Ind. Eng. Chem. Res. 2006, 45, 6153-6161
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A Microcellular Foaming Simulation System with a High Pressure-Drop Rate Qingping Guo, Jin Wang, and Chul B. Park* Microcellular Plastics Manufacturing Laboratory, Department of Mechanical and Industrial Engineering, UniVersity of Toronto, Toronto, M5S 3G8, Ontario, Canada
Masahiro Ohshima Department of Chemical Engineering, Kyoto UniVersity, Kyoto 606-8501, Japan
In this paper, we undertook an experimental and theoretical analysis of the pressure-drop behaviors of a batch foaming system with a visualization window that was designed for microcellular foaming simulation. A polystyrene (PS)-CO2 system was used in the experiment and analysis. The maximum pressure-drop rate achievable was 2.5 GPa/s from the designed system. Some experimental simulation results at high pressuredrop rates and at low pressure-drop rates are also discussed. We observed that the application of a higher pressure-drop rate results in a higher cell density (and, thereby, a smaller cell size) for plastic foams. This confirms that the pressure-drop rate is one of the most important parameters to control the cell density of plastic foams. In addition, the results show that the content of the blowing agent (CO2) dissolved into a given polymer has a significant effect on bubble nucleation and growth. Introduction In the past decade, efforts have been made to reduce cell size and enhance cell size uniformity in microcellular plastic foams, because plastic foams with a finer cell size and a more uniform distribution exhibit better mechanical and thermal properties.1-6 Park and co-workers7,8 have shown that the pressure-drop rate has a strong role in determining the cell density of foams in a continuous extrusion processes. In reality, the pressure drop is not instantaneous; it occurs over a finite time period. During the course of the pressure drop, the gas in the solution will either diffuse into the nucleated cells or nucleate new cells to reduce the free energy of the system. The higher the pressure drop, the more gas that is used for cell nucleation instead of for cell growth. Consequently, the cell density (i.e., the number of cells per unit volume of unfoamed plastic) is expected to be large. Therefore, a high pressure-drop rate is a critical factor in microcellular plastic foam processing. In addition, Park et al.9 have demonstrated that the content of the blowing agent (gas) dissolved into a given polymer has a significant effect on bubble nucleation and growth. Generally, the more gas that is dissolved into a polymer, the smaller the bubble size and the larger the bubble number. The amount of gas in a polymer is decided by the solubility of the gas in that polymer and the given saturation conditions (i.e., the initial temperature and pressure). Thus, the saturation conditions are yet other critical factors to be considered in the microcellular plastic foaming process. A fundamental understanding of how the properties of materials and the processing parameters affect the foaming processes is required to manufacture high-quality plastic foams. Experimental and computer simulation studies based on certain fundamental properties (i.e., the thermophysical and rheological properties of polymer/gas mixtures) are important to elucidate the cell nucleation and growth behaviors in foam processing. Hitherto, much research has been conducted on foaming processes, from both experimental and theoretical viewpoints. * To whom correspondence should be addressed. Tel.: +01-416978-3053. Fax: +01-416-978-3053. E-mail address: park@ mie.utoronto.ca.
Most experimental studies have focused on the use of the actual processing equipment while varying the processing conditions, material compositions, and system configurations. Efforts have been made to investigate the effects of these parameters on cell nucleation and growth behaviors. However, because of the numerous parameters that have a role in the actual foaming procedure, very limited success has been achieved in identifying the fundamental mechanisms of cell nucleation and expansion using the processing equipment.7,10-16 To better understand the nature of the various foaming mechanisms, some researchers have mounted a visual window in the mold and/or die.12-16 For example, Villamizar et al.12 performed visual observation experiments on the injection foam molding of a mixture of polyethylene and a chemical blowing agent, using a rectangular mold cavity with glass windows on both sides. Han and coworkers13,14 performed a visual observation of bubble nucleation in shear flow of a mixture of polystyrene (PS) and trichlorofluoromethane, using a light scattering technique; they reported several types of nucleation mechanisms. Xanthos et al.15 recently developed an in-line optical method that is used to generate the solubility data of inert gases in PS and poly(ethylene terephthalate) (PET) in single- or twin-screw extruders with gas injection capabilities. Shimoda et al.16 conducted visual observation experiments of polypropylene mixed with isobutene in a shear flow. All of these studies focused on visualization during the actual processing, and valuable information about bubble formation was obtained. However, because of the dynamic nature of the flow during actual processing, achieving control of the parameters in the visualization window was not easy and the information derived from the visualization window was limited. However, some researchers have focused on experimental simulations under static conditions; these situations allow for better control of the processing parameters when studying cell nucleation and growth behaviors. With the help of a high-speed camera, a microscope, and image processing, Ohyabu et al.17 designed a high-pressure visualization device to observe the early stages of bubble nucleation and growth behaviors of molten polypropylene when it is dissolved with supercritical carbon dioxide. Oshima and co-workers18-20 also designed and
10.1021/ie060105w CCC: $33.50 © 2006 American Chemical Society Published on Web 08/08/2006
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∆mvalve ) f(valve)(Pt2 - P0)Fvalve∆t 2t
(5)
The geometry parameter f(valve) is determined by the expression Figure 1. Tthermodynamic model of a high-pressure chamber and solenoid valve.
