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- Permanent Link:
- https://ufdc.ufl.edu/UFE0045866/00001
## Material Information- Title:
- Performance Characterization of Dynamic Accumulation Method for Measuring the Oxygen Transmission Rate of Polymer Films Using Fluorescence Oxygen Detection
- Creator:
- Abdellatief, Ayman A
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2014
- Language:
- english
- Physical Description:
- 1 online resource (179 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Agricultural and Biological Engineering
- Committee Chair:
- WELT,BRUCE ARI
- Committee Co-Chair:
- TEIXEIRA,ARTHUR A
- Committee Members:
- SHUKLA,SANJAY
MCLAMORE,ERIC BUTLER,JASON E - Graduation Date:
- 5/3/2014
## Subjects- Subjects / Keywords:
- Diffusion coefficient ( jstor )
Experimental data ( jstor ) Mathematical models ( jstor ) Modeling ( jstor ) Oxygen ( jstor ) Packaging ( jstor ) Polymers ( jstor ) Sensors ( jstor ) Simulations ( jstor ) Volume ( jstor ) Agricultural and Biological Engineering -- Dissertations, Academic -- UF accumulation -- dynamic -- film -- microperforation -- otr -- oxygen -- packaging -- perforated -- rate -- transmission - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Agricultural and Biological Engineering thesis, Ph.D.
## Notes- Abstract:
- Oxygen plays an important role in the preservation or degradation of many products. Rate at which oxygen transmits through a material is an important parameter in designing packaging for many products. The current method most commonly used for measuring oxygen transmission rates (OTR) of packaging materials is complex, expensive, and requires a lot of maintenance. High respiring products are a special case that has oxygen requirements that exceed what can be provided by the highest oxygen transmitting films. Therefore it becomes necessary to incorporate perforations in packaging of high respiring products. More knowledge on how oxygen is distributed in a perforated package is necessary to design a package for high respiring products. A new method for measuring oxygen transmission rate through non-perforated films was developed for this study. The new method assumed oxygen concentration in the test chamber was uniform at all times. A model was developed in order to test the validity of this assumption. The new method was then compared to the method most commonly used for measuring OTR through non-perforated films by measuring the same packaging films using both methods. Another model was developed to better understand the distribution of oxygen in a perforated package. This model predicts oxygen concentration in a perforated chamber at any given location and time. The mathematical model was developed using two methods. One method involves discretization of the perforation which was more complex and required expensive software. The other was an approximation method which was simple enough to a write a computer algorithm. Results from the new method for measuring OTR of non-perforated films were similar to results from the common method. The mathematical model that describes how the new method measures OTR of non-perforated films was validated and demonstrates that any film available with standard laboratory size equipment can be measured. Output from mathematical model that predicts oxygen concentration in a perforated chamber agreed with experimental data. However, the simpler method for this model was slightly less accurate but still gave a reasonable approximation. Therefore the loss in accuracy is justified. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2014.
- Local:
- Adviser: WELT,BRUCE ARI.
- Local:
- Co-adviser: TEIXEIRA,ARTHUR A.
