THE THERMAL CONDUCTIVITY OF POLYURETHANE FOAMS FROM ROOM TEMPERATURE TO 20K

Chung-jen Tseng*, Masahito Yamaguchi, and Takao Ohmori
*The New Energy and Industrial Technology Development Organization (NEDO)
Sunshine 60, 29F, 1-1, 3 chome Higashi-Ikebukuro, Toshima-ku, Tokyo 170, Japan

Advanced Technology Department, Research Institute
Ishikawajima-Harima Heavy Industries, Co. Ltd.
1 Shin-nakahara-cho, Isogo-ku, Yokohama 235, Japan


Abstract

The thermal conductivity of the polyurethane (PU) foam in the temperature range between 300 K and 20 K is investigated theoretically and experimentally for the development of liquid hydrogen storage tanks. An experimental set-up based on the JIS standard A1412 is built to measure the thermal conductivity of the foams under various temperature and pressure conditions. An analytical model, including the contributions from radiative heat transfer, is used to predict the thermal conductivity of the PU foams. The model is found to be able to accurately predict the thermal conductivity of the PU foams under normal and reduced pressures. The thermal conductivity of the PU foam can be reduced by as much as 70% by evacuating the gases in the foam cells. Radiative heat transfer is found to accounts for about 10 to 20 % at room temperature. The thermal conductivity data below 90 K are available for the first time.

Keywords: thermal conductivity; liquid hydrogen; polyurethane foam; insulation

Nomenclature
Aij : coefficient defined in Eq. (4)
d : mean pore diameter [m]
H : insulation thickness [m]
k : thermal conductivity [W/mK]
k' : the Boltzmann's constant [1.38 x 10-23 J/K]
ktr monatomic value of thermal conductivity
Kn Knudsen number [l/d]
l : mean free path [m]
L : heater length [m]
M : molecular weight [kg/kmole]
n : refractive index
P : pressure [N/m2]
Q : heater power [W]
q : heat flux [W/m2]
R : radius [m]
T : temperature [K]
X : mole fraction
z : axial coordinate

Greek Symbols
a : accommodation coefficient
b : extinction coefficient [m-1]
constant defined in Eq, (5)
f : porosity s : Stefan-Boltzmann constant [5.6696x10-8 W/m2K4]
x : molecule diameter

Superscripts and Subscripts
c : convection
eff : effective
g : gas
in : inner surface
out : outer surface
r : radiative
s : solid
t : total

Introduction

Hydrogen will become one of the primary energy sources in the twenty first century. It offers a non-polluting, inexhaustible, efficient, and potentially cost-effective energy system. However, large scale liquid hydrogen transportation and storage post a new challenge to engineers due to its extremely low temperature (20K). High performance thermal insulation systems are required. The polyurethane (PU) foam is chosen as one of the candidates for the insulation system because it offers a high strength-to-weight ratio, low thermal conductivity, and low cost. Currently, the thermal conductivity data of PU foams are available only for temperatures above 95 K1. This work will provide the thermal conductivity data for temperatures down to 20 K. Heat transfer mechanisms in PU foams are studied to help better understand the thermal performance of PU foams.

Heat transfer mechanisms

The PU foam used in this study has a closed-cell porous structure and a density of 32 kg/m3. The average diameter of the cells is approximately 400 mm (see Fig. 1). Heat is transferred through the PU foam via four mechanisms: (1) gas conduction, qg, (2) solid conduction, qs, (3) radiation, qr, and (4) convection, qc. The total heat flow, qt, is a result of the interactions between the four modes

(1)

Natural convection within the pores is negligible since the pore size is so small (< 0.5 mm) that the Rayleigh number is much less than the critical value (- 1000 ) for the onset of convection.

