Author's Accepted Manuscript

Heat transfer in one-dimensional micro- andnano-cellular foams

Pavel Ferkl, Richard Pokorný, Marek Bobák,Juraj Kosek

PII: S0009-2509(13)00277-7DOI: http://dx.doi.org/10.1016/j.ces.2013.04.018Reference: CES11008

To appear in: Chemical Engineering Science

Received date: 17 January 2013Revised date: 4 April 2013Accepted date: 9 April 2013

Cite this article as: Pavel Ferkl, Richard Pokorný, Marek Bobák, Juraj Kosek,Heat transfer in one-dimensional micro- and nano-cellular foams, ChemicalEngineering Science, http://dx.doi.org/10.1016/j.ces.2013.04.018

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Heat transfer in one-dimensional micro- andnano-cellular foams

Pavel Ferkla, Richard Pokornya, Marek Bobaka, and Juraj Kosek∗b

aDepartment of Chemical Engineering, Institute of Chemical TechnologyPrague, Czech Republic

bResearch Centre New Technologies, University of West Bohemia,Univerzitnı 8, 306 14 Pilsen, Czech Republic

Abstract

We investigate heat transfer in one-dimensional multi-layer models of polymerfoams and determine their equivalent thermal conductivities. The proposed modelconsiders participating gas and solid phases, i.e., combined heat transfer by conduc-tion and radiation in both phases and partial photon reflection on phase interfaces.A correction for possible non-Fourier heat conduction in small gas cells, occurringin nanocellular foams, is implemented by the reduction of gas thermal conductivitybased on the Knudsen number. The developed model was used for the optimizationof foam structure with respect to its insulation properties. We found that althoughradiation can account for more than third of the total heat flux, it can be reduced infoams with cell-sizes below 100 µm and be almost nullified by decreasing the foamcell size to sub-micron range. The presented model is a starting point for the devel-opment of an advanced model of heat transfer in spatially three-dimensional polymerfoams.Keywords: Foams, Heat transfer, Radiation, Polymer nanofoams, Thermal insula-tion, Mathematical modeling.

1 Introduction

Heat transfer in porous materials is a hot topic in present research, because it is associatedwith many applications in both construction and chemical industry. The focus of thisarticle is on polymer foams, which are common materials used for insulation purposes.

∗Corresponding author, E-mail: kosekj@ntc.zcu.cz, Phone: +420 377 634 744, Fax: +420 377 634 702

1

Two main heat transfer modes that must be considered when dealing with polymerfoams are conduction and radiation. The influence of free convection can be safely ne-glected, because the pore size in polymer foams is smaller than 3 mm (Govan et al., 1983).In materials, where the heat conduction is the dominant heat transfer mode, the heat ra-diation can be omitted and the problem simplifies significantly. The overview of analyticaland numerical methods used for conduction-only methods used in porous materials can befound in Ochsner et al. (2008). However this is not the case for high porosity foams, wherethe radiative contribution can reach up to 40 % of the total heat flux (Kaemmerlen et al.,2010).

The thermal radiation can be treated either as (i) independent or (ii) coupled with theheat conduction. The first approach is not physically correct and the temperature fieldcannot be determined. On the other hand, it can substantially simplify the numericalscheme while still being able to estimate the heat flux, which is often the most importantresult in the heat transfer modeling. For this reason, it is still often used when computingthe heat flux in complex multi-phase domains (Placido et al., 2005; Wang and Pan, 2008a).The second approach, coupled conduction-radiation in participating media, is the physi-cally correct method that moreover provides us with temperature field, and therefore withmore detailed information about the distribution of conductive and radiative heat fluxes inthe medium. On the other hand the second approach was due to the numerical complexityand computational cost so far studied mostly in homogeneous media or in materials withfairly limited number of obstacles (Wu and Chu, 1998).

The goal of this paper is to study the coupled heat transfer by conduction and radiationin one-dimensional multi-phase media representing the polymer foam and, in comparisonwith commonly employed diffusion approximation, to provide some additional insight tothe heat transfer in polymer foams like the detailed temperature profile and the interactionbetween conduction and radiation near the phase interfaces. Because the proposed modeldoes not use any empirical assumptions or diffusion approximation, it is also suitable for thestudy and understanding of the interplay of conduction and radiation on micro- and nano-spatial scales. As a result, our model can be used for the optimization of foam structure(such as porosity and cell size) and for the estimation of the heat transport properties ofmicro- and nano-cellular foams, in which the Knudsen effect is present.

2 Theory

2.1 Heat conduction

The heat conduction in polymer foams is carried out by gas molecules in pores and byphonons in solid phase. Both of these mechanisms are described by Fourier’s law in caseswhen the mean free path of heat carrier is much smaller than the characteristic length ofthe phase in material (Chen, 2005)

qcon = −k∇T, (1)

2

where qcon is the conductive heat flux, ∇T is the temperature gradient and k is the heatconductivity, which is equal to kgas in gas phase or kpol in polymer phase.