- Statement of Responsibility:
- by Ayman A Abdellatief.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Abdellatief, Ayman A. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 907379069 ( OCLC )
- Classification:
- LD1780 2014 ( lcc )
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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E8FJ9UAGD_W042XN INGEST_TIME 2014-10-03T21:58:54Z PACKAGE UFE0045866_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 PERFORMANCE CHARACTERIZATION OF DYNAMIC ACCUMULATION METHOD FOR MEASURING THE OXYGEN TRANSMISSION RATE OF POLYMER FILMS USING FLUORESCENCE OXYGEN DETECTION By AYMAN ABDELLATIEF A DISSERTATION PRESENTED TO THE GRADUATE SCHO OL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 201 4 PAGE 2 201 4 Ayman Abdellatief PAGE 3 To all those who have helped me get th is far in life. PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my advisor Dr. Bruce Welt for his friendship and support throughout the years. I would like to show gratitude towards my committee members Dr Jason Butler, Dr. Art Teixeira, Dr. Eric McLamore, and Dr. Sanjay Shukla for their advice and guidance. I sincerely appreciate the help of Billy Duckworth, Paul Lane, and Steve Feagle with the technical aspects of this work. I am very grateful to Devinder Saini, Peter Gerard, and Oxysense Inc for the funding and support for this project. I would like to thank my friends colleagues, fellow graduate students, and family for their love and support through the difficult times and my recovery from my unfortunate accident especially my mom Paula Abdellatief and my sister Samya Abdellatief I am very grateful to the people at Shands Orthopedic Center especially my hand therapist Cindy Frazier who has worked long and hard in the rehabilitation of my right hand. No words could express the appreciation I have for my ha nd surgeon Dr Robert Matthias and trauma surgeon Dr Kelia Sadasivan whose assistance helped me to survive to finish this work. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 TABLE OF CONTENTS ................................ ................................ ................................ .. 5 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ ........ 11 LIST OF SYMBOLS ................................ ................................ ................................ ...... 14 ABSTRACT ................................ ................................ ................................ ................... 18 Chapter 1 INTRODUCTION AND OBJECTIVES ................................ ................................ ..... 20 Background ................................ ................................ ................................ ............. 20 Permeation Theory ................................ ................................ ................................ 20 Steady State Theory ................................ ................................ ......................... 20 Unsteady State Diffusion ................................ ................................ .................. 23 Permeability Units ................................ ................................ ................................ ... 25 Oxygen Permeability verses Oxygen Transmission Rate (OTR) ............................ 26 Methods for Measuring OTR ................................ ................................ ................... 27 Pressure and Volume Increase Methods ................................ .......................... 27 Dynamic Accumulation Method ................................ ................................ ........ 28 Steady State Method ................................ ................................ ........................ 29 Statement of the Problem ................................ ................................ ....................... 29 Research Objectives ................................ ................................ ............................... 30 2 DESCRIPTIONS OF THE DYNAMIC ACCUMULATION METHOD AND STEADY STATE METHOD ................................ ................................ .................... 31 Steady State Method ................................ ................................ .............................. 31 Theory of Operation ................................ ................................ ......................... 31 Steady State Method Oxygen Sensing Mechanism ................................ ......... 31 Dynamic Accumulation Method ................................ ................................ .............. 32 Theory of Operation ................................ ................................ ......................... 32 Dynamic Accumulation Oxygen Sensing Mechanism ................................ ...... 34 Intensity and decay based me asurements ................................ ................. 34 Frequency Modulated Excitation ................................ ................................ 36 PAGE 6 6 3 MATHEMATICAL MODELING AND DESIGN CRITERIA FOR THE DYNAMIC ACCUMULATION METHOD WITH NON PERFORATED FILMS .......................... 38 Partial Differential Equation and Boundary Conditions for the Dynamic Accumulation System with A Non Perforated Film ................................ .............. 38 Numerical Solution for Mathematical Model of Dynamic Accumulation System with a Non perforated Film ................................ ................................ .................. 40 Design Criteria for Dynamic Accumulation System with Non perforate d Films .... 43 Calculating Oxygen Transmission Rates in Dynamic Accumulation Systems with Nonuniform Oxygen Concentrations for Non Perforated Films ................... 44 4 MATHEMATICAL MODELING OF OXYGEN TRANSMISSION THROUGH A MICROPERFORATION ................................ ................................ .......................... 45 Application for Perforated Films ................................ ................................ .............. 45 Attempts at Modeling Oxygen Transmission Through a Perforation ....................... 46 Experimental Procedure for Measuring Oxygen Transmission Through a Perforation ................................ ................................ ................................ ........... 47 Flow Through Method ................................ ................................ ...................... 47 Static Method ................................ ................................ ................................ ... 48 Mathematical Model of Dynamic Accumulation Chamber with a Perfo ration .......... 49 Numerical Approximation for Model of Dynamic Accumulation Chamber with a Perforation ................................ ................................ ................................ ........... 51 Simulation of Perforation Mode l ................................ ................................ .............. 55 Computation Time Minimization Strategies ................................ ...................... 55 Simulation Convergence versus Computational Cost ................................ ...... 56 5 EXPERIMENTAL SETUP AND PROCEDURES ................................ .................... 58 Comparison of Steady State Method to Dynamic Accumulation ............................. 58 Steady State Measurements ................................ ................................ ............ 58 Dynamic Accumulation Measurements ................................ ............................ 58 Statistical Analysis and Comparison of Steady State M easurements and Dynamic Accumulation Measurements of OTR ................................ ............. 59 Validation of Mathematical Model for Dynamic Accumulation Method for Measuring Oxygen Transmission through Non Perforated Films ........................ 59 Simulation of Mathematical Model for Dynamic Accumulation Method for Measuring Oxygen Transmission Through Hypothetical Non Perforated Films .. 60 Measuring Oxygen Transmission Through a Perforation ................................ ........ 61 Numerical Simulation of Oxygen Transmission Through a Perforation ................... 62 Finite Element Method ................................ ................................ ..................... 62 Finite Volume Method ................................ ................................ ...................... 63 PAGE 7 7 6 RESULTS AND DISCUSSION ................................ ................................ ............... 65 Comparison of OTR measurements from Steady State Method with those from Dynamic Accumulation Method ................................ ................................ ........... 65 Validation of Mathematical Model for Dynamic Accumulation with Non perforated Films ................................ ................................ ................................ ... 67 Dynamic Accumulation Model Simulations with Non Perforated Films ................... 67 Validation of Mathematical Model fo r Oxygen Transmissions through Perforations ................................ ................................ ................................ ......... 69 7 CONCLUSIONS ................................ ................................ ................................ ..... 71 8 FUTURE WORK AND RECOMMENDATIONS ................................ ...................... 72 Appendix A OXYGEN TRANSMISSION RATE MEASUREMENTS ................................ ........... 73 B VISUAL BASIC CODE FOR NUMERICAL SIMULATION ................................ ..... 154 Model of Dynamic Accumulation Chamber with a Non Perforated Film ............... 154 Model of Oxygen Transmission Through a Perforation into an Accumulation Chamber ................................ ................................ ................................ ............ 157 LIST OF REFERENCES ................................ ................................ ............................. 171 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 179 PAGE 8 8 LIST OF TABLES Table page 1 1 Oxygen Permeability Coefficients of Common Polymers ................................ ... 23 1 2 Common Permeability CoefficientsUnits and Conversion Factors. ..................... 26 4 1 Coefficients a up a down and, a p in the axial direction. Right side of equation 12, b i,j in axial direction. ................................ ................................ ..................... 