Thermal conductivity of rigid foam insulations for aerospace vehicles

Cryogenics 55–56 (2013) 12–19

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Thermal conductivity of rigid foam insulations for aerospace vehicles M. Barrios ⇑, S.W. Van Sciver National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA Mechanical Engineering Department, FAMU/FSU College of Engineering, Tallahassee, FL 32310, USA

a r t i c l e

i n f o

Article history: Received 18 January 2012 Received in revised form 6 November 2012 Accepted 25 November 2012 Available online 23 January 2013 Keywords: Polyisocyanurate foam Insulation Thermal conductivity SOFI

a b s t r a c t The present work describes measurements of the effective thermal conductivity of NCFI 24-124 foam, a spray-on foam insulation used formerly on the Space Shuttle external fuel tank. A novel apparatus to measure the effective thermal conductivity of rigid foam at temperatures ranging from 20 K to 300 K was developed and used to study three samples of NCFI 24-124 foam insulation. In preparation for measurement, the foam samples were either treated with a uniquely designed moisture absorption apparatus or different residual gases to study their impact on the effective thermal conductivity of the foam. The resulting data are compared to other measurements and mathematical models reported in the literature. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Liquid hydrogen and liquid oxygen are commonly used as rocket fuel and as a result the aerospace industry has a demand for high quality cryogenic insulation that can be applied to launch vehicles. One example of an application for porous insulating media is the thermal protection system of the external tanks of the Space Shuttle, where spray-on foam insulation (SOFI) has been used for decades [1]. This rigid foam insulation was selected for Space Shuttle application because it has a low thermal conductivity (k = 0.02 W/m K at room temperature) and high strength to density ratio (8–9 kPa/kg/m3) [2]. Extensive research has been performed on thermal transport in foam insulations, although the vast majority has been associated with near room temperature applications in, for example, the building industry. Low temperature applications are unique due to the potential for moisture absorption and other condensable gases affecting thermal transport. Mathematical models have been proposed to predict heat transport through foam insulation, but these models generally have not been tested at low temperatures. As a result, more experimental data on the foams of interest subjected to the anticipated operating conditions are required. Here we report thermal conductivity measurements using a novel apparatus that has been developed specifically to study rigid foam samples at temperatures ranging from 20 K to 300 K [3]. The effective thermal conductivity of three samples of NCFI 24-124 foam insulation was measured over the full temperature ⇑ Corresponding author. Address: Facility for Rare Isotope Beams, Michigan State University, USA. Tel.: +1 757 269 7058. E-mail addresses: [email protected], [email protected] (M. Barrios). 0011-2275/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cryogenics.2012.11.004

range. Prior to measurement, some of the samples were treated with different residual gases or water vapor to investigate their effect on the thermal conductivity. The data are compared to measurements reported in the literature and to mathematical models developed to predict the thermal conductivity of porous media. 2. Theory background Thermal conductivity is an important measure of insulation performance. However, because nearly all cryogenic insulation is inhomogeneous, the thermal conductivity of such materials is difficult to predict. Fourier’s conduction law provides the definition for the thermal conductivity, k(T), of a solid body:

q ¼ kðTÞrT

ð1Þ

When the material in question is isotropic and the temperature difference is small, the thermal conductivity can be averaged over the temperature range, and the one dimensional heat conduction of a constant cross-section sample becomes:

Q¼

 kðTÞA DT L

ð2Þ

where L is the length of the sample in the direction of heat transfer and A is the cross sectional area (perpendicular to the heat transfer  direction). Note that in the Eq. (2), kðTÞ refers to the average thermal conductivity of a solid. In the case of cryogenic insulation (e.g. MLI, foam, aerogel beads), the conditions mentioned above (isotropic and small DT) are rarely met. In practice for anisotropic materials, it is normal to measure the effective thermal conductivity, keff, of a sample. The effective

