Design of an Apparatus to Measure Gas

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1 Design of an Apparatus to Measure Gas Solubilities in Polymers Esther V. Richards A thesis submitted in conformity with the requlrements for the Degree of Master of Applied Science Department of Mechanical and Industrial Engineering University of Toronto

2 y ons and Acquisitions et 81 iographic Services services bibliographiques The author has granted a nonexclusive licence allowing the National Library of Canada to reproduce, loari, distribute or seu copies of this thesis in microform, paper or electronic formats. The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced without the author's permission. L'auteur a accordé une licence non exclusive permettant zi la BibliotMque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la fonne de microfiche/film, de reproduction sur papier ou sur format électronique. L'auteur conserve la propriété du &oit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

3 Design of An Apparatus To Measure Gas Solubilities In Polymers Esther V. Richaràs Degree of Master of Applieà Science, 2001 Department of Mechanical and Industrial Engineering University of Toronto The design for an apparatus to measure the solubility of gases in polymers is presented. The apparatus is based on a pressure decay method. and utilizes a threethamber configuation for achieving multiple experimental pressures with the same sample. The design was tested using polystyrene (PS) and (C2) at temperatures of 100 and 160 C and pressures up to 12 MPa. The pressure-specific volume-temperature (PVT) data of the PS- C2 combination were used to correct the solubility data for the swelling effect of the polymer. The results indicated that the solubility increased with increasing pressure and decreased with increasing temperature. However. the data did not compare favorably with the literature, since the typical Henry's law behavior was not observed. The main source of error was believed to be due to the pressure transducer setup in which the transducer, because of its limited tempenture range, was placed extemai to the main experimental setup.

4 To my beloved parents: Thank You, for your wisdom and guidance over the years. Thank You, for your support. Thank You, for your unconditional love. Thank you, for king my parents. Thank you. iii

5 ACKNWLEDGEMENTS First and foremost, 1 would like to express my thanks to God. Who provided me with the opportunity to obtain my Master's Degree. 1 thank You for Your continued blessings and for Your promise, that You'll never put more on me than I cm bear. To my family at home, 1 would like to say "thank you" for your continuous love and advice. May God bless you dl. This thesis has afforded me a unique leaming opportunity, for which 1 am grateful to my supervisor. Thank you Prof. Park for your vision, guidance and never-ending patience. I would also like to express my sincere thanks to Prof. Sinclair for his continuous support. Throughout my stay ai the University of Toronto, 1 have also had the honor of working with the best machinists. Thank you Mark, Patrick, Dave, Mike, and Jeff. 1 could not accomplish half of what 1 did, if it were not for your brilliance. To Mary-Rose, Teresa, and Sheila in the Receiving Department, thank you for your patience. Your always-kind reception made my studies a much more pleasant experience. Finally to my colleagues at the Microcellular Plastics Manufacturing Lab., it was a distinct pleasure working with you al]. 1 am especially indebted to Hani, Linda, Patrick, and Ghaus for your friendship, advice and silent support over the two yeûrs. To Qingping, thank you for your assistance in the final days-1 could not have finished without your help.

6 TABLE F CNTENTS.. ABSTRACT... II... Dedication...III Acknowledgments... iv Table of Contents... v List of Tables VIII List of Figures... ix.... XII Nomenclature Chapter 1: INTRDUCTIN 1.1 Preamble Foam Processing Microcellular Processing Physical Blowing Agents Thesis bjectives and Scope of Research Thesis Structure... 8 Chapter 2: LITERATURE SURVEY AND THERETICAL BACKGRUND 2.1 Solubility... II Ideal Sorption Isotherms Non-ldeal Sorption Isotherrns Solubility Measurement Meihods Gravimetric Sorption Technique Pressure Decay Sorption Technique Volume Decay Sorption Technique Piezoelectric Quartz Sorption In-Line Measurement of Gas Solubility Physicd Blowing Agents Theoretical Treatments... 29

7 Chapter 3: PRBLEM STATEMENT AND PRPSED DESIGN verview Principle of Solubility Measurement (Pressure Decay Method) Problem Statement and verall Strategy Effect of Pol ymer Dilation Background on Axiomatic Design Approach Axiomatic Design of the verall Process Design Design II Summary Chapter 4: DETAILED DESIGN 4.1 verview Pre-solu bility Calculations Theoretical Sample Size Sorption Cell Size I Thennostatted il Bath Vacuum System Solubility Calculations Summary Chapter 5: EXPERIMENTAL verview Experimental Apparatus Experimental Materials Experimental Procedure... 8? Volume Measurements Sorption Experiments Results and Discussion Error Analysis Erron in Temperature Measurement Emrs in Pressure Measurement vi

8 5.6.3 Errors due to Transducer Configuration Chapter 6: SUMMARY. CNCLUSINS AND FUTURE WRK 6.1 Summary Conclusions Future Work REFERENCES vii

9 Table 3.1 Parameters for calculating PiA and PIB Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 5.1 Table 5.2 Table 5.3 Parameters for calculating the pressure drop due to diffennt sample sizes Parameters for calculating chamber thickness Properties of air and Heat Transfer Fhid Parameters for determining the heating capacity High-pressure chamber volumes Experimental Solubility Data Error contribution of externd volume viii

10 LIST F FIGURES Figure 1.1 Microcellular foam production Figure 2.1 Schematic representation of sorption isotherms in rubbery pol ymers Figure 2.2 Schematic representation of sorption isotherms in glassy polymers Figure 2.3 Schematic of electrobalance Figure 2.4 Schematic diagram of the piezoelectric-quartz sorption apparatus Figure 2.5 Basic Principle of ultrasonic technique Figure 3.1 Schematic of pressure decay method based on three Chambers Figure 4.1 Schematic for determining the pressure drop due to solubility for different VpNc ratios Figure 4.2 Relative sample size vs. pressure drop due to 0.94% solubility at 140 C Figure 4.3 Determination of relative sorption cell sizes Figure 4.4 Stepwise Pressure Increase vs. Iteration No. for 4 chamber ratios Figure 4.5 Stepwise Pressure Increase vs. Iteration No. for 7 chmber ratios Figure 4.6 Detailed schematic of high-pressure chamber #I Figure 4.7 Detailed schematic of high-pressure chamber #2 and high-pressure chamber lids... 79

11 Figure 4.8 Detailed schematic of high-pressure chamber # Figure 4.9 Photograph of solubility measuring apparatus Figure 4.10 Photograph of pressure measurement and temperature control instrumentation...,...,...*...,...* 8 1 Figure Schematic for calculaiing solubility Figure 5.1 Schematic of solubility measuring device Figure 5.2 Gasket sealing of sorption chamber Figure 5.3 (a) Schematic representation of the measurement of intemal chamber volume Figure 5.3 (b) Schematic representation of the measurernent of intemal chamber volume Figure 5.3 (c) Schematic representation of volumes to be measured Figure 5.4 (a) Solubility of PS in C2 ai 100 C Figure 5.4 (b) Solubility of PS in C2 at 160 C Figure 5.5 Solubility of PS in C2 at 100 and 160 C Figure 5.6 Solubility error contribution from temperature fluctuation of f 1 C Figure 5.7 Solubility error contribution from pressure fluctuation of I 8.27 kpa Figure 5.8 (a) Schematic for detemining the error due to the transducer Configuration I

12 Figure 5.8 (b) Schematic for determining the error due to the transducer Configuration II Figure 5.9 Temperature variation in tube leading from heated oil bath to pressure trasducer (extemal volume) for oil bath temperatures of 100 and 160 C

13 Attenuation (db/crn) Exposed area (cm') Amplitude Hole affinity constant (atm") Concentration of a permeant Speci fic heat capaci ty (kllkg-k) Concentration of gas (cm3-~~~lcrn~-pol~) Hote saturation constant ( cm'-~~~/cm~-~ol~) Diffusivity (cm2/min) Thickness (cm) Heat of solution (cavmol) Heat flux (k~lcm') Heating capacity (kw) Henry's h w constant (cm3- atm) Diffusion distance (cm) Surface heat losses (kw) Polymer mass (g) Molecular mass (g/kmol) Amount of gas present in cell (kmol) Pressure (psi) Equilibrium gas pressure (kpa) xii

14 Initial pressure (atmospheric pressure) Required pressure Total amount of air to be removed (ft3) Energy to heat initial charge of heat transfer fluid (kj) Energy to heat steel tank (kj) Radius Universal gas constant (kjkmol-k) Degree of swelling Tangent ial stress Heating time (hours) Diffusion time (min) Temperature (K) Sound velocity (ms") Pure pol yrner specific volume (cm3) Swollen polymer specific volume (cm3) Volume (cm3) Solubility (g-gasjg-pol ymer) Compressibility factor Density (g/cm3) Design parameters Functional requirements Time delay (s) Temperature difference (K) xiii

15 Chapter 1 INTRDUCTIN 1 Prearnble It is becoming almost impossible to go through an entire day without corning into contact with an object that contains a foamed plastic part. This is a testament to the extent to which foamed polymeric materials have pervaded our everyday life. Each foamed polymeric material can be classified as either a thermoplastic or a thermoset foam. Thermoplastic foams can be recycled. whereas thennoset foams cannot be reprocessed because of the extensive crosslinking present. Different applications for foamed products may require different densities, consequently, several types of polymeric materials are foamed to various low densities for applications that require attributes such as weightreduction, insulation, buoyancy, energy dissipation. convenience and cornfort. Thermoplastic and thermoset foam products differ from solid plastic products and are identifiable by virtue of a unique cellular structure, which is created by the atmospheric expansion of a blowing agent in the polymer matrix. This cellular structure is normally characterized in terms of the ceil density and cell size. For conventiona! foams, typical cell population densities are 1o3-106 cells/cm3 with ce11 sizes of the order of 100 pm or larger, however, the ceil size and distribution can be inconsistent thereby comprornising the mechanical properties of the foam. Conversely, microcellular plastics are characterized by cet1 densities greater than 109 cells/cm3 and cell sizes smaller than 10 Pm. This unique group of plastic foms was conceived by Martini et al [Il, and is based on the idea that if a large

16 number of bubbles smaller than pre-existing flaws in the polymer are created, the material cost can be reduced without compromising mechanical properties. It has been shown that microcellular plastics possess supenor impact strength [Il, toughness [3], and fatigue life [4] compared to solid polymers. Furthemore, solid polymeric foams cm consist of either closed or open cells. Closed cell foams possess a cellular structure in which neighbouring air bubbles are entrapped within a continuous macromolecular phase. pen cell foms, on the other hand, have a cellular network in which continuous channels are available throughout the solid macromolecu lar phase. The gaseous phase in any polymeric foam material is derived from the use of blowing agents in the foam manufacturing process. There are generally iwo types of blowing agents used in foam production: chemical blowing agents and physical blowing agents. Chernical blowing agents are chemical compounds, which evolve gases under foam processing conditions through thermal degradation or chemical reactions. Physical blowing agents on the other hand, are inert gases, such as nitrogen and carbon dioxide; volatile hydrocarbons such as propane, n-butane, i-pentane; and low boiling point chlorofluorocarbons (CFCs), hydrofluorocarbons (HFCs) and hydrochlorofluoro-carbons (HCFCs). Due to the environniental hazard posed by CFCs and HCFCs, there has been a drive to replace these blowing agents with environmentdly friendly substitutes. The properties of blowing agents and their impact on the processing variables in foam production are key issues to be considered in order to find effective replacements. The specific thermodynarnic and kinetic properties that have the greatest influence on the ability to produce microcellular foams, are the solubility and diffusivity of the blowing agent in the polyrners. The solubility

17 determines the arnount of blowing agent that can be absorbed by the polymer at a given temperature and pressure, while the diffusivity determines both the rate at which the blowing agent will penetrate into the polymer matrix to form a homogenous solution, and the rate at which the blowing agent escapes to the atmosphere dunng the cell nucleation and growth processes. 1.2 Foam Processing Microcellular Processing The foamed plastics industry is a continuously developing one, which encornpasses various methods of production for many product applications. The choice of polymer, blowing agent. and production method, dictate the foam formation and its morphology and therefore its properties. Stemming from an industrial challenge to enhance polymer value by improving perforrnancelweight characteristics. the microcellular foaming concept was developed. Earlier works focused on batch processes, the concept resulted in good foam structures with ce11 sizes under 5 pm using supercritical carbon dioxide [Il. The batch process was further developed and improved by Cha et al. [SI, whereas Kumar 161, and Kumar and Suh [7,8] developed r semitontinuous pmcess derived fkom a modified thermoforming process. Further investigation by Park [9], Park et al. [10], and Park and Suh [ 1 11 led to the development of an extrusion-based process for microcellular filament as a first step in the development of a continuous process. The fundamental principle involved in the formation of microcellular polymer foarns consists of three basic steps namely; (i) polymerhlowing agent solution formation. (ii) microcell nucleation, and (iii) celt growth and density reduction as illustrated in Figure 1.1.

18 The single-phase polymer/blowing agent solution in the first step is formed by saturating the polymer with the blowing agent under high pressure. The saturation point is determined by the solubility limit of the blowing agent in the polymer, while the time required for the solution formation is determined by the rate of diffusion of the blowing agent into the polymer matrix. Micrucellular nucleation is achieved by inducing a themodynamic instability in the single-phase solution. This is usually accomplished by drastically reducing the solubility of the gas in the solution by controlling the pressure ancilor temperature of the solution [ Since the separation of the polymer and gas phases is thermodynamically more favourable, the resulting supersaturated state becomes the driving force for the nucleation of numerous microcells [Il. Continuous microcellular processing typically utilizes a rapid pressure drop to nucleate bubbles. This stage is very crucial to the overall process, because it dictates the cell morphology of the material and the resulting properties. Therefore, the solubility as a function of pressure is important for the development of the process. The final stage in the production of microcellular plastics is cell growth. After ce11 nucleation has occurred, any available gûs diffuses into the ceil and increases the cell size thereby reducing the density of the polymer matrix. Generally. ceil growth is affected by the time allowed for the cells to grow, the system temperature, the amount of gas available (state of supersaturation), the processing pressure, and the viscoelastic properties of the polymer/gas solution [ 151.

19 1.2.2 Physical Blowing Agents An important aspect of the creation of a cellular structure is the use of a physical blowing agent. Historically, CFCs and HFCs, such as CFC-11 and HCFC- 141 b, were used primwily for low-density foam production mainly because of their solubility, volatility, and non-toxic nature. However their stability and reactivity with ozone in the atmosphere raised substantial concem about ozone depletion. in an effort to address the environmental impact of these man-made compounds, the Montreal Protocol [16] was signed in 1989 by 29 countries and amended in subsequent years. The Montreal Protocol mandated the gradua1 phase-out in the production of CFCs by 2010 and HCFCs by 2030 in drveloped countries. As a result, CFC production fell from 980 metric tons in 1986 to 95 metric tons in 1996 [17]. Since the early 1990s. global warming or the greenhouse effect has become another major issue. The Intergovemmental Panel on Climate Change (IPCC) reported a scientific assessrnent on the warming potential of various compounds relative to carbon dioxide [ 181. It was reported that the global warming potential of CFCs and HCFCs is 5,000-10,000 times greater than that of carbon dioxide wi th stratospheric life cycles of years. In addition, the Kyoto Protocol on Climate Change [19] adopted by 160 nations in 1997, sets binding limits on the greenhouse gas emissions for developed countries. It strengthens the framework established by the Montreal Protocol with new policies and measures. The combined global wming and ozone depleting potential of these substances and the resulting policies, have created a void in the foam processing industry and the need has therefore arisen to replace CFCs and HCFCs with environmentally friendly blowing agents possessing good foam-blowing properties. The concems over the ozone layer and global warming represent just a few of the issues facing the foam processing industry. Disposal,

20 waste Stream control and usage of recycled plastics still require a deep understanding of foaming technology. Microcellular plastics do not utilize the conventional CFC foam blowing agents; instead, carbon dioxide and nitrogen are typically used. Although these blowing agents represent a dramatic improvement in terms of environmental hazard, they too pose their own difficulties. Carbon dioxide has a high solubility in the plastic melt, approximately 10% at 200 C and 27.6 MPa [20], and produces a very uniform cell structure. Conversely, nitrogen has a Iower solubility, about 2-3% at 200 C and 27.6 MPa, and processing requires much higher pressures to dissolve sufficient nitrogen to create a unifonn structure as explained in Section Higher processing pressures however, require more robust processing equipment and ieads to higher equipment and processing costs. ther alternative blowing agents used are liquid blowing agents such as butane and iso-pentane. To date, however, liquid blowing agents have not been successfully used to achieve a microcellular structure. ne possible barrier that exists to the use of these liquid blowing agents is the lack of information about their solubility and diffusivity at high temperatures and pressures. 1.3 Thesis bjectives and Scope of Research ne long-tenn goal of cesearchers working in the foarn-processing field is to improve the manufacturing processes for polymer foarns. Their success hinges on the properties of the blowing agent and their impact on the processing variables. Given the environmental issues surrounding the use of physicil blowing agents, the objective of this research is to develop an apparatus that will measure the solubilities of different blowing agents in various polymers over a range of temperature and pressure.

