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"19:2?! rink-é??? . aways»: 1' n ‘ ‘5 ... 2 'u. . v.4‘w-u .w ‘10». m» a.» ' ‘ ag. .m-J" ~u um. 1.32...” w , m ”-5 , tvvv‘ v' . w.-. w :7 4. ’51).“ {’2‘9‘: “I '5 ‘ I I: 3643; TATE umv as LIBRAR'ES n mm NW N u m mm M m I M 3 1293 01410 7274 This is to certify that the dissertation entitled COMPUTER SIMULATION MODEL FOR DIFFUSION OF OXYGEN INTO A PACKAGED LIQUID FOOD SYSTEM WITH SIMULTANEOUS OXIDATION OF ASCORBIC ACID presented by Jai Neung Kim has been accepted towards fulfillment of the requirements for Ph. D. degree in Food Science grim/66’ / / Major professor Date August 31, 1995 MS U i: an Affirmative Action/Equal Opportunity Institution LIBRARY 7 Michigan State,‘ University I —— PLACE N RETURN BOXto remove this chockom from your record. TO AVOID FINES mum on or before date duo. DATE DUE DATE DUE DATE Due - I M :_—H_' > » _7_V_>77__fi_» u';. ;Tl“ I} l ".- ‘ t;.\ I. a i .l 1534:; t; . A!" FEB 1 37nn7 MSU II An Afflrmdivo Adlai/E Oppommly Initiation ' w Wm: COMPUTER SIMULATION MODEL FOR DIFFUSION OF OXYGEN INTO A PACKAGED LIQUID FOOD SYSTBN WITH SIMULTANEOUS OXIDATION OF ASCORBIC ACID BY Jai Neung Kim A DISSERTATION Submitted to nichigan state University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Food Science and Human nutrition 1995 ABSTRACT CONPUTER SINULATION’NODEL FOR DIEEUSION OE OXYGEN INTO A RACXAGED LIQUID EOOD SYSTHN‘NITN SINULTANEOUS OXIDATION O! ASCORBIC ACID BY Jai Neung Kim One and three dimensional mathematical models for predicting loss of quality of packaged apple juice and model liquid system (mixture of water and ascorbic acid) were developed. The models were based on the mass transfer of oxygen through the package and into the liquid, mass transfer of ascorbic acid (indicator of shelf-life) through the product with simultaneous oxidation of ascorbic acid in the packaged liquid food product. Three computer programs were developed to solve the mathematical model, using the finite difference method, one dimensional open system (without packaging material), one dimensional closed system (with packaging material), and three dimensional closed system (with packaging material). Variables such as the permeability of the packaging material, diffusion coefficient of oxygen and ascorbic acid, and kinetic reaction constant of oxidation of ascorbic acid were included. Most input parameters for computer program were from published sources, though the kinetic reaction constant of ascorbic acid was measured. The distinctive characteristic of these models is that they take into account the three dimensional model of oxygen, ascorbic acid diffusion with simultaneous oxidation of ascorbic acid in a packaged liquid food product. The concentration of the ascorbic acid in the liquid food product was analyzed using high performance liquid chromatography. The one and three dimensional mathematical models were successfully validated by experimentation. The computer programs were also used to simulate the effect of physical parameters of a packaged liquid model food system such as diffusion coefficient of oxygen and ascorbic acid, oxygen permeability of packaging material, and kinetic reaction constant of ascorbic acid oxidation on shelf life. Copyright by Jai Neung Kim 1995 Dedicated to My Wife, Hee-jung Kim and my son, Joowon and my daughter, Jooeun ACKNONLEDGENENTS Above all, I give thanks to my God, Jesus Christ, for being my lord and God, giving me this opportunity to study at Michigan State University, guiding and leading me until now and forever. I wish to express my heart felt gratitude to my advisors, Drs, Bruce Harte and Jerry N. Cash, for kind help, inspiring guidance, consistent encouragement and critical evaluation of the whole manuscript. Also I give thanks to the CFPPR at School of Packaging for providing the financial assistance for this project. I am also deeply indebted to Dr. Gary Burgess, for his sincere guidance, assistance, encouragement, and friendship in the whole course of my graduate study. Genuine appreciation is extended to the members of the research guidance committee, Drs. Robert Y. 0foli and Jack Giacin for reviewing this manuscript. For many helpful suggestions, technical assistance, and timely help in preparation of this dissertation, I owe a special debt of gratitude to Dr. Nirmal K. Sinha. Thanks are also due to all of my lab partners at MSU, especially Young 800 Chung, who offered their invaluable help and encouragement throughout the course of present studies. Finally and for many reasons, I owe the greatest debt to my wife Hee jung, sweet son, Joowon, and daughter, Jooueun, for their help, interest and concern which enabled me to reach this stage. V TABLE OP CONTENTS Page LIST 0! TABLES ........................................ viii LIST 0! PIGURES ....................................... ix LIST OT APPENDIX ...................................... xii I. ImODUCTION ......OOOOOCOOOOOOOOO......OOOOOOOOOO 1 .5 II. Lxrmm m1" ......OOOOOOOOOOOOOOOOC0.......0 N O p... Oxygen permeability of polymeric packaging material .................................... 4 2.2 Oxygen diffusion in aqueous solution ........ 7 2.3 Oxygen solubility in water .................. 8 2.4 Ascorbic acid diffusion in aqueous solution.. 9 2.5 Ascorbic acid ............................... 9 2.6 Assay of ascorbic acid in fruit juice ....... 10 2.7 HPLC analysis ............................... 11 2.8 Ascorbic acid oxidation ..................... 12 2.9 Influence of dissolved oxygen on ascorbic acid oxidation .............................. 14 2.10 Kinetics of ascorbic acid oxidation ......... 15 2.11 Simultaneous oxygen diffusion and oxidation model ....................................... 18 2.12 Finite difference method .................... 20 III. MATHEMATICAL MODEL AND COMPUTER PROGRAM .......... 22 3.1 One dimensional diffusion ................... 23 3.2 Three dimensional diffusion ... ..... ......... 28 3.3 computer program ......CCOOOOI......OOOOOOOOC 33 Iv. ”Tnxnsmmons ......OOOOOOOCOOOOOOOCO.... 35 4.1 Oxygen partial pressure measurements .. ..... . 35 4.2 Kinetic reaction rate measurement ........... 37 4.3 Ascorbic acid assay ......................... 39 4.4 Experimental materials ...................... 41 4.5 Exposure cells .............................. 42 4.6 Statistical analysis ........................ 45 v. RBBULTBANDDIBCUBBION 000............COOOOCOOCOOO 46 5.1 Comparison between analytical solution and finite difference solution .................. 47 5.2 Experimental determination of kinetic rate constant .................................... 53 vi .3 Standard curve of ascorbic acid ............. 57 .4 Model food system analysis .................. 6O .5 Apple juice analysis ........................ 68 .6 Effects of physical properties of apple juice and packaging material on shelf life ........ 86 VI. CORCLUBION ......OOOOOCOOO......OOOOOOOOOOO0.... 93 ”szns 0.00.... ...... ......OOOOOOOOOOOOOOOOC...... 94 name‘s 00.0.0.0...0.0.000.........OOOOOOOOOO...... 124 vii Table 3.1 5.1 LIST OF TABLES Parameters needed for computer program .......... Parameters used in the one dimensional .......... open system for the model liquid system Parameters used in the one dimensional .......... closed system for the model liquid system Parameters used in the three dimensional......... closed system for the model liquid system Parameters used in the one dimensional .......... open system for the apple juice Parameters used in the one dimensional .......... closed system for the apple juice Parameters used in the three dimensional ........ closed system for the apple juice viii page 33 61 65 69 74 78 82 LIST OF EIGURES Figure page .1 Finite difference method ............... ........... 21 .1 Diagram of one dimensional diffusion cell ......... 22 (open system) 3.2 Diagram of one dimensional diffusion cell ......... 27 (closed system) 3.3 Three dimensional diffusion cell .................. 28 (Block shape) 4.1 Oxygen meter calibration diagram ................. 37 4.2 Measurement of kinetic reaction constant .......... 38 4.3 Cell for one dimensional oxygen diffusion .. ...... . 42 (open system) 4.4 Cell for one dimensional oxygen diffusion .. ....... 45 (closed system) 4.5 Cell for three dimensional oxygen diffusion ....... 45 (close system) 5.1 Comparison between analytical solution and ........ 52 finite difference solution 5.2 Linear plot presenting a first order rate equation for degradation of ascorbic acid in model liquid system ............... .......... ................. 54 5.3 Linear plot presenting a first order rate equation for degradation of ascorbic acid in model liquid system (k1)........................................ 54 5.4 Linear plot presenting a first order rate equation for degradation of ascorbic acid in model liquid system (k2)........................................ 55 5.5 Linear plot presenting a first order rate equation for degradation of ascorbic acid in apple juice.... 55 5.6 Linear plot presenting a first order rate equation for degradation of ascorbic acid in apple juice (k1) 56 5.7 Linear plot presenting a first order rate equation for degradation of ascorbic acid in apple juice (k2) 56 5.8 HPLC chromatogram of ascorbic acid for model liquid system ............................................ 58 5.9 HPLC chromatogram of ascorbic acid for apple juice. 58 5.10 Standard curve of ascorbic acid for model liquid... 59 5.11 Standard curve of ascorbic acid for apple juice.... 59 5.12 Concentration of ascorbic acid remaining at different position (3 cm, 6 cm, 9 cm depth) for the one dimensional open system................ 62 5.13 Comparison between experimental data and computer simulated data (concentration) for the one dimensional open system ........................... 62 5.14 Comparison between experimental data and computer simulated data (percent of ascorbic acid) for the one dimensional open system................ 63 ix 5.15 5.16 5.32 5.33 Concentration profile of oxygen for the one dimensional open system ........................... 63 Concentration of ascorbic acid remaining at different position (3 cm, 6 cm, 9 cm, 12 cm depth) for the one dimensional closed system.............. 66 Comparison between experimental data and computer simulated data (concentration) for the one dimensional closed system ......................... 66 Comparison between experimental data and computer simulated data (percent of ascorbic acid) for the one dimensional closed system.............. 67 Concentration profile of oxygen for the one dimensional open system ........................... 67 Comparison between experimental data and computer simulated data (concentration of ascorbic acid at 2. 6 cm) for the three dimensional closed system.... 70 Concentration profile of ascorbic acid at 2.6 cm depth at 6 hrs for the three dimensional closed system ............................................ 70 Concentration profile of oxygen at 2.6 cm depth at 6 hrs for the three dimensional closed system........ 71 Concentration profile of ascorbic acid at 2.6 cm depth at 12 hrs for the three dimensional closed system ............................................. 71 Concentration profile of oxygen at 2.6 cm depth at 12 hrs for the three dimensional closed system......... 72 Concentration of ascorbic acid remaining at different position (3 cm, 6 cm, 9 cm depth) for the one dimensional open system................ 75 Comparison between experimental data and computer simulated data (concentration) for the one dimensional open system ........................... 75 Comparison between experimental data and computer simulated data (percent of ascorbic acid) for the one dimensional open system................ 76 Concentration profile of oxygen for the one dimensional open system ........................... 76 Concentration of ascorbic acid remaining at different position (3 cm, 6 cm, 9 cm, 12 cm depth) for the one dimensional closed system.............. 79 Comparison between experimental data and computer simulated data (concentration) for the one dimensional closed system ......................... 79 Comparison between experimental data and computer simulated data (percent of ascorbic acid) for the one dimensional closed system.............. 80 Concentration profile of oxygen for the one dimensional open system ........................... 80 Comparison between experimental data and computer simulated data (concentration of ascorbic acid at 2.6 cm) for the three dimensional closed system.... 83 5.35 5.36 5.37 5.38 Concentration profile of ascorbic acid at 2.6 cm depth at 6 hrs for the three dimensional closed system ............................................ Concentration profile of oxygen at 2.6 cm depth at 6 hrs for the three dimensional closed system........ Concentration profile of ascorbic acid at 2.6 cm depth at 12 hrs for the three dimensional closed system ............................................. Concentration profile of oxygen at 2.6 cm depth at 12 hrs for the three dimensional closed system...... The effect of diffusion coefficient of oxygen in liquid food (all input data is the same as Table 5.3 except for diffusion coefficient of oxygen) ... The effect of diffusion coefficient of oxygen in liquid food (Percent remain of ascorbic is the value at 12 hours) ................................. The effect of diffusion coefficient of ascorbic in liquid food (all input data is the same as Table 5.3 except for diffusion coefficient of ascorbic acid in liquid food) ............................... The effect of diffusion coefficient of ascorbic acid in liquid food (Percent remain of ascorbic is the value at 12 hours) .......................... The effect of permeability constant of packaging material (all input data is the same as Table 5.3 except for diffusion permeability constant of packaging material) ............................... The effect of permeability constant of packaging material (Percent remain of ascorbic is the value at 12 hours) ................................. The effect of kinetic reaction constant of ascorbic acid in liquid food (all input data is the same as Table 5.3 except for kinetic reaction constant)..... The effect of kinetic reaction constant of ascorbic acid in liquid food (Percent remain of ascorbic is the value at 12 hours) ..................... ..... xi 83 84 84 87 88 88 89 90 9O LIST OF APPENDIXES Appendix page 10 computer program 000.0.....0. ..... ......00.... 94 xii I . INTRODUCTION Liquid packaged foods may undergo spoilage due to several deteriorative mechanisms. These mechanisms affect the quality due to degradation of nutrients. The presence of dissolved oxygen in liquid packaged foods can cause several deteriorative problems.