Radiative heat transfer

Radiative heat transfer refers to the transport of energy by electromagnetic waves and attenuation of radiation takes place in the forms of reflection, absorption, and scattering. Resins used for making PU foams are partially transparent in the 2 to 30 mm range of wavelengths2. Sections associated with thermal insulations, however, are normally optically thick (optical thickness >> photon mean free path). The optical thickness is defined as b H, where b is the extinction coefficient (= scattering coefficient + absorption coefficient) and H the insulation thickness. Tien and Cunnington3 stated that the extinction coefficient of a 35 kg/m3 PU foam is about 3000m-1. More recently, Stern4 used an infra-red spectrometer to measure the transmission, and then deduced the extinction coefficient of PU foams. His results ranged from 2040 m-1 for 28 kg/m3 samples to 4250 m-1 for 43 kg/m3 samples. In this work, the extinction coefficient is taken as 3000 m-1 for the 32 kg/m3 sample used.
For optically thick media, radiation can be considered as a diffuse process with an effective radiative conductivity given by5

(2)

where n is the refractive index of the insulation, s is the Stefan-Boltzmann constant (5.6696*10-8 W/m2K-4), and T is the mean temperature of the insulation. For such (optically thick) systems, radiation and conduction contributions can be effectively separated6

.

Solid conduction

Solid conduction is due primarily to two mechanisms: lattice vibrations, and translation of free conduction electrons. The free-electron contribution dominates in the energy transport in metals and the lattice vibration contribution is predominant in dielectric solids. The disordered dielectrics with no free electrons and considerable lattice imperfection are the poorest solid conductors of heat, and consequently most porous insulations are made of materials such as glass or polymeric plastics.
Solid conduction in PU foams takes place through the cell walls and membranes of the foam. Since PU resins are made up of disordered materials, there is little lattice conduction at low temperatures. The only currently available thermal conductivity data of pure PU is cited by Kikuike7 as 0.17-0.23 mW/m-1K-1 at room temperature. Figure 2 shows the thermal conductivity of three polymers, Teflon, Nylon, and polyvinyl chloride (PVC), which are available for cryogenic temperatures8,9. The curve for PVC is obtained by a polynomial fitting of available data (denoted by the '+' symbols) and the zero point. Because all three curves have similar temperature dependence trend, and the thermal conductivity of PVC at 300 K is closest to the value cited in Ref. 7, it will be used in this work as the pure solid thermal conductivity.

Gas conduction

The remaining mode of heat transfer is gas conduction in the foam cells. The PU foams used in the present study are expanded by CO2 and HCFC 141b (R141b)10. Figure 37 shows the partial pressures of the major gas components for a PU foam expanded by R11 and CO2. Initially only CO2 and R11 are present in the cells. While CO2 soon diffuses out of the cells due to its high diffusivity, R11 remains almost unchanged because of its very low diffusivity. Air has a diffusivity approximately 20 times larger than that of R11. It diffuses into the cells continuously and reaches about 50 kPa after 35 days. Eventually, the air partial pressure will reach 1 atm. No similar data is available for the CO2 and R141b system of the sample used10, however. Because the sample used in this work had been stored for more than one year at the time of measurement, the air pressure is assumed to be 100 kPa at 300K in the analytical model. The partial pressure of R141b at 300K is not known for the experimental sample. Although it is thought to be lower than the partial pressure of R11 (50 kPa) due to R141b's lower vapor pressure and diffusivity, no data is available. Several values were used in this study to best fit the measurement results.
The thermal conductivity of the major gases found in the insulating foams is shown in Fig. 4. The thermal conductivity data of R141b is obtained from a recent work by Tanaka et al.11 The conductivities of R11, N2, and O2 are obtained from8. It can be seen from the figure that while the thermal conductivity of R141b is slightly higher than that of R11, both are much lower than that of N2 and O2. The values of the thermal conductivities shown in Fig. 4 are used in the numerical calculation in this study. When a mixture of gases is present, the gas conductivity kg can be estimated by