21 Establishment of such an apparatus is not only important for finding effective replacements for hazardous blowing agents, it also forrns the basis for improving foam production methods such as thermofonning, injection moulding and extrusion. As explained in a previous section, knowledge of the solubility limit of a gas in a polymer is important for gas injection. If excess gas is injected into the molten polymer, then a single-phase polymer/gas solution will not fom and the resulting foarn will not have a unifonn cell structure: instead, big voids will be present, which will compromise the physical and rheological properties of the foam. This topic has ken of interest to various researchers for some time. Several methods have ken used by these researchers in an attempt to measure the solubility characteristics of polymer/gas systems [14, 34-72]. However, rnost of these methods rely on accurate knowledge of the volume of a polymer sample in contact with gas at a certain temperature and pressure. Under these conditions, when the blowing agent diffuses into the polymer, the resulting polymer/gas mixture has a higher specific volume than the pure polyrner, i.e., the polymer swells. Theoretical models have been applied to mode1 and predict the amount of polymer swelling that may occur. However, in the absence of reliable experimentai data, the accuracy and applicability of these models for the polyrner/gas mixture cannot be verified. Park et al developed a system, which facilitates the measurement of the specific volume of polymer/gas solutions. With this information, the volume of the polymer under specific conditions of temperature, pressure, and gas content can be known. With the knowledge of the amount of polymer swelling that can occur, corrections can be applied to the solubility data to account for the swollen polymer volume. The apparatus employed in this research is designed based on a pressure decay method [22, 231 and

22 employs a threeçhamber configuration. The basic principle involved is the detemination of the difference between the amount of gas initially in contact with the polymer sample and that remaining in the gas phase after equilibration. The gas amount is calculated using the equation of state of the gas. The volume of the high-pressure chambers is an important factor in the calculations and can lead to significant errors if not determined correctly. As a result, the volumes were determined using a modified helium expansion method as described by Koros et al. [22]. Axiomatic design principles were employed to create a robust design and an error analysis was also conducted to investigate the accuracy of the system. Finally the system was tested using polystyrene (PS) and carbon dioxide. 1.4 Thesis Structure This thesis presents the design of an apparatus for the measurement of solubility of a blowing agent in a polymer at high temperatures and pressures. Chapter 2 of this thesis explores solubility measurement in detail. The basic laws goveming solubility of most systems are explained and the work of other researchers is summarized. The various methods available for measuring blowing agent solubility are examined as well as the theoretical models that daim to predict solubility. In Chapter 3, the apparatus design is tackled. Beginning with an explmation of the principle of the pressure decay method, a design is proposed and evaluated based on an Axiomatic Design Approach. Physical and funciional coupling of the design is analyzed in detail through a hierarchical system.

23 Chapter 4 deals with solubility measurement. The principle of operation of the designed apparatus is illustrated. and the detailed design of some physical parameters such as chamber size ratio, and ideal polper size, are described in detail. The experimental section of the thesis is presented in Chapter 5. The apparatus is described, dong with the procedure followed and materiais used for the investigation. The results obtained are presented and compared to literature values to detennine the functionality of the apparatus design. An in-depth error analysis of the parmeters affecting the solubility measurements is conducted to determine the operating error range of the design. Finally the thesis concludes with Chapter 6, which includes the summary, conclusions, and suggestions for future work.

24 0 :rati; a ;ucleat a PLYMER Single-phase polyrnedgas solution Microcellular nucleation Microcellular structure Figure 1.1: Microcellular foam prodiction

25 LITERATURE SURVEY AND THERETICAL BACKGRUND 2.1 Solubility In this section of the thesis. a theoretical background of solubility is given. The various methods used to measure solubility will be explored, followed by a qualitative review of the published literature regarding the solubility for various polymer-blowing agent systems. Finally theoretical approaches to the estimation of solubility will be presented Ideal Sorption Isotherms The solubility of a penetrant (blowing agent) in a permeant (polymer) is usually described by the sorption isotherm. It correlates, at a constant temperature, the amount of sorbed penetrant to the pressure of the gas phase around the polymer. Sorption isothexms may show considerable ciifferences in shape. depending on the nature of the polyrner- penetrant system. The sorption behaviour of simple gases in rubbery polymers is linear in nature (see Fig. 2.1) and obeys Henry's law [24], which states that the concentration of a penetrant, c ( ~rn~-~~~/crn~~ol~), in a polymer is a linear function of pressure:

26 where H is the Henry's law constant (cm3-stp/c~-atm), and P is the equilibrium gas pressure (atm). Henry's law is based on an ideal solution state, where the solution is an extremely dilute one, with a low penetrant concentration. It does not take into account the interactions between the pol ymer and gas; therefore, the solu bility predicted from Henry's law normally deviates from the actual solubility especially at high pressures. Nonetheless, various researchers have used Henry's law to describe polymer/diluent systems. The temperature dependence of the Henry's Iaw constant was derived through an Arrhenius-type equation by Durrill and Griskey [ 121: In this expression, H and Ho both denote Henry's law constants (cm3-~~~/crn~-atm), Es is the heat of solution (kj/kmol), R is the universal gas constant (kjkmol-k) and Ti is the temperature (K). In order to predict the heat of solution as indicated in Eq. 2.2, Henry's law constant needs to be established at various temperatures. For high penetrant concentrations the sorption isotherm tums upwards into a convex shaped curve, as shown in Fig This isothem, which is ofien observeâ for vapors king dissolved in rubbery polymers. is well represented by the Flory-Huggins theory [25] 2.13 Non-Ideal Sorption Isotherms For amorphous or glassy polymers, i.e., polymers below the glass transition point, the simple Henry's Iaw mode1 for gas sorption and transport is not adequate to describe the

27 sorption behaviour. Instead, equilibrium sorption of gases in glassy polyrners typically exhibits concave shaped isothenns as shown in Fig In this figure, the concentration Cg represents the concentration of penetrant molecules for which the experimental temperature is equal to the glas transition temperature of the polyrner-penetrant system. Above this concentration the Rory-Huggins regime is val id. ne model that has been developed to explain the concave shape of the sorption isotherm is the Dual Mode Sorption model [26]. In this model, the glassy polymer is considered to consist of a matrix of homogeneous density in which a small amount of microcavities or holes are embedded. The volumes of these micro-cavities sum up to the total unrelaxed free-volume of the polymer. The sorption behaviour depends on the location where a penetrant molecule is sorbed [27]. In the homogeneous matrix, the sorption is assumed to occur as ordinary dissolution described by Henry's law (Eq. 2.1). However, in the holes, the sorption behaviour obeys an adsorption-type isotherm, namely a Langmuir isotherm. In this interpretation, the parameter CH (cm3-~~~/cm3-po~y) represents the gas adsorbed into the holes, while the parameters CL (C~~-STP/C~~-~~~) and b (atm-') are the hole saturation and hole affinity constants, respectively. Since the Langmuir sorption corresponds to a hole filling process, this population should not participate in the dilation of the polymer due to sorption. nly that fraction of sorbed penetrant, which is sorbed in the Henry's mode, should swell the polymer resulting in a linear relation of the sample dilation

28 with increasing applied pressure. Summation of the Henry's Law equation (Eq. 2.1) and the Langmuir isotherm (Eq. 2.3) results in the Dual Mode Sorption mode]: Further development and verification of these models can be found in the literature [ Solubility Measurement Methods There are two experirnental techniques that can be used to deteminc the solubility of a gas in a polymer, namely pemeation and sorption techniques. Pemeation experiments involve measurements of the steady state mass flow of a gas flowing through a thin membrane, whereas in the sorption kinetics techniques. the mass uptake of gas by the polymer sample is measured. The major difference between the two techniques lies in the method by which solubility is determined. Pemeation experiments rely primarily on a mathematical expression to determine the pemeability, diffusivity and hence the equilibrium solubility of the gas in the polymer indirectly when steady state flow has been attained. n the other hand, the sorption rnethod mesures directly the mass gain of the polymer due to gas dissolution and therefore represents a more direct approach to determining the equilibrium solubility. The sorption method will be used in this thesis io determine the solubility information for polymer-blowing agent combinations, and consequently, this section will focus only on a review of sorption techniques.

29 2.2.1 Gravimetric Sorption Technique The gravimetric sorption method investigates the solubility by directly measuring the mass gain of a polymer sample due to gas dissolution. ne of the earliest gravimetric techniques utilized a quartz seing measuring system referred to as a McBain Balance The balance was operated by suspending the polymer from a quartz spring in a low-pressure gas environment. As the polymer gained weight due to gas dissolution. the spring elongated. Utilizing Hooke's Law, the mass of the sample can be detemined as a function of elongation by calibrating the spring with known weight increments. The quartz spring method was used, for example. to determine the solubility data of ethylbenzene [36] and toluene [37] in polystyrene for various vapour pressures. A high pressure gravimetric technique referred to as a batch process, was utilized by Baldwin et al. [38] for the measurement of carbon dioxide solubility and diffusivity in thennoplastic polyesters. This method utilizes multiple samples, which are exposed to the gas for various time periods, and compiles the mass uptake curve by normalizing the time mis for sample thickness. An in-situ gravimetric sorption method investigates the solubility by measuring the mass gain of a polymer sarnple directly with a high-precision electrobalarife capable of measurements at high temperatures and pressures. The sensitivity of the instrumentation,! which can attain values of 1 part per million, makes the technique desirable for solubility measurement involving low solubility gases such as inert gases. This method is also suitable for measurement of solubility for polymers in either the rubbery or glassy state. A schematic diagram of the apparatus is shown in Figure 2.3. The apparatus is rnounted on a vibration free surface with the weighing unit contained in a constant temperature environment.

30 The solubility of the blowing agent is detennined from measurements of the mass increase of the sample with increasing blowing agent pressure. Wong et al. [39] reported on the use of an electronic microbalance to measure the gas solubility and diffusivity of C2 and HFC134a in PS. filled poly(vinylch1oride) - FPVC. and unplasticized poly(vinylchloride) - UPVC. An alternative method for measuring gas solubilities in polymers was presented by Chaudhary and Johns [40]. It involved the use of a magnetic suspension device similar in principle to the electrobalance. The rnost significant difference is that the weighing mechanism is;. physically decoupled from the high temperature and high-pressure environments through a magnetic suspension coupling. This equipment was used to measure the solubilities of nitrogen, isobutane, and carbon dioxide in polyethylene. More recently, Sato et al. [41] reported on the use of a magnetic suspension balance to measure the solubilities of carbon dioxide in poiy(viny1 acetate) (PVAc) and polystyrene. The high sensitivity of these types of balances dictates that mass measurements must be corrected for the change in buoyancy of the sample. Thus, knowledge of the dilation of the polymer with blowing agent uptake is also needed for solubility calculations Pressure Decay Sorption Technique The pressure decay method is used to detemine the gas solubility in a closed system by measuring the pressure drop due to gas dissolution in a polymer sample. This method is based on the assumption that al1 changes in the gas pressure are due to the mass sorption of the polyrner. Consequently, the mass uptake of the polymer is detennined indirectly by measuring the pressure decay of the fixed volume system. By accurately measuring the

31 apparatus volume and recording the pressure and temperatun of the system, the mass of the gas in the closed system is determined as a function of time using its equation of state. The solubility is then determined from the overail experiment pressure change. This technique requires rather careful calibrations and can only be used for gases whose equations of states ore accurately known. There are three rnethods for employing the pressure decay technique, namely single-, dual-, and three-chamber systems. Single-Cham ber Sorplion The single chamber system consists of a single chamber, which contains the polymer sample. The chamber is subjected to a rapid pressure increase, and the resulting pressure decay due to sorption is recorded as a function of time. Due to the rapid mass gain in the initial stages, and the thermodynamics of the gas system, a stable reading is often not recorded until several minutes after initial gas contact with the polymer. As a result, the initial pressure reading, which is needed to determine the initial mass of gas in the system, must be extrapolated from the pressure decay curve. The extrapolation can cause a significant error in determining the initial mass of gas in the system and consequently the total mass change due to gas sorption in the polymer. Dual-Chamber Sorplion The dual chamber sorption system [14,52] utilizes a reservoir chamber of known volume filled with gas to a known pressure, while a second chamber contains the polymer sarnple. By opening a valve separating both chambers, the gas is allowed to flow into the second chamber and thus into the polymer. The valve is then closed and the sorption

32 chamber is monitored for pressure decay. The reservoir chamber pressure is also measured. The mass absorbed by the polymer sample is then determined from the difference of the initial mass of gas in the reservoir chamber and the final mass of gas in both the reservoir and sorption chambers. Sorption experiments are normally performed in a stepwise manner, so that the pressure drop for each experimental pressure step is relatively small. A typical stepwise pressure decay sorption experiment begins with a low reservoir (and thus sorption) pressure. The resulting pressure decay due to sorption is rnonitored until the equilibrium pressure indicates that equilibrium mass gain has occurred. Without evacuating the chambers, new gas is introduced into the system so that the pressure is increased a step amount. The pressure decay is then monitored, and the process is repeated. The solubility is determined as a function of pressure by cumulatively adding the mass gain of each pressure step. mree-ckcimber Sorption In the three-chamber sorption system [53,54], the measurement principle is identical to the dual chamber configuration described above, Le., the mass uptake of the polymer is determined from the equation of state of the gas using measurements of chamber volume, pas pressure, and gas temperature. In the threethamber system, there are two reservoir chambers; the first reservoir is used as a pressure source for the sorption chamber, while the other is used as a source for the first reservoir. This configuration allows for multiple measurements at different pressures without introducing new gas and a new sample. This minimizes the temperature shock to the system caused by the introduction of new gas.

33 Previous Research Using Pressure Deaty Sorption Techniques Newitt and Weale [14] reported some of the earliest measurements of gas solubilities in polymers using a dual chamber system. They reported on the solubility of hydrogen and nitrogen in polystyrene over the pressure range of 8.1 to 30.4 MPa, and at elevated temperatures up to 190 C. A mercury pump was employed to attain the high pressures in the reservoir chamber. The pressure in the sorption chamber was first measured one minute after first subjecting the sample to the high-pressure gas. This delay in measurement was a result of the pressure instability produced by the initial expansion of the gas into the sorption chamber, this was further compounded since the gas was not pre-heated to match the temperature of the sorption chamber. The unstable pressure initially observed contributed to the difficulty experienced in detennining the initial pressure reading required to calculate the equilibrium solubility. To reduce the magnitude of this error, the researchers employed a large sample of grams, which was thinly cut into strips to increase the mass diffusion rate (or reduce the time required to obtain equilibrium solubility) and thus increase the magnitude of the pressure drop. Utilizing the stepwise sorption technique described earlier, the solubility was then calculated as a function of pressure. Lundberg et al. [ , and Lundberg [44] used a single chamber sorption apparatus to determine the solubility of gases in polymers at pressures between 3 and 71 MPa, and temperatures between 102 and I8S0C. The stepwise sorption expriment was used to estimate the solubility and diffusivity of a gas in a molten polymer. Dumll 1451, and Durriil and Gnskey [46,47) employed a pressure decay method with a single chamber apparatus to investigate the solubility and diffusivity coefficients of

34 nitrogen, helium, carbon dioxide, and argon in molten polyethylene, polyisobutylene, and polypropylene it pressures up to 2 MPa. Before coming into contact with the sample, the test gas was preheated in a themostatted air environment. The first pressure reading however. was not recorded until 100 seconds after the gas first contacted the polyrner sample. A stepwise sorption rnethodology was used to calculate the solubility as a function of pressure. ther researchers have utilized the pressure decay method only to measure the solubility characteristics of gases in polymers at pressures up to 2 MPa [48-50) and 8.3 MPa [511. Stem and De Meringo 1521 used a dual chamber system to measure the solubility of carbon dioxide in cellulose acetate at pressures up to 4.6 MPa. Sato et al. [53] employed a three-chamber sorption apparatus to measure the solubilities of carbon dioxide and nitrogen in polystyrene for pressures up to 20 MPa, and temperatures in the range C. The sorption chambers were controlled to within 0.05 K by a constant temperature air bath. In a later publication, Sato et al. [54] reported on the solubility of carbon dioxide and nitrogen in polypropylene, high-density polyethylene. and polystyrene. PVT measurernents of the polymer at high temperatures and pressures were conducted to provide the volume of the polymer necessary for the solubility calculations, while the swollen polymer volumes caused by gas dissolution ai different pressures and temperatures, were predicted using the Sanchez Lacombe Equation of State Volume Decay Sorption Technique As the name irnplies, volume decay sorption techniques measure the volume change of the gas due to polyrner sorption in a closed system ai constant pressure and temperature. The

35 mass uptake of the polymer, or equivalently the solubility. is indirectly detennined from measurements of the volume decay. A volume decay sorption apparatus was utilized by Rosen [58] to measure the solubility and diffusivity of acetone in cellulose acetate. methyl chloride vapour in polystyrene, and water vapour in neoprene, at sub-atmospheric pressures. The system was designed as an alternative to the quartz spnng apparatus. Mulrooney [59] used a constant pressure sorption concept based on the volume decay method to investigate the solubility and diffusivity of liquid blowing agents such as isopentane in polystyrene at elevated pressures. A positive displacement syringe pump capable of operating in a constant pressure mode was used as the constant pressure source, while the entire assembly was operated in a thermostatted air bath for constant temperature control. The reasoning was that since the system was closed. any volume changes occumng in the blowing agent were correlated to the piston movement of the pump and electronically recorded. However, the swelling effect of the polymer could not be accounted for. If the volume of the sample remained constant, then al1 the mass gained by the polymer would correspond to a,volume Ioss of the isopentane. If the sample volume increased equally as the volume of isopentane decreased, then no net change in volume would be observed. However, if the volume change of the isopentane was less than the volume change of the sarnple. the net measured volume change would be underestimated Piezoelectric Quartz Sorption This sorption technique, which is usually applied to organic solvents, measures solubility based on the principle that the vibration frequency of a quartz crystal changes in

36 response to a change in the rnass deposited on the crystal surface. The experimental apparatus involved in this measurement is shown schematically in Fig 2.4. The device essentially consists of a sorption cell containing the polymer coated piezoelectric crystal oscillator, and a solvent cell containing the gas. When the gas is introduced into the sorption cell, it is adsorbed onto the polymer and changes in the frequency of the crystal oscillator are measured with a frequency detector, recorded, and indicated on a frequency counter. The frequency change measured in sorption experiments incorporates a number of terms, namely the frequency changes due to: sorption of gas into the polymer, adsorption of gas on to the crystal, coating of polymer film, hydrostatic pressure of ideal gas, and viscous resistance of the gas. Bonner and Cheng [60, 611 experimentally detennined that the frequency of a quartz crystal oscillator varied with temperature and pressure. In order to offset the pressure dependence of the frequency in their sorption experiments, they made use of two crystal oscillators with similar pressure dependencies-a sorption crystal coated with the polymer and an uncoated reference crystal. When a reference crystal is not used, an accunte estimate must be made of the pressure dependence of the crystal oscillator at the experimental temperature. Such an estimate has been reported by Stockbridge [62] for pressures below 0.13 MPa. Utilizing an uncoated reference crystal, Masuoka et al. [63] investigated the solubilities of benzene, cyclohexane, n-hexane, toluene and ethylbenzene in polyisobtylene at low temperatures up to 65 C and low pressures. They tested the effect of polymer coating thickness, in the range of about Pm, on the solubility of the solvent in the polymer, and found that the experimental results were not affected within this range. They also

37 concluded that for polymer molecular weights of 5ûûûû and , the molecular weight had no definitive effect on the solubility. Dispensing with the reference crystal, Wang et al. [64] experirnentally determined the pressure dependence of the crystal oscillator without a polymer coating at pressures up to 10 MPa in an atmosphere of helium In-Line Measurernent of Gas Solubility In-line measurement techniques wen developed from an interest in detemining phase equilibria during the actual foaming process. These techniques usually incorporate the measurement devices in-line with the process. Phase separation (i.e. solubility limits) is then detected visual l y, or by means of sensitive instrumentation In-Line Monitoring Dey et al. [65], and Zhang et al. [66,67] reported on the use of an in-line technique to measure the gas solubility in various polymers during the foam extrusion process. The apparatus consisted of an extruder with a specially designed opiical window and flow restrictor valve positioned between the die and the end of the extruder. Through this window. the onset of bubble formation could be observed using a microscope-ccd camera- monitorhecorder system. In order to observe the appearance or disapparance of bubbles during phase L separation, a low pressure in the optical window was initially used so that a two-phase polymer-gas mixture existed. The pressure was then gradually increased until the polymer and gas appeared to be a single-phase solution. This pressure was considered to be the lowest pressure required to keep the gas in the solution under the specified conditions.

