!!!!!!A MATHEMATICAL MODEL FOR PREDICTING OPTIMAL MICRO -PERFORATED PACKAGING By Jin Zhang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Packaging -Master of Science 2015 ABSTRACT A MATHEMATICAL MOD EL FOR PREDICTING OPTIMAL MICRO -PERFORATED PACKAGING By Jin Zhan g Micro -perforated packaging has been known to maintain the quality of fresh produce by increasing gas exchange volume through polymer films. Even though by far several mathematical models have been proposed to predict gas exchange process of micro -perforated films , to the best of author's knowledge, there are no statistical methods used to analyze the models themselves . Therefore, the aim of this research was to employ inverse problem technique (ordinary least square, sequential and bootstrap methods) to model gas exchange process of micro -perforated packaging based on a different approach, and to assess the validity of the prediction. The prediction model involved hydrodynamic flow (minor head loss th eory) and diffusion flow (total pressure gradient !). To better predict hydrodynamic flow, the entra nce shape of micro perforations was analyzed under SEM. To test the prediction model, experimental data sets were evaluated together with a published data set. The predicted results had high agreement with both data groups, and all the predicted values fell within the asymptotic and bootstrap confidence and prediction bands showing the reliability of the model. Besides, the condition number (cond(J) =1) and bootstrap residual analysis further verified the validity of the model. Due to the slight d ifference from the reference boundary values, parameters in the model were also accurately estimated by inverse problem technique. Copyright by JIN ZHANG 2015!iv I dedicate this thesis to my dear parents !v ACKNOWLEDGEMENTS It took me long time to reach the end of this research. Absolutely, without the endless help from my professors, friends and family members , I would never ever finish this journey. First of all, I would like to express my sincere gratitude to Dr. Joseph Hotchkiss for his patience and guidance during the past years. With his continual help, I gradually establish ed systematic thinking for studying this research . I appreciate his trust in my research plan and his valuable support to my experiment s. Secondly, genuine thanks go to my committee members, Dr. Giles Brereton, Dr. Eva Maria Almenar Rosaleny and Dr. Randolph Beaudry, for their agreement on offering me an opportunity to initiate this research and sharing their works and experience to help me understand the fundamental knowledge regarding this research. Thirdly, s pecial thanks to Professor Kirk David Dolan in the department of Food Science and Human Nutrition for his kind help and priceless suggestions to improve my Matlab program. Fourthly, thank you very much to Professor Susan Selke. Whenever I was under great pressure, she always considerately cared about my situation and kindly motivate d me. I am lucky to have her as one of my professor s. Besides, I would like to thank my lov ely friends, Hayati Samsudin, Woranit Muangmala, Noor Zainah Adzaly, Yanzhe Wu and Yangyang Huang for standing by me, !vi encouraging me and helping me pull through all the difficulties . I feel blessed to have them accompanying me during my tough time. We shared tears and joys. We encouraged each other. I am grateful to have them as my friends. Last but not least, the deepest appreciation is extended to my beloved parents, Yayan Liu and Yang Zhang , for never que stioning my choices and always fully supporting me throughout my life. Also, I want to thank to my grandparent s on my mother Õs side . They are my spiritual motivation to keep moving forward. Jin Zhang !vii TABLE OF CONTENTS LIST OF TABLES ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.x LIST OF FIGURES ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..xi KEY TO SYMBOL SÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..xiv KEY TO ABBREVIATIONS ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..xviii CHAPTER ONE: Literature Review ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...1 1.1 Background ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..É......1 1.2 Modified Atmosphere Packaging ÉÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ.........1 1.3 Micro -perforated Packaging ..ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ........3 1.3.1 Introduction ..ÉÉÉÉÉÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ......3 1.3.2 Factors Influencing the Performance of Micro -perforated Packaging ÉÉ......4 1.3.3 Techniques for Making Micro perforations ÉÉÉÉÉÉÉÉÉÉÉÉ.......6 1.3.3.1 Mechanical Needle ÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.....6 1.3.3.2 Electrostatic Discharge É...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..6 1.3.3.3 LaserÉÉÉÉÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ......6 1.3.3.4 Mineral Filling ÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.......7 1.4 Theory of Gas flow through Micro Tubes ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.........7 1.4.1 Diffusion Flow ÉÉÉÉÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ....7 1.4.1.1 Stefan-MaxwellÕs Diffusion LawÉÉÉÉÉÉÉÉÉÉÉÉ.É.....7 1.4.1.2 Knudsen Diffusion Flow ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...8 1.4.1.3 Transitional Diffusion Flow ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ11 1.4.1.4 Ordinary Molecular Diffusion Flow ÉÉÉÉÉÉÉÉÉÉÉÉ...11 1.4.1.5 Diffusion Flow at Total Pressure Gradient !!ÉÉÉÉÉÉÉÉ..12 1.4.1.6 End Correction of Diffusion Flow ÉÉÉÉÉÉÉÉÉÉÉÉ..É13 1.4.2 Hydrodynamic Flow ÉÉÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...É..14 1.4.2.1 PoiseuilleÕs Law ..ÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...É..15 1.4.2.2 Minor Head Loss ÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...É..16 1.5 Current Works on Micro -perforated Packaging ÉÉÉÉÉÉÉÉÉÉÉÉ...É....18 1.5.1 Effective Diffusi on pathway É...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...É..19 1.5.2 The Influence of the Dimension of Micro Perforations on Gas /Vapor Exchanging Rates ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ20 1.5.3 Mathematical Models for Micro -perforation Studies ÉÉÉÉÉÉÉ..........21 1.6 MATLAB¨ program simulation ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..............27 1.6.1 Forward Problem ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..........28 !viii 1.6.1.1 Sensitivity Coefficient and Scaled Sensitivity Coefficient ÉÉ...........28 1.6.2 Inverse Problem .ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...........29 1.6.2.1 Parameter Estimation ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ........29 1.6.2.2 Ordinary least square (OLS) ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.....30 1.6.2.3 Sequential estimation ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ........30 1.6.2.4 Bootstrap ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...........31 CHAPTER TWO: Objective sÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.........32 CHAPTER THREE: Materials and Methods ÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.......33 3.1 Materials ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..............33 3.1.1 Sample Films ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...........33 3.1.2 Micro -tool ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ............33 3.1.3 Gas Measurement Container ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...33 3.1.4 Ga s Measurement Syringe ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...34 3.2 Methods ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ................34 3.2.1 Sample Preparation ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..............34 3.2.2 Micro -perforations under SEM ÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...............34 3.2.3 G as Exchange Measurement ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...........35 3.2.4 Mathematical ModelÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.......... ......37 3.2.4.1 Forward problem ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.........37 3.2.4.2 Inverse problem ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...........38 3.2.4.2.1 Nlinfit ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...........38 3.2.4.2.2 Sequential estimation ÉÉÉÉÉÉÉÉÉÉÉÉ...........38 3.2.4.2.3 Bootstrap ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..........38 3.2.5 Prediction Evaluation ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..................38 3.2.6 Effective or Equivalent Circular Diameter ÉÉÉÉÉÉÉÉÉ..................40 CHAPTER FOUR: Results and Discussion ÉÉÉÉÉÉÉÉÉ..............................41 4.1 Characterization of Micro -perfo rations ÉÉÉÉÉÉÉÉÉÉÉÉÉ...................41 4.1.1 SEM Images ÉÉÉÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ..................41 4.1.2 Evaluation of Minor Head Loss Coefficient K ÉÉÉÉÉÉÉÉ...............46 4.2 Predictive Model ÉÉÉÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ......................46 4.2.1Mathematical Model ÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ...................46 4.2.2 Parameters of interest ...ÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ..................56 4.3 Evaluation of Prediction Model ÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ.................56 4.3.1 Gas Exchange Experiment ÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ.............56 4.3.2 Assessment by Using Other Packaging Researchers Õ Experimental Data .......62 4.3.2.1 Forward problem ÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ...............63 4.3.2.1.1 Scaled sensitivity coefficient ÉÉÉÉÉÉÉÉ................63 4.3.2.2 Inverse Problem ÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉ................68 !ix 4.3.2.2.1 Nlinfit ÉÉÉÉ..ÉÉÉÉÉÉÉÉ...ÉÉÉ..................68 4.3.2.2.2 Sequential estimati onÉÉÉÉÉÉÉÉÉÉ................73 4.3.2.2.3 Bootstrap estimation ÉÉÉÉÉÉÉÉÉÉÉ.................79 4.3.3 Parameter Estimation ÉÉÉÉÉÉÉÉÉÉÉÉ.......................................89 4.4 Effective or Equivalent Circular Diameter ÉÉÉÉÉÉ..........................................91 CHAPTER FIVE: Conclusion and Future Work ÉÉÉÉ..ÉÉÉÉ.É..................96 5.1 Micro Perforations ÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ.................................96 5.2 Prediction Model ÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ....................................96 5.3 Predicted Results ÉÉÉÉ..ÉÉÉÉÉÉÉÉÉ.ÉÉÉÉ...................................97 5.4 Future Work É..ÉÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ.................................99 APPENDICES ÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..É........................100 APPENDIX A O xygen Calibration Curve of GC-TCDÉÉÉÉ...................................101 APPENDIX B N itrogen Calibration Curve of GC-TCD ÉÉÉÉÉÉÉÉÉÉ.........102 APPENDIX C E xperimental Data of Gas Exchange Research É...................................103 REFERENCE SÉÉÉÉ..ÉÉÉÉÉÉÉÉÉÉÉÉÉ.........................................105 !x LIST OF TABLES Table 1.1 Advantages of Modified Atmosphere Packaging (Sivertsvik, Rosnes, & Bergslien, 2002, Chap 4 ) ..ÉÉÉÉÉÉ........................................................3 Table 1.2 End Correction Factors Evaluated for Films With Different Thickness ............20 Table 1.3 A vailable Mathematical Models for Gas E xchange Prediction..........................22 Table 1.4 Various Micro -perforated Samples Applied in Available Models ......................24 Table 3.1 Characteristi cs of micro -perforated packaging used by Gonz ⁄lez-Buesa et al. (2009)ÉÉÉÉÉ...ÉÉÉÉÉÉ.................................................................39 Table 4.1 Diameters of Micro -perforations on HDPE and PLA films ..............................45 Table 4.2 Parameter Estimation ÉÉÉÉÉÉ.................................................................62 Table 4.3 Parameter Estimation ÉÉÉÉÉÉ.................................................................91 Table 4.4 Comparisons between Experimental and Predi cted Values (O 2).......................94 Table 4.5 Comparisons between Experimental and Predicted Values (CO 2) ....................95 Table C Experimental Data of Gas Exchange Researc h..................................................103 !xi LIST OF FIGURES Figure 1.1 Schematic of Diffusi on flow ÉÉÉ................................................................10 Figure 1.2 Schematic of Effective Length of Diffusi on Flow (Modified from Paul & Clarke, 2002 ) ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.............14 Figure 1.3 Schematic of Three Types of Entrance Conditions (Modified from Munson, Young, Okiishi, & Huebsch, 2009, Chap 8 ) ÉÉÉÉÉÉÉÉÉÉÉ..............18 Figure 1.4 Schematic of Static Method Setup ÉÉÉÉÉÉÉÉÉÉÉÉÉ.............26 Figure 1.5 Schematic of Flow -through Method Setup ÉÉÉÉÉÉÉÉÉÉ............26 Figure 3.1 Schem atic (top side view) of the gas measurement container used for testing micro -perforated films ÉÉÉÉÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ....37 Figure 4.1 Topside view of micro perforation under SEM (HDPE) .ÉÉÉÉÉÉ......41 Figure 4.2 Topside view of micro perforation under SEM (PLA) ....ÉÉÉÉÉÉ......42 Figure 4.3 Bottom side view of micro perforation under SEM (HDPE) .ÉÉÉÉÉ...42 Figure 4.4 Bottom side view of micro perforation under SEM (PLA) ÉÉÉÉÉÉ...43 Figure 4.5 Cro ss-sectional view of micro perforation under SEM (HDPE) .ÉÉÉÉ..44 Figure 4.6 Cross -sectional view of micro perforation under SEM (PLA) ..ÉÉÉÉÉ44 Figure 4.7 Schematic of PoiseuilleÕs Model and Minor Head Loss Model ÉÉÉÉÉ50 Figure 4.8 N2 Changing Process inside the Glass Jar (2 micro perforations; !, experimental value; Ð"Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) .ÉÉ...ÉÉÉÉ.ÉÉÉÉÉÉ............60 Figure 4.9 O2 Changing Process inside the Glass Jar (2 micro perforations; !, experimental value; Ð"Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) .ÉÉÉ.ÉÉ...ÉÉÉÉÉÉÉ............60 Figure 4.10 N 2 Changing Process inside the Glass Jar (3 micro perforations; !, experimental value; Ð Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) ÉÉÉÉÉÉÉÉ..ÉÉÉÉ..É.........61 !xii Figure 4.11 O2 Changing Process inside the Glass Jar (3 micro perforations; !, experimental value; Ð Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) . ..ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.61 Figure 4.12 N 2 Changing Process inside the Glass Jar (3 micro perforations; !, experimental value; Ð Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation); Ð.Ð 99% confidence band; 90% prediction band) .ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..............................62 Figure 4.13 Scaled Sensitivity Coefficient s for the Prediction of Carbon Dioxide Changing Process inside Micro -perforated Packagin gÉÉÉ.ÉÉÉÉÉÉÉ.65 Figure 4.14 Correlations Between the Parameters of Carbon Dioxide Changing Process inside Micro -perforated Packaging Prediction ( !!!!"!!!!!!!!!!!!!"!"!!!!!!!!!"!!ÉÉÉÉ.....ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ............66 Figure 4.15 Scaled Sensitivity Coefficient s for the Prediction of Oxygen Changing Process inside Micro -perforated Packaging ÉÉÉ........ÉÉÉ...ÉÉÉ............67 Figure 4.16 Correlations Between the Parameters of Oxygen Changing Process inside Micro -perforated Packaging ( !!!!"!!!!!!!!!!!!!!!!"!).ÉÉÉÉÉ...68 Figure 4.17 95% Confidence and Prediction Bands for Carbon dioxide Prediction Curve based on Each Parameter Estimation ÉÉÉÉÉ...ÉÉÉÉÉÉÉ.ÉÉ.........69 Figure 4.18 95% Confidence and Prediction Bands for Oxygen Prediction Curve based on Each Parameter Estimation ÉÉÉÉÉÉÉÉÉÉ.ÉÉÉÉÉ...É...É....72 Figure 4.19 Sequential Normalized Plots for Each Parameter (Carbon dioxide Prediction) ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.ÉÉÉÉÉÉÉ.74 Figure 4.20 Sequential Normalized Plot for All the Parameters on the Same Scale (Carbon dioxide Prediction) ÉÉÉÉÉÉÉÉÉÉÉÉÉÉ...ÉÉ................76 Figure 4.21 Sequential Normalized Plots for Each Parameter (Oxygen Prediction) .........77 Figure 4.22 Sequential Normalized Plot for All the Parameters on the Same Scale (Oxygen Prediction) ÉÉÉ...ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.É...........79 Figure 4.23 95% Bootstrap Confidence Bands and Prediction Bands for Carbon Dioxide Prediction Curve based on Each Parameter Estimation ÉÉÉÉÉÉÉÉ.........81 Figure 4.24 95% Bootstrap Confidence Bands and Prediction Bands for Oxygen Prediction Curve based on Each Parameter Estimation ÉÉÉ...ÉÉÉÉÉ......84 Figure 4.25 Residual Analysis (Carbon Dioxide Prediction) .ÉÉ...ÉÉÉÉÉÉÉ....86 !xiii Figure 4.26 Residual Analysis (Oxygen Prediction) .ÉÉÉÉÉÉÉÉÉ..................88 Figure 4.27 O 2 Changing Process inside Micro -perforated Packaging ÉÉÉ..............93 Figure 4.28 CO 2 Changing Process inside Micro -perforated Packaging ÉÉÉ............94 Figure A Oxygen C alibration Curve of GC -TCD ÉÉÉÉÉÉÉÉÉ.É.................101 Figure B Nitrogen C alibration Curve of GC -TCD ÉÉÉÉÉÉÉÉÉÉÉÉÉ.É102 !xiv KEY TO SYMBOLS !"!!" Concentration gradient of gas A (kg/ m 4) !!"! Total molar concentration of fluid species 1, 2, É, n (mol/L) D Pipe diameter (m) DAB Molecular diffusion coefficient (m2/hr) DKA Knudsen diffusi on coefficient for gas A (m 2/hr) DKB Knudsen diffusi on coefficient for gas B (m 2/hr) Dij Stephan -Maxwell diffusion coefficient (m2/hr) !!!!! Diffusion coefficient of the gas mixture of O 2 and N 2 (m2/hr) !!!!"! Diffusion coefficient of the gas mixture of O 2 and CO 2 (m2/hr) F Hydrodynamic flow (m 3/hr) Ji and Jj Individual fluxes of fluid species A i and A j (m3/hr) K Coefficient of minor head loss ! Pipe length (m) MA or M B Molar mass of component A or B (kg/mol) N Total number of micro pores in polymer films NA Diffusion flux of gas A (m3/hr ) NB Diffusion flux of gas B ( m3/hr ) NAÕ Combined diffusion and hydrodynamic flow of gas A ( m3/hr ) NBÕ Combined diffusion and hydrodynamic flow of gas B ( m3/hr ) NA* Combined diffusion and hydrodynamic flow of gas A passing through micro perforations ( m3/hr ) !xv NB* Combined diffusion and hydrodynamic flow of gas B passing through micro perforations ( m3/hr ) OTRstatic Oxygen transmission rate obtained by static experimental method (m 3/m2 "hr) OTRflow Oxygen transmission rate obtained by static flow through method (m 3/m2 "hr) ! Difference of pressure at the ends of micro perforations (dyne/cm 2) P1A and P2A Partial pressure of gas A at the top and bottom part of a micro perforation (Pa) P1B and P2B Partial pressure of gas B at the top and bottom part of a micro perforation (Pa) !P Total pressure difference between both ends of a micro perforation (Pa) P Pressure (Pa) !!"!!! and !!"!!! Partial pressures of carbon dioxide in air and inside a micro perforated packaging respectively (Pa) !!!!! and !!!!! Partial pressures of oxygen in air and inside a micro perforated packaging (or a glass jar) respectively (Pa) !!!!! and !!!!! Partial pressures of nitrogen in air and inside a glass jar respectively (Pa) Q Volume flow rate (hydrodynamic flow) (m3/m2 "hr) !!"!" The negative gradient of hydrostatic pressure (Pa/m) ! Radius of a micro perforation (m) R Gas constant (J/ K "mol) !!!! Respiration rate of oxygen consumption (m 3/kg "hr) !!!"! Respiration rate of carbon dioxide generation (m 3/kg "hr) t Time (hrs) !xvi T Temperature (Kelvin) ! Dynamic viscosity (m2/s) !! Chemical potential of species A i (J/kg) ! Rate of volume movement ( m3/hr ) !!" Average velocity of fluid flow ( m3/hr ) !! and !! Molar fraction s of fluid species Ai and A j !! The !th sensitivi ty coefficient !!! The !th scaled sensitivi ty coefficient X Length of a micro perforation (m) !! Mole fraction of gas A !!! and !!! Mole fractions of gas A at both ends of a micro perforation !!!!!"# Molar fraction of oxygen in air !!"!!!"# Molar fraction of carbon dioxide in air !!!!!"# Molar fraction of oxygen inside a glass jar !!!!!"# Molar fraction of nitrogen inside a glas s jar !!"!!!! Molar fraction of carbon dioxide in the headspace of a micro -perforated packaging !!!!!! Molar fraction of oxygen in the headspace of a micro -perforated packaging !!!!!! Molar fraction of nitrogen in the headspace of a micro -perforated packaging VA or V B Atomic diffusion volume of component A or B (m 3) Vhs Headspace volume of a micro -perforated packaging (m3) Vjar The volume of a glass jar (m3) !xvii W Product weight (kg) ! Gas flux ratio !! Dependent variable ! A parameter vector !! The !th parameter in a parameter vector ! End correction factor ! Coefficient of viscosity (poise or dyne "s/cm2) ! Fluid density or fluid concentration (kg/ m3) !xviii KEY TO ABBREVIATIONS MAP Modified atmosphere packaging Nlinfit Non -linear regression HDPE High density polyethylene OLS Ordinary least square PLA Polylactic acid SEM Scanning electron microscope ! !1 CHAPTER ONE: Literature Review 1.1 Background For fresh produce (fresh vegetables and fruits) with high respiration rat e, color change s, textur e alteration , off -flavor and odor development, mold growth, and nutrient loss are major issues occurred during transportation and storage (Valentas, Rotstein, & Singh, July, 1997 ). The rates of these deteriorating activities are significantly influenced by temperature, relative humidity, and atmosphere . To reduce the rate of deteriora tion of respiring produce, a number of approach es have been implement ed including temperature control (Fennema, Powrie, & Marth, 1973), optimization of gas composition (O 2, CO 2 and H 2O) surrounding food pro duce (Zagory & Kader, 1988 , 1989) and reduction in physical damage of fresh produc e (Montanez, Rodriguez, Mahajan, & Frias, 2010 ). Besides, packaging technology is commercial ly used to mitigate several aforementioned deteriorating activities . 1.2 Modified Atmosphere Packaging Modified atmosphere packaging (MAP) is widely applied for fresh vegetables and fruit s. It refers a sealed polymer packag ing filled with a gas mixture with its composition (the percentages of O 2 and CO 2) that is different from normal air (Robertson, 2006 ; Zhuang, Barth, & Cisneros -Zevallos, 2014 ). The modified gas mixture normally contains high percentage of CO 2 (around 10% to 15% ) (Breese et al., 2010 ) and low percentage of O2 (but can not lower than 1%) (Almenar et al., 2006 ) to slow down deterioration process and ensure quality consistency of fresh produce inside the polymer packaging . Under this scenario, the modified gas mixture prolong s the shelf life of package d produce (Salvador, !2 Jaime, & Oria, 2002 ). Except lengthening life span of fresh produce , other advantages of MAP are shown in Table 1.1. MAPs can be categorize d into two groups based on the ir gas modif ication process : active MAP and passive MAP. Active MAP technology uses a gas flux with predetermined gas compos ition to replace the air inside a packaging. Except gas replacement method, packaging processors add absorbers, scavengers or emitters into active MAPs to achieve desired gas composition (Breese et al., 2010 ). In passive MAPs, the modifi ed gas composition is resulted from respiration or other metabolic activ ities of package d fresh produce (Robertson, 2012 ). For both active and passive MAPs, permeabilit y of packaging films makes contribution to the modif ication of inside gas composition as well . Permeabilit ies of d iffer ent packaging films are not same . Generally, the permeability of O 2 of a packaging film is smaller than the permeability of CO 2 of th e film (Breese et al., 2010 ). The permselectivity (the ratio of CO 2 permeability coefficient to O 2 permeability coefficient of a polymer packaging) of a packaging film normally varie s from 4 to 6 (Al-Ati, 2002 ). For packaged fresh produce with high respiration rate, O 2 consumption and CO 2 accumulation inside MAPs progress at fast speed . Within a short time period, dramatically reduced O 2 inside the packaging and a large amount of permeated CO 2 (since the permselectivity of polymer pa ckaging ranges from 4 to 6) through the packaging create an inside environment with low concentrati on of O2 and comparatively low concentration of CO2 (Zagory, 1997 ). As a consequence , anaerobic activity will be in itiated to accelerate degradation process of the inside fresh produce and thus shorten the shelf life of the package d produce. !3 Table 1.1 Advantages of Modified Atmosphere Packaging (Sivertsvik et al., 2002, Chap 4) Advantages # She lf-life increase by possibly 50 Ð400% # Reduced economic losses due to longer shelf -life # Decreased distribution costs, longer distribution distances and fewer deliveries required # Provides a high quality product # Easier separation of sliced products # Centralized packaging and portion control # Improved presentation Ð clear view of product and all -around visibility # Little or no need for chemical preservatives # Sealed packages, barriers against product re -contamination and drip from package # Odorless and convenient packages 1.3 Micro -perforated Packaging 1.3.1 Introduction Micro -perforated packaging is classified as passive MAP. It is design ed for fresh produce with high respiration rate. By making micro perforations (size range: 40 !" to 200!!" in diameter ) (Sandhya, 2010 ) in the polymer films of MAPs, packaging processo rs create channels for exchanging O2 and CO 2 between the inside and outside packaging environment . Consequently, aerobic activity of package d fresh produce can be maintain ed. Since the ratio of gas exchange rate of CO2 to O2 thro ugh micro -perforated packaging films is improve d close to 1, wh en the concentration of O 2 inside micro -perforated packaging reach es desired low level (O 2 concentration level varie s according to specific type of food product), the inside CO 2 concentration accumulate s at a relative ly high level (within the CO 2 tolerance) (Breese et al., 2010 ; Gontard & Guillaume, 2010 ; !4 Zago ry, 1997 ). The goals concerning degradation deceleration and quality assurance for the packaged fresh produce with high respiration rate can thus be achieve d. Mechanical needle, electrostatic discharge , laser and mineral fillings are the technique s packaging processor s currently employ ed to make micro -perforated packaging . Among them, laser is widely applied due to its easiness to change the number and size of micro perforation s in polymer films . Depend ing on specific storage and marketing condition s, the number and size of micro perforation s in polymer packaging change in accordance with individual cultivar to keep produce quality. Other than developing efficient techniques to make micro perforations, packaging research ers also focus on revealing the relationship between the gas exchange volum e through micro perforations in packaging films and the number, size and shape of the micro perforations by developing mathematical models. They proposed their mathematical mod els by presenting the agreement between their experimental results and the theoretical values computed by their models regarding gas exchange process through micro -perforated packaging , but by far, there are no statistical methods presented to support the solidity of the formulae those models employed . 1.3.2 Factors Influencing t he Performance of Micro -perforated Packaging Different produce has different physiological attribute . Based on the unique characteristics, micro -perforated packag ing need s to be modif ied to meet partic ula r requirements. There is no universal micro -perforated packaging , namely a fixed number of micro -perforations with a fixed diameter for all types of produce . In most cases the following aspects are considered as major factors affecting the performance of micro -perforated packaging: !5 (1) The number and diamet er of micro perforations. These two variables directly control the volume of gases exchan ging through micro perforations in packaging films. The gas exchanging volume is proportion al to the number and diameter of micro perforation s. (2) The effective length of micro perforations. The thickness of polymer films dete rmines the length to a large extent . T o obtain more accurate value of effective gas flow distance , end correction factors of micro perforations in packaging films should be put into consideration. (3) The entrance shape of micr o perforations . Micro perforations with top ends surrounding by debris of polymers will introduce high air resistance and thus have less gas volumes passing through micro perforations compar ing to those have flat edges of entrances . (4) Consistency of the shape of micro perforations. It is hard to precisely adjust the gas composition inside a micro -perforated packaging with various si zes of micro perforations . In this case , the product quality of any fresh produce bagged into the package will be difficult to maintain . (5) Respiratory activity of produce . O 2 consumption and CO 2 generation of fresh produce influence gas exchanging process through micro perforations . To attain an optim al micro -perforated packaging for each produce, simply considering one aspect above will have great diffi culty to achieve the goal. Only with thorough consideration, one may obtain a packaging solution, which will meet the requirement s of the packaged produce . !6 1.3.3 Techniques for Making Micro perforations 1.3.3.1 Mechanical Needle Rolls of small needles are usually made of stainless steel. As polymer film s are being fed under the rolls, the needle drums press films to make micro perforations. While making perforations, the needles can be either heated or kept at room temperature (Chow, 2003). By using this method, it is more poss ible to have micro perforations with elliptical shape instead of round s hape. Besides, if micro perforatio ns with differ ent densities or different size s are required, packaging processors have to spend extra time fabricating new drum s. 1.3.3.2 Electrostatic Discharge ÒA packaging film is passed through high electrostatic voltage. Sparks are generated through the film, causing micro holes.