flux qeff is given by qeff = qs + qf + qr . (15).
• The heat fluxes for the solid qs and for the fluid qf are calculated by the temperature distribution’s first momentum in (6).
• The steady state is assumed to be reached, when the calculated effective thermal conductivity does not change more than 0.01% over the latest 10,000 time steps.

2.3. Simulation variables

• Commonly used β values for air and water vapor are 1.63 and 1.5 respectively.
• An initial estimate for the average pore size δ is given by (20) from the VDI Heat Atlas [27].
• Where d is the average particle diameter and Π the porosity of the silica.
• The scattering coefficient σs is estimated as 0.1 for the infrared regime [30].

2.4. Measurements

• In order to measure the thermal conductivity of the VIPs, a measurement system was constructed.
• The measurement system is composed of a wooden box filled with mineral wool.
• A second hot plate, placed underneath the first, is used to hinder heat loses through the bottom of the measurement system.
• Once the result reaches a stable level for 750 s, the pressure in the panel is reduced and a newmeasurement begins.
• To determine undesired heat flux to the environment or over the foil surrounding the core material, a well known standard material needs to be measured before every measurement.

2.5. Measurement uncertainty

• Calculating the propagation of uncertainty using Tables 2 and 3, together with the uncertainty for the thickness and area of the VIP, 5% and 1% respectively, in (23), a total uncertainty of 8.73% for the thermal conductivitymeasurement is obtained.
• Two pressure sensors are used, a Pirani sensor and a Siemens SITRANS P200 capacitive sensor tomeasure at low and high pressures respectively.
• The capacitive sensor has amaximumuncertainty of 0.5%, whereas the Pirani sensor is used between 0.01 and 10 mbar and has a maximum uncertainty of 15%.

3. Results

• Using the proposed method, several simulation setups were generated to evaluate the reliability and accuracy of the approach.
• The first study presented is a grid independence study used to confirm convergence of the model.
• Next, the results of varying air pressure are presented and compared to reported literature values.
• Lastly the effect of compressing the geometry on the thermal conductivity is presented.

3.1. Grid independence

• To confirm grid independence, a simplified geometry, shown in Fig. 6, considering a single sphere between two plates, is used to evaluate the effective thermal conductivity.
• The radiation intensity is calculated using the local lattice temperature and (12).
• The relative error versus the system resolution is shown in Fig.
• Thermal conductivity through air and due to radiation have a superlinear experimental order of convergence.
• The thermal conductivity through the solid is sublinear due to the abundance of curved boundaries.

3.2. Effective thermal conductivity as a function of pressure

• The radiation intensity is calculated using the local lattice temperature and (12).
• The total thermal conductivity is composed of the thermal conductivity due to the air, radiation and solid.
• As the system pressure decreases, the thermal conductivity of the air clearly decreases.
• The radiation and temperature distributions in the geometry can be seen in Figs. 9 and 10 respectively.
• While this should not be the case at very low pressures, this is essentially the coupling effect seen between particles.

3.3. Comparing with literature and measured values

• Fig. 11 shows a comparison between the simulations results and literature values provided by Fricke et al. [7] for heat conductivity of silica at various pressures.
• The radiation intensity is calculated using the local lattice temperature and (12).
• Literature values show that a constant thermal conductivity is reached below 10mBar.
• The relative error between the simulated and reference values is 6.12%.
• The simulations are also compared tomeasured thermal conductivities of VIP panels compressed under various pressures.

3.4. Effect of compression on thermal conductivity

• In order to investigate the effect of different compression forces on the thermal conductivity of the VIPs, several panels were compressed and their thermal conductivity measured.
• As the compression force increases, the thermal conductivity through the panel decreases.
• The porosity and average pore size was calculated for each geometry and then used to simulate the heat transfer through the generated geometry in OpenLB.
• The results are shown in Fig. 12b, where the compression level indicates the number of iterations applied to compress the particle geometry.
• 5 mW/mK is added to correct for thermal bridging artifacts and power losses in the measurement.

4. Conclusion

• The goal of this work was to generate a holistic method to simulate the effective heat transfer through vacuum insulation panels using procedurally generated 3D geometry of silica aggregates.
• This was achieved through combination of two open source software packages, YADE DEM and OpenLB.
• Comparisons were provided that show simulation results closely match measured effective heat transfers of real VIPs.
• The simulations are performed over a range of internal pressures and the geometry is compressed over varying degrees.
• Work has already begun developing new packing geometries which will result in cost reduction and increased lifespan of VIPs by increasing the minimum pressure required for low effective thermal conductivity.

Figures

Fig. 8. Heat conductivity breakdown due to air, radiation and through the solid material.

Fig. 10. Temperature distribution through periodic packing of silica aggregates.

Table 1 Comparison of thermal conductivity λ of high performance thermal insulation materials [3] (EPS and XPS are expanded and extruded polystyrene respectively).

Fig. 4. Thermal conductivity measurement system composed of: [1] insulated enclosure, [2] vacuum pump, [3] thermostat, [4] pressure sensor, [5] measurement electronics and laptop.