38 Combining the information with the gas flow rate and the melt throughput, the solubility was then calculated, It was found that the degree of mixing (single- or twin-screw extruder). the screw rotational speed, and polymer throughput were parameters affecting the in-line measurement of gas solubility. ne reported advantage of this method was that the solubility data could be recorded in real-time and therefore accounted for the dynamic nature of the extrusion process and the possible role played by the extrusion process in gas dissolution and bubble nucleation in the melt In-line Infrared Sensors Near infrared (NIR) spectroscopy is a technique developed for monitoring the polyrner/blowing agent mixture duting polymenc extrusion foaming processes. Infrared monitoring of the foam extrusion process is achieved through the use of dual transmission infrared sensors or probes, which transmit NIR light throügh the polymer running in a fiow cell. The probes are linked with fibre-optic cables to a fourier transfomi near-infrared spectrometer (Fï-NIR), which records the absorption spectra of the melt. The flow ce11 for NIR measurements is located at the exit of the extruder on a side stream of polymer flow taken from the main fiow stream. A gear pump is installed downstream of the flow ce11 to realize a steady flow rate. NIR spectroscopy has been reported to have a number of advantages, such as remote data collection, and ease of sample handling. It has been used for the online measurements of polymer composition [68], polymer viscosity [69] and concentration of HCFC in poiystyrene [70].

39 Nagata et al. [711 utilized on-line NIR spectroscopy to measure the C2 concentration in molten propylene for C2 extrusion foaming processes. Three different C2 concentrations and three separate flow rates were used experimentally. In order to remove the baseline of the obtained NIR spectra, the wavelet transform was employed, i.e., the given signals were represented by the linear combinations of known functions. They claimed that the experiments demonstrated a strong correlation between the NIR spectrum and the C2 concentration. By performing the wavelet transformation and removing the baseline from raw NIR spectrum, the effects of temperature and flow rate were erased. This technique however is limited in practice: if any dispersed material is present in relatively large quantities in the melt. the incident light from the probes is scattered out. The absorbance may then become too weak to be analysed precisely. Furthemore, the calibration curve must be developed whenever the polymer andfor the processing conditions art: changed. Thomas et al. [70] also investigated the ability of NIR spectroscopy to detect bubble formation in the die as a function of blowing agent concentration and pressure for the PS/HCFC 142b system. n decreasing the die pressure gradually, they observed that NiR sensors could detect degassing of the melt. The appearance of bubbles caused scattering of the light, which induced a large increase in the attenuation in the level of baseline absorbance of infrared waves. They ais0 investigated the effect of talc on the performance of NIR spectroscopy, and stated that for talc contents 4 wt%, NIR analyses were still possible. Piché et al. [72] patented a non-invasive technique involving the use of ultrasonic sensors, which probe sound propagation charactenstics through the polymer melt. These

40 sensors have ken used in-line to measure the thermodynamic properties of polper/blowing agent systems 1731, and off-line to measure the influence of blowing agents on the rheology of foams [74J. This technique is based on the measurement of the velocity of sound and the attenuation of low amplitude ultrasonic waves in the mixture. This principle is illustrated in Figure 2.5. The sample material is confined between two steel rods. Piezoelectric transducers, which are used to generate and detect ultrasonic waves, are attached at the ends of the steel buffer rods. The emitting transducer produces short burst of ultrasound that travel down the fint buffer line and arrive at the polynier/steel interface with amplitude Ao. The acoustic energy transmitted in the sample reverberates between the two interfaces producing a series of echoes. Ai, A*, A,... that are detected by the receiving transducer. The sound velocity, v (ms"), is determined from the thickness, e (m) and the time delay, At (s), between successive echoes: and the attenuation, a (db/cm), is given by the relative amplitude of successive echoes: Sahnoune et al. employed this technique [73] to mesure the thermodynarnic propeaies of pol y styrene1hcfc 1 4Zb mixtures. For phase separation measuremen ts, they observed t hat as the pressure was decreased, the velocity of sound decreased by as much as 4.51 from a

41 steady state value; this was explained to be due to the phase separation process. The attenuation on the other hand, exhibited a different trend, which was attributed to the themodyniunics state of the blowing agent. 2.3 Physical Blowing Agents Many researchers, employing some of the techniques described above, have investigated the use of physical blowing agents as a replacement for traditional CFC and HCFC blowing agents. This section will review in brief, some of the results obtained for fîuorocarbon blowing agents, inert gases such as carbon dioxide (C2) and nitrogen (N2). and long chah al tematives such as isopropanol Fluorocarbon Blowing Agents The solubilities of HCFC-142b, HFC-134a. HFC- 125 and isopropanol in PS were presented by Daigneault et al. [75] using both a vapour pressure and gravimetric method. Isopropanol was the most soluble followed by HCFC- 142b. HFC- 134a. and HFC Differences in the data acquired by the two methods were reported. especially for HFC- 134a, however, the isopropanol data were reconcilable. The PVT relationship for the PSIHCFC- 142b system was reportedly ubtained using an ultrasonic cell [72]. Boudouris et al. [76] rneasured the solubilities of HCFC-22 and HFC-152a in polystyrene (PS) and poly(methy1 methacrylate) (PMMA) using a piezoelectric quartz technique. Measurements were performed in the Iow-temperature region from 35 to 85 C and at low to moderately high pressures. They found that the solubilities of both

42 fluorocarbons were greater in PMMA, while HCFC-22 exhibited a greater solubility in both PS and PMMA when compared to HFC- 152a. They also found that for al1 systems except the PS/HFC-152a. a gas-induced plasticization of the polymers occurred characterized by convex-shaped sorption isothems Inert Gases Chaudhary and Johns [40] investigated the solubilities of Ni, isobutane and C? in low-density polyethylene (LDPE). Isobutane and C2 were substantially more soluble in LDPE than N2. The solubilities of isobuane and C2 increased with increasing pressure and decreased with increasing temperature, however Cz was found to be less sensitive to changes in temperature. Also, about 2-38 of the LDPE was dissolved in the isobutene, whereas C2 did not extract any of the LDPE. The swelling of the polymer due to gas dissolution was theoretically determined based on the assumption that the mass absorbed by the LDPE has the same mass as the polymer at the relevant temperature and pressure. Sato et al. (531 measured the solubilities of C2 and N2 in polypropylene, high-density polyethylene (HDPE) and polystyrene (PS) at pressures up to 18 MPa, and temperatures to 200 C, using a pressure-decay technique. Their results revealed that the solubilities of both gases increased aimost linearly with pressure, while those of C2 decreased with an increase in temperature. The solubility of N2 however increased with temperature. The solubilities of both gases increased in the order PS, HDPE, and PP. They also found that the solubilities of C2 in PS, HDPE, and PP were not significantly lower than the currently used fluorocarbon blowing agents.

43 The solubility of Cz, N2, and argon in PS and poly(eihylene Terephthalate) (PET) were studied by Zhang et al. 166, 671 using an in-line measurement procedure. Pressures up to 6 MPa were employed, and in both polymer systems, the solubility of the gases ranked as C2>Ar>N2. It was also shown that the solubility of al1 three gases were lower in the PET melt than in PS. 2.4 Theoretical Treatments Theoretical approaches for explaining and predicting the solubility of gas in a polymer have been performed. These theories originate from the prediction of pressure-volume- temperature relationship for a pure component, which are then expanded to polymer/solute systems. The models presented are based on a lattice fluid mode1 in which each molecule occupies r sites (an r-mer) with vacant sites present. Random mixing of the r-mers with each other and with the vacant sites is assumed. The theoretical models, which are summarized and presented in this thesis, include the Flory-Huggins theory [77, 781, Sanchez-Lacombe Equation of State [55-571, and the Simha-Somcynsky theory The Flory-Huggins (F-H) theory [77, 781 was derived from treating the polymer solution as a lauice in which a solvent molecule occupies the same lattice position as the polymer segment. It provides information about the solubility and phase relationships, and assumes that the volume and enthalpy of mixing are zero, while introducing a reduced Gibbs energy parameter, X, to correct the energetic effect of mixing. The pparameter is assumed to be independent of composition and temperature. The original F-H theory was modified by B lan ks and Prausnitz [80], w ho introduced an entropic contri bution to the X-parameter. Despite this modification, the PH theory is considered inadequate to describe polymer

44 solutions because it ignores the equation of state properties of pure components and neglects the effect of pol ymer chah architecture on the intermolecular packing. The Sanchez and Lacombe Equation of State (S-L ES) [ is a lattice fiuid model for pure fiuids and mixtures that requires three pure component parameters to characterize a pure fluid and one adjustable binary interaction parameter. The definitive equation for the S- L ES indicates that if the interaction parameter, and the PVT properties of the components at the solubility pressure are acquired, then the equilibrium solubility of gas dissolved into a polymer can be determined. ne obstacle, however, is that there is scarce information on the PVT properties of polymers and interaction parameters. if the solubility data is available in a limited range. one can determine the parameters by a non-linear regression analysis. Panayiotou and Verra (P-V)[8 11 also obtained an equation of state based on a lattice hole theory. This model differs from the S-L ES in that a constant site volume for al1 r- mers is used, and non-random mixing arising from the molecular interaction is introduced. The adjustable binary interaction parameter in the P-V ES is incorporated as a correction for the binary interaction energy. In the S-L ES, the binary interaction term modifies the characteristic pressures and thus has a different physical meaning. Much like the F-H or S-L ES theories, the Simha-Somcynsky (SS) model [79] stems from treating molecules as segments on a lattice. In the case of a mixture, the Iattice contains both species, which are divided into approximately equally sized segments. However, unlike the other theones, the SS theory allows for a pressure- and temperature-dependent fraction of vacancies or holes that are to express free-volume within the lattice, which account for molecular disorder in the lattice model. The equations derived from the SS theory include

45 temperature- and pressure-independent parameters that account for intra- and intermolecular interactions within the mixture's components. As an improvement to the lattice fluid model, Rodgers and Sanchez [82] determined that adding an empirical correlation for the interaction parameter, improved the predictive scope of the LF model. With the addition of this correlation based on Hansen's threedimensional solubility parameters [83], the LF model was reported to be able to predict solubilities in a11 types of gadpolyrner systems without the use of adjustable parameters; only the pure component equation-of-state and solubility parameters are required. The Polymer Reference Interaction Site (PRISM) theory [84,85] was used by Curro et al. to compute the sorption of a monatomic gas in a polymer liquid. This theory describes the intermolecular packing between polymer chahs and sohte using the integral equation theory of molecular liquids. in order to compute the sorption of a gas in a polymer liquid, the chernical potentials of the solute species in the polymer must be obtained.

46 Concentration of gas Pressure Fipre 2J: Schematic representation of sorption isothenns in rubbery polyrners Concentration of gas I I I I I b Pressure Glassy Figure L2: Schematic representution of sorption irorhenns in glossy polyrners

47 I - Weighing Unit 1 i II C - II 1 1 Vibration Damper Fipre 2.3: Schematic of electrobalance - Frequency Detector Frequency Counter il Recorder ~onst. ~emp Bath Fipre 2.4: Schernatic diagram of the piezoelectric-quartz sorption apparatus

48 Transmitting Ultrasonic Transducer Sample Steel Rod Receiving Ultrasonic Transducer Figure 2.5: Basic Principle of Ultrasonic technique

49 PRBLEM STATEMENT AND PRPSED DESIGN 3.1 verview In this chapter. a conceptual design of the experimental apparatus to measure the gas solubility in polymers is proposed. The pnnciples of solubility measurement based on the three-chiunber pressure decay method are first explored in detail with particular reference to the problem of the swollen polymer volume. Then, an axiomatic approach to the design process is employed to cnate a robust, functional apparatus that measures the solubility of gases in polyrners at high pressures and high temperatures. 3.2 Principle of Solubility Measurement (Pressure Decay Method) In order to measure the solubilities of gases in polymers. the pressure decay method is employed. This method involves rneasuring the amount of gas initially in contact with a polymer sample and the amount of gas remaining in the gas phase after equilibration. The physicai quantities measured are the resulting decrease in pressure as gas is absorbed into the polymer sample, the temperature of the sample and gas, and the volume of the system in which the expriment takes place. A schematic outlining the principle of measurement is shown in Figure 3.1.

50 The apparatus essentially consists of thne sorption cells of known inner volumes, a pressure sensor and a number of valves. Initiaily a polymer sample is placed in sorption ceil 3, and the apparatus is evacuated through valve 4, to remove al1 contaminants from the highpressure sorption cells. Next, the test gas is introduced at pressure Pi (kpa), into sorption cells 1 and 2 only via valves 1 and 2. When the pressure has stabilized, valve 2 is closed and the amount of gas n2 (kmol), present in ce11 2 can be calculated using Eq where Vc2 (m3) represents the inner volume of sorption ceii 2, T (K) is the temperature of the gas, and Zi represents the compressibility factor of the real gas under conditions of T and Pi, as obtained from the thennodynamic gas tables [86]. R is the molar gas constant (kj/kmolx). Valve 3 is then opened to allow the gas to flow into sorption ceil 3 and permit diffusion of the gas into the polymer sample. At the equilibrium point, when the sample has been satunted, the pressure will be relatively constant, and the amount of gas remaining in the gas phase in chambers 2 and 3, n2+3, is calculated using Eq. 3.2: In this expression Pz is the measured equilibrium pressure, Va (cm3) is the volume of sorption ce11 3, Vs (cm3) is the volume of the polymer sample, and Z2 is the gas compressibility under conditions of T and P2. Finally, the amount of gas dissolved into the

51 polyrner, np (kmol), can be determined as the difference of the initial amount of gas before sorption and the amount of gas remaining after sorption, Le: To permit further measurements at higher pressures, a stepwise pressure increase is applied. This is achieved by first closing valve 3 and opening valve 1. This allows the highpressure gas in chamber 1 to combine with the lower pressure gas in chamber 2 to produce an intemediate pressure, Pi+i (where i represents the stepwise pressure iteration number). Valve 2 is then closed. and the amount of gas available in chamber 2 is detennined by substituting the intermediate pressure, Pi+i, and the corresponding gas compressibility factor, Ziri into Eq The residual gas, n3, residing in chamber 3 is computed as: This residual gas amount is added to the gas amount in chamber 2 to obtain the new amount of gas king exposed to the polymer. Valve 3 is then opened, and ai the new equilibrium pressure, P2,ii the amount of gas remaining is calculated using Eq For this step increase in pressure. the quantity of gas that dissolved into the polyrner, n, is calculated using Eq. 3.5, while the total amount of dissolved gas, n. is computed curnulatively.

52 The total solubility x (g-gaslg-polymer), at any stage can also be obtained through Eq. 3.6 where M representr the molecular weight of the gas under consideration and mp (g) is the mass of the polymer sample in the sorption chamber. The principles described above represent a variation on a dual chamber configuration where sorption ce11 1 or 2 would be eliminated. This principle was employed by Newitt and Weale 1141, and Stern and De Meringo 1231, wliile a similar three-chamber configuration was utilized by Sato et al. [53,54]. The three-chamber configuration however, offers a number of unique advantages over a dual chamber technique. Firstly, it allows multiple measurements to be taken over a range of pressures for the same sample through the stepwise pressure method. Second, since evacuation of the system is not necessary after each measurement, the experiments can be done more quickly and efficiently. Finally, by using a fixed initial amount of gas for repeated measurements at a constant temperature, the errors that can be caused by introducing new gas at a lower temperature are elirninated. It is also worth noting that stepwise experiments using the threeshamber configuration are only valid for increasing pressures. If the experiments are perfonned in such a way that the initiai sorption pressure is higher than the proceeding sorption pressures, then desorpiion of the sample occurs due to the decrease in solubility with decreased pressure. The resulting thermodynamic insiability results in foaming of the polymer as explained in Chapter 1, and consequently in unreliable solubility data. Therefore, stepwise sorption experiments must

53 always be conducted by subjecting the polymer to a higher experimental pressure at each step. 3.3 Problern Statement and verall Strategy When a polymer sarnple is subjected to a gas above its melting point or glass transition point, the specific volume of the polymer increases due to the dissolution of gas into the polymer. The degree of the change in the specific volume depends primarily on the extent of the solubility of the gas into the polymer. With respect to the calculation of the solubility, it can be seen from Eq. 3.2, that the measurement depends on knowing the volume of the polymer. During each stage of a typical sorption expriment, the pressure and hence the amount of gas present increases. Since the solubility increases at higher pressures, then the volume of the polymer will also increase. If this change in volume is not accounted for, then the calculated solubility will be affected. The extent of the problem is investigated in the next section Effect of Polymer Dilation In order to fully understand the effect of the polymer swell on the solubility, a quantitative analysis of the polystyrene (PS)-C2 system is performed. Literature values 1211 for the PVT measurements of the PS-C2 system indicated that at a dissolved gas content of 4 wt%, 13.8 MPa (2000 psi), and 220 C, the specific volume of the PS was The specific volume of the pure PS material at 13.8 MPa and 220 C was reported as cm31g. The percentage increase in the specific volume of PS from the original condition was therefore about 4.4%.