‘ Oxygen may cause change in color, nutritional value and flavor of liquid packaged foods. The retention of L-ascorbic acid (Vitamin C) in liquid packaged foods may be important as an indicator of product quality of liquid packaged foods because it is a highly soluble compound that has acidic and strong reducing properties, but may be rapidly oxidized in the presence of dissolved oxygen (Fenema, 1986). Quality prediction models can save time and money, therefore, many prediction models have been developed since 1945. However, when many parameters are involved and the physical system is complex, these models can be difficult to use. Very few models have been developed which consider oxygen diffusion with simultaneous chemical reaction. Complicated simulation generally relies on computers and numerical techniques such as finite difference and finite element method in order to obtain good approximation values of a desired quality related parameter. Little has been done with simultaneous oxygen transfer and oxidation in liquid foods. 2 Major emphases of this research have been to develop computer programs that can mathematically predict the quality history of a liquid food, and to validate the computer program through experimentation. This research was designed to assist in evaluation of storage stability for new product package combinations. This information is helpful to the product manufacturer in selecting appropriate packaging material for new or existing products. A second use of the simulation is the capability to predict the ascorbic acid concentration at any time during storage and use of such information in labeling. A third use of the simulation is the capability to evaluate the effect of some physical properties of liquid food such as diffusion coefficients of oxygen and ascorbic acid, and kinetic reaction rate, etc. on shelf life of liquid packaged foods. For the purpose of this study, L-Ascorbic acid (Vitamin C) was selected as the quality component and apple juice was selected as the liquid food. The storage temperature in this study was maintained at 25 r 1°C. Kinetic reaction constant of ascorbic acid was measured and all other constants used in the computer simulation model came from published data. The procedure for using this information should be applicable to other liquid and solid foods, with minor modification. 3 The specific objectives of this research were : 1. To develop a mathematical model to analyze simultaneous oxygen diffusion and oxidation of ascorbic acid in a liquid food system. 2. To develop computer programs to predict loss of quality. 3. To validate the models and computer programs through experimentation. 4. To simulate the influence of physical properties, such as diffusion coefficient of oxygen and ascorbic acid, kinetic reaction rate of ascorbic acid, and permeability of packaging material, on the shelf life of a liquid food system. II. LITERATURE REVIEN 2.1. Oxygen Permeability of Polymeric Packaging Material Prolonging the consumer-acceptable shelf-life of food products is a major benefit generated through the use of packaging. The first step in defining packaging requirements is quantifying the transport properties and the critical vectors of quality loss as well as the variables that influence them (Rizvi, 1981). Two usual types of mechanisms of mass transfer through materials are capillary flow transport and activated diffusion. Capillary flow involves molecules traveling through pinholes and/or high porous media such as paper, glassine, cellulosic membranes, etc. Activated diffusion consists of solubilization of the penetrants into an effectively non-porous film at the inflow (upstream) surface, diffusion through the film under a concentration gradient and release from the outflow (downstream) surface at the lower concentration (Stannett and Yasuda, 1965). The mass transport of a penetrant basically includes three steps: adsorption, diffusion, desorption. Adsorption and desorption depend upon the solubility of the penetrant in the film, i.e. the thermodynamic compatibility between the polymer and the penetrant. Diffusion is the process by which mass is transported from one part of system to another 4 5 as a result of random molecular motion. Within the polymer matrix, the process is viewed as a series of activated jumps from one "cavity" to another. Qualitatively, the diffusion rate increases with the increase of the number of size of cavities caused by the presence of substances such as plasticizer. Structural entities such as crosslinks or degree of crystallinity decrease the size or number of cavities and thereby decrease the diffusion rate (Roger, 1985). At constant temperature and differential partial pressure, diffusion of a penetrant through a polymeric film of unit area normal to the direction of flow for a period of time leads to a steady state diffusive flux, J, J = Q / A t (1) where Q is the total amount of penetrant which has passed through surface area A in time t. Fick's first law of diffusion also applies : J = - D (dC/dx) (2) where x is the space coordinate normal to the reference place; C is the concentration of the diffusing penetrant; and D is the diffusivity, assumed independent of x, t, or C. For experimental and predictive purposes, the diffusing concentration , C, is usually related to the ambient penetrant concentration, c*, in contact with the polymer surface by the Nernst distribution function C = Kc* (3) 6 where k is the distribution coefficient and is a function of temperature. Penetrant concentration , C is also proportional to pressure p through an appropriate gas law equation. For example, when Henry's law is obeyed (CspS), it fellows that the steady state flux can be written as J = DS(p1 -p2)/L (4) where p1 and p2 are the respective upstream and downstream pressure on a film of thickness L, and S is the Henry's law solubility coefficient, assumed independent of p and C. By defining the permeability constant, Pm, as the product of solubility and diffusivity, equation 4 is used for evaluating the permeability constant, ( 5 ) PD=DS= £9 1 .Atta-pz This is the basic equation for determining the permeability constant. However, D and 8 generally vary with C,p,x,or t so that Pm will also be dependent on those variables (Rizvi and Chao, 1988) Permeation of gases and vapor in polymeric food packaging materials is of great importance due to its direct effect on the preservation of foods. There are three methods commonly used to measure the permeability of plastic films; 1) the absolute pressure method; 2) the isostatic method; and 3) the quasi isostatic method (Giacin, 1991). 7 2.2. Oxygen Diffusion in Aqueous Solution Oxygen diffusion coefficient is the key parameter necessary to determine oxygen absorption by liquid foods, and is essential whenever mathematical modeling is desired. Their magnitude is important since it may vary significantly from one food to another, and may be quite different in value when compared to non-gaseous components. There are two basic procedures to measure gaseous diffusion coefficient in liquid (Goldstick and Fatt, 1970): The steady state (liquid jet and diaphragm cell), and the unsteady state (diffusion into non-moving liquid). A major disadvantage of steady state method is that it yields only the product of the diffusion coefficient. Devices also need to be calibrated with a material of known coefficient. A great advantage of the unsteady state method is that the rate of absorption (of oxygen) by a still layer of liquid yields an absolute value of the diffusion coefficient (Crank, 1956). Gas solubility data is not required to analyze gas concentrations as a function of time. Hayduk and Laudie (1974) demonstrated that diffusivity coefficients of liquids, gases, and solids, dissolved in water, can be predicted by any of three mathematical correlations, with an averaging error of 9.5 %. Cussler (1976) reported oxygen diffusion coefficient of 2.1 x 10'5 8 cmZ/sec for water at 25°C. Sadler (1984) obtained an oxygen diffusion coefficient of 1.65 x 10'5 cm2/sec for orange juice and 1.71 x 10'5 cmz/sec for tomato juice at 25°C. Barron (1993) measured oxygen diffusion coefficient in water as 2.0 x 10‘5 cmz/sec, and for apple juice as 1.4 x 10'5 cmz/sec at 25°C. 2.3. Oxygen Solubility in later Since the experimental oxygen concentration values were obtained from oxygen partial pressures (atm), conversion from oxygen partial pressure (atm) to dissolved oxygen concentrations (mg/ml) was necessary. In this study, Henry's law constant was utilized for the conversion of oxygen partial pressure into dissolved oxygen concentrations in the liquid as follows (Battino and Clever, 1966): Pg g “9 XL p9 = HL CL C = Hc CL where pg is oxygen partial pressure of gas (mmHg), XL is mol fraction of oxygen, CL is concentration of oxygen in the liquid phase, C9 is concentration of oxygen in the gas phase, BL is Henry's law constant (mmHg), ML is Henry's law constant (mmHg)(L of solvent)/(mol of gas), and He is Henry's law constant (dimensionless constant) At 25°C, 9 4.33 x 10" mmHg 2|: II 24.6 mmHg.mL/mg =1: I" ll 32.14 (dimensionless) :1: 0 II 2.4. Ascorbic Acid Diffusion in Aqueous Solution The ascorbic acid diffusion coefficient may have a effect on the self life of juice product. Loncin (1975) reported the diffusion coefficient of ascorbic acid in water as 0.84 x 10'5 cmZ/sec at 25°C and it was compared with literature and showed very good correlation. 2.5 Ascorbic Acid L-Ascorbic acid is a highly soluble compound that has both acidic and strong reducing properties. These qualities are attributable to its enediol structure, which is conjugated with the carbonyl group in a lactone ring. The natural form of the vitamin is the L-isomer; the D-isomer has about 10% of the activity of the L-isomer and is added to foods for nonvitamin purposes (Fenema, 1986). L-Ascorbic acid is an important vitamin having a chemical structure that justifies its classification as a carbohydrate. Its physical/chemical properties, complied by Kutsky (1973), are as follows: aqueous solubility is 0.3 g/mL, melting point is 190 - 192°C, redox potential (E0) is 0.166 V at pH4, pKal is 4.17, pKa2 is 11.57, and absorption maxima is 245 nm (acid) and 265 nm (neutral) 10 2.6 Assay of Ascorbic Acid in Fruit Juice Over the last decade, numerous publications have described analytical methods for ascorbic acid in food products, and biological samples. Most are classified into 4 major sections: spectroscopic, electrochemical, enzymatic, and chromatographic methods of analysis (Pachla and Reynolds, 1985). The classical chemistry associated with spectroscopic methods can be divided into 2 general categories: (a) those using a redox indicator in its oxidized form, and (b) those involving chromogen formation by derivatization. The redox indicators that will be discussed include 2,6- dichlioroindophenol, metal ions, and other miscellaneous reagents. The derivatization of ascorbic acid includes reaction with 2,4-dinitrophenylhydrazine, diazotization, and quinoxaline formation (Pachla and Reynolds, 1985). The electro-oxidation of ascorbic acid follows an irreversible, EC type of electrode mechanism, involving the loss of 2e” and 2H+ to form dehydroascorbic acid (Perone and Keeflow, 1966). This product reacts with ascorbic acid rapidly via an irreversible hydration to form diketogluconic acid. Ascorbic acid has a rapid heterogenous rate constant with many types of electrodes; therefore, it is not surprising to find many electroanalytical techniques. Enzymatic methods using ascorbic acid oxidase relied on ll isolation of the enzyme from plant material. Few practical methods for determination of ascorbate have been reported, because enzymes were not commercially available. Liu and coworkers (1982) have used commercially available ascorbic acid oxidase to develop a suitable method for clinical serum or plasma samples. Many of the spectroscopic and electrochemical methods can not distinguish ascorbic acid, dehydroascorbic (DHAA), d-isoascorbic (IAA) or other oxidizable compounds. Chromatographic methods (paper and thin layer chromatography, gas chromatography, and liquid chromatography) can detect them successfully. 2.7. SPLC Analysis HPLC using strong anion-exchange or reversed-phase columns and ultraviolet (UV) detection has been reported to be a rapid and sensitive method for analyzing juices and other food products for ascorbic acid (Sood et al., 1976; Rouseff, 1979; Shaw and Willson, 1982; Haddad and Lau, 1984; Wimalasire et al., 1984; Wilson, et al., 1987; Zapata and Dufour, 1992). Recoveries of added ascorbic acid determined by HPLC were generally quite good as were comparisons of HPLC values with those obtained by a standard titration method (Haddad and Lau, 1984; Augustin et al., 1981; Rouseff, 1979). 12 2.8. Ascorbic acid Oxidation L-Ascorbic acid is highly sensitive to various modes of degradation. Factors that can influence the nature of the degradative mechanism include temperature, salt and sugar concentration, pH, oxygen, enzymes, metal catalysts, initial concentration of ascorbic acid, and the ratio of ascorbic acid and dehydroascorbic acid (Fennema, 1985). The loss of ascorbic acid potency in processed juice products will primarily follow two pathways; aerobic pathway ' and anaerobic pathway (Fennema, 1985). Aerobic and anaerobic degradation of ascorbic acid in an aqueous medium have been proposed by Bauerfeind and Pinkert (1970). Although the pathways in apple juices has not been studied, it is reasonable to assume that the pathways shown below would occur in apple juice. 0’0 co OOH Hog} 0 030-] 0:8 - H00 2 0:0} 0=¢ . -—> ——) OH Hg; Ht; HQOH ——> (I HocH “00.“ noon 0 CHO -CHZOH CHZOH éHzoa HF . AA DHA DKA lonoerobic \ coou g-OH HC-OH ’CHo Q-OH é-OH ' OHOH I. .I CHOH —’ OHOH’COZ—i ¢HOH “"7 (J’LCHO OHOH cHon cHou Furfurol CHZOH CHZOH- CHzOH 13 The aerobic pathway requires oxygen which may enter packaged juice products in four ways: (1) it can be dissolved in the product; (2) it can penetrate into the product through the packaging material; (3) it can dissolve into the product from the package headspace; or (4) it can come from the decomposition of residual hydrogen peroxide. The anaerobic pathway does not require oxygen and is related to the specific characteristics of the product. Predominant factor influencing anaerobic degradation is temperature (Nagy, 1980). The most important step in the aerobic pathway is the reversible formation of dehydroascorbic acid which is further oxidized to 2,3-diketogulonic acid (DKGA). This causes the loss of vitamin activity, i.e., nutritional value. Subsequent oxidation of DKGA produces brown pigments, typically associated with vitamin C loss in fruit products (Mazur and Harrow, 1971). The anaerobic pathway could also contribute to ascorbic acid degradation in the presence of oxygen, though the uncatalyzed oxidative rate is very much greater than the anaerobic rate at ambient temperatures. Therefore, both pathways may be operative in the presence of oxygen, with the oxidative pathway dominant. Storage studies (Kefford et al., 1959: Nagy and Smoot, 1977) on the loss of ascorbic acid in canned, single strength orange juice have shown an 14 initial period (about 1-2 weeks) of rapid loss of ascorbic acid caused by the presence of free oxygen. Although vacuum deaeration during juice processing substantially reduced the oxygen content of the product, there was still some oxygen dissolved in the juice (about 0.05%) (Nagy, 1980). After free oxygen is consumed, ascorbic acid degrades anaerobically at rates lower than by the aerobic process. 