# 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.

Topics: Heat transfer (62%), Thermal conduction (59%), Thermal insulation (58%), Thermal conductivity (58%), Vacuum insulated panel (57%)

Conjugate heat transfer through nano scale porous media to

optimize vacuum insulation panels with lattice Boltzmann

methods

Jesse Ross-Jones

a,b,c,

∗

, Maximilian Gaedtke

b,c

, Sebastian Sonnick

a,b

,

Matthias Rädle

a

, Hermann Nirschl

b

, Mathias J. Krause

b,c

a

CeMOS (Center of Mass Spectrometry and Optical Spectroscopy), Mannheim University of Applied Sciences, Mannheim, Germany

b

Institute for Mechanical Process Engineering and Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany

c

Lattice Boltzmann Research Group, Karlsruhe Institute of Technology, Karlsruhe, Germany

Keywords:

Rarefied gas dynamics

Nano-porous materials

Lattice Boltzmann method

Vacuum insulation

Mesoscopic methods

Nano-porous silica

a b s t r a c t

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) simu-

lation, 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.

1. Introduction

Energy efficiency is currently a very active topic of discussion and concern [1]. Buildings, homes and transport vehicles

require energy and insulation to keep their occupants and/or wares at acceptable temperatures. Vacuum insulation panels

(VIPs) are of interest to industry sectors looking to improve thermal efficiency [1]. A VIP provides significant thermal

∗

Corresponding author at: CeMOS (Center of Mass Spectrometry and Optical Spectroscopy), Mannheim University of Applied

Sciences, Mannheim, Germany.

E-mail address: j.ross-jones@hs-mannheim.de (J. Ross-Jones).

Table 1

Comparison of thermal conductivity λ of high performance thermal insulation materials [3]

(EPS and XPS are expanded and extruded polystyrene respectively).

Material Thermal conductivity in W/mK Density in kg/m

3

Polystyrene

0.034–0.036 15–30

(EPS, XPS)

Mineralwool 0.033–0.040 8–500

VIP 0.004–0.008 150–350

insulation performance (0.004–0.008 W/(mK)) [2]. Due to this very low thermal conductivity, space required by insulation

can decrease 5 to 10 times (see Table 1) when compared to using traditional thermal insulation materials. VIPs have already

found widespread use in stationary refrigeration equipment [2]. Limited adoption, in house and building construction, has

been seen due to cost, limited lifetime and risk of damage during installation and subsequent use of the panel. However, 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.

To increase the minimum required pressure inside the VIP, the geometry of the packing medium can be modified. 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.

Previous work is available which study heat transfer through VIPs. These works are divided here into two groups based

on the form in which the geometry used for the simulation is generated. The first method generates the geometry based

on idealized elementary units or fractal geometry, the second based on inhomogeneous procedurally generated geometry.

The advantage of geometry based on elementary units lies in the relative simplicity of the mathematical formulation. For

example, Enguehard et al. [4] use transmission electron microscopy (TEM) images of nanoporous silica as a basis in order

to generate elementary bricks which emulate porous properties. Similarly, Spagnol et al. [5] present a method for modeling

nanoporous silica with fractal geometry. Knowing the fractal dimension of the aggregates and agglomerates, the geometry is

modeled with procedurally generated models of the porous structure. In another work, Jae-Sung Kwon [6] presented various

methods for modeling VIPs as idealized geometry, with a focus on modeling the solid and gaseous thermal conductivity as

well as radiation. The paper discusses modeling the contact area of different packing materials and compares these values

to values reported by Fricke et al. [7]. Extending upon the models of Jae-Sung Kwon [6], Kim et al. [8] used the idealized

geometry to study the effects of conductivity and radiation. The results of the models are compared to measurements.

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). The geometry

is then used to simulate effects of diffusion and conduction. Coquard et al. [10] also look into modeling the thermal

conductivity of VIPs while also considering the effects of moisture. The geometry is modeled in the form of elementary

cubes with a particular porosity and contact surface area.

While it is possible to closely approximate the geometry using the aforementioned methods, it is not possible to make

changes to the geometry once generated, nor is it possible to produce inhomogeneous geometries. Furthermore, manual

involvement is usually needed to transfer the generated geometry to the CFD simulation. Lallich et al. [11] present a method

which uses a diffusion limited cluster–cluster aggregation algorithm, which is used to model silica aggregates with a fractal

dimension between 1.8 and 2.5. The study is limited to studying radiation transfer in a vacuum. The use of lattice Boltzmann

methods (LBM) to study heat transfer (conduction and radiation) through complex porous (foam) structures is presented by

Wang et al. [12]. Similarly, Ferkl et al. [13] presented a method for 1D simulation of porous foam media for simulation of

heat transfer with some extensions to 3D. Both these works focus on foam structures and are not able to simulate aggregate

structures.