used a similar high-pressure cell and significantly contributed to the identification of the bubble nucleation and growth behaviors. However, these simulation systems had limitations to the achievable pressure-drop rate (dp/dt < 3 MPa/s) and, thus, were effective to simulate mainly conventional foaming processes. For the purposes of this study, the pressure-drop behavior of a high-pressure chamber is theoretically analyzed to design and set up a microcellular foam simulation system that is capable of a high pressure-drop rate. The pressure-drop behavior of the designed system is experimentally verified and compared to the theoretical data. This paper will also report some foaming experimental results obtained using the designed system under high and low pressure-drop rates.
The pressure-drop rate in a foam simulation system is theoretically analyzed in this section. Figure 1 shows a thermodynamic model of a high-pressure chamber and a solenoid valve, which constitute the pressure-releasing part of the foam-simulation system. Vessel 1 and vessel 2 represent a high-pressure chamber and a solenoid valve, respectively. The gas (or liquid) in the two vessels is assumed to be in a stable state at a time t. After a time interval ∆t, some of the gas escapes from the two vessels and the gas in the two vessels will be in a new stable state at the time t + ∆t. The pressures after ∆t, in vessel 1 and Pt+∆t in vessel 2, are dependent on the Pt+∆t 1 2 pressures Pt1 in vessel 1 and Pt2 in vessel 2 at time t and the amount of gas lost during the time interval ∆t. After the pressures Pt+∆t and Pt+∆t are known, the pressure-drop rate in 1 2 each chamber can be calculated. The mass changes in each chamber are calculated using the following equations:
mt+∆t - mt1 ) ∆m1 1
(1)
- mt2 ) ∆m2 mt+∆t 2
(2)
Note that the symbols with suffixes 1 and 2 correspond to variables in vessel 1 and vessel 2, respectively. In the following equations, the symbols m, V, Q, and F represent the mass, volume, volume flow rate, and gas density, respectively. The mass change21 during the time interval ∆t for vessel 1 is
(3)
The geometry parameter f(tube) is determined by the relation
f(tube) )
4
πr 8ηL
(
)
(4)
where r is the radius of the tube, L the length of the tube, and η the viscosity of the gas. The mass lost during time interval ∆t through the valve is given as
]
Ro2 - Ri2 Ri2 ln Ro - Ro2 ln Ri Ro2 - Ri2 Ri2 + ln Ri 2 4 2 ln(Ri/Ro) (6) where Ri and Ro are the inner and outer radii of the valve flow channel (for a donut cross-section), respectively, and η is the viscosity of gas. According to fluid mechanics,21 as the pressure difference increases, the mass flow rate Q first increases; however, when the pressure difference exceeds a certain value, the mass flow remains constant. At the critical pressure difference, the mass flow attains its maximum, and the velocity is exactly equal to the sonic velocity:
a2 )
Theoretical Analysis of Pressure Decay in a Foam Simulation System
) - ∆m1 ) f(tube)(Pt1 - Pt2)Ftube∆t ∆m1tube t
[
2 2 Ro2 - Ri2 Ro2 π R o - Ri + ln Ro f(valve) ) 2ηL 4 ln(Ri/Ro) 2
dp dF
(7)
where a is the speed of sound. The speed of sound decreases as the density F increases. When the speeds of gas (calculated based on eqs 3 and 5) are greater than the speed of sound, eqs 3 and 5 cannot be used to calculate the mass flow of gas during time interval ∆t. Instead, the following equations should be applied:22
1 ) -∆m1 ) aAt2∆tFtube ∆m1tube t 2
(8)
