‘2'!- -.‘~ SOME swmas on OXYGEN PERMEATION ‘ 0F PLASTIC PACKAGES Thesis for the Degree of M . S. MICHIGAN STATE UNIVERSITY JUNICHT UNO 1 9 7 3 ABSTRACT SOME STUDIES ON OXYGEN PERMEATION OF PLASTIC PACKAGES BY Junichi Uno Nearly all packaging materials transmit gases and water vapor during long term storage periods and, especially in food packaging, this type of permeation can have deleterious effects upon product quality and shelf life. Accurate measurement of permeation rates, expecially those of oxygen, is obviously essential. This thesis consists of four chapters. In Chapter I, there are represented general cone cepts of gas permeability, and surveys of the previous efforts at measurement of gas permeability. In Chapter II, a new method for measuring the gas permeability rates of a film is proposed, the principle of which is that the difference in the permeation rates of two gases will result in a total pressure change within a sealed chamber. An example is used in which oxygen and nitrogen permeation through an unknown plastic film is shown. Junichi Uno In Chapter III, the oxygen permeability of a PVC blow moulded bottle, called 1 liter Manpak, is measured by both the gas chromatographic method and the chemical reaction method. In this test the effects of the capping torque and head space volume upon the oxygen permeation rate are investigated. In Chapter IV, the oxygen permeation rates of a single wall package and a double wall package are compared‘ using an analytical solution of a differential equation system. Several examples of this solution are given. Also, the analytical and the numerical approach to solving the differential equation system of a double wall package model are compared. SOME STUDIES ON OXYGEN PERMEATION OF PLASTIC PACKAGES BY Junichi Uno A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE School of Packaging 1973 ACKNOWLEDGMENTS The author is grateful to express his hearty thanks to Dr. James W. Goff, Dr. Wayne H. Clifford, and Mr. Istvan Gyeszli for their advice and encouragement to accomplish this study, and also thanks to the many peOple who aided him in many ways in this study. Also he wishes to express his sincere thanks to Kikkoman Shoyu Co., Ltd. for the generous financial support and material supply for this study. And lastly, special thanks are due the author's wife, Mieko, who encouraged him to complete the work when he most needed encouragement. ii Chapter I. II. III. IV. TABLE OF CONTENTS Introduction . . . . . Permeability Properties GENERAL CONCEPTS OF PERMEABILITY AND PREVIOUS WORKS . . . Methods of Permeability Measurement. . Introduction . . . . . Equation Derivation . Example - HDPE . . . . Conclusion . . . . . . OXYGEN PERMEABILITY OF A MANPAK BOTTLE . . . . . Introduction . . . . . Experimentals . . . . Oxygen Permeability of Results and Discussions Conclusion . . . . . . Introduction . . . . . Model Preparation . . iii A NEW METHOD FOR MEASURING CONSTANTS O O O O O O O l- PERMEABILITY LITER Bottles . . . . DOUBLE WALL MODEL IN OXYGEN PERMEATION . . . . . . . Page 11 23 23 24 33 42 43 43 46 62 66 69 70 70 71 Chapter Derivation of System . Example Example Example Example (I). . (II) . (III). (IV) . Comparison of Analytical Numerical Solutions Conclusion . BIBLIOGRAPHY . APPENDICES . . iv Page 74 77 82 84 88 97 102 105 112 Table 2.1 4.10 Relationship among Initial Nitrogen, Oxygen Partial Pressure.and Critical Values LIST OF TABLES An Example of Retention Time . Absorbance and Oxygen Content Calibration . . . . . . . . . Specification of Bottles . . . Average Permeability and Correlation Coefficient . . . . . . . . . Pressure Changes in Example (I) Pressure Changes in Example (I) Thickness Example Pressure Changes in Example (II) Thickness Example Thickness Example Thickness Example Thickness Example Thickness Example Thickness Example and Critical Point in (I) o o o o o o o o o and Critical Point in (II) o o o o o o o o and Critical Point.in (III) 0 o o o o o o o and Critical Point in (III) 0 o o o o o o o and Critical Point in (III) 0 o o o o o o o and Critical Point in (III) 0 o o o o o o o and Critical Point in (III) 0 o o o o o o o Page 37 54 60 67 68 79 81 83 85 86 89 89 9O 90 91 Thickness and Critical Point in Example (III) . . . . . . . . . Thickness and Critical Point in Example (III) . . . . . . . . . Thickness and Critical Point in Example (III) . . . . . . . . .3 Thickness and Critical Point in Example (III) 0 o o o o o o o 0 Thickness and Critical Point in Example (III) . . . . . . . . . Pressure Changes in Example (IV) Analytical Solution and Numerical Solution . . . . . . . . . . . vi Page 91 92 92 93 93 98 103 LIST OF FIGURES Concentration Increase Method Volume Increase Method . . . . . . Pressure Increase Method . . . . . . Oxygen and Nitrogen Permeation Chamber . Pressure Change in Case 1 . . . . . Pressure Change in Case l . . . . . Pressure Change in Case 2 . . . . . Pressure Change in Case 3 . . . . . . Pressure Change in Case 3 . . . . . . Permeation Cell . . . . . . . . . . f(x) = x ln(x) - (x-b) ln(x—b) . . . . . Approximation Procedure for f(x) = O . 1 liter Manpak PVC Bottle . . . . . . Details of Sampling Hole . . . . . . Dual Column/Dual Detector System . Schematic for Separation of Oxygen and Nitrogen . . . . . . . . . . . . . . . Gas Chromatograph Output for Calibration Gas Chromatograph Output for Sample Gas. Beckman DB Spectrophotometer . . . . Calibration Curve . . . . . . . . . vii Page l3 l3 14 25 30 30 32 32 36 36 40 4O 45 47 49 49 51 52 58 61 Figure Single and Double Wall Package Models. Example (I) . . . . . . . . . . Effect of Dimension on Critical Point. Double Wall Package, Example (IV) viii Page 73 80 94 95 LIST OF APPENDICES Appendix Page 2.1 Calculation of eq(2.22) and eq(2.23) . . . . 112 2.2 Calculation of Permeability Constant of Unknown Plastic Film . . . . . . . . . 113 2.3 Numerical Solution of x ln(x)-(x-b) ln(x-b) = 0 . . . . . . . . . . . . . . . 114 3.1 Regression for Calibration Curve . . . . . . 115 3.2 Original Data and their Calculated Values. . 119 4.1 Laplace Transform Applied to Solve eq(4.6) and eq(4.7). . . . . . . . . . . . 132 4.2 Pressure Changes in Example (I) . . . . . . 135 4.3 Pressure Changes in Example (I) . . . . . . 136 4.4 Thickness and Critical Point in Example (I). 137. 4.5 Pressure Changes in Example (II) . . . . . . 138 4.6 Thickness and Critical Point in Example (II) . . . . . . . . . . . . . . . 139 4.7 Thickness and Critical Point in Example (III) . . . . . . . . . . . . . . 140 4.8 Pressure Changes in Example (IV) . . . . . . 141 4.9 Comparison of Analytical, Euler's and Runge—Kutta Methods . . . . . . . . . . . 142 ix CHAPTER I GENERAL CONCEPTS OF PERMEABILITY AND PREVIOUS WORKS Introduction As Raphael (53)* wrote in his book concerning the three major functions of the packaging and generally the term "3 1'3 of Package" means Information, Interaction and Isolation, a package has to protect its contents from the environment or the environment from its contents. Nearly all packaging materials transmit water vapor, gases, and light during long term storage periods, especially in food packaging, these permeation phenomena have very important roles in determining the shelf-life of packaged food. In the general sense in food packaging, the term "Permeability" is considered in connection with passage of water vapor and gases through protective film. But in the broader sense, the term has to include the passage through films of liquids such as water, fats and oils, and of gaseous materials imparting flavor and odor to foodstuffs, and, finally, passage of electromagnetic waves constituting both the visible and invisible spectra. * Footnote numbers refer to reference entries. The most important permeability factors in food packaging are, however, water vapor and gases permeation through the packaging material, especially, oxygen and water vapor permeation. Two permeation mechanisms are known by which permeation occurs: one consists of gases and water vapor dissolving in one side of the packaging material, migrating through it, and re-evaporating from the opposite side of it, first prOposed by Grahm in 1866, and the other by which permeation occurs is that gases and water vapor pass through pores or interstices in the packaging material, for example, the permeability of aluminum foil. The former mechanism depends on the kinds of gases and the molecular structure of the packaging materials and also in this mechanism the solubility characteristics due to the combina- tion of gases and packaging materials effects the permeation phenomenon. As mentioned above, light penetration is also included in permeation phenomena. In food packaging, visibility is very desirable from both sales and consumer viewpoints, however transparent packaging materials are usually permeable to a wide range of wavelengths of light which almost certainly results in undesirable quality changes of the contents. Light promotes discoloration of certain foodstuffs, such as its bleaching effects upon butter, cheese, and potato chips, or may cause a darkening in color, as in sausages, boiled ham, and other cured meats. Light lowers the vitamin C and riboflavin content of foods, milk being particularly affected. Charlton (10) reported that while light of below 325 millimicron wavelengths has a marked effect, light in regions above 460 millimicron has comparatively little effect. On the other hand, it has been suggested that it might be feasible to package meat or cheese in a film which would be permeable to the ultraviolet germicidal wavelength of 2537 X in order to reduce bacterial contamination on the surface of the product. Ikawa (29) reviewed the effect of light on foodstuffs containing tar- colorant, natural colorant and vitamins. He also reported the deterioration of plastic containers caused by light, i.e. almost all plastics except acrylics will degrade in light. Aoki (4) reported the effect of sunlight upon the deterioration of soy sauce, called Shoyu in Japan. Holmes (27) tested the effect of an ultraviolet absorber in cellulose acetate film. Another permeability problems in food packaging are grease and oil permeability: the staining of a food package with grease, or the actual appearance of an oily film on the outside of a package. These detract greatly from aesthetic appeal which is a definite factor in customer acceptance. In this thesis, however, only the permeation problem of gases, especially oxygen permeation is considered. Many cases exist in food packaging where gas permeation should either be completely avoided or be facilitated to some extent during the storage period. Three examples which have been considered by others, are Shoyu, meat, and cheese packages. Aoki (4) reported that oxygen accelerates degradation of Shoyu and that if Shoyu is bottled in PVC bottles it trends to increase in butylaldehyde formation in comparison with the situation where glass bottle is used. Therefore, oxygen permeation should be avoided as much as possible in Shoyu packaging. Sacharow (54) mentioned that in meat packaging, control of oxygen permeation requires a compromise between development of ideal color and prevention of oxidative degradation reactions, i.e. oxygen is needed for "bloom" in red meats, but it will promote rancidity in fats, particularly in pork fat. Charlton (10) reported that in natural cheese packaging, Parakote* which uses cellophane as a base sheet and is coated with a rubber-wax composition is suitable because this film has high carbondioxide permeation and low oxygen permeation properties. During the course of cheese aging it evolves carbondioxide, and low oxygen permeability prevents cheese from dark ring formation. Other examples in relation to the gas permeation of packages and the shelf * Cheese Wrapper by Marathon Corp. U.S. Patent 2,339,242. . life of the product have been given as follows. Galbraith (23) developed a graphical method to estimate the shelf life for an aqueous product. Hu(28) tested packages described as inside foil laminated pouches for freeze dried raw pork loin and potato granules. As to prediction of the product shelf life inside the package, Brown (7) developed two categories between permeability and shelf life: a simple prediction of shelf life and a complex one. Daoud (11) compared an aluminum laminate with a Mylar-Saran- polyethylene* laminate for freeze dried red bell pepper packaging and concluded that aluminum laminate provides better resistance to oxygen permeance than the all-plastic laminates. Heiss (26) stated the relationship between moisture content and shelf life, with examples of lean meat, dehydrated foods, roasted coffee and other low- moisture foods. Therefore, the degree of acceptable permeability depends on the product to be packaged. Permeability Properties In order that methods of measuring permeability of package and the packaging materials may be discussed, a general acquaintance with the permeability properties of packaging materials is of great help. * No. 3800, 50 M/V28P, 0.5 mil Mylar, 0.2 mil Saran, and 2 mil polyethylene. Paine (47) stated about preliminary concepts of permeability. Briston (5) and Davis (16 & 17) summarized general factors affecting permeation phenomena, for example, permeation area, thickness of films, temperature, pressure, relative humidity, and pinholes. Li (36) tested the effect of temperature and pressure on solubility and permeation rates of several kinds of films. Stannett (60) wrote the details of the permeability theory and factors affecting permeation phenomena. Lesse (35) generalized the diffusion equation for polymers. Peterlin (51) calculated the permeant gas current in both directions and their ratio at a constant pressure difference. Storstrgm (62) explained the general phenomena of gas and water vapor permeation in paper-polymer composites. Let me summarize the permeation theory, using oxygen permeation through plastic film