(3)

where xi's are mole fractions of the species, and ki's are the thermal conductivities of the corresponding species. Eq. (3) usually overestimates the mixture thermal conductivity, especially for nonpolar-polar and polar-polar gas mixtures, such as air-R141b system in the present case. For such cases Reid et al.12 recommend to use Mason and Saxena's modification of the Wassiljewa equation.
The Wassiljewa equation takes the form

(4)

where Aij is a function to be specified. Mason and Saxena suggest the following form for Aij

(5)

where M is molecular weight, ktr is the monatomic value of the thermal conductivity, and E is an empirical constant. The detail can be found in Reid et al.12 Due to the very low temperature involved in this work, the partial pressures of individual gases may be low enough that pressure dependence of the gas conduction needs to be considered. The voids could be cryopumped for temperatures near 20 K. The various regimes of gas conduction can be defined according to the Knudsen number,Kn=l/d where l is the molecular mean free path, and d the characteristic length of the physical system (the pore size in the present case)3 .

1. Kn> 10, free molecule
2. 0.1 < Kn < 10, transition,
3. 0.01 < Kn < 0.1, temperature-jump (slip) region,
4. Kn < 0.01, continuum.

The mean free path can be calculated by

(6)

where k' is the Boltzmann's constant (1.38 x 10-23 J/K-1), x is the molecule diameter, and P is the pressure of the gas. Due to its extremely complicated nature, gas conduction in PU foam insulations defies any rigorous quantitative description and semiempirical representation becomes necessary. A convenient semiempirical technique is based on the use of an effective mean free path3, which takes into account both the molecule-to-molecule and molecule-to-wall collisions,

(7)

The effective gas thermal conductivity at reduced pressures is then

(8)

where a is the accommodation coefficient. The accommodation coefficient is a measure of the efficiency of thermal energy interchange that occurs when a gas molecule collides with the surface. It may vary between 1 (complete accommodation, diffuse re-emission) and 0 (specular re-emission).

Effective thermal conductivity

By considering the effects of void space and solid walls, Russell13 proposed the following equation for calculating the effective (apparent) thermal conductivity, keff, of a porous medium:

(9)

where F is the porosity of the foam.
No data of f of the PU foams used in the experiment is available. The manufacturer [Nichiasu Co., Ltd.] suggests to use the density of the PU foam and that of pure PU resins (- 1100 kg/m3) to estimate the porosity10. Therefore, for the 32 kg/m3 foams, F is approximately 0.97.
The above equations do not take into account of radiation. As mentioned previously, conduction and radiation can be treated independently for this optically thick medium. Therefore the radiation contribution can be accounted for by using the radiative thermal conductivity, kr, as defined in Eq.(3).
The final form of the equation for calculating keff can be expressed as

(10)

Thermal conductivity measurement

This section describes the experimental setup and procedure used to measure the thermal conductivity of PU foams from room temperature down to liquid hydrogen temperature (20K) under atmospheric pressure and evacuated condition.

Experimental setup

An experimental apparatus has been built to measure the keff of the PU foams. The steady cylindrical method is employed. It is based on the JIS standard A1412. As shown in Fig. 5, the system consists of a test section which is placed in a vacuum chamber, a vacuum pump, a two-stage GM refrigerator, a digital recorder, a temperature controller, and a computer. The test section is a sealed vessel placed horizontally. The initial pressure in the test chamber can be set at various values.
Details of the test section is shown in Fig. 6. The test samples are hollow cylinders of 400 mm long with inner and outer diameters of 30 mm and 70 mm, respectively. A film heater is stuck to the inner surface of the test sample with GE varnish. It is joule heated by a DC power supply. The film heater is made of Nichrome foil of 0.1 mm thick and 300mm long. High conductive alumina powder is used to fill the space inside of the heater, so that uniform temperature can be realized on the heater surface. At both ends of the test sample, fiber glass insulator and the same PUF are used to reduce heat loss. Copper made cooling plates are used to enclose the outer surface of the sample which is connected to a constant power GM refrigerator. The temperature of the cooling plates is controlled by adjusting the temperature of the refrigerator with a heater embedded in the cold head. The inner and outer surface temperatures are measured by using Celnox sensors (Lake Shore Cryotronics, Inc.). These sensors are of resistant thermometer type and has a wide operating temperature range ( about 1 to 325 K) with an accuracy of 0.02 K at 20K and 0.14K at 300K. Four sensors are placed on each surface. A radiation shield is also placed in the vacuum chamber and is connected to the first stage of the GM refrigerator, and thus reducing radiative heating from outside of the test section.