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Nomenclature q Q k A T l

heat flux heat energy thermal conductivity cross sectional area temperature length of material in the heat transfer direction, cell dimension effective thermal conductivity effective solid conductivity effective gas conductivity effective radiative conductivity Knudsen number mean free molecular path characteristic spacing of a medium pressure molecular mass density

keff ks kg kr Kn k L p M

q

 thermal conductivity can replace kðTÞ in Eq. (2), but is assumed to include additional heat transfer modes such as gas conduction, radiation heat exchange, and in some cases convection and contact resistance. Mathematical models exist that estimate these components of keff under different conditions. Most models found in the literature combine solid and gas thermal conductivity using geometric simplification of the internal structure of the foam [4–10]. Radiation is assumed to occur in parallel with the conduction modes and is added to the solid and gas conductivity to arrive at an estimate for the overall effective thermal conductivity. In the following section, we review various attempts to model keff for porous media. The purpose is to provide basic understanding of the mechanisms and to show where additional theoretical effort is necessary. 2.1. Solid and gas conductivity Maxwell [7] was one of the first to examine conduction in heterogeneous media. He developed a fairly simple expression for the electrical conductivity of a material consisting of spherical inclusions within a medium. This model can be adapted to describe the effective thermal conductivity as,

 keff ¼ ks

2kg þ ks þ Pðkg  ks Þ 2kg þ ks  2Pðkg  ks Þ

 ð3Þ

where ks is the thermal conductivity of the solid matrix material and kg is gas conductivity. The porosity (P) is defined as the ratio of the pore volume to the total volume of the material. Use of Eq. (3) depends on knowledge of the thermal conductivity of the solid matrix and contained gas and it ignores any contribution due to radiation heat transport. Determining the solid conductivity contribution to the effective thermal conductivity requires knowledge of the thermal conductivity of the solid matrix material, ks. If the thermal conductivity of the solid polymer that makes up the polyurethane (PU) or polyisocyanurate (PI) foam is known for the temperature range in question it can be used in conjunction with the internal geometry to determine the effective contribution from the solid matrix, km. However, for some porous media, there is little data available for the thermal conductivity of the solid material. Tseng et al. [11] approached this problem by assuming the thermal conductivity of PU foam to be equal to that of nylon, a similar material. Still this approach is limited by the incomplete knowledge of the internal structure of the PU material. Alternatively, Wu et al. [12] estimated

CV c

constant volume specific heat average molecular speed Boltzmann constant accommodation coefficient indices empirical function for use with gas mixtures total number of gases in a mixture, refractive index mole fraction of a gas in a mixture constant used in the Maxon–Saxena relation porosity thermal conductivity of a solid matrix fraction of solid in the struts of a porous material extinction coefficient extinction coefficient of solid polymer foam density solid polymer density

j a i,j Aij n y

e P km fs b bs

qf qs

the thermal conductivity of the solid matrix by subtracting the calculated radiation contribution from effective thermal conductivity of the porous medium measured in vacuum. This latter method may be reasonable for open cell foams, but for closed cell foams, performing measurements in vacuum does not ensure that the residual gas has been entirely removed from the cells. Obtaining the gaseous contribution to the thermal conductivity, kg, can also be difficult because it is dependent on the composition and pressure of the contained gas. For most gases at pressures above about 1 Pa, the thermal conductivity is mainly a function of temperature. However, at low pressures (p < 1 Pa) and for many types of porous media, the mean free path of the gas molecules becomes larger than the pore size. In this case, the gas thermal conductivity is mainly function of the accommodation coefficient, a, a quantity that determines how well the gas molecules transfer energy to the solid material. For this reason, it is most desirable to know the gaseous pressure and temperature within the porous media even though this may not be achievable for closed cell foams. Springer [13] and Tien and Cunnington [14] considered four different regimes of gaseous heat conduction based on the value of the Knudsen number:

Kn ¼

k l

ð4Þ

where k is the mean free path of the molecules and l is the pore size. The four gas regimes are: free-molecule (Kn > 10), transition (10 > Kn > 0.1), temperature-jump (slip) (0.1 > Kn > 0.01), and continuum (Kn < 0.01). Springer estimated the gaseous heat conduction for the above regimes for simple geometric configurations (parallel plates, coaxial cylinders, and concentric spheres) and small boundary temperature differences. From kinetic theory the thermal conductivity of a gas in the continuum approximation can be expressed by:

kg /

1 cqC V k 3

ð5Þ

where q is density, CV is the heat capacity at constant volume (J/ kg K), and c is the average molecular speed:

c ¼

rffiffiffiffiffiffiffiffiffiffiffi 8kB T pM

ð6Þ

where kB is the Boltzmann constant. However, for the residual gases present in porous insulating media the small voids can cause the

