Thermal Conductivity of Plastic Foams
`
`R. J. J. WILLIAMS* and C . M. ALDAO**
`lnstitute of Materials Science d7 Technology (INTEMA)
`University of Mar del Plata
`(7600) Mar del Plata, Argentina
`
`A simple equation enabling the prediction of the thermal
`conductivity of plastic foams, without the aid of adjustable pa-
`rameters, is proposed. The equation is based on a recurrent
`method, previously developed, that showed reasonable agree-
`ment with experimental results. Ways of decreasing the
`thermal radiation contribution are shown. In particular, the
`influence of cell size, radiation transmission through solid
`membranes, and low-emissivity boundary surfaces are ana-
`lyzed. Errors involved in steady techniques of measuring the
`thermal conductivity associated with radiation are discussed.
`
`INTRODUCTION
`ellular materials, and particularly plastic foams, are
`
`C widely used in insulation in the low-ambient tem-
`
`perature range. The analysis of the modes of heat trans-
`fer through the foam should enable one to state condi-
`tions for minimizing the apparent thermal conductivity.
`This would permit one to reduce insulation costs and/or
`energy requirements.
`The aim of this article is to expand our previous results
`( I ) , with the following objectives:
`Derive a simple analytical expression without ad-
`justable parameters to predict the apparent ther-
`mal conductivity of a plastic foam (In our previous
`analysis (I), the radiation contribution could not
`be expressed in a simple form).
`Analyze the contribution of the thermal radiation
`to the total heat transport, stating conditions en-
`abling it to be a minimum.
`Discuss errors involved in steady experimental
`techniques of measuring thermal conductivity in
`relation ‘lo the contribution of thermal radiation.
`THE THERMAL CONDUCTIVITY OF A PLASTIC
`FOAM
`As previously shown (l), the heat flux through a plas-
`tic foam may be expressed as:
`9 = &~&f + (&?id + qrad)(l - f)
`(1)
`where f is the fraction of transversal area occupied by
`solid (e.g., 2/3 of the solid mass), and (1 - f) is the trans-
`verse area corresponding to alternating layers of gas
`cells and solid layers (e.g., 1/3 of the solid mass). As
`shown in Fig. 1, the specimen thickness L is assumed to
`be composed of n gas cells of thickness L, and n solid
`membranes of thickness L,:
`
`e
`
`q
`
`!2
`I
`I
`I
`I
`I
`I
` I
`I
`I
`I
`I
`I
`L
`
`I
`Fig. 1 . Geometric model of the plastic f w m ;
`
`~
`
`L = n(L, + LR)
`Also, since the area A, occupies 213 of the polymer,
`(3)
`LA, = 2nL,A,
`The foam density is given by:
`~ . ~ L A , P , + nLd,p,
`L(A, + A,)
`” =
`From Eqs 2 to 4 , the following equations arise:
`As
`- - (Pf - P,)
`(Pf - P,)
`f = A, + A,
`1 . 5 ( ~ , - P,)
`1.5P.s
`
`(4)
`
`(5)
`
`* Research Member of the Consejo Nacional d e Investigaciones Cientificas y
`Tknicas (National Research Council), Argentina; to whom all correspondence
`should h e addressed.
`** Member of the Comisi6n de Investigaciones Cientificas de la Provincia de Buenos
`Aires, Argentina.
`
`ps and ps are, the densities of the plastic material and
`the gas filling the cells, respectively. Thus, the area
`fractionf, the number of interposed solid layers per unit
`
`POLYMER ENGINEERING AND SCIENCE, APRIL, 1983, Vol. 23, No. 6
`
`293
`
`BOREALIS EXHIBIT 1069
`
`Page 1 of 6
`
`

`
`R. J . J . Williams and C. M . Aldao
`
`specimen length, n/L, and the thickness ofa solid mem-
`brane, L,, may be easily calculated as a function of
`known densities and the average thickness of a gas cell,
`L.V.
`The heat flux conducted through the solid is given by:
`(8)
`q%i = k.JT/L
`while the energy conduction through the gas may be
`approached by:
`
`@% = k,AT/L
`(9)
`where k, and k, are the thermal conductivities of the
`polymer and the gas, respectively.