2.9. Influence of Dissolved Oxygen on Ascorbic acid Oxidation In aqueous solutions, ascorbic acid degrades quickly, especially in the presence of dissolved and headspace oxygen. One of the earliest investigations on the influence of dissolved oxygen on ascorbic acid degradation was by Guthrie et al. (1938). They observed that the destruction of ascorbic acid was caused by the dissolved oxygen in the milk. Sharp et al. (1940) developed a continuous milk deaerating unit to prevent oxidized flavor and ascorbic acid oxidation. The degradation of ascorbic acid during storage was observed by Beattie et al. (1943). They noted that in several fruit juices the changes in color occurred concurrently with progressive losses of ascorbic acid during storage. Nebesky et al. (1949) reported that heat and oxygen content were the most specific accelerating agents responsible for deterioration of color during storage. Deaerated juices stored at 21.1 to 26.7°C retained their 15 color and exhibited very little change during storage for six months. Bayes (1950) studied the influence of oxygen on ascorbic acid in aqueous standardized solutions stored for 6 weeks at 35°C. He obtained fairly good agreement between the experimental and theoretical values of ascorbic acid and found that they were directly proportion to the amount of available oxygen in the containers. 2.10 Kinetics of Ascorbic Acid Oxidation The kinetics of the oxidative breakdown of ascorbic acid to dehydroascorbic acid and other further breakdown products are not well understood. In addition to being dependent on light , pH, and trace metals the rate of ascorbic acid oxidation appears to be dependent on the dissolved oxygen concentration. Theoretically, 1 mL of oxygen reacts with 15.7 mg of ascorbic acid (based on one mole of ascorbic acid combining with one atom of oxygen) (Bayes, 1950). This is equivalent to reaction of 3.3 mg of ascorbic acid with 1 mL of air. Literature on the order of the reaction with respect to ascorbic acid concentration is conflicting. Barron et al. (1936) described the reaction as zero order; Schuemmer (1940) reported the reaction to be pseudo-unimoelcular; Weissberger at al. (1943) expressed results as first order 16 constants; Peterson and Walton (1943) reported the reaction to be first order for 50 to 80% of the reaction below pH 7 and above pH 11. Dekker and Dikinson (1940) used first order constants, but noted a drift. Joslyn and Miller (1949) indicated that the reaction was first order. Timberlake (1960) studied the stability of ascorbic acid in black currant juice and reported that the reaction appeared to be first order and this was confirmed by the independence of the half life period on the initial ascorbic acid concentration. Khan and Martell (1967), and Blaug and Hajratwala (1972), respectively, reported that the oxidation of ascorbic acid followed first order reaction. Singh, (1976) stated that the overall reaction of ascorbic acid and oxygen was confirmed to be a second order reaction under limited dissolved oxygen pressure in infant formula. Eison- Perchonok and Downs (1982) reported a second order and first order reaction with respect to vitamin C and oxygen respectively. In this experiment, oxygen was replenished continuously through gas flow to maintain dissolved oxygen content. The oxygen dependence of ascorbic acid oxidation has been reported to be first-order above 0.2 atm oxygen (Khan and Martell, 1967). Shtamm and skurlator (1974) have reported a half-order dependence on oxygen, which has been supported by the results of Jameson and Blackburn (1975). Jameson and Blackburn (1975) also suggested that the oxygen dependence data of Khan and 17 Martell (1967) more adequately fits half-order kinetics. First order rate constants were directly proportional to oxygen when oxygen partial pressures ranged from 0.4-1.0 atm. The rate constants were nonlinear with respect to oxygen below 0.4 atm partial pressures oxygen. The second order rate constant was calculated to be 0.568 M31 sec'1 for 0.4 - 1.0 atm partial pressures of oxygen (Blaug and Hajratwala, 1972). Robertson and Samaniego (1986) studied the effect of initial dissolved oxygen level on the degradation of ascorbic acid and browning of lemon juice at 36°C under dark conditions. They observed a rapid disappearance of oxygen in the juice samples, reaching a low level after three days (0.12-0.15 mg/ml). They reported first order constants of 1.67, 1.65, and 1.8 x 10'2/day for juice samples with dissolved oxygen levels of 0.41, 1.44 and 3.74 mg/mL respectively. Barron (1993) reported first order reaction and zero order reaction rate constant in the apple juice with respect to dissolved oxygen concentration. Kinetic rate constants for oxygen were determined liquids supplemented with vitamin C, at 25°C. In a closed system, with initial oxygen levels of 0.21 atm, kinetic rate constant for oxygen yielded a first order rate constant of 0.0049 min."1 for water, and 0.0073 min'1 for apple juice. A zero order rate constant (1.567 x 10" atm/min) was obtained for water with a vitamin content of 11.35 mM. In general, kinetic data for ascorbic acid is expressed 18 as a first order constant. The kinetic reaction constants depend on many factors and are quite variable, therefore, kinetic reaction constants have to be measured for each specific condition. 2.11 Simultaneous Oxygen Diffusion and Oxidation Model Literature citations of oxygen absorption and diffusion, coupled with a chemical reaction is limited in liquid foods. Van de Vusse (1961) dealt with the mathematical analysis of zero order reactions coupled with gas transfer. His theoretical work was because on the fact that most oxidation reactions of liquid hydrocarbons with oxygen belong to zero order type. The resulting mathematical solution to the diffusion equation expressed the concentration of gas (oxygen) and the nongaseous reacting component as a function of time and location. The diffusion and reaction equations were also solved for a semi infinite system by Brian (1964). His proposed general kinetics order "n" was assigned to the gas (oxygen) and order "m" was assigned to the non-gaseous component. Orders "n” and "m" included zero and first order chemical reactions. His mathematical treatment of the diffusion equations illustrates the effect of the chemical reaction kinetic equation upon the gas absorption rate. It also demonstrates its usefulness for determining reaction 19 kinetics from absorption rate measurements (Gilliland, 1958). He concluded that any chemical reaction which appears to have a reaction order less than unity must change at very low concentrations to a reaction order greater than or equal to unity. Hikita and Assi (1964) derived approximate solution equations (semi infinite system) for gas absorption, coupled with an irreversible reaction of order ”n" with respect to a reactant solute. The proposed equations were applicable to a reaction of any order, integer or fraction , and to any system whose reaction velocity is a power function of the concentrations of the reactants. Singh (1976) developed a computer model, based on finite differences, to simulate successfully the oxygen and vitamin C diffusion in an infant liquid formula packaged in glass bottles. Earlier studies present a more detailed mathematical analysis of simultaneous diffusion and reaction in simple solution. Sadler (1984) treated the coupled phenomena as a semi infinite medium in which simultaneous oxygen absorption, diffusion and a first order chemical reaction between dissolved oxygen and vitamin C took place. Oxygen diffusion in the packaging plastic membrane and in the liquid foods were compared in a single weighted diffusion coefficient. Sadler used this approximated coefficient to predict oxygen concentrations in foods. 20 Baron (1993) treated simultaneous oxygen diffusion and oxidation in a packaged apple juice using a commercialized finite element computer program and reported first order and zero order kinetic reaction rate constants for ascorbic acid oxidation in packaged liquid juice. No literature citations have been found in reference to the three dimensional analysis of simultaneous oxygen transfer and oxidation of food components in a packaged liquid food system. 2.12 Finite Difference Method Exact solutions to differential equations governing moisture movement can be obtained for simple geometries; sphere, slab or cylinder (Crank,1975). The solutions can be obtained for each fixed geometry, but when the total package and product of package are considered at the same time, the solution becomes more complicated and often does not exist. Numerical techniques can be utilized for more complex and closer to real life situations. Among numerical methods available to solve the governing differential equations for oxygen transfer are the finite difference and finite element methods. Even though the finite difference method often results in long computation times, it is easier to use and the results are more accurate for simple geometric shape such as sphere and brick types than the finite element method. The finite difference method of analysis is based on approximation by the difference derivative at a point. 21 From Figure 4, the first order differential equation and the second order differential equation can be expressed as follows: (Crank, 1975) Cx+AL *4— Cx-AL / x+AL x x+AL Figure 4. Finite different method The first order differential equation is §£=A€=M£e£§ (a) ax AX x+AL-x AL The second order differential equation can be obtained from eqn.(6). Cx+AL'Cx _ Cx" C'x-AL 62 C a AX A L = char-2 Cx" CX-AL (7) 62:2 AL AL2 into the two main sections, three dimensional model. two sub-sections , III. MATHEMATICAL MODEL AND COMPUTER PROGRAM The mathematical models used in this study are divided a one dimensional model and a The one dimensional model contains one for the open system package and the other for the closed system package. Mathematical models were developed based on the following assumptions: 1) 2) 3) 4) 5) 5) The temperature outside the package is constant. Oxygen is transferred through the polymer packaging material and liquid food product by molecular diffusion for one and three dimensional closed system. Three dimensional unit has a block type. There is no diffusion on top area of three dimensional unit. Initial concentration of oxygen and ascorbic acid in liquid food system are homogeneous. The diffusion coefficient of oxygen and ascorbic acid in liquid food system and oxygen permeability of packaging material depend only on temperature. 22 2 3 3 . 1 . ONE DIMENSIONAL DIPPUSION 3.1.1. WITHOUT PACKAGING MATERIAL (Open system) L x Liquid db Figure 3.1. Diagram of 1-D diffusion cell IH'!IO I J . ! . J . ! 1 Oxygen diffusion Mass balance on elemental volume for oxygen diffusion is; (Rates of mass transfer) - (Consumption) = ( Gain ) into slice Ax rate rate D°A(— 660° x|,,- 96—0 xIN”) -AkC°Ax=A 959° Ax (8) where D° is diffusion coefficient of oxygen (cm2/sec), A is cross section area (cmz) , C° is concentration of dissolved oxygen in liquid (mg/m1), x is distant from liquid surface (cm), L is the depth of liquid in container (cm), K is reaction constant (first order) (sec'l), and t is time (sec). Divide by AAx, (9) 32__q_° -"kC° aC" D 6x231: 24 Boundary condition: t > 0, at x - L, aC°/6x 8 O at x - O, C°=Pout°ln°= constant Initial condition : t = 0, at all x, C° = Ci" , (23° -= constant Where pout° is oxygen partial pressure in air (0.21 atm) , H° is Henry's constant for oxygen (24.6 atm.ml/mg), Ci° is initial concentration of dissolved oxygen in liquid (mg/ml), and (23° is concentration of oxygen in layer in contact with air (mg/ml) . Ascorbic acid diffusion According to first order oxidation of ascorbic acid with oxygen, one mol of oxygen reacts with half mol of ascorbic acid, e.i. , 1 mg/ml oxygen reacts with 11 mg/ml ascorbic acid. Therefore, the kinetic reaction constant of ascorbic acid (Kn) is 11 times higher then that of oxygen, KA = 11 x K The ‘mass balance on. elemental volume for ascorbic acid diffusion is; (Rates of mass transfer) - (Consumption) = ( Gain ) into slice Ax rate rate -DMA ( 53L” |,,-fl |,,,,,,> -11AkC°Ax=AQCfi Ax (1°) 3x 6x at where D"m is diffusion coefficient of ascorbic acid (cmZ/sec)and CAB is concentration of ascorbic acid (mg/ml). Divide by AAx, 25 MyCAA_ K 0: 60M (11) .D —5;3- 11 C? -—52- Boundary condition : t > 0, at x s L, acfih/ax = O at x = O, dam/6x = 0 Initial condition : t =- 0, at all x, CM = C1” where C1“ is initial concentration of ascorbic acid in liquid (mg/m1) - rinits diffsrsncs solution General solutions using the finite difference method are; 62C 3 Conx-z Cx+Cx-Axl (12) ax2 Ax2 6 6C_ Cx+Ax-Cx (13) 55—“ It Be C' . -C E: tact clx (14) oxygen diffusion Applying equations 12,13, and 14 to equation 9 and rearranging, (15) Cox. coAt=G[Cox*Ax, c+C°x-Ax. :1 + (l-ZG-kA t) Cox. 1: where G=D°At/Ax2 . At t > 0, at x = L, ac°/ax = o and at x = o, c° =cH°. 26 Ascorbic acid diffusion Applying equations 12,13, and 14 to equation 11 and rearranging, 16 ch. c+Ac=Q [ CMxOAx, c+CMx~Ax. :1 + (1-20) Cux. t-llkA tcox, t ( ) where Q==D“At/Ax2. At x = L, BCM/ax = O and at x = O, acM/ax = 0. Calculation of the amount of vitamin C consumed by oxidation The rate of consumption per unit volume located at x at time t is; 17 .kC°(x,t) ( ) where k is oxygen reaction constant (sec’l), and CP(x,t) is concentration (mg/ml) of oxygen in liquid at time t. The rate of consumption for the whole liquid at time t is Aka°(x, t) dx (18) Therefore, the amount consumed (Wt) as a function of time is 3L Wt=Afka°(x, t) dxdt (19’ 0 0 where A is cross section area (cm2) and L is the depth of liquid. 27 3 . 1 . 