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gas to enter the free-molecule regime at low pressures. For such complex void geometries, Tien and Cunnington [14] suggested the use of an effective mean free path to develop the following empirical relation for effective gas thermal conductivity in the transition and temperature-jump regimes: 0

kg ¼ akg ½l=ðl þ kÞ

ð7Þ

In the free-molecule regime, the apparent thermal conductivity can be estimated using Knudsen’s formula:

kFM ¼ a  plDT



cþ1 c1

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 8pMT

n X i¼1

y ki Pn i j¼1 yj Aij

ð9Þ

where kg is the thermal conductivity of the gas mixture, ki is the thermal conductivity of gaseous component i, n is the total number of gases in the mixture, yi and yj are the mole fractions of components i and j, and Aij is a function to be specified (Aii = 1). For nonpolar gas mixtures, Poling et al. [15] suggest the Mason–Saxena relation:

h

Aij ¼

e 1 þ ðki =kj Þ1=2 ðMi =Mj Þ1=4 ½8ð1 þ Mi =Mj Þ1=2

16n2 rT 3 3b

i1=2 ð10Þ

where M is molecular weight of the gaseous components and e is a constant near unity. Once values for ks and kg have been determined, it is then necessary to account for the tortuous path that heat conduction follows within the structure of the porous media. This is especially important in evacuated media, when gas conduction is minimized. Several heat transfer models [4–6] developed with heat conduction through loose fill insulations can be simplified for use with rigid foam calculations. These models involve the additional complexities of surface contact, packing arrangement, and porosities on different scales (i.e. hollow glass spheres, aerogel beads) which can generally be neglected in the case of rigid foam. Other models have been developed specifically for rigid foam [7–10]. Many approaches found in the literature analyze a unit cell within the microstructure of the porous media. The unit cell is defined by Dul’nev [4] as the smallest volume whose effective heat conductivity coincides with the effective heat conductivity of the disperse system. The unit cell can be represented in many forms: cubic inclusion, spherical inclusion, cubic skeleton with interconnecting pores, etc. Different methods are used to find the thermal conductivity of the unit cell chosen to represent the internal structure of the system. Geometric parameters are then used to relate the dimensions of the unit cell to the porosity of the media. The effective thermal conductivity of the media is then found as a function of the solid and gas conductivities and the porosity. 2.2. Radiative conductivity Radiation heat transfer can significantly contribute to the effective thermal conductivity of a porous medium. For optically thick media, Siegel and Howell [16] suggested an effective radiative conductivity given by:

ð11Þ

where n is the refractive index of the porous media, r is the Stefan– Boltzmann constant, and b is the extinction coefficient, the inverse of the mean penetration distance of radiation in the medium. Unfortunately, the extinction coefficient is another property that can vary with polymer type, morphology, and temperature. Glicksman [10] analyzed the structures of cell walls and struts to find the following relation for the extinction coefficient for PU and PI foams:

rffiffiffiffiffiffiffiffiffiffiffi  ffi

ð8Þ

A further complication in the calculation of thermal conductivity occurs when there is a mixture of gases present. In many cases when calculating the thermal conductivity of a mixture of gases, a simple molar average of the thermal conductivity of the components is insufficient [15]. In mixtures involving two gases with different molecular weights, however, the thermal conductivity tends to be lower than expected. The Wassiljewa equation can be used to account for this correction:

kg ¼

kr ¼

fs q f

b ¼ 4:1

qs

l

 þ bs

ð1  fs Þqf



qs

ð12Þ

where qf is the density of the foam, qs is the density of the solid polymer, fs is the fraction of solid in the struts of the foam, and bs is the extinction coefficient of the solid polymer. The first term on the right side of Eq. (12) is the contribution to the extinction coefficient due to the struts. Because the struts are typically much thicker than the cell walls, they can be considered opaque. Therefore, the extinction coefficient of the solid material is not included in the strut contribution. Although there has been a considerable effort applied toward modeling thermal transport in porous media containing gases, the results are qualitative at best. In the present case of SOFI material at low temperature, the effective thermal conductivity is complex because the internal structure of the material is not well characterized, the thermal conductivity of the solid matrix material is not well known and the material is a closed cell foam, with the gas composition and pressure within the foam cells not well known. One additional complexity comes about as a result of the low temperature application of SOFI. That is, the possibility that in its application some of the contained gas may condense at low temperature further complicating the effort to calculate the gas contribution to the effective thermal conductivity. All these factors combine to make prediction of the effective thermal conductivity very challenging and thus demanding experimental measurements to properly characterize the material property. 3. Thermal conductivity measurement apparatus The experimental apparatus developed for measurement of the effective thermal conductivity of flat plate samples of SOFI or other solid materials at low temperatures has been described previously [3]. Here we provide a summary of the important characteristics of this device so that the reader can best understand the experimental methods and results. The ASTM standard C177, ‘‘Standard Test Method for SteadyState Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus,’’ [17] was used as a template and modified to accommodate the additional requirements for operation in a cryogenic environment. The apparatus design uses a single-sided guarded-hot-plate with only one cold plate and one specimen. This approach greatly simplified the design of the cooling plate, which was connected directly to a cryocooler as a heat sink. Also, a double-sided design would require a more complicated thermal link and thermal stabilization between the two cold sides. Fig. 1 is a schematic of the experimental chamber. The apparatus was calibrated using a disk of nylon with a known thermal conductivity as a control sample. Through this process, it was found that contact resistance between the faces of the sample and the hot and cold plates was a limiting factor, even when copper grease was applied to the faces of the sample. The thickness (25.4 mm) of the samples and the diameters of the