Recently, there has been an effort to implement Fourier’s heat conduction by Lattice-Boltzmann Method (LBM), which was originally developed for the calculation of the fluidflow (Chen and Doolen, 1998). LBM was used to solve also the coupled conduction-radiation heat transfer by Mishra and Roy (2007). Later the same problem was studiedon non-uniform meshes (Mishra et al., 2009). Combination of radiation with non-Fourierflow was considered in Mishra et al. (2008) and Nabovati et al. (2011). The speed of LBMfor heat transfer is comparable to the more traditional method like Finite Volume Methodand depends on the implementation and the solved problem. However, to the best ofour knowledge, LBM was not developed yet for the calculation of combined conduction-radiation heat transfer in porous media. Moreover, since the LBM is inherently transient,it is not optimal for steady-state simulations.

For the heat flow inside nanofoams, the mean free path of gas molecules Λmean can becomparable to or smaller than the characteristic dimensions δcell of the system (cell size)(Baetens et al., 2010). The Fourier law is in this case not valid, because the maximumheat flux carried out by gas molecules in a narrow gap is limited. The maximum heat fluxqmax, which can be transported across a narrow gap is proportional

qmax ∝ ZkB(TA − TB), (2)

where Z is the frequency of collisions of gas molecules per unit area of a solid wall, kB is theBoltzmann constant and (TA−TB) is the temperature difference of solid walls. Let us notethat in difference from eq. (1) the heat flux is independent of the gap width. Therefore,the thermal conductivity of the gas kgas in narrow cells is smaller than the bulk valuekbulk. The reduced value of kgas can be, in the simplest case of the two plates separated bythe distance L, calculated from the kinetic theory of gases, according to equation (Zhang,2007)

kgas =kbulk

1 +Kn2− αTαT

9γ − 5

γ + 1

, (3)

where γ = cp/cV is the heat capacity ratio, αT is the thermal accommodation coefficient,which determines the efficiency of energy transfer when gas molecules hit the solid phaseand Kn is the Knudsen number, which is defined as

Kn =Λmean

δcell. (4)

The mean free path of gas molecules Λmean can be estimated from the collision theory(Kuhn et al., 2009) as

Λmean =kBT√2πd2gp

, (5)

where dg is the diameter of gas molecules and p is the pressure.

3

Experimental data for the values of the thermal accommodation coefficient αT arescarce. Heat accommodation phenomenon is usually studied in high vacuum for cleanmetal surfaces. The closest parameters of such studies to our system were measured byAmdur and Guildner (1957) for oxygen on wolfram αT = 0.905, on nickel αT = 0.862and platinum αT = 0.853. Baetens et al. (2010) uses values of thermal accommodationcoefficients between 0.85 and 1 for the study of vacuum insulation panels.

At atmospheric pressure the gas conductivity calculated from eq. (3) is practically equalto bulk gas conductivity if the cells are larger than 10 µm but it is significantly reduced dueto the Knudsen effect in cells thinner than 1 µm. An alternative approach to determinethe thermal conductivity kgas in small cells is to use Monte Carlo method (Shan and Wang,2013).

In this paper, we don’t consider the reduction of polymer heat conductivity kpol invery thin polymeric walls, because this reduction would be speculative as it can’t be basedon experimental data obtained with thin polymer films. However, significantly reducedheat conductivities were experimentally measured for sub-micron layers of Cu (Nath andChopra, 1974) or SiO2 (Liu and Asheghi, 2004) and were also theoretically explained, e.g.,by Wang et al. (2011).

2.2 Heat radiation

The exact formulation of the thermal radiation heat transfer is derived using the Stefan-Boltzmann law, the Beer law and the balance of radiative energy without further approx-imations. However, exact formulation leads even in the simplest case of one-dimensionalheat transfer in homogeneous media to nonlinear integro-differential equation, which iscomputationally-demanding to solve (Modest, 2003). For this reason, many different ap-proximations of radiative heat flux were developed. The most commonly used ones areP1-approximation, discrete ordinates method, zonal method and diffusion approximation.Many papers were published on comparing different approximative techniques for radia-tion, e.g. Ravishankar et al. (2010), Modest and Yang (2008), Asllanaj et al. (2007) etc.In participating media each volume element is absorbing and emitting heat radiation.

If we consider spectral variation of the extinction coefficient we must choose from one ofthe so-called nongray methods (Modest, 2003). These include narrow band method, line-by-line integration, stepwise gray box model, wideband model, weighted-sum-of-gray-gases,narrowband k distribution or full spectrum k distribution. Because some of these methods(e.g., line-by-line integration) are computationally too expensive for ordinary engineeringpurposes, we chose the gray box model, which is simple and the most commonly usedmethod (Mazumder, 2005).

In this paper we don’t consider interference (superposition) effects of heat radiation asa wave phenomena. Furthermore we don’t consider photon tunneling or phonon-polaritontunneling effects. We believe that omission of these effects is not affecting results becausethe importance of heat transfer by radiation in micron and submicron-sized cells is small.Moreover, omission of these effects greatly simplifies the problem.