`The net fraction of radiant energy sent forward by a
`solid membrane of thickness L,s is calculated in Appen-
`dix 1. It is given by
`
`The fraction (1 - TN) is sent back. The coefficient r is the
`fraction of incident energy reflected by each gas-solid
`interface. It is related to the refractive index of the plas-
`tic, w, by:
`
`The coefficient t is the fraction of radiant energy
`transmitted through the solid membrane, as given by
`Bouguer's law:
`
`(12)
`t = exp (-&)
`The absorption coefficient, a, is taken as the average
`value over the range of wavelengths of the radiant
`energy.
`Assuming that the foam is placed between two black
`surfaces at temperatures TI and T,, it is necessary to ana-
`lyze the net radiant energy transmitted through the set
`of n solid layers. This calculation, which is shown in Ap-
`pendix 2, leads to
`
`- T = (TI + T2)/2 is the average temperature. Thus, the
`apparent thermal conductivity of a plastic foam, defined
`as k = qL/AT, is given by:
`
`For plastic foams used in thermal insulation, pf <<
`ps, implying f +=
`0 and n + L/L,. Then, E q 14 reduces
`to:
`
`k = k, +
`
`= k, + k, (15)
`
`4aT3L
`1 + (L/L,)(1/TN - 1)
`Equation 14 (or 15) enables one to calculate the
`thermal conductivity of a plastic foam without the use of
`adjustable parameters. This equation is the analytical
`expression of our previous recurrent method (l), which
`was shown to give a reasonable agreement with experi-
`mental results.
`
`THE CONTRIBUTION OF THERMAL
`RADIATION
`The thermal radiation contribution to the overall heat
`transfer will be illustrated for a typical polystyrene foam
`- (PS), with air filling the cells, at an average temperature
`T = 283 K. For this case, the results (1) are:
`k,, = 0.02132 Kcal rn-"C-'h-',
`pS = 1052.5 Kg/m3, w = 1.6, a = 7.53 10-3pm-1
`Commercial materials have average cell sizes in the
`range of Lg = 50 to 300 pm. From E q 7, the thickness of
`a solid membrane, L,, varies in the range 0.2 to 7.3 p m
`for foam densities in the range pf = 10 to 50 Kg/m3. In
`fact, there is an inverse correlation between pfand L,(2),
`narrowing the actual L, range. Equation 12 gives trans-
`mission coefficients t varying from 0.95 to 1. Using
`these values and Eqs 10 and 11, the coefficient TAT lies in
`the range 0.88 to 0.90. This narrow range suggests an av-
`erage value of TN = 0.89. Thus, solid membranes of typi-
`cal polystyrene foams allow 89 percent of the incident
`radiation to pass through them, while 11 percent is sent
`back.
`Figure 2 shows the ratio k,/ k as a function ofthe speci-
`men thickness L, and diffeient cell sizes L,. The radia-
`tion contribution increases with the specimen thick-
`ness, attaining an asymptotic value. For L > 0.5 cm,
`radiation accounts for 7 to 34 percent of the heat trans-
`port through a polystyrene foam at = 283 K. This frac-
`tion varies with temperature and with the nature of the
`gas filling the cells.
`
`0.4
`
`kr
`-
`k
`
`0.3
`
`0.2
`
`0.1
`
`300
`
`200
`
`50
`
`C
`3
`1
`2
`L icm]
`Fig. 2 . Relative contribution vfradiativn to the ouerall thermal
`conductivity f o r . PS fvam, as a function of the specimen thick-
`ness L, and dqferent cell sizes L,(T = 283 K ) .
`
`I
`
`294
`
`POLYMER ENGINEERING AND SCIENCE, APRIL, 1983, Vol. 23, No. 6
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`Page 2 of 6
`
`

`
`Thermal Conductiuity of Plastic Foams
`
`It is interesting to analyze the possibilities of decreas-
`ing thermal radiation. An obvious way is to reduce the
`average cell size, requiring modifications in the foam-
`ing process. However, in the manufacture of low-
`density foams, an inverse correlation between foam
`density and cell size is often observed (2). Then, special
`techniques must be developed to produce microcellular
`low-density foams.