54 4 2 Coefficients a in a out and, a p in the radi al direction. Right side of equation 12, b i,j in radial direction. ................................ ................................ .......................... 55 5 1 Expected OTR range of samples tested. ................................ ............................ 58 5 2 Number o f each type of node ................................ ................................ ............. 59 5 3 Hypothetical Chambers simulated in model ................................ ....................... 60 5 4 Dimensions and parameters inputted into the model ................................ .......... 62 5 5 Perforation radius, effective diffusion length, simulation times, initial discretization and re discretization used in the second method. ......................... 64 6 1 Measured OTR values using dynamic accumulation and steady state methods. ................................ ................................ ................................ ............. 65 6 2 Differences of means at a 95% Confidence Interval between the Dynamic Accumulation method a nd the Steady State Method. ................................ ......... 66 6 3 Film permeability 10,000 cc mil/m2/day (2.94 x 10 12 m2/s) measured on 100 m hypothetical chamber ................................ ................................ ............... 68 6 4 Comparison of final %O 2 at sensor location achieved by experiment and model predictions using Finite Element (FEM) and Finite Volume Methods (FVM) for each orifice for the times shown in Table 5 5. ................................ .... 69 A 1 Steady State Measurements 10,000 cc/m 2 /day ................................ .................. 73 A 2 Steady State Measurements 1,000 cc/m 2 /day ................................ .................... 74 A 3 Stea dy State Measurements 10 cc/m 2 /day ................................ ......................... 75 A 4 Dynamic Accumulation Measurements 10,000 cc/m 2 /day ................................ .. 76 A 5 Dynamic Accumulation Measure ments 1,000 cc/m 2 /day ................................ .... 77 A 6 Dynamic Accumulation Measurements 10 cc/m 2 /day ................................ ......... 78 PAGE 9 9 A 7 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 1 .............. 79 A 8 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 2 .............. 80 A 9 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 3 .............. 81 A 10 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 4 .............. 82 A 11 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 5 .............. 83 A 12 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 6 .............. 84 A 13 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 7 ............. 85 A 14 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 8 .............. 86 A 15 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 9 .............. 87 A 16 Oxygen Concentrat ion versus time, Data 10,000 cc/m 2 /day Sample 10 ............ 88 A 17 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 11 ............ 89 A 18 Oxygen Concentration versus time, Data 10,000 cc/m 2 /day Sample 12 ............ 90 A 19 Oxyg en Concentration versus time, Data 1,000 cc/m 2 /day Sample 1 ................ 91 A 20 Oxygen Concent ration versus time Data 1,000 cc/m 2 /day Sample 2 ................ 93 A 21 Oxygen Concentration versus time, Data 1,000 cc/m 2 /day Sample 3 ............... 95 A 22 Oxy gen Concentration versus time, Data 1,000 cc/m 2 /day Sample 4 ............... 97 A 23 Oxygen Concentration versus time, Data 1,000 cc/m 2 /day Sample 5 ................ 99 A 24 Oxygen Concentr ation versus time, Data 1,000 cc/m 2 /day Sample 6 .............. 101 A 25 O xygen Concentration versus time, Data 1,000 cc/m 2 /day Sample 7 .............. 103 A 26 Oxygen Concentration versus time, Data 1,000 cc/m 2 /day S ample 8 .............. 105 A 27 Oxygen Concentration versus time, Data 1,000 cc/m 2 /day Sample 9 .............. 107 A 28 Oxygen Concentration versus time, Data 1,000 cc/m 2 /day Sample 10 ............ 109 A 29 Oxygen Concentration versus time, Data 1,000 cc/m 2 / day Sample 11 ............ 111 A 30 Oxygen Concentration versus time, Data 1,000 cc/m 2 /day Sample 12 ............ 113 A 31 Oxygen Concentration versus time, Data 10 cc/m 2 /day Sample 1 ................... 115 PAGE 10 10 A 32 Oxygen Concentration versus time, Data 10 cc/m 2 /day Sample 2 ................... 118 A 33 Oxygen Concentration versus time, Data 10 cc/m 2 /day Sample 3 ................... 121 A 34 Oxygen Concentration versus time, Data 10 cc/m 2 /day Sample 4 ................... 124 A 35 Oxygen Concentration versus time, Data 10 cc/m 2 /d ay Sample 5 ................... 127 A 36 Oxygen Concentration versus time, Data 10 cc/m 2 /day Sample 6 ................... 130 A 37 250 m precision orifice, perforation radius = 1.24475 x 10 4 m ...................... 133 A 38 200 m precision orifice, perforation radius = 1.02445 x 10 4 m ....................... 136 A 39 100 m precision orifice, perforation radius = 5.0065 x 10 5 m ........................ 139 PAGE 11 11 LIST OF FIGURES Figure page 1 1 Unsteady State Diffusion in a Polymer Film with Uniform Initial Distribution and Surface Concentrations ................................ ................................ ...................... 24 1 2 Concentration versus time curve of a gas in a polymer ................................ ...... 25 1 3 Gas Transmission Rate properties from a packaging film specification sheet for breathable film manufactured by Cryovac Division of Sealed Air Corporation. ... 27 1 4 Dynamic Accumulation Apparatus ................................ ................................ ...... 28 1 5 Oxygen accumulation over time in purged chamber ................................ ........... 28 1 6 Cell arrangement ASTM D 3985 ................................ ................................ ......... 29 2 1 A ccomplished oxygen accumulation versus time ................................ ............... 34 2 2 Jablonski diagram showing the process of fluorescence. ................................ ... 35 2 3 Fluorescence Decay with and without oxygen. ................................ ................... 36 2 4 Sinusoidal Curves for Frequency Modulated Excitation ................................ ..... 37 3 1 Diagram of dynamic accumulation chamber with film sample. ........................... 39 3 2 Node setup for Dynamic Accumulation Model ................................ .................... 41 4 1 R epresentation of the flow through method for measuring oxygen transmission through a perforated film. ................................ ................................ ................... 48 4 2 R epresentation of the static method for measuring oxygen transmission through a perforated film. ................................ ................................ ................................ 49 4 3 Profile of Oxygen Accumulation Chamber with centered perforation. ................. 49 5 1 Representation of Experimental Oxygen Accumulation Chamber ...................... 61 5 2 Accumulation chamber and perforation discretized into meshes by COMSOL Multiphysics ................................ ................................ ................................ ........ 63 5 3 Oxygen accumulation chamber discretized into concentric cylindrical discs surrounded by toroids ................................ ................................ ......................... 63 6 1 Mean OTR and 95% Confidenc e Interval for A) low transmitter, B) moderate transmitter, and C) high transmitter film for both the dynamic accumulation and the steady state method ................................ ................................ ..................... 66 PAGE 12 12 6 2 Validation of model of dynamic acc umulation with non perforated films. ............ 67 6 3 OTR ratio vs time ratio. ................................ ................................ ....................... 68 6 4 Curve of Oxygen Concentration at Sensor versus time generated by experimental data and model using Finite Element and Finite Volume Method for A) 250 m, B) 200 m, and C) 100 m precision orifice mounted on accumulation chamber. ................................ ................................ ................................ ............ 70 A 1 Accompl ished Oxygen Plot 10,000 cc/m 2 /day Sample 1 ................................ .. 144 A 2 Accomplished Oxygen Plot 10,000 cc/m2/day Sample 2 ................................ 144 A 3 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 3 ................................ .. 144 A 4 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 4 ................................ 145 A 5 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 5 ................................ .. 145 A 6 Accomplished Oxygen Plot 10,000 cc /m 2 /day Sample 6 ................................ 145 A 7 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 7 ................................ 146 A 8 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 8 ................................ .. 146 A 9 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 9 ................................ .. 146 A 10 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 10 ............................... 147 A 11 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 11 ................................ 147 A 12 Accomplished Oxygen Plot 10,000 cc/m 2 /day Sample 12 ................................ 147 A 13 Accomplished Oxyg en Plot 1,000 cc/m 2 /day Sample 1 ................................ .... 148 A 14 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 2 ................................ .... 148 A 15 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 3 ................................ .... 148 A 16 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 4 ................................ .... 149 A 17 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 5 ................................ .... 149 A 18 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 6 ................................ .... 149 A 19 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 7 ................................ .... 150 A 20 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 8 ................................ .... 150 PAGE 13 13 A 21 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 9 ................................ .... 150 A 22 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 10 ................................ .. 151 A 23 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 11 ................................ .. 151 A 24 Accomplished Oxygen Plot 1,000 cc/m 2 /day Sample 12 ................................ .. 151 A 25 Accomplished Oxygen Plot 10 cc/m 2 /day Sample 1 ................................ ......... 152 A 26 Accomplished Oxygen Plot 10 cc/m 2 /day Sample 2 ................................ ......... 152 A 27 Accomplished Oxygen Plot 10 cc/m 2 /day Sample 3 ................................ ......... 152 A 28 Accomplished Oxygen Plot 10 cc/m 2 /day Sample 4 ................................ ......... 153 A 29 Accomplished Oxygen Plot 10 cc/m 2 /day Sample 5 ................................ ......... 153 A 30 Accomplished Oxygen Plot 10 cc/m 2 /day Sample 6 ................................ ......... 153 PAGE 14 14 LIST OF SYMBOLS A Area of Transmission normal to gas flow through polymer Coefficient associated with oxygen transfer from below Coefficient associated with oxygen transfer from inside Coefficient associated with oxygen transfer from outside a p Coefficient associated with a particular control volume Coefficient associated with oxygen transfer from above b i,j constant, right side of algebra ic equation for a particular control volume c Concentration of gas in film Ambient oxygen concentration in atmospheric air Average oxygen concentration in accumulation chamber with continuous film C chamber (r,z,t) oxygen concentration in accumulation chamber with perforation as a function of position and time Concentration of Oxygen in non perforated film Oxygen Concentration at a particular node and time in model with non perforated film Oxygen concentration ins ide the accumulation chamber with non perforated film D Diffusion Coefficient across film Diffusion coefficient of oxygen air f time step weight coefficient PAGE 15 15 Intensity of emitted light from fluorophore at a particular oxygen concentration Intensity of emitted light from fluorophore in the absence of oxygen J Diffusive Flux Stern Volmer constant l Thickness of film L Distance from film edge to sensor or accumulation chamber effective Effective diffusion path length in a perforatio n n i,j number of moles of oxygen in a control volume Number of nodes inside non perforated film moles of oxygen Nr number of discretized radial positions in accumulation chamber with perforation Number of no des inside accumulation chamber with non perforated film Nz number of discretized axial positions in accumulation chamber with perforation OTR Oxygen Transmission Rate Oxygen Transmission Rate inputted into model of dynamic accumulation chamb er with continuous film Oxygen Transmission Rate calculated from model output of dynamic accumulation chamber with continuous film p Partial pressure of gas PAGE 16 16 Permeability of Film Oxygen Permeance of the film partial pressure of oxygen in the ambient environment outside the accumulation chamber partial pressure of oxygen in the accumulation chamber Q Amount of Gas Passing through film R gas law constant R chamber radius of accumulation chamber with perforation r i radial position in accumulation chamber with perforation inside radial boundary of a control volume outside radial boundary of a control volume r perforation radius of perforation s Solubility coefficient of film t Time T absolute temperature Time scale for oxygen in non perforated film Time ratio for oxygen in film to oxygen in accumulation chamber Time scale for oxygen in accumulation chamber with non per forated film V i,j Volume of a control volume in accumulation chamber with perforation Volume of accumulation chamber with non perforated film Position inside the dynamic accumulation system including non perforated film and the accumulation cham ber PAGE 17 17 bottom axial boundary of a control volume z j axial position in accumulation chamber with perforation top axial boundary of a control volume Time step Node width inside film Nod e width inside accumulation chamber Dimensionless property of film Dimensionless property of interface on film side Dimensionless property of accumulation chamber Dimensionless property of interface on accumulation chamber s ide Decay time of emitted light from fluorophore at a particular oxygen concentration Decay time of emitted light from fluorophore in the absence of oxygen PAGE 18 18 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PERFORMANCE CHARACTERIZATION OF DYNAMIC ACCUMULATION METHOD FOR MEASURING THE OXYGEN TRANSMISSION RATE OF POLYMER FILMS USING FLUORESCENCE OXYGEN DETECTION By Ayman Abdellatief May 201 4 Chair: Bruce Ari Welt Major: Agricultural and Biological Engineering Oxygen plays an important role in the preservation or degradation of many products. Rate at which oxygen transmits through a material is an imp ortant parameter in designing packaging for many products. The current method most commonly used for measuring oxygen transmission rates (OTR) of packaging materials is complex, expensive, and requires a lot of maintenance. High respiring products are a s pecial case that has oxygen requirements that exceed what can be provided by the highest oxygen transmitting films. Therefore it becomes