4

2.3 Coupled heat conduction and radiation

When dealing with porous media, the most popular approach is to treat the heat radiationindependent of conduction. First, mean material properties are calculated, i.e., effectivethermal conductivity and effective extinction coefficient. These parameters incorporatethe entire morphology information of original material, i.e., the volume fraction of pores,the distribution of pores and the material properties of all phases. They are used later asthe input data for combined conduction-radiation heat transfer simulation in homogeneousmedia of the same size as the original material. Analytical methods for the calculation ofeffective conductivity were reviewed in Wang and Pan (2008b). Numerically the effectiveconductivity can be calculated by Finite Volume Method (FVM) as was shown, e.g., inCoquard and Baillis (2009) or by using Lattice-Boltzmann Method (Wang et al., 2007).The effective extinction coefficient is usually obtained from experiment. The Ray-Tracingmethod applied to 3D meshes was used by Coquard et al. (2011) to determine effectiveextinction coefficient.

Assuming the porous material is highly absorbing, the diffusion approximation of ra-diative flux can be used. This procedure stands on the assumption that the total heat fluxqtot can be split into heat fluxes due to the radiation qrad, due to the conduction in poresqpor and due to the conduction in solid qsol as

qtot = qrad + qpor + qsol. (6)

Under the diffusion approximation, all of the aformentioned fluxes are proportional to thenegative temperature gradient (−∇T ). Thus, we can write

ktot = krad + kpor + ksol, (7)

where kpor and ksol are heat conductivities related to the conductivities of the bulk gas orthe solid phase through geometrical relationships that can be found in Placido et al. (2005)

kpor = εkpor,b, (8)

ksol =1

3(1− ε)(2− fstrut)ksol,b, (9)

where kpor,b is the bulk gas conductivity, ksol,b is the bulk solid conductivity, ε is theporosity and fstrut is the strut volume fraction. The radiative conductivity krad in thediffusion approximation can be expressed as

krad =16σT 3

m

3κR, (10)

where σ is the Stefan-Boltzmann constant, Tm is the mean temperature, and κR is theRosseland mean absorption coefficient. Rosseland mean absorption coefficient κR is a typeof gray absorption coefficient, but it is calculated differently from the Planck mean absorp-tion coefficient used in this paper (Modest, 2003).

5

Literature about calculating the coupled conductive-radiative heat flux without thediffusion approximation in large multi-phase domains is still fairly limited. To the bestof our knowledge it is still restricted to studying multilayer materials. A model withopaque internal interfaces (representing highly reflecting foil) was studied by Bai and Fan(2006) and Pasztory et al. (2011). A multilayer model with semitransparent interfaceswas published by Tan et al. (2003a) and was further developed in Tan et al. (2003b), themodel was however mainly developed for composite materials like layers of semiconductorsor glass and the extension of this model to spatially three-dimensional structures would bevery difficult.

3 Numerical implementation

In this paper we opted for a P1-approximation of radiative flux which is common andsuitable for engineering purposes (Modest, 2003).

The energy conservation equation for the one-dimensional steady-state heat flux byconduction and radiation can be written as

dqtotdx

=dqcondx

+dqraddx

= 0, (11)

where qcon and qrad is the conductive and the radiative heat flux, respectively. The diver-gence of the heat flux by conduction is derived from Fourier’s law (see eq. (1)) as

dqcondx

= −kd2T

dx2. (12)

For the stepwise gray box model, the divergence of the radiative heat flux can be writtenas (Modest, 2003)

dqraddx

=B+1∑k=1

κk(4Ebk −Gk), (13)

where B is the number of gray boxes, κk and Gk are the absorption coefficient and theincident radiation in wavelength interval defined in the k-th spectral box, respectively. Theincident radiation Gk is defined as

Gk =

λk,u∫λk,l

Gλ dλ, (14)

where λk,l and λk,u are the lower and upper boundaries of the k-th gray box (see Fig. 1),respectively. In this paper, the absorption coefficient κk is equal to the extinction coefficientsince we neglect scattering in homogeneous phases. The summation in eq. (13) is to B+ 1since we chose B intervals (the so called gray boxes) in the spectrum which don’t necessarilyhave to follow directly one another. The wavelengths which don’t belong to any of these

6

wavelength A

bso

rpti

on

coeffi

cien

t

κ1 κ2

Δλ1

Δλ2

λ1,l λ1,u

Figure 1: Schematic illustration of the stepwise gray box model.

boxes are then summed up in the (B + 1)-th variable. The blackbody emissive power Ebkis given by

Ebk =

λk,u∫λk,l

Ebλ dλ = [f(λk,uT )− f(λk,lT )]n2σT 4, (15)

where n is the index of refraction and f(λT ) is the so-called fraction of the blackbodyradiation that represents a portion of emitted energy at temperature T , which is containedin photons with wavelength smaller than λ. It can be calculated according to Siegel andHowell (2002) as

f(λT ) =15

π4

∞∑p=1

[exp(−pζ)

p

(ζ3 +

3ζ2

p+

6ζ

p2+

6

p3

)], (16)

where ζ = C2/λT . Here C2 = 0.014387752 m K is constant in Planck’s spectral energydistribution. Summation of the first 30 terms in eq. (16) was deemed satisfactory for precisecalculation of the fraction of blackbody radiation f . Substitution of eqs. (12) and (13) intothe eq. (11) yields

kd2T

dx2=

B+1∑k=1

κk(4Ebk −Gk), (17)

which is in our case subjected to isothermal boundary condition

T (0) = TA, (18a)