as an example. Assume a plastic film, through which oxygen permeates from a higher pressure side of oxygen partial pressure to a lower pressure side, whose thickness is L mm and permeation area is A sq.cm, and consider that the oxygen transmission is of an activated diffusion type, i.e. the concentration gradient of oxygen in the film causes oxygen diffusion. It is obvious that after a comparatively short period steady state will be reached and oxygen will permeate through the film at a constant rate, providing the pressure difference between the two sides is maintained. Here Fick's law of diffusion can be applied, dc Q = - D A a; (1.1) where, Q = oxygen flux through the film (cc(STP)/hr.) D = diffusion constant (cc(STP)mm/sq.cm hr. mol/lit.) A = diffusion area (sq.cm) 3% = concentration gradient in the direction of gas permeation (mol/lit./mm) If D is independent of concentration and position, eq(1.1) can be integrated between c=cl and c=c2, x=0 and x=L, as follows, Q = £32 (cl - c2) (1.2) where c1 and c2 are the oxygen concentrations at the sur— faces of higher oxygen partial pressure side (mol/lit.) and lower side, respectively. Generally, the oxygen concentration at the surface is expressed as follows if Henry's law holds, c = S p (1.3) where, S = solubility constant of oxygen in the film (mol/lit./atm) oxygen partial pressure at the film surface (atm) Plug eq(l.3) into eq(1.2), then, ._11 _ where p1 and p2 are higher and lower oxygen partial pressure, respectively. In eq(l.4), D S is referred to as permeability constant P , i.e. v = = Q_E_____ PV D s A(pl’pz) (1.5) where PV has the dimension of (cc(STP)mm/(sq.cm hr atm)) and is called the volumetric permeability constant. If Q is measured in molar units (mol/hr), the permeability constant is defined as Pm and has the dimension of (mol mm/(sq.cm hr atm)) and is called the molar permeability constant. These two constants have the following relation- ships, providing that oxygen obeys the gas laws under the conditions of temperature and pressure, RTO m = P (1.6) PC V P where T0 = 273 deg. K and P0 = 1 atm, R is gas constant, in this case R = 0.082 atm lit./deg. K mol. If a container has the surface area of A(sq.cm), inside volume of V (cu.cm) and thickness of L (mm), and the oxygen partial pressure inside the container pl (atm) is lower than that of outside p2 (atm), oxygen permeates from outside into inside at the rate of, _ A _ Qm - f Pm (p2 pl) (1.7) From the gas law, pl V = n R T (1.8) where n = no. of moles in the container Differentiate eq(l.8) with respect to time t, then, dp l _ dn "d't"" V " d"_t R T (1'9) .Since 92 = Q e (l 9) becomes dt m’ q ' ' dp 1 _ A EE’ - VIE Pm (P2 ‘ P1) R T (1'10) Plug eq(1.6) into eq(l.10), then, dpl = P A po (p2 - pl) R T dt v V L R TO dp 1- A I _ as" — Pv v—r Po To (92 P1) (1°11) (p = 1 atm) If p2 and T are constant, eq(l.11) can be integrated. This will be done in Chapter IV. Temperature Effect on Permeability Constant. Pv has been shown experimentally as follows, PV = Pov exp(-Ep/RT) (1.12) where, PV = permeability constant at temperature T deg. K POV = temperature independent constant Ep = activation energy for permeation process R = gas constant Davis (16 & 17) discussed Ep as follows, Ep is the sum of the activation energy of the diffusion process, Ed, and 10 the heat of solution, H. Ed is always positive, and the diffusion coefficient increases with increasing temperature. H is small and positive for permanent gases, but negative for easily condensable vapors. Hence, solubility increases slightly with increasing temperature for permanent gases, but decreases for vapors. The permeability of a specific polymer-penetrant system, therefore, may increase or decrease with increases in temperature, depending upon the relative effect of temperature on the solubility and diffusion coefficients of the system. For this reason, permeability values of different types of film determined at one specific temperature may not be in the same relative order at other temperatures. As suggested by eq(l.12), the effect of temperature on permeability may be studied conveniently from a plot of log permeability versus the reciprocal of the absolute temperature. In many cases such a plot gives a straight line which enables permeability values at temperatures other than those studied to be estimated with reasonable accuracy. Pressure Effect on the Permeability Constant. As shown in eq(l.4), the permeation rate is directly proportional to the difference in pressure or partial pressure on the Opposite surfaces of the film. But deviation has been noted in experiments with some vapors. 11 As to this deviation, Briston (5) stated that it could be caused by pressure-dependent solubility coefficient, i.e. failure to obey Henry's law. Effect of Thickness on Permeability Constant. Lockhart (37) reported in his thesis that most investigators had verified the empirical result that the permeability constant is independent of the thickness of the film being permeated. But Briston (5) showed that the permeability constant is not independent on film thickness, i.e. in the case of water vapor transmission a thicker film has a higher permeability constant. On the other hand, Charlton (10) showed that an increase in film thickness resulted in a decrease in permeability constant. Methods of Permeability Measurement There have been reported many experiments concerning the measurement of gas permeability, but they may be classified into the following three categories as Taylor (64) reported, 1. Concentration increase method 2. Volume increase method 3. Pressure increase method Landrock (34) discussed in detail the various methods which have been used for gas permeability measurements. Stocker (61) discussed the three kinds of methods mentioned 12 above. In the first method, shown in Fig. 1.1, the total pressure on each side of the test film is approximately equal. Using a different gas on each side of the film, both at atmospheric pressure, the difference in the partial pressure of the test gas between the two sides is one atmosphere. This causes it to flow through the film, where it is quantitatively analyzed. In the second and the third methods, as shown in Fig. 1.2 and Fig. 1.3, the amount of gas which permeates is measured either by an increase in pressure at a constant volume, or by an increase in volume at a constant pressure. In these cases, the pressure difference is obtained by evacuating one side, or increasing the pressure of one side, or by a combination of both. There have been developed many modified procedures from the above three basic methods. In the current ASTM D 1434-66 (76), two methods are proposed: Method M (Manometric) and Method V (Volumetric). Method M corresponds to the pressure increase method, and Method V corresponds to the volume increase method. Gas transmission rate (GTR) is defined, cu.cm (STP) GTR = 24 hr. sq.m atm (1.13) And gas permeability coefficient (P) is, = cu.cm (STP) cm of thickness sec. sq.cm cmHg WI at 23 deg.C (1.14) As to WVTR (water vapor transmission rate), ASTM E 96-66 (77) shows that the test film is fastened over the 13 Test Gas ( atmospheric pressure ) / I / l Test Film Other Gas ( atmospheric pressure ) Concentration Increase Measured Fig. 1.1 Concentration Increase Method Test Gas ( atmo he ic r Z/// elevated pressure ) Test Film I F: Vacuum or Test Gas ( atmospheric pressure ) r1 A{Volume Increase Measured p-._r__i_.at Constant Pressure Fig. 1.2 Volume Increase Method l4 convex omsuhosH ohdmmohm mssao> psdpmsoo pd consume: unwohosH thmmohm sszow> m.H .mHm a —\ afim pmo‘a\ J \ % K A whammohm cahonmmospw v new pmoe 15 mouth of a cup containing either a desicant or water and after a certain time under a controlled atmosphere the weight gain or loss of this unit tells WVTR. The units are defined as follows, Rate of water vapor transmission = 24 firgréq m (1.15) Water vapor permeance = 24 hrl gg'm Hg (1.16) Water vapor permeability = 1 gr. cm (1.17) 24 hr. sq.m mmHg And there are defined the following six procedures, Procedure A: desicant method at 23 deg. C Procedure B: water method at 23 deg. C Procedure BW: inverted water method at 23 deg. C Procedure C: desicant method at 32.2 deg. C Procedure D: water method at 32.2 deg. C Procedure E: desicant method at 37.8 deg. C Besides these physical methods, chemical methods, as can be said one of the application of concentration increase method, has been proposed by Calvano (8) and Mack (40). As to flavor and odor permeability, Muldoon (44) used methyl furoate which is absorbed in caustic solution and analyzed spectrophotometrically as furoic acid. E. G. Davis (15) used NH3-sensitive paper to detect minute pores or pinholes in the packaging material, for example, PVDC and PVDC coated papers. Eichhorn (19) applied the chemical reaction, 2Na+2H20=2NaOH+H2 16 to evaluate water vapor transmission through polyethylene electrical insulation. Fites (21) tested the drug-perme- ability properties of several water insoluble film, cellulose acetate, ethyl cellulose, methyl hydroxyprOpyl cellulose and polymethylvinylether maleic anhydride. In the concentration increase method, Davis method (12 & 13) has been used or modified which can be classified as an Orsat gas analyser, gas Chromatograph, thermal conductivity, mass spectrometer, weight increase (gravimetric), humidity sensitive detector or radio active tracer methods. Dean Milk Company (2) used the modified Orsat device for determining oxygen content in milk powder pouch. Stahl (59) compared a gas chromatographic method and the Orsat method and reported that a gas chromatographic method required less than 1 ml gas sample to be analyzed and was more rapid and quite versatile as compared to an Orsat method. There have been many reports utilizing gas chromatographic methods. Vosti (66) detected micro— quantities of nitrogen, oxygen, carbondioxide and hydrogen by means of the equipment modified to incorporate a two stage column and two thermal conductivity detectors using argon gas as the carrier gas. Fricke (22) used F & M gas Chromatograph Model 500 to measure the oxygen, carbon- dioxide and nitrogen permeation of cellophane, polyethylene, Mylar, polyester, polypr0pylene, polystyrene and PVC. 17 Karel (32) measured oxygen and carbondioxide permeation in several packaging films, including polypropylene, Mylar and fluorocarbon films. Jeffs (30) used a modified Fisher Gas Partitioner, Model 25, equipped with a single 6 ft molecular sieve 13 X column and thermistor detectors to measure the oxygen permeability constants of low density polyethylene, polypropylene and other films, and concluded that the gas chromatographic method has nine advantages and that the biggest advantage offered by this method is that the package itself and not just the material is tested. Gilbert (24) and Caskey (9) used a gas Chromatograph to measure the oxygen permeability, and Smyser (58) measured the oxygen transmission rates at various humidity using a gas Chromatograph. Loudenslagel (38) tested the whole- package transmission of many gases through high density polyethylene, rigid PVC and other polymers. Kanitz (31) used Varian Aerograph 1520-C gas Chromatograph for measuring the nitrogen and methane permeability of graft copolymers of polyethylene and teflon FEP film. As to the thermal conductivity application for gas analyzer in the concentration increase method, Ziegel (74) measured hydrogen isotope permeation in polyvinyl fluoride films and natural rubber using the differences in their thermal conductivities. Pasternak (49) developed an efficient instrument, called the Polymer Permeation Analyzer Model PPA-l to determine the permeability constants 18 of carbondioxide, nitrogen, oxygen and hydrogen. Ahlen (1) measured a sorbed water vapor permeability coefficient through paper and cellulose films using a thermal conduc- tivity cells. Yasuda (73) used the thermal conductivity method to measure the nitrogen, oxygen, carbondioxide, helium and hydrogen permeability of polymers, such as polydimethylsiloxane, polyphenylene oxide, low density polyethylene, high density polyethylene and PVC, and found they are comparable to conventional vacuum type methods. Maneval (42) measured carbondioxide and oxygen permeation through rigid PVC and vinylchloride copolymer using a mass spectrometer to analyze oxygen, carbondioxide, and nitrogen in argon gas. As to the weight increase method (gravimetric method), Laine (33) measured the permeability of poly- ethylene film to 21 organic vapors at 21, 38 and 49 deg. C using a dynamic sorption method, i.e. the weight increase of silica gel in a plastic pouch. The water vapor trans- mission of silicon rubber, epoxy film, polyethylene, nylon 6 and polystyrene was measured by Osburn (46) with the Laine method. Winrich (69, 70 & 71) prOposed the methods of the water vapor transfer measurement through sheet materials, and also he (68) tested R.H. senser, St. Regis - Honewell WVTR tester, for determining the water vapor transmission rate of Kraft/Wax/Kraft, Glassine/Wax/ Glassine and polyethylene. 19 Stocker (61) reported that C14 was used to investi- gate the permeability of elastomers to carbondioxide, and Deterding (18) used the radioactive isotope of hydrogen, tritium, for measuring the permeability of plastics, paint films and rubbers to water or hydrocarbon liquids. The volume increase or decrease method has the advantage that the pressure difference over the film is constant. Todd (65) designed an apparatus using the volume decrease method, and Major (41) developed a very simple apparatus and showedthat the results correlated well with those from the ASTM method. The pressure increase method was summarized by Stocker (61) up to 1963. Recently, Williams (67) used this method and measured the gas permeability and WVTR of polyoxymethylene. Takizawa (63) measured the water vapor permeability of hydrophilic cellophane, hydrophobic polyethylene film and the composite membranes. Nakagawa (45) measured the transport parameter of PVC film. Both works were conducted by the pressure increasing method (vacuum method). Woodgate (72) measured leak rates of nitrogen, oxygen, helium, hydrogen, methane, argon and neon through the commercial polycarbonate film (Kimfol) by pressure increase method and compared these values with previous values available. Morrow (43) tested carbondioxide permeation of 10 oz. plastic bottle of PVC, acryionitrile terpolymer and low density polyethylene, using the pressure decreasing method. 