Experimental procedure

In this work, a 32 kg/m3 sample was used in the measurement. Two sets of measurements were performed. In the first set, the test section (inner chamber) was cooled from 300 K at atmospheric pressure down to 20 K gradually, and then back up to 300 K again. In the second set, the inner chamber was connected to the outer chamber by drilling several holes on the inner chamber, so that the inner chamber could be evacuated. The same sample was evacuated for three weeks at 300 K before being cooled down. There were no observable difference (cracks, fallen pieces, etc.) of the sample between before and after the experiment.
The procedure starts with setting the refrigerator temperature first. The heater power is then set to a value so that the temperature difference between the inner and outer surfaces of the sample is approximately 10 K. The temperatures are obtained by measuring the resistance of the thermometers, and then converting the resistance data to temperatures using the resistance-temperature calibration data. The computer calculates the temperatures continuously. If the change in the temperatures is less than 0.1K within a 30-minute span, steady state is assumed and the temperatures and the heater power data are recorded. Eleven reading are recorded for each steady-state temperature. It usually takes additional 30 minutes to log the data. The refrigerator is then set to another temperature, and the process is repeated.
The whole process is controlled by the computer so that the measurement can be performed 24 hours a day. It usually takes 6 to 12 hours to obtain one data point.

Data deduction

The steady cylindrical method is adopted in this measurement. The thermal conductivity can be calculated by the following equation

(11)

where Q is the heater power, L is the length of the heater, Rin and Rout are the radii of the inner and outer surfaces respectively, and Tin and Tout are the average temperatures of the inner and outer surfaces respectively.
The heat loss through both ends of the sample can be estimated by solving the two-dimensional steady-state heat conduction equation :

(12)

where the conductivity is a function of position. The control volume approach14 is employed to numerically solve the above equation with Q and outer surface temperature values from the experimental measurement. The inner surface temperature is used as a way to verify the numerical model.
The uncertainty of the measurement can be estimated from the error propagation equation of Eq. (11):

(13)

The uncertainty of the thermal conductivity measurement, dk/k , is found to be approximately 10%.