2 . ON! DIMENSIONAL DIFFUSION (with packaging material) Open to air , Packaging T I material L x 1 Liquid Figure 3.2. Diagram of 1-D diffusion cell Wu Mathematical model (Boundary condition) The only difference in the mathematical models for the open system (no packaging material) and the closed system (with packaging material) is in the boundary condition at x=0 in equation 9 for oxygen diffusion. For ascorbic acid diffusion, there is no difference, therefore equation 11 and its boundary condition are valid. Using the permeability constant, the mass balance is; [ Oxygen that diffused ] = [ Oxygen that leaves ] into packaging material packaging material dm=PAAp°=_DOA 6C°| (20) dt TH ax ”’0 where P is permeability (mg.cm/cm2.atm.sec) of packaging material, m is mass of permeant (mg), TH is thickness of packaging material (cm), and Ap° is difference of partial pressure of oxygen (atm) on the inside and outside of the package. 28 P (P ooutside-p Oinside) = _Do Colx-O -CO lx-Ax 2?! Au: (21) where, Col ".0 is concentration of oxygen (mg/ml) at x=o, C o x-Ax pinside=H°CO Ix-O therefore, at x=o, boundary condition is ._ (p ooutside) P(AX) +D°m 0, at x = Lx, 6C°/ax = 0 at x = 0, Co _ pooutside FAX + TH Do Colxan TH D° + P H° Ax at y = Ly, 6C°/ay = at y = O, CO pooutside PAY + TH 00 CLolycAy TH D° + P H° Ax at z = Lz, aC°/az = 0 at z = O, BCO/Bz = 0 Initial condition: t = 0, at all x,y,z, C° 8 C1° where Lx,Ly,Lz is the length, width, depth of rectangular shape container. 3 O 3 . 2 . 2 . Ascorbic acid diffusion lass balance on elemental volume : ( Rates of mass transfer ) - (Consumption) = ( Gain ) into all 6 sides rate rate 60“ ac“ BC“ 6CM 'DMAYAZ ( 1x- If? ImAx) ’DMAXAZ ( 7",- If—ay— In”) -D‘“AxAy( gag; |,--aa£—: IN") -11kC'°AxAyAz 60M (27) at: AxAyAz where C“(x,y,z) is concentration of ascorbic acid in liquid at time t (mg/ml); Divide by AxAyAz, AA AA AA AA D‘“[ 63C + 620 + 32C ]-llKC°= 6C (28) 6x2 Byz 622 at D“V2C“-11KC°= BC“ (29) at Boundary condition : t > 0, at x = L, 6C9A/6x = 0 at x = o, aCM/ax = 0 Initial condition : t = 0, at all x, CM = C1“ 3.2.3. Calculation of the amount of vitamin C consumed by oxidation The rate of consumption per unit volume located at x,y,z at time t is; kC°(x.y. Z) (30) 31 where k is 02 reaction constant (sec'l), and CP(x,y,z,t) is concentration (g/ml) of 02 in liquid at time t. The rate of consumption for the whole liquid at time t is; 1'15ny ffka°(X.y, z, t) dxdydz 000 (31) Therefore, the amount consumed (Wt) for time for block shape is; c‘héflh Wc=f[ffka°(x,y,z, t) dxdydz] dt 0 000 (32) 3.2.6. Finite Difference Hethod For Three Dimensional model General solutions of finite difference method are; 62C: CX’AX,y,z-2Cx'y’z+cx-Ax'y'ZI (33) 3X2 Ax2 ‘ 62C: CX.Y*A}'.z—2Cx,y,z+Cx,y-Ay,zI (34) 3Y2 Ay2 ‘ 62C: Cx'Yv3’A8-2C1:yaz+cx.y.z-Ax I (35) 622 A22 ‘ 3C- Cc+Ac-Ccl (36) E. A t xv)“: Oxygen diffusion Applying’ equations 33,34,35, and 36 to equation. 28 and rearranging, 32 o .. o o o o o C x,y,s,t+At-G [C x+Ax.y,z,t+C x—Ax,y,z,t] +0 [C x,yOAy.s,t o o o +Cox.y-Ay,s,t] +W [C x.y,s¢As,t+C x.y.z-Az.t] + [1-2G°-2o°-2 W°-kA (5] card“, t (37) where G°=D°At/Ax2, Q°=D°At/Ay2, W°=D°At/Az2 Ascorbic acid Applying equations 36,37,38, and. 39 to equation 31 and rearranging, Cuxm. s, t+Ac=GM [Cuonm y, z. t+CMx-Ax.y, z. t] +9“ [Cu ,yoAy, z, t +CMx.y-Ay. z. (:1 +WM [Cuxdn z+As. c+CMx.y. z-Az. :1 + [1-2GM-2QM-2 w“) c'M -11kA tC° ( 33) XoYoZoc x,y,z,: where G“=D“At/Ax2, QAA=DMAt/Ay2, WREDMAt/Azz. 33 3.3. Computer program 3.3.1. Parameters needed for the computer program The mathematical model and finite difference computer program require several input parameters. Most have been published and they were used for program. These parameters are shown in Table 3.1. Table 3.1. Parameters needed for the computer program Parameters References Diffusion coefficient of oxygen in water Cussler, 1976 Diffusion coefficient of oxygen in apple Barron, 1993 juice Oxygen solubility in water Battino, 1966 Diffusion coefficient of ascorbic acid Loncin, 1975 ‘ Oxygen permeability of packaging material Barron, 1993 Kinetic reaction constant of ascorbic acid measured oxidation Initial oxygen concentration in liquid measured Thickness of packaging material measured Initial ascorbic acid concentration in decided liquid Package dimension decided 3.3.2 Description of Computer Program These computer programs were written in Fortran language for each of the different.mathematical model; one-dimensional open system, one-dimensional closed system, and three- dimensional closed system. Main program calls subroutine INPUT to read the input 34 parameters, subroutine INITIAL to set the initial conditions, subroutine BOUND to set the boundary conditions, and subroutine CALCUL to calculate oxygen concentration profile, ascorbic acid concentration profile, and calculate shelf-life of apple juice. IV. NATSRIALS AND NBTNODS Experiments were conducted to validate the computer program based on finite difference method. Therefore, experiments were divided into two sections, one for one- dimensional diffusion experiment and the other for three- dimensional diffusion experiment. The one-dimensional diffusion experiments were conducted to confirm whether the published parameters used in mathematical model explain the simultaneous oxygen and ascorbic acid diffusion and ascorbic acid oxidation mechanism. They also were divided into two sub-experiments, one for the open system to insure all parameters used in the computer program, the other sub- experiment for the closed system to confirm the permeability constant for the given system. The three-dimensional diffusion experiment was conducted to estimate shelf-life of model liquid food system and apple juice packaged in a three dimensional shape. The storage temperature was held constant (at 25 i 1°C) and all samples were stored in the dark. 4.1. Oxygen partial pressure measurements The MI-73O oxygen-electrode (Microelectrodes, Inc.) was used to measure initial oxygen partial pressure in equilibrium with the physically dissolved oxygen in model food system and apple juice. The electrodes have a total 35 36 length of 6.3 cm with an outer diameter at the tip of 0.3 cm, and a response time of less than 20 seconds. The reference electrode was a Ag type. The oxygen electrode is contained in an acrylic housing with a teflon membrane incorporated in the housing tip. Preparation of the electrodes includes addition of the electrolyte solution (provided by Hicroelectrodes, Inc.) into the acrylic housing. Caution was taken to avoid creation of air bubbles that may affect detection. Signal output from the oxygen electrode was transmitted to a biological ox-z oxygen meter (Microelectrodes, Inc.) , which has direct read out in atm or percentage of oxygen. 4.1.1. Calibration of Hicro oxygen electrode Two separate calibration chambers were used, one for the 0% oxygen and another for the sloping gas such as 21%. When setting up the calibration chamber initially, 30 minutes were required to flush each chamber to obtain a steady state oxygen level and a constant temperature. The bubbling rate was carefully regulated (3-6 bubbles per second). Calibrating standards and samples were maintained at the same temperature for accurate oxygen measurements. The tip of the electrode was immersed into the 0% oxygen standard and adjusted to zero after a stable reading was obtained. The electrode was removed from the first standard and placed into the second standard (21% oxygen). 37 Oxygen Oxygen electrode electrode "‘ '1 inlet ‘———————- inlet tube ——————— tube water 0% oxygen 21% oxygen Figure 4.1. oxygen meter calibration diagram The calibration control adjusted to the value of the second standard. This procedure was repeated between the two standards until stability and reproducibity were achieved. 4.2 Kinetic reaction rate measurement The kinetic reaction constant of ascorbic acid oxidation depends on many factors, such as temperature, concentration of ascorbic acid and dissolved oxygen in the liquid etc. No published kinetic reaction constant of ascorbic acid for the model liquid system used (dissolved oxygen concentration; 0.07 atm, ascorbic acid concentration; 0.06mg/ml, at 25°C) was available, therefore, the kinetic reaction rate constant was measured. The oxidation reaction involving ascorbic acid and 3S dissolved oxygen in model food system and apple juice, was expressed in changes in oxygen. In this study, a rate constant in terms of oxygen concentration and first order reaction expression were used because the mathematical model was developed based on kinetic reaction rate expressed in first order of oxygen. . Changes in oxygen were measured in Figure 4.2. In order to prevent oxygen transfer from air, a glass bottle used to contain the model food system and apple juice was immersed into water. Concentration of ascorbic acid in model system was 0.06 mg/ml, initial dissolved oxygen concentration was 0.07 atm, temperature was 25 i 1°C Concentration of ascorbic acid in apple juice was 0.077 and 0.7 mg/ml, initial dissolved oxygen concentration was 0.05, 0.06, 0.14 atm, temperature was 25 i 1°C Oxygen Meter Oxygen Probe Water _J Model Food Apple Juice Figure 4.2. Measurement of oxygen concentration to get kinetic reaction constant 39 4.3. Ascorbiancid Assay The standard method for measuring ascorbic acid in foods is by titration with 2,6-dichlorophenol-indophenol (AOAC, 1990). For juice product, the method can be limited by substances in juice matrix that obscure end point determination (Wilson, 1987). Also this method is limited when the amount of the sample which can be taken is very small (u unit) because it is impossible to measure endpoint. In our studies, the amount of sample is very small (u unit) in order not to cause headspace in diffusion cell. Therefore, the ascorbic acid assay was conducted by highs performance liquid chromatographic (HPLC) method which uses very small sample (n unit) but give very accurate results. 4.3.1. Reagents Analytical grade ascorbic acid (Boxter,IL) and ammonium phosphate monobasic (Boxter,IL) were used. HPLC-grade acetonitrile (EM science Gibbstown, NJ), phosphoric acid 85% (Mallinckrodt, KY), and ultra pure water (Milli-Q water purification system, Millipore, HA U.S.A) were filtered through a membrane filter (Hillipore GV type, average pore size, 0.22pm) and vacuum degassed before use. 40 4.3.2. Chromatographic Conditions The method of analysis was identical with that described by Wimalasiri and Wills (1983). The HPLC system consisted of a M-45 Solvent delivery system (Waters, Milford, MA) , model 06K universal liquid chromatograph injector (Waters Associates), a guard uBondpack C18 pre- column 10pm particle size (waters Associates), uBondpack C18 (3.9 mm x 300 mm) 10 um and 125 A analytical column (waters Associates), a V4 variable wavelength absorbance detector (Isco, Inc, lincoln, NB). Sensitivity of detector was 0.1, rising time 0.8 (sec), wave length 254 nm, and a Peaksimple Data System (SRI instruments, NV) was used with IBM computer for data analysis. Two different mobile phases were used, acetonitrile- water (70:30 v/v) containing 0.01 M ammonium dihygrogen phosphate (pH 4.3 adjusted with orthophosphoric acid) at 2mL/min for model system, 0.1% H3PO4 at 2 mL/min for apple juice. An aliquot of 20 uL was injected for each analysis. 41 4.4. Experimental Materials 4.4.1. Liquid Food 4.4.1.1. Model Liquid System Model liquid system was distilled water and analytical grade ascorbic acid (Boxter, IL) mixture (concentration of ascorbic acid 0.06 mg/mL) to prevent interference of other components in apple juice, such as sugars which may prevent ascorbic acid oxidation in liquid food. 4.4.1.2. Apple juice Glass bottled and pasteurized apple juice was purchased in a local grocery store, carefully selected according to production date and code number to insure sample homogeneity. Juice was kept at room temperature (25 i 1°C) to approximately maintain the environmental shelf conditions found in the grocery store. After opening for sampling, bottle was stored at 5°C to avoid other degradations. 4.4.2. Packaging Material (High density polyethylene) High density polyethylene (HDPE) was used as a packaging material because HDPE has oxygen permeability that allows oxygen to transfer into juice quickly. It was obtained from company, and stored at a constant temperature. 42 4.5. Bxposure Cells 4.5.1. Cell for one dimensional oxygen diffusion (Without Package : open system) The one dimensional oxygen (open system) experiments were performed using an open plastic container (Figure 4.3), 5.5 cm in length, 5.5 cm in width, and 15 cm in depth, with 3 injection ports at 3 cm, 6 cm, and 9cm depth from the bottom. The container was filled with model liquid system and apple juice to known volume (9cm depth). The amount of ascorbic acid was measured using HPLC. Initial low levels of dissolved oxygen were achieved by bubbling nitrogen gas in the samples, prior to initiation of the experiments. During the diffusion experiments, the top surface of the liquid food was in direct contact with air (0.21 atm). The purpose of this experiment was to insure that the experimental data and computer simulated data agreed with each other. Open to air + : Injection 3cm port -—- 3cm + —- 3cm + —- 3cm + —- 3cm Figure 4.3. Cell for one dimensional oxygen diffusion (Without Package : open system) 43 4.5.2. Cell for one dimensional oxygen diffusion (With Package : Closed system) After all parameters (except for permeability of packaging material (HDPE)) were confirmed by open system experiments, the one dimensional oxygen diffusion, closed system experiments were performed using the model liquid food and apple juice, covered with HDPE film (thickness 0.85 mil) in a permeation cell (Figure 4.4) 5.5 cm in length, 5.5 cm in width, and 15 cm in depth, with 4 injection ports at 3 cm, 6 cm, 9cm, and 12 cm depth from the top. HDPE film was tightly glued to the top of the cell to prevent oxygen ingress. To measure the amount of ascorbic acid, two samples (20 uL) were taken at each time and evaluated. The amount of ascorbic acid was measured using HPLC. Initial low levels of dissolved oxygen were be achieved by bubbling nitrogen gas in samples, prior to initiation of the experiments. During the diffusion experiments, the top surface of the juice was in direct contact with the HDPE film with no headspace between the two layers. The purpose of this experiment was to determine if the experimental data and computer simulated data agreed with each other. 44 Packaging Material (HDPE) + : Injection 3cm port + -- 3cm + —~ 3cm + —-- 3cm + —» 3cm Figure 4.4. Cell for one dimensional oxygen diffusion (With Package : Closed system) 4.5.3. Cell for three dimensional oxygen diffusion (With Package : closed system) After confirming all parameters by the one dimensional open and closed system experiments, the three dimensional oxygen diffusion experiment with package (closed system) was performed using model food system and juice packaged in a HDPE film (thickness 0.85 mil) bag, (Figure 4.5), 5.6 cm length, 5.6 cm depth, and 5.6 cm width. In order to maintain the rectangular shape, a plastic frame was configured into the cell. HDPE film was tightly glued to the frame to prevent oxygen leakage. To measure ascorbic acid, two samples (each 20 uL) were taken at each sampling time and the amount of ascorbic acid was measured using HPLC. Initial low levels of dissolved oxygen were achieved by bubbling nitrogen gas into the apple juice sample, prior to initiation of the experiments. During the diffusion 45 experiments, the top surface of the juice was in direct contact with the HDPE film and no headspace was permitted between the layers. The purpose of this experiment was to determine if the experimental data and computer simulated data agreed with each other. 