20 Applied Fluidics Inc. (3) reported that applied fluidics could be used for detecting pinhole leaks and other defects in plastic bottles before filling. Elschnig (20) investigated the gas and water vapor permeability imparted by polyvinylidene chloride coatings on polyethylene, polypropylene, polystyrene, PVC, polyester and polyamides. Schrenk (55) summarized properties of multilayered films, including polyethylene, polypropylene, polystyrene and Mylar film, and concluded that permeabilities of multilayer films can be calculated from layer geometry and individual component properties. Paul (50), PetrOpoulos (52) and Pasternak (48) described a dynamic method for investigating the mechanism of the gas permeation and diffusion through polymers. Siegel (56) proposed a method to check an error in diffusivity obtained from permeation experiments using the time lag technique. Loudenslagel (38) showed one approach by analog computer simulation to solve gas permeation problems. And Gyeszli (25) used an analog computer to solve permeation problems. As mentioned above, there have been reported numerous methods of measuring or estimating permeabilities, and it is difficult to choose the best one among them as they all have their advantages and disadvantages. Taylor (64) used the three basic methods mentioned above and compared their results as follows: 21 Permeability Correlation Coefficient P : P 0.980 c v P : P 0.980 C P P : P 0.997 p v where, Pc is the permeability constant obtained by the concentration increase method (Orsat analyzer) Pv is the permeability constant obtained by the volume increase method Pp is the permeability constant obtained by the pressure increase method. But unfortunately, there exists a considerably big difference in permeability data by different investigators, for example, the oxygen permeability constant of low density polyethylene ranges from 6,000 to 9,000 cc(STP) mil/day sq.m atm, and Landrock (34) reported as low as 4,090, on the other hand, Simril (57) reported as high as 13,300. The reasons for this variation have not been established, but it is supposed the result of: 1. Variation between different batches of the film 2. Difference between the usage of additives, i.e. stabilizer, plasticizer, colorant and so on 3. Experimental differences due to Operational conditions, i.e. process temperature, pressure and time 4. Inherent differences in measurement methods 22 In the case of hydrOphilic plastic films, such as generated cellulose, glassine and some of the polyamide film, permeabilities are influ- enced by the relative humidity level (RH) at which the measurements are made. Davis (12) reported the effect of relative humidity Davis (14) mentioned about the effect of heat «processing on the gas permeability of the same materials. CHAPTER II A NEW METHOD FOR MEASURING PERMEABILITY CONSTANTS Introduction As described in CHAPTER I, there exist many methods for measuring permeability Constants. Here I propose one original method--using the differences in the gas permeability constants of two gases for a plastic film. For example, consider the combination of oxygen and nitrogen permeation through a plastic film, i.e. as shown in Fig. 2.1, the chamber volume is V cc, the permea- tion area is A sq.cm, the thickness of the film is L mm and this chamber is placed in Open air of constant temper- ature, T deg. K. Consider the case that the.initial oxygen content (or the initial oxygen partial pressure, poi atm) is less than that of air, p0a = 0.21 atm and the sum of the initial partial pressure of oxygen, p and oi' nitrogen, p atm. And assume that there is no ni’ ls pti interaction between the oxygen and nitrogen permeations. 23 24 Equation Derivation The partial pressure changes of oxygen and nitrogen in the chamber with time are expressed from eq.(l.ll) as follows: dp ____°— E_A_ _ dt _ on TO V L (poa p0) (2'1) dp _n_ $_A_ _ dt — Pvn TO V L (pn pna) (2'2) where, p0 = oxygen partial pressure in the chamber (atm): pn = nitrogen partial pressure in the chamber(atm) (2.3) p0a = oxygen partial pressure in air: 0.21 (atm) ? pna = nitrogen partial pressure in air: 0.79 (atm) (for the sake of simplicity) PVO = volumetric oxygen permeability constant of W the film (cc(STP) mm/sq.cm hr. atm) Pvn = volumetric nitrogen permeability constant of the film (cc(STP) mm/sq.cm hr. atm) >(2.4) T = room temperature (deg. K) To = standard temperature (deg. K) t = time (hr.) _J Put, T A_ T —Vf — k (2.5) and assume that k, on’ and Pvn are constant during the experiment. nonsmso soapsoshom cowohpwz can sawhuo H.m .mwm 25 s4; . n.<.> s o 1» 9+ at a wsm+aomuflpe s as 1. e .a Iv on Hon puppw ouppm 26 Then, these two equations eq(2.l) and eq(2.2) can be integrated between t = 0 and t = t (corresponding pOi and p0, pni and pn) P0 = poa - (p0a - poi) exp(-onk t) (2.6) Pn = pna + (pni - pna) exp(-Pvnk t) (2.7) The total pressure inside the chamber is the sum of p0 and P nl pt = pO + pn (2.8) Take the derivative of Pt with respect to time, in order to follow the change of pt with time, EEE = 332 + 332 (2 9) dt dt dt ° p and pn have been obtained with respect to time in eq 0 (2.6) and eq(2.7), therefore, these two equations can be differentiated analytically as follows: dp o = _ _ dt— (poa poi) onk exp( PVOk t) (2.10) dpn dE7'= -(pni - pna) Pvnk exp(-Pvnk t) (2.11) Plug eq(2.10) and eq(2.ll) into eq(2.9), then we get, dpt dt = (poa - poi) onk exp(-onk t) - (p - pna) Pvnk exp(-Pvnk t) (2.12) ni The change of the total pressure in the chamber with time can be determined by eq(2.12) as follows: 27 dp if EEE 2 0, pt does not decrease with time and otherwise, dpt If at_ < 0, pt decreases With time. Let us consider the following three cases, case 1 PVO > Pvn (2.13) case 2 P = P (2.14) vo vn case 3 Pvo < Pvn (2.15) Practically case 2 is unlikely to happen. For case 1, eq(2.12) can be rearranged, dp P - - p t _ _ _ _ n1 na dt_ _ (poa poi) onk (exp( onk t) (p - p .) oa 01 vn §—— exp(-Pvnk t)) (2.16) vo In eq(2.16), (p0a -poi), PVO and k are positive, and also define f(t) as follows: P -'P P f(t) =exp(-P k t) - n1 na vn exp(-P k t) (2.17) V0 poa-poi vo vn . . dpt If f(t) lS p051tive, aE—-is positive. Assume that f(t) is positive, then, since eXp(-Pvnk t) is positive, exp(—onk t)- > pni - pna vn exP(-Pvnk t) Poa - p01 on P - ‘ P P ex kt P - P > “1 na V“ (2.18 p( ( vn VO)) poa - poi on ) 28 P - ‘ P P Then, k t (Pvn - PVC) > 1n( “1 _ na Pvn) oa poi vo (2.19) We have eq(2.13), so, ln(pni - pna Pvn) t < poa ' poi on (2.20) k (Pvn - on) Because t should be positive, on pni < Pna +‘§- (poa - poi) (2.21) vn Also, above derivation gives us the root of f(t) = 0 as follows: pni - pna Pvn 1n(p _ p . P ) _ 0a 01 v0 t1 — k (P _ P ) (2.22) vn vo Therefore, when 0tl, f(t) is negative i.e. this means that for 0tl pt decreases with time, and pt has the maximum value dp . t _ _ 1, Since dt_ — 0 at t — t1. 9) Fr H" II t w r‘. w II Olpti=p'+p 01 ni and as t goes to infinity eq(2.6) and eq(2.7) give, exp(-onk t) 0 II o exp(-Pvnk t) 29 so, (p ) = poa (p )m = pna (pt)oo = p0a + pna The maximum value of pt is given; (pt) = p - (p max oa - poi) exp(—onk t1) oa +p + (p na - pna) exp(—Pvnk t1) (2.23) hi The rough change of pt with time can be figured out as - p .), then, shown in Fig. 2.2. If pni>p 01 na P 08. dp eq(2.16) gives us, 3E2 is negative for all positive t. Therefore, pt decreases all the time. Fig. 2.3 shows pt change for this case. For case 2, eq(2.12) gives, (PVO = Pvn = PV) dpt dt_ = pvk exp(—ka t) (poa - poi - pni'+pna) (2'24) = ka exp(-ka t) (l - pti) (2.25) Since pOa + pna = 1 atm, and pti means the initial total pressure in the chamber, we obtain, '1 dpt lfpti>lr F . if pti 1 dt 0 (2 26) dpt lfpti 1n k P - P 2.27 (poa _ poi on) / ( vn VO) ( ) vo and pni > pna + P__ (poa — poi) (2'28) vn then, we get, f(t) > 0, therefore, if 0tl, f(t)>0 where, t1 = 1n(pni : pna :Vn)/k (pvn - on) (2.29) poa poi vo i.e. this means that during Otl, pt increases with time, and pt has the minimum value dp _ t _ = at t - tl because dt — 0 at t t1. And also at t = 0, pti = p . + p and as t goes infinity, (pt) = p + p (X) The minimum value of pt is given, (pt)min = poa - (poa — poi) exp(-onk t1) + pna + (pni - pna) exp(-Pvnk t1) (2.30) The rough change of pt with time is shown in Fig. 2.5. P vo < ——— - . pna + P (poa p01 vn If p ), eq(2.16) gives us ni Pressure Pti pni 1.00 0.79 pti pni 0.21 pno \ 32 pt \ X Pn Time Fig. 2.4 Pressure Change in Case 2 Pressure Pti 1.00 pni ,1’ 51: Dn . 0.79 I 0.21 _ po poi 1;1 Time . P Fig. 2.5 Pressure Change inCase 3 (pni> pna+ PILO (pas-poi) ) vn 33 dp dt Fig. 2.6 shows pt change for this case. t >0 for all t>0, therefore, pt increases all the time. For case 1 and 3, the maximum or minimum value of pt, and the time t which gives the maximum or minimum 1 value of ptm are determined by the initial values of oxygen and nitrogen partial pressure, p and pni' and the oi permeability constants of oxygen and nitrogen, pVO and pvn’ providing that the other experimental constants, A, V, L and T do not change during the experiment. and t Conversely speaking, if (pt) min 1 max or (pt) can be measured by the experiment, the permeability constants of oxygen and nitrogen can be determined numerically. Example - HDPE Let us calculate the actual values of (pt)max or (Pt)min and t1 for the ordinary plastic film. For example, the extruded high density polyethylene film has the permeability constants given as follows: P = 185 (cc.mil/100 sq. inch day atm) V0 (2.31) Pvn = 42 (cc.mi1/100 sq. inch day atm) from "Guide to Plastics" (75). And the experimental constants are as follows: T = 273 + 30 = 303 deg.K T0 = 273 deg.K v = 200 cc t (2.32) L = 0.05 mm A = 500 sq.cm 34 For the sake of simplicity, it is assumed that the initial total pressure (pti = p + poi) is 1 atm. ni The calculated values of (pt)max and t1 by eq(2.22) and eq(2.23) are shown in Table 2.1. As predicted by rule of thumb, the lower the initial oxygen partial pressure is, the higher the maximum value of pt becomes, but tl remains constant. In the case that pni is 1.0 atm and pOi is 0.0 atm, i.e. initially there exists only nitrogen in the chamber, the maximum value of pt is 1.10502 atm and this value is obtained at t1 = 113.90 hr. (Appendix 2.1) The difference between (pt)max and atmospheric pressure is 0.10502 atm and this value can be measured easily by a routine pressure gauge, for instance, a Bourdon tube, diaphram gauge, U-manometer or strain gauge. Fig. 2.7 shows the concept of this chamber, called the permeation cell, which is very similar to Karel's (32). Let us continue the simplest case, i.e. pni = 1.0 atm (2.33) pOi = 0.0 atm Suppose that we get the values of (pt)max and t1 from the experiment, and let us consider how to calculate the values of PV and Pvn under the condition (2.33) and p0a = 0.21 0 atm pna = 0.79 atm hold. 35 Prerure 1.00 r r Pti Pt 6 pni N““*~' iqg====._____ 0.79 0.2 ‘ fir- po poi. Time P Fig. 2.6 Pressure Change in Case 3 (pni$pna+ P39- (poa—p ) ) vn oi Pressure Gauge /1 I \ 45E:) \ .i | \ ‘Ill / \ i I ‘1 '1:— ~ I ' Gas Flow Tube Gas Flow :Tube & Valve . & Valve «15- Gas Flow Tube Gas Flow Tube & Valve & Valve Fig. 2.7 Permeation Cell 36 moo.a om.maa om.o om.o omo.a om.mHH mH.o mm.o mmo.H om.mHH oa.o om.o omo.a om.mHH mo.o mm.o moa.H om.maa oo.o oo.H lapel smelter A.Hec He lapel Hoe lapel see .mmsam> HMUHuHHU paw musmmmum Hmfluumm smmzxo .swmoauflz HMHuHsH msofim mflanOHumamm H.N canoe 37 We get, TO L Pvn t1 = §--§é-ln (§-)/(Pvn - on) (2.34) V0 on vn (Pt)max = l - 0.21(exp(P _ P 1n P__) V0 vn vo Pvn Pvn ‘eXP(P _ p ln 5——)) (2.35) vo vn vo T A. Put, — —— = k TO VL l vn k t1 = p — p 1“ 5-" = a (2.36) VD V0 V0 (p ) -1 t max _ _ _ _ 0,21 — -eXp( onk tl)-+exp( Pvnk tl)-b (2.37) i.e. a and b can be determined eaSily from t1 and (pt)max. Plug eq(2.36) into eq(2.37), then we get, exp(-a on) - exp(-a Pvn) = - b (2.38) and from eq(2.36), exp(-aP ) P exp(a(Pvn _ on)) = ex (-ano) = P23 (2'39) p vn vo Here, the new variables, x and y are introduced such as, exp(-aP ) = x V“ (2.40) exp(-aPVO) = y and rewrite eq(2.38) and eq(2.39), y-fX=-b (2.41) P X = .22 x P VO 38 Using the following relationship to the above equation, -aP = 1n x vn -aPv0 = 1n y and since a # 0, then we get, y = 1n x . x In y (2.42) Plug eq(2.41) into eq(2.42), we obtain, x - b _ 1n x x — 1n x-b) (2'43) If x # 0 and x - b # l, we get, x 1n x - (x-b) ln(x-b) = 0 (2.44) , i.e. We want the roots of eq(2.44) in order to get Pvn if we have one root of eq(2.44), x1, we can calculate Pvn as follows: exp(-ann) = x1 so, -ann = 1n xl a is known from t1 and k which can be obtained from T,TO,A,V and L, therefore, Pvn = (-1n K1)/a (2.45) And also we get, Y1=x1’]D so, on =(-ln (xl - b))/a (2.46) Let us consider how to solve eq(2.44). It is impossible to get the roots of eq(2.44) analytically. Let us consider the numerical solution of eq(2.44). 