Heat transfer in open cell polyurethane foam insulation

Heat transfer in open cell polyurethane foam insulation

Abstract This paper study systematic investigates the combined conductive and non-gray radiative heat transfer of open cell polyurethane (PU) foam in the pressure range between 760 and 0.02 Torr. Direct transmission measure-ments are also taken using Fourier transform infrared (FTIR) spectrometer. In doing so, experimental results are obtained for the spectral extinction coef®cient from 2.5 to 25 lm. In addition, the P-3 approximation method along with the box model is employed to calculate the non-gray radiative heat ¯ux. The diffusion approximation method is also applied to calculated the radiative conductivity. Also tested herein are three samples with different cell sizes ranging from 330 to 147 lm. According to those results, the spectral extinction coef®cient increases with a decrease of cell size, leading to a decrease of thermal conductivity. Moreover, evacuating the gases in the foam cells can re-duce the thermal conductivity of the PU foam by as much as 75%. Furthermore, radiative heat transfer accounts for about 4% of total heat transfer at 760 Torr and increases to 20% at 0.02 Torr.

List of symbols

fv solid volume fraction

Ibk spectral blackbody intensity, [W/(m2lm)] kc solid conductivity, [W/mK]

kr radiative conductivity, [W/mK] L thickness of the medium, [m] qc _{conductive heat ¯ux, [W/m}2_] qr _{radiative heat ¯ux, [W/m}2_] qt _{total heat ¯ux, [W/m}2_]

T1,2 temperature of hot wall & cold wall, [K] x coordinate Greek symbols k wavelength, [lm] e1;2 wall emissivity jk extinction coef®cient, [m)1] w0 incident radiation

w1 radiative heat ¯ux

w2 second moment of incident radiation w3 third moment of incident radiation r Stefan-Boltzmann constant, 5.667 ´ 10)3_[W/(m2_K4_)] Subscripts b blackbody c conductive r radiative

1,2 hot wall and cold wall 1

Introduction

Rigid polyurethane foam, consisting of a highly porous but solid body with a cellular structure, has found diverse applications: from buildings to refrigerators. Heat trans-port occurs via gaseous and solid thermal conduction as well as radiation. Radiative heat transfer profoundly in-¯uences the design of many engineering systems, partic-ularly in predicting the heat transfer rates through an absorbing, emitting, and scattering material. However, realistic predictions, even in extremely simple geometrical systems, are dif®cult to obtain owing to the frequency and temperature dependent radiative properties of the media.

Thermal insulation has received extensive interest in recent decades. Aronson et al. [1] estimated the absorption and scattering coef®cients using the geometrical optics theory and the Rayleigh approximation by separately considering coarse and ®ne ®bers. In a related work, Tong and Tien [2] demonstrated extent to which thermal radi-ation in¯uences in ®brous insulradi-ation. That investigradi-ation also reviewed the analytical models capable of predicting radiative heat transfer in ®brous media based on a simple conductive and more elaborate radiative approach. Later, Tong and Tien [3] and Tong et al. [4] analytically and experimentally studied radiative heat transfer in light-weight ®brous insulation. In their investigations, two-¯ux and linear anisotropic scattering models were used to predict the radiant heat ¯ux; the radiative properties of ®brous insulation were calculated as well. For this pur-pose, they examined the radiative heat transfer in this anisotropically scattering material using an infrared spectrophotometer and a guarded hot-plate apparatus.

Received on 20 April 1998 J.-W. Wu, H.-S. Chu

Department Mechanical Engineering National Chiao Tung University Hsinchu, Taiwan, Republic of China Correspondence to: H.-S. Chu

The authors would like to thank the National Science Council of the Republic of China for ®nancially supporting this research under Contract No. NSC86-2221-E-009-047. The National Center for High-Performance Computing (NCHC) is appreciated for performing the numerical calculations.