`Another factor that may be analyzed in the transpar-
`ency of a solid membrane to radiation, as given by the
`coefficient TN. Let us take its limiting values. The re-
`fractive indices of plastic materials lie in the range 1.338
`to 1.71 (3), giving reflection coefficients r = 2.09 *
`to 6.86 -
`Limiting t values are 0 for a completely
`opaque material and 1 for a completely transparent ma-
`terial. Then: TN = 0.466 for a plastic foam made from an
`opaque polymer with the highest refractive index, and
`TN = 0.959 for a plastic foam made from a transparent
`polymer with the lowest refractive index.
`In order to quantlfy the influence of TN, the ratio of
`asymptotic thermal radiation contributions is taken:
`- (l/TNmax - 1) = 4.275 *
`-
`(UTN - 1)
`(l/TN - 1 )
`kr(TNmaJ
`Figure 3 is a graphical representation of Eq 16. At high
`TN values, the parametric sensitivity of thermal radia-
`tion to TN is very significant. For example, a 50 percent
`reduction in the thermal radiation contribution of a
`polystyrene foam my be achieved if TN is lowered from
`0.89 to 0.80. Additives increasing the average absorp-
`tion coefficient a (i.e., showing strong I.R. absorption
`
`(16)
`
`1
`
`I
`
`I
`
`bands in the same wavelength range of the radiation)
`may be useful for this purpose. Also, black pigments
`may be tried. For black bodies, TN = 0.5 (r = t = 0), in
`which case, Fig. 3 shows that radiation may be neglected
`for practical purposes.
`There is still another way to decrease the thermal ra-
`diation mode of heat transfer in commercial materials.
`For example, PS foams used in insulation are frequently
`covered with surfaces acting as vapor barriers. If these
`surfaces have low emissivity values, el and e2, thermal
`radiation will be attenuated. The analysis of the overall
`radiation from the boundary at TI to the boundary at T2
`is shown in Appendix 3. The result is
`u m - T.3
`1 + n(l/TN - 1) + ( l / e l + l / e 2 - 2)
`
`Qrnd =
`
`(17)
`
`Then:
`
`kr =
`
`4uT3L
`1 + ( L / L g ) ( l / T ~ - 1) + ( l / e l + l / e 2 - 2)
`(18)
`Figure 4 shows the decrease in k, for a PS foam of aver-
`age cell size Lg = 200 pm, at ?? = 283 K, and el = e2. A
`relevant effect is shown when e 5 0.2 However, in-
`creasing L or decreasing L, lowers the relative incidence
`of using low-emissivity boundaries.
`
`EXPERIMENTAL ERRORS ASSOCIATED WITH
`THERMAL RADIATION
`For plastic foams showing a significant thermal radia-
`tion contribution, an experimental error is introduced in
`usual steady techniques, when hot and cold plaques are
`not black bodies. In this regard, standard methods usu-
`ally prescribe blackening the plaques, although no
`quantification of this effect has been given. Equation 18
`
`kr d
`
`Lg = 2 0 0 fim
`
`Fig. 3. Influence of the transparency of a solid membrane to ra-
`diant energy, Tfi, upon the thermal radiation contribution (PS :
`polystyrene foam).
`
`1
`2
`L (cm) 6
`0
`4
`Fig. 4 . Radiation contribution to the overall thermal conductiv-
`ity for a P S foam, as a function of the specimen thickness L, and
`different emissivities of the boundary surfaces (T = 283 K).
`
`POLYMER ENGINEERING AND SCIENCE, APRIL, 1983, Vof. 23, No. 6
`
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`Page 3 of 6
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`

`
`R. 1.1. Williams and C . M . Aldao
`original value (i.e., T x > 0.8); this might be
`achieved by including additives or pigments in the
`formulation;
`c) using boundary surfaces with low emissivity
`values (i.e., e s0.2)) for slabs with medium or
`great cell sizes.