1 ) aAv2Fvalve∆t ∆m2valve t 2
(9)
where At and Av are the cross-sectional areas of the tube and the valve, respectively, whereas Ftube and Fvalve represent the densities of gas in the tube and in the valve, respectively. The net mass change during time interval ∆t for vessel 2 is
- ∆mtube ∆m2 ) ∆m2valve 1t t
(10)
Because m ) FV, eq 2 can be rewritten as
- Ft2)V2 ) ∆m2 (Ft+∆t 2
(11)
Equations 10 and 11 can be solved simultaneously for Ft+∆t . 2 Because the density F is a function of the pressure P and temperature T, the pressure Pt+∆t in vessel 2 after time interval 2 ∆t can be calculated using the equation of state (EOS) of the gas. Similarly, for vessel 1, Ft+∆t can be derived; the pressure 1 in vessel 1 after ∆t can then be calculated. Pt+∆t 1 Based on the pressures Pt1 and Pt+∆t , the pressure-drop rate 1 in vessel 1 is obtained using the following equation:
Pt+∆t - Pt1 Pt1 - Pt+∆t 1 1 dP )) dt ∆t ∆t
(12)
Following the same procedure, the pressures P1t+2∆t, P1t+3∆t, ..., P1t+n∆t respectively at time intervals 2∆t, 3∆t, ..., n∆t can be calculated until the pressure in vessel 1 becomes the atmospheric pressure. The pressure-drop rates at these 2∆t, 3∆t, ....., n ∆t can be obtained using eq 12.
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Figure 2. Schematic of batch foaming-simulation system.
Figure 3. Relationship between light, magnification, and camera frame rates in the visualization system for (a) low magnification and (b) high magnification.
The temperature has a significant influence on the density for compressible fluids. In the aforementioned calculation, the temperature change must be considered. Assuming this system is adiabatic during the short time period of gas release, the system energy was equal before and after a quantity ∆m of gas was released from the chamber:
Qt + Wt + mt(ht + gz) ) Qt+∆t + Wt+∆t +
(
where Cp is the isobaric specific capacity of gas. Therefore, the expression for the temperature change for liquid (gas) is
Tt+∆t - Tt )
)
where Q and W represent the heat and the work, respectively, that the system acquires from outside; ht+∆t is the enthalpy per unit mass of gas at the time t + ∆t; and υ is the speed of the escaping fluid (note that the maximum value of υ is the speed of sound, a). Because the system is assumed to be adiabatic during the release of gas,
∆Q ) Qt+∆t - Qt ) 0
(14)
∆W ) Wt+∆t - Wt ≈ 0
(15)
∆m ) mt+∆t - mt
(16)
υ1 )
υ2 )
∆m1tube t ∆tFtubeAt
∆m2valve t ∆tF
valve
Av
) f(tube)(Pt1 - Pt2)/At or υ1 ) a1 (20)
) f(valve)(Pt2 - P0)/Av f(tube)(Pt1 - Pt2)/Av or υ2 ) a2 (21)
The temperature in vessel 1 is calculated to be
Tt+∆t - Tt )
Tt+∆t - Tt ) (17)
In terms of thermodynamics,
(18)
∆m1tυ12 2mt1Cp
(22)
The temperature in vessel 2 is calculated to be
Thus,
ht+∆t - ht ) Cp(Tt+∆t - Tt)
(19)
The flow speeds for vessel 1 and vessel 2 are, respectively,
υ2 mt+∆t(ut+∆t + Pt+∆tV + gz) - ∆m ht+∆t + + gz (13) 2
∆mυ2 mt(ht+∆t - ht) ) 2
∆mυ2 2mtCp
∆mt2υ22 2mt2Cp
(23)
Determination of Chamber Size and Simulation of Pressure Decay. In this theoretical analysis, carbon dioxide (CO2) was used as an example. The viscosity of CO2 was assumed to be 3 × 10-5 Pa s in its gas and supercritical states, and
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Figure 4. Experimental results of pressure decay and pressure-drop rates at various saturation pressures and room temperature.
9 × 10-5 Pa s in its liquid state.23 The time required to open the solenoid valve fully was 0.1 s, according to the manufacturer.