39 First of all, figure out the shape of y = f(x) = x 1n x - (x-b) ln(x-b), where b is positive. Fig. 2.8 shows the rough shape of y = f(x), and apparent from Fig. 2.8, if b.) 1, f(x) = 0 has no root at all. In the case of b Cu——> Cu (3.5) Cu + Cuti——e> Cu+ (3.6) Cu+ + 2 NH-3—§>Cu(NH3)2+ Colorless (3.7) Cu++ + 4 NH3-€> Cu(NH3)4++ Blue (3.8) 56 The reaction (3.5) means the copper oxidized by oxygen and the reaction (3.6) means that if there exists a large excess of the metallic copper in the bottle the cupric (Cu++ ) complex is reduced to cuprous (Cu+) complex. The reaction (3.7) tends to keep the solution colorless as long as the metallic copper exists. When the solution in the cuprous complex state is drawn from the bottle, i.e. removed an excess of the metallic copper, atmospheric oxygen completes the oxidation of cuprous ions to cupric state, then the solution takes blue, i.e. the reaction in the bottle is, 4H++ 02+ 4Cu———>4Cu++ 21120 (3.9) After removal from the bottle, the reaction is, 2 H 0 + 0 + 4 Cu+-———4>4 Cu++ 2 2 + 4 0H (3.10) And the overall reaction is, + ++ - 4 H + 2 0 + 4 Cu—-> 4 Cu + 4 OH (3.11) 2 Stoichiometrically, each gram of the metal copper converting to cuprous ions consumes 88.1 cc of oxygen (STP) by the reaction (3.9), and cuprous ions to cupric ions consumes 88.1 cc of oxygen (STP) by the reaction (3.10). As a totaL the metallic copper to cupric state consumes 176.2 cc of oxygen (STP) by the reaction (3.11). There are two methods to determine cupric ions in the solution: volumetric analysis and colorimetric analysis. Because of its simplicity, and because it uses 57 a small amount Of the sampling liquid, the colorimetric analysis is used here. Fig. 3.7 shows Beckman DB Spectro— photometer* and wavelength 625 millimicron is used. The sample liquid used here is 1 ml and this sample is placed Open for over 30 minutes to permit complete oxidation before it is measured. The instrument indicates absorbance which is proportional to the concentration Of Cu++. This absorbance reading can be equated linearly to the concen- tration Of oxygen in the solution by the following calibra- tion. The cupric nitrate salt and the cupric sulfate are used for the calibration, because Of its quickness and accuracy. Known amounts Of these salts are dissolved in amonia reagent which is the same as the solution used in the test, and these solutions are mixed with oxygen com- pletely before measuring their absorbance. Take Cu(N03)2 3H20 for example, 327.7 mg Of the cupric nitrate is weighed by a chemical balance and dissolved in 50 ml Of ammonia solution. This solution is diluted into several concentrations Of the cupric nitrate with ammonia solution. By the reaction (3.11), 327.7 mg Of Cu(N03)2 3H20/50 ml solution is equivalent to 30.4 cc oxygen (STP)/100 ml solution, because 327.7 mg = 327.7/241.6 = 1.358 m mol Of Cu(N03)2 3H 0, and l m mol of 2 *By Beckman Instrument Inc. Scientific and Process Instruments Division, 2500 Harbor Boulevard Fullerton, California, Model DB, Cat. NO. 1401, Serial NO. 304636, 115 V 1 Amp 50/60 Hz. 58 Fig. 3.7 Beckman DB SpectrOphotometer 59 Cu(N03)2 3H20 corresponding to 11.2 cc oxygen (STP). Table 3.2 presents the calibration data and Fig. 3.8 shows the calibration curve. As can be seen in Fig. 3.8, excel- lent agreement was Obtained, except for high range absor- bance. The linear regression analysis gives the following relationship: for Cu(N03)2 3 H20, 02 (cc (STP)) = 15.582 X absorbance - 1.147 (3.12) /100m1 correlation coefficient = 0.998 for CuSO4 5 H20, 02 (cc(STP)) = 14.449 X absorbance — 0.762 (3.13) L/lOOml correlation coefficient = 0.999 It can be concluded that there is no significant difference between Cu(NO3)2 3 H20 and CuSO4 5 H20, therefore, 02 (cc(STP)) = 15.112 X absorbance - 0.972 (3.14) [100ml correlation coefficient = 0.998 The regression equation (3.14) can be used for the evalua- tion Of the data. (Appendix 3.1) The test is conducted by placing copper turnings whose weight corresponds to 45 gr/lOOOml ammonia solution into the PVC bottle. After purging the air in the bottle by blowing the nitrogen gas, the bottle is filled with a defined amount Of ammonia solution and immediately plugged. Some bottles are filled with copper powder instead Of 60 Table 3.2 Absorbance and Oxygen Content Calibration Dilution Cu(N0 ) 3H 0 CuSO 5H 0 Ratio 3 2 2 4 2 327.7 mg/SOml 272.0 mg/50ml Absorbance 02 content Absorbance 02 content CC(STP) CC (STP) 100 ml 100 ml 1/1 0.755 30.40 0.750 24.40 1/2 0.688 15.20 0.685 12.20 1/3 0.710 10.12 0.615 8.14 1/4 0.570 7.60 0.475 6.10 l/5 0.467 6.09 0.390 4.88 1/6 0.400 5.06 0.339 4.07 1/8 0.323 3.80 0.272 3.05 1/9 0.300 3.38 0.235 2.71 1/10 0.270 3.04 0.209 2.44 1/20 0.155 1.52 0.144 1.22 Oxygen Concentration in Liquid (cc(STP)/100 ml ) 61 30 keys : --”- Cu(NO3)2 3H20 -——x——- CuSO4 5H20 20 x 10 . 5 1 x ! i , / I ! 1 i l 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Absorbance Fig. 3.8 Calibration Curve 62 turnings. By using the Spring Torque Tester,* the bottle is capped at the defined torque. After capping the bottle the sample gas and sample liquid are drawn out from the bottle by syringes, and these samples are measured for oxygen content by the gas Chromatograph and their absorbance by the colorimeter. These bottles are stored in a constant temperature room, and at certain time interval the sample gas and sample liquid are measured. Oxygen Permeability Of Bottles Let us consider that the bottle is empty and the oxygen partial pressure inside the bottle, p2, is lower than that outside, p0, i.e. oxygen permeates into the bottle from outside. Then,eq(1.ll) gives us the rate form as, dpz A dt_ = PVO VL (p0 - p2) (3'15) and the integral form as, p - (p). 1n (p: _ p: 1) = on %0 §% t (3.16) where, PVO = volumetric permeability constant Of oxygen (cc(STP) mm/sq.cm hr. atm) * Owen Illinois Glass Company, Serial NO. 25—461. 63 A = surface area Of the bottle (sq.cm) V = volume of the bottle (cu.cm) L = thickness Of the bottle (mm) T = room temperature (deg. K) To = standard temperature (273 deg. K) p0 = oxygen partial pressure outside the bottle (atm) p2 = oxygen partial pressure inside the bottle (atm) t = time (hr) Practically, A and L are hard to be measured and as shown in Fig. 3.1, the thickness Of the bottle is not uniform. Therefore, it is useful to express on' A and L together, as the permeability of the bottle, -- _ .4. Permeability Of Bottle — Pv0 L (3.17) From the experimental data Of p2 and t, PVO % can be cal- culated as follows: (p0 ' (92)1 T (t2 ‘ t1) p _ (p2)2)/(fO-——:V——— ) (3.18) P VO = 1n tflv 0 where, (p2)1 and (p2)2 correspond to the p2-values at time t1 and t2, respectively. In this case, the partial pressure of oxygen is expressed by the volume percent Of oxygen. The next step is to consider that the bottle is filled with liquid Of volume Vl (ml) and the head space volume is V2 (cc). From eq(l.4), the oxygen permeation flux is given as follows: _ A _ Q — f on (p0 p2) (3.19) 64 Integrate eq(3.l9) between t1 and t2, then the total amount of oxygen permeated during t1 and t2 is given by, t t 2 A 2 V = Q dt = — ( p - p ) dt 02 .1; L voJ{t O 2 1 1 =9 5mm (t -t> (320) v0 L 1m 2 1 ° where, (Ap)lm = logarithmic mean of (pO-(p2)1) amd(pO-(p2)2) _ (p2)2 - (p2)1 _ (3.21) 190 - (p2)l 1n( ) pO (p2)2 Then, the oxygen permeability Of the bottle can be Obtained from eq(3.20) and eq(3.21), Vo P %,= 2 (3.22) (Ap)1m (tz-tl) The total amount Of oxygen in the bottle is the sum Of oxygen in gas phase and in liquid phase. Oxygen in gas phase is Obtained from gas Chromatograph data as (02)% and oxygen in liquid is obtained from absorbance data of the colorimeter by eq(3.l4) as (C) cc(STP)/100 ml solution. Therefore, the total amount Of oxygen in the bottle, T0 cc(STP) is expressed as follows: T0 = v (02) TO (C) 2 166‘ T’ + V1"fit (3.23) During t and t , V is given by, l 2 O2 65 V02 = (T0)2 - (TO)l (3.24) SO, we can calculate the permeability of the bottle as follows: (Ap)1m (t2-t1) A _ P f — (3.25) VO In Appendix 3.2, two cases are shown: local Pv0 %-and overall PVO %-with considerable fluctuation. . A . Concerning the overall PVO f' if ((TO)2 - (T0)l)/(Ap)lm and tZ-tl are plotted on (Y,X) coordinate, the data should form a straight line which is, Y = a X (3.26) where, (T0)2 - (T0)l Y = (AP)1m (cc(STP)/atm) X = t2—tl (hr) tfl> a = P vo (cc(STP)/atm hr) If the least square procedure is applied to determine a in eq(3.26), we put A as follows: A =z(ax - Y)2 = 2(ax)2 - ZZaXY + 2Y2 In order to make A minimum, differentiate A with a, dA 2 a; 2aZX 22(XY) Put, 66 then, we get, :(xxr) Z X a = (3.27) In this case, we get the correlation coefficient as follows: r = nX(XY) - ZXZY (3.28) /(nZX2-(ZX)2)(nZY2-(ZY)2 Results and Discussions 31 bottles Of 1 lit. Manpak were tested; some were filled with ammonia-ammonium chloride sOlution and metallic copper, and others were empty. The specifications Of the bottles are shown in Table 3.3, and the original data and calculated permeabilities of bottles (overall and local values) are listed in detail in Appendix 3.2. Table 3.4 summarizes the average permeabilities of bottles and their correlation coefficient given by eq(3.37) and eq(3.38). In Appendix 3.2 , Henry constants Of solution in the bottles are tabulated. As shown by the permeability Of the bottles (total) in Appendix 3.2, the initial data are somewhat different from the other, because the equilibrium state had not been established at that time, i.e. initial Henry constants deviated unusually from the others. Therefore, a better result is Obtained if the first data are chancelled, shown in Appendix 3.2. High correlation coefficients were Obtained except for bottle rmn 5, 18, 19 and 20, and the Table 3.3 Specifications of Bottles. 67 Bottle NO. Capping Torque Liq. Vol Head Space Vol. Remarks \DCDNO‘U‘IAWNH 994.0cc 994.0 996.0 996.0 496.0 994.0 994.0 996.0 996.0 994.0 994.0 994.0 996.0 996.0 996.0 494.0 494.0 496.0 000000000 000000000 65.8 cc 66.0 61.1 49.2 491.5 67.0 66.6 54.4 47.6 54.8 56.8 59.3 46.7 56.0 57.1 564.5 557.4 550.5 552.4 551.6 565.6 556.7 1050.0 1050.0 1050.0 1050.0 1050.0 1050.0 1050.0 1050.0 1050.0 Turnings Turnings Powder Powder Turnings Turnings Turnings Powder Powder Turnings Turnings Turnings Powder Powder Powder Turnings Turnings Turnings Turnings Turnings Turnings Turnings NO NO NO NO NO NO NO NO NO copper copper copper copper copper copper copper .copper copper 68 Table 3.4 Average Permeability and Correlation Coefficient Bottle NO. Permeability Of Bottle Correlation Coefficient (cc(STP)/hr. atm.) (-) 1 0.084 0.922 2 0.104 0.956 3 0.026 0.745 4 0.019 0.455 5 0.018 -0.088 6 0.064 0.976 7 0.062 0.973 8 0.032 0.706 9 0.016 0.713 10 0.074 0.966 11 0.090 0.982 12 0.070 0.989 13 0.023 0.677 14 0.028 0.793 15 0.033 0.625 16 0.030 0.910 17 0.060 0.854 18 0.034 -0.345 19 0.052 -0.501 20 0.045 0.282 21 0.051 0.839 22 0.042 0.886 23 0.125 0.984 24 0.097 0.916 25 0.064 0.859 26 0.104 0.918 27 0.078 0.908 28 0.056 0.929 29 0.050 0.859 30 0.100 0.941 31 0.078 0.908 69 bottles filled with copper powder give comparatively lower correlation coefficient than those filled with copper turnings. It could be considered that the c0pper powder settled on the bottom Of bottles and made poor contact with the solution and oxygen, or the mixing of the solution made powder become suspended and made it to hard to draw sample liquid without copper powder. Therefore, copper turnings are best suited for this test. Conclusion From Table 3.3 and 3.4, it can be seen that the permeability of the bottles of this type ranges from 0.01 to 0.15 cc(STP)/(hr. atm) and the capping torque and the head space volume have no significant effect on the per- meability Of the bottles. CHAPTER IV DOUBLE WALL MODEL IN OXYGEN PERMEATION Introduction In actual packaging, the double wall packaging design has become popular, mainly because it is desirable to keep the content as fresh as possible before it is consumed, especially in the case of a hygroscopic or oxygen sensitive content, i.e. a small package (inside package) prevents the content from being degraded by moisture or oxygen in air, even though a larger package (outside package) is Open. The Other advantage Of the double wall package is that in the case Of oxygen permeation, the increase in oxygen partial pressure is slower than that Of a single wall package until a certain time, providing that the total thickness or total amount Of the double wall package are equal to those Of the single wall package. This can also be said Of water vapor transmission from outside the package and vise versa. Gyeszli (25) wrote in his thesis that this time limit tc is dependent on properties Of the packages (materials, surface area, thickness and the space between double walls) and that tc is directly proportional 70 71 to the difference Of the external and internal partial pressure at time t = 0. A more quantitative treatment Of these properties is interesting. It is quite important tO compare the analytical solution and the numerical solution Of the double wall model, because if the model can be solved as a linear problem, it is not necessary to consider about the numerical solution, but in a practical packaging problem it is quite important to give an account Of the model in a nonlinear problem, in this case the numerical method becomes the only way of solving the problem. Model Preparation Let us consider the single and double wall package models shown in Fig. 4.1. pl, p2 and p0 show the oxygen partial pressure Of inside the inner package, between the walls and outside the package, respectively. A and A 1 2 are the surface areas Of the package, and L1 and L2 are the thicknesses, and V1 and V2 are the volumes, where the suffix 1 is to the inner package and suffix 2 is to the outer package. PVO is the volumetric oxygen permeability constant of the film and T and T0 are the room and the standard temperatures. No—suffix in p, A, L and V means that of the single wall package. For the single package, the oxygen partial pressure change is given from eq(l.11), 92: 11.2 _ t on VL TO (p0 p) (4'1) 72 For the double wall package, the similar concept leads to, dp A dt2 = PVO V L % (pO-p2)_PVO El——:: (p -p ) (4 2) 2 2 O V L T 2 l ’ 2 l O dp A l 1 T ——— = P — (p - P ) (4.3) i.e. providing PVO and T are constant, eq(4.2) and eq(4.3) are linear systems with constant coefficients. And the initial condition is, at t = 0, p1 = plo (4.4) p2 = 920 Use the constants kl’ k2 and k3 as follows: A l k = P 2 3 1 VO V2L2 TO A l T k =1. _ L (4.5) 2 VO V2Ll TO k = 9 A1 3 3 VO VlLl TO J then, the system becomes, dp2 EE’ — k1 (p0 - p2) - k2 (p2 - pl) (4-6) dp1 EE‘ ‘ k3 (p2 ‘ pl) (4'7) This system can be solved by the following two procedures, 1. Operators 2. Laplace Transform 73 maeees emmxeem Has; mangen use mameem H.e .mam mmmxosm Ham; mansom omsxosm Ham; mamsflm CL N Fla H e >» O {1. in < e >. 