Energy Conversion and Management 48 (2007) 481–485 www.elsevier.com/locate/enconman

Thermal expansion of polyurethane foam at low temperature C.G. Yang *, L. Xu, N. Chen Institute of Refrigeration and Cryogenics Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 13 December 2005; accepted 26 June 2006 Available online 26 September 2006

Abstract The mean linear thermal expansion coefficient of a newly developed polyurethane foam with a blowing agent of HFC245fa was measured from the liquid nitrogen temperature to room temperature by means of a relative comparison method in this paper. An automatic temperature measuring system, a vacuum system and an electrical heating system were adopted in the experiment. The relative comparison method has been analyzed, and according to the Kelvin model, a model describing thermal expansion of the foams was introduced. A very low thermal expansion with a slowly increasing trend versus temperature rising from 77 K to 230 K was shown in the result. In addition, it was indicated that the thermal expansion rate along a direction perpendicular to the pore rise is a bit higher than that along a parallel direction.  2006 Elsevier Ltd. All rights reserved. Keywords: Polyurethane foam; Low temperature; Thermal expansion; Measurement

1. Introduction Polyurethane (PU) foam has many properties suitable for thermal insulation such as low density, low thermal conductivity, low cost and high strength to weight ratio, all of which make it a useful insulating material in refrigerated vehicles, vessels for refrigerated cargo, pipelines, liquid gas tanks for LNG and LPG and cryogenic wind tunnels [1]. As an important parameter, the thermal expansion of PU foam has great significance in terms of thermal insulation design. Under low temperature, the thermal stress triggered by a significantly different expansion rate between PU foam and stainless steel will cause the foam to crack and damage the insulation layer. As a result, the thermal insulation performance will turn bad. Different techniques are available to measure the thermal expansion of PU foam. All these techniques can be divided into two categories. The first one is an absolute method and the other is a relative method. In the former, linear changes of the samples are directly measured at var-

*

Corresponding author. Tel./fax: +86 21 629 33 251. E-mail address: [email protected] (C.G. Yang).

0196-8904/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.06.016

ious temperatures, while in the latter, thermal expansion is determined by comparison with a known thermal expansion rate [2,3]. All these methods are adapted to different standards [4,5]. In this paper, the relative method was used and a military standard was adopted to measure the mean linear thermal expansion coefficient of PU foam [6]. 2. Experimental 2.1. Sample The PU foam has been newly developed to meet the need of heat insulation systems for carrier rockets. HFC245fa was used as a substitute to replace CFCs, which have high ozone depletion potential (ODP). Some properties and a scanning electronic microscope (SEM) picture of the newly developed foam are shown in Table 1 and Fig. 1, respectively. 2.2. Experimental setup The dilatometer used for the experiment has many advantages including automatic control, convenient operation and high precision etc. (see Fig. 2).

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Table 1 Some properties of the sample Properties

Test direction

Data

Parallel to pore rise Parallel to pore rise Parallel to pore rise

40 ± 2 95% 2–3 60.19 P0.278 0.67 P0.430

3

Density (kg/m ) Closed cell rate (%) Self extinguishing (S) Water absorption rate (kg/m2) Compressive strength (N/mm2) Poisson ratio Tensile strength (N/mm2)

avoided. In addition, helium gas has a very small, density, so the sample was less influenced by the gas weight. (3) Electrical heating system. The electrical heating system consisted of a thin copper tube, manganin heating wire (0.2 mm in diameter), thin copper wire and a direct current source (30 V, 0.5 A). The manganin heating wire was coiled bifilarly around the copper tube. The copper wire was connected with the direct current source through an O-ring under a glass cover. The temperature was controlled by the power from the direct current source [7]. (4) Vacuum system and low temperature environment. In order to obtain the mean linear thermal expansion coefficient of PU foam in a wide temperature range, the sample was required to be heated from 77 K to 293 K. In addition, to avoid a frost phenomenon on the surface of the sample under low temperatures, a protective tube was separated from the liquid nitrogen by a stainless steel case. The protective tube and the stainless steel case were evacuated by a vacuum pump, and then, helium gas with a pressure of 0.01 Pa was charged into the protective tube. Liquid nitrogen could ensure the low temperature environment. The increment in the sample length was read by a micrometer. The temperature fluctuation was controlled to less than 0.4 K difference. 3. Theory and model

Fig. 1. SEM of the sample foam.