Fig. 9. Radiation distribution through periodic packing of silica aggregates.

Table 2 Measurement uncertainty for temperature sensors.

Fig. 5. Schematic diagram of measurement system. The power required to maintain the top hot plate at 40 ◦C is proportional to the effective thermal conductivity.

Table 3 Measurement uncertainty for power sensor.

Fig. 1. Geometry generated with YADE before and after compression, aggregates shown in different colors.

Fig. 2. The two domains for LBM, Ωs and Ωf for the solid and fluid respectively, with periodic boundaries Γp and temperature boundaries Γt and Γb .

Fig. 11. Comparison of effective thermal conductivity to reference values [7]. Simulated effective thermal conductivity decreases at low pressures due to solid conductivity being influenced by the globally defined mean free path.

Fig. 12. (A) Measurement of the effect different compression forces on the thermal conductivity of the VIP. (B) Simulation results of the effect of different compression forces on the thermal conductivity of the VIP (3.5 mW/mK is added to correct for thermal bridging artifacts in the measurement).

Fig. 6. Simplified geometry used to calculate experimental order of convergence. Top and bottom boundaries are held at constant temperature and the other 4 boundaries are periodic.

Fig. 7. Relative error as a function of simulation resolution.

Fig. 3. Schematic representation of the speed directions according to D3Q7 [25].

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Conjugate heat transfer through nano scale porous media to optimize vacuum insulation panels with lattice boltzmann methods" ?

This aggregate packing is then imported into OpenLB ( openlb. net ) as a fully resolved geometry, and used to study the effects on heat transfer at the microscale. The infrared intensity distribution is solved and coupled with the temperature through the emissivity, absorption and scattering of the studied media using the radiative transfer equation by means of LBM. This new holistic approach provides a distinct advantage over similar porous media approaches by providing direct control and tuning of particle packing characteristics such as aggregate size, shape and pore size distributions and studying their influence directly on conduction and radiation independently. 

Q2. What is the thermal transfer due to radiation at higher pressures?

At higher pressures, the number of air particles also increases and thus absorption of radiation increases and the heat flux due to radiation decreases slightly. 

Q3. What is the coupling term for the gas?

The coupling term arises from the increased heat transport bridging between neighboring particles or fibers on the micro-scale and increases as the heat conductivity of the gas and solid increases. 

Q4. What is the process for generating the geometry?

Since the entire process for generating the geometry is procedural and parameter driven, multiple geometries can be generated based on the same base geometry. 

Q5. What is the way to understand the effects of different packing geometries on the individual?

In order to understand effects of different packing geometries such as porosity and aggregate size on the individual contributions that make up the heat transfer through the VIP, a high fidelity model of the packing geometry is required. 

Q6. What is the effect of the pressure inside the VIP on the heat transfer?

As the pressure inside the VIP decreases, the heat transfer through the fluid decreases inversely proportional to the Knudsen number in (18) [7,27]. 

Q7. What is the equilibriumfunction for the heat transfer through a VIP?

(12)Heat transfer through a VIP is composed of the heat transfer through gas λG (convection), through solid λS (conduction), through radiation λR and a coupling term λC . 

Q8. What is the first method of generating the geometry?

The first method generates the geometry based on idealized elementary units or fractal geometry, the second based on inhomogeneous procedurally generated geometry. 

Q9. What is the thermal conductivity of a standard polystyrene board?

To calculate thermal conductivity as a function of temperature and density the following equation from Zarr et al. [31] holds for the standard material. 

Q10. What is the advantage of the three dimensional geometry generation method?

The three dimensional geometry generation method implemented provides a distinct advantage over other porous media approaches by allowing direct control and tuning of particle packing characteristics such as aggregate size, shape and pore size distributions and studying their influence directly on conduction and radiation independently. 

Q11. What is the holistic approach to silica?

Their holistic approach is composed of two parts, the first generates the silica particle geometry with YADE, the second simulates the effective thermal conductivity through the geometry usingOpenLB 

Q12. What is the effective heat conductivity of the solid qs and the fluid qf?

The effective heat conductivity λeff though the resolved packing is calculated byλeff = qeffL ∆T(14)where L is the cell length,∆T is the temperature difference between the upper (Γt ) and lower boundary (Γb) and the effective heat flux qeff is given byqeff = qs + qf + qr . (15)The heat fluxes for the solid qs and for the fluid qf are calculated by the temperature distribution’s first momentum in (6). 

Q13. What is the way to determine the thermal conductivity of a test plate?

To confirm grid independence, a simplified geometry, shown in Fig. 6, considering a single sphere between two plates, is used to evaluate the effective thermal conductivity. 

Q14. How is the thermal conductivity of the VIP panels measured?

The VIP panels compressed with 25 and 30 bar closely follow the simulations with compression levels 0 and 2 respectively, both with a relative error of 3.7%. 

Q15. What is the method for generating VIP nanostructure geometry?

Rochais et al. [9] also present a method for procedural generation of VIP nanostructure geometry based primarily on the fractal dimension and repetitions of periodic base structures (square-shaped, diamond-shaped, brick-shaped).