54 In order to detemine the effect of the polymer swelling, it is possible to separately evaluate the pressure drop caused by the gas solubility first with the assumption of swelling and then with an assumption of no swelling. Case A: Swelling For a polymer sample of volume Vpl placed in a sorption cell of volume Vc at T, and pressurized with gas to pressure PIA, the number of moles of gas present in the gas phase just before sorption occurs. ni, can be approximated using Eq where Vgi represents the volume occupied by the gas phase in the original instance, ZIA represents the gas compressibility factor at conditions of PI* and T, and R is the universal gas constant. After sorption has occurred, it is assumed that the polyrner volume has increased to VpZ, and thus the gas volume is reduced to Vg2. The equilibrium sorption pressure, Pz, cm be detennined from Eq. 3.8.

55 In this equation, ns represents the amount of gas dissolved into the polymer, and Z2 is the compressibility factor at Pz and T; ns can be calculated as follows: where x and p represent the solubility of the gas into the polymer, and the density of the polymer respectively, at conditions of T and P2, and M is the molar mass of the gas. Substituting Eqs. 3.7 and 3.9 into Eq. 3.8, results in a new expression for the final sorption pressure (Eq. 3.10). The expression for Vgi is obtained as shown in Eq Assuming that the ratio of the polyrner volume to the chamber volume is taken to be equal to 0.5. then:

56 Vg2 is obtained by considering the chamber volume and the swollen polymer volume, VP2 i.e., vg2 = vc - V P~ (3.13) In Eq. 3.14, S represents the degree of swelling of the polymer and is obtained from the PVT data of the PS-C2 system. S cm be expressed as the ratio of the swollen polyrner specific volume (vn) to the pure polymer specific volume (vpi): The implicit assumption made in Eq is that the mass of the swollen polymer is equal to the mass of the pure polymer. In reality, the mass of the swollen polyrner is greater than the mass of the pure polymer because of the infusion of gas, however, in order to simplify the calculations, the assumption is made. Substituting Eqs into 3.10 and solving for Pt A yields the following relatiomhip for the initial pressure:

57 Case B: No Swelling The second case considers the effect of the gas solubility on the pressure drop without assuming that the polymer swells due to the diffusion of gas. In this case, Eq is the starting point for the equilibrium sorption pressure expression and can be rewritten as Eq for case B. The essential difference between the two cases lies in the expressions for VgI and Vg2. Without swelling, the polymer volume and hence the gas volume rernains unchanged, Le., Vg2 = V,, =.5Vc (3.18) Substituting Eq and 3.18 into Eq and solving for Pis gives us the relationship represented by Eq Taking the difference of P2 and PiA or PiB gives the pressure drop to due gas solubility for case A or case B respectively. It should be noted that in both cases, the final sorption pressures are taken to be equal. Table 3.1 gives a list of the parameters used in the final calculation and their values and shows the final cdculated values of PIA and PlB.

58 Based on the calculated values of PIA and Pis. the pressure drop due to gas solubility when the swelling is considered is 2503 kpa, however when no swelling is assumed the pressure drop is almost twice as much at 4545 kpa. The error with respect to the pressure drop when swelling is considered is approximately 81%. If the solubility error is to be minimized, it is therefore necessary to determine the degree of swelling which occurs when the polymer is subjected to gas at various temperatures and pressures (PVT data). Despite the advantages of the pressure decay method, researchers in the past were limited by the availability of information pertaining to the volume of the polymer in the swollen condition or the PVT data. Theoretical models such as the FIory-Huggins theory [77, 781, and the Sanchez Lacombe Equation of State [ were used to predict the specific volume of the polymer at specific temperatures and pressures. However, in the absence of experirnental data. the accuracy of these models to predict specific volume information cannot be verified. With advances made by Park et al. [21], the swollen polymer volume can now be determined as a function of pressure, temperature and dissolved gas content using a foaming dilatometer. With the PVT data available, the overall approach based on the rneasurement principle of the pressure decay apparatus outlined above, will be to apply the specific volume information for the polymer into Eq. 3.2 at al1 temperatures and pressures under consideration. Applying this information will give more accurate values of the gas solubility. 3.4 Background on Axiomatic Design Approach The axiomatic design concept is a systematic method used for guiding the design process. This approach, developed by Suh (861, consists of five basic steps: i) formulation of

59 the desired needs; ii) conceptualisation of design solutions; iii) analysis of the proposed solutions; iv) selection of the most suitable design from the proposed designs, and v) implementation of the chosen design. In the context of this thesis, the basic goal of the axiomatic approach is to establish a scientific foundation for the design process, and thecefore provide a fundamental basis for the creation of a solubility measuring apparatus. The fint step in the design process is to arrive at a set of functional requirements (FRs) that satisfy a perceived need for a product. These FRs are representative of what we want to accomplish with the product. In order to satisfy these FRs, design parameters (DPs) are chosen. DPs are tangible entities in the physical dornain that satisfy the stated FRs with the least expenditure of resources. The axiomatic design approach enibles identification of good designs by satisfying two design axioms: the independence Axiom, which guarantees the independent control of each FR, and the Information Axiom which minimizes the information content of the design and assures design simplicity. The mathematical representation of the Independence Axiom is a simple design matrix shown in Eq. 3.20, which also highlights the relationships between FRs and DPs. The left side of the equation represents what we want to achieve and the right side represents how we hop to satisfy the FRs: or equivalently:

60 FR1 ' FR2 FR, FR,, b = The elements Au in matrix A can be either "X" or '*", where element "X" is indicative of a strong relationship between the FR and corresponding DP, and "0 represents a weak or non-existent relationship between the respective FR and DP. When the diagonal elements of the design matrix are zero, the Independence Axiom is satisfied and the design is said to be uncoupled. An uppet or lower triangulat matrix signifies a decoupled design which also satisfies the axiorn. In al1 other cases, the design is coupleci, ohich means that the FRs cannot be controlled independentl y. There are two very important facts about the design and design process, which should be mentioned. Firstly, the FRs and DPs have hierarchies and they can be decomposed into lower levels. Secondly in such a decomposition, an ith level FR cannot be decomposed into the next level of the FR hierarchy without first finding a DP to satisfy the ith level FR. This hierarchal structure simplifies the design pmcess one level at a tirne. 3.5 ~xiomatic Design of the verall Process Design 1 Using the axiomatic principles outlined in the previous section, an apparatus was designed in the earlier stages of this research, to measure the gas solubility in polymers. The design basically employed a dual chamber configuration in a thenostatted air bath. Although the design did not hinction as expected, it was analysed to provide useful

61 information on the performance of various components in order to facilitate further improvements. It allowed us to identify critical design features, which were previously not considered. Some of the parameters, which were identified as trouble spots were: i) Temperature control: the use of a thermostatted air bath did not provide adequate temperature control; temperature fluctuations were frequent and difficult to avoid. ii) Pressure measurement: the pressure decay due to solubility was found to be very low because of the chamber size and configuration. This pressure decay could not be detected at times, since it was within the error range of the measuring equipment, and this correlated to huge errors and unreliable results. Subsequently, this first design was modified to produce a secûnd design, which is presented in the following section Design II First Level FRs and DPs The parental FR for the design is defined as "a solubility measuring device", while the corresponding DP is "a modified pressure decay apparatus". The FRs and DPs at the pnmary level of the axiomatic design hierarchy are then chosen as foiiows: FRl = Maximize experimental efficiency FR2 = Vary and control experimental temperature FRs = Vary and control experimental pressure FR4 = Compensate data for polymer swelling due to gas dissolution

62 DPl = Multiple chamber configuration DP2 = Heating and cooling system DP3 = Gas pumping system DP4 = Memurement of PVT data The relationships between these first level FRs and DPs are best described in the matrix shown in Eq. 3.22: X X X FR4 00x0 0 X In order to improve the overall equipment performance and reduce the experimental time (FRl), a multiple-chamber configuration is chosen for the apparatus layout (DP,). The experimental efficiency depends on the ease with which both the pressure and temperature of the system can be varied and controlled (DP2 and DP3). Consequently, FRl is also a function of DP2, and DP3. For a functionaily robust apparatus, it is necessary to perform solubility experiments at various experimental temperatures (FR2) within the material processing temperature range. To accomplish ihis task, a heating and cooling is chosen to maintain accurate and uniform experimental temperatures. The pressure also plays a similar role in the versatility of the appmtus. To investigate the solubility behaviour at high pressures (FR3), an adequate gas delivery system should be established (DP3), which is capable of achieving high gas pressures. Any auxiliary

63 components must also be designed to support such high pressures. Although the temperature control function will be independent of the pressure control, it is expected that by changing the gas pressure (DP3), the gas temperature (F&) will also experience a slight change because of the isentropic compression of gases. Hence there is a relationship between FR2 and De3. Changing the experirnental temperature (DP2) will also affect the gas pressure (FR3), however, sorption experiments are normally conducted isothermally, such that for any series of experiments the temperature is maintained constant for various pressures. As a result, FR3 does not depend on DP2. Since the pol ymer dilation or volume change that accompanies gas dissolution is unavoidable, the PVT information must be determined for the specific polymer and gas combination, in order to correct the solubility data. This requires a series of PVT experiments which can be obtained from the literature or which can be performed using an extruder-based foaming dilatometer [211. Ultimately, obtaining the PVT information is done independently of the solubility-meilsuring device; as a result, this FR can be independently achieved. The design evaluation results in a decoupled design (lower triangular matrix) as shown in Eq. 3.22, consequently decomposition into the next level FRs and DPs can proceed Lower Level FRs and DPs For the second level of the proposed design, the following functional parameters are detived from the first level FRs. The corresponding DPs were then chosen to uniquely satism the particular FRs.

64 Decomposition of FR,: FRll : Isolate gas prior to sorption FR,, : Allow sample to absorb gas FR,, : Perforrn multiple experiments FR,, : Minimize equipment expansion FRl5 : Minimize or prevent leakage X X X X X X X X DP,, : Pressure chamber # I DP,, : Pressure chamber #2 De, : Pressure chamber #3 DP,, : Equipment sizing De, : Metal to metal seal Decomposition of FR1 yields the decoupled relationship shown in Eq Since the accurate calculation of the solubility depends on the determination of the initial amount of gas (see Eq. 34, it is necessary to be able to evaluate the total amount of gas prcsent before sorption takes place. This is accomplished by isolating the gas (FR, 1) using a high-pressure nce the quantity of gas is detemined, sorption (FRl2), is facilitated using another high-pressure chamber (DPi2) which contains the sample. In order to repeat the sorption process without having to introduce new gas or replace the sample (FRi3), a third high-pressure chamber (DPi3), which essentially acts as a high-pressure gas reservoir, is implemented. The ability of this chamber to act as a reservoir obviously depends on the presence of the previous two chambers. Cowquently, FRi3 is a function of DPi i, DPi2 and DP13. T reduce the expansion of the equipment at the high temperatures and pressures involved (FRl& the appropriate choice of materials must be made (DPI4). This in turn influences al1 thne high-pressure chambers (DPl1, DPI2 and DI+,). Finally, in ordet to prevent leakage of the high-pressure gas to the environment, a metal-to-metal seal is employed.

65 FR, : Minimize expansion of tubing DP,,, : Chamber thickness DP,, : High - pressure valves DP,, : Thick - walled tubing 1 Due to the high temperatures and pressures involved in the measurement of solubility, it is necessary to consider the expansion of the components as this can change the volume of the system and ultimately produce errors in the final result. Thus, in order to minimize the expansion of the high-pressure chambers (FR14i), and to ensure the high-pressure durability of the apparatus, the chambers are manufactured using mild steel with an appropriate thickness (DPi41). Special ty high-pressure valves (DP1d2) are also preferred to minimize the expansion and provide complete shut-off at the high pressures involved (FRl42). Thick walled, srnall diameter high-pressure tubing (IN') is used throughout the apparatus to control the tubing expansion. FR,, : Provide heating source FR,, : Maximize heat transfei FR, : Minimize heat iosses FR2, : Control temperature FR, : Detect temperature FR, : Maintain T unifomity DP,, : Electrical heater DP,, : Heat transfer fluid DP?, : Insulation DPZ4 : Temprature Controller DPs : Them~couples DP,: Circulating fan

66 The decomposition analysis of the second FR is decoupled and is as follows: In order to increase the temperature of the system (FR21), a heating source is required in the fom of an electrical heater. The upper limit of the temperature range for the system was defined to be approximately 200 C. Consequently, the electric heater was chosen to cover this temperature range. The other components of the system will also be designed for this maximum operat ing temperature. To maximize the heat transfer (FRz2) between the heat source and the components (chambers. valves, etc.) of the system, a heat transfer fluid (DP22) is needed. However, the chatacteristics of the fluid will also influence the temperature distribution or temperature uniformity within the system (FR3j). In order to minimize the heat losses from the system (FRu), insulation is provided (DP23). Temperature control of the system (FRu) is achieved through the use of dedicated thennocouple controllers (DP24), while temperature detection (FRa) is accomplished with the use of thermocouples (DPz). Finally, the temperature uniformity of the system (FRza) is rnaintained via circulating fans (DPZ6). Decomposition of FRJ: ] [ ]( FRN : Varypressure X X DP,, : Syringe pump FR3, : Maintain pressure = X DP32 : Shut off valves FR,, : Detect pressure X Dq3 : Pressure transducer In order to obtain different experimental pressures (FR,,), a positive displacement pump is while high-pressure shut-off valves (DPn2) are used to prevent the escape of the gas and thus a loss of the system pressure (FRJ2). To detect the pressure of the

67 system during experimentation (FR33), a high accuracy pressure transducer is used (DP33. The resulting second level analysis of FR3 is decoupled. It should be noted that most commercially available high-precision transducers normally do not have high operating temperatures; as such the transducer must be physically separated from the high-temperature system, while king connected to the test gas via 1/8" tubing. 3.6 Summary The pressure decay principle utilized for solubility measurement was explored in this chapter. Based on this principle, accurate measurement of the solubility depends on knowing the volume of the polymer king used in the experiments. The extent of the problem was investigated in a theoretical approach, utilizing PVT data reported in the literature [211. It was found that the pressure drop error due to the solubility could be as much as 81% if the swelling of the polymer is ignored. Consequently, the PVT data for the polymer-gas system should be applied in order to correct the solubility. An Axiomatic Design Approach was used to design a solubility measuring apparatus. The design was shown ta be either decoupled or uncoupleci at al1 leveis, however, the design needs to be tester or verified with some preliminary experiments.

68 Table 3.1: Parumefers for cakufating PIA and Pie - Parameter -- Value

69 - Sorption ceil2 To vacuum Sorption cell 1 Sorption ceil3 PumP Figure 3.1: Schematic of pressure decay rnethod based on three chambers

70 Chapter 4 DETAILED DESIGN 4.1 verview Chapter 4 presents the design details of the solubility-measuring device. The chapter begins with a detailed design of the essential elements for the solubility apparatus, such as the sorption cell size, ideal sample size, thermostatted oil bath size, and vacuum pump capacity. Finally, each feature of the solubili ty-measuring device is h ighl ighted. 4.2 Pre-solubility Calculations In this portion of the thesis, calculations will be presented for parameten that need to be determined in order to constnict the solubility apparatus. The axiomatic design process resulted in a qualitative selection of the components; however, some quantitative analysis must be done, in order to determine the specific requirements for these components. Some of the calculated parameters include the sorption chamber sizes, the theoretical sample size, the oil bath heiting requirements, and the vacuum pump capacity Theoretical Sample Size As part of the design of the solubility-measuring device, it is necessary to determine the relative theoretical sample size that will produce a certain pressure decay due to solubility, based on an assumed solubility. This parameter is important because if the sample size is too

71 small, then the pressure drop due to solubility may be too low to be measured. The rationale for determining the theoretical sample size (Figure 4.1 ) assumes that the polymer does not swell due to the gas dissolution, and is based on the equation of state of the gas under consideration. if a polymer sarnple of volume Vp (m3), is placed in a sorption chamber of intemal volume Vc (m3), at temperature T (K), and pressure PI (kpa), then before sorption begins, the amount of gas, ni (kmol), occupying the remaining volume can be cdculated as: where Zi is the gas compressibility at Pl and T, and R is the universal gas constant (kl/kmolk). After sorption has occurred, the equilibrium pressure, Pz, can be detemined from Eq where ns represents the amount of gas dissolved into the polymer: In Eq. 4.3, x represents the weight-percent solubility of the gas into the polymer with respect to the weight of the polymer, p is the density of the and M is the molar

72 mass of the gas (g/kmol). Substituting Eqs. 4.1 and 4.3 into 4.2 and simplifying the resulting expression yields the expression shown in Eq Finally, extracting Vc from Eq. 4.4 and using y to represent the ratio of the polymer volume to the sorption chamber volume (Vflc), the following equation for the equilibrium pressure is obtained: Since we are interested in finding the sample size, which will produce a reasonable pressure drop nt the lowest solubility, the lowest value for the solubility, x, and the comsponding pressure and temperature obtained from the literature were used for the calculation. It is expected that at higher solubilities, the pressure drop will be greater. The parameters used in the calculations and their values are summarized in Table 4.1. The resulting variation of the pressure drop due to solubility with different relative sample sizes is depicted graphically in Figure 4.2. From Figure 4.2, it can be seen that the pressure drop due to a solubility of 0.94 wt% increases gradually as the relative sarnple size increases from 0% to 70% of the chamber size. Further increase in the sample size beyond 70% will lead to a higher pressure drop, however there are a number of limitations to using a larger sample size. First, the polymer is known to swell or increase in volume due to gas dissolution; for some polymerlgas systems, this

73 increase has been shown to be as high as 20% [87], consequently having samples that occupy tw great a portion of the sorption chamber may physically restrain the polyrner sweliing during gas dissolution and thus impede the solubility measurements. Second, for larger polymer samples. it will take an increasingly long tirne to saturate the samples with gas at any particular pressure due to the increased diffusion time. The diffusion time, t ~ can, be estimated from the relationship represented in Eq. 4.6: where ID is the diffusion distance and D represents the diffusivity. In this relationship, the diffusion time decreases as the diffusion distance decreases and as D increases. In general. it is desired to reduce the experimental time; to this end, it would be advantageous to use as thin a sample as possible, which reduces the diffusion distance and hence reduces the diffusion time required to reach the equilibrium solubility. In addition, if the experimental time is too long, then premature decomposition of the polymer may occur, before the equilibrium sorption point is reached, particularly at high temperatures. In order to reduce the diffusion time while using a sample volume as high as 50% of the sorption chamber volume, the sorption chamber (chamber #3) should be designed to have a large intemal diameter to height ratio, as shown in Figure 4.8.