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To Cryocooler Copper Stem

Stainless Steel Plate

Hot Plate

Insulation Sample

Cold Plate Phenolic Guard

Compression Springs

Aluminum Guard

Vacuum Chamber

Fig. 1. A schematic of the experimental chamber for measurement of the effective thermal conductivity of solid foam samples.

temperature sensors (3.175 mm) created a large enough uncertainty in the gauge length to eliminate the possibility of embedding the sensors within the sample. It was thereafter decided that, in order to minimize the contact resistance, the foam sample should be bonded to copper plates using Stycast 2850 epoxy. The Stycast provides excellent thermal contact with the faces of the sample. Temperature sensors were then mounted to the copper plates. The added thermal resistance of the Stycast layers and copper plates was determined to comprise less than 1% of the overall sample resistance. The experimental cell is attached to a cryocooler (Cryomech model PT-810) at the top of the cold plate by means of a copper stem that penetrates the stainless steel plate and a braided copper thermal link. The cryocooler is capable of providing 14 W of cooling power to the apparatus at 20 K. A heater is mounted to the

head of the cryocooler to control the overall temperature of the apparatus. Stainless steel tubes are welded to the top of the cold plate to provide mechanical supports. The apparatus is placed inside an evacuated cryostat with a liquid nitrogen shield to isolate it from ambient. Fig. 2 is a schematic of the entire apparatus. The effective thermal conductivity of the SOFI samples was measured by recording the temperature on either side of the sample for a specific heater power after steady state was achieved. Prior to installing a sample in the apparatus, the diameter and thickness were measured at various locations. These measurements were then used to calculate the surface area and the average thickness of the sample. The heater power was measured by recording the voltage across the heater and the voltage across a known resistor in series at room temperature to yield the current through the circuit. The temperatures were measured using Cernox

Fig. 2. A schematic of the apparatus for measurement of the effective thermal conductivity of SOFI.

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sensors on the copper plates attached to the specimens and on the aluminum guard. Before calculating the effective thermal conductivity, a correction was made to account for any heat leak between the hot plate and aluminum guard. The typical accuracy of the temperature sensors was ±40 mK. This translates to an uncertainty of less than 1% in temperature measurement. The heater voltage resolution on the Lakeshore 340 temperature controller is 1.25 mV. The estimated minimum required operating voltage is 553 mV. This corresponds to a maximum heating power uncertainty of 0.2%. Width and area measurement uncertainty were limited to below one percent simply by performing several measurements with a micrometer. Thermal contraction measurements from the literature vary, but do not exceed 2% between room temperature and 80 K. The sum of the errors listed above is reflected in the overall 6% error bars seen in the results. For most measurements a DT of 10 K across the sample was chosen to make it possible to determine the effective thermal conductivity with an instrument error below 5%. 4. Measurements and discussion The thermal conductivity of three samples of NCFI 24-124 foam was measured. This foam has a porosity of 97% and an average pore diameter of 0.3 mm. Table 1 lists the three samples and shows the conditions under which the thermal conductivity was measured. The first NCFI sample (#1) was tested ‘‘as received’’ by installing it within the experimental chamber under room temperature and atmospheric pressure of air (296 K and 102.6 kPa) and sealing the chamber. The thermal conductivity was then measured at various temperatures between 296 K and 30 K. The gas pressure in the experimental chamber varied from 102.6 kPa at 296 K to 0.267 kPa at 30 K. Subsequently, in order to examine the effect of different residual gases on the thermal conductivity of the foam, sample #1 was tested under vacuum and with helium gas added to the experimental chamber. Fig. 3 shows the thermal conductivity of sample #1 in air and under vacuum. The thermal conductivity shows a clear decrease with temperature down to 80 K followed by a sharp increase. As is discussed below, this increase in keff probably results from the condensation of the lower thermal conductivity heavy gases onto the cell walls thus increasing the molar ratio of higher thermal conductivity trace gases such as helium. Tseng et al. [11] observed a similar phenomenon for newly sprayed polyurethane foam between 230 K and 280 K. In this temperature range, the R141b gas present in the cells condenses and the air remaining in the cells dominates the thermal transport. In the present case, helium was inadvertently introduced into the foam during a leak checking process. As a result during the measurement, the air present in the cells condensed below 80 K and the remaining helium gas dominated the heat transport dramatically increased the effective thermal conductivity of the SOFI sample. The data from Fesmire et al. [18], shown as the solid circles in Fig. 3, are the effective conductivity of the same material as our sample, recorded at average temperatures of 297 K and 185 K, respectively. Their data appears slightly below that found in the present study. Fesmire et al. measured a sample aged in a climate