1

at ¼

14 2 3 13 12

4 5 6 7 8 9 10 11

The linear thermal expansion coefficient of a solid at is defined as

1. glass cover 2. O-ring 3.flange 4. evacuation nib 5. push rod 6. protective tube 7. liquid nitrogen Dewar 8. sample 9. heater 10. stainless steel case 11. supporter 12. liquid nitrogen inlet 13. evacuation nib 14. micrometer

Fig. 2. Schematic diagram for the measurement of thermal expansion.

1 dL L0 dT

ð1Þ

where L0 is the original length. The entire length change between two temperatures is a function of at, target temperature T, initial temperature T0 and original length of the sample L0. The following equation can be obtained: Z T  Z T LT at dT ¼ ln or LT ¼ L0 exp at dT ð2Þ L0 T0 T0 If the linear thermal expansion coefficient is constant and is a stable function of the temperature, at can be defined as the mean linear thermal expansion coefficient am, so the equation is LT  L0 ½1 þ am ðT  T 0 Þ

ð3Þ

or (1) Material of push rod and protective tube. Among nonmetal materials, quartz has a very small thermal expansion and good performance in dimensional stabilization, so it has been used to make the push rod and protective tube in order to ensure a precise result. (2) Gas environment. The sample was in a helium environment through which heat could be transferred from the heater to the sample and, consequently, a negative impact on the sample, triggered by condensed water, was

DL  am ðT  T 0 Þ ¼ am DT L0 where DL = LT  L0 and DT = T  T0. In the experiment, 293 K was regarded as the initial parameter (0 state). According to the Kelvin model [8], which was used to calculate the foam compressive Young’s modulus, some conclusions can be obtained in the system of the closed cell

C.G. Yang et al. / Energy Conversion and Management 48 (2007) 481–485

foam. The total tensile strain e in each cell face, triggered by the change of temperature DT = T0  T, consists of two parts: eh and eE, which are the elastic and thermal strain, respectively. e ¼ eh þ eE

ð4Þ

eh ¼ ap DT

ð5Þ

Assuming the polymer is linearly elastic, eE is rf eE ¼ ð1  vÞ E

ð6Þ

where ap is the linear thermal expansion coefficient of the polymer; rf is the biaxial tensile stress; v is Poisson’s ratio and E is Young’s modulus. Considering the detailed structure of a unit cell in the Kelvin model, rf can be described as follows: rf ¼

3ðpr  pÞ 2R

ð7Þ

where pr is the relative pressure, p is external pressure and R is the foam relative density, which is defined as the foam density divided by the polymer density. Taking the definition of the thermal expansion coefficient and combining Eqs. (4)–(7), the following equation can be written: af ¼

e 3p ð1  vÞ ¼ ap þ r DT 2ERDT

ð8Þ

where af is the linear thermal expansion coefficient of the foam.

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dard literature [6]. The maximum difference is 6.2%. Additionally, repetition of the data is very clear. 5. Discussion 5.1. Results analysis The thermal expansion along three directions was measured. Sample dimensions for three directions were (X · Y · Z, unit: mm): 25.10 · 8.24 · 7.86, 7.58 · 24.40 · 7.10 and 7.52 · 8.63 · 24.86. X and Y were parallel and perpendicular to the direction of pore rise, respectively. The elliptical shape of the cell structure leads to anisotropy and produces a different effect on thermal expansion along the various directions. The mean linear thermal expansion coefficient and relative length changes of the three directions are shown in Figs. 3–5, respectively. During the process of the experiment, no obvious anomaly of the thermal expansion was observed. It can be seen in Fig. 6 that the thermal expansion along the direction perpendicular to the pore rise is a bit higher than that along the parallel direction. Compared with PVC or nylon, PU foam has a lower linear thermal expansion [9,10], which is not higher than that of phenolic foam and glass fibre reinforced nylon. The lower thermal expansions with negligible differences along the various directions justify that PU foam has good dimensional stability when used in low temperatures.