0 Injection port ; , ; é Plastic ‘ ‘ Film Plastic frame Figure 4.5. Cell for three dimensional oxygen diffusion (With Package : open system) 4.6 Statistical Analysis The experiment was designed as a two factor randomized model (replication x time). All determinations were made in triplicate. The mean, standard errors, mean square errors, one factor ANOVA, correlation, and interaction of main effects were done using Super ANOVA software (Abacus Concepts, Inc., Berkeley, CA). Mean separations were performed using LSD with the mean square error term at the 0.05 and 0.1 level of probability. V. RESULTS AND DISCUSSION Results and discussion is divided into several section. The first section is a comparison between finite difference solution and analytical solution. Analytical solution was developed to determine the accuracy of the finite difference solution. The second section provides the results for kinetic reaction constants for model liquid system and apple juice. The third section presents the results for the standard curve for determination of ascorbic acid by HPLC in model liquid system and apple juice. The fourth section presents the results for simultaneous oxygen diffusion and ascorbic acid oxidation of the model liquid system. It is separated into three sub-sections. The first sub-section is a presentation of results for the one dimensional diffusion open system, without package. It also contains the experimental results necessary to confirm specific published and measured input parameters in computer program. The second sub-section presents the results for the one dimensional diffusion closed system, with package. It also contained the experimental results necessary to confirm the permeability of packaging material, which was needed in the computer program. The third sub-section is a presentation of the three dimensional simultaneous diffusion and oxidation experimental results necessary to estimate the 46 47 shelf life of a rectangular shaped packaged liquid food system. The fifth section is a presentation of results for the simultaneous oxygen diffusion and ascorbic acid oxidation in apple juice. It is also separated into the same three sub- sections. The sixth section presents the results related to the effects of physical properties of apple juice such as oxygen diffusion coefficient, ascorbic acid diffusion coefficient, along with permeability of packaging material on the quality of apple juice. 5.1 Comparison between analytical solution and finite difference solution 5.1.1 Analytical solution development The equation describing the simultaneous oxygen diffusion and oxidation in a one dimensional analysis was given in eqn. (9). ac"/X(x) = (1/0) T(t)'/T(t) = -12 then, X(x)"/X(x) - -12 (42) (1/0) T’(%)’mn (57’ 5.1.2 Analytical solution to solve the equation 9 t: c’ -0.34-1 q -0.36v‘*1 'Ir"'l" ‘l 150 180 210 240 270 Time (min) Figure 5.4. Linear plot presenting a first order rate equation for degradation of ascorbic acid (0.06mg/ml) in model liquid system (k2) ln(P/Po) .m\ ~0.1‘ K1 -0.3" i '\ K2'——i -0.5" -0.7 I'I IfiIfiI‘Ifil‘W I/VI/ U 30 60 90 120 150 ISO 210 240 270 300 TuneOnko Figure 5.5. Linear plot presenting a first order rate equation for degradation of ascorbic acid (0.077mg/ml) in apple juice (showing two K values) 56 ln(P/PO) 0.0 43' ’ y . - 7.7057e-3 - 2.518693): R42 8 0.997 -0J‘ 412- \\\\‘J 413- ' K; -4.1 977111 OA(-S)secA(-1) \ '004 ‘- I/ ' ll ' fi/ v 1 fi *1 0 3 O 6 0 9 0 1 20 1 50 Time (min) Figure 5.6. Linear plot presenting a first order rate equation for degradation of ascorbic acid (0.077mg/ml) in apple juice (k1) In(P/Po) -O.45 I I I I y = - 0.24339 ~12007e-3x M2 = 0.993 -0.” - E\ 1 \\ ~O.60" K2=2.0x10*(-5)sec*(-1) l l -o,ss ....,...../....j/....,..... ISO 210 240 270 300 330 'nmeomm) Figure 5.7. Linear plot presenting a first order rate equation for degradation of ascorbic acid (0.077mg/ml) in apple juice (k2) 57 5.3. standard Curve of Ascorbic Acid 5.3.1 For model food system In figure 5.8 is showed the chromatographic resolution for ascorbic acid which has a retention time of 1.35 min under the conditions of this study. Standard curves were made for the model liquid system (mobile phase-acetronitril : water=70:30 v/v) using nine different standard ascorbic acid solutions (0.06, 0.05, 0.04, 0.03, 0.02, 0.01, 0.007, 0.005, 0.0001 ug/uL). Figure 5.10 shows the standard curve for ascorbic acid determination, R2 for this curve was 0.999. Concentration of ascorbic acid (mg/ml) was calculated using the following relationship. WW= Meow re+ -6 (61) ascorbic acid 52261 5.3.2 For apple juice In figure 5.9 is shown the chromatographic resolution for ascorbic acid which has a retention time of 2.01 min under the conditions used in this study. Standard curve was made for apple juice (mobile phase-0.1% H3PO4) using five different standard ascorbic acid solutions (0.10, 0.08, 0.06, 0.04, 0.02 ug/uL). Figure 5.11 shows the standard curve for ascorbic acid determination. R2 for this curve was 1.00. Concentration of ascorbic acid (mg/ml) was calculated using the following relationship.' = W (62) ascorbic acid 49320 58 0 Ce E 1- “E: {1.000 S: I 3 2- '3 c 1 .93 m 3- 0: 4 Figure 5.8 Chromatogram of ascorbic acid by HPLC for model food system 0 g 1 E 1- o I g «¥ c 2.. #2, 2 00 c 1 2 a) 3- CI 4 Figure 5.9 Chromatogram of ascorbic acid by HPLC for apple juice 59 Peak Area 40001 - 24.663 + 5.2261014): 8‘2 = 0.999 3500-. y 3000 '- 2500 - 2000 "' 1500- I 1000- ' 1 500- 0 ‘7' V l U I U I f 1 U ‘ 1 1 I I 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Concentration of ascorbic aiod (mg/ml) Figure 5.10. Ascorbic acid standard curve for model food system Peak Area 5000 - l l ml ‘ y e - 30.097 + 41.9me 3A2 e 1.000 / 4000- 3000‘ 2000 ‘ / 1000- 0 f I T I 2 I fi * I 0.00 0.02 0.04 0.06 0.03 0.10 Concentration of ascorbic acid (mg/ml) Figure 5.11. Ascorbic acid standard curve for apple juice 60 5.4. MODEL 100D SYSTEM ANALYSIS 5.4.1 One-Dimensional Diffusion 5.4.1.1 Without Package Open System Initial ascorbic acid concentration was 0.06 mg/ml and initial oxygen concentration in model food system was 0.057 t 0.003 atm. Figure 5.12 shows that there was no significant concentration difference between different depths (3cm, 6cm, and 9cm from bottom) (p<0.1). This means the concentration of ascorbic acid during storage was homogeneous and the initial oxygen concentration was high enough to prevent a concentration profile from developing due to oxygen diffusion into the model liquid system. All experiments were done three times. Table 5.1 shows the input data for the computer program. The computer program simulated the data and is compared with the experimental data. Figure 5.13 shows the comparison between experimental data and computer simulated data. Figure 5.14 shows the comparison between experimental data and computer simulated data as percent of ascorbic acid remaining in the model liquid system. Among the 2 measured kinetic reaction constants, k1 value is most reasonable. The k1 value has good agreement with experimental data in Figures 5.13 and 5.14 (not significantly different at p<0.1). Figure 5.15 shows the simulated concentration profile of oxygen in model 61 liquid system, which gradually decreased due to oxidation of ascorbic acid . Therefore, all published and measured inputs used for one dimensional open system computer program were confirmed. Table 5.1. Parameters used in the one dimensional open system for the model liquid system. Parameters Values References Diffusion coefficient of 2.1x10‘5 Cussler, 1976 , oxygen in water cmz/sec Oxygen solubility in water 24.6 Battino, 1966 atm.ml/mg Diffusion coefficient of 0.84x10'5 Loncin, 1975 ascorbic acid cmZ/sec Kinetic reaction constant of 0.00002853 measured ascorbic acid sec"1 Initial oxygen concentration 0.057 atm measured Initial ascorbic acid 0.06 mg/ml set-up concentration depth of liquid in container 12 cm set-up 62 Concentration of ascorbic acid (mg/ml) 0.06-II 1 \. -—6— Experiment (8cm) 0. O 5 _ "\\ ‘-0- experiment (6cm) -—l— experiment (3cm) n 0.04 - n \ (L03- , . . . II N. Not significantly dmerent (p<0.05) 8: Significantly diflenent (p<0.05)‘ “I . I n 002 . K' . 1’ . i/ 1 37 0 6 1 2 1 8 2 4 Time (hrs) Figure 5.12. Concentration of ascorbic acid remaining at different position (3 cm, 6 cm and 9 cm depth) for the one dimensional open system. Concentration of ascorbic acid (mg/ml) 0,05 nzNot significantly different (p411) —9— Experiment (90m) ' I » s:Signiiicantiy diflerent (p<0.1) —e— Computer (9cm) §o —l— experiment (6cm) 0 05 _ . -—0— Computer (6cm) . ' —I— experiment (3cm) "\ fl ‘ —O—- Com uter 3cm \ @— P I ) 004~ ‘\\;:::‘~e ‘ n \. § 0.03 - n s 0 02 ' i T I ‘ I ' 4 0 6 1 2 1 8 2 4 Time (hrs) Figure 5.13. Comparison between experimental data and computer simulated data (concentration) for the one dimensional open system. 63 Percent 01 ascorbic acid remaining (16) .. l ‘00 q 11: Not significantly ditterelnt (p< 0.1) s: Significantly diflerent (p< 0.1) . l , 80- \ l—O— Experiment n -—0— Computer \ 30: n \\ 40- WI 8 20 j j v I v j j fl 0 6 12 13 24 Time(hrs) Figure 5.14. Comparison between experimental data and computer simulated data (Percent of ascorbic acid) for the one dimensional open system. Concentration of oxygen (mg/ml) 0.0020 4 —6— Computer (9cm) J —0-— Computer (6cm) I —II-— Computer (3cm) 0.001 0 - 0.0000 . , . , j / ‘ I7 0 6 1 2 1 S 2 4 Time (hrs) Figure 5.15. Concentration profile of oxygen in model liquid system (computer simulated data) for the one dimensional open system. 64 5.4.1.2 With Package Closed System Initial ascorbic acid concentration was 0.06 mg/ml and initial oxygen concentration in model food system was measured to be 0.057 t 0.002 atm. Figure 5.16 shows that there was no significant concentration difference between different depths (3cm, 6cm, 9cm, and 12cm depth from bottom) at p<0.05 . This showed that the concentration of ascorbic acid during storage was homogeneous and initial oxygen concentration was high enough to prevent a concentration profile due to oxygen diffusion into the model liquid system. Table 5.2 shows the input data for computer program. Figure 5.17 shows the comparison between experimental data and computer simulated data in ascorbic acid concentration. Figure 5.18 shows the comparison between experimental data and computer simulated data as percent of ascorbic acid remaining in model liquid system.. Among the 2 measured kinetic reaction constants, k2 is reasonable. The K2 simulated data has good agreement with experimental data in Figures 5.17 and 5.18 (not significantly different at p <0.05). Figure 5.15 shows the simulated concentration profile of oxygen in model liquid system, as it gradually decreased due to oxidation of ascorbic acid. Therefore, all published parameters and measured parameters used for the one dimensional open system computer program were confirmed. 65 All experiments were done triplicate. Table 5.2. Parameters used in the one dimensional closed system for the model liquid system. Parameters Values References Diffusion coefficient of 2.1x10'5 Cussler,1976 oxygen in water cmZ/sec Oxygen solubility in water 24.6 atm.ml/mg Battino,1966 Diffusion coefficient of 0.84x10's Loncin, 1975 ascorbic acid cmz/sec Permeability of packaging 10593.1cc.mil/ Barron, 1990 material In2 . atm day Kinetic reaction constant of 0.00001317 measured ascorbic acid oxidation sec"1 Thickness of packaging 0.85 mil measured material Initial oxygen concentration 0.057 atm measured Initial ascorbic acid 0.06 mg/ml set-up concentration Depth of liquid in container 15 cm set-up 66 Concentration of ascorbic acid (mg/ml) 0.03 J" I 1 —a— Experiment (12cm) —0— Experiment (9cm) --I— Experiment (6cm) 0.05 _ 7 Experiment (3cm) " 4 0.04 - 11: Not significantly different (p<0.05) 3: Significantly dillerent (p<0.05) . I l n "I (L03 - {’ . 34’ . g’ . j . a7 O 6 1 2 1 8 2 4 3 0 Time (hrs) Figure 5.16. Concentration of ascorbic acid remaining -at different position (3, 6, 9 and 12 cm depth) .for the one dimensional closed system. Concentration of ascorbic acid (mg/ml) —e— Experiment (12cm) 0. 06 _Ill —0— Experiment (9cm) \ —I— Experiment (6cm) °\. —0—- Experiment (3cm) :\ -—l— Computer (12cm) 11 —0— Computer (9cm) 0.05 - \\ -—e— Computer (6cm) 004- “\\\“. fl \\ ?\ ‘ 11: Not significantly dillerent (p<0.05) \ , 9: Significantly ditlerent (p<0.05) s \ I I 3 “ (L03 1 {’ e g/ . i/ . 1’ 1 3’ 0 6 12 13 24 30 Time (hrs) Figure 5.17. Comparison between experimental data and computer simulated data (concentration) for the one dimensional closed system. I \\I -—6— Computer (3cm) n \ 67 Percent of ascorbic acid remaining (16) m fl I I 100' n: Not significantly different (p< 0.1) s: Significantly different (p< 0.1) 80 - , . n —-9- ExperimentL [—0-- Computer 60" n C 50 v I 1 fi fi I 3 I I I/ 0 6 12 18 24 30 Time (hrs) Figure 5. 18. Comparison between experimental data and computer simulated data (Percent of ascorbic acid) for the one dimensional closed system. Concentration 01 oxygen (mg/ml) 0.003 - # I —e— Computer (12cm) I —0-— Computer (9cm) d\ + Computer (6cm) 0.002 d -—o—- Computer (3cm) \ 0001- Q ' 0 .000 . . ~ 1 . a . u . :7 0 3 12 13 24 30 Time (hrs) Figure 5.19. Concentration-profile of oxygen in model I liquid system (computer simulated data) for the one dimensional closed system. 68 5.4.1.3 Three dimensional closed System Initial ascorbic acid concentration was set as 0.06 mg/ml and initial oxygen concentration in model food system was measured to be 0.12 1 0.001 atm. All experiments were done four times. Table 5.3 shows the input data for computer program. Figure 5.20 shows the comparison between experimental concentration profile and computer simulated concentration profiles at 2.6 cm depth from top. There was no significant difference (p<0.05) between them. Among the 2 measured kinetic reaction constants, the k2 is reasonable. The computer simulated data with k2 value has good agreement with experimental data. Figures 5.21 and 5.23 show the three dimensional computer simulated concentration profile of ascorbic acid at 2.6 cm depth from the top after 6 hours and 12 hours. These Fiqures indicate that the four corners of the container had the lowest concentration of ascorbic acid and the center of the container has the highest concentration of ascorbic acid. Figures 5.22 and 5.24 show the three dimensional computer simulated concentration profiles of oxygen at 2.6 cm depth from the top after 6 hours and 12 hours. These Figures indicate that the four corners of the container have the highest concentration of oxygen and the center of the container had the lowest concentration of oxygen. 