Originals Heat and Mass Transfer 34 (1998) 247±254 Ó Springer-Verlag 1998

Another method attempts to reduce gas conduction through the porous insulation by use of ultra-®ne powders with diameters of the order of 10 nm. In pioneering work, Scheuerp¯ug et al. [5] investigated the consolidated ultra-®ne powder insulation in the form of rigid SiO2-aerogel tiles for double-plane window application. Chu et al. [6] systematically studied, for the ®rst time, the spectral ra-diation heat transfer through the ultra-®ne powder insu-lation Aerosil 380 with particle diameters close to 7 nm. Gas conduction, which is characterized by the molecular mean free path, can be signi®cantly reduced if the pore size between solid particles is less than the mean free path. In a related work, Heinemann et al. [7] theoretically and experimentally investigated the thermal transport in low density silica aerogel over a wide range of optical thickness and ratio of radiative to conductive heat transfer. That investigation also proposed a high precision numerical method to calculate the temperature pro®le and the total heat ¯ux in these semi-transparent, scattering, non-gray media.

The heat transfer modes in polyurethane foams are determined by gaseous conduction within the pores, conduction via the solid structures of the foams, and by radiative transfer. Among the related numerical and ex-perimental studies: Glicksman et al. [8] concentrated on the radiative contribution towards heat transfer in foam. Foams scatter radiation due to the interaction of the ra-diation with struts and walls such that radiative transfer can be modeled as a diffusion process. Kuhn et al. [9] thoroughly investigated polystyrene (PS) and a polyure-thane (PU) foams and the contribution of each thermal transfer modes. In particular, they theoretically and ex-perimentally studied radiative transfer. In a similar work, Doermann and Sacadura [10] presented a predictive model for thermal transfer in open-cell foam insulation as a function of foam morphology, porosity, thermal properties of solid and gaseous phases, and optical properties of the solid phase. Caps et al. [11] recently measured the thermal conductivity of polyimide foams in the temperature range of 173±323 K under different gas pressures and using different gas types (CO2and Ar). Their investigation also developed a quantitative model to accurately predict the thermal conductivity of polyimide foam as a function of density, gas pressure and temperature. Hahn et al. [12] presented a model to calculate the transient combined radiative/conductive heat transfer in heterogeneous semi-transparent materials at elevated temperatures. Those investigators also applied the three ¯ux model to solve the integrol-differential equation for an 8-band-simulation. Tseng et al. [13] more recently investigated the thermal conductivity of polyurethane foam in the temperature range between 300 and 20 K for the development of liquid hydrogen storage tanks. In general, heat transfer occurs in foam by natural convection, gas conduction, solid-solid conduction and radiation. In addition, the dominant heat transfer modes in foam insulation are thermal radiation and solid-solid conduction if the foam insulation systems are evacuated. Moreover, a low apparent thermal con-ductivity can be achieved in an evacuated volume because the gas conductive and convective heat transfer modes do not exist in a vacuum.

This study investigates the heat transfer of evacuated open cell polyurethane foam with the cell size ranging from 330 to 147 lm. This study focuses primarily on de-termining the heat transfer mode in this system and pre-dicting the radiative thermal conductivity. A Fourier transform infrared spectrometer is experimentally used to measure the spectral extinction coef®cient in the wave-length range of 2.5 to 25 lm. In addition, effective thermal conductivity is measured with a guarded-hot-plate system. The albedo developed by Kuhn et al. [9] is employed for the analysis. The stepwise gray or box model [14] is also applied to incorporate the effects of the non-gray charac-teristics. Moreover, the radiative heat transfer is calculated using the P-3 approximation, which has been demon-strated to effectively generate accurate approximate solu-tions to the gray problem [15].

2

Theoretical analysis

Consider a non-gray absorbing, emitting and isotropically scattering polyurethane foam that transfers energy through parallel plates as depicted in Fig. 1. Both plates have the same wall emissivity and are maintained constant at different wall temperature. The left bounding plate is hotter than that of the right one. The physical properties of the foam are assumed to be independent of temperature, particularly the evacuated solid thermal conductivity. Under these conditions, the energy equation can be ex-pressed as kcd 2_T dx2 ÿ dqr dx ˆ 0 …1†

associated with the following boundary conditions:

T…0† ˆ T1; T…L† ˆ T2; …2†

The radiative heat ¯ux qr_{can be obtained by integrating of} radiative intensity I, which can be determined by solving the equation of transfer. Herein, the P-3 approximation is used to calculate radiative heat transfer. The spectral equations for the P-3 approximation [16] can be written as:

Fig. 1. Physical model and coordinate system

dw0k dx ˆ ÿ3jk…1 ÿ xf1†w1k‡ 14 3 jk…1 ÿ xf3†w3k ; …3a† dw1k dx ˆ 4pjk…1 ÿ x†Ibkÿ jk…1 ÿ xf0†w0k ; …3b† dw2k dx ˆ ÿ73…1 ÿ xf3†w3k ; …3c† dw_3k dx ˆ ÿ 8 3p…1 ÿ x†Ibk‡ 2 3…1 ÿ xf0†w0k ÿ5₃…1 ÿ xf2†w2k ; …3d†

where the value of f1,2,3depend on the type of scattering being modeled and by de®nition, w0…x† and w1…x† denoted the incident radiation and the radiative heat ¯ux, respec-tively. In addition, w2…x† and w3…x† represent the second and third moment of incident radiation respectively. Moreover, Ibk is the blackbody intensity. The boundary conditions for Eqs. (3a) to (3d) are

at x ˆ 0, e1w0k‡ 2…2 ÿ e1†w1k‡₄5e1w2k ˆ 4pe1Ibk; e1w0k‡12₅ …2 ÿ e1†w1k‡5₂e1w2k‡8₅…2 ÿ e1†w3k ˆ 4pe1Ibk ; …4a† at x ˆ L, e2w0kÿ 2…2 ÿ e2†w1k‡₄5e2w2kˆ 4pe2Ibk; e2w0kÿ12₅ …2 ÿ e2†w1k‡5₂e2w2kÿ8₅…2 ÿ e2†w3k ˆ 4pe2Ibk ; …4b†

where e1;2denotes the hemispherical emissivity at the wall for x ˆ 0, and x ˆ L, respectively. Simultaneously solving Eqs. (3a)±(3d) using the BVPFD subroutine of a com-mercially available software package called IMSL allows us to obtain the incident radiation w0k deemed necessary for the energy equation and the radiative heat ¯ux qr_requisite for heat-transfer evaluation. Herein, P-3 approximation is applied to non-gray radiation, by dividing the participat-ing bands into a number of subbands. The extinction co-ef®cient of each subband is assumed to remain constant, as solved by using box model. By using the radiation in-tensity calculated in Eqs. (3a)±(3d), the derivative of the radiative heat ¯ux can be written as

r
qr_ˆXN iˆ1

jki‰4p…1 ÿ x†Ibkiÿ …1 ÿ xf0†w0kiŠ
…5† The total heat ¯ux across the PU foam is calculated from qt _{ˆ ÿk}

cdT_dxÿ krdT_dx …6†

where kr represents the radiative thermal conductivity Equations (1), (3a±3d) and (5) are solved using the ®-nite-difference method to determine the radiative heat ¯ux

and temperature distribution. The solution procedure is outlined as follows:

1. Guess the temperature and intensity distributions; 2. Solve the energy equation by a point-by-point

itera-tive method to obtain a new temperature distribution. The conductive heat ¯ux can be obtained from the temperature distribution;

3. With the guessed intensity ®elds and the new tem-perature, solve the spectral equations to obtain the new intensity distribution and radiative heat ¯ux; and 4. Repeat steps (2) and (3), until the convergence

cri-terion for heat ¯ux on each point is satis®ed. The convergence criterion for all calculations is set at

Tnew i ÿ Tiold

  _{ 1  10}ÿ7 _7†

to satisfy all mesh points. This iterative procedure is continued until the convergence criterion is reached. 3

Extinction coefficient measurement

Samples with cell sizes of 330, 214 and 147 lm are marked as ``sample A'', ``sample B'', and ``sample C'', respectively. In this study, the direct transmission measurements are taken in the wavelength range from 2.5 to 25 lm using a Fourier transform infrared spectrometer (Perkin-Elmer Spectrum 2000). Each sample is measured after removing the moisture effects. In addition, transmission data are used to calculate the spectral extinction coef®cient jk, according to Beer's law, Ik…x†=I0kˆ exp…ÿjkx†. Figure 2 presents the extinction coef®cients jkof samples A, B, and C. According to this ®gure, the extinction coef®cient increases with a decrease of the cell size.