74 4.2.2 Sorption CeII Size Reiative Chamber Sizes The idea behind utilizing three sorption chambers or cells dictates that the best chamber volume ratio (ce11 1:cell 2: cell 3) must be found in order to maximize both the number of experiments that can be performed before having to replace the sample, as well as the stepwise pressure increase in each stage of the expriment (as explained in Section 3.2). Consequently, the most effective chamber volume ratio is evaluated by comparing the maximum achievable pressure and the number of stepwise pressure increases for different chamber ratios. The methodology employed to determine the best chamber ratios is based on the equation of state of an ideal gas, Eq. 3.7: where P = gas pressure (kpa) V = chamber volume (m3) n = number of moles of gas (kmoles) R = molar gas constant (kj/kmol K) T = gas temperature (K) This theoretical method, which is illustrated in Fig is based on the assumption that no sample is present in the chambers, and therefore no sorption occurs. The calculation procedure for an assumed volume ratio is described below.

75 i) Beginning with an evacuated system, cells 1 and 2 are filled with gas at an arbitrary temperature and pressure, Le., 20 C and 13.8 MPa; i i) the amount of gas in chamber #l and #2 is determined; iii) iv) with valve 1 closed, the amount of gas in chamber #2 (n2) is then determined; with valve 2 opened to allow the gas to expand into chamber #3, the resulting theoretical pressure (P2+3) is calculated, and valve 2 is then closed; v) from the remaining arnount of gas in chambers 1 and 2, the resulting pressure (Pl+*) is catcu lated; vi) vii) valve 1 is closed again, and the amount of gas in chamber #2 (II*) found; with valve 2 opened, the combination of gas in chambers #2 and #3 produce a new pressure (P2+3,i) which is found; this pressure reprrsents the ith stepwise pressure increase over the initial chamber pressure P2+3; viii) valve 2 is closed and the procedure is repeated from step (iv), until the stepwise pressure increase is less than approximately 0.7 MPa (100 psi). At this point, the pressures in al1 three chambers are approximately equal. This iteration was performed for 4 different chamber ratios, and the results obtained are depicted graphically in Figures 4.4. As iiiustraied, the chambers ratio 1 :S: 1 gives the largest range for the stepwise pressure increase, ranging from about 32 % of the initial pressure value on the first iteration to a maximum of 6096 of the initial pressure for the sixth iteration. The other chamber ratios investigated were not so favourable. Based on this ratio, another set of calculations was performed. The chamber #2:chamber #3 ratio was kept constant at.sa or equivalently 1:2, and the chamber #I value was varied from 1 to 12. Since chamber #I acts as a pressure reservoir, then the larger the reservoir, the pater the

76 obtainable pressure range for stepwise experiments. Figure 4.5 depicts the result. As the chamber #l value is increased from 1 to 12, the maximum achievable stepwise pressure dramatically increases at first and then more gradually. Further increases in the size of chamber #1, beyond a chamber ratio of 10:1:2, does not significantly affect the maximum achievable stepwise pressure. As a result, a relative chamber ratio of 10: 1:2 was chosen for the sorption cells. It should be noted at this point, that the relative chamber volumes discussed thus far refer only to the volume occupied by the gas. The absolute chamber volumes were detennined based on the overall space requirements of the system, and were chosen to be as follows: chamber #1-200 cm3. chamber #2-20 cm3. and chamber #3-80 cm3. The absolute value for chamber #3 was obtained by considering both the relative gas volume and the relative volume of the sarnple (50% Vc3) as determined in Section Absolute Chamber Sizes There are a number of formulas available to aid in the design of a cylinder to withstand a certain intemal pressure. However, in the design process. a nurnber of factors must generally be taken into consideration, namely, (i) the kind of material king used whether ductile or brittle, (ii) the construction of the cylinder ends (whether open or closed), and (iii) the classification of the cylinder, Le., thin- or thick-walled. Generally, a cylinder is considered to be thin-walled when the ratio of the wall thickness to insider diarneter is 0.1 or less, and thick-walled when this ratio is greater than 0.1. For this design, thick-walled cylinders were chosen. For a thick-walled cylinder of intemal radius Ri, and extemal radius Rz, subjected to an internai pressure Pi, the tangential

77 stress, S,, developed in the cylinder at any radius, R, can be expressed by Lamé's Equations 1881: Similarly, the radial stress, Sr, can be expressed as: An investigation of Lm6's equations for intemal pressure reveals that the maximum stress, which occurs. is the tangential stress acting on the inner surface at radius Ri, Le., Solving Eq for the intemal radius, Ri, gives Eq. 4.11, which is used along with the parameters listed in table 4.2, to calculate the minimum extemal radii for the high-pressure chambers. Al1 of the design details of the chambets cm be found in Figures

78 4.2.3 Thermostatted il Bath The first design explored for the measurement of solubility, utilized air as the heating medium to provide temperature uniformity for the system. However, this design was found to be inadequate especially at high temperatmes. To improve the heat trmsfer from the heat source to the system's components, a heat transfer tluid (Hm was chosen. Table 4.3 compares the heat capacities and heat transfer coefficients of air and the HTF. It can be seen that the HTF has a higher heat capacity than air, which means that it takes more energy to heat to the same temperature. However, it has a higher heat transfer coefficient than air, which makes it more efficient as a heat transfer medium. The HTF is one component of the overall temperature control system. The other components of this system include a stainless steel bath, immersion heaters, type J thermocouples, and temperature controllers. This remainder of this section presents the calculations perfomed to determine the heater size required. The kilowatt heating capacity required for the immersion heater was determined from the following equation: w here QF = energy to heat initial charge of heat transfer fluid (kl) Qc = energy to heat steel tank (id) Ls = surface heat Iosses (kw) t = heating time (hours)

79 QF for the heat transfer fiuid was cdculated using its density-pp, the volume-vf, the specific heat capacity-cf, and the temperature difference for heating-at (Eq. 4.13). Qc for the stainless steel tank was determined in a similar manner using the corresponding values for stainless steel. (Eq. 4.14). Finally, the surface heat losses from the system at the steady state were determined based on the exposed area, A. At the temperature difference for heating. a value for the heat flux, h, from the insulated surfaces was found using heat loss curves for insulated metal surfaces. The heat loss is then given by: The values used in this calculation are surnrnarized in Table 4.3. Based on a heating time of 2 houn and a maximum temperature of 200 C, the calculations indicated that a total heating capacity of 4.5 kw would be needad. Two immersion heaters (Chromalox MT- 330AM40) rated at 3 kw each were chosen and installed for a total heating capacity of 6 kw. The initial heating time to increase the bath temperature from 25 to 200 C would therefore be reduced to a little less than 2 hours.

80 4.2.4 Vacuum System Prior to perforrning the actual solubility experirnents, it is necessary to completely evacuate the system to remove any contaminants such as air. This provides a consistent reference point for al1 the experiments, and also ensures that the measured solubility value is due solely to the gas under consideration. in order to determine the capacity of the vacuum pump required to cornpletely evacuate the solubility-measuring device and provide an acceptable vacuum level of 29.û"Hg within a reasonable time frame, the following equation was used: in which Q = total amount of air to be removed (ft3) V = volume of chambers plus connecting piping (ft3) Pl = initial pressure P2 = required pressure This equation is based on the assumptions that no rnoisture is present and no leakage occua. From the design of the chambers (Figures ), the total volume of the high pressure chambers is approximately 320 cm3. To account for the interconnecting fittings and tubing an extra 20% of this value is added, i.e., the total volume to be evacuated is taken as 384 cm3 ( ft3). The required vacuum level is 29."Hg. Based on these values, the total amount of air to be removed to reduce the pressure inside the system from atmospheric

81 pressure to a vacuum level of 29.WHg is 1396 cm3 (0.049 ft3). The corresponding pump capacity is calculated as: Volume of air to be removed (ft3) Capacity (ACFM) = Evacuation time (min) In order to evacuate the system within one minute, the corresponding pump capacity was calculated to be ACFM. Consequently a vacuum pump (Gast: DAA-VI 75-EB) with a free air capacity of ft3/min (CFM) was used. 4.3 Solubility Calculations Based on the apparatus developed, the solubility of gases in polymers cm be calculated by following the principle outlined in Section 3.2 with some slight modifications. A schematic of the measurement principle is shown in Figure 4.9. With the polymer sample in chamber #3, the gas is introduced into chamber #l and #2 only. At the equilibrium condition, valve 4 is closed and the amount of gas, n2.1 (i=l) present in cell#2 is calculated as follows: where Piai = gas pressure in chamber #2 Vc2 = chamber #2 volume T = system temperature Zii = gas compressibility factor at Pi and T

82 R = gas compressibility factor Vo = outside volume To = average temperature of outside volume & = gas compressibility factor at Pl and T Subscript i refers to the stepwise pressure iteration number After opening valve 5 to allow expansion of the gas into chamber #3, the gas starts to dissolve into the polymer. At the equilibriurn condition, when the pressure remains relatively constant, the amount of gas remaining in the gas phase in chambers #2 and #3, n?+~,is calculated using Eq. 4.19: where: Pri = equilibrium sorption pressure Vc3 = chamber #3 volume Vs = swollen polymer volume Z2,i = gas compressibility factor ai P2 and T 2: = pas compressibility factor at P2 and To The amount of gas dissolved into the polymer cm then be calculated as the difference between Eqs and 4.19, i.e..

83 It is important to note that the swollen polymer volume is estimated from the PVT data of the PS-CI system. The initial swollen polymer volume is taken as the specific volume of the PS- C2 at the temperature under consideration and at zero pressure. After calculating the solubility with this volume, the resulting solubility value is then used to obtain the next estimate of the swollen volume. This iteration is performed until the solubility value converges. For stepwise pressure increases (i >1), chamber 2 is isolated and the amount of gas is calculated as in Eq However, there is now a new component, which is the residual amount of gas in chamber #3,( 113,i.~) from the previous stepwise pressure. This is calculated using the data from the previous stepwise pressure iteration: the solubility is then found as: np.i = n2,i + n3.i-~ - n?+3,i The overdl solubility at each step is calculated as the sum of dl the np,i up to that step. 4.4 Summary The parameters needed for the construction of the solubility-measunng device were presented in this chapter. A theoreticai sarnple size was calculated as a ratio of the chamber size, such that a reasonable pressure drop would be achieved at low solubilities. A very small sample size may produce too smail a pressure drop, while a large sample size will

84 reduce the space available for the polymer to swell with gas dissolution. Consequently a sample size of 508 of the chamber size was determined to be ideal. The chamber size ratio was also calculated in order to optimise the stepwise pressure experiments. A ratio of 10: 1:2 for the chamber #1: chamber #2: chamber #3 ratio was selected. The designed temperature control system consisted of a heat transfer fluid bath heated with immersion heaters. The heating time to 200 C was found to be 2 hours. Finally. the procedure for calculating the solu bility using the designed system was illustrated.

85 Table 4.1: Parameters for calculating the pressure drop due to different sample sizes Parame ter Value Varied from 3.45 kpa to kpa Depends on Pi and T [86] Depends on Pz and T [86] 0.94 wt% [53] i.s x id g/m kj/kmol* K K [53] gkmol varied from. 1 to 0.7

86 Table 4.2: Parameters for calculating chamber thickness Value Parame ter Chamber #1 Chamber #2 Chamber #3 st 406 MPa 406 MPa 406 MPa p I 69 MPa 69 MPa 69 MPa RI 32 mm 18 mm 54 mm Minimum R2 38 mm 21mm 64 mm

87 Table 4.3: Properties of Air and Heu? Transfer Fluid Propexty 1 Air 1 Heat Transfer Fluid Heat Capacity (kj/kg K) Heat transfer coefficient (W/m K)

88 Table 4.4: Parameters for determining the heating capaciîy - Parameter Value 868 kg/m m kj/kgk 175 K 8000 kg/m x 10' m 0.5 kj/kgk 20 wlftz 12 ft2 2 hours

89 SRPTIN BEFRE SRPT IN AFTER SRPTIN Figure 4.1 : Schematic for detennining the pressure drop due to solubility fur different VpWc ratios Pressure (MPa) Figure 4.2: Relative sample size vs. pressure drop due to 0.94% solubility ot 140 C

90 ......m... '.'...*...'...*.' S......*... Valve 2.W '.*.*.'.*.'.'.*.'.*.*.' a I f I Y Figure 4.3: Determination of relative sorption ce11 sires Figure 4.4: Stepwise Pressure Increase vs. lteration No. for 4 chamber ratios

91 Stepwiss Pmseure Iteration No. Figure 4.5: Stepwise Pressure Increase vs. Iteration No. for 7 chamber ratios

92 DFTnlL B (Sealing Surface) DETAIL x l/bm Fernale NPT C SECTIN AA 1 Figure 4.6: Detailed schematic of high-pressure chamber #l

93 7"" ' 7 112' NPT Female LID #3 Quantity: 1 LID #1 Quantity: 1 Figure 4.7: Detailed schettaric of high-pressure cha~nber #2 and hîgh-pressure chamber lids

94

95 Figure 4.9: Photograph of solubility measuring apparatus Flpn 4.10: Photograph of pressure meusurement and temperature control instrumentation

96 I 1 AVERAGE I I EXTERNAL i TEMPERATURE ' valve 3 Valve 4 Valve 5 Valve 6 ' I I - 1 i I I 1 % 'fc1 Chamber #2 I I 1 I Chamber #1 Chamber #2 I va va THERMSTATTED IL BATH AT TEMPERATURE T I I Figure 4.11: Schernatic for calculating solubility

97 5.1 verview The design concepts developed in Chapters 3 and 4 identified the major components necessary for the development of a solubility-measuring apparatus. In this chapter, a complete description of the experimental apparatus is presented. In addition. o description of the experimental materials used. and the experimental procedure adopted to determine the solubility information are also presented. The chapter concludes with a summary of the results obtained, a discussion, and an in-depth error analysis of the system. 53 Experimental Apparatus The sorption apparatus developed in this thesis to investigate the solubility parameters of blowing agents in polymer melts is shown schematically in Figure 5.1. Actual photographs of the system can be found in Figure 4.9 and For descriptive purposes, the apparatus can basically be divided into two systems: a temperature control system and a pressur measurement system. The sorption apparatus utilizes a thermostatted oil bath to achieve and maintain experimental temperatures up to 200 C. A stainless steel tank (HL2" x L24" x D24") was I 83

98 manufactured using grade 304 stainless steel with a thickness of mm (1/8"), and filled with approximately 90 litres of heat transfer fluid (Calflo AF). Two lightweight oil immersion heaters (mega: MT-330/240) are employed to provide a heat source for heating the oil to the desired temperature. The heaters provide a total power of 6 kw with a watt density of 24 w/in2. When the desired temperature is reached, temperatun control is independently achieved at the set point for each heater using temperature controllers (Fuji: PXZ4-TAYI-4V). The temperature controllers are PD Autotune controllers that employ fuzzy logic algorithms. The inputs to the controllers are Type J ungrounded thermocouples (mega: GJQSS-1 MG-24), which are positioned in the oil bath close to the heaters to prevent overheating and premature bumout of the heating elements. To investigaie the interna1 response of the temperature within the high-pressure chambers, and compare with the extemal oil bath temperature, a thermocouple is also positioned within the first highpressure chamber. This themocouple is used for indication purposes only and not for controlling the oil bath temperature. It is used to monitor temperature fluctuations that cm occur within the high-pressure chambers dunng experimentation due to the isentropic expansion of high-pressure gas within the system. These fluctuations will cause a similar response in the temperature controllers and the oil bath temperature would reflect the temperature variation. In order to minimize temperature gradients in the system and improve the temperature distribution, shaft-mounted propellers driven by a motor are submerged in the oil bath. To reduce the heat loss from the system to the surrounding environment, the tank walls are insulated with fibreglass insulation. Stainless steel sheet covers are also provided for the top

99 of the tank to impedc excessive heat loss through evaporation. As a result, the temperature of the system can be controlled to within f 1 C using this thermostatted oil bath. Pressure Measurement The measurement portion of the sorption device consists of a positive displacement syringe pump (Isco: 260D) and controller (Series D), a high-accuracy pressure transducer (Mensor: 6010) with pressure range psia and RS-232 wtput, and a pressure transducer (Dynisco: PT462E-M 10) with digital readout (Dynisco: 1290). Due to the limitations of most commercially available high accuracy pressure iransducers at high temperatures, rneasurement must be perfonned at lower temperatures. In this design. measurement is accomplished by positioning the transducer (Mensor: 6010) at some distance away from the heated oil bath, such that the surrounding temperature is within the operating temperature range of the transducer. n the high temperature measurement side a pressure transducer (Dynisco: PT462E-M I) and digital readout (Dynisco: 1290) are used to provide an estimate of the system pressure. With a combined accuracy and repeatability of f 120 psi, this transducer is inadequate for making accurate measurements and is therefore used only to provide a preliminary indication of the pressure range within the high-temperature system. The overall high-pressure durability of the sorption apparatus is attained by selecting high-pressure valves (Sno-Trik: SS-410-FP-G) rated to 7,500 psi at high temperatures up to 450 C. Stainless steel components are extensively used due to the outstanding corrosion resistant properties and high strength capabilities. The mm (1/8") tubing used has an allowable working pressure of 10,900 psi (75.2 MPa). The fittings used are Swagelok

100 compression tube fittings with metal-to-metal contact that provided excellent leak resistance under the operating conditions. AI1 other treaded connections utilize 1/8" or 1/4" nominal pipe thread (NPT), which provides a superior seal when the threads are wrapped with teflon tape prior to tightening. Each of the sorption cells was machined to specifications from 4140 pre-heat treated mild steel, and are submerged into the thermostatted oil bath. Mild steel was chosen because of its good thermal properties. which allows for fûster temperature equilibration, and its highpressure durability. The high-pressure integrity of two of the chambers was assured by utilizing metal-to-metal contact on al1 sealing surfaces and avoiding polymeric -rings, which can absorb trace amounts of blowing agents and produce sorption measurement errors. A copper~rin gasket was manufactured and provides sealing by means of edge contact (Figure 5.2). The gasket is positioned between the chamber and the polished chamber cover. The two concentric grooves on the chamber provide an edge contact against the copper gasket, which is pressed against the cover, thereby deforrning the copper gasket and providing a high-pressure seal. In order to ensure a consistent seal around the entire circumference of the chamber. the chamber bolts are al1 torqued equally during experimentation to 294 Nm (217 ft-lb) using an electric impact wrench. The third chamber employed an 1/2" NPT fitting to seal its opening. Details of each chamber can be found in the previous chapter (Figures ). To maintain the specimen chamber free of molten polymenc material, sample pans were manufactured from aluminium. Aluminium was specifically chosen because of its high thermal conductivity, which allows good heat conduction between the sample and the smple pan. The sample pans can be reused and are

101 cleaned by placing them in an oven at high temperature to get nd of any residual polymeric material. 5.3 Experimental Materials The experimental polymeric material utilized in this investigation was polystyrene grade 101 supplied by Nova Chemicals. The pellets, which are approximately 1 Smm in diameter, were used as received withuut further modification. Commercial grade carbon dioxide (99.5% purity) was used as the blowing agent, and was supplied by Matheson Gas Company. 5.4 Experimental hocedure Volume Measurements Due to the importance of each sorption chamber volume of in the solubility calculations, it is necessary to determine. as accurately as possible, the volumes of these chambers including the interconnecting tubing and fittings. The method used to accomplish this task is adapted from a helium expansion method 1221, and utilizes a positive displacement syringe pump filled with distilled water. The syringe purnp has two features, which makes it suitable for measuring the volume in-situ. Firstly, it can be operated in a constant pressure mode, Le., it can maintain a constant pressurized environment, and secondly, it is capable of indicating both its intemal volume and the gas flow rate.