Ó (Chow, 2003 ) Controlling the density of micro -perforations is the toughest job since packaging processors lack the accessibility to manually manipulate the electrostatic discharge process. 1.3.3.3 Laser Compared to the approaches above, laser perforation is a lately introduced method. While perforating , a beam of laser light heats a polymer film until the temperature high enough to melt and eventually vapori ze the film to make a micro perforation . This entire process happens within a second t o obtain a micro perforation with both ends that are similar to round shape. Also, the density of micro perforations is controllable by activating laser light for different times to pierc e through polymer films. Exce pt swift process and controllable density of micro perforations , this laser technic also can make micro holes with difference sizes . Based on the demand of a specific produce, packaging !7 processors only need to adjust the intensity of laser light to perfora te a pa rticular size of perforation . In order to study the improvement and influence of micro -perforated packaging on fresh perishables with high respiration and transpiration rates, laser perforation became a popular choice of packaging researchers . Only a few packaging researchers s elected mechanical needle or electrostatic discharge as t he way making their micro -perforated films for experimental uses (Ghosh & Anantheswaran, 2001 ; Samsudin, 2010 ) . 1.3.3.4 Mineral Filling According to a patented film making technology (obtained by Albert Fisher , Inc.) , a large amount of mineral fillers, su ch as cr ushed calcium carbonate , are added during film extrusion process , as a result, the embedded fillers not only create micro channels or micro perforations in the polymer films, but also they change polymer characteristics, such as transparency and stiffness (Gates, 2010 ; Mangaraj, Goswami, & Mahajan, 2009 ). 1.4 Theory of Gas flow through Micro Perforations Gas flux passing through micro perforations involve s two general mechanic s: diffusi on flow and hydrodynamic flow. 1.4.1 Diffusi on Flow Several models of diffusion flow through micro perforations have been developed depending on specific condition s: 1.4.1.1 Stefan -MaxwellÕs Diffusion Law Stefan-Maxwell diffusion law is a complicated model. Scientists James Clerk Maxwell and Josef Stefan aimed at using this system to predict diffusi on phenomenon of !8 a gas mixture of n species (Maxwell, 1868 ). In other words, this theory explains fluid diff usi on flow with multicomponent , and i t can be defined by Equation 1.1 (Bothe, 2011 ). !!!!"!"#$!!!!!!!!!!!!!!!"! !!"!!!!!!!!!!"#!!!!!!!!! (Equation 1.1) Where !!!"#!!!! Molar fractions of fluid species !!!!"#!!!; !!! Chemical potential of species !!; !!!!"# !!! Individual fluxes of fluid species !!!!"#!!!; !!"! ! Total molar concentration of fluid species 1,2,É,n; !!"! Stephan -Maxwell diffusion coefficient; !!!!! Gas constant; !! Temperatur e; 1.4.1.2 Knudsen Diffusion Flow As shown in Fig ure 1.1, airflow (compo sed of gas A and B) diffus es though a short micro perforation , P1 and P 2 are constant total pres sures at both ends of the perforation , D and L are the diameter and the thickness of the perforation respectively. If the total pressures P 1 and P 2 are equal to each other, an d the ratio of the perforation diameter to the mean free path of a gas molecule , !, is less than 0.2, Knudsen diff usion flow will occur . This type of diffusion describe s the phenomenon that rather than collide with each other, gas molecules frequently collide with internal wall of perforations under !9 zero total pressure gradient , and it can be defined by Equation 1.2 and 1.3 (Knudsen, 1909; Knudsen & Fisher, 1910 ; Youngquist, 1970 ): !!!!!!"!"#!!!!!!!!! (Equation 1.2) !!!!!!"!"#!!!!!!!!! (Equation 1.3) Where !!! Diffusion flux of gas A; !!! Diffusion flux of gas B; !!"! Knudsen diffusion coefficient for gas A; !!"! Knudsen diffusion coefficient for gas B; !!!!!"#!!!!! Partial pressure of gas A at top and bottom part of the tube; !!!!!"#!!!!! Partial pressure of gas B at top and bottom part of the tube; !! Gas constant; !! Temperature in Kelvi n; !10 Figu re 1.1 Schematic of Diffusi on flow To determine mean free path s of gas molecules under specific conditions , some researchers have computed the value s for a variety of gas species . Also, Roy et al . (Roy, Raju, Chuang, Cruden, & Meyyappan, 2003 ) established a formula for resear chers to determine the value s (Equation 1.4) : !!!"!!!!!"# (Equation 1. 4) Where !! Dynamic viscosity; !! Fluid density; !!!! Gas constant; !11 !! Temperature 1.4.1.3 Transitional Diffusion Flow If keep the environmental conditions of Knudsen diffusion, but change the gas species or the diameter of the micro perforation until the ratio of !! falls between 0.2 and 20, then, transitional diffusi on flow will take place. The diffusion flux of gas A , thus, can be defined by Equation 1.5 or 1.6 (Scott & Dullien, 1962 ): !!!!!!"!!!!"!!!!!!!!!!!!!!"!!" (Equation 1. 5) Or !!!!!!!!!!!!"!!!!"!!!!" (Equation 1. 6) Where !!"! Molecular diffusion coefficient ; !!!!"! Concentration gradient of gas A ; !!! Mole fraction of gas A ; ! Gas flux ratio; 1.4.1.4 Ordinary Molecular Diffusion Flow If keep the environmental conditions of Knudsen diffusion, but change the gas species or the diameter of the micro perforation until the ratio of !! greater than 20, !12 then, ordinary molecular diffusi on flow will happen . The diffusi on flux of gas A, in this case, can be describe d as in Equation 1.7 or 1.8 (Youngquist, 1970 ): !!!!!!"!!!!"!!!!!!!!!! (Equation 1. 7) Or !!!!!"!!"#$ !"!!!!!!!!!!!!!!! (Equation 1.8) Where !!!!"#!!!!! Mole fraction s of gas A at both ends of a micro perforation ; 1.4.1.5 Diffusi on Flow at Total Pressure Gradient !! When pressure difference is introduced to the top and bottom ends !!!!!!) of t he micro perforation in Fig ure 1.1, diffus ion flow and hydrodynami c flow should be considered together to evaluate gas flux through the perforation . The formulations used for determine transitional diffusi on flow c an be applied here to assess diffusi on flow under total pressure gradient !!. Considering hydrodynamic flow and dif fusi on flow occur at same ti me, gas flux A can be defined by Equation 1.9 (Youngquist, 1970 ): !!!!!!!"!!!!"!!!!"!!"!!!!!!"!!!!!"!!" (Equation 1.9) !13 And gas flux B can be determined by Equation 1.10 : !!!!!!!"!!!!!!!!!"!!!!"!!"!!!!!!"!!!!!"!!" (Equation 1.10) Where !!!! Combined diffusion and hydrodynamic flow of gas A; !!!! Combined diffusion and hydrodynamic flow of gas B; !! Hydrodynamic flow ; To determine a gas flux ( !!! or !!!) in a binary diffusion flow, the gas flux ratio, !, should be inversely proportional to the square root of its molecular weight (Kosov, 1982 ; Spiegler, 1966) 1.4.1.6 End Correction of Diffusion Flow StefanÕs Law of 1881 states that while airflow diffusing through small aperture s, the diffusion is not only proportional to the diameter of the aperture s, but also, it is proportional to the difference in density measured between the underside of the pore and a point some distance away from it (Willmer & Fricker, 1996 ). This statement re flects the fact that normally the density of dif fusi on flow can be known at a point some distance away from ends of micro perforations , it is not reasonable to evaluate the diffusi on flow based on the length an d diameter of micro perforations only . To examine diffusi on gas flux , Ô effective diffusive pathway (Leff )Õ should be applied . The Ôeffective diffusiv e pathwayÕ refers to a micro perforation with the same cross sectional area as the actual perforation but greater length (Figure 1.2). The difference !14 between Ôeffective diffusive pathwayÕ of the micro perforation and its actual length is called Ô end correction Õ. It s value is controlled by t he diameter of the micro perforation . (Meidner & Mansfield, 1968 ). Figure 1.2 Schematic of Effective Length of Diffusi on Flow (Modified from Paul & Clarke, 2002 ) 1.4.2 Hydrod ynamic Flow Hydrodynamic flow present s when total pressure difference along the micro perforation in Figure 1.1 exists !!!!!!). It is studied for understanding fluid flow , driven by total pressure gradient, moving a long tubes or pipelines. !15 1.4.2.1 PoiseuilleÕs Law In 1840, French physiologist and physician, Poiseuille Jean -L”onard -Marie, developed PoiseuilleÕs law . The purpose of developing this law was to understand blood movement circulating vessels . During the experiment , Po iseuille found that the rate of fluid flow was a result of pressure difference al ong blood vessels, and to determine the velocity of the fluid flow moving along the long vessels , the value of the radius of vessels to the power of four (!!) must be known . Since then, PoiseuilleÕs law became well kno wn due to it s application in evaluat ing laminar flow through cylindrical and capillary tubes (Equation 1.11, K irkham, 2004 ): !!!"!!!!" (Equation 1.11) Where !! Rate of volume movement ; !!!Difference of pressure at the ends of the tube; r = Radius of the tube; X = Length of tube; !!!Coefficient of viscosity ; If convert the unit of coefficient of viscosity by applying NewtonÕs law, PoiseuilleÕs law can also be stated as in Equation 1.12 (Kirkham, 2004 ): !!!!!"!!!!!!!"!"! (Equation 1.12) !16 Where r = Radius of the capillary tube ; !! Viscosity coefficient of the solution ; !!!"!"! = The negative gradient of the hydrostatic pressure ; In the field of fluid mechanics, engineers modif ied PoiseuilleÕs law to study laminar flow passing through pipes as in Equation 1.13 (Munson et al., 2009 ) : !!!!!!!!"#!! (Equation 1.1 3) Where !!!Volu me flow rate (hydrodynamic flow) ; !! Pipe diameter; !! = Total pressure difference at each end of pipe; !! Dynamic viscosity ; !! Pipe length ; 1.4.2.2 Minor Head Loss Hydrodynamic flow going through tubes generally causes two types of head loss: major head loss denoted as hLmajor and minor head loss denoted as hLminor . Frictional effect influences major head loss. M inor head loss results from entrance shape and/or area changes of flowing path. The classification s of ÒmajorÓ and ÒminorÓ do not mean that one type of head loss is always important than the other one. When hydrodynamic flow goes along long and straight tubes, m ajor head loss dominates total h ead loss; If !17 hydrodynamic flow travel s through short tubes , then minor head loss will play crucial role (Fox, McDonald, & Pritchard, 2003 ). The formula established for determining minor head loss is defined as in Equation 1.14 (Munson et al., 2009 ): !!"!!!!!!!! (Equation 1.14) Where !!"! Average velocity of fluid flow ; !! = Total pressure difference at each end of pipe; ! Concentration of fl uid flow; !! Coefficient of minor head loss ; There is no universal value for loss coefficient, K, fit ting in any situation when minor head loss dominates . K can be determined by conducting experiment s for each case, and it depends on the geometric pattern o f entrance shape of a short tube . Figure 1.3 shows three typical entrance geometries. !!!!! when the tube entrance patter n belongs to the category of ÔReentrantÕ; !!!!! if the tube entrance shape is grouped into Ôsquare -edgedÕ; For the third condition, if the tube entrance has slightly rounded shape from cross sectional view , then K ! 0.2, and if the entr ance has well -rounded shape, K ! 0.04. All the estimations above are derived from experimental data (Munson et al., 2009 ). !18 Figure 1.3 Schematic of Three Types of Entrance Conditions (Modified from Munson et al., 2009, Chap 8 ) 1.5 Current Works on Micro -perforated Packaging Several p ackaging researchers have spent years on examining and evaluating the properties of micro -perforated packaging . Some of them focused on gas exchanging rates across micro perforated films with different thickness; Some of the m assessed the influence of micro perforations with different sizes on gas exchang ing rates through micro perforated films; The others aimed at developing mathematical models for produce to predict gas concentration change inside micro perforated packaging with time increasing . !19 1.5.1 Effective Diff usion pathway During past twenty years, some packaging researchers conducted experiments to determine Ôeffective diffusive pathwayÕ for micro perforations in polymer films . They introduced Ôend correction factor sÕ into their computation s to establish the relationship between diameters of micro perforations and Ôend correctionÕ (the product of a Ôend correction factor Õ and a diameter of a micro perforation is equal to Ôend correction Õ). In 1994, Renault et al. (Renault, Houal, Jacquemin, & Chambroy, 1994b ) emphasized the importance of introducing Ôend correction fa ctorsÕ into the prediction of gas exchange rates through micro perforated packaging. They explained that the air resistance near the ends of micro perforations , resu lted from accumulated polymer deposit s around the ends of micro perforations , was the primary factor leadi ng to the difference between predicted and experimental values of gas exchang ing rates through micro perfora ted packaging. O nly when introducing an Ôend correction factor Õ into their evaluation, could they obtain predicted results withi n acceptable variance . Table 1.2 listed experimental values of Ôend correction factorsÕ proposed by packaging researchers . The y determined the factors by testing perforations in various sizes. Among the following evaluations, the size of perforations tested by Gonz ⁄lez et al. in 200 8 fell into the rang e of micro perforations made by packaging processors for current commercial uses. !20 Table 1.2 End Correction Factors Evaluated for Films With Different Thickness Film Thickness (L) and Diameters (D) of Micro Perforations End Correction Factors ! References !!!"