Fig. 2. Speci®c absorption coef®cient for three different cell sizes

4

Results and discussion

This study veri®es the in¯uence of grid size on the nu-merical results by performing a grid-independence test for ®ve different grid sizes. Table 1 summarizes the results of temperature, heat ¯uxes (at middle plane) and cpu time. This table reveals that the temperature and heat ¯uxes only slightly differ between grids 51 and 71. Hereinafter, the uniform grid system of 51 is selected for all the calcula-tions. Next, an attempt is made to demonstrate that the number of spectral bands much represent the spectral variation of extinction coef®cient with suf®cient accuracy. Table 2 compares the temperature and heat ¯uxes (at middle plane) results for different band numbers. Ac-cording to this table, variations in temperature and heat ¯ux are only slight when the band number exceeds 30. Therefore, thirty bands of extinction coef®cients between 2.5 to 25 lm are used to calculate the non-gray radiation heat transfer, as described in the following.

Table 3 displays the effective thermal conductivities of three samples measured by a guarded-hot-plate system. The fact that the radiative conductivity is independent of evacuation pressure allows us to obtain the conductive conductivity by substracting the radiative conductivity from measured effective thermal conductivity. In addition, according to our results, the variation of effective thermal conductivity becomes only slight when the pressure falls below than 0.02 Torr. Hence, the gaseous heat transfer can

be reasonably neglected when the evacuation pressure is 0.02 Torr. To produce the sample in this study, the foam is put into the laminated ®lm bag and was evacuated to low pressure of 0.02 Torr. Figure 3 illustrates how cell size in¯uences the heat ¯ux at T1ˆ 301 K, T2ˆ 271 K, e1ˆ 0:1, e2ˆ 0:1, L ˆ 0:01 m, and at an evacuation pres-sure of 0.02 Torr. The differences in cell size are attributed to the in¯uence of the open cell walls and the struts having formed at the junctions of the cell walls. Such an in¯uence is owing to that the extinction coef®cient decreases with an increase of the cell size; the radiative heat ¯ux increases with an increase of cell size as well. According to our re-sults, the conductive heat ¯ux also exhibits the same trend.

Table 4 compares the radiative and conductive heat ¯uxes (at mid-plane) for three samples at evacuation pressures ranging from 760 to 0.02 Torr. As expected, the radiative heat ¯uxes only negligibly differ for the different pressure conditions of the three samples were small. This table also reveals that the conductive heat ¯ux of sample A at 760 Torr is around ®ve times larger than that at evac-uation pressure 0.02 Torr. Based on above results, we can conclude that gaseous heat transfer is the dominant heat transfer mode at atmospheric condition. As mentioned earlier the gaseous heat transfer can be reasonable

ne-Table 1. Comparison of the results for different grid sizes of sample A at x ˆ 0:005 m

Grid no. Temp. (K) qr_(W/m2_{) q}c_(W/m2_{) Cpu time (min.)}

31 286.318 5.2359 22.6932 11.31

51 286.301 5.2108 22.5890 42.07

71 286.288 5.2032 22.5853 91.73

91 286.243 5.1980 22.5487 124.87

111 286.251 5.1965 22.5401 227.84

Table 2. Comparison of the results for different band numbers of sample A at x ˆ 0:005 m

Band no. Temp. (K) qr_(W/m2_{) q}c_(W/m2₎

10 286.30 5.1943 22.5901

20 286.30 5.2062 22.5786

30 286.30 5.2108 22.5890

40 286.29 5.2118 22.5907

50 286.30 5.2102 22.5786

Table 3. Summary of experi-mental data for three different samples at Tmˆ 286 K Parameter Sample A B C fv 0.037 0.038 0.042 Cell size (lm) 330 214 147 760 Torr keff(mW/mK) 34.2 33.87 33.4 9 Torr keff(mW/mK) 33.4 32.5 31.3 1 Torr keff(mW/mK) 23.5 24.94 13.9 0.1 Torr keff(mW/mK) 22 16.7 12.7 0.02 Torr keff(mW/mK) 9.5 8.35 7.3 0.014 Torr keff(mW/mK) 9.4 8.2 7.2