Wang et al. [14] presented a geometry generation algorithm to simulate the formation of porous media, based on

four parameters, core distribution probability, growth directionality, porosity and phase interaction growth probability.

While this method can model many different types of porous media, it does not facilitate modification of the geometry,

e.g. compression. Once the geometry is generated, LBM is used to simulate conduction through the media, radiation was

assumed to be negligible. However, radiation becomes a major component of the thermal conductivity and at low pressures

can contribute around 40% of the heat flux [15]. Additionally, Qu et al. [16] present a method for simulating heat transfer in

Silica Aerogels with LBM and compare their simulation results to published literature values. Using stochastically generated

porous geometry, heat transfer due to gas and solid heat conduction with radiation is simulated. While this method allows

direct control over pore size and overall porosity of the generated geometry, the method is implemented only in two

dimensions. Falcucci et al. have investigated the use of LBM in the transition Knusden regime ( 0.1 > Kn > 1) and introduce

a ‘sputtering’ boundary condition which redistributes outgoing lattice velocities to fit a probability distribution [17]. The

authors were able to show fair agreement between their results and theoretical/experimental values for complex nanoporous

geometries [18,19]. In the present study, convection is assumed to be negligible when the pore size is smaller than 3 mm [20]

and an average Knudsen number is computed at the start of each simulation and used for the whole domain.

The advantage of LBM is, on the one hand, the use of simple grids in combination with an efficient parallel algorithm,

which allows resolving complex geometries with greater ease than with comparable methods. On the other hand, no special

interface consideration is needed in LBM to assure temperature and heat flux continuities at the solid–fluid interfaces.

In this work a method for procedurally generating a compressed packing of 3D spherical silica and thereafter simulating

the effective heat transfer through VIPs without manual intervention, is presented. 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. Radiation and conduction are coupled in order to

simulate heat transfer at the nanoscale and for a range of pressures, from 1 bar down to 1 mBar. This enables an in depth

understanding of the heat transfer mechanism at the nanoscale and leads to both an improvement in the lifetime and

reduction of the manufacturing costs of VIPs. Simulation results are compared to measurements taken of VIPs, produced

with precipitated silica and pre-compressed under different pressures, as well as to literature values for silica VIPs. OpenLB

is used as it is an open source implementation of LBM with a parallel execution capable of handling complex geometry [21].

Our aim is to generate one holistic tool which can be used to generate silica geometry, and then automatically simulate the

thermal conductivity through the generated geometry. The simulations will be performed over a range of internal pressures

and the geometry compressed over varying degrees. We will therefore be able to increase the minimum pressure required

for a low effective heat transfer, which yields both an improvement in the lifetime and reduction of the manufacturing costs

of VIPs.

This work begins by presenting the different thermal transfer LBM components simulated in VIPs, including the effects

of low pressure and small pore size. Subsequently, the method used to generate the silica packing geometry is introduced.

Following, the measurement system used to measure the thermal conductivity of the VIPs is presented. Next, convergence

of the simulation results as a function of resolution is discussed. Lastly, the simulation results are compared to measured

thermal conductivity of VIPs.

2. Methods

Our 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 using OpenLB. We 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].

The process is divided into three steps, generating the aggregates, generating a loose packing of said aggregates and lastly

compressing the packing. This geometry is passed automatically to OpenLB to simulate the effective thermal conductivity as

described in Section 2.2.

The first step uses the YADE function makeCloud, which creates a random loose packing of spheres inside a parallelepiped.

Parameters for particle size distribution, aggregate size, and aggregate shape are provided to makeCloud to create a set of

aggregates. Each aggregate is generated with different random seed values so as to create a randomized set of aggregates.

These aggregates are saved as individual text files defining particle locations and radii.

In the second step the YADE function makeClumpCloud is used with parameters defining an initial bounding box and the

set of aggregate to be used. The function makeClumpCloud creates a loose periodic packing of aggregates chosen equally from

the input set to fill the volume. This volume is then saved to a text file defining the particle locations, radii as well as an index

defining to which aggregate the particle belongs.

Lastly, in step 3, the volume in step 2 is compressed periodically on all three axes using the YADE compression engine

PeriTriaxController, shown in Fig. 1 (The YADE compression engine produces compacted geometry and should not be confused

with gas compressibility). Of note is that the particles within an aggregate are not internally compressed, only the aggregates

themselves to one another. A maximum strain can be defined as a stopping criterion for the compression engine. Once the

compression is completed, the geometry is exported to an XML file containing particle locations and radii. This geometry

can then be directly imported into OpenLB in order to simulate the thermal conductivity as described in Section 2.2.