69 Table 5.3. Parameters used in the three dimensional closed system for the model liquid system. Parameters Values References Diffusion coefficient of 2.1x10'5 Cussler,1976 . oxygen in water cm2/sec Oxygen solubility in water 24.6 atm.ml/mg Battino,1966 Diffusion coefficient of 0.84x10's Loncin, 1975 ascorbic acid, cmz/sec Permeability of packaging 10593.1cc.mil/ Barron, 1990 material m2.atm day Kinetic reaction constant of 0.00001317 measured ascorbic acid oxidation sec'l Thickness of packaging 0.85 mil measured material Initial oxygen concentration 0.12 atm measured Initial ascorbic acid 0.06 mg/ml set-up concentration Depth of diffusion cell 5.6 cm set-up Length of diffusion cell 5.6 cm set-up Width of diffusion cell 5.6 cm set-up 70 Concentration of ascorbic acid (mg/ml) I I 0.03 11: Not significantly dillerent (p<0.05) I ‘\“~§§u szspnmaMMquemmtnxoos) - n \ --9— Experiment (Average) \ —0— Computer 005- ”TI” ‘ II 0.04- n W 3 Time (hrs) Figure 5.20 Comparison between experimental data and computer simulated data (concentration of ascorbic acid at 2.6 cm depth from the top) for the three dimensional closed system. Concentration 01 ascorbic acid (mg/ml) N .— .— 8 a! v' in Length(cm) Figure 5.21. Concentration profile of ascorbic acid in model liquid system at 2.6 cm depth from the top at 6 hrs, for the three dimensional closed system. 71 ,_ - 3% 43 ‘IV fi 4" Concentration 01 oxygen (mwhfl Ummhkm) Figure 5.22. Concentration profile of oxygen in model liquid system at 2.6 cm depth from top at 6 hrs, for the three dimensional closed system. > 224 112 Width (cm) Concentratlon of ascorbic acid (mg/ml) N "1 .. 3.36 9 IQ Q' 10 Umcmldn) Figure 5.23. Concentration profile of ascorbic acid in model liquid system at 2.6 cm depth from top at 12 hrs, for the three dimensional closed system. 72 2.24 Width (cm) Concentration 01 oxygen (mg/ml) 1 3.36 0 ‘7. Length (cm) " Figure 5.24. Concentration profile of oxygen in model liquid system at 2.6 cm depth from top at 12 hrs, for the three dimensional closed system. 73 5.5. APPLE JUICE ANALYSIS 5.5.1 One-Dimensional Diffusion 5.5.1.1 Without Package Open System Concentration of ascorbic acid in glass bottled pasteurized apple juice was too high for detection by HPLC, therefore, the ascorbic acid in these samples was destroyed by oxidation in light and a known concentration (0.77 mg/ml) was introduced into the juice. Initial oxygen concentration in apple juice was measured to be 0.07 i 0.002 atm. Figure 5.25 shows that there was no significant concentration difference between different depths (3cm, 6cm, and 9cm from top) at p <0.05. This shows that the concentration of ascorbic acid during storage was homogeneous and the initial oxygen concentration was high enough to prevent from the concentration profile due to oxygen diffusion into the apple juice. All experiments were done three times. Table 5.4 shows the input data for computer program. Figure 5.26 compares the experimental data and computer simulated data in terms of concentration profiles of ascorbic acid . Figure 5.27 compares the experimental data and computer simulated data in terms of percent ascorbic acid remaining in the apple juice. Among the 2 measured kinetic reaction constants, k2 value is reasonable. The k2 value has good agreement with experimental data in both Figure 5.26 and 5.27 (not significantly different at p 74 <0.05). Figure 5.28 shows that the simulated concentration profile of oxygen in apple juice gradually decreased due to oxidation of ascorbic acid . Therefore, all published parameters and measured parameters used for the one dimensional open system computer program were confirmed. Table 5.4. Parameters used in the one dimensional open system for the apple juice. Parameters Values References Diffusion coefficient of 1.4x10'5 Barron, 1990 oxygen in water cm2/sec Oxygen solubility in water 24.6 Battino, 1966 atm.ml/mg Diffusion coefficient of 0.84x10'5 Loncin, 1975 ascorbic acid cm2/sec Kinetic reaction constant of 0.0000200 measured ascorbic acid sec'1 Initial oxygen concentration 0.07 atm measured Initial ascorbic acid 0.077 mg/ml set-up concentration depth of liquid in container 12 cm set-up 75 Concentration of ascorbic acid (mg/ml) 1103 I . . I . I ' 11: Not significantly drilerent (p<0.05) ‘ 8: Significantly diiierent N05) 0.07 - _ —a— (9 cm Depth) \ —+— (s cm Depth) 0.03 - " O, x- , oo-c0 x-L, dC /cix-o I.c : t-O, all x, 80'9“ 1.2. Ascorbic acid diffusion PDE : DA (dzc /de) -kco . dCA/dt a.c : t>0, x- , dCA/dxso st, dc /dx-O I.c : t-O, all x, O -c, it*************.*ttfittflttflfi ii************t*************.* I‘fifififififlififififififififiifififii t t i t t t * * t t t * t t * t t t * * t++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ * MAIN PROGRAM t++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ i * Put declaration, dimension and open statements here. INTEGER I,J,N,NR,22,F1,TT DOUBLE PRECISION O,A,DO,DA,ICO,ICA,HO,XO,CPAC,AR, SL,SOCL,SACL,MOL,MAL,PAC, MOOA,TMOOA,TMOL,TTMOL,MAOA1,MAOA2 + L,GO,GA,DX,DT,VL,F,F2,Z,Zl,Z3,Y,R, + POUT,PIN,DIPF1, + IMOL,IMAL, + + PARAMETER (NR8200,Z-200,N312) DIMENSION O(NR,N+7),A(NR,N+7),PAC(NR), SOCL(N+7),SACL(N+7),TT(NR),SL(NR), MOL(NR), MAL(NR),MOOA(NR),TMOOA(NR),TMOL(NR), TTMOL(NR),MAOA1(NR),MAOA2(NR) +-++-+ OPEN (UNIT-6,?IL88'SOL-10',FORMS'FORMATTED', + SIATUSI'UNKNOWN') warts (6,10) 10 FORMAT ('THIS PRGRAM Is FOR l-D DIFFUSION WITHOUT', + ' pus (OPEN sxsrau)°,//) ttit*ifittfitiittfiitfiifitfiitfit't**********tit‘ttfiiiii’tti’tfitttfl********* * INITIAIION * t0*it.*titfitttfifittfittttflfitittflt*‘i*fittflfittfi********.************** DO 30 I-1,NR,1 PAC(I)-0.00 MOL(I)=0.00 TMOL(I)-0.00 TTMOL(I)80.OO 95 MAL(I)-0.00 MOOA(I)-0.00 TMOOA(I)-0.00 MAOA1(I)-0.0 MAOAZ(I)-0.0 SL(I)-O.OO TT(I)-0.00 80CL(J)-0.00 8ACL(J)-0.00 20 CONTINUE 30 CONTINUE ***t***fi****i**i****i*****t*******.fi***************************** * CALL SUBROUTINE * ******************tfl****itiitttttititfittifit.********************* CALL INPUT (DO,DA,HO,RO,POUT,PIN,ICO,ICA,CPAC,AR,L, + GO,GA,DT,DX,R.VL,N,Z) CALL TIME (2,21,22,23,Y,DT,N,NR) CALL INITIL (O,A,ICO,ICA,I,J,N,NR) CALL BOUNDY (O,A,ICO,IMOL, + VL,ICA,IMAL,J,N,NR, + SOCL,SACL,TMOOA,MOL,MAL,TMOL) CALL CALCUL (O,A,GA,GO,KO,DT,SOCL,SACL,MOL,AR, + DX,MOOA,TMOL,TMOOA,TTMOL,MAL,IMAL, + DIFPI,PAC,CPAC,SL,HO,POUT,IMOL, + I,J,N,NR,P,2,TT,P1,P2,MAOA1,MAOA2) STOP END *###f##f#f######if##f##fI!iIt!################f######f#f#f#ff * SUBROUTINE *###########f####ff###########if#i##f############I######f#f## CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE 8 INPUT C C PURPOSE : TO INPUT DATA C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE INPUT (DO,DA,HO,KO,POUT,PIN,ICO,ICA,CPAC,AR,L, + GO,GA,DT,DX,R,VL,N,Z) DOUBLE PRECISION DO,DA,HO,RO,POUT,PIN,ICO,ICA,CPAC,AR,L, + GO,GA,DT,DX,R,VL,2 INTEGER N titttt*ttt*******tt*ttttttttttttt*tttt*ttttttfitttttttttttttt * Enter the data * *************fl*******iti.**************************fi******** 110 FORMAT(F10.5) PRINT *,'Enter the diffusion coefficient of oxygen ', +'in liquid in x cm*2/sec' READ(5,110)DO PRINT *,'Enter the diffusion coefficient of ', +‘ascorbic acid in liquid in cm‘2/sec' READ(5,110)DA 96 PRINT *,'Enter the Henry constant of oxygen into liquid ', +‘(atm.mL/mg : p-Ro x CL)‘ RERD(5,110)HO PRINT *,'Enter kinetic reaction constant Of oxidation ', +‘of ascorbic acid in l/sec' READ(5,llO)RO PRINT*,'Enter the outside oxygen partial pressure ', +'in atm.‘ READ(5,110)POUT PRINT *,'Enter the intial concentration of oxygen in ', +‘liquid in atm' READ(5,110)PIN PRINT *,'Enter the intial concentration Of ascorbic acid', ‘ +'in liquid in mg/mL' asan(s,110)Ica PRINT*,'Enter critical t loss of ascorbic acid ', +‘in liquid in t' READ(S,110)CPAC PRINT*,'Enter the cross section area liquid in cm‘2 ' READ(5,110)RR PRINT*,'Enter the depth of liquid in cm' 1 READ(5,llO)L f. PRINT*,'Enter the time period for calculation in hours' PJ READ(5,110)Y t************************i***it*fltfifi*****fi****************** * #ff CHECK THE HENRRI'S CONSTANT FOR DIPFUSION OCCURANCE * t***********tfl******tt*******************.*******i********** IOOanN/ao R-ICO/POUT IF (R.GT.HO) THEN PRINT *,'There is no diffusion from haedspace', + 'into liquid because initial concentration' PRINT *,'in liquid is lower than that in headspace' STOP ENDIF Dx-L/N VL-AR*L co-I*DO*36OO/(z*(nx**2)) OTsOO*(Ox)**2/OO GA-DA*DT/(DX)**2 *eeeeeeeeeeseseeeeaestateseasseeeeeeseeeeeeeeeeeeeeeseeeeeas * DISPLA! OF ALL ENTERED DATA AND DIMISSIONLESS CONSTANT * *************iii!********it******fi************************** PRINT *,'Rll data are checked successfully lll' PRINT *,'Re-check all data entered' PRINT *,' ', + ' v NRITE(6,401)DO*100000 401 FORMAT('Diffusion coefficient of oxygen in liquid', + ' in x 10*-S cm‘2/sec -',F15.7) WRITE(6,402)DA*100000 402 FORMAT('Diffusion coefficient of ascorbic acid' , + ' in liquid in x lot-5 cm‘2/sec8',F10.7) WRITE(6,403)HO 403 FORMAT('Henry constant of oxygen into liquid',/, 97 + ' in atm*mL/mg (p-Ho x CL) -',F15.7) WRITE(6,404)RO 404 FORMAT('Oxidation reaction rate of ascorbic acid in', + ' 1/sec (RO) - ',F15.10) wRITs(6,405)POUT 405 FORMAT('Outside oxygen partial pressure in atm', + ' -',F15.7) WRITE(6,406)PIN 406 PORMAT('Initial concentration of oxygen in liquid', + 'in atm -',F15.7) WRITE(6,407)ICO 407 FORMAT('Initial concentration of oxygen in liquid', + 'in mg/mL -',P15.7) WRITE(6,408)ICA 408 FORMAT('Initial concentration of ascorbic acid in', + ' liquid in mg/mLs',FlS.7) WRITE(6,409)AR 409 FORMRT('Cross section area liquid in cm‘2 -',F15.7) WRITE(6,410)L 410 FORMAT('Depth of liquid in cm s',FlS.7) WRITE(6,413)VL 413 FORMAT(‘Volume Of liquid in cm‘3 s',F15.7) PRINT *,' ‘, + I O PRINT *,' Hit any key to continue.....' PRINT *,' ', + ' v PRINT *,'DIMENSSIONLESS CONSTANT' PRINT *' . 0' + ' - WRITE(6,505)GO,GA,DT,DX,N 505 FORMAT('GOI',E10.4,/, +'GA.' pElOe4plg 'DT.' [81°04plp 'DX" '310e4plp ' N-' '15) PRINT *' . 0' + ' . PRINT *,' Hit any key to continue ......' RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : TIME C C PURPOSE : TO DECIDE TIMESTEP TO PRINT C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE TIME (Z,21,22,23,Y,DT,N,NR) DOUBLE PRECISION 2,21,23,Y,DT INTEGER 22,N,NR PRINT *,' .' + ' v 510 WRITE (6,508)NR,NR 508 PORMAT(5X,'Enter the number Of time to calculate', +' concentrations at once.',/,8x,'') 98 PRINT *,' 'r + I WRITE(6, 550) 550 PORHRT(////////////////////////////l 21:2/Nn 22-INT(21) z3-a35(zI-z2) IF((Z.LT.NR).OR.(23.GT.0.0000001)) THEN PRINT *,‘Calculation error happened lll' PRINT *,'Re-check no. of time step.‘ GOTO 510 ELSE Y-z*DT/3600 wRITs(6,512) 512 FORMAT(20X,'CONCENTRRTON PROFILE IN PRODUCT') WRITE(6,513)Z,Y 513 FORMAT(12X,'Each step is',FlO.2,1X,'steps',' (',F10.S, +lx,'hrs)') WRITE(6,514)N 514 FORMAT(14X,'<# of the divided shell of product is', + I4,'>') PRINT*,' ', + ' . PRINT*,'MOL -nass of present 02 (mg)' PRINT*,'MOOA cmass of 02 consumed (mg)' PRINT*,'TMOOA=total mass of OZ consumed (mg)' PRINT*,'DIFF =11*MOOA-MAOA (mg)' PRINT*,'RATE -MAOA/MOOA' PRINT*,'MAL -mass of present R.R (mg)' PRINT*,'MAOA -mass of A.A consumed (mg)' PRINT*,'TMAL stotal mass of A.A consumed (mg)' PRINT*,'t Remain- t conc. of A.A remaining (%)', + ' MOL + TMOOA - IMOL (at t=0)' PRINT*,'S.L- shelf-life of liquid food (hrs)', MAL + TMAL - IMAL (at tsO)‘ ; “Ba—2 ". ‘i la. ~I . ... PRINT*, PRINT*,' Concentration profile in juice (mg/mL) ', + + Calculation' PRINT*, + PRINT *,' COO CO3 C06 C09 MOL ', +' MOOA TMOOA DIFP RATE' PRINT *,' CAO CA3 CA6 CA9 MAL ', +' MAOA TMAL tRemain S.L(hr8)' PRINT*,' ', + U ' ENDIP RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUEROUTINE : INITIL C C PURPOSE : TO INITIATE CONDITION AT T80 C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 99 ***************fi*********************fi*************.***fi********i * INITIAL CONDITION * eeeeeeeeeeeeeeaseeseaeeeeeesaasasteaseassesseseeeeeeeeeeeeeeeeeee SUEROUTINE INITIL (O,A,ICO,ICA,I,J,N,NR) INTEGER I,J,N,NR DOUBLE PRECISION O,A,ICO,ICA DIMENSION O(NR,N+7),A(NR,N+7) DO 30 I-1,NR,1 DO 20 J-1,N+7,1 O(I,J)-IOO A(I,J)-ICA 20 CONTINUE 3o OONTINUE RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : BOUNDY C C PURPOSE : TO GIVE BOUNDAR! CONDITION C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC fi**fi*********tit**ttttii**tifi************************************ * BOUNDARY CONDITION * **********ti***********fi***************iflifiiti*t***************** SUBROUTINE BOUND! (O,A,ICO,IMOL, + VL,ICA,IMAL,J,N,NR, + SOCL,SACL,TMOOA,MOL,MAL,TMOL) INTEGER J,N,NR DOUBLE PRECISION O,A,ICO,IMOL, + VL,ICA,IMAL, + SOCL,SACL,TMOOA,MOL,MAL,TMOL DIMENSION O(NR,N+7),A(NR,N+7),SOCL(N+7),SACL(N+7),TMOOA(NR), + MOL(NR),MAL(NR),TMOL(NR) DO 600 J-2,N+l,1 O(1,J)-Ico A(l,J)-ICA 600 CONTINUE IMOL-ICO*VL IMAL-ICA*VL O(l,1) sIOO O(1,N+2)=IMOL O(l,N+3)-0.0 O(1,N+4)-o.o O(l,N+5)=0.0 O(1,N+6)-0.0 A(l,l) -ICA A(1,N+2)-IMAL A(1,N+3)-0.0 A(1,N+4)=0.0 A(l,N+S)-100.0000 A(l,N+6)-0.0 100 SOOL(1)-o.o SACL(l)-0.0 TMOOA(l)-0.00 MOL(l)-IMOL MAL(1)-IMAL TMOL(I)-O.oo waITE (6,1000)(O(l,J),J-1,N,N/4), + (O(l,J),J-N+2,N+6) waITE (6,1000)(A(1,J),J-1,N,N/4), + (A(1,J),J-N+2,N+6) 1000 FORMAT (rs.5,1x,rs.s,Ix,ra.s,Ix,ra.s,1x,rs.5,Ix,r8.s,1x, + ra.2,1x,ra.2,1x,rs.2,1x) RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : CALCUL C C PURPOSE : TO CALCULATE CONC. AND SHELF-LIFE C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC ***t****************fi****it*******i*************t***i************ * CALCULATION OF CONCENTRATION OP 02 AND A.A IN LIQUID * *******t**************i***********t**********************fi******* SUBROUTINE CALCUL (O,A,GA,GO,RO,DT,SOCL,SACL,MOL,AR, + DX,MOOA,TMOL,TMOOA,TTMOL,MAL,IMAL, + DIPFI,PAC,CPAC,SL,HO,POUT,IMOL, + I,J,N,NR,P,2,TT,P1,P2,MAOA1,MAOA2) INTEGER I,J,TT,P1,N,NR DOUBLE PRECISION O,A,GA,GO,RO,DT,SOCL,SACL,MOL,AR, DX,MOOA,TMOL,TMOOA,TTMOL,MAL,IMAL, DIPFI,PAC,CPAC,SL,HO,POUT,IMOL, Z,P,E2,MAOA1,MAOA2 +++ DIMENSION O(NR,N+7),A(NR,N+7),PAC(NR), + SOCL(N+7),SACL(N+7),TT(NR),SL(NR), + MOL(NR),MAOA1(NR),MAOA2(NR), + MAL(NR),MOOA(NR),TMOOA(NR),TMOL(NR), + TTMOL(NR) TT(l)-l 2000 DO 700 I82,NR,1 DO 710 J-2,N,1 O(I,J)-GO*(O(I-l,J-l)+O(I-1,J+1))+ + (1-2*GO)*O(I-l,J)-RO*DT*O(I-1,J) A(I,J)-GA*(A(I-l,J-l)+A(I-1,J+l))+ + (l-2*GA)*A(I-l,J)-ll*RO*DT*O(I-1,J) .IF(0.00.GT.A(I,J)) THEN A(I,J)-0.00 ENDIF 710 CONTINUE O(I,N+1)=O(I,N) 101 A(I:N+1)'3(I:N) A(I,l)-A(I,2) O(I,1)-O(I,2) ***********************iffli.*tfitfiit********************.**** * It: CALCULATION or CONCENTRATION or 02 IN HEADAPSCE ### * **********ti************ti...*ttifliittttitt...