74 Derivation Of System Using the Operator, D = g% eq (4.6) and eq(4-7) become, (D + k3) pl - k3p2 = 0 (4.8) -k2p1 + (D + k1 + k2) p2 = klpO (4.9) Eliminate p2 from above equations, thus we get, (D2+(kl+k2+k3)D+klk2)pl = klk3p0 (4.10) eq(4.10) can be solved as follows, i.e. a general solution Of the homogeneous differential equation Of eq(4.10) is, p1 = Clexp(a t) +C exp(b t) (4.11) 2 where Cl and C2 are integral constants, and define a and b are the roots Of the following equation, 2 D + (k1 + k + k3) D + k k = 0 (4.12) 2 1 2 (D - a)(D - b) = 0 a and b are negative real differnt numbers, since the determinant Of eq(4.12) is, Det 2 (k1 + k + k3) - 4 k k 2 1 3 2 2 3) + k2 + 2(klk (kl — k 2 + k2k3) > 0 for kl' k2 and k3 are pOSltlve. The particular solution Of eq(4.10) is given, 75 _ k1k3po p1 (D-a)(D-b) 2 -l _ l a+b Q_ '— a b (1 + ab D + ab) k1k3p0 ' (4'13) Take taylor's expansion and neglect the terms D, D2, ' ', then we get the particular solution as follows, ._ 1 p1 ‘ ab k1k3po Since, ab = kl 3, then, P1 = P0 (4.14) Therefore, we Obtain the solution of the original differen- tial equation (4.10) as follows: p1 = Cl exp(at) + C2 exp(bt) + pO (4.15) It is reasonable that as t goes to infinity, pl becomes pO since exp(at) and exp(bt) gO to zero. Next step is to determine integral constants, C l and C2. We have the initial condition that at t = 0, p1 = p10 1 dpl > (4.16) (dt ) = k3(p20 - plo) Therefore, ‘ c1 + c2 + p0 = plo (4.17) a C1 + b C2 = k3(p20 - p10) (4.18) From eq(4.17) and eq(4.18) we get, = b(Pio'Po) ’ k3(on’Pio) (4.19) 1 (b-a) C 76 a(p - p ) - k (p - p ) C2 = 10 O 3 20 lO (4.20) (a - b) Plug these constants into eq(4.15), thus, p = b(plo-po) ' k3(on'Plo) exp(at) 1 (b - a) a(Pio‘po) ' k3(on'Pio) + (a _ b) exp(bt) + pO (4.21) Plug eq(4.21) into eq(4.7), we get, 1 dpl P2 - k3 (EE’) + P1 (4°22) or, _ a 9 p2 — Cl (1+E3)exp(at) + C2(1+k3)exp(bt) + p0 (4.23) As shown in Appendix (4.1), if Laplace Transform is applied to solve eq(4.6) and eq(4.7), we get, plo(a+kl+k2) + p20k3 + bp p1 = O exp(at) (a - b) plo(b+kl+k2) + p20k3 + ap ° exp(bt) + pO (4.24) (b - a) p (a+k ) + p k + p (b+k ) p2 _ 20 3 lo 2 o 1 exp(at) (a - b) p20(b+k3) + plok2 + po(a+kl) exp(bt)+pO (4.25) (b - a) Of course, eq(4.21) is equivalent to eq(4.24), since a + k1 + k2 = - b - k3 and also eq(4.23) is equivalent to eq(4.25). 77 To compare the change of oxygen partial pressure in the single wall package and that of the double wall package, holding that the experimental constants are same and the sum of wall thickness or the total amount Of packaging materials of the double wall package is equal to that of a single wall package, let us consider the analytical solu- tion of eq(4.l). For the single wall package, this has been solved by Gyeszli (25), i.e. _2P_= _LT Po'p on v L To dt (4.26) If the experimental constants do not change, eq(4.26) can be integrated between pi and p, 0 and t as follows; (Po - pi) ln (p0 _ p ) = k4 t (4.27) where, pi = initial value Of p — £L.I and k4 ‘ on VL T O 1.6. p = pO - (pO-pi) eXp(-k4t) (4.28) Example (I) Al < A2, L = Ll + L2 Let us compare the changes of p1 and p with time given by eq(4.21) and eq(4.28). Take the experimental constants as follows for example, 78 1. PVC = 42 cc mil/100 sq.inch day atm A1 = 500 sq.cm A2 = 2,000 sq.cm A = 500 sq.cm Vl = 500 cu.cm V2 = 1,000 cu.cm V = 500 cu.cm ? (4.29) L1 = 0.1 mm L2 = 0.1 mm T = 273 + 30 = 303 deg.K p0 = 0.21 atm plo = 0.01 atm p20 = 0.01 atm pi = 0.01 atm L = 0.1 + 0.1 = 0.2 mm J Table 4.1 shows the values of p1, p2 and p calculated by eq(4.21), eq(4.22) and eq(4.28), respectively, and Fig. 4.2 shows these changes, i.e. p2 is always higher than p1, and pl, p2 and p go to p0 as time t goes to infinity. Until t = 1,670 hr., p is higher than p1 and this moment when pl becomes higher than p is called the "critical point". (Appendix 4.2) Table 4.2 shows the case that the initial partial pressure of oxygen between walls is atmospheric pressure, i.e. without replacing the space with nitrogen. Obviously, in this case, pl is always higher than p. (Appendix 4.3) 79 Table 4.1 Pressure Changes in Example (I), p10=0.01, p20=0.01. Time (hr.) pl (atm.) p2 (atm.) p (atm.) 0 0.010 0.010 0.010 100 0.011 0.038 0.018 200 0.014 0.061 0.025 300 0.018 0.080 0.032 400 0.023 0.096 0.038 500 0.029 0.110 0.045 600 0.036 0.121 0.051 700 0.042 0.130 0.057 800 0.049 0.139 0.063 900 0.056 0.145 0.068 1000 0.063 0.151 0.074 1100 0.069 0.157 0.079 1200 0.076 0.161 0.084 1300 0.082 0.165 0.088 1400 0.089 0.169 0.093 1500 0.095 0.172 0.097 1600 0.100 0.174 0.102 1670 0.10432 0.176 0.10438 1680 0.10488 0.176 0.10479 80. —X— p ‘/ ‘/ /‘/ 0.19 ./’r Pressure (atm.) I / / ' W” o 1 ‘7 1 155’; / ’/7 /././ / */'/'/ a/ ' ./ / : ./ ./ 0.05 j/ ‘:// f/// /)/ ,/ . /* ./ . / ./ ,/ / ./ 41:".a’"’/’ 500 1000 1500 1670 Time (hr.) LaL +1) Fig. 4.2 Example (I) A1< A2 9 1 2 81 Table 4.2 Pressure Changes in Example (I), plo = 0.01, p20 = 0.21. Time (hr.) pl (atm.) p2 (atm.) p (atm.) 0 0.010 0.210 0.010 500 0.069 0.190 0.045 1000 0.107 0.188 0.074 1500 0.133 0.191 0.097 2000 0.152 0.195 0.117 2500 0.166 0.198 0.133 3000 0.177 0.201 0.146 82 Let us consider the effect Of the combination of L1 and L2 on the critical point. Table 4.3 shows the critical points at the several combinations of thicknesses and the combination, L1 = 1.5 mm and L2 = 0.5 mm, gives the maximum critical point value Of 1,808 hr. (Appendix 4.4) Example (II) A 2 + —— L 1 Al 2 Again, consider eq(4.28), providing that the Al < A2, L = L packaging material costs Of the double wall package and the single wall package are equal, i.e. AlLl + A2L2 = A L (4.30) where, A = A1 Then, the thickness of the single wall package is given by, A2 L = L + -—»L (4.31) 1 Al 2 or, the outer thickness of the double wall package is given by, A1 L2 = i: (L - Ll) (4.32) Put, — AI k5 " on VL TO where L is given by eq(4.3l). Then, we can calculate the pressure in the single wall package whose total amount Of the packaging material is 83 Table 4.3 Thickness and Critical Point in Example (I). Ll (mm) L2 (mm) Critical Point (hr.) 0.01 0.19 313 0.02 0.18 577 0.03 0.17 802 0.04 0.16 994 0.05 0.15 1159 0.06 0.14 1300 0.07 0.13 1421 0.08 0.12 1522 0.09 0.11 1606 0.10 0.10 1674 0.11 0.09 1728 0.12 0.08 1767 0.13 0.07 1793 0.14 0.06 1806 0.15 0.05 1808 0.16 0.04 1802 0.17 0.03 1789 0.18 0.02 1774 0.19 0.01 1759 84 equal to the double wall package by the following equation, P = P0 - (pO - pi) exP(-k5 t) (4.33) In this case, A < A then L + AZ/Al L 1 2' 1 >L+L 2 l 2’ therefore, the pressure calculated by eq(4.28) is always higher than that by eq(4.33), i.e. the critical point becomes smaller compared with that Of eq(4.21) and eq(4.28). Table 4.4 shows these values calculated eq(4.29) holds, i.e. the critical value for L = Ll + L2 is 1,670 hr and forL==Ll + A2/Al L2 is 340 hr. (Appendix 4.5) Let us consider the effect Of the combination of L1 and L2 on the critical point. In this case, it is convenient to compare p and p1 providing that L is fixed to 0.2 mm and L2 is given by eq(4.32). Table 4.5 shows the results of the calculation. (Appendix 4.6) It can be said that in this case the thicker the inside package becomes, the longer the critical point becomes, and in the region Of L > L/2 less influence on 1 the critical point is Observed. Example (III) 2 + —— L 1 Al 2 A < A 1 L = L 2’ Let us consider the effects Of the dimension Of the double wall package on the critical point. Take a cylin— drical package for example, with dimensions Of, 85 Table 4.4 Pressure Changes in Example (II). Time (hr.) pl (atm.) p2 (atm.) p (atm.) 0 0.010 0.010 0.010 50 0.010 0.025 0.012 100 0.011 0.038 0.013 150 0.012 0.050 0.015 200 0.014 0.061 0.016 250 0.016 0.071 0.018 300 0.018 0.080 0.019 340 0.020 0.087 0.020 350 0.021 0.088 0.020 86 Table 4.5 Thickness and critical Point in Example (II). Ll (mm) L2 (mm) Critical Point (hr.) 0.01 0.047 37 0.02 0.045 72 0.03 0.042 106 0.04 0.040 137 0.05 0.037 176 0.06 0.035 195 0.07 0.032 221 0.08 0.030 244 0.09 0.028 266 0.10 0.025 285 0.11 0.022 302 0.12 0.020 317 0.13 0.018 329 0.14 0.015 338 0.15 0.013 344 0.16 0.010 348 0.17 0.008 349 0.18 0.005 349 0.19 0.003 349 87 Diameter D1 = 10 cm Height HI = 15 cm Wall thickness Ll mm and suppose that the outer package is symmetrical to the inner package and its dimensions are, D2 — D1 (1 + K/10) cm H2 = H1 (1 + K/lO) cm K = l, 2, 3, - — - L2 mm For the single wall package, the dimensions are as follows: D = 10 cm H = 15 cm A2 L = L1 + KI L2 = 0.2 mm In order to late, Al V1 A2 V2 And, L2 apply eq(4.21), eq(4.22) and eq(4.28), we calcu- 1 3.14 D 1 (H1 + Dl/Z) 3.14 D 2 1 Hl/4 3.14 132 (H2 + 92/2) ) (4.34) A (1 + K/10)2 l 2 3.14 D2 V 2 l V 1 ((1 + K/10)3 - 1) (4.35) (L - L1) W 88 For K = l to K = 10, the critical point and the pressures at that point are calculated and shown from Table 4.6 to Table 4.15. > ' (Appendix 4.7) Similar to Example (II), the thicker inside package makes the critical point longer for the relatively small outside package, as shown in Table 4.6. But as the outer package becomes larger, there appears the longest critical point (maximum point) at some combination Of the thickness, for example, in Table 4.11 the combination Of L = 0.14 mm 1 and L2 = 0.023 mm gives the longest critical point, 2,900 hr. Generally, a larger outer package makes the critical point short, and these relationship are shown in Fig. 4.3 schematically. Example (IV) Al > A2, L = Ll + L2 Consider another case: Al > A2, as in Fig. 4.4. 12 cylindrical plastic bottles whose this can happen diameters are 3 cm and whose heights are 20 cm are packed in a plastic film case in 3 x 4 x l, which is sealed hermetically. In this case, the mass balance Of oxygen leads to, fl£§=P A23(p-p)-P A3-T—(p-p) dt vo V2L2 TO O 2 VO V2L1 TO 2 l (4.35) dp1_P Al dt _ VO ———— (p r p ) VlLl 2 l 89 Table 4.6 Thickness and Critical Point in Example (III). V 390 cu.cm 1178 cu.cm V 1 2 A1 = 628 sq.cm A2 = 760 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.149 1100 0.04 0.132 1900 0.06 0.116 2400 0.08 0.099 2700 0.10 0.083 3000 0.12 0.066 3200 0.14 0.050 3400 0.16 0.033 3400 0.18 0.017 3500 Table 4.7 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 858 cu.cm A1 = 628 sq.cm A2 = 905 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.125 1100 0.04 0.111 1800 0.06 0.097 2300 0.08 0.083 2700 0.10 0.069 3000 0.12 0.056 3200 0.14 0.042 3300 0.16 0.028 3400 0.18 0.014 3400 90 Table 4.8 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 1410 cu.cm A1 = 628 sq.cm A2 = 1062 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.107 1000 0.04 0.095 1700 0.06 0.083 2300 0.08 0.071 2700 0.10 0.059 2900 0.12 0.047 3100 0.14 0.036 3200 0.16 0.024 3200 0.18 0.012 3200 Table 4.9 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 2055 cu.cm A1 = 628 sq.cm A2 = 1231 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.092 1000 0.04 0.082 1700 0.06 0.071 2200 0.08 0.061 2600 0.10 0.051 2900 0.12 0.041 3000 0.14 0.031 3100 0.16 0.020 3100 0.18 0.010 3100 91 Table 4.10 Thickness and Critical Point in Example (III). V1 = 1178 cu.cm V2 = 2798 cu.cm A1 = 628 sq.cm A2 = 1414 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.080 900 0.04 0.071 1600 0.06 0.062 2100 0.08 0.053 2500 0.10 0.044 2700 0.12 0.036 2900 0.14 0.027 3000 0.16 0.018 3000 0.18 0.009 3000 Table 4.11 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 3647 cu.cm A1 = 628 sq.cm A2 = 1608 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.070 900 0.04 0.063 1500 0.06 0.055 2000 0.08 0.047 2400 0.10 0.039 2600 0.12 0.031 2800 0.14 0.023 2900 0.16 0.016 2900 0.18 0.008 2800 92 Table 4.12 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 4610 cu.cm A1 = 628 sq.cm A2 = 1816 sq.cm L1 (mm) ‘ L2 (mm) Critical Point (hr.) 0.02 0.062 800 0.04 0.055 1400 0.06 0.048 1900 0.08 0.042 2200 0.10 0.035 2500 0.12 0.028 2700 0.14 0.021 2800 0.16 0.014 2800 0.18 0.007 2700 Table 4.13 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 5692 cu.cm A1 = 628 sq.cm A2 = 2036 sq.cm Ll (mm) L2 (mm) Critical Point (hr.) 0.02 0.056 800 0.04 0.049 1300 0.06 0.043 1800 0.08 0.038 2100 0.10 0.031 2400 0.12 0.025 2600 0.14 0.019 2600 0.16 0.012 2600 0.18 0.006 2600 93 Table 4.14 Thickness and Critical Point in Example (III). Vl = 1178 cu.cm V2 = 6902 cu.cm A1 = 628 sq.cm A2 = 2268 sq.cm L (mm) L2 (mm) Critical Point (hr.) 0.02 0.050 700 0.04 0.044 1300 0.06 0.039 1700 0.08 0.033 2000 0.10 0.028 2300 0.12 0.022 2400 0.14 0.017 2500 0.16 0.011 2500 0.18 0.006 2500 Table 4.15 Thickness and Critical Point in Example (III). V1 = 1178 cu.cm V2 = 8246 cu.cm A1 = 628 sq.cm A2 = 2513 sq.cm L1 (mm) L2 (mm) Critical Point (hr.) 0.02 0.045 700 0.04 0.040 1200 0.06 0.035 1600 0.08 0.030 1900 0.10 0.025 2200 0.12 0.020 2300 0.14 0.015 2400 0.16 0.010 2400 0.18 0.005 2400 94 venom Heespaeo no seemeeeHn me eeemem m.e .mee Assv HA omdxomm possH mo mmosxoflga ABo.amv Nd omwxosm popdo Ho mmmm\oowMHSm A.pev venom Hmeepapo 95 Dimension : mm Scale : non Fig. 4.4 Double Wall Package, Example (IV) 96 where, '~ A1 = 3.14 D H + 3.14 02/2 2 v1 = 3.14 D H/4 A2 = 2 (4 D H + 3 D H + 12 D2) , (4.36) v = 12 D2 H - 12 v 2 1 A3 = 12 A1 The solutions Of eq(4.35) are similar tO eq(4.21) and eq (4.22). Since the packaging material costs of the double wall package and the single wall package are equal, we get, 12 AlLl + A2L2 = 12 A L (4.37) where, A = A1 A2 So, L = Ll + A_ L2 (4.38) 1 where, A2 = 1,056 sq.cm 12 A1 = 1,210 sq.cm i.e. A2 L1 + Ii‘X' L2 < L1 + L2 1 In order to compare with the symmetrical double wall pack- age, put, L = L + L l (4.39) 2 And, eq(4.38) and eq(4.39) are used in