#!!!!!"" ! !!!!! Fishman et al. (1996) !!!!"#$!!"!!!!!!"#!!" !!!"#$!!"!!!!!!"#!!"! !!!!! Paul et al.(2002) !"#!!!!!"#!! !"!!"#!!! !!"#!!!"!!!! !!!!!!!!!!!!!!!"!!"# !!!! !!!!!"!!"# !!"!! Gonz ⁄lez et al.(200 8) 1.5.2 The Influence of the Dimension of Micro Perforations on Gas/Vapor Exchanging Rates In 2003, Del-Valle et al. (Del-Valle, Almenar, Lagaron, Catala, & Gavara, 2003 ) concluded that gas flux going through one micro perforation with diamete r of 2d could not have the same gas exchanging rate as the flux going through four micro perforations with diameter of d. This implied that gas exchanging rat e through micro perforations was related to perimeter of mic ro perforations rather than area . Also, they stated that this relationship would be more noticeable when the diameter of micro perforat ions is larger than the thickness of micro perforated films . In 2010, Gates (Gates, 2010 ) mentioned that no water vapor flo wing through micro perforations with diameter smaller than 200 !" could be detected. Possibly, if increase d the diame ter of micro perforations up to 250 !" or even larger, water vapor through micro perfo rations would be detected. However, he did not offer any detailed explanation or experimental data to support his point of view, and his conclusion was contrary to the !21 experiment al results obtained by Del-ValleÕ group. This research group did successfully detect water vapor exchanging through micro perforation s with 225 !" in diameter . 1.5.3 Mathematical Models for Micro -perforation Studies Since 1991, packaging research ers started conducting experiments to study gas composition changing process inside micro -perforated packages (Del-Valle et al., 2003 ; Emon d, Castaigne, Toupin, & Desilets, 1991 ; Fishman, Rodov, & Ben -yehoshua, 1996 ; Ghosh & Anantheswaran, 2001 ; Gonz⁄lez, Ferrer, Oria, & Salvador, 2008 ; Gonz⁄lez, Ferrer., Oria, & Salvador, 2009 ; Hirata, Makino, Ishikawa, Katsuura, & Hasegawa, 1996 ; Renault, Souty, & Chambroy, 1994a ). By employ ing theoretical laws regarding diffusi on flow and hydrodynamic flow , the y established mathematical models to predict gas exchang ing rates through micro -perforated films . Table 1.3 list ed 7 predic tion model s. Even though different laws applied in these models , all the predicted values were reported having agree ment with corresponding experimental results . !22 Table 1.3 Available Mathematical Models for Gas E xchange Prediction Applied Law and Model Nomenclature Reference Modified FickÕs law: !!!!!!!!!!!!!!!!!!!!!! xt and x 0 are volumetric fractions of gas ÒiÓ inside the package at time t and time 0 respectively; !! is volumetric fraction of gas ÒiÓ outside the package; !!! is effective permeability of gas ÒiÓ through perforations; V is the package total free volume; Emond et al. (1991) Stephan Maxwell law: !!!!!!"!!!!!"!!!!!"!!!!"!!!! P is total gas pressure; Yi and Y j are molar fraction of gases ÒiÓ and ÒjÓ; !pi and !pj are molar fluxes of gases ÒiÓ and ÒjÓ though a perforation; Dij is binary diffusion coefficient ; Renault et al. (1994) GrahamÕs law and FickÕs law: !!!!!!!! !!!!!!!!!! !!!!!!!!!!!!! S is total area of the perforations; L is film thickness; A is effective area of the film; P is gas permeability coefficient; p1 and p 2 are gas pressures at both sides of polymer films; !!!!!"#!!! are the amount of gas molecules colliding with unit area per unit time at the gas pressures p 1 and p 2. Hirata et al. (1996) !23 Table 1.3 (cont Õd) !!!!!!!!!!!! D is diffusion coefficient of gas in air; C is partial concentration of the examined gas in packaging; CA is oxygen concentration in ambient atmosphere; Lh is the sum of film thickness and radius of the perforation; Fishman et al. (1996) Diffusion flow and PoseuilleÕs law: !!!! !!!!"!!"!!!!"# !!!!!! !!!!!!!!!!!!!!!!!!"!!!!!!! L is the thickness of polymer film; d is the diameter of micro pores; DA, mix diffusivity coefficient of gas ÒAÓ in a gas mixture; !!!!! molar fraction of gas ÒAÓ outside the packaging, at time 0; !!!!!!! molar fraction of gas ÒAÓ inside the packaging, at ti me t; !!!!! is net diffusional flow; Del-Valle et al. (2003) FickÕs law: !!!!!!!"#! !!!!!!!!!!"#!!!!!!!! Ci and C i,out are the volumetric fraction of gas ÒiÓ inside and outside the package; TRi is gas transmission rate; Gonz⁄lez et al. (2008) Diffusion flow and PoiseuilleÕs law: !!!!! !!!!!!!!!!!!!"!!!!!!!!!"#!!" !!!!! pi and p i,out are partial pressures of gas ÒiÓ inside the package and outside the package respectively; P is pressure; Jh,i is flow of gas ÒiÓ through the holes; TRi is gas transmission rate; Jp,i is hydrodynamic flow of gas ÒiÓ (based on the principle of PoiseuilleÕs law; Gonz⁄lez et al. (2009) All t he researchers referred above , except Hirata et al. , agreed that temperature, polymer thickness, the diameter of micro perforations did influence gas flux travelin g trough micro perforations . Hirata et al. used GrahamÕs law to prove that the contribution !24 of diameter of micro perforation s could be Òassumed as naughtÓ. Therefore, they did not introduce diameter as a variable into their evaluation . Among these researche rs, Hirata et al., Emond et al., and Renault et al. didnÕt solely take gas flux through micro perforations into consideration, along with this, they included gas permeation across polymer films into their researches as well. As for the sizes of micro -perforated samples tested for above models , both macro and micro perforations were used . Table 1.4 included the sizes of perforation s tested in all those models. Acc ording to the table, Emond et al. and Fishman et al. selected the sample sizes that were far more bigger than the range of micro perforations (from 40 to 200 !!! in diameter) currently used in packaging field, while the other researchers did evaluate micro perforatio ns within the range from 40 to 200 !!! in diameter . Table 1.4 Various Micro -perforated Samples Applied in Available Models Micro -perforation Sizes (Diameter) Reference From 6000 to 11000 !! Emond et al. (1991) From 85 to 250 !! Hirata et al. (1996) 2000 !! Fishman et al. (1996) 80 !! Renaul et al. (1994) From !"!!"!!"!!"#!!!" !! Gonz ⁄lez et al. (2008) From !"!!"!!"!!""!!"" !! Gonz ⁄lez et al. (2009) 225!!! Del-Valle et al. (2003) To validate pre dicted values estimated by math ematical models, researchers, in most cases, preferred s tatic than flow -through method to collect experimental data w ith respect to the gas changing process inside micro -perforated containers . As shown in Figure 1.4 (the ap paratus setup might be modified by different packaging researchers), in static method , a micro -perforated film was attached at the top !25 of a container . On the body of the container, there were three ports : inlet port, outlet port and sampling port. The inlet and outlet ports were designed for allowing sweeping gas (nitrogen or helium) to flush out air inside the c ontainer. The sampling port was used to withdraw sample gas from the container at each predetermined time interval. At the beginning of a test , sweeping gas was used, then , the in let and outlet ports were blocked immediately , after every predetermined time interval, a small a mount of gas inside the container was sampled from the sampling port by using a syringe for gas composition analysis . In flow -through method ( Figure 1.5), a micro -perforated film separated a container into t wo compartments. Both compartments have a pair of inlet and outlet ports. The upper pair was used for flushing tested gas (oxygen and/ or carbon dioxide) through the upper compartment, the lower pai r was used for allowing sweep ing gas (nitrogen or helium) to continuously take sample gas o ut of the lower compartment . For each test, the gas composition of sweeping gas mixed with sample gas was analy zed at every predetermined time spot . Compared to flow -through method, s tatic approach provided more actual values , but the time spent for a complete test (around several days ) was considered as its downside. Flow -through method did significantly cut down the time (around 2 ho urs), however , experimental data with less preciseness w ere observed (Ghosh & Anantheswaran, 2001 ). !26 Figure 1.4 Schematic of Static Method Setup Figure 1.5 Schematic of Flow -through Method Setup !27 To obtain precise values in much shorter tim e span, Ghosh and Anantheswaran in 2001 developed a regression equation to interpret the relationship between static and flow -through gas exchanging rates (Equation 1.15) . !"#!"#"$% !!!!"!"#!"#$ !!!"# (Equation 1.15) They proposed that it was timesaving to run flow -though system firstly to get exp erimental data, and then to apply the regression equation to obtain accurate experimental values. By testing micro perforations within the range of 96 to 247 !! in diameter (film thickness: 50.7±3.23 to 76.9±1.61 !!), Ghosh and Anantheswaran proved that their predicted gas exchange rate s had agreement with their experimental data determined by Eq uation 1.15. By far, there were no other researc hers further verify that Equation 1.15 was applicable for testing micro perforations in polymer films . 1.6 MATLAB¨ program simulation Matlab¨ is a numerical analysis softwar e. It paves a n efficient and reliable way to study mathematical models . To provide m ore insights for researchers to evaluate their models , Matlab¨ modeling programs are coded into t wo parts: forward problem and inverse problem. In forward problem part, the method of sc aled sensitivity coefficient is incorporated. By employing this , researchers will know the relationship between the dependent variables of their mode ls and the involved parameters, and the approaches for parameter estimation in inverse problem part c an be thus determined . In inverse problem part, numerical analysis methods such as ordinary least square , sequential estimation and bootstrap are included for parameter estimation and dependent variable prediction. Aside !28 from these, each method analyzes mathema tical models from different aspect to indicate the reliability of tested models. In previous works, there is no publication report ed employing Matlab¨ software to study modeling regarding micro -perforated packaging . Therefore, this research introduced Matl ab¨ int o the field of micro -perforated packaging for gaining more insights. 1.6.1 Forward Proble m 1.6.1.1 Sensitivity Coefficient and Scaled Sensitivity Coefficient A mathematical model typically incorporates one or more parameters for establish ing a connection between independent and dependent variables. In order to further understand the mathematical model, taking the first derivative of dependent variable with respect to involved parameter (s) is strongly recommended. This first derivative is named a s sensitivity coefficient (Equation 1.16 , Beck & Arnold, 1977 ): !!!!!!!! (Equation 1.16) Where !!! The !th sensitivity coefficient ; !! Dependent variable ; !! A parameter vector ; It functions as an indicator to present the magnitude of reactivity of the dependent variable to perturbing parameter s. The larger the magnitude, the more sensitive the parameter s to the dependent variable are . The notable perturb ation helps researchers to !29 know within whic h range of independent variable, more u seful exper imental data can be collect ed accordingly while conducting experiments (Grijspeerdt & Vanrolleghem, 1999 ). If two or even more sets of sensitivity coefficients are found correlated to eac h other, then it is impossible to estimate all the parameters at the same time, more efforts and time have to be spent on parameter estimation. To observe and compare the reactivity of each parame ter together at the same scale, a scaled se nsitivity coefficient plot is needed , and the scaled sensitivity coefficient is defined as in Equation 1.17 : !!!!!!!!!!! (Equation 1.17) On this plot, a scaled sensitivity coefficient curve with the maximum response to the dependent variable implies that the parameter this curve represents should be the most accurate one (small est relative error) estimated by the approaches in in verse problem part. Correlated sensitivity coefficient curves indicate the identical responses of depend ent variable to the parameters . A s a result, the difficulty of parameter estimation will be significantly increase d (Dolan & Mishra, 2013 ). 1.6.2 Inverse Problem 1.6.2.1 Parameter Estimation Parameter estimation is a statistical evaluation process to determine values of constants included in a mathematical model by using experimental data (values of dependent variable). This process is commonly known as inverse problems (Beck & Arnold, 1977 ). In most cases, manually estimating parameters is complicated and time consuming. It requires researchers to search and check potential statistic methods , test the !30 methods and then determine the most suitable approa ch for their calculation s. For modeling study, researchers may need to run extra experiments to determine parameter of interest first . Even in some situation s, researchers can find ref erence values for their parameters, the available data may not reflect the real cond itions of the researchers Õ experiments . Using Matlab¨ softwa re to program modeling saves researcher s from this time consuming and complicated situation . It provides an efficient way for researches to obtain the estimations . Parameters of interest can be ev aluated together with prediction process of dependent variable. The built -in statistical coding of Matlab¨ software helps researche rs compute the values in a reliable manner. 1.6.2.2 Ordinary least square (OLS) OLS is one of the common methods in Matlab¨ for parameter estimation. It uses non-linear regression (known as Nlinfit for short) command built in Matlab¨ software. Except generating the values for parameters, standard errors and standard deviations , confidence band, prediction band, and an evaluatio n value called condition number cond (J) reflect the reliability and the validity of a mathematical model. 