Fig. 3. The thermal conductivity versus average temperature for NCFI 24-124 sample #1 in the ‘‘as received’’ and evacuated cases compared with literature data [18]. The increase at low temperature in the ‘‘as received’’ case is probably the result of residual helium gas. The error bars correspond to a 6% uncertainty determined by the temperature sensors, heater, sample width and area measurement, and thermal contraction.

controlled area for 18 months. Our sample was aged in the same area for 18 months, and stored in a sealed plastic bag for 4–5 years before testing. The discrepancy in the data could be due to aging, diffusion of the low thermal conductivity blowing agent out of the foam over time, or the presence of the higher thermal conductivity helium gas. For the vacuum case, sample #1 was placed in the experimental chamber and evacuated, using a turbo-molecular vacuum pump, for one week at room temperature. At this point the pressure inside the chamber was 138.3 Pa, decreasing at less than 0.4 Pa per hour. The cryocooler was then turned on and the sample cooled to 30 K. As expected, the measurements show a clear decrease in effective thermal conductivity compared to the as received case with helium contamination. The data for the evacuated case follow a similar trend to that of the as received case at high temperature, but do not reproduce the anomaly below 80 K. In order to further examine the effect of residual gas on the thermal conductivity of the foam, sample #1 was placed in the experimental chamber, evacuated overnight and then purged with 101 kPa of helium gas at ambient temperature three times. After the third filling, the cryocooler was then turned on and the thermal conductivity measurement repeated. Due to the 97% high porosity of the foam, the residual gas can dominate the effective thermal conductivity. This is shown in Fig. 4, where the effective thermal conductivity of the foam with helium as a residual gas is compared to that of the evacuated and as received cases. The effective thermal conductivity of the helium case is roughly three times larger than that of the as received case. In this regime, the thermal conductivity of helium gas is at least six times larger than that of air and the thermal conductivity of the solid material is greater than that of helium gas, so it is surprising that the effective thermal conductivity measured is lower than that of pure helium gas. This

Table 1 NCFI 24-124 samples and the conditions under which their thermal conductivity was measured. Sample #1 (‘‘as received’’, x) was subjected to an unknown amount of helium gas during the leak check process, which is believed to have affected the thermal conductivity. Sample number

As received

#1 #2 #3

x

Water vapor conditioned x

x

Helium purged

Under vacuum

Helium/air mixtures

x

x x

x

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Fig. 4. The thermal conductivity data of NCFI 24-124 sample #1 for the helium purged case, the ‘‘as received’’ case, and the evacuated case. The thermal conductivity of the conditioned sample #2 ( ) is shown for comparison.