T (L) = TB, (18b)

where TA and TB are the temperatures of the bounding isothermal plates in the distanceL. The emissive power Ebk in eq. (17) is for the sake of numerical stability linearized as

Ebk = E∗bk + E′∗bk(T − T ∗), (19)

7

L

gas polymer

TA TB

x0 δ wall

δ cell

Figure 2: A model of one-dimensional foam.

where the asterisk denotes the value calculated in the previous iteration and E′

is given by

E′

bk =

λk,u∫λk,l

dEbλdT

dλ = [h(λk,uT )− h(λk,lT )] 4n2σT 4. (20)

Here h(λT ) can be calculated as

h(λT ) = f(λT ) +15

4π4

(C2/λT )4

exp(C2/λT )− 1. (21)

The incident radiation Gk with the neglection of scattering is given by P1-approximation(Modest, 2003) as

d2Gk

dx2= 3κ2k(Gk − 4Ebk), ∀k = 1, 2, . . . , B + 1, (22)

subjected to boundary condition at x = 0 and x = L

dGk

dx

∣∣∣∣0

=3

2

ε02− ε0

κk[Gk(0)− 4Ebk(0)], ∀k = 1, 2, . . . , B + 1, (23a)

− dGk

dx

∣∣∣∣L

=3

2

εL2− εL

κk[Gk(L)− 4Ebk(L)], ∀k = 1, 2, . . . , B + 1, (23b)

where ε0 and εL are emissivities of the bounding materials.To preserve the heat flux at phase interfaces (I) (see model of foam in Fig. 2), the

temperature and the incident radiation is subjected to interphase conditions

T+∣∣I

= T−∣∣I, (24)

k+dT

dx

∣∣∣∣+I

= k−dT

dx

∣∣∣∣−I

. (25)

8

In this paper, we consider non-absorbing, partially transmitting, partially reflecting phaseinterfaces. Boundary condition for such interface leads to (Siegel and Howell, 2002)

2

3κ+k

dGk

dx

∣∣∣∣+I

=1− ρ+

1 + ρ+Gk|+I −

1− ρ−

1 + ρ+Gk|−I +

2

3κ−k

1− ρ−

1 + ρ+dGk

dx

∣∣∣∣−I

, ∀k = 1, 2, . . . , B+1,

(26)where ρ− and ρ+ are the total hemispherical reflectivities, which can be calculated fromthe Fresnel’s equations. The reflectivity for the direction from the material of lower indexof refraction n1 to the optically denser material of higher index of refraction n2 can becalculated as (Siegel and Howell, 2002)

ρ1 =1

2+

(3n+ 1)(n− 1)

6(n+ 1)2+n2(n2 − 1)2

(n2 + 1)3log

n− 1

n+ 1

−2n3(n2 + 2n− 1)

(n2 + 1)(n4 − 1)+

8n4(n4 + 1)

(n2 + 1)(n4 − 1)2log n, (27)

where n = n2/n1. Reflectivity in the opposite direction (from n2 to n1) is calculated as

ρ2 = 1− 1− ρ1n2

. (28)

Equations (17) and (22) can be also formally written in the matrix form as suggestedby Mazumder (2005) for shorter calculation time. They can be solved using standardnumerical methods like finite volume method or finite difference method.

Once the temperature T and the incident radiation G fields are calculated, the totalheat flux can be calculated at any location x ∈ (0, L) as (Modest, 2003)

qtot = −kdT

dx−

B+1∑k=1

1

3κk

dGk

dx. (29)

The second term in eq. (29) represents the radiative heat flux integrated over the entirewavelength spectrum.

The equivalent thermal conductivity, the most important property of insulating mate-rial, is evaluated from its definition as

keq =qtotL

TA − TB. (30)

It corresponds to the conductivity of the material with the same width that would transferthe same amount of heat only by conduction.

4 Results and discussion

Numerical experiments were performed on the model example of polystyrene foam con-taining carbon dioxide as the blowing agent. We calculated with thermal conductivities of

9

Abs

orpt

ion

coef

ficie

nt κ

PS (m

-1)

0

105

2×105

3×105

4×105

Wavelength λ (μm)0 5 10 15 20 25

Figure 3: Absorption spectrum of pure polystyrene measured on 38.1 µm thick film.

polystyrene kPS = 0.15 Wm−1K−1 (Sakakibaba et al., 1990; Pasquino and Pilsworth, 1964)and of carbon dioxide kCO2 = 0.015 Wm−1K−1. Absorption spectrum of CO2 (see Fig. 4)was taken from webbook.nist.gov. It was measured on a grating–equipped spectrometerin 10 cm KBr cell. The absorption coefficient was adjusted to partial pressure of carbondioxide pCO2 = 1 bar. Absorption spectrum of polystyrene (see Fig. 3) was measured onNicolet FTIR spectrometer. The sample was 38.1 µm thick film of polystyrene, which iscommonly used as calibrating standard for FTIR spectrometers. Gray absorption coeffi-cient of carbon dioxide κCO2 = 26 m−1 was taken from Modest (2004). Gray absorptioncoefficient of polystyrene κPS = 7.5×103 m−1 was calculated from the absorption spectrumin Fig. 3, obtained by our measurement. The index of refraction of PS and CO2 was takenas nPS = 1.6 and nCO2 = 1.0, respectively. The reflectivity for the direction from CO2 toPS was calculated from eq. (27) as ρ1 = 0.106 and the reflectivity in the opposite directionwas calculated from eq. (28) as ρ2 = 0.651. In this paper we assumed the emissivity of theboundaries at x = 0 and x = L to be ε = 1. The representative values for the thermalaccommodation coefficient and the heat capacity ratio were chosen to be αT = 0.9 andγ = 1.32, respectively.