1.6.2.3 Sequential estimation According to Beck (1977) and Dolan (2013), the sequential estimation method formed by applying the matrix inversion lemma based on the Gauss minimization method. Other than computing the same type of data generated by OLS, the crucial reason for employing this a pproach is that this method can examine experimental duration packaging researchers spent on their t ests by analyzing parameters fluctuation trend within the range of independent variable . This examination helps researches schedule enough experimental time, that is, they need to perform their tests to the time !31 when the fluctuation trend of parameters rea ches a constant. This valuable analysis is very helpful for researche s to optimize their experiment design. 1.6.2.4 Bootstrap Bootstrap is built based on Monte Carlo methods. To estimate the value of dependent variable, Bootstrap method random ly sample s numbers of sets of data to form fic tional data group s. Normally, 1000 times of random sampling is suggested for predicting dependent variable with more precise value . Compared to nlinfit, the narrowed confidence and prediction band s of dependent variable are considered more accurate by reason of the large quantity of sample data (Mishra, Dolan, & Yang, 2011 ). !32 CHAPTER TWO: Objectives The goal s of this research were : (1) Develop a predicti ve model of gas flow throu gh polymer films containing micro perforations (within !"!!""!!m in diameter ) used for fresh produce packaging. (2) To test the predicti ve model using micro -perforated films. (3) To test the predicti ve model by applying experimental data published by other researchers . !33 CHAPTER THREE: Materials and Methods 3.1 Materials 3.1.1 Sample Films High density polyethylene, HDPE, film was provided from P rintpack, Inc . (Jackson, TN, specification number: 190 B559 -722 (Z294) ). This film was specially designed for food packaging. The thickness of the film was !"!!!!!!!!"# !", determined by averaging five measure ments wit h a TMI digital micrometer (Ronkonkoma, NY, model number: 49 -70-01-001). Biaxially oriented p olylactic acid, PLA, film (Ingeo TM biopolymer 4042D) was obtained from NatureWorks, LLC (Minnetonka, MN ). The thickness of this film was !"!!"!!!!!"# !", determined by averaging five measurements wit h a TMI digital micrometer (Ronkonkoma, NY, model number: 49 -70-01-001). 3.1.2 Micro -tool The Micro -tool used for making micro -perforations was acquired from Ted Pella, Inc. (Redding, CA). This instrument had two components: a stainless steel micro -tool handle (product number: 13675, length: 120mm, color: gold), and a long micro -needle (product number: 13061) mounted in an anodized tool cone . The micro -needle was made from first grade hardened tool steel with a shank size of 120 !!". 3.1.3 Gas Measurement Container Cylindri cal glass containers (260 ml Mason jars with silver vacuum seal lid, wide mouth, Collection Elite) were purchased from Ball Broth ers Glass Manufacturing !34 Company (Broomfield, CO). These jars were used for examin ing gas exchanging process through micro -perforate d sample films. To make gas inlet and outlet ports on the lids of the glass jars, r ubber vial stoppers were acquired from Zicis Group LLC (product model: 13RBS -Red -100, outer rim -to-rim diameter: 13mm, color: red). This type of stopper was designed for serum vial s with 7mm mouth in diameter. 3.1.4 Gas Measurement Syringe To exam ine gas c omposition inside the glass jars at each predetermined time interval, a Supelco 100 µ l g astight syringe was ordered (product number: 509531, needle size: 23 gauge) . This syringe was mad e by Sigma -Aldrich Corporation (Bellefonte, PA ). 3.2 Methods 3.2.1 Sample Preparation A HDPE and a PLA sample film with the surface area of !!!!!"!! m2, a polystyrene board (thickness: !!!!!"!! m) with the area of !!!"!!"!! m2, and the micro-tool were conditioned at room temperature for 12 hours before making micro perforations . After conditioning, th e sample film s were attache d to the polystyrene board by using pushpins and transparent tape , then, the micro -tool was vertically held to slowly pierc e through the sample fil ms to make approximately 35 micro perforations on each film at random location s. 3.2.2 Micro perforations under SEM The sample films were prepared for SEM observation by : (1) At room temperature, five small pieces of films with the area of !!!!!"!! m2 each were cut from each sample film by using utility knife , and at least four !35 micro holes were contained on each small film . Among the five small pieces of HDPE or PLA films , three were prepared for cross sectional observation and two were prepared for top and bottom side observation s. (2) Three aluminum SEM specimen stubs (diameter: 25mm) were placed into liquid nitrogen for around ten minutes . (3) To mi nimize deformation of the cross -section al shape of micro perforations , three small pieces of HDPE films and three small pieces of PLA film s were immediately placed on the SEM specimen stub s after removing from liquid nitrogen then cutting micro perforations through the center by using razor blade . (4) The last step was adding conductive coating on polymer film . All sample films were treated by NEOC -AT Osmium Coater (Meiwafosis Co., Ltd., Osaka, Japan). O smium plasma coating was chosen to provide finer resolution. SEM photos were achieved by placing both the cut and uncut small films into the chamber of JEOL JSM 6610LV. 13 (JEOL, Tokyo, Japan). The top , bottom and cross sectional views of micro pore s were visualized under accelerating voltage at 10 kV. 3.2.3 Gas Exchange Measurement In the gas exchange experiment, glass jars were used as gas exchange containers. On the lid of each container, there were three ports. Two ports in same size (6 mm in diameter ) were design ed as gas inlet and gas outlet ports. In order to flush the container and take gas samples by using the gastight syringe, two rubber vial stoppers were pla ced into the ports, and waterproof silicon sealant were added at the rims of the stoppers to !36 prevent air leakage. The third port (13 mm in diameter ) was used for attaching the micro -perforated HDPE film with epoxy glue ( Figure 3.1). At the beginning of th e experiment, the micro -perforated films were fully covered by non-porous aluminum foils. Pure nitrogen was then injected from gas inlet port for flushing the glass jars until the gas composition inside the containers was ! 100% nitrogen . To prevent high g as pressure accumulated inside the jars and to provide exits for flush gas, the lids of the containers we re loosely closed, in addition, a 18-gauge -syringe needle was insert ed into gas outlet port . After flushing, the 18 -gauge -syringe needle was immediately removed and the lids were tightly closed. The containers were finally kept in a temperature (23¡C) and relative humidity (50%) controlled chamber under 98300 Pa environmental pressure . At every predetermined time interval (8 hours) , 100 µ l gas inside each glass jar was taken from the gas outlet port. The composition of each gas sample was analyzed by a TRACEª GC Ultra gas chromatograph (Fisher Scientific Inc, Waltham, MA ) with a thermal conductivity detector (TCD) and a Supelco Carboxen 1010 PL OT capillary column (Size: 30 m (L) ! 0.53 mm (I.D.), material: fused silica ). Initially, the temperatur e of the GC oven was maintained at 45 ¡C for 4 minutes. It then climbed up to 190¡C at a ramp of 60 ¡C/min. The temperature stayed at 190 ¡C for 1.3 minute. 200 ¡C and 250 ¡C were set as t he temperatures for the GC inject or and de tector. To quantify the gas composit ion of each sample, previously prepared calibration curves were employ ed (See appendices A and B) . The final experimental results pr esented are the average values of three replicates. !37 Figure 3.1 Schematic (top side view) of the gas measurement container used for testing micro -perforated films 3.2.4 Mathematical Model A model of gas tran smission through micro perforations in packaging films was developed using MATLAB ¨ R2012b (version 8.0, the MathWorks, Inc., MA, USA) . To predict gas exchang ing process through micro -perforated packaging, the MATLAB¨ modeling program was coded by following steps: 3.2.4.1 Forward problem Scaled sensitivity coefficient was employed to evaluate the magnitude of reactivity of the dependent variable to the pertur bing parameter. By doing this, the parameter with Gas inlet port Gas outlet port Mirco-perforated film !38 the most accurate estimation was revealed , the useful range of experimental time for research ers to collect data was known, and the correlation between sensitivity coefficient curves and the dependent variable was presented to indicate the methods need ed to be applied in the second part of MATLAB¨ modeling program . 3.2.4.2 Inverse problem In this part, three mathematical methods were employed to determine the parameters of the gas transmission model, to predict gas -changing process inside micro -perforated packaging, and to evaluate the reliability of the mathematical model: 3.2.4.2.1 Nlinfi t Purpose s: parameter estimation, prediction of gas tra nsmission and evaluation of reliability of the prediction model according to computed condition number cond (J) and computed confidence and prediction band s of prediction curves. 3.2.4.2.2 Sequential estimation Purposes: estimating parameters and evaluati ng stability of the parameters of interest within the range of independent variable . 3.2.4.2.3 Bootstrap Purposes: based on 1000 loops of computation to further estimate the modeling parameters, to predict gas transmission process and to evaluate the reliability of the prediction model according to generated asymptotic confidence and prediction bands and 95% of bootstrapping confidence and prediction band s of prediction curves. 3.2.5 Prediction Evaluation There were two sets of data used for the predictive modeling program : a) a set of experiment al data obtain ed from a gas exchange experiment , and b) a set of experiment al !39 data published by other packaging researchers . To test the reliability of the prediction model by using other packaging researchers Õ experimental results, a group of observed values published by Gonz ⁄lez et al. in 2009 were selected. In this experiment, t he known values were : # ÔCalanteÕ peach was selected as the evaluated produce . For this product, the respiratory properties in terms of carbon dioxide generation and oxygen consumption rates wer e directly adopted from the published paper (Gonz⁄lez et al., 2009 ). # Product weight was 0.25 kg. # The storage t emperature for the produce was assumed to be steady at 277.15 K . # The storage relative humidity was 80%. # At initial stage , the gas composition inside packaging was identical as that of the atmosphere. # Table 3.1 lis ted the characteristics of m icro -perforated packaging used by Gonz ⁄lez et al. . Table 3.1 Characteristics of micro -perforated packaging used by Gonz ⁄lez et al. (2009) Film Thickness (m) Headspace Volume (m3) Environmental Pressure (Pa) Number of Micro Perforations Micro Perforation Dimension (!"! HDPE !!!!!!"!! !!!"!!"!! 98792 2 !"!!" !40 3.2.6 Effective or Equivalent Circular Diameter Other than evaluating the validity of the prediction model, the influence s of perimeter and area of oval -shaped micro perforations on the prediction of gas exchange volume through micro perforations were evaluat ed. For either O 2 increasing or CO 2 decreasing process inside micro -perforated packaging, t wo prediction curves under the s ame environmental condition were plotted together with the experimental data of the ÔCalanteÕ peach . One of the prediction curve s was generated based on the effective diameter of micro perforations with the same perimeter as the actual micro perforations ( !"!!" !"), the other prediction curve was generated based on the effect ive diameter of micro perforations with the same area as the actual perforations. Then, at the same time spots, the predicted values on both prediction curves were compared to the corresponding expe rimental points . Finally, a conclus ion regarding the dependence of the prediction of gas exchange volume on perimeter or area of micro perforations was drawn. !41 CHAPTER FOUR: Results and Discussion 4.1 Characterization of Micro Perforations 4.1.1 SEM Images Figure 4.1 to Figure 4.4 showed the SEM images of both top and bottom e nds of the micro perforations in the HDPE and PLA sample films. On Figure 4.1 and Figure 4.2, it was obvious that the issue of heavy polymer deposit around topside of micro perforations had been avoided by applyin g the methods described in chapter three, however, this approach did not success ful ly p revent polymer deposit piling surround the bot tom end of micro perforations, e specially, on the HDPE films this phenomenon was evident ( Figure 4.3). Despite this, these micro perforation s were more uniform and less deform ed than those micro perforations reported in references (Ghosh & Anantheswaran, 2001 ; Gonz⁄lez et al., 2008 ). Figure 4.1 Top side view of micro perforation under SEM (HDPE) !42 Figure 4.2 Topside view of micro perforation under SEM (PLA) Figure 4.3 Bottom side view of micro perforation under SEM (HDPE) !43 Figure 4.4 Bottom side view of micro perforation under SEM (PLA) Figure 4.5 and Figure 4.6 showed the cross -section s of micro perforation s i n HDPE and PLA sample films under SEM. At the central of the figures, t he images visualized in light gray scale were the cross sections of micro perforations , and the polymer film s located at the left and right sides of th e cross sectional part s. Both images further proved that polymer residuals did exist at the bottom end s of micro perforations resulted in lengthened pathway of transmitted gas flux, even though there wa s no polymer deposit around the topside entrances. For the micro perforations in HDPE films, t he tube length was elongated to !"!!!!!!"!!! (the thickness of polymer deposit accumulati ng at the bottom ends included) . For the micro perforations in PLA films, the tube length was stretched to !"!!!!!!"