discrepancy is almost certainly due to insufficient purging of the sample. Because of the closed cell nature of the foam it is very difficult to replace all of the R141b gas and air originally present in the cells with helium gas. In any case, the trend in the thermal conductivity of the helium purged case approaches that of the sample #1 ‘‘as received’’ case below 50 K. This is further evidence that in the as received case, the sample was contaminated with helium gas. Also seen in Fig. 4 are the data from the conditioned NCFI 24-124 sample #2. Sample #2 was tested in the same fashion as sample #1; but it was first subjected to a launch pad conditioning process. The conditioning process subjected the foam to a typical environment expected on the launch pad in Cape Canaveral (T = 34 ± 2 °C, Relative Humidity >75%) for 8 h. This conditioning process causes moisture absorption in the foam, potentially affecting its thermal properties [19]. Above 70 K, the thermal conductivity of the conditioned sample is nearly the same as the ‘‘as received’’ sample. This shows that the conditioning process and moisture absorption have little effect on the thermal conductivity of the foam. To quantify the effect of helium intrusion on the thermal conductivity of the foam at low temperature, sample #2 was first evacuated for several weeks and the thermal conductivity then measured between 30 K and 290 K. This provided a baseline for the thermal conductivity of the foam with minimal contribution from the residual gas. The chamber was then filled with air to a pressure of 85.3 kPa. A period of 24 h was then given for the air to diffuse into the foam during which time additional air was added to the chamber to maintain the pressure constant. The thermal conductivity of the sample was then measured from 20 K to 120 K. The sample was then warmed to near room temperature and a small amount (267 Pa partial pressure) of helium gas was added to the 85.3 kPa of air. This brought the mole fraction of helium gas to 0.3% at room temperature. At low temperature the helium gas would then be in the transition gas conduction regime. Lastly, sample #2 was warmed to room temperature and more helium was added up to a partial pressure of 16 kPa, which brought the total gas pressure to 101 kPa at room temperature and the mole fraction of helium to 16%. This mixture was chosen to maintain the continuum gas regime over the entire temperature range 20–300 K. As discussed previously, the continuum regime occurs when the Knudsen number is less than 0.01, meaning the mean free path

17

Fig. 5. The thermal conductivity vs. temperature for sample #2 with various residual gas mixtures. Lines through the data are guides to the eye.

of the molecules is much shorter than the characteristic dimension of the cells within the foam. Fig. 5 shows that the thermal conductivity of the 85.3 kPa of air case for sample #2 follows the thermal conductivity of the conditioned case closely until around 80 K where it begins to approach the thermal conductivity of the evacuated case. This is expected as uncontaminated air will condense and freeze below 80 K and leave a near vacuum environment within the cells. The presence of 0.3% helium gas should cause the thermal conductivity of the sample to be larger than that of the air and evacuated cases at temperatures below 80 K. It can be seen in Fig. 5 that, in fact, the thermal conductivity begins to increase as the temperature is lowered below 100 K. This marked increase shows that even a small amount of helium gas produces an increase in the effective thermal conductivity of the foam at lower temperatures. As expected, the thermal conductivity of the sample was much higher due to the higher mole fraction of helium (see Fig. 5). However, as the temperature of the sample increased from 40 K to 60 K an unexpected drop in effective thermal conductivity was observed. Near the observed pressure at 40 K, the sublimation line nitrogen is crossed. This could cause some of the nitrogen to leave the surfaces of the foam. As the nitrogen pressure increases, it mixes with the helium gas and impedes the overall heat transport, thus lowering the effective thermal conductivity. However, a corresponding increase in pressure is not recorded in the measurements possibly because the pressure increase is small and the time necessary for the gas to diffuse out of the sample had not elapsed. The sample #2 effective thermal conductivity measurements were then compared to the Maxwell model, Eq. (3). This model requires knowledge of the porosity and thermal conductivity of the solid material and the gas. Values for the porosity and solid thermal conductivity at room temperature were taken from the literature [20,21]. Below room temperature, the solid thermal conductivity was extrapolated by reproducing the relationship between temperature and thermal conductivity seen in similar materials, such as PVC and nylon. The cell dimensions were determined from average values measured using SEM images [19]. The residual gas thermal conductivity, required in all of the models, is a difficult parameter to estimate. In order to do so accurately, the mixture of gases and their partial pressures must be known. Also, at low temperatures the condensation of gases must be taken into account (e.g. in a mixture of nitrogen and helium, the nitrogen will condense at higher temperatures than helium, causing the molar ratio of helium in the mixture to increase). At low pressures, as

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Fig. 6. The effective thermal conductivity data for the evacuated NCFI 24-124 sample #2 compared to predictions based on the Maxwell model, Eq. (3).