`Blackening the plaques of experimental devices
`used to measure the thermal conductivity of plas-
`tic foams. At steady conditions, this is a reasonable
`precaution. However, in most cases, it will not in-
`troduce any modification in the experimental re-
`sults (assuming el, e2 > 0.5).
`
`gives an estimation of the errors introduced in a particu-
`lar case. The relative error may be defined as:
`
`Figure 5 shows E% for a low-density PS foam, with L =
`3 cm, at T = 283 K, and e , = e2 = e. The relative error
`depends on the average cell size. For L, 5 100 pm, an
`emissivity greater than 0.1 will suffice to give results ly-
`ing within the precision range of the experimental tech-
`nique. In general, for PS foams at T = 283 K, an e > 0.5
`will decrease the experimental error to less than 5 per-
`cent.
`It may be concluded that, although blackening the
`plaques is a reasonable precaution, in most cases, it will
`not introduce any modification in the experimental
`results.
`
`CONCLUSIONS
`A simple equation (Eq 14)enabling the prediction
`of the thermal conductivity of plastic foams
`without using adjustable parameters was pro-
`posed. This equation is the analytical expression of
`a recurrent method developed in a previous article
`(1). A reasonable comparison with experimental
`results was already shown (1).
`A significant thermal radiation contribution may
`be expected for several types of foams. In the case
`of PS foams at T = 283 K, radiation accounts for 7
`to 34 percent of the overall heat transport. This
`fraction may be decreased by
`a) diminishing the cell size through a modification
`in the foaming process;
`b) lowering the coefficient T~v when it has a high
`
`-"I
`
`E
`7-
`
`APPENDIX 1
`Net Fraction of Radiant Energy Sent Forward by a
`Solid Membrane
`The situation is depicted in Fig. 6. A fraction r of the
`energy incident upon each air-solid interface is
`reflected. From the energy radiated through the mate-
`rial, a fraction t reaches the second interface. Then, if Si
`is the net radiation flux at the positions indicated in the
`figure:
`S2 = Slr + S4t(l - r)
`(1.1)
`S3 = S1(l - r) + S4tr
`(1.2)
`(1.3)
`S4 = S3tr
`(1.4)
`S j = S,t(l - r)
`The energy fraction transmitted through the solid
`may be obtained from Eqs 1.1 to 1.4, as:
`t(1 - r)'
`1 - ft2
`
`t&. = SJS, =
`
`solid
`
`s3
`
`,
`
`s1,
`+
`
`s2
`
`1
`s4
`
`1
`e
`Fig. 5. Relutioe errvr in the erperimentul determination of the
`thermal conductivity using a steady technique, as a function vf
`the emissiuity vf the boundary surfaces, and different cell sizes
`L,, (PS foam ut T = 283 K ) .
`
`Fig. 6 . Radiant energy through a solid membrane.
`
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`
`POLYMER ENGINEERING AND SCIENCE, APRIL, 1983, Vol. 23, No. 6
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`
`

`
`Thermal Conductivity of Plastic Foams
`
`r N = SJS, =
`
`(1.6)
`
`(1.7)
`
`The energy fraction that is reflected is given by:
`r[l + f(l - 2r)l
`1 - P12
`The fraction that is absorbed is:
`aN = (1 - t ) ( S , + S4)/Sl = (1 - r)(l - t)/(l - rt)
`Obviously, it is verified that:
`t N + r N + a N = 1
`(1.8)
`At steady state, the absorbed energy is re-emitted. As
`the temperature of each solid layer is assumed to be uni-
`form, the emission is the same for both directions. Thus,
`the net fraction of radiant energy sent forward by a solid
`membrane is given by:
`
`Then,
`
`From Eq 2.6,
`UT;P
`9Li = 1 + n(l/TN - 1)
`
`and
`
`(2.12)
`
`(2.13)
`
`APPENDIX 3
`Net Radiant Energy between Two Grey Boundaries
`with Emissivities el and e2 with a Set of n Interposed
`Solid Layers, Each One Transmitting a Fraction TN of
`the Incident Energy
`The situation is depicted in Fig. 7b. From Eq 2.14 of
`Appendix 2, the net transmission of the set of n solid lay-
`ers is given by
`
`T =
`.