102 For measurement, the pump is connected to the evacuated volume to be measured and operated in a constant pressure mode at a certain pressure as shown in Figure 5.3(a). The volume indicated on the digital readout of the pump is noted (VPI) when the Row rate reaches zero, which is taken as the equilibrium point. Valve A is then opened (Figure 5.3(b)) and the water is allowed to flow into the sorption chamber. As a result the pressure decreases, however, since the pump is operating in a constant pressure mode, it maintains the same pressure by adjusting its intemal piston position. This adjustment should equal the volume added to the system when the valve was opened, and is refiected as a change in the indicated volume (Vm) at zero flow rate. Since the system is a closed one, then any intemal change in the system volume can be found by monitoring the syringe pump volume. Simply subtracting Vm from Vpl gives the corresponding change in the volume from valve A to valve B inclusive of the connecting tubing and fittings. Using this method, the volume of each sorption chamber from its inlet valve to its outlet valve can be systematically determined. The volumes to be measured are shown schernatically in Figure 5.3(c). Since the method is an imitation of the actual solubility experiments in terms of the valve opentions, then the volumes obtained are identical to the volumes needed at each stage of the solubility calculations. The volume measurements were also conducted at high pressure (20.68 MPa) to investigate the effect of high pressure on the volume. The results obtained are presented in Section Sorption Experiments The experimentai procedure followed during the sorption experiments is outlined below with particular reference to Figure 5.1 :

103 Before beginning any experiments. the syringe pump was refilled with the carbon dioxide by opening valves VI and V2 and closing valves V3 and V8. With the refrigerated chiller switched on, the syringe pump is operated in the refill mode. After refilling, the pump is isolated by closing valves V1 and V2. The mas of the polymer sample was rneasured to within I 0.01g, recorded and inserted in a specimen pan and placed in sorption cell3. After sealing sorption ceil 3 and filling the oil bath with the heat transfer fluid, the entire system was evacuated by closing valves V3 and V8, opening valves V4 - V7 and switching on the vacuum pump. After the desired vacuum level of 29.û"Hg was reached, V6 was closed followed by vaives V3 - V5. The oil bath was then switched on and allowed to operate until the set temperature was achieved. For temperatures up to 160 C, this took approximately 2 hours. The system was then evacuated again by opening valves V4 - V7, to remove any further vapours that may have been released from the polymeric sample. After the second evacuation, V4 - V7 were closed again. While in the vruum state, al1 of the pressure readouts were zeroed, to provide a standard pressure baseline for the experiments. Sorption cells 1 and 2 were then pressurized by opening V3 and V4, refilling the syringe pump, and pressurizing the pump in a constant pressure mode to the desired operating pressure. V3 was then closed and the gas allowed to reach thermal equilibrium with the oil bath temperature. After equilibration, V4 was closed to isolate the gas in sorption cell #2,

104 and the pressure noted as indicated by the pressure transducer, PT2. This pressure was 1 o. Il. designated as the first (i= 1 ) stepwise pressure, Pi+i. V5 was then opened to allow the gas to access the sample in sorption ceil #3. As the gas dissolved into the polymer, the pressure in sorption cells 2 and 3 decreased; When the pressure decay due to the gas solubility remained relatively constant for about 20 minutes, then the equilibrium state was achieved and the next stepwise pressure increase was implemented. 12. For the next stepwise pressure, Pi+i (i=2), V5 was closed and V4 opened. The gas in both sorption cells 1 and 2 combined to produce a new pressure as indicated by the pressure transducer PT After pressure and temperature equilibration, V4 was closed to isolate sorption cell#2. Steps 10 through 13 were then repeated until the stepwise pressure increase became insignifiant, Le., until the new stepwise pressure did not change greater than Ipsi over the previous pressure. 15. If f'urther experiments were needed at higher pressures, then new gas was introduced from the pressurized syringe pump by closing V4 and opening V After pressurization to the new desired pressure, V3 was closed. Steps 9 through 14 were then repeated. Two temperatures were used for the solubility experiments: 100, and 160 C, and a new sample was used for each temperature. The entire procedure outlined above was followed for each new sarnple at each temperature. ne set of experiments was conducted at each temperature and the results obtained are presented in the next section.

105 5.5 Results and Discussion Volume ~easukents The volume measurement results are summarized in Table 5.1. As mentioned earlier, the volume measurements wen conducted at 6.90 MPa (1000 psi) and MPa (3000 psi). It can be seen that the volume remained relatively constant over this range of pressure. An average value was therefore computed for each chamber and these values were used for al1 the solubility calculations. Sorption Experiments The raw experimental data obtained using the methodology described in Section are tabulated in Table 5.2, and the analysed data are illustrated in Figure 5.4. The calculated solubility is plotted as a function of the sorption pressure for each of the two temperatures and compared to the literature values reported by Sato et al. [SI]. In Figure 5.4 (a), the solubility is plotted for a temperature of 100 C. It cm be seen that as the pressure increased, the solubility also increased. Within the sorption pressure range of 0-12 MPa, a cornparison of the two linear regression lines reveals significant deviations (about 34%) in the values obtained in this work with respect to the values reported by Sato et al. Assuming the literature to be. correct, the deviations observed may be attributed to differences in the polystyrene material grade. The discrepancy may also be due to a number of other factors. First, the solubility data were cdculated by incorporating the specific volume (PVT) infornation of the polymer under the respective pressure, temperature, and gas content conditions. This PVT information was obtained from Park et al. [21] for the PS-C2 combination. n the other hand, Sato et al. predicied the specific volume using the Sanchez-

106 92 Lacombe equation of state (S-L ES) [ This difference in approach for measuring the specific volume of the polymer-gas combination may account for the observed discrepancy. Second, the experimental PVT information reported by Park et ai. was only available for a limited temperature range ( C), a limited pressure range ( MPa), and a limited gas content range ( - 4 wt%). In order to obtain the PVT information outside these ranges. a non-linear regression analysis was conducted on the data. Since there is no certainty about the behaviour of the pol ymer-gas combination other than at the conditions investigated, a non-linear regression may not provide a tme representaiion of the data. At 160 C, a similar trend of increasing solubility with increasing pressure was observed (Figure 5.4 (b)). The deviation from Sato's data is seen to be about 6%. In addition, the data does not exhibit the typical Henry's law behaviour, as was demonstrated in the literanire. The reasons for this discrepancy are unknown at this point; however, it is suspected that differences in the material characteristics between this work and Sato's work may be responsible. In addition, as mentioned earlier, different information about the specific volume of the plymer in the swollen condition can lead to significantly different results. Unfortunately, detailed information of the specific volume of the swollen polymer was not reported in the Iiterature. Consequently, it is not possible to examine both sets of data to verify any anomalies. Further investigation needs to be conducted to detemine the source of the observed deviations. In Figure 5.5, the solubility of PS in C2 is compued at both of the experimental temperatures investigated. At each temperature, the data are curve fitted. The highest solubility is observed to occur at 1 0 C, followed by 160 C. Based on Eq , an inverse relationship normally exists between the solubility and the temperature at low

107 pressures, Le., as the temperature increases, the solubility decreases. The general trend obtained in this work therefore conforrns to the theory in that respect. With respect to the pressure variation of the solubility, there were some deviations observed from the reported values as noted earlier. The data did not show a trend consistent with Henry's law, rather the data followed the Flory-Huggins mode1 (Figure 24, Le., the solubility increased disproportionately relative to pressure. At the temperatures investigated, the solubility obtained with this device exhibited lower values compared to the reported values (Figure 5.4). For microcellular foam processing, if the solubility is underestimated, a microcellular structure may not be obtained since enough gas will not be dissolved into the polymer melt to nucleate a sufficient number of cells. 5.6 Error Analysis Based on the measurement principle outlined in Chapter 3 for the measurement of solubility, it can be deduced that the solubility coefficients are very strong functions of the pressure, system volume, and experirnental temperature. This section provides some detailed analysis of the effects of each of these parameters, Le., pressure, volume, and temperature. on the solubility data Errors in Temperature Measurement Errors in the measurement and control of the system temperature &se from the degree of accuracy of the instrumentation. Based on the observations made during the course of the experiments, it can be stated that the temperature of the system was controlled to within f

108 I C of the set point temperature. The ermr caused by a temperature fluctuation of i I C is shown in Figure 5.6 for 100 and 160 C. The maximum error in the solubility is seen to occur at 160 C with a value of wt% Errors in Pressure Messurement The pressure transducer selected for the solubility-measuring device is capable of measurements with an ovenll accuracy of f 0.02% of the full-scale range of the transducer. This translates into an accuncy of f 8.27 kpa. The effects of this equiprnent error range on the solubility at the two temperatures, and over the range of pressure investigated, are presented in Figure 5.7. The Iargest errcrs are observed to occur ût high pressures Errors. due to Transducer Configuration The unique configuration of the sorption device (Figure 5.1) in which the transducer is situated outside of the heated oil bath creates some unavoidable errors. In calculating the solubility, it is necessary to achieve a themodynamically well-defined state, Le., a state where the temperature, pressure, and volume of the system remain constant. By introducing a situation where a portion of the test gas is at a different temperature, temperature gradients are also introâuced. Further anaiysis is therefore needed to quantify the ems associated with this equipment configuration. If we consider a system at equilibrium as shown in Figure 5.8 (a), the amount of moles of gas, na, occupying the volumes shown, at pressure Pi. and temperature Tl, can be calculated as:

109 where Va, VC3, and Vo are the volumes of chamber #2, chamber #3 and the external volume respectively, and Zi represents the gas compressibility factor of C2 at Pi and Ti. It should be noted that in reality, the extemal volume consists only of the 1/8" tube and the interna1 volume of the pressure transducer. If a portion, Vo, of the total gas volume is now placed at a different temperature T2 (see Figure 5.8 (b)), then, if the pressure remains the same, the number of moles of gas, nt,, under these conditions can be approximated as: where Z2 represents the gas compressibility factor at Pi and T2. The resul ting change in the amount of gas from the first case to the second can be estimated from the difference of Eqs. 5.1 and 5.2: This amount of gas also represents the excess amount that would dissolve into the polyrner on transition from the first to the second case. The resulting excess solubility, xc, can be therefore be calculated as:

110 where M represents the molar mass of C2, and m, is the mass of the polymer present in the volumes VT and VI. The solubility error can then be calculated with respect to the rneasured value by dividing Eq. 5.4 by the measured solubility, xm. It should be noted that the ternperature T2 in Eqs. 5.2 to 5.4, represents the average temperature existing in the tube leading from the heated oil bath to the pressure transducer. Measurements of the temperature profile within this tube show that the temperature has a distribution dong the length of the tube as shown in Figure 5.9 for oil bath temperatures of 100 and 160 C. At any particular oil bath temperature, the average teriqerature inside the tube was calculated by dividing the appropriate curve into linear segments and finding the average temperature of those segments. The error results obtained using Eq. 5.4 are tabulated in Table 5.3. At each temperature, the maximum error occurred for the highest sorption pressure. The maximum solubility errot calculated was very high at 20.45%. From Eq it can be seen that the error is directly related to the volume ernerging from the oil bath and inversely reiated to the average temperature within this volume. It would therefore be advantageous to reduce this volume as much as possible in order to decrease the contnbuting error. ln addition, if the average temperature can be elevated (within the operating range of the transducer), the error contribution will dso be reduced.

111 Table 5.1: High-pressure chamber voiumes Pressure IMPa) Volume (cm3) ' Chamber 1 1 Chamber 2 1 Extemal 1 Chamber 3 AVERAGE

112 Table 5.2: Experimental Solubility Data Temperature T (K) Initial Pressure P2 (MPa) Final Pressure P2+3 (MPa)

113 Table 5.2: Error contribution of extemal volume il Bath Temp (KI I I 1 I Average - Sorption Measured Extemal Temp. Pressure Wubility P (MPa) % T2 (KI (wtoh) Mass of Pdymer mp (g) Gas Cornpress. Factor 21 Gas Cornpress. Factor 22 Cakulated Excess Salubility Xc (Wh) Error ("w

114 Vent I I I V J V J vo t t I I I Ta 1 vacuum Sample gas Refrigerated Syringe cylinder chiller PumP Figure 5.1: Schematic of solubility mecsuring device Bolt-hole Potished surface Figure 5.2: Gasket sealing of sorption chamber

115 il Piston Chamber Valve 6 1 ciosed Constant P Volurne=Vpi Syringe pump Figure 5.3 (a): Schematic representation of the measurement of internal chamber volume Measured Volume (VP,-VPP) Valve A opened Constant P Volume=Vp2 Syringe pump Figure 5.3 (b): Schematic representation of the measurement of internal chamber volume

116 w Transducer High Accuracy Pressure Chamber Chamber 1 #3 1-1 Chamber / & 1 #1 I I I I Figure 5.3 (c): Schematic representation of volumes tu k measured Vcr = Chamber #I volume VC2 = Chamber #2 volume Vc3 = Chamber #3 volume Vo = Externul volume

117 a This work Sato et al. A. 14 -{ - - Linear (Sato et al.) Pressure (MPa) Figure 5.4 (a): Solubility of PS in C2 al 100 C This Work Estimated (Sato et al,) - Linear (This Work) 14 h g I Pressure (MPa) Figure 5.4 (b): Solubility of PS in C2 at 160 C

118 Pressure (MPa) Figure 5.5: Solubility of PS in Ct at 100 and 160 C

119 .,.,.,. 1 ' I Pressure (MPa) Fi pre 5.6: Solubility error contribution from temperature fluctuation of il C Pressure (MPa) Figure 5.7: Sdrlbility error conttibution fm pressure fluctuation off 8.27 kpa

120 I 1 I t I I vo I Valve 7 I 1 Valve 3 pened Valve 5 Valve 6 1 closed opened opened closed, 1 r I vc3 I I Vc1 vc2 i I Chamber #3 I Chamber #1 I Chamber #2 I 7 I I I I l I I I Fipre 5.8 (a): Schematic for determining the error due to the transducer confguration I I I I I AVERAGE I I EXTERNAL I I TEMPERATURE 1 I t 1 1 vo I, l I, Valve 3 Valve 6, 1 I closed r Vct Chamber # 1 opened T2 vu Chamber #3 closed I l I I I Fipre 5.8 (b): Schematic for detennining the error due to the transducer configuration II

121 Distance Frorn Hot il Surface (cm) Figure 5.9: Temperoture variation in tube leading from heated oil bath to pressure transduce r (extemal vol urne) for oil bath tempe ratures oj 100 and 160 C

122 CHAPTER 6 SUMMARY, CNCLUSINS AND FUTURE WRK 6.1 Summary The work presented in this thesis focused primarily on the design and construction of an apparatus to measure the solubility of gases in polymers. An important factor in the calculation of solubility is the volume of the polymer after gas has diffused into it. Since, the polymer swells with the absorption of gas, and since the solubility is inversely related to the volume of the polymer, it is necessary to compensate the calculated solubility for the change in the polymer volume. The prînciple employed in this work to calculate the solubility of gases into polymers was based on a pressure decay sorption technique, in which the pressure decay due to gas dissolving into a polymer sample was measured and correlated to the number of moles of gas dissolving into the polymer. The apparatus was designed using the principle of axiomatic design and essentially consisted of three highpressure chambers immersed in a thermostatted oil bath. The threethambet design was chosen over a two-chamber design because of the advantages that it offered in terms of increasing the experimental efficiency. The experimental efficiency of the systern was further improved by calculating a number of parameters. First, the relative ratio of the three chambers was calculated in

123 order to (i) maximize the number of experiments that cm be performed before having to replace the polymer sample. and (ii) maximize the stepwise pressure increase in each stage of the experiment. In addition, an ideal sample size was calculated which would produce a relatively large pressure drop at very low solubilities. After constmction, the system was experimentally verified using polystyrene (PS) and C2 and the results were compared to the reported literature values for the PS-C2 system. 6.2 Conclusions Based on the work presented in this thesis, the following conclusions can be made: An experimental apparatus was designed and constmcted using the Axiomatic Design principles, to measure the solubility of gases in polyrnea at high temperatures up to 200 C and pressures up to MPa based on a pressure decay method. The secondary functions of the system were experimentally verified with polystyrene (PS) and C2 as the blowing agent, at temperatures of 100 and 160 C, and pressures up to 17 MPa. The prirnary function of the system. which was to measure the gas solubility in polymer, was not satisfactody verified due to the âeviations observeci in the obtained data. As the experimental pressure increased, the solubility was found to increase. However, the typical Henry's law behaviour was not observed. Instead, the data points followed a second order trend. As the sorption temperature increased, the solubility of C2 in PS was found to deciease. The system configuration, in which the pressure transducer measures the pressure of the high-temperature system at room temperature, was not an optimum one due to the errors

124 induced. Ideally, the pressure measurement should be done at the same temperature as the sorption process. The situation arose from difficulties in sourcing high accuracy pressure transducers capable of openting at high temperatures. 7. The effect of errors in the temperature measurement on the solubility was investigated for a temperature fluctuation of f I C. It was found that the errors tended to increase at higher pressures and at the higher temperature. 8. The cffect of errors in the pressure measurement on the solubility was investigated for a pressure fluctuation off 8.27 kpa. 9. The errors in the solubility introduced by the transducer configuration, in which the transducer is placed external to the heated oil bath, were investigated. The errors involved ranged from about 4% to 20%. It is believed that this configuration is the biggest single source of error, which also contributes to errors in the pressure measurement. 6.3 Recommendations and Future Work The following suggestions are made for the direction of future research on the measurement of the solubility of gases in polymm: 1. The apparatus developed in this thesis can be used to provide useful insights into the measurements of diffusion coefficients. Such coefficients can be combined with the solubility information to provide appropriate design strategies for the development of polymer foam processing technologies.