4.1 Temperature profile

Temperature profile in 1D semitransparent media that is transferring heat by conductionand radiation is generally a nonlinear function. Although the steady-state temperatureprofile in polymer foams is usually piecewise linear, it becomes nonlinear within individualcells in high porosity foams, in which radiative heat flux is important and whose cell sizeis comparable to the reciprocal value of the gas absorption coefficient. The example of

10

Abs

orpt

ion

coef

ficie

nt κ

CO

2 (m

-1)

0

25

50

75

100

125

150175

Wavelength λ (μm)0 5 10 15 20

Figure 4: Absorption spectrum of carbon dioxide at 1 bar. Taken from NIST database(webbook.nist.gov).

such nonlinear temperature profile is displayed in Fig. 5, which is calculated for the foamwith large cell size of 1.2 cm. The globally nonlinear steady-state temperature profile,as reported for example by Modest (2003) for homogeneous media whose conduction-to-radiation parameter is smaller than one, was not observed in results of our simulations.

The diffusion approximation approach would provide always piecewise linear temper-ature profile regardles of the value of absorption coefficient. However, conditions for ne-glecting of free heat convection are not met in Fig. 5.

The prediction of the fraction of the total heat flux transferred by radiation (qrad/qtot)for a foam with porosity 95 % is plotted in Fig. 6. If polymeric walls are thicker than 100µm, the thermal radiation is significantly decreased in the polymeric phase (see Fig. 7).Alternatively, the radiative heat flux can be reduced by addition of highly absorbing sub-stance like carbon soot into the polymer, resulting in higher absorption coefficient of thesolid phase. Results of our simulations show that the dependence of heat flux transferredby radiation on the spatial coordinate is highly non-uniform in foams with fewer cells andoptically thicker walls. The ratio of radiative to total heat flux in Figs. 6 and 7 has a risingtrend from left to right because of the increasing temperature in that direction. In opticallyvery thin materials, the ratio of radiative to total heat flux would be constant along theposition x. In optically very thick materials, ratio qrad/qtot would be approximately pro-portional to x3 (see eq. (10)). Polymer foams are closer to optically thick materials. Thus,although the variation of the qrad/qtot ratio with position seems linear in Figs. 6 and 7 it isonly because the difference between boundary temperatures was only 10 K (TA = 300 K,TB = 310 K).

The decrease of the radiative heat flux at the boundaries in Figs. 6 and 7 is caused

11

Tem

pera

ture

T (K

)

300

302

304

306

308

310

Position x (m)0 0.025 0.05 0.075 0.1 0.125

Figure 5: Steady-state nonlinear temperature profile in PS-CO2 foam with large cells.Foam has porosity 99 % and contains 10 cells.

by the fact that the model foam has polymer phase at both ends and by the black wallboundary condition, i.e., all photons are absorbed at the boundary. These two picturesjust illustrate the fraction of heat flux transferred by radiation in foam, but a systematicstudy is presented below.

Although the spatial profile of (qrad/qtot) inside the foam is often very nonhomogeneous,further in the text we will refer mainly to its mean value. However note that the local ratioof radiative to total heat flux in the center of thick polymer walls can be very differentfrom the mean value. In Fig. 7 approximatively 15 % of energy is transferred by radiation.The limiting value of the fraction of heat flux transported by radiation in 1 cm wide gapfilled with gas but without any polymer is 79 %.

4.2 Equivalent heat conductivity

Figure 8a shows the dependence of the equivalent conductivity keq on porosity ε for twocases: (i) when the cell size is varied and (ii) when the wall size is varied. Figure 8bshows the dependence of the ratio of spatially averaged radiative heat flux to total fluxon porosity for the same two cases. As Fig. 8a indicates, an optimal porosity of polymerfoams often exists, at which the foam exhibits best insulating properties. This fact can beeasily understood if we point out that polymers have higher conductivity than gases, so theincreasing porosity leads to the reduction of the equivalent conductivity (see Fig. 8a). Onthe other hand, if the porosity is too large, the radiative heat flux will increase rapidly andthus the equivalent conductivity again increases. If the porosity is changed by enlargingthe size of the cells at constant wall size of 7.5 µm (solid line in Figs. 8a and 8b), the high

12

Ratio

qra

d / q

tot

0.076

0.078

0.080

0.082

0.084

0.086

0.0880.090

Position x (m)0 0.02 0.04 0.06 0.08 0.1

Figure 6: The ratio of radiative to total heat flux (qrad/qtot) for PS-CO2 foam of 10 cmwidth. Foam has porosity 95 % and contains 500 cells (cell size is 190 µm).