The volume of the solenoid valve was measured to be 26 cm3. Using the equations derived in the previous section, a series of simulation was conducted at various saturation pressures while varying the dimensions of the chamber and tube. When the tube length and the volume of chamber were chosen to be 0.8 m and 7.0 cm3, respectively, very high pressure-drop rates that can simulate microcellular foaming were achieved. The simulation results are compared to the experimental data in the “Results and Discussion” section. Design of a Microcellular Foaming Simulation System. Based on the combination of the high-pressure chamber and the solenoid valve discussed in the previous section, a microcellular foaming simulation system was designed. The simulation system consists of a high-pressure, high-temperature
Figure 5. Comparison of pressure decay and pressure-drop rates between experimental and theoretical results obtained at various saturation pressures.
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Figure 6. Images of foaming and comparison of bubble nucleation and growth under two different pressure-drop rates for the same saturation conditions (T0 ) 180 °C and P0 ) 15 MPa).
chamber, a pressure-drop rate-control system, a data acquisition system for pressure measurement (ACAD board), a gas supplier (gas tank, syringe pump, and valves), an objective lens, a light source, and a high-speed CMOS camera. A schematic of the setup is shown in Figure 2. Many requirements were considered for the design of microcellular foaming simulation system. As mentioned previously, a high pressure-drop rate was the first requirement for the chamber. Under the higher pressure-drop rate, the foaming process will be completed in a few microseconds, because of the higher nucleation rate;24 this system therefore necessitates a camera with very high frame rates that would be fast enough to capture the changes in bubble nucleation and growth during the foaming process. A highly sensitive camera with a high frame rate (up to 128 000 fps) (Photron USA, Inc., Model ULTIMA APX FASTCAM) was selected, which could record the rapid foaming process. The bubble size of microcellular foams is very small (109 cells/cm3 can occur when the pressure-drop rate is as high as 220 MPa/s under a saturation pressure of 21 MPa (equivalently with 12% CO2, according to the solubility of CO2 in PS25) and a saturation temperature of 180 °C; this confirms that the current system is utilizable for conducting simulation experiments for the microcellular foaming process. Effects of Gas Content on Bubble Nucleation and Growth. Figure 8 illustrates subsequent foaming images taken under the
same pressure-drop rates (50 MPa/s) but different saturation pressures (10 and 15 MPa) for the PS-CO2 system. In Figure 8b, the nucleation rate under the higher saturation pressure (15 MPa) was higher than that under the lower saturation pressure (10 MPa); the bubble could grow to a larger size under the lower saturation pressure. The reason is that, under a higher saturation pressure, more gas was dissolved into the polymer. It is wellknown as the gas concentration increases in the polymer, the critical radius (Rc) is reduced, and, therefore, the nucleation rate increases.26,27 According to the solubility data of CO2 in PS,25 the dissolved CO2 contents for 10 and 15 MPa at 180 °C are 4% and 8%, respectively. Similarly, the solubility of CO2 in the other experimental conditions can be determined, and, based on these data, the cell density can be obtained as a function of CO2 content at various pressure-drop rates, as shown in Figure 9. Other Notable Effects on Bubble Nucleation and Growth. Figure 10 shows subsequent foaming images taken on two different layers of sample under the same pressure-drop rate (6 MPa/s) and the same saturation conditions (T0 ) 180 °C, P0 ) 15 MPa). The term “middle layer” refers to the case in which the microscope lens was focused on the middle across the 0.5 µm thickness, whereas the term “interface” suggests that the microscope lens was focused on the bottom layer of the sample that was touched by the sapphire window. In Figure 10, the number of cell nucleations that occurred at the interface layer was larger than that of the middle layer; the bubble size was also smaller. One possible reason for this is that the interface