125 This study was confined to a preliminary test of the polystyrene and carbon dioxide system. However, once optimised. the solubilities of other blowing agents (whose equations of state are defined) in other thermoplastics materials can be examined. f particular interest, would be the investigation of potential environmentally benign replacement blowing agents. Further experiments need to be conducted in order to accurately rneasure the PVT data of polymer-gas solutions. so that the degree of swell can be compensated for more accurately in the solubility measurements. Due to the errors involved in the pressure transducer arrangement, and the difficulties in finding accurate pressure transducers with high temperature ranges, the use of a magnetic balance to measure gas solubilities in polymers (as reported in the literature) should be ex~lored as an alternative to provide more accurate solubilitv data.

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134 1 value B set Numbe r Figure 52 - Test 2 / First der Merhod Weigltt Convergence History UPPER Value UPPER I S Set Number LYER Figure 53 - Test 2 / First rder Method Stress Cons traint Histor y

135 DTMIZATfN UYMAX LbZR UPPER 320 Value 240 'i \ ', \., UPPER LYER set Number nsys Wingbox Analysis Figiwe 51 - Test 2 / Firsr rder Method Displacetrteiir Constraht History Genetic Algorithm Four GA tests were performed with different genetic operators. The population was held constant in al1 cases. The fitness (weight) convergence history of a11 test cases is shown on Figure 55. All tests converged after 50 generations and took approximately 4000 CPU seconds. The best resuhs were obtained wit& the uniform crossover and 2-bit mutation. Even though in the convergence graph they present higher weight values. the results are fully constmined cornparrd to the other gnetic operator tests where the final weight is lower but the designs do not meet the constraint requirements. The 2-point crossover I 2-bit mutation combination of genetic operators produced good results but do not completely meet the penalty constraint imposed for the optimization.

136 The obtained results indicate that the wtcome is directly dependent on the amount of information that can be exchanged between the memben of the population over pnerations. Unifom crossover and two bit swapping had higher rates of information exchange and allowed to find better results with relatively the same convergence time than other genetic operaton combinat ions. Moreover, the high rate of exchange of information allowed the GA to search further the design space for better alternative solutions avoiding a premature convergence into local optimums. However, the main drawback of the use of this high information exchange operators is the long time required for the process. GA ptimizatim Mcthod Comoarison Tests Stitistics I I 1 I 1 85ûû - 1 i Poinî(Cmssver) - 1 Bit (MutatJon) Pdnî(Crorsver) - 1 Bit (Mulition) 2 Point(Cmsvsr) - 2 Bit (Mutation) - - Unifom(Crossver) - 2 Bit (Mutation) Figure 55 - Genetic Algorith Test Results - Merhods Contparison Compared to the tradi tional methods the GA requires more time to search the design space but the quality of the results are much better in most of the cases. Furthemore, the use of an evolving population of points (repmenting different design combinations) leads to better results over time, not requiring the specification of initial points near the feasibie region.

137 AlIowing the GA io evolve for more gmcrations and nhhing the genetic information of the population (adding new randomly selected individuals), will lead to better results than other deterministic methods that stop when a convergent solution is found. even if it is a local minimum. A numencal comparison of the results obtained between different optimization tests is shown below (Table 17). The GA with uniform crossover and Zbit mutation obtains the lower weight constraint results. The First rder exact method leads to similar results as the GA approach, but it is dependent on the starting point indicated by the second test performed with this method (which present a highly unconstraint local sub optimum). ther GA tests with different genetic operators showed mixed results, the?-point crossover, 2-bit mutation leads to a good solution with a small unconstraint value. ther operaton lead to inferior solutions due to its lower information exchange. The sub-problem approximation method leads to the poorest solution found in ternis of weight and constraints. This method requires high computational time and requires a high number of functior, evaluations to adequately approximate the design space. Table ptirnitaiion Metbds Coinparison l Top Skin Thickms (in) Baitom Skin Thictims (in) SubProblem First rder First der Genetic Cenetic Genetic Cenetic Approximation Exact Exact Algorithm Algorithm Algorithm Algorithm Initial Point Initial Point Initial Point 1 I ptni L'munw 2 n i w 12 ~bini ~nisrnrn [unilbrm Cimovrr II, II 4. II. J ~. i1.r. 1i.a ii.tl c ~7.t1.71 a 1-Bli hfuiatw % MuQl*ml?-Dit Muiaihlnl slu(.hl Fmm Spr W& Thicknrss (in) Back Spw Wcb Thickms (in) Riôs Thickms (in)

138 7.7.2 Full Complexity GA-FEA Design ptimizetion Taking advantage of the GA variable handling charactenstics, two tests were performed using continuous and discrete variables to represent the main structural panmeters of the wing. The optimization procedure history of the best population memben is shown in Figiou 56. Notice that both cases converged after 40 generaiions for a period of CPU seconds. The process was allowed to continue funher until pneration 90. but very slight changes of the optimized solution was observed after convergence. Both test cases met the imposed constraints criteria. The use of uniform crossover and 2-bit mutation resulted in the lower total wing-box weight; the use of?-point crossover and 2-bit mutaiion led to a constraint wing-box result with 2% higher weight. Full Compleuty Wing BX GA plimiztttion Statistics log I I I I r j d Point(Crorsver) - 2 Rit (Mutation) - Uniform(Crossver) - 2 Bit (Mutation). loa ; 3 - I \ I 1 t I ' _ * 1 1 l t l I I t Genention nunber Figure 56 - Full Cotoytexity GA optimization Resitlts

139 ne interesting aspect obtained from the test is the indication of a fast global convergence. It can be see in Figure 30 that both procedures reduce the fitness values. 4 orders of magnitude in the fint 10 generations. Since the process starts randoml y. the fint generation designs do not necessuily lie near the optimum region of space, reflecting in very high penalty functions. With the exchange of genetic information of the system (exchange of design charactenstics) the process is able to rapidly find the region of the space where the best combination of design parameters lie. and further search in the Iocality of those regions for better designs. This explains the 4 orders of magnitude reduction in the fint genentions where the maximum number of information exchange occurs with the subsequent gradua1 change of fitness values at Iater generations. The above point can also be seen in Figure 57 where the maximum. mean, and minimum fitness values of the uniform crossover 2-bit mutation ;ire plotted. This figure shows how the Wrst generation variability is high. and with time. thrt variability is reduced due to the exchange of infornation that produce better off-springs (better designs) more localized in the best regions of the design space. x 10'~ Full Compfexity WlnpBox GA p(lmizaliori StrUlcs 5 L, i I T 1 I - -, Mein Mw Min - Figure 57 - Fimess Range For Full Complexity Wing-Box ptimization 129

140 Table 18 - Wing-Box Full Contplexity ptirniiatioii Tests Co~iparisoii Top Skin Thickness (in) 0.63 Cenetic Algonthm [2 Point Crossover 1 & 2-Bit Mutation] Genetic Algorithm [Uniform Crossover 1 & 2-Bit Mutation] Boitom Skin Thickness (in) I Spar Web Front niickness (in) S par Web Back Thickness (in) Ribs Thickness (in) Front Top Spar Cross Section Front Bottom Spar Cross Section Back Top Spar Cross Section Back Bottom Spar Cross Section Top Stringers Cross Sections Boitom Sirinpers Cross Sections T Section , ] L Section [ 1, I.,09] L Section [ l.s. 1.5,0.093] T Section [ j T Section (1.75, 1.5, L Section [ 1.75, 1.S Top Skin Material 1 Stainless Steel (15-SPH) 1 Titanium (6AI-3V) 1 Botiom Skin Material Front Top Spar Material Front Bottom Spar hlaierial Back Top Spar Material Back Boitom Spar Maienal Spür Web Front Maienal Spar Wcb Back Maierial Ribs Material Top Stringers Maierial Back Stringers Material Winp-Box Weipht b) Steel (AMS 6,221 ) Siainless Steel (AM AMS 5547) Magnesium (HM2 1 A-Tû) Magnesium (HK3 1 A-HN) Aluminium (2048) Aluminium (2219 7'62) Steel (AISI 4340) L Seciion [ j T Section (1.75,0.875,0.094] - L Section [0.875, ] L Section [ 1.25, 1.25,0.0937) C Section [ ,0.078] L Section [S, 0.7,0.0625] Siainless Steel (AM355 - AMS 5517) Mûgnesium (HK3 1 A-) Steel (AMS 6437) Aluminium (2024-TB61 ) Titanium (6AI4V) Aluminium (2024-T6) Aluminium (7 178-T6) Aluminium (7005-T53) c~nvcrprd? Yes Faible Consmints? 1 YCS 1 Yrr 1 m 1 Note: Aluminium (7075-l76) Aluminium (7050-T ) Magnesium (HK3 1 A-) Yes T Section Dimensions [Top kngih. Base Hcighi, Thicbs] L Section Dimensions [Base Icngih. Heighi. Thiclincss] - C Section Dimensions [Bu Lrngth. Heighi. Top kngth. Thickncss] Steel ( AMS ) Aluminium (2024-T86 1 ) I The cornparison between the two tests is shown in Table 18. It is interesting to see how the optirnized solution uses L and T cross-sections, with moderate lengths and reduced thickness, which provide a Iight structure and positively affect the moment of inertia of the wing-box and improve its shear strength characteristics. The materiais selected Vary considerable in each test, but the GA selected typical materials used for aerospace structures.

141 Aluminium is the main material type chosen by the GA. but certain mdtoffs appearrd in the use of materials and thickness especially for the skins panels and spar webs. ne cm also see the use of aircraft steel in the spars for uniforrn crossover case. This is typical for most real aircraft wing-boxes to improve bending nsistance. Since the process does not take into considention the manufactunbility factors. it selects rnatenals like magnesium. stiinless steel and titanium dloys due to their structural properties and weight. Enhancements could be done to include machinability, oxidation resistance and even cost constraints to determine the best feasible structure. For example. the use of titanium and stainless steel improves the structural characteristics and weight of the structure but make the structure expensive to build. Considering al1 the above facts. both test cases resuited in wing-boxes with weight in the range of real aircrafts and fully constrained to the specified load buckling, deflection and material yielding cri teria. hsys Wingbox Analysis Figure 58 - Full Complexity Wing-Bar ptirniration Displacement Results

142 Figure 58 shows the deflection charactenstics of the best-found solution for the wing-box optimization including continuous and discrete variables. Notice that the maximum deflection of 32 inches below the aliowable total deflection imposed of 7% half of the span (443 inches) was obsewed after the analysis. Figure 33 and 34 shows the Von-Misses and shear stress distributions. It can be seen that al1 structural elements are constrained and do not yield under the applied load for the selected materials and geometncal properties. Figure 59 - Full Con>plexiry Wing-Box Von-Misses Stress Results

143 Figute 60 - Full Compluiry Whg-Box Shccir Sttess Results

144 7.7.3 Simplify Full Cornplexity GA-FEA Design ptimization As discussed before, design engineers generally know what type of materials could be used not only from a suuctural point of view but also from a manufacturing point of view. For the final test the material types used by the program were limited to those commonly use for aircraft structural design. The convergence history for the best population member of each generation is shown below. For this test the optirnization process was stopped for the case when the best member found do not change significantly for the 1st 25 generations. The process converged to a solution relatively fast, at genention 10'~. x ld Simplify Fun Cornplexity Wng-Box GA ptimization Statistics I i 1 1 I 1 I Besl] Ttsted case: Unifm Crossowr / 2-Bit Mutation l Figure 61 - Simpli' Full Cmtplexiiy Wing-Box ptimization History

145 Numerical results for this case are shown in Table 19. Aluminium is the material of choice except for the main bottom front spar where the GA selects steel. ne interesting aspect lies in the selected thickness of the cross-sections due to the limitation of materials. It can be seen that the GA selected thicker sections compared to the previous tests which improve the structural behaviour and avoid stmctural failure under the applied loads. Table Sin,plih Fidl Conlplexity Wiilg - Box prinr iration Numerical Resulrs -- Genetic Algorithm (Uniform Crossovcr Top Skin Thickncss (in) Botiom Skin Thickness (in) Spar Web Front Thickness (in) Spar Web Back Thickness (in) Ribs Thickness (in) Froni Top Spar Cross sectionp Froni Bottom Spar Cross Section Back Top Spar Cross Section Back Bottom Spar Cross Seciion Top Siringers Cross Sections Bottom Stringers Cross Sections Top Skin Material Botiom Skin Maicrial Front Top Spar Material Front Bottom Spar Material Back Top Spar Material Back Bottom Spar Material Spar Web Front Material Spar Web Back Material Ribs Material Top Stringers Material Back Stringers Matenal WinpBox Weight (Ib) bsibk Constnints? Note: I L Section [ ,0.1875] T Section [l.75, ) T Section [ 1.S L Seciion [ ] C Section [ 1.S. 1.S ] I Section [2.125,2.125, ] Aluminium (7075-T6) Aluminium (7475-T85) Aluminium (7075-T76) Steel (AMS 6437) Aluminium (2024-Tû6 1 ) Aluminium (7475-T85) Steel (AMS 41 30) Aluminium (20 14-T4) Aluminium (2024-Tû6 1) Aluminium (7050-ï76) Aluminium (2014-T4) B Yes - T Section Dimensions [Top kngth. Base Height, Thictincss] - L Section imcnsions [Base Icngth, Height. Thickmss] - C and I Sections Dimensions [Base Lrngth, Hcight. Top Lrngth. Thickness]

146 The structure is fully constrained in buckling and displacement criteria but had a 2% unfeasibility in the sass requinment cntena. This could be solved by reducing the safety factor imposed to the structure or allowing the GA to evolve further. An alternative way is to reksh the genetic information of the population and include new random designs that could lead to further improvement over tirne. The obtained weight of the structure is much lower than the previous case. mainly due to the limited number of materials considered in the optimization and the displacement constra.int used. As a cornparison. the weight of a normal aluminium structure for the type of aircraft analysed can be calculated using analytical methods. Tonnbeek [69] provided an approximate weight of 9100 lb for the basic structure which is in accordance to similar real aircrafts types like the MD82 or Boeing 7 Li which wing-box weight is approximately 8950 Ib. Based on this values the obtained wing-box structure had a relative weight savings of 4%. Notice that further manufacturing considenitions and cost should be included and will affect directly the outcome of the optimization procedure. The displacement. stress and shear smss characteristics of the optimum structure are shown below (see Figure 62 to Figure 64). DISPLACMENT STEP* 1 SUD *1 FEB J:35:20 Figure 62 - Simplifjl Full CompIe.rity Wing-Box primizution Displacement Resuifs 136

147

148 Chapter 8 - Conclusions In this study, a Soft Computing tool for design and optimiration is developed for aerospace design puiposes. The code integrates a Genetic Algorithm (GA) and Simulateci Annealing (SA) techniques with additional improvement using an Artiticial Neural Network (ANN) for design space mapping which is feed from searched designs stored in a database. The developed code was initially validated using simple truss structural optimization tests where found results overpass other literature solutions based on different traditional algorithrns methods. Additionally. an insight in the value of ANN as an approximation technique of the design space is shown. Application of the develop framework for MD0 optimization at the conceptual level design was tested for the aircraft design optimization problematic. Sets of aircraft analysis twls were developed taking into account the rnulti-disciplinary linki ng between the anal ysis and the design variables. These analyses were integrated into the framework where two aircraft optimization tests were performed. An initial test compared the different SC techniques for a business aircraft cost optimiration; it is found that the GA technique lead to better results as compared to the SA method. The ANN-GA improvement technique provides reliable results as compared to the normal GA optimization but with less computational cost due to the ANN ability to rnap the design space. A second analysis evaluates funher the best-found technique (GA) testing which genetic operaton lead to improved optimization results. The test is performed for a commercial aircraft jet with weight as an optimization goal. Results found, show that unifom crossover -adaptive penalty function combinat ion Iead to better resul ts of al1 genetic operators combinations. The obtained results show feasible design configurations as cornpared to a real aircraft with similar design profile. Lower takeoff weights and reduced constraint validation values show the ability of the procedure to integrate multiple variable type including discrete aircraft charactenstics and find optimum design combinations in presence of heavy multi-disciplinary interactions.

149 Finally, the developed optimization framework was used for a preliminary design phase, testing the capabilities of the GA technique to handle configurations where interdisciplinary interactions are reduced but the complexity of the optimization problem is higher (higher number of variables and constraints). The SC optimization framework was tested using only a GA based on previous expenences as the best SC optimizer. The framework is integrated with ANSYS Finite Element and RUENT CFD software to produce an accurate and robust structural optimization module. A wing-box mode1 optimization example was developed based on a results from the previously aircraft design analysis. According to given wing-box geometry. and loading conditions provided by a CFD analysis. three different optimization cases were examined. The first case compared the methodology with more traditional methods showing the potential benefits of the technique finding not only better solutions but also avoiding the complications involved in traditional techniques that select the initial optimization points and obtain the accurate solutions wit hout afiy design space approximation. A second tested was conducted to take advantage of the GA capabilities in handling continuous, intepr and discrete variables at the same time. It was shown that the GA accepts the wi ng-box main physical parameters including skin thickness, spars and stringers crosssectional areas and materials for al1 structural elements as input. Tests revealed good potential solutions for the wing-box optimization under given displacement. buckling and stress constraints. In the final test, we used a reduced specific number of matenals, compared to the previous test, to partially reflect manufacturing considerations. Results showed a feasible wing-box configuration with a 4% weight savings similar to real aircraft configurations. In general, ii was shown the implemented SC methodo~ogy find better global solutions compared to traditional methods for different range problems. In addition, it cm handle problems with high multi-disciplinary links involving continuous, integer and discrete variables, as well as more detailed preliminary complex optimization problems. The GA- ANN is found to be as g d as the best optirnization alternative using unifonn crossover and an adaptive penalty function approach.