`
`1
`1 + n(l/TN - 1)
`
`(3.1)
`
`Then,
`
`S,' = e l u c + (1 - e,)S;
`(3.2)
`S ; = (1 - T)S,' + TS,+1
`(3.3)
`S,' = TS; + (1 - T)Si+,
`(3.4)
`(3.5)
`S;+, = (1 - e2)S,'
`(3.6)
`Si+l = e2S:
`The net radiation emitted from the grey surface at TI,
`and absorbed by the grey surface at T2 is:
`+
`qrrrd = S:+,
`(3.7)
`Solving Eqs3.2 to3.6 to obtain S,f+,, and substituting
`into Eq 3.7, gives:
`
`(a)
`
`T1
`
`1
`
`b
`
`n
`
`grey
`81
`
`grey
`0 2
`
`(b)
`Fig. 7 . (a) Radiant energy through a set of n solid layers with
`black boundaries. ( b ) Radiant energy through a set of n solid
`layers with grey boundaries.
`
`The fraction (1 - TN) is sent back.
`
`APPENDIX 2
`Net Radiant Energy Through a Set of n Solid Layers,
`Each One Transmitting a Fraction TN of the Incident
`Energy
`The situation is shown in Fig. 7a, where S: is the net
`radiation flux leaving the generic plane i to the right,
`and S; is the corresponding radiation flux sent back to
`the left. It can be stated that:
`S t = TNSt-1 + (1 - TN)Si+l
`(2.1)
`Si+i = (1 - 2")s: + TNSG+p
`(2.2)
`Ski = TNS: + (1 - TN)S,,
`(2.3)
`From Eqs 2.1 to 2.3, the following recurrent law may
`be obtained:
`St/&L1 = 142 - S?+,,/Si+) , for i < n
`(2.4)
`For i = n, the radiation S,' is entirely absorbed by the
`black surface a? TZ. Then:
`(2.5)
`S,'/S,'-l = TN
`The net radiation arising from the black surface at TI
`and, reaching the black surface at Tz, is given by:
`9&(i = S,' = uT;PS,f/S,' = uT: fl (S$/St-1)(2.6)
`n
`i = l
`where u is the Stefan-Boltzmann constant.
`From Eqs 2.4 and 2.5:
`S$-i/S;-2 = 1/(2 - S,'/S:-:_,) = 1/(2 - TN) (2.7)
`Then, the recurrent Eq 2.4 gives:
`s,'-Js:-, = 1/{2 - [1/(2 - TN)]} = (2 - T N ) / ( ~ - ~ T N )
`(2.8)
`s,'-ds$-, = (3 - 2TN)/(4 - ~ T N )
`(2.9)
`S:-$s,'-, = (4 - 3T~)/(5 - 4TN)
`(2.10)
`s:/s: = [(n - 1) - (n - 2 ) T ~ y [ft - (?I - 1 ) T ~ l
`(2.11)
`
`POLYMER ENGINEERING AND SCIENCE, APRIL, 1983, Vol. 23, No. 6
`
`297
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`

`
`R. J . J . Williams and C. M . Aldao
`
`9 A l d =
`
`uTf
`1/T + (l/el + l/ez - 2)
`The net radiant energy is then given by:
`
`(3.8)
`
`q n r d = 9&1d - 9 L
`-
`U(T? - 2’;)
`-
`1 + n(l/TN - 1) + (l/e, + l/ez - 2)
`
`(3.9)
`
`REFERENCES
`1. J . H. Marciano, A. J. Rojas, and R. J. J. Williams, Eur. I . Cell.
`Plast., 3, 102 (1980).
`2. E. E. Pasqualini, Proceedings of the I Latino American Con-
`gress on Heat and Mass Transfer, La Plata, Argentina (1982).
`3. J. Brandrup and E. H. Immergut, eds., “Polymer Hand-
`book,” 2nd Ed,, Vol. 3 , p. 241, John Wiley & Sor~s, New York
`( 1974).
`
`298
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