4. Confirmatory experiment of the equipment

8 relative length change mean linear thermal expansion coefficient

60

7

50

(dL/L0) × 104

6

40 5

30 4

20

αm × 105 K-1

In order to ensure the reliability of the experiment, a blank experiment and a confirmatory test of oxygen free copper was examined, and the results are shown in Table 2. (1) Blank experiment. During the process of temperature decreasing, the differences of the length changes between the protective tube and the quartz rod were within 0.5 lm. This could justify that they had an approximately equal value of contraction. (2) Confirmatory test of oxygen free copper. The dimension of the oxygen free copper was /6.50 mm · 52.38 mm. Repetitive experiments were done, and the results were compared with the standard values. In Table 2, it is clear that the thermal expansion of oxygen free copper obtained from the experiment is close to that mentioned in the stan-

3

10

0 60

80

2 100 120 140 160 180 200 220 240 260 280 300

Temperature / K Fig. 3. Results along X-direction versus temperature.

Table 2 Repetitive experiment data of oxygen free copper LT L273 L273

 100

The first test The second test The third test Mean value Standard value Deviation of mean value from standard value/% Maximum deviation from standard value/%

T/K 78

123

173

223

273

298

323

373

0.2842 0.2834 0.2857 0.2844 0.2696 5.5 6.0

0.2314 0.2367 0.2324 0.2335 0.2228 4.8 6.2

0.1598 0.1623 0.1614 0.1612 0.1555 3.6 4.3

0.0827 0.0821 0.0812 0.0820 0.0801 2.3 3.2

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0408 0.0417 0.0404 0.0410 0.0414 0.9 2.4

0.0857 0.0864 0.0832 0.0851 0.0836 1.8 3.3

0.1676 0.1786 0.1752 0.1738 0.1699 2.3 5.1

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C.G. Yang et al. / Energy Conversion and Management 48 (2007) 481–485 80

8 relative length change mean linear thermal expansion coefficient

70

7 60

40

5

30 4

αm × 105 K-1

(dL/L0) × 104

6 50

case, the different batches of PU foam must be tested, respectively, in order to get accurate data. Thermal expansion of stainless steel was also measured in the experiment, which is shown in Fig. 7. Stainless steel has far less thermal expansion than that of PU foam. The different dimension changes between stainless steel and an insulation layer made of PU foam must be considered during heat insulation design to avoid foam cracking. 5.2. Error analysis

20 3 10 0 60

80

100 120 140 160

As for a single experiment, the quadratic equation is often adopted to estimate error [11]. Assume y is a function of n independent variables, i.e. y = f(x1, x2, . . . , xn). Then, the uncertainty dy of y can be shown through the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 ffi dy oy oy oy ¼ ð9Þ dx21 þ dx22 þ    dx2n y ox1 ox2 oxn

2 180 200 220 240 260 280 300

Temperature / K Fig. 4. Results along Y-direction versus temperature.