!!! (the thickness of polymer deposit accumulating at the bottom ends included) . These extended p athways of micro perforations implied that it should be not correct to only consider packaging film thickness as the distance of gas es flow ing through micro perforations. The actually pathway of gas flux was longer. !44 Figure 4.5 Cross -sectional view of micro perforation under SEM (HDPE) Figure 4.6 Cross -sectional view of m icro perforation under SEM (PLA) !!!45 Film While trying to cut micro perforations to observe cross sections, the SEM specimen stubs pretreated by liquid nitrogen temporarily froze the sample films placed on them. This proposed method did effectively avoid film distortion , but two problem s were emerged. The techni c indeed enhanced the stiffness of polyme r films , nevertheless , it leaded to significantly hazed polymer surface . I n this situation, it was tough to cut the films right at the middle of every targeted micro perforation, unqualified sample perforations were thus detected und er SEM. Besides, the razor blade deformed a coup le of cross sections, qualified sample s were, in this case, lessened . In future works , an advanced method may be need ed for better understanding the geometric properti es of micro perforations . Table 4.1 recorded the average diameter s of micro perforations in the HDPE and PLA sample films. Due t o the issues explained above, only 4 micro perforations could be counted in the PLA samples , and 12 cross sections were identified in the H DPE sample s. Table 4.1 Diameters of Micro -perforations on HDPE and PLA films HDPE PLA Perforation Numbers 12 4 Average Diameter 54.92 !! 110.5 0 !! Standard Deviation 8.46!!! 4.60!!! The significant variance between the d iameters of micro perforations i n the two types of films indicated that surface tension of polymer film , polymer processing procedure, !46 orientation of polymer films or additives in polymer films may influence the size of micro perforations and thus the volume of gas transmission . 4.1.2 Evaluation of Minor Head Loss Coefficient K Since the entra nce shapes of micro perforations presented i n Figure 4.5 and 4.6 had no exact counterp art s shown in Figure 1.3, it was mere ly reasonable to infer that the values of minor head loss coefficient s, K s, for both types of the micro tubes should be around 0.04 to 0.2. The MATLAB¨ model ing program could further determine more precise value s while predicting gas transmission s volume through the micro -perforated polymer films . 4.2 Predictive Model 4.2.1 Mathematical Model In 1994, Renault et al. (Renault et al., 1994a ) recorded that during their experiment of micro -perforated packaging , the pressure difference (be tween the surrounding environme nt and inside packaging) caused by a small volumetric decrement of examined packaging could be frequently observed. This phenomenon indicated that during the gas transmission process through micro -perforated packaging, hyd rodynamic flow and diffusi on flow existed at the same time. This inference agreed with the standp oint proposed by Paul et al. (Paul & Clarke, 2002 ) and Del -Valle et al (Del-Valle et al., 2003 ). On account of this, the formulae used for determining gas diffusi on flow at nonzero pressure gradient (Equation 1.5, 1.6 and 1.7) should be applied for this research. While considering hydrodynamic flow and diffusion flow together as the total gas flux traveling through micro perforations in packaging films, Equation 1.9 and 1.10 were regarded as the suitable formula e: !47 For gas A in a gas mixture (Youngquist, 1970 ): !!!!!!!"!!!!"!!!!"!!"!!!!!!"!!!!!"!!" (Equation 1.9) And for gas B in the gas mixture : !!!!!!!"!!!!!!!!!"!!!!"!!"!!!!!!"!!!!!"!!" (Equation 1.10) For this research, w hen gas transmission process of mi cro -perforated packaging reached steady state, Equation 1.9 and Equation 1.10 could be integrated as Equation 4.1 and 4.2 : For gas A in a gas mixture : !!!!!!!!!!!"!!"!!!"!"!!!"!!!!"!"!!!!!!!!!!!!!!"!"! (Equation 4.1) For gas B in the gas mixture : !!!!!!!!!!!"!!!!!!!"!!!"!"!!!"!!!!"!"!!!!!!!!!!!!!!"!"! (Equation 4.2) Where !!!!! Combined diffusion and hydrodynamic flow of gas A passing through micro perforations ; !48 !!!!! Combined diffusion and hydrodynamic flow of gas B passing through micro perforations ; !! Volume flowrate ! !! End correction factor ; In 200 8, Gonz ⁄lez et al . (Gonz⁄lez et al., 2008 ) experiment ally determined Ôend correction factors Õ for micro perforations within the range of !"!!"!!!!!!!"#!!!"!!!. They proposed that their Ôend correction factors Õ (0.56 for O2 and 0.46 for CO2) resulted in better prediction curves of gas transmission flux through micro perforations. As the dimension of the micro perforations Gonz ⁄lezÕs group evaluated fell within the dimension range of the micro perforations commercially used in packaging films (!"!!!!""!!!), the v alue of Ôend correction factor Õ for O2 suggested by Gonz ⁄lez et al. was thus introduced into this predicti on model for th e evaluation . For the Ôend correction factor Õ for CO2, 0.46 was used for evaluating gas exchange experiment and 0.44 ( Gonz ⁄lez et al. proposed in 2009) was used for evaluating Gonz ⁄lezÕs experiment al data. To determine the hydrodynamic flow F in Equation 1.9 and 1.10, the gas law about minor head loss fits better in the case of micro -perforated packaging than PoiseuilleÕs law. PoiseuilleÕs law is only valid for laminar fluid flow. It focuses on fluid flow going through long and straight pipes. The ratio of the length of a pipe (L) to its diameter is no less than 20 (Marquardt, 2009 ). As the schematic presented in Figure 4.7a, when a gas flux travel s through a long pipe, the fluid f low gradually forms a fully developed parabolic velocity profile owing to gas viscosity and shear stress. Under this circumstance, the flowing gas flux at pipe wall has the smallest speed (zero), and its !49 velocity increases to the largest rate at the center line of the pipe , Eventually , the flow possesses characteristics of laminar flow (Fox et al., 2003 ). Unlike the model of Poiseui lleÕs law, the tube type applie d in the study of minor head loss is different. Like the graphic shown in Figure 4.7b, the diameter o f tubes in this case is comparatively larger than the length of the tubes, hence, gases flowing through such tube s or pipe s are possibly hard to have velocity profiles with fully developed par abolic shape due to the limited pathway . Talking about the micro perforati ons in packaging films, the length of the micro perforations is limited. It is normally less than the diameter of the micro perforations (!"!!!!""!!!). Because of this fact, the theory of minor head loss was employed into this mathematica l model for predicting hydrodynamic flow . !50 Figure 4.7 Schematic of PoiseuilleÕs Model and Minor Head Loss Model a. PoiseuilleÕs Pipe Model (Munson et al., 2009 ) b. Minor Head Loss Model For the convenience , the equation regarding using minor head loss (Equation 1.14) to determine hydrodynamic flow through one short tube can be modified as Equation 4.3 : !!!!!!!!!!!" (Equation 4. 3) If combine Equation 4.1, 4.2 and 4. 3, gas flux going through micro -perforated polymer films could be defined as: !51 For gas A in a gas mixture : !!!!!!!!!!!"!!"!!!!!"!!!"!!!!"!"!!!!!!!!!!!!!!!"!!!!!!!"!"! (Equation 4. 4) For gas B in the gas mixture : !!!!!!!!!!!"!!!!!!!!!!!!!"!!!"!!!!"!"!!!!!!!!!!!!!!!"!!!!!!!"!"! (Equation 4. 5) To evaluate the experiment al data obtained from the gas exchange experiment , it was assumed that O 2 and N2 were the targeted gases, uniformly spread inside the entire glass jars . The hydrodynamic flow together wit h the diffusi on flow though micro perforations in the HDPE films contributed 100% of gas exchange volume between outside environment and inside environment of the glass jars . In other words, the gas exchange volume across polymer films was regarded as negligible. Hence, Equation 4.4 and Equation 4.5 could be defined as: For gas flux of O 2 through micro perforations : !!!!!!"!!!!!!!!!!!!!!!!!!!!!"!!!!!"!!!!!!!!!"!!!!!!!!!!!!"!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!! (Equation 4. 6) !52 For gas flux of N2 through micro perforations : !!!!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!"!!!!!!!!!"!!!!!!!!!!!!!!!!!"!!!!!!!!!!"!!!!!!!!!!!!!!!"!! (Equation 4. 7) Where !!!!!!! Molar fraction of oxygen in headspace ; !!!!!!! Molar fraction of nitrogen in headspace ; !!!!! and !!!!!! Partial pressures of oxygen in air and inside the glass jars respectively ; !!!!! and !!!!!! Partial pressures of nitrogen in air and inside the glass jars respectively ; !!!!!! Diffusion coefficient of the gas mixture of O 2 and CO 2; !! Total number of micro pores in polymer films ; Finally, gas -changing process inside the glass jars could be defined as: For the gas dynamics of O 2: !!!!!"# !!! !!!"# !!!!!!! !!!!!"!!!!!!!!!!!!!!!!!!!!!"!!!!!"!!!!!!!!!"!!!!!!!!!!!!"!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!! (Equation 4.8) !53 For the gas dynamics of N2: !!!!!"# !!! !!!!!!"# !!!!!!! !!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!!!"!!!!"!!!!!!!!!"!!!!!!!!!!!!!!!!!"!!!!!!!!!!"!!!!!!!!!!!!!!!"!!! (Equation 4.9) Where !!"# !!The volume of the glass jar ; !!!!!"# ! Molar fraction inside the glass jar ; !!!!!"# ! Molar fraction inside the glass jar ; !! End correction facto r for nitrogen passing through micro perforation s; !! Time ; Similarly, t o evaluate Gonz ⁄lez et al. Õs experiment al data , it was assumed that O 2 and CO2 were the targeted gases, uniformly spread inside the entire package. The hydrodynamic flow together with the diffusi on flow though micro perforations in packaging films contributed 100% of gas exchange volume between outside environment and inside environ ment of packaging. In other words, the gas exchange volume across polymer films was regarded as negligible. If considering respiratory act ivity of a produce inside micro -perforated packaging, then Equation 4.4 and Equation 4.5 could be defined as: !54 For gas flux of O2 through micro perforations : !!!!!!"!!!!!!!!"!!!!!!!!!!!!!"!!!!!"!!!!!!!"!!"!!!!!!!!!! !!"!!!!!!!!!!!!"!!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!! (Equation 4. 10) For gas flux of CO 2 through micro perforations : !!"!!!!!"!!!!!!!!"!!!!!!!!!"!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!! !!"!!!!!!!!!!"!!!"!!!!!!!!!!"!!"!"!!!!!"!!!!"!!!! (Equation 4. 11) Where !!"!!!!! Molar fraction of carbon dioxide in headspace ; !!"!!! and !!"!!!! Partial pressures of carbon dioxide in air and inside the packaging respectively ; !!!!"!! Diffusion coefficient of gas mixture of O 2 and CO 2; !!!!! Respiration rate of oxygen consumption ; !!!"!! Respiration rate of carbon dioxide generation ; !! Product weight ; !55 Finally, ga s-changing process inside micro -perforated packagin g for fresh produce could be defined as: For the gas dynamics of O 2: !!!!!!!!! !!!!!!!!!!"# !!!!!"!!!!!!!!"!!!!!!!!!!!!!"!!!!!"!!!!!!!"!!"!!!!!!!!!! !!"!!!!!!!!!!!!"!!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!! (Equation 4. 12) For the gas dynamics of CO2: !!"!!!!!!! !!!!!!!!!!"!!!"# !!!!!"!!!!!!!!"!!!!!!!!!"!!!!!!!!!!!!"!!!!!!!!!!!!!!!"!!"!!!!!!!!!!!!!!! !!"!!!!!!!!!!"!!!"!!!!!!!!!!"!!"!"!!!!!"!!!!"!!!!! (Equation 4. 13) !56 Where !!!! Headspace volume of micro -perforated packaging; !!!!!"# ! Molar fraction of oxygen in air ; !!"!!!"# ! Molar fraction of carbon dioxide in air ; 4.2.2 Parameters of interest In Equation 4.8 and 4.9, the parameters of interest are: # !"!!, Knudsen diffusion coefficient of O 2 # !"!!, Knudsen diffusion coefficient of N2 # !!!!!, Diffusion coefficient of the gas mixture of O 2 and CO 2 # !, Minor loss coefficient # !, End correction facto r for nitrogen passing through micro perforation s. In Equation 4.12 and 4.13, the parameters of interest are: # !"!!, Knudsen diffusion coefficient of O 2 # !"!"!, Knudsen diffusion coefficient of CO 2 # !!!!!"!, Diffusion coefficient of gas mixture of O 2 and CO 2 # !, Minor loss coefficient 4.3 Evaluation of Prediction Model 4.3.1 Gas Exchange Experiment In the gas exchange exper iment, gas transmission s through 2 and 3 micro perforations in the HDPE sample films were tested respectively (See appendix C for experimental data of the gas exchange experiment) to evaluate Equation 4. 8 and 4. 9. Figure 4.8 and 4.10 showed N2 changing process inside the glass jars, and Figure 4.9 and Figure 4.11 presented O2 changing process inside the glass jars. According to the se !57 Figure s, the mathematical model (Equation 4.8 and 4.9) predicted that at 88th hour, the gas composition inside the glass jars with 2 micro perforations in the sample films would reach steady state. This agreed with the experimental results. For the glass containers with 3 micro perforations in th e HDPE films , the experimental observations proved that the mathematical mod el predicted the correct time for O2 reaching steady state, but there was a small gap between the observed and the predicted times for N2 getting to steady state. The theoretical value lagged around 3-hour behind. Also, different dropping speed s of the N 2 volume inside the glass jars with 3 micro holes in the HD PE sample films were noticed in Figure 4.10 (0 to 20 hours). While testing the micro perforated films, every amount of examined gas ( target volume : 100 !") withdrawn from the glass jars, sampling spe ed at each time (comparatively faster sampling speed could cause unexpected fluctuation of pressure gradient along the micro perforations), and consistency of the GC analysis were considered as major aspects brought about variance s, which might explain the predicted quicker speed of N 2 depletion and other differences between the experimental and predicted results in Figure 8 to Figure 11. In spite of the se facts , the 90% prediction band computed by the mathematical model incorporated all the observation data, and the predicted N 2 reducing curve locate d at the centerline of the 99% confidence band showing its small difference from the sample mean of N 2 data (Figure 4.12). The estimations of the parameters of interest (Table 4.2) further pro ved the reliability of the mathematical model (Equation 4.8 and 4.9). The analyzed result of minor head loss coefficient, K, supported aforementioned estimation Ðthe coefficient of the micro perforations in HDPE sample films should be between 0.04 and 0.2. As the estimation !58 was closer to 0.04, it further implied that the entrance shape o f the micro perforations could be categorized into well -rounded group. In 1966 , Fuller et al. (Fuller, Shettler, & Giddings, 1966 ) proposed a method to determine diffusion coefficient for gas mixture at low pressures (Equation 4.14). This well known equation indicated that the diffusion coefficient , !!", should be proportion al to temperature , T, and inversely proportion al to pressure, P. Based on this, the dif fusion coefficient, !!!!!, in Equation 4.8 and 4.9 equaled to 7.60e -01!!!!! which was close to the corresponding estimation (7.22e-01 !!!!!) computed by the prediction model . !!!"