*****O*******i DO 780 J-2,N SOCL(J)-SOCL(J-1)+O(I,J) SACL(J)-SACL(J-1)+A(I,J) 780 CONTINUE MOL(I)-AR*DX*((O(I,l)+O(I,N+l))/2+SOCL(N)) MOOA(I)-KO*DT*MOL(I-l) TMOL(I)-MOL(I)+MOOA(I-l) TMOOA(I)sTMOOA(I-1)+MOOA(I) TTMOL(I)=TMOOA(I)+MOL(I) MAL(I)-AR*DX*((A(I,1)+A(I,N+1))/2+SACL(N)) MAOA1(I)-MAL(I-1)-MAL(I) MAOA2(I)-11*MOOA(I) DIFFl-MAOA2(I)-MAOA1(I) PAC(I)-MAL(I)*lOO/IMAL TT(I)-1+TT(I-l) SL(I)-(TT(I)-1)*DT/3600 O(I,I)-pOUT/MO O(I,N+2)-MOL(I) O(l,N+3)-MOOA(I) C(I,N+4)=TMCOA(I) O(I,N+5)-DIFF1 O(I,N+6)-MAOA1(I)/MOOA(I) A(I,N+2)-MAL(I) A(I,N+3)-MAOA1(I) A(I,N+4)-IMAL-MAL(I) A(I,N+5)8PAC(I) A(I,N+6)=SL(I) IF(PAC(I).GT.CPAC) THEN GOTO 700 ELSE GOTO 5800 ENDIE 700 CONTINUE *fit.*ifi***i***********ti***********ifi***************i******* * if! PRINT THE CONCENTRATION PROFILE IE! * *********fi****fi**************fi********************fi*t******* 2500 F-TT(NR)/z Fl-INT(F) F2-ABS(F-Fl) IF(P2.GT.1/Z) GOTO 3000 WRITE (6,1000)(O(NR,J),J'1,N,N/4), I (O‘NRIJ)IJ'N+21N+6) 102 wRITE (6,1000)(M(NE,J),J-I,N,N/4), + (A(NR,J),J-N+2,N+6) 1000 FORMAT (P8.5,IX,E8.S,IX,PB.S,IX,P8.5,1X,PB.S,1X, +r8e5'1x’r8es'1x'r8e4'1‘pr802) *0***************************t**t***********0*************** * CALCULATION SHELF-LIFE * ****t***fi****************ti**.*****************.********tflit 3000 IF((PAC(NR).GT.CPAC).AND.(PAC(NR).NE.CPAC)) TEEN TT(l)-TT(NR)+1 TMOOA(1)-TMOOA(NR) DO 4100 J-l,N+l O(1,J)-O(NE,J) A(1,J)=A(NR,J) 4100 CONTINUE GOTO 2000 ELSE GOTO 6000 ENDIP ********t********i***t**fi*************t*****t************t** * SHELF-LIFE * **t*fi******t**t***************************t***************** 5800 WRITE (6,1000) (O(I-1,J),J-1,N+1,N/3), (O(I-1,J),J-N+2,N+4, 2) WRITE (6,1000) (A(I-l,J),J-1,N+l,N/3), 6000 PRINT*,' ', + ' ' wEITE(6,6100)SL(I-1),SL(I-1)/24 6100 PORMAT(ZX,'SHELF-LIFE OF THIS PRODUCT IS',1X,P10.S, +'hrs',2x,'[',FlO.5,'daysl') RETURN END ‘ rll r'll Itigb‘flm’w L 103 ti...tit.Gift.*.***.t*i**ttt.........ifitiifittit***tt**...... COMPUTER PROGRAM TO ESTIMATE THE SHELF LIFE OF JUICE One Dimensional Closed system This is a program for l-d diffusion mathematical model with packaging material (closed system) 2.1. Oxygen diffusion PDE : DO (dzcoldxz) -kco - ace/at E.C : t>0, x . 0 o o c° -p mm. PAx 4» TH D C°|qu x-L, dc /dx-O I.C : t-o, 611 x, Eb-CO, 2.2. Ascorbic acid diffusion PDE : DA (dzc /dx2) -kCo - «lg/at B.C : t>0, x3 , dCA/dx-O x-IL, (10 [(12:80 I.C : tao, all x, 6 BC aaseeeuseseeeseaseseesaeee«*0ngseeeeeeeeeeeeeeeseeeeeeeees * i!!! *l-Uitfi UI'UI’IIPU fil'iltfi 0 t t R t t C Q m . I M TED°+FE°Ax a t l I Q a s U fl m t t++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ * MAIN PROGRAM t++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ * Put declaration, dimension and Open statements here. INTEGER I,J,N,NR,22,F1,TT DOUBLE PRECISION O,A,DO,DA,ICO,ICA,NO,NO,CPAC,AR, L,MWO,MWA,GO,GA,DX,DT,VL,P,E2,2,21,23,Y,R, POUT,MAOA,DIFE1, IMOL,IMAL, SL,SOCL,SACL,MOL,MMOL,MAL,MMAL,PAC, MOOA,TMOOA,TMOL,TTMOL, PIN,TH,PP +-+4-+-++ PARAMETER (NR-200,2-200,N-15) DIMENSION O(NR,N+6),A(NR,N+6),PAC(NR), SOCL(N+6),SACL(N+6),TT(NR),SL(NR), MOL(NR),MMOL(NR), MAL(NR),MMAL(NR),MOOA(NR),TMOOA(NR),TMOL(NR), TTMOL(NR),MAOA(NR) +-++-+ OPEN (UNIT36,FILE='SOL-3',FORMt'EORMATTED', + STATUS='UNKNOWN') WRITE (6,10) 10 FORMAT ('THIS PRGRAM IS FOR 1-D DIFFUSION WITH', + ' PKG (CLOSED SYSTEM)',//) ***itittttfi*t*t***t******ttt***ttiii.*tit.*t*****t*********tt**** * INITIATION . * tit...ttittit.*t*t****ttiti***fl*fifiifltttit*********C*.C*********** DO 30 I=1,NR,1 PAC(I)80.00 . p r 6' 104 MOL(I)-0.00 MMOL(I)-0.00 TMOL(I)-0.00 TTMOL(I)-0.00 MAL(I)-0.00 MMAL(I)-0.00 MOOA(I)=0.00 MAOA(I)-0.0 TMOOA(I)-0.00 sL(I)-0.00 TT(I)-0.00 DO 20 JsI,N+6,1 SOCL(J)-0.00 SACL(J)-0.00 20 CONTINUE 30 CONTINUE ************************tt**tiflt***********fi********************* * CALL SUBROUTINE * ***************************************************************** CALL INPUT (D0,DA,HO,NO,POUT,ICO,ICA,CPAC,AR,L,Z, + GO,GA,DT,DX,R,VL,N,MWO,MWA,TH,PP,PIN) CALL TIME (Z,21,22,Z3,Y,DT,N,NR) CALL INITIL (0,A,ICO,ICA,I,J,N,NR) CALL BOUNDY (0,A,ICO,IMOL, + VL,ICA,IMAL,J,N,NR, + SOCL,SACL,TMOOA,MOL,MAL,TMOL) CALL CALCUL (O,A,GA,GO,K0,DT,SOCL,SACL,MOL,AR, + DX,MMOL,MWO,MOOA,TMOL,TMOOA,TTMOL,MAL, + MMAL,MAOA,MWA,DIPP1,PAC,CPAC,SL,NO,POUT, + I,J,N,NR,P,Z,TT,F1,E2,IMAL,TH,PP,DO,PIN) STOP END *#f##########I####II#######fi#############################f## * SUBROUTINE *############i#######II###################################### CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : INPUT C C PURPOSE : TO INPUT DATA C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE INPUT (D0,DA,HO,KO,POUT,ICO,ICA,CPAC,AR,L,Z, + GO,GA,DT,DX,R,VL,N,MWO,MWA,PP,TH,PIN) DOUBLE PRECISION DO,DA,HO,RO,POUT,ICO,ICA,CPAC,AR,L, + GO,GA,DT,DX,R,VL,MWO,MWA,TH,PP,PIN,z INTEGER N ********.*O******ttifi*ttttt********************************* * Enter the data * ********t******************tit*ttttititt******************** MWO-32 MWA-l76.1 ...—mi“. Fii'l—L “KT-T m 1 105 110 FORMAT(F10.5) PRINT *,'Enter the diffusion coefficient of oxygen ', +'in liquid in cm‘2/sec' READ(S,110)DO PRINT *,'Enter the diffusion coefficient of ', +'ascorbic acid in liquid in cm‘2/sec' READ(5,110)DA PRINT *,'Enter the Henry constant of oxygen into liquid ', +'(atm.mL/mg : p-Ho x CL)‘ READ(5,110)HO PRINT *,'Enter the oxidation of ascorbic acid reaction ', +'rate in 1/sec' READ(5,110)xO PRINT*,'Enter the outside oxygen partial pressure ', +'in atm.‘ READ(S,110)POUT PRINT *,'Enter the intial concentration of oxygen in ', +'liquid in atm' READ(5,110)PIN PRINT *,'Enter the intial concentration of ascorbic acid', +'in liquid in mg/mL' READ(5,110)ICA PRINT*,'Enter the permeability of PEG in mg.cm/cm‘2.atm.sec' READ(5,110)PP PRINT*,'Enter the thickness of PEG in cm' READ(S,110)TH PRINT*,'Enter critical t loss of ascorbic acid ', +'in liquid in t' READ(5,110)CPAC PRINT*,'Enter the cross section area liquid in cm ' READ(S,110)AR PRINT*,'Enter the depth of liquid in cm' READ(5,110)L PRINT*,'Entet the time period for ccalcualtion' READ(S,110)Y **********************t**********i*i************************ * ##f CHECK THE HENRRY'S CONSTANT FOR DIFFUSION OCCURANCE * ****************************************************fi******* ICOcFIN/EO R-ICO/POUT IF (R.GT.HO) THEN PRINT *,'There is no diffusion from haedspace', + 'into liquid because initial concentration' PRINT *,'in liquid is lower than that in headspace' STOP ENDIF Dx-L/N VL-AR*L 00-DO*3600*I/(z*(Dx**2)) DT-GO*(DX)**2/DO GA-DA*DT/(DX)**2 **t*******************i**t*i**********t********fi************ * DISPLA! OF ALL ENTERED DATA AND DIMISSIONLESS CONSTANT * it********t**************t**t************************fi****** PRINT *,'All data are Checked successfully 111' PRINT *,'Re-check all data entered' PRINT *" " 106 + I I NRITE(6,401)DO*100000 401 FORMAT('Diffusion coefficient of oxygen in liquid', + ' in x 10*-5 cm‘2/sec -',F15.7) WRITE(6,402)DA*100000 402 FORMAT('Diffusion coefficient of ascorbic acid' , + ' in liquid in x 10*-5 cm‘2/sec-',F10.7) NRITE(6,403)HO 403 FORMAT('Henry constant of oxygen into liquid',/, + ' in atm*mL/mg (p-Ho x CL) -',F15.7) WRITE(6,404)KO 404 FORMAT('Oxidation reaction rate of ascorbic acid in', + ' l/sec (KO) -',F15.10) WRITE(6,405)POUT 405 FORMAT('Outside oxygen partial pressure', + ' in atm -',F15.7) WRITE(6,407)ICO 407 FORMAT('Initial concentration of oxygen in liquid', + ' in mg/mL -',F15.7) WRITE(6,408)ICA 408 FORMAT('Initial concentration of ascorbic acid in', + ' liquid in mg/mLc',F15.7) WRITE(6,409)PP 409 FORMAT('Permeability of PKG in mg.cm/cm‘2.atm.sec', + ' -',F15.l3) WRITE(6,410)TH 410 FORMAT('ThiCkness of PKG in cm -',F15.7) WRITE(6,411)AR 411 FORMAT('Cross section area liquid in cm -',F15.7) WRITE(6,412)L 412 FORMAT('Depth of liquid in cm -',F15.7) WRITE(6,413)VL 413 FORMAT('VOlume of liquid in cm‘3 -',F15.7) PRINT *,' . + ° . PRINT *,' Hit any key to continue.....' PRINT *,' ', + ' v PRINT *,'DIMENSSIONLESS CONSTANT' PRINT *,' ', + O O WRITE(6,SOS)GO,GA,DT,DX,N 505 FORMAT('COs',E10.4,/, +.GA" '810e4p/p 'DTs. '81004'I' 'DX" '81004’I' ' N3. '15) PRINT *" I, + I O PRINT *,' Hit any key to continue ......' RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C SUBROUTINE : TIME PURPOSE : TO DECIDE TIMESTEP TO PRINT C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE TIME (2,21,22,23,Y,DT,N,NR) '_-'_. e— I 4 tum- - ., 107 DOUBLE PRECISION 2,21,23,Y,DT INTEGER 22,N,NR PRINT *,' .' + I I 510 WRITE (6,508)NR,NR 508 FORMAT(SX,'Enter the number of time to calculate', +' concentrations at once.',/,8x,'') PRINT *' . I ' + ' . WEITE(6, 550) 550 FORMAT(////////////////////////////) zI-Z/NR zz-INT(21) 23-AES(Zl-z2) IF((z.LT.NR).OR.(23.GT.0.0000001)) THEN PRINT *,'Calculation error happened 111' PRINT *,'Re-check no. of time step.’ GOTO 510 ELSE Y-z*DT/3600 WRITE(6,512) 512 FORMAT(20X,'CONCENTRATON PROFILE IN PRODUCT') WRITE(6,513)Z,Y 513 FORMAT(12X,'Each step is',F10.2,lx,'steps',' (',F10.5, +lx,'hrs)') WRITE(6, 514)N 514 FORMAT(14X,'<# of the divided shell Of product is', + I4, '> ) PRINT*, ' + ' . PRINT*,'MOL Imass of present 02 (mg)' PRINT*,'TMOOA-total mass of 02 consumed (mg)' PRINT*,'DIFF 811*(02 consmd) -(A.A consmd) (mg)' PRINT*,'RATE -A.A consumed] 02 consumed' PRINT*,'MAL -mass of present A.A (mg)' PRINT*,'TMAL stotal mass Of A.A consumed (mg)' PRINT*,'tRemain-percent of A.A remaining (t)', + ' MOL + TMOOA IIMOL (at t-o)' PRINT*,'S.L Ishelf life of liquid food (hrs) ', + ' MAL + TMAL IIMAL (at t-0)' PRINT*,' ' + I I PRINT*,' Concentration profile in juice (mg/mL) ' + ' Calculation' . PRINT*,' ° + I I PRINT *,' COO C03 COG C09 0012', +' MOL TMOOA TMOL RATE' PRINT *,' CAO CA3 CA6 CA9 CA12', +' MAL TMAL tRemain S.L(hrs)' PRINT*,' , ' + ' v ENDIF 108 RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE 8 INITIL C C PURPOSE : TO INITIATE CONDITION AT T30 C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC *****************************i********t****fi******t*******O.***** * INITIAL CONDITION * *eeeeseaseseceasesseeeeeeeeeeeeeeaeeeeeeceeeeeeeeeeeeeeeeeeeesees SUBROUTINE INITIL (0,A,ICO,ICA,I,J,N,NR) INTEGER I,J,N,NR DOUBLE PRECISION 0,A,ICO,ICA DIMENSION 0(NR,N+5),A(NR,N+5) DO 30 I31,NR,1 D0 20 J'1,N+5,1 I 0(I,J)SICO A(I,J)8ICA 20 CONTINUE 30 CONTINUE RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : BOUNDY C C PURPOSE 8 TO GIVE BOUNDARY CONDITION C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC ****fi********i*it******t*.it.************************************ * BOUNDART CONDITION * ************fi**************************fl************************* SUBROUTINE BOUND! (0,A,ICO,IMOL, + VL,ICA,IMAL,J,N,NR, + SOCL,SACL,TMOOA,MOL,MAL,TMOL) INTEGER J,N,NR DOUBLE PRECISION O,A,ICO,IMOL, + VL,ICA,IMAL, + SOCL,SACL,TMOOA,MOL,MAL,TMOL DIMENSION O(NR,N+6),A(NR,N+6),SOCL(N+6),SACL(N+6),TMOOA(NR), + MOL(NR),MAL(NR),TMOL(NR) DO 600 J-2,N+1,1 O(1,J)sICO A(l,J)-ICA 600 CONTINUE IMOL-ICO*VL IMAL-ICA*VL O(1,1)-ICO O(1,N+2)-IMOL O(l,N+3)-0.0 O(1,N+4)-0.0 109 O(I,N+5)-0.0 A(l,l) -ICA A(1,N+2)=IMAL A(1,N+3)-o.0 A(1,N+4)s100.0000 A(1,N+5)-0.0 SOCL(1)-0.0 SACL(1)-0.o TMOOA(1)-0.00 MOL(1)-IMOL MAL(1)-IMAL TMOL(1)-0.00 WRITE (6,1000)(O(1,J),J-1,N,N/4), + (O(1,J),J-N+2,N+5) WRITE (6,1000)(A(1,J),J-1,N,N/4), + (A(l,J),J-N+2,N+S) 1000 FORMAT (F8.s,1x,Fs.s,1x,F8.s,Ix,Fa.s,1x,Fa.s,1x,Fa.5,1x, + F8.5,1x,Fa.4,1x,F8.2,1x) RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE : CALCUL C PURPOSE : T0 CALCULATE CONC. AND SHELF-LIFE C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC *******************it*ttfittiiflfitfi*****t************************** CALCULATION OF CONCENTRATION OF 02 AND A.A IN LIQUID * *******ifl**iQ...**********t************************************** SUBROUTINE CALCUL (0,A,GA,GO,KO,DT,SOCL,SACL,MOL,AR, + DX,MMOL,MWO,MOOA,TMOL,TMOOA,TTMOL,MAL, + MMAL,MAOA,MWA,DIFF1,PAC,CPAC,SL,BO,POUT, + I,J,N,NR,F,Z,TT,F1,F2,IMAL,TB,PP,DO,PIN) INTEGER I,J,TT,F1,N,NR DOUBLE PRECISION O,A,GA,GO,KO,DT,SOCL,SACL,MOL,AR, + DX,MMOL,MWO,MOOA,TMOL,TMOOA,TTMOL,MAL,IMAL, + MMAL,MAOA,MWA,DIFF1,PAC,CPAC,SL,HO,POUT, + 2,F,F2,PP,DO,PIN,TN DIMENSION O(NR,N+6),A(NR,N+6),PAC(NR), + SOCL(N+6),SACL(N+6),TT(NR),SL(NR), + MOL(NR),MMOL(NR),MAOA(NR), + MAL(NR),MMAL(NR),MOOA(NR),TMOOA(NR),TMOL(NR), + TTMOL(NR) Do-0.000021 pp-0.000000041 TH-0.002159 TT(1)-I 2000 D0 700 132,NR,1 D0 710 J-Z,N,1 0(I,J)3GO*(0(I-1,J-1)+0(I-1,J+1))+ + (1-2*GO)*O(I-1,J)-KO*DT*O(I-1,J) A(I,J)-CA*(A(I-1,J-1)+A(I-1,J+1))+ 110 + (1-2*GA)*A(I-l,J)-11*KO*DT*O(I-1,J) IF(0.00.CT.A(I,J)) THEN A(I,J)-0.00 ENDIF 710 CONTINUE O(I,N+1)-O(I,N) A(I,N+1)-A(I,N) A(I,1)-A(I,2) **********************ttt****iit**************************** * ### CALCULATION OF CONCENTRATION OF 02 IN BEADAPSCE Eff * ****************************flt******i***t*****i*t**i***fi**** DO 780 J-2,N SOCL(J)=SOCL(J-1)+O(I,J) SACL(J)-SACL(J-1)+A(I,J) 780 CONTINUE MOL(I)-AR*DX*((O(I,1)+O(I,N+l))/2+SOCL(N)) MMOL(I)=MOL(I)/(1000*MWO) MOOA(I)-KO*DT*MOL(I) TMOL(I)-MOL(I)+MOOA(I-1) TMOOA(I)sTMOOA(I-1)+MOOA(I) TTMOL(I)=TMOOA(I)+MOL(I) MAL(I)-AR*DX*((A(I,1)+A(I,N+1))/2+SACL(N)) MMAL(I)=MAL(I)/(1000*MWA) MAOA(I)-MAL(I-1)-MAL(I) DIFFl-ll*MOOA(I)-MAOA(I) FAC(I)-MAL(I)*100/IMAL PIN-O(I,l)*HO TT(I)-1+TT(I-1) SL(I)-(TT(I)-1)*DT/3600 O(I,1)-(PP*POUT*DX+TH*DO*O(I-l,2))/(PP*DX*HO+TH*DO) PIN-0(I,1)*HO O(I,N+2)=MOL(I) O(I,N+3)=TMOOA(I) O(I,N+4)=DIFF1 O(I,N+5)=MAOA(I)/MOOA(I) A(I,N+2)=MAL(I) A(I,N+3)-IMAL-MAL(I) A(I,N+4)-PAC(I) A(I,N+5)-SL(I) IF(PAC(I).GT.CPAC) THEN GOTO 700 ELSE COTO 5800 ENDIF 700 CONTINUE *****************fi****************************************** 111 . it: PRINT THE CONCENTRATION PROFILE ##i * ******ti*****t***fi*t***tiiit*tttttfiit.********i**titt******* 2500 F-TT(NR)/z FI-INT(F) F2-AES(F—F1) IF(F2.CT.1/z) GOTO 3000 WRITE (6,1000)(O(NR,J),J-1,N,N/4), + (O(NR,J),J-N+2,N+5) WRITE (6,1000)(A(NR,J),J-1,N,N/4), + (A(NR,J),J-N+2,N+S) 1000 FORMAT (Fa.s,1x,Fe.s,1x,Fa.s,1x,Ps.s,1x,Fa.s,1x, L +F8.5,1X,F8.5,lX,F8.4,lX,F8.2) .*1 ************************************************************ * CALCULATION SHELF-LIFE * ********t****t***t****fi*t**t********************tt********** 3000 IF((PAC(NR).GT.CPAC).AND.(PAC(NR).NE.CPAC)) THEN Vfifififi' TT(1)-TT(NR)+1 TMOOA(1)-TMOOA(NR) DO 4100 Js2,N+1 O(1,J)-O(NR,J) A(1,J)-A(NR,J) A(1,1)-A(NR,2) 4100 CONTINUE GOTO 2000 ELSE GOTO 6000 ENDIF ***t************************t***fl*************fi************* * SHELF-LIFE * ****fi******************t*******t*i*********fi*************fi** 5800 WRITE (6,1000) (O(I-l,J),J-l,N+l,N/3), I (O(I-1!J)IJ.N+2IN+412) WRITE (5,1000) (A(I-1,J),J81,N+1,N/3), I (“(I-IIJ)IJ.N+20N+4I2) 6000 PRINT*,' '1 + I I WRITE(6,6100)SL(I-1),SL(I-1)/24 6100 FORMAT(2X,'SHELF-LIFE OF THIS PRODUCT Is-,1x,F10.s, +'hrs',2x,'[',FlO.5,'daysl') PRINT*,' ', + ' . PRINT*,'INITIAL CONDITION' PRINT*,' ', + I I WRITE(6,6150)POUT,ICON,ICO,IMOL,IMMOL,ICA,IMAL,IMMAL 6150 FORMAT ('Initial partial pressure and conc. of 02 in air-’, /,F20.13,' atmn',lx,F20.13,' mg/mL',/, 'Initial conc. ,mass and mol of 02 in juice -', /,F20.13,' mg/mLs',1x,F20.l3,' mg-‘,lx,F20.13,' ”01'1/0 'Initial conc. ,mass and mol of A.A in juice 2', +-+4-+‘+ 112 + /,F20.13,' mg/mL-',lx,F20.l3,' mgs',lx,r20.13,' mol') PRINT*,' 0' + ' . PRINT*,'FINAL CONDITION' PRINT* , ' . , + ' - WRITE(6,6155)MOL(I-l),MMOL(I-l),MAL(I-l),MMAL(I-l), IMAL-MAL(I-1), TMOOA(I-1),PAC(I-l) 6155 FORMAT ('Final total mass and mol of 02 in juice -', /,F20.13,' mg-',F20.l3,' mol',/, 'Final total mass and mol of A.A in juice 3', /,F20.l3,' mg-',F20.13,' mol',/, 'Final total mass of A.A consumed by oxidation -', F20.13,' mg',/, 'Final total mass of O2 consumed by oxidation 3', F20e13' . mg.'