eq(4.28), i.e. 97 _ AT k5 _ on VL'T O _ AT k6_onfiT 0 Then, P1 = P0 - (p0 - pi) exp(-k5 t) (4.40) P1 = P0 - (PO - pi) eXp(-k6 t) (4.41) Table 4.16 shows the comparison of the double wall package and the single wall package whose thicknesses are given by eq(4.38) and (4.39), i.e. the case of eq(4.39) gives the lower oxygen partial pressure than the case Of eq(4.38). I (Appendix 4.8) Comparison Of Analytical and Numerical Solutions It is very important to compare the pressures of the double wall package and the single wall package, obtained by the analytical method and the numerical method solving eq(4.2) and eq(4.3L because the solutions in previous parts are Obtained under constant temperature, i.e. linear equations, but practically it happens that the temperature varies with time, i.e. non-linear equations, in such cases it is impossible to solve the system by the analytical method. Therefore, it is very meaningful to check the amount Of error that occurs with the numerical method. Let us consider two numerical methods, 98 Table 4.16 Pressure Changes in Example (IV). él‘hime) Pl (atm) p2 (atm) p (atm) p (atm) r. L—L1+L2 L-L1+L2A2/12Al 0 0.010 0.010 0.010 0.010 100 0.012 0.037 0.021 0.025 200 0.015 0.053 0.031 0.038 300 0.020 0.063 0.040 0.051 400 0.024 0.070 0.049 0.063 500 0.029 0.075 0.058 0.074 600 0.034 0.080 0.066 0.084 700 0.039 0.084 0.074 0.093 800 0.044 0.088 0.081 0.101 900 0.049 0.092 0.088 0.109 1000 0.054 0.095 0.094 .0.117 1100 0.058 0.098 0.101 0.124 1200 0.062 0.102 0.106 0.130 1300 0.067 0.105 0.112 0.136 1400 0.071 0.108 0.117 0.141 1500 0.075 0.111 0.122 0.146 1600 0.079 0.114 0.127 0.151 1700 0.082 0.116 0.131 0.155 1800 0.086 0.119 0.135 0.159 1900 0.090 0.122 0.139 0.163 2000 0.093 0.124 0.143 0.167 2100 0.096 0.127 0.147 0.170 2200 0.100 0.129 0.150 0.173 2300 0.103 0.131 0.153 0.176 2400 0.106 0.134 0.156 0.178 2500 0.109 0.136 0.159 0.180 2600 0.112 0.138 0.162 0.183 2700 0.115 0.140 0.164 0.185 2800 0.117 0.142 0.167 0.186 2900 0.120 0.144 0.169 0.188 3000 0.123 0.146 0.171 0.190 99 l. Euler's method 2. Runge-Kutta method (4th order) Let us consider eq(4.6) and eq(4.7). dpz. 3E"= kl(po - p2) - k2(p2 - Pl) (4-6) dp1 HE‘ = k3‘92 ‘ Pi) (4'7) Putting p2(t) and pl(t) as solutions Of eq(4.6) and eq(4.7), and apply the Taylor's expansion Of p2(t) and pl(t) with respect to time, we get, 2 . p2(ti+l) = P2(ti) + hlfl(tilp2(ti)) + 2! h13fi'(ti,p2(ti)) + + - - - (4.42) 3! where, ti+1 = t1 + h1 h1 = time increment and, 2 l h2 f2 (ti,pl(ti)) pl(ti+l) _ Pl(ti) + h2f2(tilpl(ti)) + 2' h 3f "(t (t )) . .,p . + l 2 1 1 1 + — - (4.43) 3! where, t - t + h 1+1 7 i 2 100 h2 = time increment dp fl(ti:P2(ti)) = 2 (at—)t=t. (4°44) 1 dpl f2(ti’p1(ti)) = (fig—)t=ti (4.45) Euler method, Take the first two terms Of eq(4.42) and eq(4.43) and neglect higher terms, and take hl and h2 as unit valve, for instance, h1 = h2 = 1 hr. Then we Obtain the following approximations, p,(ti+1) = p2(ti) + fl(ti,p2(ti)) (4.46) P1(ti+1) = p1(ti) + f2(ti.pl(ti)) (4.47) 121 Apply the initial conditions of eq(4.46) and eq(4.47) as follows: P2(tl) = P20 Pl(tl) = P10 The iteration procedure gives us the values Of p2 and p1 with time. Runge-Kutta method (4th order). In order to solve eq(4.6) and eq(4.7), the following approximations are Obtained by this method, h _ l — p2(ti) + 7r(ml+2m +2m +m4) (4.48) 92(ti+1) 2 3 101 where, W i Z l ti+1 = ti + hl’ h1 = time increment m=f(t+ih p(t.)+lhm) 2 l i 2 1' 2 1 2 1 l ( m=f(t+l'-h p(t)+lhm) (449) 3 1 i 2 1’ 2 i 2 l 2 ° m=£(t+lh (t)+lhm) 4 li 21'p2i 213 J function fl is given by eq(4.44). and, h2 pl(ti+l) = p1(ti) + 7T(nl+2n2+2n3+n4) (4.50) where, i Z 1 1 ti+l = ti + h2, h2 = time increment nl = f2(ti'pl(ti)) n=f(t+lh p(t')+lhn) (451) 2 2i 22' 1:1 221 f ' n—f(t+ih (t)+—l-h) 3 ‘ 2 i 2 2' pl i 2 2H2 n=f(t+lh p(t)+-]-'-hn) 4 2 i 2 2' 1 i 2 2 3 . function f2 is given by eq(4.45) Taking hl and h2 as unit values, for instance, h1 = h2 = 1 hr. 102 and apply the initial conditions into eq(4.48) and eq(4.50) as follows: 92(t1) = on Pl(tl) = P10 The iteration procedure gives us the values Of p2 and p1 with time. Table 4.17 shows the values Of p2 and pl, calculated by the analytical method, Euler method and Runge-Kutta method. (Appendix 4.9) As shown in Table 4.17, the maximum error in the pl value does not exceed 0.02% bvauler method after 3,000 iteration procedures, and 0.004% by Runge-Kutta method. Therefore, the numerical approximation can be said to be accurate enough for the nonlinear system Of double wall packaging problems. Conclusion In order to compare the oxygen permeation properties Of a double wall package and a single wall package, the system Of differential equations was solved simultaneously, and the critical points Of four special cases were calculated by using CDC 6500 digital computer. The effects Of thick- ness combination and dimensions of the double wall package on the critical point were investigated. On the other hand, the numerical approach to solving this system was the use Of two methods:7 Euler's method and Runge-Kutta method. It was proved that there exist no 103 Table 4.17 Analytical Solution and Numerical Solution Time (hr.) pl (atm.) pl (atm.) pl (atm.) Analytical Euler's Runge-Kutta 0 0.010000 0.010000 0.010000 500 0.029180 0.029169 0.029179 1000 0.062672 0.062677 0.062673 1500 0.094646 0.094663 0.094649 2000 0.121189 0.121212 0.121193 2500 0.142130 0.142154 0.142134 3000 0.158305 0.158328 0.158309 104 significant errors in applying the numerical methods to this system. Therefore, it seems more complex non-linear permeation problems may be solved using these methods. BIBLIOGRAPHY 10. 11. 12. BIBLIOGRAPHY Ahlen, A. T. "Study Of Diffusion Of Sorbed Water Vapor Through Paper and Regenerated CelluloseFilms," Tappi, 53 (6) June (1970) 1320-1326- Anon. "Using Nitrogen in Packaging one Pound Pouches," Package Engineering, 1 (12)(1956) 14-16. Anon. "Applied Fluidics Announces Simplified Leak Detector," Aerosol Age, 15 (8) Aug. (1970) 78. Aoki, S. et.al. "On the Quality Of Soy Sauce During Storage in a Bottle," Chomi Kagaku, 11 (6) Nov. (1970) 201-214. Briston, J. H. "Plastics in Packaging Part II & III," Flavor Ind., 1 (2) (1970) 105-107 & (11) (1970) 201-214. Brown, L. R. "The Utilisation Of Gas Permeability Data," Evaluation.of Package Performance, Printing, Packaging & Allied Trades Research Association Randalls Road, Leatherhead, Surrey (1963). Brown, L. R. "Permeability and Shelf Life," Modern Packaging, Oct. (1964) 184. Calvano, N. J. et.a1. "0 Permeation of Plastic Con— tainers," Mod. Packaging, Nov. (1968) 143. Caskey, T. L. "Dynamic Gas Chromatographic Method for Measuring Gas Permeability Of Film," Modern Plastics, 4§_(4) Dec. (1967) 153. Charlton, F. S. et.a1. "Foods and Permeability," Mod. Packaging, March (1956) 227. Daoud, H. N. et.al. "Packaging Of Foods in Laminate and Aluminum-film Combination Pouches IV. Freeze— dried Red Bell Pepper," Food Technology, 21 (3a) (1967) 339-343. Davis, D. W. "Isostatic Method for Determining Gas Permeability Of Sheet Materials," Paper Trade Journal, 123, NO. 9 (1946) 33-40. 105 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 106 / Davis, D. W. "Gas Permeability - an Isostatic Test Method," Mod. Packaging, May (1952) 131. Davis, E. G. et.al. "Heat Processing vs. Permeability," Mod. Packaging, March (1960) 208. Davis, E. G. "Rapid Test for Pinhole in Packaging Film," CSIRO Food Research Quarterl , 29 (2) (1969) 35-39. Davis, E. G. "Protective Properties of Food-Packaging Materials," CSIRO Food Research Quarterly, 31 (3) (1971) 47-52. Davis, E. G. "Protective Properties Of Food Pack- aging Materials," Australian Packaging, Feb. (1972) 25-30. Deterding, J. H. et.al. "Radioactive Tracer Techniques for Measuring the Permeability Of Materials to Water or Hydrocarbons," Paper presented at the corrosion conference 1970. Eichhorn, R. M. "Measurement Of Water Vapor Trans- mission Through Polyethylene Electrical Insula- tion," Polymer Engineering and Science, 10 (1) Jan. (1970) 32-37. Elschnig, G. H. et.al. "PVDC Stops up Film Permea- bility," Paper Film Foil Converter, 43 (9) Sept. (1969) 58—59. Fites, A. L. et.al. "Controlled Drug Release Through Polymeric Films," Journal of Pharmaceutical Sciences, 22.(5) May (1970) 610—613. Fricke, H. L. "Gas Chromatograph Measures Film Permeability," Package Engineering, Dec. (1962) 51-55. Galbraith, A. D. "Estimating Aqueous-Product Life," Mod. Packaging, Sept. (1962) 160-164. Gilbert, S. G. et.al. "Find New Way to Measure Gas Permeability with This Step-by—Step Method You can Check Gas Permeability over a Wider Range Of Humidities and Temperatures than Previous Possible on Two Films at Once,” Package Engineering, 14 (1) Jan. (1969) 66-69. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 107 Gyeszli, I. "Gas and Vapor Permeability Of the Double Wall Compared to Single Wall Plastic Packages," Michigan State University, Master Thesis, 1971. Heiss, R. et.al. "Moisture Content and Shelf—Life," Food Manufacture, May/June (1971) 53-65/37-42. Holes, R. J. et.al. "Testing for Ultraviolet Effect," Mod. Packaging, Oct. (1960). Hu, K. H. et.al. "Gas & Water Vapor Transmission Can Spoil Your Product Inside Foil Laminated Pouches," Food Technology, 22 (9) (1968) 61-64. Ikawa, F. "Food Packaging and Light," Shokuhin Kogyo, Z (1970) 101-109. Jeffs. K. D. "A Chromatographic Method for Measuring the Gas Permeability of Packages," Evaluation of Package Performance, Printing, Packaging & Allied Trades Research Association Randalls Road, Leather- head, Surrey (1963). Kanitz, P. J. F. et.a1. "The Permeation Of Gases Through Modified Polymer Films IV. Gas Permea- bility and Separation Characteristics Of Graft Copolymers Of Polyethylene and Teflon FET Films," Journal of Applied Polymer Science, 22 Jan. (1971) 67-82. Karel, M- et.a1. "Application Of Gas Chromatography tO the Measurement Of Gas Permeability Of Packaging Materials," Food Technology, 21_March (1963) 91-94. Laine, R. et.al. "Permeability Of Polyethylene Film to Organic Vapors," Journal Of Applied Polymer Science, 2§_Feb. (1971) 327-339. Landrock, A. H. et.al. "The Simultaneous Measurement of Oxygen and Carbondioxide Permeabilities of Packaging Materials," Tappi, 2§_(6) (1952) 241-246. Lesse, P. F. "Generalizations Of the Diffusion Equation," Journal Of Polymer Science A-2, 2 April (1971) 755-758. Li, N. N. et.al. "Permeation Through Plastic.Films," A.I.Ch.E.Journal, 2§_(l) Jan. (1969) 73-78. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 108 Lockhart, H. E. "Deisgn and Application of an Isostatic Permeabiliterest Method for the Study of Cellulosic and Plastics Barrier Materials, Packages and Closures," Michigan State University, Ph.D.Thesis, 1965. Loudenslagel, K. D. et.al. "Analog Simulation of Permeation,“ Modern Plastics, March (1969). Loudenslagel, K. D. et.a1. "Whole-Package Transmission Test," Mod. Packaging, 22 (9) Sept. (1970) 78—80, 82. Mack, A. C. "OZ-Penetration into FOOd Tubs," Mod. Packaging, Dec. (1971) 52-54. Major, C. J. "Compact Gas Permeability Apparatus," Mod. Packaging, Jan. (1963) 119. Maneval, M. V. "Gas Permeability Of Vinyl Films," SPE Journal, 22 (11) Nov. (1969) 31-33. Morrow, D. R. et.al. "Losing Fizz in Plastic-Bottled Beverage," Package Engineering, Aug. (1971) 58-62. Muldoon, T. J. et.a1. "The Transmission Of Organic Vapor Through Packaging Material," Food Technology, 2 (11) (1951) 449-452. Nakagawa, T. et.al. "Transport Of Fixed Gases in Radiation-Stabilized Poly(Vinyl Chloride)," J. Applied Polymer Science, 22 Jan. (1971) 231-245. Osburn, J. O. et.al. "Vapor Permeabilities by Dynamic Sorption," J. Applied Polymer Science, 22 March (1971) 739-746. Paine, F. A. "Fundamental Of Packaging," Blackie & Son Limited, London 1962. Pasternak, R. A. et.al. "A Dynamic Approach to Diffusion and Permeation Measurements," J. Polymer Science A-2, 2 (3) March (1970) 467-479. Pasternak, R. A. et.al. "New Instrument for Measuring Transmission Through Plastics," Mod. Packaging, £2 (5) May (1970) 89-94, 98. Paul, D. R. et.a1. "Determination of Penetrant Solubility from Transient Permeation Measurements," J. Polyper Science A—l, Z_Aug. (1969) 2031-2035. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 109 Peterlin, A. "Vapor and Gas Permeability Of Asymmetric Membranes," J. Applied Polymer Science, 15 (12) Dec. (1971) 3127- 3136. PetrOpoulos, J. H. et.a1. "Determination of Solubility Coefficients from Diffusion Time Lags," J. Polymer Science A-2, 2 (8) (1970) 1411-1412. Raphael, H. J. "Packaging: a Scientific Marketing Tool," Michigan State University Book Store, 1969. Sacharow, S. & Griffin, R. C. "Food Packaging," The AVI Publishing Company, Inc. 1970. Schrenk, W. J. et.a1. "Some Physical Properties Of Multilayered Films," Polymer Engineering Science, 2 (6) Nov. (1969) 393-399. Siegel, R. D. et.a1. "Errors in Diffusivity as Deduced from Permeation Experiments Using the Time- -Lag Technique," J. Applied Polymer Science, 14 Dec. (1970) 3145- 3149. Simril, V. L. et.al. "Permeability of Polymeric Films to gases," Mod. Plastics, 21 (11) July (1950) 95, 96' 98’ loo-R20 Smyser, H. D. "Now; New Quick-Time Tests Check Film's Gas Transmission," Package Engineering, l2, (5) May (1970) 71-73. Stahl, W. H. et.al. "A Gas Chromatographic Method for Determining Gases in the Headspace of Cans and Flexible Packages," FOOd Technology, 12 (1) (1960) 14-15. Stannett, V. "Diffusion in Polymers (Crank & Park)," Academic Press. London and New York 1968. Stocker, J. H. "Measurement Of Gas Permeability Of Packaging Films," Evaluation of Package Performance, Printing, Packaging & Allied Trades Research Association Randalls Road, Leatherhead, Surrey 1963. ll Storstrom, H. "Mechanisms for Gas and Water Vapor Permeation in Paper-Polymer Composites," Svensk Pappers Tidning, 12 (8) April (1971) 244-247. Takizawa, A. et.al. "Water Vapor Permeability of Composites Membrane," Kohbunshi Kagaku, 22 (309) Jan. (1971) 24-30. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 110 Taylor, A. A. et.a1. "Measurement of Oxygen Permea- bility," Mod. Packaging, 3; June (1960) 131. Todd, H. R. "Apparatus for Measuring Gas Transmission Through Sheets and Films," Paper Trading Journal, 118 (10) (1944) 32. Vosti, D. C. et.a1. "Analysis of Head Space Gases in Canned Foods," Food Technology, lé (1961) 29. Williams, J. L. et.a1. "The Diffusion of Gases and Water Vapor Through Grafted Polyoxymethylene," J. Applied Polymer Science, l4 Aug. (1970) 1949- 1959. Winrich, K. M. "Dynamic WVTR Testing, an Experience Report of the St. Regis Honeywell Tester," Tappi 54 (8) Aug. (1971) 1302-1304. Winrich, K. M. "Dynamic Measurement of Water Vapor Transfer Through Sheet Materials: Proposed New Suggested Method T523," Tappi, 53 (7) July (1970) 1371-1372. Winrich, K. M. "Gravimetric Determination of Water Vapor Transmission Rate of Sheet Material at High Temperature and Humidity Proposed Revision of T 464-45 as a Suggested Method," Tappi, §4_(5) May/(7) July (1971) 792-794/1169-1171. Winrich, K. M. et.a1. "Water Vapor Transmission Rate of Sheet Material at Normal Temperature," Tappi, £4 (7) July (1971) 1169-1171. Woodgate, B. E. "Leak Rates of Thin Plastic Films," Journal of Physics E: Scientific Instruments, Volume 4 (1971). Yasuda, H. et.al. "Isobaric Measurement of Gas Permeability of Polymers," J. Applied Polymer Science,_l4 (11) Nov. (1970) 2839-2877. Ziegel, K. D. et.al. "Measurement of Hydrogen Isotope Transport in Poly(Vinyl Fluoride) Films by the Permeation-Rate Method," J. Polymer Science A—2, 1 May (1969) 809-819. Guide to Plastics, by the Editors of Modern Plastics Encyclopedia, McGraw-Hill, Inc. 1970. 76. 77. 78. 79. 111 "Standard Methods of Testing for Gas Transmission Rate of Plastic Film and Sheeting," 1972 Annual Book of ASTM Standards Part 27, pp 460-472. "Standard Methods of Testing for Water Vapor Trans- mission of Materials in Sheet Form," 1972 Annual Book of ASTM Standards Part 27, pp 815-822. Badger, W. L. "The Determination of.Oxygen by the C0pper-Ammonia-Ammonium Chloride Reagent," The Journal of Industrial.and Engineering Chemistry, l2 (2) Feb. (1920) 161-164. Cooper, C. M. et.a1. "Performance of Agitated Gas- Liquid Contactors," Industrial and Engineering Chemistry, 16 (1944) 504. APPENDICES APPENDIX 2 . 1 Calculation of eq(2.22) and eq(2.23) 112 . (.UUH Comv CoCCU (ca U€oL Coficm (oncm (.FFQ . (It “cwotauu.r.coun.CIH.thOCu SCH AWoCHuchciucu CC“ Afimoanquvacqon_vficoa_VRv\w*~h\nF*H0NI U$0QHUQ U*QQHGQ Acm.&*¢mom*.ccw*.m ZOHquDUJdU >P~J~1 *.\.C-.*u 1 *o\o(u~o*u m *oCIfiVFGEQCU w acfi.cmumvkqfiaou U 2<.D.~.¢ FZaOQ —C~ cam Ch 06 2xc *\r.c~u.2n xqzlu *\m.csu.*u uz~P*oc1~vkzzaou ucm x\\\*cho~.H.umo C2:.n >m fawn“ no WFZFHJHGXO$¢~IHVFQEHCU "C0 .m.c~uu.k<2acu ccn QZ>Oonc>Qo XCEFnuonF oKCO, FZNUQ <\«&D «\nuc alucuu< F Iflflac nz<.E.FCCQ JJalfiz>ac\~k\.HO>Q\FZ>QVCCJ *.\.CLH.*H 7 *.\.:HHo*n M *oCIHVFqZflCH ( ntflocmu VFQEQCU U A*OZ~ om; WCOIPLK twhfimmuflfl .1 .... Zmic_«...--__wj..vJF~J~ldu...0...fl*o “I“ VF H+Dnfi QC“ (LU CF 0U IQHALVNX W+Hu~ CC“ «(1.1cfl..;tw Cxcvut H-H CF CC AL.UJonZ..ocpc_x Ecumzuzhc . A.5_._T:-.._.:C.F:02~ vuovyt f(CCCOO . .CCJ 0.2-run. APPENDIX 3 . 1 Regression for Calibration Curve ~19! - 1. I ll. VII... ,lyl.f 115 czu .\\h.o~u.*u sz_u~uumoc Foukqswqaoo*.orfick *oCIfivkdzaou QC" a\\.*lll H ms zosuuuaowa aqmz_4 *.\\\.m.umm.*n r *.\om.u~u .sn.<*.cI~+k3 m_m>4qz< zoskoqu sqolslio sou zo_ku*>m|>>m*xzckcosuc\Axm*xmlxxm*achmvu< Axm*xmuxxm*nzck<04uc\1>u*xuu>xu*AZCP*AHCX+>XWH>XU - -. , . . A~c>*a~v>+>>vu>>v a~c>+>Mu>w AH 531— vx+xxmnxxru Aacx+xmuxm Z._u_ OH CC CoCn>XU 1 . 5 . 5 ;.Ii, sC-Cu>}¢.lr Cocu>w CQCHXXU c.0uxu mCdDCU, DO Sim... CFC ..~.xcc.qcfikz~nc .._VXvu.CCH C400 Zocnw FZHOC ZowCHC/NUO ..culx EQNmzmslc AFDOFDCotfizhv . 5 C00 BCDO kfl EQQ oCCJ QZFM 11,6 ammo. 4 an.“ <<.& “Pom Wcom, hCod- cued Foe ifiom KW.“ Vac-m Kn.m mum bc.m oc.m €05 (H sew-C cco.c Ufifioc «Fa-C Cmfi.c OK-C Uhqoc W~Ioc mmyoc bn.c m.c .marro soc .wm<.c FUoC «Foc @— 11V n\\.*ll" mu ZOHmmwflwwfl Um .mowwum.cIHckJm*>ml>>m*azck404uc\axm*xmlxxm*.z.h<04u.Chacmtmua kzwuusuuwou zofiqumncou mo ZCHFmVu< axm*xmlxxm*aZcFm*xml>Xux.ZCFQOAHVHE 70HP*.N.x+>xmu>xm .~c>*x~.>+>>mn>>m A~V>+>mu>m .ch*a~.x+xxmuxxm .nex+xmnxm Z.Hu_ c“ CO Ooou>xm C.Cu>>m c.on>m c.ouxxm o.cuxm mfiaDCw no 220 07¢ «Ego no 23w uC ZOHquDUJqu nz.fiu~.xx~.>.nhcxcv.qcfiszoo qhqc PDDFDC AZ.HuH.An_C>..chv..co~ o..om.x zo_mzw5Ho .PDQFDO.FDOZLvoomyazzqaooau . ra.ccr.0\§\c .‘ll - ,II on . .CCJ OFFU. ...U o COL: o C../:. 118 nmofl. Ug—ao sq.m Gum.c ab.m- : Mmmwc, flCom NDOOC rc.~:.;l©rr.c UCoq CK.C "of the-C dyom5 UwfoC dem Eur-C dcom hmoc .mm.m. m.o Com Vance mc.m goo Coco hedge COP Fmoo Nu.od sh.o - ‘.. m QZm A\\FOCHKQ*H FZU~UHELMOU ZOHFQJHCUCCX QEQCH PC" 0 “xx F.V.CHH.#+ *omocut.an > TIHC¥ 00k 00" *.CIHC¥..w.H.o>.dp\op.s.sc«\.s.H.oo..H.~su.s.H.» an~ o» co .°.os~-.ou..a.n.oo.uH «2.3um mm ca «2.5"» an co wathzoo om maznbzou mm o.==~-u.a.u.z>a o.oo~uu.s.ucc> mdw m~ ch ou o.-~-n.n.u.z>a mc~ mm o» no .5.H.o>\.s.wcouu.w.~.r>a oo~ o» co .o.s.cw..s.aco>.uH o.~\.~so.ou.s.H.q>.~««.ma.u.s.H.o> mHN o» oo .o.oo«-.cu..w.H.¢>.uH «2.5uH mm on ~z.«uq ow oo mzzmpzoc as MDZHEBO m.“ o.oo~uu.s.u.ou saw mg on oc AN¢+«¢.\o.ooa.«au.s.H.oo .wooz:\.5.scyz.o.ohu~q .scocr\.w.~.oz.o.fi~u«q mom saw eh ow .c.oaau.cw..s.H.2xcuH gem mo~ o» oo osN oh oo .c.os«-.cu..s.u.ox.uu 22.5uH ma ca «2.3": as on UTZH pZOC HH o.so«-u.s.a.q> Na «z.n~uH NH on «2.2": 5H oo .Nz.fiuw..H.ho..mad aqua .Hz.«uu...wcm>..H.H>...mad mama .Nz.dus..Nm.suu..s.H.<>...o«H aqwm .~2.««H...H.uz:..H.ooz...mo« name .Nz.«us..«z.«nH...5.H.21..5.H.orcc..oad aaoz Nz.«z.HcH aqua m.:m~ufi» a.n-ua» .o~.dn.:>a..o~.dn.4>a..dm.m>a..«n.q>a..aw.«n.mc..o~.dn.2>a..e~.da ncx>a..a~.am.a>a..o~.»o..o~.dn.»..o~.dn.<>..c~.Hm.c>..snows..«n.d>u ..c~.«n.>a..o~.an.oo..o~.oz:..o~.ooz..o~.Hn.zx..o~.«m.ox onmzwzHo .p:n»:o.»:azH. mooyo zqyeona nsanxao «uhao NONJIoon> 2km 5°30 coo wooxa zqywoon om m: a: on mN ow ma ca 120 moan onm.mm.m~. mz.aufi «z.auH om oc mm 00 szH»zou szthoo c.=a¢un.fi.u.a>a mm or ow A.afiuhOIaw+fiuhout2;<.\A95.nvhtaa+5.mvhuu.5.H-a>a «c.oonafi.uvoono.«mvnzsa mm o» co «o.o....«+5.H.oouo.«~.\..s.a.oouo.a~..uosax..5.H.oo-.a+s.ucou.nzsa mo: o» oo 19.9.cm.oou.uH .5.H.oc-.«+s.ucooucoo oo: c» on .o.ocnu.cw..s.H.~.uH «z.auH mz.«ufi fl! .~z.~uw..«n.«~uH..a.H.>a...mcs amm .Nz.~us.As~.a«uh..s.uc>a...ems DNN .Nz.~us..=H.«nH..a.H.>a...oea «.5 mm 00 cm 00 Nzunz szaa szma bzwma szmQ thxa bZHmd tzszoo tzHbzoo o.coalu.fi.mv>a a: C» 0O aaavha on.fivho.\.H.~>lo\oho.a..5.Hvoouo.HN_\.Aa.HVOOIc.«Nvvooqdvunfi.uv>a . com 0h 00 Ao.o.uz..Hvd>qu az.dnH Nz.~ua .~2.«uw..an.«~uu..q.H.:>a...moa DNN .Nz.«us..e~.H«uH..n.a.:>a...omfi o- .Nz.«us..oa.aua..s.u.:>a_..oafi sow .«z.auH..Nz.«uw..a.wczx...mow mks .«z.duH..~2.Huw..s.wcor...mca ops .«z.HuH..~z.Hus..5.H.¢>...omN mmw .Nz.«ua..Hm.a~uu..s.H.oo...mafi oNN .Nz.«us..o~.««uu..w.H.ou...ssa omm sz.duw..s«.HuH..n.H.ou...535 ems .Nz.«nq..«n.«~uu..s.H.o>_..mmfi cNN .~z.«us..o~.HHuH..a.H.o>...oma smN .Nz.uus..ofi.dnH..a.H.o>...sma mma nsxan\ao auboo ~0~J|Q.M> Zhu 0:30 CDC a: 00 m: co hzuym szma szxa bzuma hzuma Csza thma szmd thoa hZumn szaa thma szwa bzuan byHaa thaa szmd hzuma thwd szyo szoa bzuma »2Hma szmn om mm cod @0 mo; m: a: can oomva mad m0 cm Wt cc m» on m6 :0 rammoma 121 mean .nm.mn.m~. ns\«M\ao o.acmouafi.H.4>o o.com-u.s.ch>a can ms as ou As.H.mc\.w.H.z>au.s.Hcssa n~u.s.»ou.w.w.mo :4ax.-u.a.~.p.u.a.mczsa ms as.o..«~-o.«~.nzsa em» on ch as do.o..1.3.Hcoo-s.s~.\.aN-o.«~.coosax.fi~-.s.m.occnx3< om. oy om .o.a.wm.zco.uu w~-.5.w.cuuzou was .mcbuNMN .m.H.hu~N .m.H.ocu«~ :55 c» om .o.w3.s.uu ask mos as on .mchouMN .~.H.»u- .~.H.ouufiN ask 0» co .~.ou.x.um .x.~.mwo 43cc Nz.muw wk oo «2.5uH s. cc .nz.«un..«n.amuH..5.H.x>o...mmw szma cmw szau .mz.aua..o~.aaun..q.H.x>a...Nmm szaa smm Prhoa .nz.auq..ed.duu..s.H.z>a...mmw szma «MN hrHua .nz.«us..am.«~uH..q.H.a>a...mofl thma omm thaa .mz.aua..o~.«HuH..5.H.a>a...oofl szwa omw bzmya .mz.fiuq.105.5uw..s.H.q>n...cm~ szma «ad pzsma w:zpbzoc mo wnszzou om o.oo:-u.w.ucx>a mm oo 0» oc ..n~u.w.3.»o..234.\.--.fi+n.~.».u.s.nsz>a a. «0.9..«~-Q.«~.uzsc :m mm o» oo so.o....«.1.~.oono.«~.\.«~-o.s~.coasq\.fi~-.s.5.a.oo.uzsq am 0» oo .s.o.cw.zoe.uH HN-.«.5.H.ounxec mm .m.hcnm~ .m.~.pu- Am.H.coufiN mm oh oo .:.ms.s.ua H. mm o» oo .flspuum» .a.H.»umN 1«.H.conam am 0» co .m.cm.y.uH .x.H.umc 5340 «upuo ~c~4-o.n> zhu case coo aocya Idorcaa mca oca mma oma o:« mma oma mfiw 122 mean nnxan\wo .xxxomco 2H woqwzwoyma wznso> zmo>xo..srdcha:you .xxxnouaaHs 2H oo as“ you upm-oo urnso> rooyxo..axdvpax¢ou .xxx..ahm-oo. wshhom wt» 2H zmm>xo s«»o»..«r«.»a..H.«>a.H...omm szca msw hznaa mzthzoo c.0onnn.u.m>a on o» co .mmmm.»mcm \.>mmnxmmn»xmmt.a+mzu~z.econuou.H.y>u mm oh ow .o.o.ow.vvuu.uH .ymm.»mmu»>mm..domzuwchqosu...xwm.xmm-xxmm..H.mz-wr.»aosu.ummmm c.2cm-u.cc¢>a me o» oo xxmm\»xmmu.uca>a «o e» co .c.o.nw.xxum.uH mothzoo .w.H.z>n..w.H.z>n.>>wmu»>mm .3.vaot.5.H.mo+xxvmnxxmm 15.H.mo..s.fic2>a¢>xmwu>xmm .3.H.z>a+>mmu>mv .3.H.mo.xmmnxmw m7.mzuo me 0: o.cu>>mw c.9uxxmm =.cnxmm o.ou»mm o.ou>xum moszrou «um? «c o» oo sum: 5. o» co 11.0w.x.uH .x.H.uwc Doro flz.aup cc co .Nz.muw..an.fimua..n.H.s>a...mmd szya oNN szaa .mz.mus..om.«suH..3.H.s>o...cra thaa owm hrfion .nz.muw..OH.HHH..3.H.5>Q...owq szma osm pzhao maz~hzoc m:szzoo n~-13.»uu.u.w.mo ama um“ cma «3H :3" mm“ and wwu cma ¢aa mad OHH mo“ «ca and on r0 No «v mm mm at on ms «upon Nomguo.n> Zhu @930 000 Taryn mam mam mod odd mad mug aha 123 moan C?u .\\\..4z oofixoocxzpa Czahmzoo >azex..fiz«.hczaoa new . 1:.cauma.ora.»craou omN .\\\ozoyoaz 4a: mmo pa wozamaomma..«ra.»azmou mmm .c«.omum.afiH.oxdc»crwou cam .xxxohzmHoHuumoo onpqswwwoo 024 «Fan hmfi a owhousomz . sakes . msppom no >p~swmawxowa moaam>q..sxsc»pHSqumrmma..ercbqryou saw .m.~«uad.=zdc»axxou mum .m.~«uo«.oxa.»azaou mmw .\\\.. sake» . ushpom my» no >pHsHmawymuao.axH.pqzaou «mm .\\\\.orficpqzoou owm .m.~«ua«.crs. kczwcu mos .\\\.. scoos . washom or» ac >hHsquwrmwa..axH.»qrwou sod .m.~fluca.o:5.»arwou can .m.~duaa.ar«. hqroou was A\\\.1wsh»om shazm. usppom or» no >pH4Hmamxmuaw.fixachazaou Hm“ .m.~«uoa.oracpqzocu e.” x\\\.hzoaux yawn zwoomhuz..uxachqroou mks .xxxnpxoawx xcma zmorxo..ax«.»qzmou ass .«.ouwd.oracharwou mm. .rm.mm.mm. nnxdmxao auhao Nwmqoo.n> Zhu mono Coo admya Iaovoao cam mnm omm wmm 12L! moan .rm.mm.mmo ns\«M\ao Huhao Nomquo.r> zhu 05:0 000 ozu zoapwy any zwobwa «ny coo c» 00 .sa.cm.H.uH and c» on .ma.cm.H.uH coo o» oo .ma.aw.H.uH com o» oo .«H.cw.H.uH :06 o» co .aa.cw.H~uH ooo or ow .u.cu.H.uH coo oh om .m.cm.H.uH com o» oo am.cw.H.uH ace o» oo .«.ow.~.uw coo c» oc .«~.uo.H.uH .y.HCuuo msznowmnm cog muo qupncmcDm UH ca 125 cccc.cc«o cccc.ccal cccc.cc«! cccc.ccwl cccc.ccal cccc.ccal cccc.ccao cccc.ccal cccc.cc«0 cons. cmmc. cc::. cmn:. ccaa. cc::. cc::. cmnm. comm. ccmw. cchm. cmc:. cmmn. chm. ccqm. cccn. cncm. cmnm. cmwm. ccmm. cco:. cco:. mHvH cccc.ccaa cccc.ccal cccc.cc«n cccc.ccau cccc.ccuu cccc.cc«n cccc.ccal cccc.ccdu cccc.ccan ccn:. omJJ. cow:. c«::. cm::. cnn:. cm~:. coaJ. csnc. cnu:. caNM. cccn. ccan. ccH:. cmcs. ammn. ckmm. con:. cc::. cmns. Gown. comm. NVNH cccc.ccal cccc.ccao cccc.cc«o cccc.cc«t cccc.ccdu cccc.ccal cccc.ccaa cccc.cc«o cccc.cc«| co~:. csac. cmMJ. cnma. owns. cows. cc~:. cm~:. c-:. cows. cowm. coon. cmwm. cams. cmm:. cwwm. cmcm. ccn:. can. cNMJ. cmnm. ccnn. mmoa cccc.ccau cccc.ccac cccc.ccal ooso.sos- ossa.os«- osso.ao«- asoe.coau ssoa.ao«- osso.oo«- sm~.. owns. 99”.. on».. owns. sm~:. am... so... om~.. smma. oosw. ann. amkN. owns. smns. scam. omom. omns. sm~.. cons. .aokm. cmmn. 5mm cccc.ccal cccc.cc«t cccc.ccau cccc.cc«o cccc.ccau cccc.ccan cccc.cc«o cccc.ccal cccc.cc«| cmn:. cowt. co::. cn~:. coma. cnwa. cc~:. ccaa. cn~:. cmn:. c:m~. csow. cmxm. cams. cmn:. cmcw. cmcw. ccne. ccm:. cmwa. cmmn. cmom. mwh AI ccnc.ccao cccc.ccwl cocc.ccat cccc.ccat cccc.ccwo cccc.ccal cccc.cc«i cccc.ccal cccc.ccdl cNNM. csum. cmcn. ccwm. cccm. econ. can”. cmmm. ccmn. ckmm. comm. comN. co:~. camm. cmcn. cmmw. cnmN. cHNJ. cows. cmwc. cwsw. Now uIv OEHB cccc.oc«| cccc.ccal cccc.cca| cccc.cca| cccc.ccw: cccc.ccal cccc.ccal cccc.cc«l cccc.ccuo c~:n. ccsn. cnam. cdcm. coon. c:cm. cmc:. coon. ccHJ. cc«:. ccnw. comm. camw. cows. csca. cmew. cccm. cca:. cm~:. c~«:. cccm. crwm. mom cccc.cc«o cccc.ccal cccc.ccaa cccc.ccao cccn.coan cccc.cca' cccc.cc«u cccc.cc«o cccc.cc«u cc:m. comm. coco. coda. cccc. comm. omnm. camm. cwan. cram. cwcd. caca. cos“. cme. chm. ckoa. cacm. Oomc. csmm. corn. cmmn. ammo. mmm cccc.ccuv cocc.cc«| ccoc.cccu cccc.ccal cccc.cc«u cccc.ccau cccc.cc«n cccc.ccac cccc.ccq: cm3n. c«:m. cccc.ccct cccc.cc«a cccc.ccal cmwn. ccur. cccc.mcun cccc.cc#0 cccc.cccu cmxw. ems“. cmma. cccc.cc«u cccc.ccdo ccawo cccm. cccc.cccu cccr.ccal cccc.cndn coca.ccfll occa.ccal cccc.ccau ccmc.cc~| cccc.ccwu cccc.ccal cccc.ccau cccc.cc«u cccc.ccau comm. cmmn. cccc.cc«| cccc.ocal cocc.ccHI cman. cmmn. occc.ccal cccc.ccwo cccc.ccal c~:a. cc:«. chA. cccc.ccan occc.ccwo cum“. cwra. cccc.cc«u cocc.ccwu cccc.cc«u nccc.ccfll cccc.ccau ccoc.ccal ccnc.ccal cccc.ccal cccc.cccl cocc.ccuu cccc.ccau cccc.cca| cmmm. cccc.ccac cccc.cc«' cccc.ccal cmcm. c¢cm. ccoc.cc«I cncc.cc«o cccc.ccdl cmmw. ccma. chad. cocc.ccal cccc.ccw0 cow". cuaw. ccoc.ccat acco.cc«| cccc.ccau cur“. ccxa. an cccc.cc«o Hm cccc.ccau on cccc.ccal mN cccc.cc«| mm cccc.ccdt hm cccc.oc.—I wN cccc.ccdi mm cccc.ccdo vN cccc.cca|.mm ammu. NN con“. MN cccc.ccul om cccc.ccunmd ccccoccau ma acma. ha cord. wH cccc.ccal ma nccc.co«! VH cccc.oc«IMH ammo. NH cmwc. Ha cmac. 0H cccc.cc«om cccc.ccaoc cfioc. h cage. 0 cceo.ccaam cccc.ccalv cccc.ccdum gumc. N mama. H o .02 mauuom LCZGEUOWQG 126 :.m m.~ o.« m.« c.~ @.a o.” m.~ c.: :.H r.N m.~ sow o.~ n.: c.n J.“ c.c n. «.H n.~ «.N H.m c.N o.« 3.“ u.« m.~ 5.“ m. N.“ m.“ h.~ o.« ~.« «.~ c.« c.« m.“ «.w o. n.« H.N w.« H.~ «.N n.« m.« c.~ c.« o.~ 0.~ H.N o.« ~.~ c.« N.~ c.« m.« e.« hVNH NmOH «.N a.« cow n.~ c.~ «.N o.u 0.“ a.~ ~.~ m.« c.~ k.a mwh cow «.N Now A.H=V OEAB ~.a mmm c.c c.c c.c c.ccfil c.ccau c.ccal c.rcal c.ccwn c.ccel c.cnan c.ccpl c.c~au «.N mom c.ccwl c.ccal c.ccan c.c :. c.ccan c.ccwo c.ccel m.m o.m «.m c.ccan c.ccut c.ccal c.ccwu c.ccal o. n.