150 Chapter 9 - Future Work The main challenge in using SC techniques lie in two main aspects: the first one is to improve the computational effort involved as compared to more traditional methods, which decrease the attractiveness of these methods if the problem is cornplex, involve mixed-variable types, or when very fast solutions are needed. The second is the determination of what combination of setting values for the different SC techniques lead to improved global optimization convergence, lmprovement for the SC optimization techniques could include the addition of a Hill Climbing algorithm which could acts as a stochastic local optimization routine. since it searches the optimum combinations of continuous, discrete and integer variables in a bit-wise fashion. This hybrid concept could accelerate converge of the system where the SC method search globally and other methods refine the search in local regions taking advantages of both optimization worlds for local and global optimization. The feasibility and implementation of this idea into the SC framework to decrease the computational time should be studied further. Future studies for the GA optimization procedure could include the use of adaptive pnetic operaton as well as multi-objective hündling. Similarly. funher analysis in the optimumcwling scheme for the SA procedure could lead to faster convergence rates and better feasible results. For the aircraft conceptud design analysis, more variables with interdisciplinary couplings -1ike discrete and structural variables- could be integrated. providing a more detailed representation of the aircraft design process; these variables will offer a suitable base for a more detailed multidixiplinary analysis. Ultimately, refinements in the performance and stability and control (including dynarnic stability) analyses will permit improved results for the andyzed variables. ther considerations for funher development as part of wing-box optimization includes the development of an automatic template creation subroutine based on specified suuctural goals and variables, the inclusion of non-linear analysis, more detailed buckling constraints, the possibility to include composite materials in the process, as well as manufacturability considerations as rnaterials. machinability, and structural cost.

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163 Appendices

164 Appendix 1 - CF0 Resultr 1.1 Feb 22. 2al1 FLUENT 5.4 (3d, ssgrsgatbd, SA) Plot CFD Symetry Plane and Wing Mesh Scrlad Re8idual8 ~ RUENT 5.4 (3d. sogmgawd, SA)

165 Convergence History Plot ConlowJ d Stak Pressure (pascal) Feb 22,2001 f LUENT 5.4 (3d, segrqated, SA) Wing Static Pressure Distribution

166 I Conlwn of Wall Shear Stress (pnscall Dec 03,2000 RUENT 5.4 (3d, dp, segregated, SA) f Wing Wall Slieat Stress Distribution

167 Appendix 2 - Defined Materials and Cross Sectional Databases Appendix Materials Database Id I I 1 I Moicrial Namc Sid A-7 (STRUCTURAL) s icsl AMS (250 Gndc Maping 900F) Stccl AMS (280 Gndc Manging 9ûF) Steel AMS 6510 (2.50 Gmde Manpinp 900F) "o"~''' Shear Modulus, Madulus, E G 'Msi) tml AMS (280 Gnde Mamginp 900F) 26.5 Srccrl AMS 6524 QurnchMemprcd Stwl AlSI I 025 SrrrrlAISl4130 Srcxl AlSI QucnchecVTemperird Stccl AIS Sicel AIS QucnchcdTTenipml Siwl AISI 4340 Quenchr~emprcd fiiel AMS 6485 QuencWempcd Steel AMS 6437 QurncWcmpcd Steel AMS 6523 QuonchMcmpcrcd 28.5 $ (Msi) I. l 10. I 10.1 l 1-1 I I I I I I I I I I I I Il II 11.1 o h n Eati0, Dcnsity, - (lbfin3) Temile Strengih Ultimate (Ksi) û 26û 190 Tensilc Stmngth icld ksi) Shcsr Sircngth Ultimaie (Ksi) Shcir Simngth Yicld [Ksi) 20. IW me ' h n R IwHB-SE 6-5 Zn, MI-HB-SE S k 1987) 1-HB-SE 5.4 v987) 5 MI-HB-SE (1987) 1-HB-5E S. $7, ) 6 MI-HB-SE i 1987) 6.5 MI-HB-SE r 1987) 1-HB-SE MI-HB-SE 11-HB-SE 1-HB-SE 1-HB-SE F987) 1-HB-SE,16 Srcvl AMS 6526 QucncWcrnprcd 283 l1.l p AM-350 (AMS 5SJ8) ankmd (SCT 850) btcel. Suinlas. AM-355 (AMS 5547) 19 barthcd c SCT 8-50) tœl. Si;iinlcss. Cusrom 450, Ha Tm rai. Srainless. Custom 455. Ha Tmt 2' PH9.T) II I I I I p5 1-HB-5E Kg871 1-HB-SE Kg87 1-H&5E 5-) Kg ial. Stainlcss. PH 13-8Mo. Hat Tmt 28.3 I t b I t8

168 c_ Treat (H900) ~l.StainIess.I5-5PH.H~tT~t(Hl~)-,~~~ 26 o 'RATPLATE b ,7 Steel. Swinlcss. IS-SPH. Hat Tmt (H935) - 10.l -.8 CASTING ,g p.282 Steel. Stainless. PH 15-7Mo. (TH 1050) Steel. Stainlcss. 174PH. Hat Trat (H900)- FRGING. PLATES [eel. Stainlcss. 174PH. Heat Treat th la)) b.282 IS LSTING Steel. Stainless. 17-7PH. Hear Treat (TH 1050) - PLATE. SHEET Siecl. Slainlcss (AISI 30 1 ), Annealed Stcel. Stainless AISI l/2 Hard Steel. St;linlrss (AISI 301). Full Hard Aluminum I (Annealed) Aluminum 2014.T4 Alunlinum T6 - PLATE Aluminum FRGED o, J J IM l I S I HB-5E HB-SE '9 11-HB-SE lf1987) I J I I ~.(I.HB-SE 8.6 ' 1987) MI-HB-SE ) 8,6 MI-HB.SE 11987) 13. l l3 11-HB-SE e987),?, bll-hb-se 11987) 39 Alurninum 2017-T J Aluminum 2024-T J J Aluminum 2024-TJ Aluminum 2024-TS6l Aluminum 2025.T6 Aluminum S $ Aluctrinum 2124-T85l Aluminum T62 Aluminum 2219-T87 Aluminum T61 - FRGED Aluminum Aluminum 5053H I AS bfi*hb.s~ k 1987) 12.3 pi-hb-se 1987) I-HB-5E I' Kg I-H B-5 E K987) Aluminum, Aluminum..HIS6, H 191 Foil I I S 12.6 I-HB-SE E987) Aluminum.MS6-H343 Aluminum 6061-T " I-HB-SE Kg I-HB-5E 18 2 r987) $5 Aluminum T Aluminum T

169 57 Aluminum 7001-T Aluminum 7010-T Aluminum 7049-l Aluminurn 7Sû-n6SI I J HB-SE l ) Aluminum 7075-T6 - SHËET. PUTE Aluminum 7075-T6 - EXTRUStN Aluminum 7075-Ti3 Aluminum 7175-ï71 - FRGING J , s $8 JS J Ml-HB-SE ) 1-HB-SE 12-3 k987) 1-HB-SE I3 k87) 12.9 b5 Aluminurn 7475-T6S Aluiiiinum 7475-Ti Aluminum A201.-T7 - CAST Aluminum T6 - CAST Aluminum T6 - CAST U HB-SE l3 E987),, pl-hb-se 12 1( 1987) 1-HB-SE E987) p ' 76 h L10 8 I 82 Aluminurn C355.hT6 - CAST Aluminum A356.0-T6 - CAST Aluminum A357.0-T6 4AST Aluminum TJ - CAST Aluminum TS - CAST agncsium. AZt I B. - EXTRUSIN. k RGING Mapncsium AZt SHEET. PLATE Mrignesium. A= 1 B-HZ6. - SHEET. PLATE agnesiu~n. AZ6 1 A-È~EX~RUS IN. ~RGING Mapnrsium. AZ9 1 C-TJ. - CASTING M i - A29 1 C-T6, - GISTING Mognssium. AZ92A. - CASTING Mapnesium. U9tA-TJ. - CASTING w U $0 31 J I I 16 I I l I I I 16 I , I n hl 1-HB-SE (1987) Ml-HB-SE il 1987) 1-HB-SE l3 1987) 1-HB-SE 13*7 K987).I-HB-SE I4 l4 14 I4 l4 l rl987) 1-HB-SE E987) 1-HB-SE Kg871 1-HB-SE %a71 1-HE-SE y9871 I-HB-SE k Magncsium HK3 1 A-. - SHEET. PLATE Mapium HK3 1 A-HZ1. - SHEET. PLAT Mapsium HM2 1 A-lü. - SHEET. PLATE Moenc~ium QEA-T6. - CASTING pihb-5e 14 W 1987) MI-HB.SE : 1987) Mo~ium ZK60A-T5. - EXTRUSINS. FRGINGS 7ii~iiumPum.Anneoleû 6g A HB-SE ' Kg871 1-HB-SE Kg

170 ' ïïtinium Ti-SAl-2.5Sn. Anneilcd - PLATE. SHEET 90 ïitûnium Ti-5Al-2.5Sn. Anneûled - FRGED 15.5 bi Tiiiniurn Ti-6Al-?Sn 4%-ZMo. Duplex Annwl iwnium Ti.6AI-LV. Annwld - PLATE. itanium Ti-6Al-4V. Solution Teai & Age -, i kitanium Ti4AI4V. Annealcd BAR If4 lu) EXTRUSIN iianium Ti-6Al4V. Solution Tmt & Agc ' Io :AR. EXTRUSIN Titanium Ti-6Al-6V-,Sn. Annealed - PLATE a Io BHEET htinium TidAl-6V-2Sn. Solution Trni & Agc - PLATE. SHEET p.31 LIU Tianiutn Ti-#AI- I Mo- I V, Single Anncal Tiianium Ti-8Al- I Mo- I V. Double Anncal Titanium Ti-l ISn-SZr-2Al - IMo, Annealcd Titanium Ti-13V.I ICr-3AI, Anmdd iwnium Ti-13V-1 ICr-3AI. Solution Tmt & 103 Timnium Ti- 1SV-3Cr-3Sn-3AI. Solution Trcat iunium Ti- ISV-3Cr-3Sn-3Al. Solution Tmt 'o Agc ls.s b IN U I ; , MI-HB-SE (1987) MI-HB-SE (1987) HB-SE il HB-SE IM I-HB-SE : 1987) 77 J'9 MI-HB-SE :1987) 87 1-HB-SE 80 1-HB-SE 32 I-HB-SE %87) 78 1-HB-SE :987) 59 1-HB-SE HB-SE E987) MI-HB-SE ) 55 1-HB-SE 8 1 Additional Aerospace Materials Includrd in the Database for Advance Applications Bcrylliuni-Coppcrr (C Hd) &q4lium-copper (C Hûd) Bcryllium. QMV Bnss: Yellow (Hani) Bonlyn E 1ST6 Bonl yn ES-T6 Bonlyn HS-T6 Bodyn H 15-T6 Bmlyn HST M 0.05% 0.W 0.0% Il Io 8.2 '" Ml-HB-SE (1987) MI-HB-SE (1987) MI-HB-SE ( 1987) MI-HE-.= 11987) MI-HB-SE (19871 MI-HB-SE (1987) MI-HB-SE (19871 M I-HB-SE 11987) MI-HB-SE ( 1 987) Bonlyn HM-773 Cas1 Iton (pariitic mllmble) % MI-HB-SE (1987) 1 MI-HB-SE, (1987) 1

171 Coppr DHP (Hard Temper) Invar Lad. pure h d ï r~ll~d) Niobium (Nb- I ZR) s Ml-HB-SE i 1987) MI-HE-SE (1987) MI-HB-SE ( 1987) ht 1-HB-SE (1987) Ml-HB-SE 19S7,.. Niobium tc- 103) Niobium (C- 139Y) Niobium (Cb-752) Molybdcnum (Mo-0.5 Ti) Molybdcnum (TZM, or Mo-.ST i-0.1 Zr) Plainuin Silvrr. pure Shape Memory Alloy Nickcüïitwiurn (Ausrenilcl Shapc hlcniory Alloy - Nickc~ituiiuin (Mancnsitc) Shopc Mcmory Alloy - Co~r/Zinc/Aluminurn (Beta-Phase) Shapc Memory Alloy - Copper~Zinc/Aluminuin (Mûncnsi te 1 Shapc Mcmory Alloy - CopprrIAluminumlNickcl (Beta-Phase) Shapc Mcmary Alloy - Copptr/Alurninum/Nicl;ci (Mariwsiie) Tontalum Thorium. induction mli Tungun (M'HA 1598 Q9Co) 90W-9Ni-2Co Zirconium (Zr-2.SNb. gndc 705) IL I, I I J l I MI-HE-SE ( 1987) MI-HB-SE i 1987) MI-HB-SE ( 1987) MI-HE-SE t 1987) hl 1-HB-SE (1987) MI-HE-SE ( 1987) MI-HB-SE i 1987) MI-HB-SE ( 1987) MI-HB-SE ( 1987) MI-HB-SE i 1987) hll-hb-se (1987) MI-HB-SE ( 1987) MI-HB-SE ( 1987) MI-HB-SE ( 1987) hll-hb-5e (1987) MI-HB-SE ( 1987) Ml-HB-SE t1987)

172 ' For the Medium Complexity test case a modification of the database was done to only take into consideration typical materials used due to it structural characteristics and manufacturability cbkiderations. Medium Complcxity Test Material Proprtics hietals Id I -? hlaierial Nam Sicel AMS 6437 Qucnchrd/Tempcrcd Steel AIS Stecl AlSI 4 I 30 Quenched~Tcmpttd Sirel. Sininlcss, AM.350 ( AMS 5548) Hardencd (SCT 8.50) Siel. Sininlas. AM455 (AMS 5547) H;irdemcâ t Sm 8501 Aluniinum 7014-T4 Alun1inun12014-T6 - PLATE Aluminum 321-T3 Aluminum 2074-TJ Aluminum $024-Tb Youns's Shar mk,, m,,&.v, blodulus Modulus. * E,c IMsi) (hlsi) ~in3) 30 Il I I I I II Il J I I Aluminutn ZIJ-t86l LI 12. Aluminurn I 13 Aluminunr ZI 14-T8SI Aluminum ' Ici Aluminum 2219-T ' Aluminum 318-7'61 ' Aluminum Aluminum 7005-T MI-HB-SE ( 1987) 19 Aluminurn 7010-T765 1 I 20 II -- $9 L LI 24 Aluminum 7039-T61 Aluminum 70$9-T7351l Aluminum 7050-T Aluminum 7075-T6 - SHEET, PLATE Aluminum EXTRUSIN j MI-HB-SE t19871 Ml-HE-SE ( 1987) MI-HB-SE MI-HB-SE î 1987) MI-HB-5E ( 1987)

173 15 Aluminum 7075-T ,, 1 MI-HB-SE 26 Aluminum 7075-Ti MI-HB-SE y Aluminum T6 Aluminum T65 1 Alurninum 7475-l761 hiagncsium. HK3 I A-. - SHEET. PLATE M~gmium. HKt l A-H24. - SHEET. PLATE Magnaiun HM21 A-T8. - SHEET. PLATE Tiianium Ti-6Al4V. Annald - PLATE SH EET TilJRum Ti4AI4V. Solution Tiwt k Age - PLATE. SHEET Tiiînium Ti-6A14V. Anncded - BAR. EXTRUSIN Titanium Ti-6A14V. Solution Tmi & Arc - BAR. EXTRUSIN J 6.s 16 l , W l l l '9 '" 4'9 MI-HB-SE (1987) M 1-HB-SE (1987) MI-HB-SE (1987) MI-HB-SE ( 1987) MI-HE-SE (1987) MI-HB-SE ( 1987) MI-HB-SE ( 1987) MI-HB-SE ( 1987) MI-HB-SE (1987) MI-HB-SE (1987)

174 Appendix Cross-Sections Database Cross section's geometric definitions are based on ANSYS's geometry system; a brief description of the different geometry arrangements ands variables is included below. Channel Beam Cross Section w1, w2, w3, tl, t2, t3 m. w2 = Lengths of the flanges w3 = verall depth tl, t2 = Flange thicknesses t3 = Web thicknesses I Beam Cross Section Wl, W2, W3, tl, t2, t3 m. ~2 = Width of the top and bottom flanges w3 = verall depth tl, t2 = Fiange thicknesses t3 = Web thicknesses

175 Z Beam Cross Section m, W2, W3, tl, t2, t3 wz, w2 = Flange lengths w3 = verall depth cl, c2 = Flange thicknesses t3 = Stem thicknesses L Beam Cross Section wl, w2 = Leg lengths ci. t2 = k g thicknesses

176 T Beam Cross Section W2, W2, tl, t2 wl = Flange width w2 = verall depth cl = Flange thicknesses t2 = Stem thicknesses HAT Beam Cross Section w1, w2, w3, w4, tl, t2, t3, t4, t5 m, w2 = Width of the brim w3 = Width of the top of the hat w4 = verall depth tl, t2 = Thickness of the brim t3 = Thickness of the top of the hat t4, t5 = Web thicknesses

177 -&Pc d 4 L Sections 1 As b I > , LA A L* I 0.7s l.IZj 12 I.Ni25 L- I 0, I I I I - ' A o L- -. w2 w3 -w4 11 t Z offset v offset z ksignation I.lS I ,- 1.7s L ,A t.75, h C s &/ 0.87s s fl I i 0625 / , 4 l. _ I , o I Iz 0,~s ~ ~ s lz p, L

178

179

180

181

182

183

184 330, 331, , , I 342, , 345-3% 347,, , 3.W , M I ? I8 I IN , Ml Z5981~ XV,0.6-W213. 1, C %063 0.W CU XU WS ? M = Z I024 I , LW I. ILW2 1.I % M % M M J ~ I ~ s I 0.05 I , W _ I.Ma BAC 1498 BAC 1498 BACI BACI BAC _ 0 BAC BAC1498 IW BAC 1498 I4 I 0 BAC BAC 1498 I 55 0 BAC t BAC 1498 I 24-0 BAC , 0 BAC BAC BAC BAC BAC BAC BAC BAC BAC BAC

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