Since the entire process for generating the geometry is procedural and parameter driven, multiple geometries can be

generated based on the same base geometry. For example, different levels of compression can be applied to the procedurally

generated aggregates, allowing direct control over the aggregate size, shape and pore size distributions of the geometries

generated. The end result is a set of similarly packed geometries of slightly varying pore and particle sizes. This allows

investigation into how small changes in aggregate pore size influence the thermal heat transfer. Furthermore, the effect of

aggregate shape and particle size distribution can also be investigated.

2.2. Thermal conductivity simulation with lattice Boltzmann methods

As Wang et al. [14] state, LBM is well suited for simulating heat transfer in resolved porous media. Two lattice Boltzmann

equations have been combined to solve the conjugate heat transfer through the particle geometry. One lattice is used to

solve the diffusion equation for the temperature as described in Section 2.2.1, and a second lattice for radiative transport

described in Section 2.2.2. The two lattices are coupled to compute heat flux and effective heat transfer through the porous

media.

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

Fig. 2. The two domains for LBM, Ω

s

and Ω

f

for the solid and fluid respectively, with periodic boundaries Γ

p

and temperature boundaries Γ

t

and Γ

b

.

2.2.1. LBM for energy equation

To solve the energy equation the domain is composed Ω = Ω

s

∪ Ω

f

, where Ω

f

and Ω

s

define the fluid and solid

domains as shown in Fig. 2. Following Wang’s method for conjugate heat transfer [23] we solve two diffusion equations

for the temperature T

ρ

f

c

p,f

∂T

∂t

= λ

f

∇

2

T in Ω

f

, (1)

ρ

s

c

p,s

∂T

∂t

= λ

s

∇

2

T in Ω

s

, (2)

where ρ is the density and the indices f and s denote the fluid (air) and solid material (silica), respectively, t is the discrete

time and λ the thermal conductivity.

The diffusion equation can be solved by means of the LBM through

f

i

(x + ∆x, t + ∆t) = f

i

−

1

τ

(f

i

− f

eq

i

), (3)

where f

i

denote the discrete probability function in the discrete direction i, at discrete space coordinate x and discrete time

t. ∆x and ∆t denote the grid spacing and time step width, respectively, and τ denotes the relaxation time. The equilibrium

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

function f

eq

i

is given according to e.g. [24].

f

eq

i

= w

i

T (x, t), (4)

wherein w

i

are the discrete lattice direction’s weights and T (x, t) is the local temperature. In this work, we use a three

dimensional velocity set with seven discrete directions, referenced as D3Q7 and shown in Fig. 3, with the discrete weights

w

i

=

⎧

⎪

⎨

⎪

⎩

2

8

, red direction

1

8

, green directions.

(5)

The local temperature T (x, t) and heat flux q(x, t) are calculated from the distribution function’s moments by

T (x, t) =

∑

i

f

i

(x, t) and q(x, t) =

τ −

1

2

τ

6

∑

i=0

c

i

f

i

(x, t) (6)

with the discrete velocities c

i

.

Through a Chapman–Enskog expansion [24], the relation between the relaxation time and the thermal diffusivity α =

λ

ρc

p

is

τ =

α

c

2

s

+

1

2

(7)

with c

s

=

1

√

4

denoting the lattice speed of sound.

This diffusion equation for the energy distribution is solved for the solid and fluid material on a single lattice with respect

to different temperature conductivities through space variant relaxation times τ |

Ω

s

= τ

s

and τ |

Ω

f

= τ

f

. As mentioned by

Wang et al. [14], this approach is valid for ρ

solid

· c

p,solid

= ρ

fluid

· c

p,fluid

. To fulfill this condition, we choose the fluid’s density

in lattice units ρ

∗

f

= 1 and the solid’s density in lattice units ρ

∗

s

= ρ

∗

f

c

p,f

c

p,s

.

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 a mesoscopic sink term. Based on the assumptions related to the P1-approximation – homogeneous participating

media, spatially constant scattering and absorption parameter σ

s

and σ

a

– the RTE can be transformed into the diffusion–

reaction equation for the light intensity density φ

1

c

∂φ

∂t

=

1

3(σ

a

+ σ

s

)

∇

2

φ − σ

a

φ. (8)

Here, c denotes the speed of light and the artificial diffusion coefficient D is defined by

D(σ

a

, σ

s

) =

1

3(σ

a

+ σ

s

)

. (9)

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##### References

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••

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[...]

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