Results and discussion

By using the numerical simulation, end heat loss is found to be approximately 7.5% of total power input. This end heat loss has been accounted for in the results presented hereafter.
Figure 7 shows the thermal conductivity measurement results along with the theoretical predictions for the case that the sample was cooled from 300 K at atmospheric pressure. The uncertainties of the measurement data are indicated by vertical bars. At around room temperature, the thermal conductivity decreases with temperature as expected. However, it increases as temperature further decreases from about 270 K due to the condensation of the lower conductivity gas, R141b (which, in turn, increases the mole fraction of the higher conductivity gas, air). The R141b completely condenses at about 230 K, and the thermal conductivity decreases as temperature lowers further. At around 50 K, air starts to condense, and solid conduction becomes the only mode of heat transfer below 40 K.
As for the theoretical model, results for three different initial R141b partial pressures are shown in the figure for comparison. The accomodation coefficients are assumed to be constant with values of 1.0 and 0.6 for R141b and air, respectively. As mentioned previously, air pressure at 300K is assumed to be 100kPa for all three cases. All three cases converge below 230 K when R141b completely condenses. Between 220 and 250 K, all three cases underestimates the thermal conductivity. Above 260 K, the 30kPa-R141b case fits the measurement data quite well while the 10kPa-R141b case overestimates and the 50kPa-R141b case underestimates. This is because R141b has a lower thermal conductivity than air. The fact that the 30kPa-R141b case fits the measurement results best is in accordance with the estimation that R141b's initial partial pressure is lower than 50 kPa discussed previously.
Figure 8 shows the thermal conductivity results for the evacuated case. Clearly, most cell gas were evacuated as the thermal conductivity were reduced sharply when compared with the previous case. However, small amount of residual gas remains in the foam cells. This can be seen from the higher measured values than the completely evacuated values predicted by the model (shown by the dashed line). The thermal conductivity decreases monotonically from about 14 mW/m-1K-1 at 300 K to about 6 mW/m-1K-1 at 150 K, and about 1 mW/m.K at 20 K. The reversal in 230K-280K region of the previous case shifts to 150K-180K region in this case because of the much lower pressure in the cells (and hence the condensation temperatures).
For this low pressure case, the mean free path at 250 K for N2, and O2 are 600 mm, and 2600 mm, respectively. Therefore, they are limited by the pore size of 400 mm. The thermal conductivity of N2 and O2 are thus further reduced. On the other hand, the mean free path at 250 K for R141b is about 6 mm, much smaller than the pore size. Hence the thermal conductivity of R141b is almost unaffected by the pore size limitation. So R141b is the dominant gas in the foam cells in this temperature region. The R141b starts to condense at about 200K. However, it still dominates in this region, so the thermal conductivity of the foam continues to decrease. At about 170K, the R141b condenses to such an extent that it no longer dominates, and hence the reversal occurs again. The R141b completely condenses at about 150K, and the thermal conductivity decreases with temperature again. The theoretical model slightly overpredicts the thermal conductivity from 50K to 130K.
The contribution of radiation is indicated in the figure by the dashed line. It is negligibly small below 150K, but accounts for about 20% of the effective thermal conductivity at 300K.
Due to the various processing conditions, methods, and blowing agents used in expanding PU foams, the thermal conductivity data in the high temperature region (above 200K) may not be applicable to other samples. However, the data in the low temperature region should be applicable to most 32 kg/m3 closed-cell PU foams because conduction by air and solid dominates in this region. These data are very useful for cryogenic applications.

Conclusions

The heat transfer mechanisms in a PU foam are reviewed and discussed. An experiment has been set up to measure the thermal conductivity of PU foams for temperatures down to 20 K. An analytical model is proposed to estimate the thermal conductivity of the PU foam. The results are compared with the measured values. Heat loss in the experiment is analyzed numerically. Based on the results, the following conclusions can be drawn:

1.The theoretical model proposed in the study predicts the thermal conductivity quite well. The discrepancy in the high temperature region can be ascribed to either the uncertainty of the gas composition inside the cells, the pure solid conductivity, or the radiative properties of the PU foam.
2.Gas conduction in the closed cells accounts for approximately 60-80% of the total heat transfer. Evacuating the gas can greatly improves the thermal performance of the PU foam.
3.Radiative transfer is not important in the low temperature region, but it accounts for about 10 to 20 % in the room temperature region.
4.To enhance its thermal performance at room temperature, PU foams should be blown with as a high percentage of R141b as possible. However, it should be blown with as much CO2 as possible if it is to be evacuated such as when used in the thermal insulation system of liquid hydrogen storage tank because the diffusivity of CO2 is much higher than that of R141b.
5.The thermal conductivity data of pure PU solid is needed for more accurate theoretical prediction of the thermal conductivity of PU foams.

Acknowledgment

This study was performed as a part of the WE-NET (International clean energy network using hydrogen conversion) project which is launched by NEDO (the New Energy and Industrial Technology Development Organization) under the support of the Agency of Industrial Science and Technology.
The author, Chung-jen Tseng, would also like to acknowledge the help and support from the Engineering Advancement Association of Japan (ENAA) and Ishikawajima-Harima Heavy Industries Co., Ltd. (IHI).

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