!!!!"!#!!!!"!!!!!!!!!!!!!!!!!!!!!!!!!!! (Equation 4. 14) Where T = Temperature; MA or MB = Molar mass of component A or B; P = Pressure; VA or VB = Atomic diffusion volume of component A or B; While checking the standard errors and standard deviation s (Table 4.2) , the small numbers indicated the narrowed distribution of sample values of parameters, and the small variance between the estima ted values and the true mean value s of the corresponding population s of the parameters . In this case, the reliability of the predicte d model itself was, again, proved. According to elementary kinetic theory , the Knudsen diffusion coefficient of a gas in a binary gas mixture is inversely proportional to the square root of its molecular weight !59 (Kosov, 1982 ; Spiegler, 1966 ). On account of this, the theoreti cal ratio of !!!! to !!!! should be 0.93, which was close to the corresponding ratio (i.e. 0.95) of the estimated coefficient s. Directly estimating end correction factor of diffusi on flow , !, while generating prediction curves for tested gases is another advantage of this MATLAB¨ model ing program . In the gas exchange experiment, the end correction factor of N2 diffusi on flow was an unknown value. There was no reference number could be used. If employ ed other modeling method s to predict gas exchange process , preliminary experiment would have to be done to determine the value of ! first . The MATLAB¨ prediction coding of this research d id not need the extra work. P arameter estimation and prediction curve genera tion were completed during the same time. For this experiment , the value of ! directly calculate d from the prediction of N 2 increasing process inside the glass jars with 3 micro perforations in the HDPE sample films. T he standard error and standard deviation were used for checking accuracy of the estimation . !60 Figure 4.8 N2 Changing Process inside the Glass Jar (2 micro perforations; !, experimental value; Ð"Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perfo ration)) !!!!Figure 4.9 O2 Changing Process inside the Glass Jar (2 micro perforations; !, experimental value; Ð"Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) !!61 Figure 4.10 N2 Changing Process inside the Glass Jar (3 micro perforations; !, experimental value; Ð Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) Figure 4.11 O2 Changing Process inside the Glass Jar (3 micro perforations; !, experimental value; Ð Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation)) !62 Figure 4.12 N2 Changing Process inside the Glass Jar (3 micro perforati ons; !, experimental value; Ð Ð predicted value; + experimental value of control sample (HDPE film with 0 micro perforation); Ð.Ð 99% confidence band; É 90% prediction band) Table 4.2 Parameter Estimation Parameter !!"#" !!!!!"! !!!! !!!!!"! !!!! !!!!!"! ! ! Estima ted Value 7.22e-01 7.67e-04 8.11e-04 4.51e-02 8.91e-01 Standard Error 2.70e-04 6.94e-09 6.77e-10 4.18e-06 7.84e-09 Standard Deviation 1.01e-03 2.60e-08 2.53e-09 1.57e-05 2.93e-08 4.3.2 Assessment by Using Other Packaging Researchers Õ Experimental Data To examine the reliability of the prediction model (Equation 4.12 and 4.13) by using other packaging researchers Õ experimental data , a group of experiment al result published by Gonz ⁄lez et al. (Gonz⁄lez et al., 2009 ) was used. Not only computed predict ed values, !63 the programed mathematic al model also estimated the parameter s of i nterest and proved the reliability of the model itself based on statistical analysis . 4.3.2.1 Forward p roblem 4.3.2.1.1 Scaled sensitivity c oefficient Figure 4.13 and 4.1 5 show ed the plots of scaled sensitivity coefficient for CO 2 and O2 changing process inside the micro -perforated packaging . In Figure 4.13, t he total span of the dependent variable s along y-axis , !!"!, was around 8!!"!!. Among all the curves of scaled sensitivity coefficient , the maximum magnitude of !!!!!"! (around 90th hour) was 93.75% of the total span of !!!"!. Given this fact, Figure 4.13 referred that the estimation for !!!"! by the prediction model should be the easiest one with most accuracy . If took absolute value of the magnitude , t he second easiest one shown in Figure 4.13 would be !!!!"! (because the coefficient curve of !!!!"! was in the opposite direction to the direction of the prediction curve). In Figure 4.15, the estimation with the smallest difficulty should be !!!! (!!!!!! was 63.80% of the total span of !!!), !!!!"! still ranked at the second place. Besides , to estimate the parameter s of interest in the prediction model with smallest error and hence to obtain optimal prediction curves, the scaled sensitivity coefficient suggested the micro -perforated experiment should be lasted around 90 hours (the time when the parameters of interest present the maximum magnitude to the prediction curves) . When calculated the ratio of each pair of the scaled sensitivity coefficients (Figure 4.14 and 4.16), correlations between parameters were found. The small range of the ratios indicated that during the gas changing process happen ing inside the micro -perforated packaging, the hydrodynamic flow and diffusion flow pass ing through the micro !64 perforations in the polymer films influenced each other . This indication supported by fact . From the point of time t = 0 hou r, the produce inside the micro -perforated packaging started consuming O 2 and generating CO 2. In this way, the respiration activi ty induced O 2 and CO 2 concentration differ ences across the micro -perforated packaging film . Due to the concentration differ ences, diffusi on flow s of O 2 and CO 2 exchanging through the micro perforations were introduced to compensate the differ ence s. In spite that the pressure surrounded the produce inside the micro -perforated packaging kept at constant, pressure gradient exist ed near the ends of micro per forations in the packaging film . Hence, hydrodynamic flow was also initiated to adjust the gas composition fluctuation . As time increasing, the gas compositions of O2 and CO2 gradually reached steady state, and the gas composition fluctuation eventually de creased to a negligible aspect. Owing to this correlation fact , simultaneously estimating all the parameters of interest by the prediction model would be hard . T he solution to deal with the correlated scaled sensitivity issue was to estimate the parameters separately by employing the programed prediction model of this research . !65 Figure 4.13 Scaled Sensitivity Coefficient s for the Prediction of Carbon Dioxide Changing Process inside Micro -perforated Packagin g !66 Figure 4.14 Correlations Between the Parameters of Carbon Dioxide Changing Process inside Micro -perforated Packaging Prediction ( !!!!"!!!!!!!!!!!!!"!"!!!!!!!!!"!!! !67 Figure 4.15 Scaled Sensitivity Coefficient s for the Prediction of Oxygen Changing Proces s inside Micro -perforated Packaging !68 Figure 4.16 Correlations Between the Parameters of Oxygen Changing Process inside Micro -perforated Packaging ( !!!!"!!!!!!!!!!!!!!!!"!) 4.3.2.2 Inverse Problem 4.3.2.2.1 Nlinfit As the first part of inverse problem , Nli nfit was programed to generate predic tion curves for gas composition changing process inside the micro -perforated packaging (Figure 4.17 and 4.18) and to estimate parameters of interest in Equation 4.12 and 4.13 (Table 4.3). Along with the prediction curves, 95% prediction bands for the experimental data and 95% confidence bands wer e computed . In either of the figures , the predicted curves !69 highly agreed with the experimental results. A ll the prediction bands incorporated all the observed data, and every predicted curve located at the centerline of the confidence bands showin g its small difference from the mean values of the corresponding experimental data . Different from sequential and bootstrap methods, Nlinfit could compute a reference value called condition number, cond(J) (J referred Jacobian matrix) , to indicate the accuracy of the predicted gas changing curves computed by the prediction model . For thi s prediction, the condition num ber was 1. Because the value was much small er than 1 million, it implied that the Jacobian matrix o f the prediction model was well conditioned, in other words , the mathe matical model itself could be relied . Figure 4.17 95% Confidence and Prediction Bands for Carbon dioxide Prediction Curve based o n Each Parameter Estimation a. DKO2 !70 Figure 4.17 (ContÕd) b. DO2CO2 c. K !71 Figure 4.17 (ContÕd) d. DKCO2 !72 Figure 4.18 95% Confidence and Prediction Bands for Oxygen Prediction Curve based on Each Parameter Estimation a. DKO2 b. DO2CO2 !73 Figure 4.18 (ContÕd) c. K 4.3.2.2.2 Sequential estimation Other than evaluating parameters of interest (Table 4.3) , sequential method was used for examining the stability of the parameter of interest within the experimental duration. Figure 4.19 and 4. 21 presented the parameters fluctuation trends during O 2 decreasing and CO 2 increasing process. Compared to other parameters, !!"!! had the largest range of oscillation while the volume of CO 2 climbing inside the micro -perforated packaging, and !!!! had the greatest fluctuation while the volume of O 2 reducing inside the micro -perforated packaging. When plotted all the sequential normalized curves on the same scale (Figure 4.20 and 4. 22), it was clearly to see that all the fluctuation trends finally approached to 1.0 at the time t = 120 hours, which evidenced the enough experimental time the packaging researchers arranged for this experimental. !74 Figure 4.19 Sequential Normalized Plots for Each Parameter (Carbon dioxide Prediction ) a. DKO2 b. K !75 Figure 4.19 (ContÕd) c. DKCO2 d. DO2CO2 !76 Figure 4.20 Sequential Normalized Plot for All the Parameters on the Same Scale (Carbon dioxide Prediction ) Dko2 Do2Co2 K Dkco2 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0 22.5 45 65 95 120 Parameter Time (hrs) !77 Figure 4.21 Sequential Normalized Plots for Each Parameter (Oxygen Prediction ) a. DKO2 b. K !78 Figure 4.21 (ContÕd) c. DO2CO2 !79 Figure 4.22 Sequential Normalized Plot for All the Parameters on the Same Scale (Oxygen Prediction )! 4.3.2.2.3 Bootstrap estimation The parameter s of interest evaluate d by bootstrap method had the same estimation s as the values computed by nlinfit approach . Like aforementioned , the primary advantage of bootstrap method was its large sample pool created by employ ing Monte C arlo theory. For this pred iction , the number of sample data was increase d up to 6000 data points (1000 groups of sample data and 6 points per group). In this case, the computed 95% bootstrap predictio n and confide nce bands should be closer to the true values of the corresponding populations , because the enlarge d sample pool decre ased the variance between the sample mean and the true mean of the gas exchanging data . In other words, bootstrap method had the ability to generate narrow er prediction and confide nce bands than the bands shown in Figure 4.17 and 4.18. Dko2 Do2co2 K 0.87 0.89 0.91 0.93 0.95 0.97 0.99 1.01 0 22.5 45 65 95 120 Parameter Time (hrs) !80 Figure 4.23 and 4.24 were the bootstrap plots. Together with these plots, the 95% prediction bands and 95% asymptotic confidence bands in Figure 4.17 and 4.18 were drawn for comparison. Even though the b ands were narrowed, the prediction bands still incorporated all the observed data, and the predicted curves were within the confidence bands to evidence that even compared the true mean values , the prediction curves still had high reliability . To further examine the reliability of the prediction model of this research, residuals between the predicted (based on Monte Carlo theory) and sample data were calculated and a nalyzed. Figure 4.25 and 4.26 showed the residual plots and analyzed results for both O 2 and CO 2 predictions. In the scattered plot s, the residuals were widely spread, and no systematic fan -shaped trend could be noticed . If more proof needed to confirm the absence of residual trend, small time interval (<20 hours) used for testing the micro -perforated packaging would be suggested to collect more sample data . In the histogram charts, the variances and means of the residuals were very close to zero (shown in residual analysis) and the value s of log likelihood were high. These results verified th e nearly normal independent distributed residuals and further escalated the reliability of the prediction model itself from statistical aspect . !81 Figure 4.23 95% Bootstrap Confidence Bands and Prediction Bands for Carbon Dioxide Prediction Curve based on Each Parameter Estimation a. DKO2 !82 Figure 4.23 (ContÕd) b. DKCO2 c. DCO2O2 !83 Figure 4.23 (ContÕd) d. K !84 Figure 4.24 95% Bootstrap Confidence Bands and Prediction Bands for Oxygen Prediction Curve based on Each Parameter Estimation a. DKO2 b. DO2CO2 !85 Figure 4.24 (ContÕd) c. K !86 Figure 4.25 Residual Analysis (Carbon Dioxide Prediction) a. Histogram Chart (based on bootstrap data points) Distribution: Normal Log Likelihood: 63847.2 Domain: - Inf < y < Inf Mean: 6.52994e Ð 07 Variance: 3.34787e Ð 11 b. Residual Analysis !87 Figure 4.25 (ContÕd) c. Scattered Plot (based on experimental data points) !88 Figure 4.26 Residual Analysis (Oxygen Prediction) a. Histogram Chart Distribution: Normal Log Likelihood: 67848.8 Domain: - Inf < y < Inf Mean: 9.96042e Ð 08 Variance: 8.82028e Ð 12 b. Residual Analysis !89 Figure 4.2 6 (ContÕd) c. Scattered Plot 4.3.3 Parameter Estimation Table 4. 3 listed the estimations of parameters of interest of the mathematical model (Equation 4. 12 and 4. 13). Although three statistical approaches were employed to estimate the parameters, there was no significant deviatio n among the estimated values computed by each statistical method . The smallest standard error and standard deviation of !!!"!(Table 4.3) verified the indication of the scaled sensitivity coefficient plot in forward problem part -the estimation of !!!"! (Knudsen diffusion coefficient of CO 2) should be the most accurate number computed by the prediction model . The small standard deviation revealed a narrow sample distribution of !!!"!, and thus the evaluated val ue was very close to the sample mean of !!!"!. Also, the small standard !90 error served as an estimator to prove that there was a tiny difference between the estimated !!!"! and the population mean of !!!"!. Followed !!!"