Ratio

qra

d / q

tot

0.00

0.05

0.10

0.15

0.20

Position x (m)0 2×10−3 4×10−3 6×10−3 8×10−3 0.01

Figure 7: The ratio of radiative to total heat flux (qrad/qtot) for PS-CO2 foam of 1 cmwidth with nine thick inner polymer walls of 270 µm thickness.

13

porosity foams have very large size of cells, in which the radiation is increasingly important.This increase in radiative heat transfer causes the increase in the equivalent conductivityof the foam, therefore for foams with wall thickness of 7.5 µm exists a minimal equivalentconductivity at porosity 0.945. At this porosity the foam gains the best heat insulatingproperties. However, if the porosity is changed by narrowing the polymer walls at constantsize of the cells of 200 µm (dashed line in Figs. 8a and 8b), the increase in the radiativeheat flux is not sufficient to offset the decrease of the conductive heat flux, and thus theequivalent conductivity of the foam monotonously decreases over the whole studied rangeof porosity. The minimal equivalent conductivity (and thus best insulating properties)has the foam with porosity ε ≈ 1. From these observations, we can conclude that theradiative heat flux is significantly reduced in foams with smaller cells. This is because oftwo reasons: (i) the foam with smaller cells has larger number of phase interfaces on whichthe photons can be reflected and (ii) the radiative heat flux is generally higher in largerdomains containing low-absorbing homogeneous phase.

The conductive heat flux is proportional to the negative gradient of the temperature(−∇T ). The radiative heat flux is proportional to the negative temperature gradient onlyin the optically very thick materials. However, in the optically very thin materials, theradiative heat flux is proportional to the difference (T 4

A − T 4B) and is independent of the

foam thickness. Thus, if the polymer foams are optically thin and the radiative heat flux isnonnegligible, the equivalent foam conductivity should be dependent on the foam thickness.This is simply caused by the fact that the conductive heat flux is proportional to reciprocalvalue of foam thickness (qcon ∝ 1/L) and radiative heat flux is independent of the foamthickness (qrad 6= f(L)), and thus the resulting equivalent conductivity keq increases withsample thickness L (see eq. (30)).

The dependence of equivalent heat conductivity on foam thickness was experimentallymeasured (Tleoubaev, 1998). Hollingsworth (1980) found that the variation of equivalentheat conductivity with foam thickness can constitute between 1.5 and 3 % of equivalentconductivity in building insulations. Pelanne (1980) observed that this effect appears onlyin high porosity materials with density smaller than ∼ 30 kg/m3. The same trends can beseen in Fig. 9, which shows the dependence of the calculated equivalent conductivity on thefoam thickness and porosity. It suggests that the polymer content in thin (less than 1 cm)foam samples with porosity higher than 97 % is insufficient to absorb all photons, and sopart of the photons travel uninterrupted from one boundary to the other. Let us pointout, that this phenomenon wouldn’t be observed if we used the diffusion approximation inour model.

The use of nongray absorption coefficient (i.e., of the IR absorption spectra of phases)is by many authors stressed as very important for the simulation of radiation heat transfer(Modest, 2003). We use the gray-box model, in which the absorption coefficient is assumedconstant over the chosen wavelength interval (i.e., a gray box). The widths of the boxes werechosen manually to approximate individual bands or their parts or intervals in Figs. 3 and 4,where both polystyrene and carbon dioxide absorption coefficients are roughly constant.The absorption coefficients of individual boxes were calculated as the mean value of theabsorption coefficient over the interval. Figure 10 shows the dependence of the calculated

14

Equi

vale

nt co

nduc

tivity

k eq (

Wm

-1K

-1)

0.016

0.018

0.020

0.022

0.024

0.026

Porosity ε0.85 0.875 0.9 0.925 0.95 0.975 1

fixed wall sizefixed cell size

(a)

Ratio

qra

d / q

tot

0

0.1

0.2

0.3

0.4

Porosity ε0.85 0.875 0.9 0.925 0.95 0.975 1

fixed wall sizefixed cell size

(b)

Figure 8: Dependence of (a) the equivalent thermal conductivity and (b) the average ratioof radiative to total heat flux on porosity for PS-CO2 foams. Spectral properties wereassumed gray. Fixed wall thickness is 7.5 µm; fixed cell width is 200 µm.

15

Equi

vale

nt co

nduc

tivity

k eq (

Wm

-1K

-1)

0.0165

0.0170

0.0175

0.0180

0.0185

0.0190

Foam thickness L (m)10−3 0.01 0.1

porosity 98.5 %porosity 97 %porosity 95 %

Figure 9: Dependence of equivalent conductivity on foam thickness for PS-CO2 foams.The thickness of polymer walls was 7.5 µm in all foams. The corresponding cell sizes are142.5 µm (porosity 95 %), 242.5 µm (porosity 97 %) and 492.5 µm (porosity 98.5 %).

Table 1: Geometrical parameters of foams in Fig. 10

number porosity width (m) walls std. dev. of keq ×103

1 0.95 0.1 100 0.552 0.95 0.01 100 0.573 0.9 0.01 100 0.314 0.9 0.1 100 0.23

equivalent conductivity on the number of gray-boxes used for four different foams describedin Table 1. The standard deviation (see Table 1) of equivalent conductivity in Fig. 10increases for foams with high porosity, where the influence of radiation is more important.But even in those foams, it was found that the resulting equivalent conductivity dependsmore on the exact specification of the individual boxes than on foam properties.