Fig. 7. The thermal conductivity data for the NCFI 24-124 sample #2 with air as a residual gas at 85.3 kPa at 300 K plotted against predictions from the Maxwell model, Eq. (3).

continuum gas heat transfer transitions into the Knudsen regime, the gas thermal conductivity becomes even more difficult to estimate. The mathematical models do not take these factors into account, and it is left to the analyst to determine the appropriate value for gas thermal conductivity at each point. Fig. 6 is a comparison between Maxwell’s model and the data from the evacuated case for sample #2. The model tends to under predict the thermal conductivity at temperatures above 40 K. The sharp increase in thermal conductivity from 30 K to 50 K is possibly due to sublimation of nitrogen inside the cells of the foam. The nitrogen increases the thermal conductivity of the foam, but since the foam is closed cell the increased pressure inside the foam is not recorded by the pressure gauge, and is therefore not reflected in the model. Fig. 7 is a comparison between the Maxwell model and the data from the 100% air sample. Again, the model fails to predict the increase in thermal conductivity between 35 K and 55 K. This is a further indication that the gauge pressure does not accurately depict the pressure inside the cells at low temperatures. Fig. 8 compares the measurements of the 84% air/16% He sample to the model. In this case, the model predicts the increase in thermal conductivity from 20 K to 40 K, but greatly over predicts the conductivity from 50 K to 110 K. Again, this is due to fluctuating gas pressures inside the closed cells, causing the thermal conductivity to decrease as nitrogen sublimates.

Fig. 8. The thermal conductivity data for the NCFI 24-124 sample #2 with an initial mixture of 85.3 kPa air and 16.0 kPa helium (84% Air/16% He) at 300 K plotted against predictions from the Maxwell model, Eq. (3).

Fig. 9. The effective thermal conductivity for two NCFI 24-124 samples. Sample #2 is launch pad conditioned and sample #3 as received.

Comparisons between the effective thermal conductivity results and the Maxwell’s model make it clear that in order to make accurate predictions of keff in SOFI it is necessary to know the types of residual gases present in the foam. Furthermore, it is important to understand how the pressures of the gases change with temperature. The large surface areas present in the foam can cause gases to condense at temperatures higher than would normally be expected. The effective thermal conductivity of a third NCFI 24-124 sample #3 was measured ‘‘as received’’ to check reproducibility and obtain data without helium gas contamination. These results are compared to the launch pad conditioned sample #2 in Fig. 9. As can be seen in the figure, the conditioning process has little effect on the effective thermal conductivity of the foam resulting in at most a 10% increase. 5. Conclusions The goal of this work was to measure the effective thermal conductivity of flat plate samples of spray-on foam insulation (SOFI) with different preparation and at temperatures ranging from 20 K to 300 K. A single sided guarded-hot-plate apparatus was developed for this purpose.

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For sample #1 as received, the data show a clear decrease in thermal conductivity with decreasing gas pressure in the experimental chamber with the data for the evacuated case following a similar trend to that of the as received sample. In any case, the trend in an anomaly in the effective thermal conductivity below 50 K appears to have been caused by helium gas contamination. The effective thermal conductivity of the sample #2 was then studied to explore the effect of launch pad conditioning and helium gas contamination. The conductivity was first measured to determine the effect of conditioning. Following this measurement, the sample was first subjected to 85.3 kPa of air at 300 K and its the thermal conductivity closely followed the previous measurements down to 80 K where it approached the thermal conductivity of the evacuated case. This is expected because as the temperature decreases, the air will adhere to the surfaces of the foam and leave a vacuum. When a partial pressure of 0.267 kPa of helium was added to the 85.3 kPa of air at room temperature, the effective thermal conductivity increased above the base case below 100 K. This marked increase shows that even this small amount of helium gas affects the thermal conductivity of the foam at lower temperatures. As more helium is added to the residual gas mixture an increase in thermal conductivity can be seen over the entire temperature range. Measurements below 120 K show that the effective thermal conductivity does not correlate with Maxwell’s model. This is probably due to the inability to know the partial pressures of the gas mixtures within the cells at low temperature. These effective thermal conductivity results stress the importance of residual gases present in foams like SOFI. Although launch pad conditions may slightly increase the thermal conductivity of the foam, exposure to high thermal conductivity gases such as helium or hydrogen can have a much greater effect. In the case of hydrogen storage, exposing the foam to venting or leaking hydrogen gas can cause a significant increase in the effective thermal conductivity of the foam.

Acknowledgements This work was supported by NASA Kennedy Space Center and the Florida Center for Advanced Aero-Propulsion (FCAAP). The National High Magnetic Field Laboratory is supported by the NSF and the State of Florida.

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