/, 'Total percent loss of ascorbic acid ='. F20.13,' t') ... TF— . . An‘:-s-.4‘g I: ‘ :E .a_ . “...—m . l + +-++-+ +-++-+ PRINT* , ' . ' RETURN END 113 *******t**************t*fi*tfitt*t*****tfi.****fi***tt*fi******tt COMPUTER PROGRAM TO ESTIMATE THE SHELF LIFE OF JUICE Three Dimensional Closed System N DION DION ii}! QIIU I156 Slit ill! 5!}! Ilfltfi DIED I!!! I!!! i This is a program for 3-d diffusion mathematical model with packaging material (Rectangular shape) 3.1. Oxygen diffusion PDE : D01 (dzco/dxz)+(d2CO/dy2)+(d2Co/dzz) ]-kC°-dCo/dt B.C : t > 0, at x - Lx, aCP/ax - 0 I.C PDE B.C I.C C° Ip° outside PAx + TH D° C"|,‘,Ax TH D° + P M° Ax at y = Ly, 6C°gay - O '9 outside PAY + TH D°C TH D° + P E° Ax atzst, 8C°az-O c0 'P outside PA: + TH Do colz-Az c°| l. y'Ay TH D° + P H° Ax xaLx/2, dC oldx-O y=Ly /2, dColdy-O z =Lz/2, dC /dz-O t-O at al x,y,z, C° 8 Ci° 3.2. Ascorbic acid diffusion D A[(d ZCA/deH-(dzc ldyoz)+(d2CA IdEZ) ]-kCo-dCA/dt t>O, x-O, dCA y-O, dC A/dy-O z-O, dCA/dz-O x-Lx/2, dCA/dx-O y-Ly/2, dC A/dy-O chz/2, dc A/dz=0 t-O, all x,y,z, CA-CAi i DION DIED I 01.6 *1}! DIIQIID Pl}, 561*!!! 61.61}. Ii'i I . t***************************t***********t******************** t++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ * MAIN PROGRAM *tt++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ * Put declaration, dimension and open statements here. ++++ ++++ INTEGER T,N,X,Y,Z,NR,M2,TT,F1 DOUBLE PRECISION DO,DA,ICO,ICA,HO,K,CPAC,P,TH,POUT,PIN, LX,LY,LZ,DX,DY,DZ,DT,V,GO,GA,QO,QA,WO,WA, R,F,F2,M,M1,M3,B,IMAL,IMOL,ICOL, O,A,OT,AT,RCO,RCA,TMO,TOL,TTA,PAC,SL, PARAMETER (NR-10,M=10,N-10) DIMENSION O(N+1,N+1,N+1),A(N+1,N+1,N+1), OT(N+1,N+1,N+1),AT(N+1,N+1,N+1), RCO(N+1,N+1,N+1),RCA(N+1,N+1,N+1), TMO(NR),TOL(NR), TTA(NR),PAC(NR),SL(NR),TT(NR) TMOL,TMOLL,TMAL,TMALL,SOCL,SOCLL 114 OPEN (UNIT'6,EILE8'WP3-2-5',EORM8'EORMATTED', + STATUSI'UNENOWN') no 10 T-l,NR,l 130(r)-o.oo TOL(T)-0.00 TTA(T)-0.00 PAC(T)-0.00 8L(T)-0.00 wr(r)-o.oo 10 CONTINUE DO 20 X-1,N+1,1 DO 20 Y-1,N+1,1 DO 20 2'1,N+1,1 C(x,!,2)-o.oo A(X,Y,Z)-0.00 ow(x,!,2)-o.oo AT(X,Y,Z)-0.00 RCO(X,Y,Z)-0.00 RCA(X'Y'Z).OCOO 20 CONTINUE CALL INPUT (DO,DA,ICO,ICA,HO,K,CPAC,P,TH, + POUT,LX,LY,L2,DX,DY,DZ,DT,V, + GOIGA:00:01:W0:WAoR:N:HoPIN) CALL TIME (M,M1,M2,M3,B,DT,N,NR) CALL INITIL (ICOL,P,POUT,DX,TB,DO,ICO,ICA,IMOL, + IMAL,V,O,A,X,Y,Z,N) CALL BOUND! (O,A,ICA,ICO,X,Y,Z,N) CALL CALCUL (T,N,X,Y,Z,NR,TT,P1,CPAC,P,TE,DO,EO,R, + POUT,DX,DY,DZ,DT,O,A,OT,AT, + GO,GA,QO,QA,WO,WA,ROO,RCA,TMO,TOL,TTA, + PAC,SL,TMALL,E,F2,M, + IMAL,TMOL,TMOLL,TMAL,SOCL,SOCLL) STOP END *tiffti########f######l######itii#i###########Iff#####i###### * SUBROUTINE *######f##i#####III#I##II###I##fi#f##f##ff#ff#I###I##II#I#I### CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : INPUT C C PURPOSE : TO INPUT DATA C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE INPUT (DO,DA,ICO,ICA,HO,K,CPAC,P,TH, + POUT,LX,LY,LZ,DX,DY,D2,DT,V, + GO,GA,QO,QA,WO,WA,R,N,M,PIN) INTEGER N DOUBLE PRECISION DO,DA,ICO,ICA,EO,R,CPAC,P,TE, + POUT,LX,LY,L2,DX,DY,D2,DT,V, 115 + GOIGAImIQAImImIRIHIPIN **********************itt*******************************t*** * Enter the data * tittitit.*t*******ttttttttttttttttttt*tttttttttttttttttttttt MWOI32.O MWA-176.1 110 PORMAT(PlO.S) PRINT *,'Enter the diffusion coefficient of oxygen ', +'in liquid in cm‘Z/sec' READ(S,110)DO PRINT *,'Enter the diffusion coefficient of ', +'ascorbic acid in liquid in cm‘Z/sec' READ(5,110)DA PRINT *,'Enter the Henry constant of oxygen into liquid ‘, +'(atm.mL/mg : p-Ho x CL)‘ READ(5,110)HO PRINT *,'Enter the oxidation of ascorbic acid reaction ', +'rate in l/sec' READ(5,110)K PRINT*,'Enter permeability of PKG in mg.cm/cm*2.atm.sec ' READ(5,110)P PRINT*,'Bnter thickness of PKG in cm ' READ(5,110)TH PRINT*,'Enter the initial oxygen partial pressure ', +'in headspace in atm.’ READ(5,110)POUT PRINT *,'Bnter the intial concentration of oxygen in ', +'liquid in atm' READ(5,110)PIN PRINT *,'Enter the intial concentration of ascorbic acid', +'in liquid in mg/mL' READ(5,110)ICA PRINT*,'Enter critical t loss of ascorbic acid ', +'in liquid in t' READ(S,110)CPAC PRINT*,'Enter length of PKG in cm ' RBAD(5,110)LX PRINT*,'Enter width of PKG in cm ' READ(5,110)LY PRINT*,'Enter depth of PKG in cm ' READ(5,110)L2 PRINT*,'Enter the time step for calculation' READ(5,110)Y ***********t*************fi********************************** * #f# CHECK THE HENRRY'S CONSTANT FOR DIFFUSION OCCURANEE * *ttttfltittt*tti**********.***tfi************************t**** ICONIPOUT*HO*IOOO/(45.61*(77+460)) Ico-PIN/ao R-ICO/POUT IF (R.GT.HO) THEN PRINT *,'There is no diffusion from haedspace', + 'into liquid because initial concentration' PRINT *,'in liquid is lower than that in headspace' STOP 116 ENDIF Dx-LX/N DY-LY/N Dz-LZ/N v-Lx*LY*Lz co-!*3600*nol(u*(nx**2)) nr-co*(nx)**2/no GA-DA*DT/(DX)**2 QOIDO*DT/(DY)**2 QA-DA*DT/(DY)**2 wo-no*nr/(02)**2 WA-DA*DT/(DZ)**2 *****i***************i*********************t**********.***** * DISPLA! OF ALL ENTERED DATA AND DIMISSIONLESS CONSTANT * ****************************t******************************fi PRINT *,'All data are checked successfully 111' PRINT *,'Re-check all data entered' PRINT *,' ', + O I WRITE(6,401)DO 401 PORNAT('Diffusion coefficient of oxygen in liquid', + ' in x lO“-5 Cfl‘Z/BOC",F10.7) WRITE(6,402)DA 402 PORNAT('Diffusion coefficient of ascorbic acid' , + ' in liquid in x lO“-S cm‘2/sec-',FlO.7) WRITE(6,403)HO 403 FORMAT('Henry constant of oxygen into liquid',/, + ' in atm*mL/mg (p-Bo x CL)-',325.7) WRITE(6,404)P 404 PORNAT('Permeability of PKG in mg.cm/cm‘2.atm.sec ', + ' -',FlS.lO) WRITE(6,4OS)K 405 FORMAT('Oxidation reaction rate of ascorbic acid in', + ‘ l/sec -',F15.lO) WRITE(6,406)TH 406 FORMAT('Thickness of PKG in cm I ',F15.7) WRITE(6,407)POUT 407 FORMAT('Initial oxygen partial pressure', + ' in headspace in atm =‘,F15.7) WRITB(6,408)PIN 408 PORNAT('Initial concentration of oxygen', + ' in atm -',P15.7) WRITE(6,409)ICO 409 PORNAT('Intial concentration of oxygen in liquid', + ' in mg/mL -',Pls.7) WRITE(6,410)ICA 410 PORNAT('Intial concentration of ascorbic acid in', + ' liquid in mg/mL-‘,Fls.7) WRITE(6,411)LX 411 PORNAT('Length of liquid in cm =‘,Pls.7) WRITE(6,412)LY 412 FORMAT('Width of liquid in cm -',P15.7) WRITE(6,413)L2 413 PORNAT('Depth of liquid in cm -',P15.7) WRITE(6,414)V 414 PORNAT('Volume of liquid in cm*3 -',P15.7) PRINT *,' ', ”-..“— errm-nuu “m-—' .- ' " I' 117 + ' ' PRINT *,' Hit any key to continue.....' PRINT *" 0' + v - PRINT *,'DIHENSSIONLESS CONSTANT' mm *" " ... 0 n WRITE(6, 505)nr,n, co, CA, or ,QO,QA,DY,WO,WA,DZ sos FORMAT('DT",F10.5,31, N-‘,I4,/, + 90- ,E10. 5, ax,'CA-',P10. s, 3x,'nx--,r1o. s, /, + Co— ,FlO.S,3X,'QA-',!10.S,BX,'DY-‘,P10.5, /, + 'wo-',P10.s,3x,'WA-',Plo.s,3x,'nzs',P10.5) PRINT *,' .' + ' v PRINT *,' Hit any key to continue ......' RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : TIME C C PURPOSE 3 TO DECIDE TIMESTEP TO PRINT C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC SUBROUTINE TIME (M,M1,M2,M3,B,DT,N,NR) INTEGER M2,N,NR DOUBLE PRECISION M,M1,M3,B,DT PRINT *,' '1 + ' . 510 WRITE (6,508)NR,NR 508 PORNAT(5X,'Enter the number of time to calculate', +' concentrations at once.’,/,8X,'') PRINT *'. I' + O O WRITE(6, 550) 550 FORMATUI/l/II/l/l/l/l/l/l/) Ml-M/NR M2-INT(M1) M3-ABS(Ml-M2) IP((N.LT.NR).OR.(M3.GT.0.0000001)) THEN PRINT *, 'Calculation error happened 111' PRINT *,'Re-check no. of time step.‘ GOTO 510 ELSE B-M*DT/3600 WRITE(6,512) 512 FORMAT(ZOX,‘CONCENTRATON PROFILE IN Paonucr') 118 WRITE(6,513)M,B 513 PORNAT(121,'Each step is',P10.2,lx,'steps',' (',PIO.5, +lx,'hrs)') WRITE(6,514)N 514 PORNAT(14X,'<# of the divided shell of product is', + 14,'>') PRINT*,' ‘ ', + ' - PRINT*,'NOL sTotal mass of O2 in juice (mg)' PRINT*,'THOL-Total mass of 02 consumed in juice (mg)' PRINT*,'TMAL-Total mass of A.A consumed in juice (mg)' PRINT*,'MAL -Total mol of A.A in juice (mg)' PRINT*,'S.L -Shelf life (hrs) ' PRINT*,'\ Loss- t conc. of ascorbic acid (t)' PRINT*,' '. + I ........ I PRINT*,' Concnetration profile in juice (mg/mL) ' PRINT*,' '. + O --------- O PRINT *,' c01 003 cos 007 009', +' COll' PRINT*,' '. + ' ' PRINT*,'Timo I 0.00' ENDIF RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : INITIL C C PURPOSE : TO INITIATE CONDITION AT T30 C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC tittt*****************t*******t*t*fltttit**t***i**i************** * INITIAL CONDITION * *****itt****i***************i**fitttt***fl************************ SUBROUTINE INITIL (ICOL,P,POUT,DX,TB,DO,ICO,ICA,IMOL, + IMAL,V,O,A,X,Y,2,N) INTEGER X,Y,Z,N DOUBLE PRECISION ICOL,P,POUT,DX,TH,DO,ICO,ICA,IMOL, + IMAL,V,O,A DIMENSION O(N+1,N+1,N+1),A(N+1,N+1,N+1) ICOL-P*POUT*DX/(TH*DO)+ICO IMAL-ICA*V IMOL-ICO*V no 30 x-1,N+1, 1 no 30 Y-1,N+l,l no 30 z-I,N+1,1 O(X,Y,2)-ICO A(X,Y,Z)-ICA 30 CONTINUE RETURN 119 END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : BOUND! C C PURPOSE : TO GIVE BOUNDARY CONDITION C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC ********iiififliit*iii.t******t***********fi*********************** * BOUNDARY CONDITION * *ttttttitttttttttttitt***titit9*******t*t****t*******t********** SUBROUTINE BOUND! (O,A,ICA,ICO,X,Y,Z,N) INTEGER X,Y,2,N DOUBLE PRECISION O,A,ICA,ICO DIMENSION O(N+1,N+1,N+l),A(N+1,N+1,N+1) DO 50 z-1,N+1,1 DO 50 Y‘l,N+1,1 DO 50 X31,N+1,1 O(l,Y,2)-ICO O(N+l,Y,Z)-ICO O(x,l,Z)-IOO O(x,N+I,z)-ICO O(X,Y,1)-ICO O(X,Y,N+l)-ICO SO CONTINUE PRINT*,' PRINT*,' This is conc. profile of O2 in liquid' PRINT*,' no 55 z-1,N/2 WRITE (6,800) 2 no 55 Y-l,N/2 WRITE (6,1000)(O(X,Y,2),x-l,N/2) 55 CONTINUE PRINT*,' PRINT*,' This is conc. profile of ascorbic aicd in liquid' PRINT*,' DO 58 2-1,N+1 WRITE (6,800) 2 800 PORNAT('z-',I3) DO 58 Y-1,N+1 WRITE (6,1000)(A(x,r,z),x-1,N/2) 1°00 FORMAT(F8.5,1X,F8.5,1X,FB.5,1X,F8.5,1X,F8.5,1X, + P8.5) 58 CONTINUE PRINT*,' PRINT*,' This is initial condition' PRINT*,' WRITE(6,67)ICA,IOO 67 PORNAT('Initial cone. of ascorbic acid in juice -',P15.lO,/, 120 + 'Initial conc. of oxygen in juice -',PlS.lO,/, + 't loss of ascorbic acid - 0.00',/, + ' Time -0.0') PRINT*,' . RETURN END CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C SUBROUTINE : CALCUL C C PURPOSE : TO CALCULATE CONC. AND SHELF-LIFE C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC ********************t******i*************t********t******** * #ff CALCULATION OF CONCENTRATION OF 02 AND A.A IN LIQUID ********************************************t************** SUBROUTINE CALCUL (T,N,X,Y,Z,NR,TT,F1,CPAC,P,TH,DO,EO,R, POUT,DX,DY,D2,DT,O,A,OT,AT, GO,GA,QO,QA,WO,WA,RCO,RCA,TMO,TOL,TTA, PAC,SL,TMALL,F,F2,M, IMAL,TMOL,TMOLL,TMAL,SOCL,SOCLL) +-++-+ “___: *T‘“ "‘1' INTEGER T,N,X,Y,2,NR,TT,F1 DOUBLE PRECISION CPAC,P,TH,DO,EO,K, POUT,DX,DY,DZ,DT,O,A,OT,AT, GO,GA,QO,QA,WO,WA,RCO,RCA,TMO,TOL,TTA, PAC,SL,TMALL,F,F2,M, IMAL,TMOL,TMOLL,TMAL,SOCL,SOCLL +-++-+ DIMENSION O(N+1,N+1,N+1),A(N+1,N+1,N+1), oT(N+1,N+1,N+I),AT(N+1,N+I,N+1), RCO(N+1,N+1,N+1),RCA(N+1,N+1,N+1), TMO(NR):T0L(NR): TTA(NR),PAC(NR),SL(NR),TT(NR) +-++-+ TT(1)-1 2000 DO 700 T82,NR,1 DO 710 X-2,N,1 DO 710 Y82,N,1 DO 710 282,N,1 90*(O(XoY’loz)+°(x:Y+1oz))+ "0*(O(Xrer'1)+°(xoYcz*1))+ (l-2*(GO+QO+WO))*O(X,Y,2)-K*DT*O(X,Y,Z) +-++ 95*(A(XIY’102)+A(xIY+1Iz))+ NA*(A(X,Y,z-1)+A(X,Y,2+l))+ (1-2*(GA+QA+WA))*A(X,Y,2)- ll*R*DT*O(x,Y,2) +-++-+ IF(0.000.GT.AT(X,Y,Z)) TEEN AT(X,Y,Z)-0.00000 ELSE ENDIP 710 CONTINUE 121 D0 510 X31,N+1,1 DO 510 Y'1,N+1,1 DO 510 2-1,N+1,1 OT(1,Y,z)-(P*POUT*DX+TH*DO*OT(2,Y,2))/(TH*DO+P*EO*DX) OT(X,1,Z)-(P*POUT*DY+TB*DO*OT(X,2,2))/(TE*DO+P*BO*DY) c OT(X,!,l)-(P*POUT*DZ+TB*DO*OT(X,Y,2))/(TB*DO+P*BO*D2) OT(X,Y,l)-OT(X,Y,2) OT(N+1,Y,2)'OT(1,Y,Z) ’ OT(x,N+1,z)-OT(x,1,z) OT(X,Y,N+1)8(P*POUT*DZ+TB*DO*OT(X,Y,2))/(TE*DO+P*EO*DZ) AT(l,Y,Z)-AT(2,Y,2) AT(x,1,2)-AT(x,2,2) AT(X,Y,l)-AT(X,Y,2) AT(N+1,Y,2)-AT(N,Y,2) AT(x,N+1,z)-AT(x,N,2) AT(x,I,N+I)-AT(x,I,N) 510 CONTINUE TNOLL-o.o TNALL-o.o SOCLL-O.o ******************fi*****tt******************************t*** * ##t CALCULATION OP TOTAL LOSS or AA AND 02 IN LIQUID * ************************it***iii.**********.**************** DO 780 z-l,N/2 DO 780 Y-l,N/2 DO 780 x-1,N/2 RCO(X,Y,Z)-(OT(X,Y,2)+OT(x+l,Y,Z)+ OT(X,Y+1,2)+OT(X,Y,z+l)+ 0T(x+1,y+1,2+1)+OT(X,Y+1,z+l)+ OT(X+1,Y+1,2)+OT(X+1,Y,Z+1))* DX*DY*DZ/8 +-++-+ TNOLsTNOLL+RCO(x,Y,z) TNOLLcTNOL SOCL=SOCLL+K*DT*RCO(X,Y,Z) SOCLL-SOCL RCA(X,Y,2)-(AT(X,Y,2)+AT(X+1,Y,2)+ AT(X,Y+1,Z)+AT(X,Y,Z+1)+ AT(x+l,Y+l,z+l)+AT(X,Y+l,z+l)+ AT(x+l,Y+l,2)+AT(x+l,Y,2+1))* DX*DY*DZ/8 +«++-+ TMAL-TMALL+RCA(X,Y,Z) TNALLeTNAL 780 CONTINUE TMO(T)-8*TMOL TOL(T)-8*(TMOL+SOCL) TTA(T)-8*TNAL PAC(T)-TTA(T)*lOO/IMAL TT(T)-1+TT(T-l) SL(T)-(TT(T)-l)*DT/3600+0.1 IF(PAC(T).GT.CPAC) THEN 122 no 790 z-I,N+I,1 DO 790 Y-1,N+1,1 DO 790 x-I,N+I,1 O(X,I,z)-OT(I,I,z) A(X,Y,Z)-AT(X,Y,2) 79o CONTINUE ELSE GOTO 5800 ENDIP 700 CONTINUE 2500 F'TT(NR)/M FICINT(F) F28ABS(F-F1) IE(P2.CT.1/N) GOTO 3000 fl**i*******ti**i******************i************************* * it! PRINT THE CONCENTRATION PROPILE ### * *******fl**t********i***************i**fi**i**********t******* WRITE (6,80)SL(NR) so PORNAT ('Time -',P20.13,' Hours',/, + + ,/,' This is conc. profile of 02 in liquid') PRINT*, no 950 z-I,N+I WRITE (6,800)z eoo PORNAT ('z =-,13) DO 850 I-I,N/2 WRITE (6,1000)(O(X,Y,2),X=l,N/2) eso CONTINUE 950 CONTINUE PRINT*, PRINT*,‘This is conc. profile of ascorbic aicd in liquid' PRINT*, no 900 z=1,N+1 WRITE(6,SOO)2 DO 900 Y-l,N/2 WRITE (6,1000)(A(X,Y,2),X8l,N/2) 900 CONTINUE 990 CONTINUE 1°00 FORMAT (F8.S,1X,F8.5,1X,F8.5,1X,FB.5,1X,F8.5,1X,F8.5) PRINT*,' WRITE (6,IIIO)PAC(NR),INAL,TTA(NR),TNO(NR),SL(NR) 1110 FORMAT ('Pecent loss of acorbic aicd 8',P10.5,' t' ,/, + 'Initial mass of ascorbic aicd -',P10.5,'mg',/, + 'Pinal mass of ascorbic aicd =‘,P10.S,' mg', /, + 'Total mass of 02 in liquid =‘,P10.5,' mg', /, + 'Shelf-life =‘,P10.4,' hrs' ) PRINT*, ************************************************fl*********fl* * CALCULATION SHELF-LIFE * ***********Qt*********fi****t************t******************* 123 3000 IP((PAC(NR).CT.CPAC).AND.(PAC(NR).NE.CPAC)) TEEN TT(I)-TT(NR)+1 DO 550 x-I,N+I,I no 550 Y-1,N+l,l DO 550 z-1,N+1,I OT(X,Y,2)-O(X,Y,2) AT(X,Y,2)-A(X,Y,2) sso CONTINUE GOTO 2000 ELSE GOTO 6000 ENDIF ************************************************************ * SHELF-LIFE * tt*****tt*****t***t*****fi**fi****t**i****.****.****.**t****** seoo SL(T)-(TT(T)-l)*DT/3600 WRITE (6,80)SL(T) DO 370 z-I,N+I WRITE (6,800)! 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