The results obtained with only one gray absorption coefficient (zero boxes in Fig. 10)differ from ”correct” ones calculated with 14 gray boxes by only few percents and can bereached approximatively 20 times faster. The use of gray absorption coefficient for polymerfoams is thus satisfactory.

16

Equi

vale

nt co

nduc

tivity

k eq (

Wm

-1K

-1)

0.025

0.030

0.035

0.040

0.045

0.050

Number of gray-boxes0 2 4 6 8 10 12 14

Foam 1Foam 2Foam 3Foam 4

Figure 10: Dependence of equivalent conductivity keq on number of gray boxes. Foam pa-rameters are written in Table 1. In this numerical experiment reflection on phase interfaceswas not considered.

4.3 Micro- and nano-cellular foams

Vacuum insulation panels have very low equivalent conductivity (Baetens et al., 2010).Although the bulk heat conductivity of gases is almost independent of pressure in largesystems, the heat conductivity in smaller systems (vacuum panels) can be considerablyreduced. This is the result of the increase of gas mean free path (see eq. (5)), whichbecomes comparable to the mean cell size (see eq. (4)). The dependence of the equivalentconductivity on the pressure inside the polymer foam with cell size of 200 µm is plottedin Fig. 11. This figure is just illustrative because open-cellular polymeric foams are moresuitable for vacuum insulation panels.

Heat conduction can be also decreased by the reduction of the cell size to sub-micronrange (see eqs. (3) and (4)). Although only few experimental studies on nanofoams areavailable (Miller and Kumar, 2011), this decrease is already observable in other nano-structured materials, such as aerogels (Wei et al., 2011). However, compared to polymerfoams, the main disadvantage of aerogels is their brittleness and hygroscopicity.

Figure 12a shows the dependence of the equivalent conductivity on both the porosityand the cell size. Regardless of the porosity, the equivalent conductivity decreases withdecreasing cell size. In foams with cells larger than 10 µm, it is mainly because of thereduction of the radiative heat flux. In foams with cells smaller than 10 µm, it is mainlybecause of the reduction of the gas conductivity due to the Knudsen effect. The minimum inthe dependence of the equivalent conductivity for the foams with large cells is at relativelylarge porosity (optimal porosity for foams with 1 cm cells is ε = 0.968), but as the radiation

17

Equi

vale

nt co

nduc

tivity

k eq (

Wm

-1K

-1)

0.006

0.008

0.010

0.012

0.014

0.016

0.0180.020

Porosity ε0.85 0.875 0.9 0.925 0.95 0.975 1

p = 105 Pap = 104 Pap = 103 Pap = 102 Pa

Figure 11: Dependence of the equivalent conductivity on porosity for PS-CO2 foams havinga different pressure in the cells. The size of cells was 200 µm in all foams.

passing through cell walls becomes more important (see Fig. 12b), it is shifted towards lowerporosities (optimal porosity for foams with 2 mm cells is ε = 0.939). However, when theconduction becomes the dominant heat transfer mode, the optimal porosity is almost equalto one (optimal porosity for foams with 1 mm cells is ε = 0.99998) similarly to the casediscussed in Section 4.2 (cf. dashed line in Fig. 8a). Let us point out that in real polymerfoams the minimum of equivalent conductivity on porosity was found even in foams withcell size lower than 1 mm. Typical cell size of closed-cell polystyrene foam is 100–200 µmand typical wall thickness is 1–10 µm. We want to stress the fact that quantitatively goodagreement with experimental data can be obtained only by the three-dimensional model.

Figure 13a shows the dependence of equivalent conductivity on porosity and wall thick-ness. In comparison with Fig. 12a the minimal equivalent conductivity is in this caseusually found for smaller porosity. This is caused by the fact that high porosity foam inFig. 13a is obtained by enlarging the cell size, which dramatically increases the amount ofenergy transferred by radiation (see Fig. 13b). The ratio qrad/qtot in Fig. 13b is for high-porosity foams larger in foams with thicker polymer walls. However, it does not imply thatthe local ratio of qrad/qtot in the center of polymer walls is also higher in foams with thickerwalls.

The proposed model and its results from Figs. 12a and 13a for PS-CO2 foams can beconveniently used for optimization purposes. If the manufacturer is, for example, capableof producing foams with 100 µm cells but variable porosities, our model can be used asa tool for determining the optimal foam morphology, so that the final foam has desiredproperties.

18

10−8

10−6

10−4

10−2

0.90.92

0.940.96

0.9810

0.05

0.1

Cell size (m)Porosity

Equ

ival

ent c

ondu

ctiv

ity (

Wm

−1 K

−1 )

(a)

10−8

10−6

10−4

10−2

0.90.92

0.940.96

0.9810

0.2

0.4

0.6

0.8

Cell size (m)Porosity

Rat

io q

rad/q

tot

(b)

Figure 12: Dependence of (a) the equivalent conductivity and (b) the average ratio ofradiative to total heat flux on porosity and cell size for PS-CO2 foams. The red line marksthe minimal equivalent conductivity for a given cell size.