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perhaps facilitated bubble nucleation, based on heterogeneous nucleation.28 Conclusion A batch foam simulation system with high pressure-drop rates was developed. This system can be used to investigate the cell nucleation and growth behaviors in microcellular foaming processes that use high pressure-drop rates. The derived theoretical simulation model effectively described the pressure decay and pressure-drop rates in the chamber. The experimental results show the pressure-drop rate and the content of a blowing agent dissolved in the polymer melt have significant effects on the foaming behaviors (i.e., cell nucleation and growth). Acknowledgment The authors are grateful to the Consortium for Cellular and Micro-Cellular Plastics (CCMCP), National Sciences and Engineering Council of Canada (NSERC), and AUTO21 for their financial support of this research. Literature Cited (1) Baldwin, D. F.; Suh, N. P. SPE ANTEC, Tech. Pap. 1992, 38, 1503. (2) Collias, D. I.; Baird, D. G.; Borggreve, R. J. M. Polymer 1994, 25, 3978. (3) Collias, D. I.; Baird, D. G. Polym. Eng. Sci. 1992, 35, 1167. (4) Seeler, K. A.; Kumar, V. J. Reinf. Plast. Compos. 1993, 12, 359. (5) Doroudiani, S.; Park, C. B.; Kortschot, M. T. Polym. Eng. Sci. 1998, 38, 1205. (6) Matuana, L. M.; Park, C. B.; Balatinecz, J. J. Polym. Eng. Sci. 1998, 38, 1862. (7) Park, C. B.; Baldwin, D. F.; Suh, N. P. Polym. Eng. Sci. 1995, 35 (5), 432. (8) Xu, X.; Park, C. B.; Xu, D.; Pop-Iliev, R. Polym. Eng. Sci. 2003, 43 (7), 1378. (9) Park, C. B.; Cheung, L. K. Polym. Eng. Sci. 1997, 37 (1), 1.
(10) Lee, S. T. Polym. Eng. Sci. 1993, 33 (7), 418. (11) Naguib, H. E.; Park, C. B.; Reichelt, N. J. Appl. Polym. Sci. 2003, 91. (12) Villamizar, C. A.; Han, C. D. Polym. Eng. Sci. 1978, 18 (9), 699. (13) Han, C. D.; Villamizar, C. A. Polym. Eng. Sci. 1978, 18 (9), 687. (14) Han, J. H.; Han, C. D. Polym. Eng. Sci. 1988, 28 (24), 1616. (15) Zhang, Q.; Xanthos, M.; Dey, S. K. J. Cell. Plast. 2001, 37 (4), 284. (16) Shimoda, M.; Tsujimura, I.; Tanigaki, M.; Oshima, M. J. Cell. Plast. 2001, 37, 517. (17) Ohyabu, H.; Otake, K.; Hayashi, H.; Inomama, H.; Taira, T. In Proceedings of the Polymer Processing Society Annual Meeting (PPS19), Melbourne, Australia, July 7-10, 2003. (18) Taki, K.; Nakayama, T.; Yatsuzuka, T.; Oshima, M. In Proceedings of the Polymer Processing Society Annual Meeting (PPS18), Guimara˜es, Portugal, June 16-20, 2002. (19) Taki, K.; Nakayama, T.; Yatsuzuka, T.; Oshima, M. J. Cell. Plast. 2003, 39 (3), 155. (20) Taki, K.; Yanagimoto, T.; Funami, T.; Okamoto, M.; Oshima, M. Polym. Eng. Sci. 2004, 11 (6), 1004. (21) Currie, I. G. Fundamental Mechanics of Fluids, 2nd Edition; McGraw-Hill, College Division: New York, 1993. (22) Frisch, K. C.; Saunders, J. H. Plastics Foams; Marcel Dekker: New York, 1972. (23) Vukalovich, M. P.; Altunin, V. V. Thermophysical Properties of Carbon Dioxide; Collets: London, 1968. (Translated from Russ. into Engl. by D. S. Gaunt.) (24) Leung, S. N.; Li, H.; Park, C. B. In Proceedings of the Polymer Processing Society Annual Meeting (PPS21), Leipzig, Germany, June 1923, 2005, Paper No. 3824. (25) Li, G.; Wang, J.; Park, C. B.; Moulinie, P.; Simha, R. SPE ANTEC, Tech. Pap. 2004, 50, 2566. (26) Tucker, A. S.; Ward, C. A. J. Appl. Phys. 1975, 46 (11), 4801. (27) Leung, S. N.; Park, C. B.; Li, H. SPE ANTEC, Tech. Pap. 2005, 51, 2592. (28) Colton, J. S.; Suh, N. P. Polym. Eng. Sci. 1987, 27, 485.
ReceiVed for reView January 23, 2006 ReVised manuscript receiVed May 31, 2006 Accepted June 30, 2006 IE060105W