80

8

relative length change mean linear thermal expansion coefficient

70

7

60

(dL/L0) × 104

5

40 30

αm × 105 K-1

6 50

4 20

where daa is the relative error of the mean linear thermal 0 expansion coefficient. d is the error of each variable. dL L0 dðdLÞ and dL represent the errors owing to the measurements Þ of length and its increment of the sample. dðdT consists of dT errors induced by temperature measurement, as well as the temperature distribution in the sample. In the experiment, the vernier caliper had a smallest reading of 0.02 mm. The minimum length of the samples was 24.4 mm. That is to say: dL0 = ±0.02 mm and L0 = 24.4 mm. The accuracy of the micrometer used to measure the length increment of the sample was ± 0.001 mm and the minimum length increment was 1.6 · 102 mm. Thus,

3 10 0

60

80

100

120

140

160

180

200

220

240

260

2 280 300

Temperature / K

Fig. 5. Results along Z-direction versus temperature.

8 relative length change along direction perpendicular to pore rise relative length change along direction parallel to pore rise

7

70 60 50 40

5

30 4

60

(dL/L0) × 104

αm × 105 K-1

6

where dx1, dx2, . . . , dxn are the uncertainties of the independent variables x1,x2, . . . , xn, respectively. Consequently, the relative error of the thermal expansion coefficient can be described as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2ffi  2  da dL0 dðdLÞ dðdT Þ ¼ ð10Þ þ þ a dL dT L0

50

20 10 0

2 60

80

100 120 140 160 180 200 220 240 260 280 300 320

T/K

Fig. 6. Difference between parallel direction and perpendicular direction.

In addition, due to the diversities of amount, distribution, dimension, morphology and porosity of cells in PU foam, different batches always have different thermal expansions. The results sometimes vary by 50–60%. In this

40

(dL/L0) × 104

mean linear thermal expansion coefficient along direction parallel to pore rise mean linear thermal expansion coefficient along direction perpendicular to pore rise

0.8

relative length change mean linear thermal expansion coefficient 0.6

30 0.4

αm × 105 K-1

3

1.0

20 0.2

10

0 50

100

150

200

250

0.0 300

Temperature / K

Fig. 7. Thermal expansion of stainless steel at low temperature.

C.G. Yang et al. / Energy Conversion and Management 48 (2007) 481–485

d(dL) = ±0.001 mm, dL = 1.6 · 102 mm. The accuracy of each thermocouple was ±0.5 K. During the process of the experiment, d(dT) = 0.5 K and dT=10 K. The relative error of the linear thermal expansion coefficient is as follows: da ¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2 2  0:02 0:001 0:5 þ þ 24:40 10 1:6  102

¼ 8:0% References [1] Anton Demharte. Polyurethane rigid foam, a proven thermal insulating material for applications between +130 C and 196 C. Cryogenics 1998;38(1):113–7. [2] Kanagaraj S, Pattanayak S. Measurement of the thermal expansion of metal and FRPs. Cryogenics 2003;43:399–424.

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[3] Ribeiro MCS, Reis JML, Ferreira AJM, Marques AT. Thermal expansion of epoxy and polyester polymer mortars—plain mortars and fibre-reinforced mortars. Polym Test 2003;22:849–57. [4] ASTM E 228. Test method for linear thermal expansion of solid materials with a vitreous silica dilatometer. Annual book of ASTM standards; 1989. p. 14.02. [5] ASTM C 531. Standard test method for linear shrinkage and coefficient of thermal expansion of chemical-resistant mortar-grouts, monholitic surfacing and polymer concretes. Annual book of ASTM standards; 2000. p. 04.05. [6] Military Standard GJB 1875-94. Test methods for the thermophysical properties of rigid cellular plastics, PR China. [7] Deng DQ, Xu L. Measurement of thermal expansion coefficient of phenolic foam at low temperatures. Cryogenics 2003;43:465–8. [8] Mills NJ, Zhu HX. J Mech Phys Solids 1999;47:669–95. [9] Titow WV. PVC technology. 4 ed. London; 1984. [10] Barucci M, Bianchini G, Del Rosso T. Thermal expansion and thermal conductivity of glass-fibre reinforced nylon at low temperature. Cryogenics 2000;40:465–7. [11] Kline SJ et al. Describing uncertainty in single-sample experiment [J]. Mech Eng 1953;75(1):3–8.