Fig. 3. Effects of cell-size on heat ¯ux for three different cell sizes with T1ˆ 301 K, T2ˆ 271 K, e1ˆ 0:1, e2ˆ 0:1, L ˆ 0:01 m, and

an evacuation pressure of 0.02 Torr

glected when the evacuation pressure is 0.02 Torr. In ad-dition, the conduction contribution can be regarded as solid conduction only at 0.02 Torr. The relative contri-butions of solid, gaseous and radiative modes to the total heat transfer at evacuation pressure 760 Torr can be pre-dicted to be 22.2%, 72.6% and 5.2%, respectively. With a decreasing pressure, the radiative contribution increases and contributes about 18.7% of the total heat transfer when the pressure is 0.02 Torr. Table 5 compares the values of the radiative and conductive heat ¯uxes (at middle-plane) for three different samples with different mean temperatures at an evacuation pressure of 0.02 Torr. In general, radiative heat transfer becomes even more critical at higher temperatures. By raising the mean tem-perature from 260 to 356 K, an increase in the radiative contribution to total heat transfer in sample A is raised from 14.2% to 31.9%. When the mean temperature is ®xed, the relative radiative contribution increases with an in-crease of cell size. Table 6 compares the effects of thick-ness on the values of radiative and conductive heat ¯uxes (at middle-plane) for three samples. This table also reveals that heat ¯uxes linearly decrease with an increase of thickness.

Figure 4 illustrates the effects of evacuation pressure on radiative heat ¯ux for sample A and C. For the gray ra-diative heat transfer, the extinction coef®cient is taken as one band averaged value by the box model. The values of extinction coef®cient for sample A and sample C are 3757.98 m)1_{and 8585.37 m})1_{, respectively. At an} evacua-tion pressure of 760 and 0.02 Torr, the gray and non-gray methods appear to signi®cantly differ because the gains in

radiant energy due to radiation emitted by the gray me-dium are larger than those of the non-gray one. For sim-plicity, foam is usually considered a gray body with a constant value of extinction coef®cient over all wavelength. In this case, the gray radiation method overestimates the

Table 4. Comparison of the heat ¯uxes in three different samples at x ˆ 0:005 m

Pressure (Torr) Sample A Sample B Sample C

qr_(W/m2_{) q}c_(W/m2_{) q}r_(W/m2_{) q}c_(W/m2_{) q}r_(W/m2_{) q}c_(W/m2₎ 760 5.2268 96.6616 3.3244 97.9124 2.2378 97.7634 9 5.2264 94.2523 3.3244 94.0843 2.2378 91.4619 1 5.2251 64.5697 3.3244 71.1156 2.2401 39.2693 0.1 5.2249 60.0744 3.3243 46.4067 2.2400 35.6621 0.02 5.2103 22.5872 3.3236 21.3495 2.2429 19.4634

Table 5. Comparison of the heat ¯uxes in three different samples at x ˆ 0:005 m

Tm(K) Sample A Sample B Sample C

qr_(W/m2_{) q}c_(W/m2_{) q}r_(W/m2_{) q}c_(W/m2_{) q}r_(W/m2_{) q}c_(W/m2₎

260 3.4470 20.5825 2.1743 19.3759 1.4655 17.6471

286 5.2103 22.5872 3.3236 21.3495 2.2429 19.4634

318 8.1912 25.0224 5.2573 23.7380 3.5558 21.6664

356 13.1091 27.9332 8.2862 26.3951 5.6414 24.2120

Table 6. Comparison of the heat ¯uxes in three different samples at x ˆ 0:005 m

Thickness (m) Sample A Sample B Sample C

qr_(W/m2_{) q}c_(W/m2_{) q}r_(W/m2_{) q}c_(W/m2_{) q}r_(W/m2_{) q}c_(W/m2₎

1.0 0.0525 0.2277 0.0333 0.2140 0.0224 0.1944

0.1 0.5249 2.2769 0.3331 2.4792 0.2241 1.9444

0.05 1.0493 4.5519 0.6662 4.2792 0.4481 3.8885

0.01 5.2103 22.5872 3.3236 21.3495 2.2429 19.4634

Fig. 4. Effects of evacuation pressure on radiative heat ¯ux with T1ˆ 301 K, T2ˆ 271 K, e1ˆ 0:1, e2ˆ 0:1, and L ˆ 0:01 m

radiative heat ¯ux by around 10.5% because the extinction coef®cient is underestimated by using a constant value. The radiative heat ¯ux for sample A is larger than sample C because the extinction coef®cient of sample A is much smaller than sample C. Figure 5 depicts how evacuation pressure in¯uences radiative conductivity, as de®ned by Eq. (6). As mentioned earlier, the radiative conductivities are nearly independent of evacuated pressure. For sample A, the radiative conductivity by diffusion method overes-timated around 12% and the gray radiation overesoveres-timated around 7.6%, respectively.