19

10−8

10−6

10−4

0.60.7

0.80.9

10

0.02

0.04

0.06

Wall size (m)Porosity

Equ

ival

ent c

ondu

ctiv

ity (

Wm

−1 K

−1 )

(a)

10−8

10−6

10−4

0.6

0.8

10

0.5

1

Wall size (m)Porosity

Rat

io q

rad/q

tot

(b)

Figure 13: Dependence of (a) the equivalent conductivity and (b) the average ratio ofradiative to total heat flux on porosity and wall size for PS-CO2 foams. The red line marksthe minimal equivalent conductivity for a given wall size.

20

4.4 The need of 3D model

In this contribution, we define the porosity as the volume fraction of the gas phase inthe foam. Please note that if we had made a three-dimensional model of the foam withthe same cell and wall sizes as in the one-dimensional foam, the porosity of the three-dimensional model ε3D would be proportional to the porosity of the one-dimensional modelε approximately through

ε3D ≈ ε3. (31)

Although we developed the current model with the extension to three-dimensions inmind, several obstacles need to be addressed. For example, appropriate numerical schememust be chosen, because highly porous foams cannot be properly discretized on simpleequidistant meshes. The development of the 3D model is needed for validation, because theone-dimensional model cannot provide results quantitatively comparable with real polymerfoams. Furthermore the one-dimensional model cannot be used to study several phenomenalike the foam anisotropy or the influence of the polymer struts on the insulative properties.

Conclusions

We developed a spatially one-dimensional model of coupled heat transfer by conductionand radiation in heterogeneous media. Compared with existing models, our method usesonly the structural information about the medium and the physical properties of purephases. Furthermore, it doesn’t make any empirical assumptions about the geometry orthe propagation of radiation. Compared to classical approach represented by diffusionapproximation, our method also provides additional information about the heat transfer,such as the variation of the radiation heat flux inside the foam. Also, it makes possible tostudy the dependence of the equivalent conductivity on the foam thickness.

The results of our model are in a good agreement with previously pulished experimentaldata and show that thermal radiation is an important factor in the heat transfer in polymerfoams. The importance of the radiation heat flux increases as the porosity and the mean cellsize increases. For the modeling of the heat transfer in polymer foams, we have shown thatthe absorption coefficient of individual phases can be assumed as a constant independentof wavelength with only a minor effect on the resulting equivalent conductivity. Thecalculated temperature profile inside polymer foams is globally almost linear, the deviationfrom the linear temperature profile was observed in highly porous foams with large cell size.Finally, our numerical predictions showed that the lowest equivalent heat conductivity canbe achieved by the reduction of the pore size to nano-range. With these results we are ableto determine the optimal foam structure with respect to heat insulation properties.

This work has described the first step towards a full 3D model of coupled conduction–radiation heat transfer in polymer foams. The current 1D model is formulated in a waythat ensures easy implementation in 3D, which is needed to obtain results quantitativelycomparable to real experiments and to optimize the real 3D structure of polymer foams.

21

Acknowledgments

The support of the Czech Grant Agency (project 106/12/P673) and financial supportfrom specific university research (MSMT No. 21/2012) are gratefully acknowledged. JurajKosek thanks the CENTEM project (CZ.1.05/2.1.00/03.088).

22

Nomenclature

B number of gray boxes, [-]cp heat capacity at constant pressure, [Jkg−1K−1]cV heat capacity at constant volume, [Jkg−1K−1]C2 constant in Planck’s spectral energy distribution = 0.01439 m Kdg diameter of gas molecules, [m]Eb blackbody emissive power, [Wm−2]f fraction of blackbody radiation, [-]G incident radiation, [Wm−2]k thermal conductivity, [Wm−1K−1]keq equivalent thermal conductivity, [Wm−1K−1]kB Boltzmann constant = 1.3807× 10−23 JK−1

Kn Knudsen number, [-]Λmean mean free path, [m]L foam thickness, [m]n real part of refractive index, [-]p pressure, [Pa]qcon conductive heat flux, [Wm−2]qrad radiative heat flux, [Wm−2]qtot total heat flux, [Wm−2]T temperature, [K]x space coordinate, [m]Z frequency of collisions per unit area, [s−1m−2]αT thermal accommodation coefficient, [-]γ heat capacity ratio, [-]δcell cell size, [m]δwall wall size, [m]ε emissivity, [-]ε porosity, [-]κ absorption coefficient, [m−1]κR Rosseland absorption coefficient, [m−1]λ wavelength, [m]ρ reflectivity, [-]σ Stefan-Boltzmann constant = 5.670× 10−8 Wm−2K−4

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Manuscript with title Heat transfer in one-dimensional micro- and nano-cellular foams by authors Ferkl P., Pokorný P., Bobák M., and Kosek J.  

Highligths: 

• In polymer foams more than 30 % of heat can be transferred by radiation.  

• One‐dimensional plane‐parallel model of polymer foam is developed. 

• Foams are treated as participating media with reflection on phase interfaces. 

• Knudsen effect in foams with sub‐micron cells is studied. 

• Foam geometry is optimized with regard to insulating properties.