« c.c c.H c.c c.c c.c c.c c.ccan c.ccwv c.ccal c.c m. c.ccao c.ccuu c.m c.ccau c.ccan v.4 c.c c.ccwu c.ccao c.ccat w.r c.u Ah c.c an c.c on c.c mm m. aw o.“ hm c.c o~ c.c m~ c.c vN c.c n~ n.“ - s.a AN c.cea- om s.ooa- ma s.oo«u ma ~.~ pH m.~ ma o.oo«- ma c.csu- vs s.os«- «H a... ma a... HH 9..“ ca o.sss- a °.os«- m m... n ..m« o o.s=«o m s.csd- e o.as«- m m.c. m m... H o .02 oauuom hIUHuY )qml 2w0>x0 127 m.h~ 0.05 0.05 m.0s 0.0N n.0s ~.0s ~.sn 0.0K c.hs w.ss N.~h mdva c.c» 3.05 c.cs hvma c.0n c.h~ c.cc ~.c0 Nmoa m.co u.«m h.~c c.cs moan ~.:c c.cca c.cca c.0n 0.5x c.cn ~.on c.ccu c.ccu ~.m0 0.50 n.s0 ~.c0 c.cca c.o0 ~.:0 :.cs «.00 0.50 N.o0 005 c.cca c.cc« c.50 c.cca «.«s ~.e0 M.L0 N00 A.H=v GEMS m.«e c.cca N.o~ c.cca c.ccH o.ns c.cca c.ccw ~.:~ N.«~ c.ccu 0.00 0.00 c.:0 5.30 m.mm mmm c.cm 0.0x 0.0x c.m¢ c.ccw c.cca c.ccau c.ccwl c.ccun c.m0 c.ccuu n.30 c.00 c.ccau c.cnao c.ccal c.cca «.00 c.cca c.ccH c.ccau c.ccan c.ccau c.cca c.cn c.cccn c.ccfiu c.ccfin c.ccwc c.ccal c.ccau m.an klcsmI c.cca c.cca c.ccau c.ccan c.ccal c.cca ‘0 .ms c.co«t c.ccao c.ccdl 0.10 «.60 c.ccal c.cca' c.oca an c.cca on c.ccu mm :.mo mm «.«c hm c.cca 0N c.cca 0N c.cca «N c.cca MN «.mn NN ".05 HN c.ccav cN c.ccdu ma c.ccdo 0H c.- 5H c.0s 0H c.ccul 0H c.ccul vH c.ccnl ma 0.«0 NH ¢.m0 Ha m.:0 ca c.ccau m c.ccal m s.n0 n 0.m0 0 c.ccdl m c.ccuo v c.ccau m m.:0 m w.»m H 0 .oz mauuom unuc 2wOOUbHZ 128 conch. odomn. chaos. cNNm0. «swan. 00ccc. c33h0. 0000c.- ccccc.cc~0 ccccc.cc~I ccccc.ccml cccco.chI om mnmkm. wussr. mncmc. n0ccn.« mmn:s.« n0nom.a chase.“ :ammc.m c:m:s.m H:c~:.n 0m0mm.~ :omm~.:aai OH Nacm0. ncmm0. ~:mm0. ccsco. 03:30. «kcma. «scan. mussw.0 ccccc.cc~| ccccc.cc~u ccccc.cc~n cccoc.ccm| mH cmmmm. mm0ms. ocmm0. swoas. woman. Nnccc. cJQOJ. Navcm.« coccc.ccm| ccccc.comu coccc.cc~| ccccc.cc~| m cmncm. Nnmkm. 0030:. msooo. mmmms. Nm:wc. somco. c~c«:.«« ccccc.cc~t ccccc.chI ccccc.cc~- ccccc.cc~t 0H :m0ms. 0Ncas. Mcmas. 00oa0. :3Nm0. 0:H00. «scum. cmacm. ccccc.cc~n ccccc.ch| ccccc.cc~I ccccc.cc~| m cmaw0. 0knmn. c0mss. nn~0h. «ann. 00cm». ccccc.c «03:». mumon. ccccc.c ccccc.c murkm.: NH Hmcon. kaJN. ncmcs. HO0MN. mrmo~.a n0s~:.« ~amm:.« csm~¢.a camcm.~ c0n¢~.« omm:n.s «ccam.=mnu h 000m0. 0Nsmm. cs0sd. :KcNN. ccccc.c ccccc.c ccccc.c ccccc.c avesd. ccnoa. mamsmo 0:cH@.: 0H «neon. mscwo. ommca. 0:00c.a ma:mm.u cmwc~.a tnaao. anc~«.« ccrcm.w c~«:~.~ 0m~¢m.0 Nadom.0ca| 0 .oz mauuom c0nmm. 0acmm. anNm. 30000. 0ccnc.« «couc.« 660cc.« nc0c0.n ccccc.cc~0 ccccc.cc~I ccccczchI ccccc.co~| ma «0NHO. Ncwno. 0:~mc. 0s~:s. McOcm.« 0mc~c.a cmuwo. o«~mn.~aan ccccc.ccmu ccccc.ccml ccccc.cc~| ccccc.ccmu m :mNON. mmomc. bwcom. mscOK. :n000. m:cam. c0004. «cmum. ccccc.ccmt ccccc.ccmt ccccc.cc~I ccccc.cht vd :n«c:. mnxm0. o:cmm. 3:0om. mmc0m. nmaan. sammo. m¢¢~m. ccccc.cc~o ccccc.ccmn ccccc.ccm| ccccc.ccmu v aka¢m. omcme. :skoo.« mava .35.». :en... madam. hv~a Noam». mmwkm. «coon. Nmoa :«mmm. «mace. occam. hmm Hanna. cocam.a sooca.d awn amnoc.a mmocm.d saon:.« ~oo Nakam. Hanan. aumoo.~ mom ~s0mm. mmcmm.« nessc. mmm ccccc.ccmn :cn:h.n rmm0«.~ mwm ccccc.ccmn cc-0.m mcmcn.: and cccoc.oc~u :mcmm.~ momcm.n« an ooooo.ocmn oaw:o.om~- sommm.o.«.o ma NH Ha «scam. aswmm. meamo. mava wands. asawc. osmmm. hvma murmn. wages. suukk. mmoH wmsao. kcfime. -N«~. nmm mmmfio. Macao. measc.« won somnm. omomu.w .cnkm.s mom snows. «_sdm.s 7mm...“ mom HAAJm. «omrk. .mmom.a mmm Decca.oo~- .mwmo.d moomo.m mom ocmoc.cam- :o.mm.a AHJQH.N aha oooao.aom- omca~.¢ moomo.o an cacao.oomu mrsko.cmwn. mmwwm.;e:o m w H A.umc mafia .1: sowxcocxth hthwroc >azmx 129 ccccc.cc~I cccccoccwo ccccc.ch| ccccc.cc~I ccccc.cc~I ccccc.cc~v ccccc.cc~I ccccc.cc~u ccccc.ch0 ccccc.cc~0 ccccc.chu ccccc.cc~I Hm ccccc.cc~I ccccc.cc~| ccccc.cc~| ccccc.cc~o ccccc.cc~0 ccccc.cc~| ccccc.cc~I ccccc.cc~I ccccc.cc~| ccccc.cc~I ccccc.cc~o ccccc.cc~t om ccccc.cc~c ccccc.cc~| ccccc.cc~| ccccc.ccmu ccccc.chI ccccc.cc~I ccccc.cc~I ccccc.ch| ccccc.ch| ccccc.cc~o ccccc.cc~| ccccc.cc~n mm ccccc.cc~n ccccc.cc~t ccccc.cc~I ccccc.chv ccccc.cc~o ccccc.cc~t ccccc.chI ccccc.cc~o ccccc.cc~I ccccc.cc~| ccccc.ch0 ccccc.cn~| 0N ccccc.ccmi ccccc.ccml ccccc.cc~0 ccccc.ccau ccccc.ccml ccccc.ch| ccccc.cc~| ccccc.cc~0 ccccc.cc~l cccccochI ccccc.cc~| ccccc.cc~| 5N ccccc.cc~I ccccc.cc~| ccccc.cc~o ccccc.cc~I ccccc.cc~n ccccc.cc~| ccccc.chI ccccc.cc~I ccccc.cc~0 ccccc.cc~I coccc.cc~0 ccccc.cc~u 0N ccccc.cc~I ccccc.cc~I ccccc.cc~0 ccccc.cc~0 ccccc.cc~l ccccc.chI ccccc.cc~l ccccc.ccmt ccccc.cc~I ccccc.cc~0 ccccc.cc~1 ccccc.cc~t 0N .oz mauuom ccccc.cc~l ccccc.cc~I ccccc.cc~| ccccc.ccmu ccccc.chI ccccc.ccuu ccccc.ccml ccccc.ccml ccccc.cc~| ccccc.ccmi ccccc.ccwn ccccc.ccmo .m. ccccc.cc~l ccccc.chI ccccc.chu ccccc.cc~I ccccc.ccmu ccccc.ccmu cccnc.ccmn ccccc.ccmu ccccc.ccm| ccccc.cc~I ccccc.cc~I ccccc.cc~o mw 0:00». :vesa. ccccc.c ccccc.c ccccc.c ccccc.c ccccc.c ccccc.c ccccc.c ccccc.c ccccc.c hcao:.~ NN :cncm. 3000c. armom. ccccc.c ccccc.c coccc.c ccccc.c ccccc.c ccccc.c 605mm. ccccc.c anukc.n HN QHVH hVNH Nmoa 5mm 005 «00 mom can moN Aha an o 7.3: made macuc. masoc. 00000. 00000. 0N5mc. Nwonu. 0~n««. ~300c. c~cnc. nvouao an 130 Nmuvc. aacrc. cnmvc. ~4cNH. hownd. mo«:w. c5maa. wnuaw. «n0ma. 00000. on «ammo. cmcrc. 0004c. :5000. :0030. c«:«~. ccccc.cnml ccccc.ccm| ccccc.ccml ccccc.ccmu ON 0505c. nna0c. nca0c. :smsc. mcccc. «0000. aaaoo. wcssc. 0000“. movso. ca 00000. 03000. «nomc. ccscc. 00050. 00000. ccccc.c ccccc.c ccccc.c ccccc.c 0N ga~nc. cam:c. 00cmc. 00000. 0000“. ¢0~mn. ccccc.ccmu ccccc.ccUI ccccc.ccw| ccccc.ccmu 0H «aa:c. 000cc.| c:acc.u :oaac. ~::«c. cma0u.c ccccc.ccm| ccccc.ccm| ccccc.ccmu ccccc.ccw| a 0045c. 0003c. 0«¢:c. 05000. s0mmc. 00050. mmscc. 00500. ~::~c. ~00cc.| 0N :::~c. nccmc. 5mawc. 4000c. 0000“. 0m0-. ccccc.ccm| ccccc.ecmv ccccc.ccm| ccccc.ccm| 0H :oa0c. «Nacc. 0mauc. nwmwc. 00uac. 00cc«.9 ccccc.ccmn ccccc.ccm| ccccc.ccm| ccccc.ccmc moswat 00500. 00cmc. anomc. NcONc. 00000. aucmc. nocnc.u 0:000. 5030c.- 5N Nm~mc. 0««:c. 0005c. camcc. ch0c. 0-sc. cmNnc. c5wcc. :c0««. sssac. 5H 0550c. ncrmc. 0cmmc. 3000c. «0050. scanc. :wc5c. 0000c. momcc. 05000. 5 :mswa. oakna. «cnmc. :nncu. scccc. «5000. c~0:c. ccccc.c domed. ccccc.c 0N 000:0. «Huac. mnoac. 0000c. 0mcuc. 00ccc. mawuc. mommc.l osnuc. 0:«cc.| 0H sac0c. Nma0c. cammc. 0050c. «0000. 00000. 00500. 0000c. cwkcc. «mono. 0 002 «tomc. 0050c. 0:500. 0:000. «0000. nn0ca. c000a. 00coc. mscwa. m0ocu. mm @5500. 000cc. condo. 000nc. c000c. 00000.! ccccc.ccm| ccccc.ccml ccccc.ccml ccccc.ccml ma ¢~«~c. mo:ac. mnmcc. c«0cc.u «tuna. :mccd. ccccc.ccml cccc=.ccmo :cccc.com: ccccc.ccmn m mauuom cwscu. 0««cc. wcwcw. 0u05c. 0050“. n«:0u. coccc.c ccccc.c ccccc.c cccce.c vN scch. 000cc. «06cc. chuc. 0cmNc. ncuoc.l ccccc.ccmu ccccc.ccma ccccc.ccmi ccccc.ccmn vH «00:0. 000ch 00ccc.n mcccc.l Nnrcc.: mmsma. ccccc.ccu0 ccccc.ccml ccccc.cor| ccccc.ccmn v n «510 50 .n OuwaJfinz 0myma. mamsc. «3mma. mMJmc. mmuwa. 0560:. 50nd". manna. 5050“. 000:0. 50060. Nrimc. 5:000. mommc. 000¢c. :axac. ccccc.c «N450. 00000.0 “:05“. MN NN 00000. 50000. wwarc.| Nmnwo. 00:90. ownsc. 5m~uc. 00000. cmmmc. 05000. Nnvcm.l 5000c. cccpc.ccou mrcrc. ccccc.ccwu 30mmc. ccccc.ccm| rsmcfl. 00000.0001 0000“. ma NH 000:0. 5500a. ”Nauc. 0cucc. 0:000. 00000. 5amac. cmmcu. 3000:. 00000. N0ruc. “~03“. ccccc.ccUI 0H000. ccccc.ccml «000“. cccco.ccmu :rmva. ccccc.ccm| «nmsr. N ~4c505004550E LC r0~mc. nmcmc. emccc. ummsao Huumc. «chc. hccmc. oasxc. 0050“. 0nn0n. AN :nsxc. 00:00. mc:ca. 00000. wkwmc. maacc. .ctcca. ncacu. wksoo. mmw~c. “omuc. camsc. 4000c. kcaoc. roama. ”0cm“. rc~Ua. H mHVH 5vNH Nmoa 500 005 «00 non 000 000 H5H advd 5vNH cha 500 005 N00 mom 000 m0~ H54 cave 5VNH NO0H 500 005 N00 mom 000 000 H5H A.u:c OEHB >hwchrdbtdw0 131 h7uuouuuwoo 70ubaqwumcv @JJFNQUNDO. NdJthmeaw. 0N5:0¢:omo. chroacocmm. H0accwwsco. ocamMNJcam. okmon¢:cmo. mnownsc0am. 5cc000m300. «acm:om0em. rnnndncm.m. moccwcmmrm. lmssmopdou. 00rawcmmam. cawsmsonmr. :ONmmcham. ockcacammo. 00mcmmmmok. na5wconss0. ccmmwnaocm. :«00000aca. ccccc0c000. 05c0~mr~a~. “macrfiooos. 0::Nsscwuo. ommwcnsmsm. crummeNrc. Nccawanmmc. ~«oanow::~. emflmccacmc. ommmcmmaww. .wuooo coHunaouuoo 02¢ «516 km .— CuwaJmel. . Jakob . NJJOmcarhc. MKONchcca. JawmmONcmc. c0nwaam00c. «mccvumssc. Ncm5nnccca. mmwoomhnoo. nc~03000¢c. 03w~¢coswa. ~330om0a40. mmdarcnwmr. .rmrkmameo. NNudnoMmuo. Pcuncwnwre. omucwmac0c. mmnxchcMU. osooknwnmc. a~p~cmflmmc. nncnsahwwc. kmmmnooowc. arcvaoccac. pammcawrsc. cmms—uvccc. c.rurmnwmc. 0w.a+mwmwc. «“60000000. NMernanac. wmcwmwucae. acrwrmmvwc. mcacrem:cn. caun05Umxc. A > ¢ m «m cm 0N 0m 0N mm mm rm aw cm 0“ «a 5a 0a .r lf .oz mauuom 04550: UL >bwgpcdmenkn uwgmw>q APPENDIX 4 . 1 Laplace Transform Applied to Solve eq(4.6) and eq(4.7) APPENDIX 4.1 Laplace Transform Applied to Solve eq(4.6) and eq(4.7) Take Laplace transforms of eq(4.5) and eq(4.7), we obtain, where, dp 2 ._ _ _ _ dp1 EE‘ ’ k3 (92 ’ Pl) (4'7) Sp1 ' plo = k3P2 ’ k391 (4'52) k P — _ l O - - — _ 392 p20 ‘ s klpz kzpz + k2pl (4°53) S = parameter —— _ -® -st P_ - L(p ) = °op e-Stdt 2 2 O 2 From eq(4.52) and eq(4.53L'we can get 5: and E; as follows: 2 p10S + S(plo(k1+k2) + pzok3) + k1k3po 2 s( s + (kl + k2 + k3)s + klk3) (4.54) 2 ons + 3(onk3 + pokl + plokZ) + k1k3p s( s2 + (k1 + k 0 (4.55) 2 + k3)s + klk3) 132 133 Define that a and b are the roots of the following equation, 2 s + (k1 + k + k3) s + k k = 0 (4.56) 2 1 3 a and b are real and different numbers, because the determinant of eq(4.56) is positive. The inverse Laplace transform theorem is applied, (Heaviside's expansion theorem), i.e. f(s.) -l f(s) .___.:=z _____ . . L (g(s)) g'(si) exp(slt) (4 57) where, si are the roots of g(s) = 0 we obtain from eq(4.54), _ —1 — 2 = k1k3po + ploa +(plo(kl+k2)+p20k3)a+klk3p «x (at) ab a(a - b) C p 2 plob +(plo(kl+k2)+p20k3)b+klk3pO (4.58) + b (b _ a) exp(bt) _ -1 — 2 k1k3po + pzoa +(p20k3+plok2+pokl)a+klk3po = ___... exp(at) ab a (a - b) 2 p b +(p k +p K +p k )b+k k p + 20 20 g (to-2a)o 1 1 3 o exp(bt) (4.59) From eq(4.56), we have, ab = klk3 (4.60) Therefore, pl 134 eq(4-58) and eq(4.59) become, p (a+k +k ) +p k + p b = p0 + lo la 2 b 20 3 o exp(at) plo(b+kl+k2) + p20k3 + poa. + b _ a exp(bt) (4.24) p (a+k ) + p k + p (b+k ) 20 3 -1g 2 o 1 exp(at) :13 + p20(b+k3) + p10k2 + po(a+kl) + b - a exp(bt) (4.25) a txMC-fllnl"rifu :‘M q ' ' :. I I'M- _ p . . . ._ J4 ‘ - APPENDIX 4 . 2 Pressure Changes in Example (I) (.LU “(c.cnuq.n.c~u.(—_. 10.542ccu rcu n\\\GO(—ko*" F *\R a .Cfiu.*u can *\moc_u.:u Cwu *\mucfiu.tu OJ #\nocfiu.*u _Jt. I~055_J_F0 SmeOOO JJcL leCOC uC ZO—FDJOW JJ.mc.~<.tl.~<~ #2006 CC" FLHOC 1 mp‘nCQ'Cmcvflfit n<.\~<4a<flanQI QJ4+~Jc\c<*nv\«<#a<10nnl Ck\5*.nchL>.\md*Uqu< .5:05:C.5;C7~ccczxu anccoc _ « FC— cc" 2 .CUJ 0.IL.u APPENDIX 4 . 3 Pressure Changes in Example (I) 136 Czu ~c_.couo.m.c~u.c~_. Incp *\«.C~u.*u ~> *\nocnufi .*u «q *\m.c—u.tn _a *\U.cfiu.2u qurmzou >+~J~mm memCUQ JJdB MJmDoc no ZOHFDJOM JJ.~>.n<.—<.Ca.—L— 52.00 can kz_aa -- - _UI.C0lccavan - - .<c\aqtaqaauc1 n p.. i. «44+.qufl4c Uk\k*n—de">~\—<*Q.\H<¢OqaquI rk\k*nch*m>.\mdtQ (oC(C0uO< r\O.(..r\rU" —< (OWFAHCF c.mcru» Concufic abo~.C~.Umr 53:“.7 Lchccu 4443 HJLZCC uncezt7t_40u Jqu_5>4<2< u C. .05."...5392 0003.... .2...ch0 oCCJ 0.4:?“ APPENDIX 4 . 7 Thickness and Critical Point in Example (III) C.K_ C.m U—qfiom Lu.c «c.( “c.c c.rhm C.ncm Cons . . CZk .\\.$h mwuaut.xm~.*m mmua&*.XN_~ .*~ VULQQ:.XF.*.>1 .flmJioaa JJdB quDOQ “Q ZOvhfiJOw J<¢~P>J.\~<#Q.\vV\~<#H.\n<#O.~<.s>.5.m(— szen (CH F.1—nunu 03+UctnttaCoC~\A¥vFaGJu+Cow.wn:m<0»0.cuk< w;i+#n:*nCoCH\nxch «itcxc. - C~o~fl¥ WU CC FUtIOnQdfiQ AoCnfiJd .cu.m*qu.m#.u("icogmt.ccc_.\c.mmnru I.&.~40.CQ.C&Q.C~0.0F.P.$O.~C~ C(Nfl «Fo~.¢mokUO 023.7 EmJJCQO JJ<3 meDOO no ZO~PDJO¢ Jd<2c . U .FDQ»:C.FDuz~.coayn zqacoca , . .004 C .‘nru APPENDIX 4 . 5 Pressure Changes in Example (II) 138 CID .c_.cuuc.n.c.u.c__. I_.5<:acu rob .\\\«.C~u.*u k *\m— .cflu.*n can *\m.c~u.*u o_n *\m.ccu.*u m4 *\Q.c~u.¢u _Jt. I_.»F~J_mm fiuJUOOQ JJ<3 meDOC no ZO—PDJOW JJ.n<.~<.ma. c— hZ—Gfl cc— 5200a —ulacnlc~0.umt A<.\~.\u<*aC\—<*a.\~<*&.\mqta CP¢CWO~> C.LCCfinR< c.comuc. ounce!» c.nguao m5o—.««.Uwo 023.3 fiuJ..:u 54x: MJGDOD no ZC_»DJOU Jdu~u>Jtz< U .FDQFDO.5:QZFCCCC>C :qnucoc oCCJ .FU APPENDIX 4 . 6 Thickness and Critical Point in Example (II) 139 020 oc~.cmum.m.c—u.c~_.m.c_um.cruch<2aou ac. . .\\\&.C~u.*u 5 *\u— .ouu.#u CNQ *\N.o~u.*n ocu * .Cluchqsaou «Cc .m.c~u.*u n) *\n.n—u.tu ~> *\m.o~ua .tu Rd *\N.C_u.*u H... *\mu.CH.¢.?u 524.50.200 >F~J:mdwfiawnu#.CI~0.55230“... uCu .\\\.#m50~ .&~ .Umo — 023 .7 >m EmJCOHO JJ<3 mJEDOO no ZO_FDJOU JJC\~<*c.\—c\u<*adr0u01 CF\P*ARJ<#0>0\0<‘&._>.n<._q.me.ficu »z_an CC" FZ—OO rvzflflnu<fic AQU.N#<30N80CL~#C.€04.-.Q~c\:.UnflvL —n.CuCC ~C.Cu(k0 "c.cucuo n.CumJ< c.cgc_un> C.CCUI~> c.cccnum< C.CCUU_< (.mhnuCF Corcrnk c.ccnco N50“.Naouuo 023.5 EUJJOQQ JJd? MJEDCO uC ZC~5:JCm J¢U_5>J #\&.C~u_ .#u «c x\n.0~u.*u ~< t\u.C—u.*u FZ5MJ~flI EHJEOCQ JJQE HJCIUC uO ZOFFDJOU JJdZutwuu~N (o€k\afivF.n>.n<._<.ma.~x~ 52000 : cc" hZ_aQ -I I _Ulaoalouacumu acclfldv\.ACflCI£w00#MII.CQIOuQc#m.\~<*Oc\—<fia.\.v\—<*Q.\R1Lon—ll<.n$*P. Conufl) n&**0*(. C¢+Z...#L#(oQ—.*O.Cfl0:<, 1.»o*¢u*~ ,1<+ac*axcdntc.nnwu U~Q~offl~<fl . sld ConFNnOF Concrnk CONQHUU RFO~.NH.UU_L CID...) ....|J..JFOC{ JJd.... LWJEQLC .00 72:25:40... JJd?d U AoSJFDC.5:21_.CCahQ IQCCCCS .CUJ .ka APPENDIX 4 . 9 Comparison of Analytical Euler's and Runge-Kutta Methods ' 'I'I'. 142 abo~.mm.om \. (L 02307 I Jul . II ‘ 5' . II ..I n~¥<*m.c+c~00unfil< QHHRNY< "mum_¥< “HQI<.H~I<.Nu.~u.uZDE 4440 "myqau.o+cmaau2ml< ~0¥<*m.o+0~aauH~I< . . ; muufiaxq «mummy< “c032.cacu.mu.fiucoz:u JJv\wc\~dta C.CCUu~> c.000cumq coconufiq (.mhnNCk c.mcmuk o.ngucc XMJCCQQ JJqB LJCICC uo ZOLFDJOU JJ E . Coccmu~> c.cccuum4 c.ccmn~< otmhnuCh Cumomnh Coucuflfl gmmaonfla.mu.~uv023u uZ_FDOcmDm Czu Acfiofiuufiomocflkokuo Ifivkdiflou fiCw A\\~ *\N.C~u.*n AJ.#\uocfiuo*u «4*. I~.F42acu nc~ .m.o_m.*n m> *\m.c~u.#u d> *\m.oHL~ Rd *\mocfiu.*u ~< *\mo(~u.*u hXQFMZOU >F~J_m£ ramJIOflQ I733 “Jul—$300 n+0 ZOHFDJOW. JJV\H<$nv\~d*eduaumI (k\..r x. a _,;.J<.rxc>v\0< xC.quLu «I "17'lfiflififl'flflfi’tmflllmflfim WE ITS