!, !!!! was the second accurate value. When calculated the ratio of the evaluated !!!! to !!!"!, the result (i.e. 1.15) agreed with the corresponding theoretical value (i.e. 1.17) based on elementary kinetic theory (Kosov, 1982 ; Spiegler, 1966 ). To check the accuracy of the evaluate d diffu sion coefficient of gas mixture , !!!!"!, Equation 4.1 4 was again employed . The theoretical value was 5.25e -01 !!!!!, which was same as the value estimated by Sequential method, and only had 2.00e -02 difference compared with the results evaluated by Nlinfit and Bootstrap approaches. The estimation of minor head loss coefficient , K= 1.99e-01 or 2.00e -01, clarified that the cross sectional shape of the micro perforations tested in Gonz ⁄lez et al. Õs research belonged to slight -rounded group. Wherefore, if tested Gonz ⁄lez et al. Õs micro -perforated samples together with the micro -perforated samples use d in previous gas exchange experiment (Assume except the cross sectional shape of both types of the micro perforations , the other experimental values were same), the flow rate of hydr odynamic flow through the micro perforations used in the gas exchange experiment would be around 3.4 times more than the corresponding rate of Gonz ⁄lezÕs samples . Hence, cross sectional shape of micro perforations in polymer films would significantly influence hydrodynamic flow through micro -perforated packaging . !91 Table 4.3 Parameter Estimation !!!! (!!!!") OLS Sequential Bootstrap Value 7.62e-03 7.63e-03 7.62e-03 Standard Error 1.72e-05 Standard Deviation 4.20e-05 !!!"! (!!!!") OLS Sequential Bootstrap Value 6.63e-03 6.63e-03 6.63e-03 Standard Error 1.26e-07 Standard Deviation 3.09e-07 !!!!"! (!!!!") OLS Sequential Bootstrap Value 5.23e-01 5.25e-01 5.23e-01 Standard Error 7.83e-04 Standard Deviation 1.92e-03 K OLS Sequential Bootstrap Value 1.99e-01 2.00e-01 1.99e-01 Standard Error 4.42e-04 Standard Deviation 1.08e-03 4.4 Effective or Equivalent Circular Diameter The diamete r of micro perforation s is a crucial variable to predict gas exchanging process . Due to the limitation of current technology, packaging research ers, in some cases, obtain elliptic al micro perforation s in polymer sample films instead of ro und micro perforations . To predict gas exchanging rate through those elliptic al micro perforation s, packaging research ers need to determine effective or equivalent circular of the elliptic al micro perforation s, and then to use the effective or equivalent circular diameter for prediction s. Effective or equivalent circular refers to round mic ro perforation s with same area or perimeter as the elliptical micro holes. !92 How to determine the effective or equivalent circular? Shoul d packaging researchers use same area or same perimeter to determine the effective or equivalent circular? Which one wil l help packaging researchers predict more accurate values, circular with same area or c ircular with same perimeter? In 2003, Del -ValleÕs group (Del-Valle et al., 2003 ) proposed that perimeter of micro perforat ions rather than area of micro perforations affected the gas transmission rate through their micro -perforated samples , but no other references discussed the effect in detail. As the last part of this research, both types of effective circular diameter (based on same area and same perimeter) were used to predi ct the gas changing process for Gonz ⁄lezÕs experimental data (Figure 4.27 and 4.2 8). Figure 4.27 showed that the prediction curve generated by using effective circular diameter based on the same area matched the experimental results from 0 to 22 hours. After 22 hours, there was no predicted value close to the experimental data, whereas, the prediction curve generated by using effective circular diameter based on the same perimeter agreed with most of the experiment al results. If compare d the estimated values to the observed data (Table 4.4), the predict ed values based on same perimeter had 7.57% deviation in average to the obse rved data, the predict ed values based on same area enlarge d the deviation up to 20.10%. Particularly at the last data point, the predict ed values based on same area had the deviation as great as 57.32%. The differ ence caused by these two types of effective circular diameter was even apparent in the predictions for CO 2 (Figure 4.28 and Table 4.5 ). Since the very beginning, the prediction based on same area underestimate d CO 2 changing proces s inside the micro perforated packaging . No predict ed value closed to the experiment al data, and the !93 average deviation was three times than the deviation of predict ed values based on same perimeter to the observed data. According to above analysis, effective or equivalent circular based on same perimeter of elliptical micro perforation s in polymer films was suggested to determine effective circular diameter. Using e ffective circular diameter based on same area of micro holes might lead to underestimate d or overestimate d prediction . Figure 4.27 O2 Changin g Process inside Micro -perforated Packaging !94 Table 4.4 Comparisons between Experimental and Predicted Values (O 2) Time (hrs) Oxygen Volume inside Micro Perforated Packaging (m3) Ratio Experimental Data Predicted Value (Same Perimeter) Predicted Value (Same Area) !!"#$!!!"#$ !!"#$ !!"#$!!!"#! !!"#$ 0 9.43E-05 9.43E-05 9.43E-05 0.00% 0.00% 22.5 6.21E-05 5.51E-05 6.21E-05 11.27% 0.00% 45 4.14E-05 3.72E-05 4.72E-05 10.14% 14.01% 65 3.38E-05 3.50E-05 4.09E-05 3.55% 21.01% 95 3.11E -05 2.93E-05 3.99E-05 5.79% 28.30% 120 2.39E-05 2.74E-05 3.76E-05 14.64% 57.32% Average 7.57% 20.10% Figure 4.28 CO2 Changing Process inside Micro -perforated Packagin g !95 Table 4.5 Comparisons between Experimental and Predicted Values (CO 2) Time (hrs) Carbon Dioxide Volume inside Micro Perforated Packaging (m3) Ratio Experimental Data Predicted Value (Same Perimeter) Predicted Value (Same Area) !!"#$!!!"#$ !!"#$ !!"#$!!!"#! !!"#$ 0 1.71E-07 1.71E-07 1.71E-07 0.00% 0.00% 22.5 2.81E-05 1.41E-05 8.69E-06 49.82% 69.07% 45 4.78E-05 4.62E-05 2.79E-05 3.35% 41.63% 65 6.38E-05 6.73E-05 4.28E-05 5.49% 32.92% 95 7.59E-05 7.71E-05 5.45E-05 1.58% 28.19% 120 8.10E-05 7.96E-05 6.16E-05 1.73% 23.95% Average 10.33% 32.63% !96 CHAPTER FIVE: Conclusion and Future Work 5.1 Micro Perforations In this research, t he approach used for making micro perforation s improve d the geome tric shape of the topside entrance . I n this case , air resistance reduced while air flux from outside environment enter ed into micro perforations in the sample films. H owever, the lengthened pathway of gas flux still existed due to the perceived polymers deposited around the bottom ends of the micro perforations . To correctly predict hydrodynamic flow through micro -perforated films , cros s sectional view of micro tubes is needed. The cross -sectional view of micro perforations could hint packaging researchers to assess the range of minor head loss coefficient , K, and thus to obtain close r predic tions to the experimental data by using the mathematical model of this research. While m easuring micro perforations under SEM, significant difference between the diameters of micro perforations in the HDP E and PLA films was noticed. As the same experimental conditions and treatment applied for making micro perforations in both types of films, surface tension of polymer films, processing procedure, orien tation of polymer films and types of additives mixed into polymer films were considered as the factors, which might result in diffe rent sizes of micro perforation and thus could influence gas exchange volume . 5.2 Prediction Model Matlab¨ program , at first time, was introduced into packaging field to predict gas changing process inside micro -perforated packaging. Unlike the other mathematical !97 models , the programed prediction mod el of this research not only computed prediction curves for gas changi ng proce ss inside micro -perforate d packaging, it also proved the reliability of the mathematical model itself based on various data generated by different statistical analysis methods. Besides, the prediction model of this research could be used to examine experimental d uration. With a n analyzed experimental duration, packaging researchers could reasonably operate their experiment s to collect enough data for their research es. More importantly, this Matlab¨ prediction model saved research ers time on performi ng extra experiment s to determine unknown parameter s. Generating predict ed gas changing curv es and estimating unknown parameter s could be completed during the same process , research ers do not need to design prelim inar y works especially for parameter estima tion , and the computed statistical data, like standard error, standard deviation, etc., could be used for checking the accuracy of parameter estimation . 5.3 Predicted Results In this research , a new mathematical model was established for predicting gas changing process inside micro perforated packaging by using Matlab¨ program. To test the reliability of the prediction model, experiment al data obtain ed from a self -perform ed experimen t and from other packaging research ers Õ experiment were adopted. In the self -perform ed experiment , almost all the predict ed data agreed with experiment al values. Higher agreement existed between the observed data and the predict ed gas changing process inside t he glas s jars with 2 micro perforation s in the HDPE sample films. The prediction curves cor rectly anticipated the time span used for the gas composition reaching steady state. A small variance was noticed between the predict ed data of N 2 changing process inside the glass jars with 3 micro perforation s in !98 the HDPE sample films. The amount of examined gas withdrawn from the containers at each time, the sampling speed, and the consistency of GC analysis were considered as the major aspects brought about the variance. While computing prediction curves , the mathematic al model also estimated the correct values of parameter s of interest. The small standard errors and standard deviation s further proved the parameter estimation and the prediction model. To manife st the prediction model is applicable for other micro -perforated tests, experiment al data publi shed by other packaging research ers were used. The prediction curves agreed with the experimental data, and the estimated parameter s shown similar values as theo retical values. In addition , the reliability of the prediction model itself was verified by generating the following information : # Scaled sensitivity coefficient. The plot of scaled sensitivity coefficient revealed the correlatio n between hydrodynamic flow and diffusion flow, which could be supported by theoretical concepts. # Nlinfit. This statistical method a) correctly estimated the parameter s of interest , b) verified the reliability of the prediction model by computing the condition number of Jacobian matrix, and c) computed 95% prediction and 95% as ymptotic confide nce bands, which supported the prediction values . # Sequential. Except correctly estimated the parameter s of interest, this mathematic al method examine d the proper experiment al duration spent on testing the micro -perforated samples. # Bootstrap. This statistical section employ ed Monte Carlo theory to a) correctly estimate the parameter s of interest, b) analyze and present a non -trend residual plot !99 proving the reliability of the prediction model, and c ) compute more precise prediction and confide nce bands . E ven in this case, the prediction curves computed by the prediction model still shown their small deviations . Also, the predicted results explained that to predict gas flux through elliptical micro perforations in packaging films, diameter of effective circular with the sa me perimeter as the micro perforations was recommended to obtain better estimations . 5.4 Future Work In this research , a gas exchange prediction model was develop ed for micro -perforated packaging . Also, most cruc ial factors, which could significantly affec t the prediction of gas exchange process through micro perforations in polymer films , were studied . However, limitations and missing parts still exist and may need to be addressed in future work. # Although clear cross -sectional view of micro perforations in polymer films was observed under SEM by employing the proposed method of this research , some micro -perforated samples were destroyed during the sample preparation process, hence, an advanced technic will be applauded to observe cross sections of micro perforations without leaving any deformation . # To further veri fy the reliability of this gas exchange prediction model, experiments regarding differen t types of fresh produce bagged with micro -perforated packaging under different sto rage conditions need to be performed and analyzed. # Other variables , such as ethylene production of fresh produce and water vapor content inside micro -perforated packaging, may be included in to the prediction model to generate gas exchanging curves closer to real packaging situation s. !100 APPENDICES !101 Appendix A Oxygen C alibration Curve of GC -TCD Figure A Oxygen C alibration Curve of GC -TCD y = 34473x R! = 0.964260 500000 1000000 1500000 2000000 2500000 3000000 3500000 4000000 0 20 40 60 80 100 120 Response Area Oxygen Concentration (%)!102 Appendix B Nitrogen C alibration Curve of GC -TCD Figure B Nitrogen Calibration Curve of GC -TCD y = 110498x R! = 0.999730 2000000 4000000 6000000 8000000 10000000 12000000 0 20 40 60 80 100 120 Response Area Nitrogen Concentration (%)!103 Appendix C Experimental Data of Gas Exchange Research Table C Experimental Data of Gas Exchange Researc h a. HDPE Sample Films with 2 Micro Perforations Time Inside O 2 Inside N 2 Control Sample for O 2 Control Sample for N2 (hrs) (m3) (m3) (m3) (m3) 0 0.00E+00 0.00E+00 0.00E+00 2.60E-04 8 2.08E-03 5.41E-07 1.66E-06 2.58E-04 16 4.16E-03 1.08E-06 1.74E-06 2.58E-04 24 6.24E-03 1.62E-06 1.96E-06 2.58E-04 32 8.32E-03 2.16E-06 2.25E-06 2.58E-04 40 1.04E-02 2.70E-06 2.40E-06 2.58E-04 48 1.25E-02 3.24E-06 2.47E-06 2.58E-04 56 1.46E-02 3.79E-06 2.71E-06 2.57E-04 64 1.66E-02 4.33E-06 2.69E-06 2.57E-04 72 1.87E-02 4.87E-06 2.87E-06 2.57E-04 80 2.08E-02 5.41E-06 2.90E-06 2.57E-04 88 2.29E-02 5.95E-06 3.05E-06 2.57E-04 96 2.50E-02 6.49E-06 3.39E-06 2.57E-04 104 2.70E-02 7.03E-06 3.66E-06 2.56E-04 112 2.91E-02 7.57E-06 3.60E-06 2.56E-04 120 3.12E-02 8.11E-06 3.98E-06 2.56E-04 !104 Table C (ContÕd) b. 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