Figure 6 presents how mean temperature in¯uences radiative heat ¯ux at an evacuation pressure of 0.02 Torr. The differences between sample A and sample C increase with an increase of mean temperature. However, the non-gray effects are only slight. Figure 7 displays the mean temperature on radiative conductivity. For higher tem-peratures, the radiative conductivity becomes even more prominent than the solid thermal conductivity. According to this ®gure, the pro®les of gray radiation and diffusion methods tend to merge together with a decrease of the mean temperature. On the other hand, the pro®les of gray and non-gray radiation methods tend to accumulate to-gether with an increase of mean temperature. In sum, the relative error between the diffusion method and non-gray radiation method increases from 3% to 58% with a decrease of the mean temperature from 356 to 156 K. Therefore, the gray radiation and diffusion methods cannot accurately predict the radiative conductivity at a low mean temperature.

Figure 8 illustrates how thickness in¯uences radiative heat ¯ux at an evacuation pressure of 0.02 Torr. Appar-ently, the radiative heat ¯ux calculated by gray method is approximately 10.4% larger than that by non-gray method for L ˆ 0:01 m and 13.1% for L ˆ 0:05 m. This observa-tion is owing to that a larger thickness not only prevents the propagation of radiation emitting and scattering for-ward but also prevents the conductive heat transfer. Figure 9 depicts how thickness in¯uences radiative con-ductivity. According to this ®gure, the radiative conduc-tivity calculated by three methods is initially maintained constant with a decrease of the specimen's thickness. For sample A, the radiative conductivity calculated by diffu-sion method is around 9.4% larger than the non-gray method and 8% larger than by gray radiation, respectively.

Fig. 5. Effects of evacuation pressure on radiative conductivity with T1ˆ 301 K, T2ˆ 271 K, e1ˆ 0:1, e2ˆ 0:1, and L ˆ 0:01 m

Fig. 6. Effects of mean temperature on radiative heat ¯ux with e1ˆ 0:1, e2ˆ 0:1, L ˆ 0:01 m, and an evacuation pressure of

0.02 Torr

Fig. 7. Effects of mean temperature on radiative conductivity with e1ˆ 0:1, e2ˆ 0:1, L ˆ 0:01 m, and an evacuation pressure of

0.02 Torr

Interestingly, a decrease of the specimen's thickness de-creased to a critical value causes a decrease of the radiative conductivity. With a decrease of the specimen's thickness, the heat transfer mode is transformed from optical thick media to optical thin media. For the optical thin case, the conduction tends to be signi®cant. Moreover, the radiative conductivity by diffusion method does not display the same tendency as the other two methods.

5

Conclusions

This study investigated the thermal performance of evac-uated polyurethane foams with three different cell sizes. Samples were designed as 100% open cell rigid bodies. The box model was used to incorporate the characteristics of non-gray bands and coupled with the P-3 approximation method for calculating radiative heat transfer. The energy equation was solved using fully implicit ®nite-difference method. Experimental results indicated that solid and ra-diative contributions are independent of pressure, thereby allowing us to obtain gaseous conductivity at higher pressures by subtracting this measured result from the total heat transfer. In addition, the spectral extinction coef®cient decreased with an increase of cell size; the ra-diative conductivity increased with an increase of cell size as well. Moreover, the radiative conductivity calculated by gray radiation method was around 8% higher than that by non-gray method, but nearly 58% higher in a low mean temperature case. On the other hand, the radiative con-ductivity calculated by diffusion method was approxi-mately 17% higher than that by non-gray method for the optical thin case and 58% higher for the low mean tem-perature case.

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pressure of 0.02 Torr

Fig. 9. Effects of media thickness on radiative conductivity with T1ˆ 301 K, T2ˆ 271 K, e1ˆ 0:1, e2ˆ 0:1, and an evacuation

pressure of 0.02 Torr

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