necessary to incorporate perforations in packaging of high respiring products. More knowledge on how oxygen is distrib uted in a perforated package is necessary to design a package for high respiring products. A new method for measuring oxygen transmission rate through non perforated films was developed for this study. The new method assumed oxygen concentration in the tes t chamber was uniform at all times. A model was developed in order to test the validity of this assumption. The new method was then compared to the method most PAGE 19 19 commonly used for measuring OTR through non perforated films by measuring the same packaging fil ms using both methods. Another model was developed to better understand the distribution of oxygen in a perforated package. This model predicts oxygen concentration in a perforated chamber at any given location and time. The mathematical model was develop ed using two methods. One method involves discretization of the perforation which was more complex and require d expensive software. The other was an approximation method which was simple enough to a write a computer algorithm. Results from the new method f or measuring OTR of non perforated films were similar to results from the common method. The mathematical model that describes how the new method measures OTR of non perforated films was validated and demonstrates that any film available with standard labo ratory size equipment can be measured. Output from mathematical model that predicts oxygen concentration in a perforated chamber agreed with experimental data. However, the simpler method for this model was slightly less accurate but still gave a reasonabl e approximation. Therefore the loss in accuracy is justified. PAGE 20 20 CHAPTER 1 INTRODUCTION AND OBJECTIVES Background Gas transmission through a polymer is an important characteristic for packaging applications. Oxygen transmission, in particular, is critical i n packaging design due to Maintaining oxygen levels around fresh fruits and vegetables in a specific range below ambient atmosphere could result in shelf life extension (Barth, 2012) (Kader, 1986) (Kim J. L., 2005) (Laties, 1978) (Yaptenco, Lacao, Esguerra, & Serrano, 2010) Some products such as meats often require an oxygen free atmosphe re (Arritt, Jahncke, Pierson, & Williams, 2007) (Arvanitoyannis & and Stratakos, 2012) (Dufresne, Smith, Liu, Tarte, Blanchfield, & Austin, 2000) (Lagerstedt A., 2011) (Limbo, et al., 2013) (Serheim & Hoy, 2013) Flexible organic LEDs (O is another example of a product who has a limited lifetime due to degradation of the organic material in the presence of oxygen (Mocon, 2011). M aterials used for packag ing must meet their products oxygen requirement. Therefore, it is necessary to measure oxygen transmission through packaging materials. Permeation Theory Steady State Theory Oxygen transmits through a polymer film by permeation. Permeation is a combination of processes involving adsorption, solubility of the permeant in the material diffusion of the permeant through the material, and desorption. A gas will permeate through a polymer in the net direction from high concentration to low concentration limited by the solubility and diffusion of the gas in the polymer. The diffusive flux, J, of a PAGE 21 21 gas in a polymer is the amount (Q) passing through a surface of area (A) normal to the direct ion of flow during time (t), i.e. (1 1 ) The diffusive flux of a gas through a film is directly proportional to the concentration gradient, through film introduction of the diff usion coefficient, D, as a constant of proportionality. For many thin film packaging applications, the diffusion coefficient may be assumed to be constant. However, polar polymers, such as ethyl vinyl alcohol (EVOH) and polyamide (PA) the diffusion coeffi cient is dependent on the moisture content of the ambient environment For multi layer film s each unique material layer has an associated diffusion coefficient Equation 1 2 ( 1 2) Equation 1 2 can be integrated from the concentration at one surfa ce, c1, to the opposite surface c 2 across a film of thickness l and arranged to: ( 1 3) T he right hand side of equation 1 3 can be subs tituted for the diffusive flux J which yields. ( 1 4) At sufficiently low concentrations as ( 1 5) PAGE 22 22 Where S is the solubility coefficient and p is the partial p ressure of the gas. Equation 1 5 can be substituted into equation 1 4 which gives. ( 1 6) The product is the permeability coefficient and is represented by yielding ( 1 7) Which can be rearranged to ( 1 8) There are four assumptions made in the above treatment of permeation; the diffusion is in a steady state condition, concentration is a linear function of distance within the polymer, diffusion takes place in only 1 dimension, and D and S are independent of concentration (Robertson, 1993) Table (1 1) shows the oxyge n permeability coefficients of some common polymer s (Lange & Wyser, 2003) PAGE 23 23 Table 1 1 Oxygen Permeability Coefficients of Common Polymers Oxygen permeability at 23 C 50% or 0% RH Polymer [cm 3 mm/(m 2 day atm)] Poly(ethylene terephthalate (PET) 1 5 Polypropylene (PP) 50 100 Polyethylene (PE) 50 200 Polystyrene (PS) 100 150 Poly(vinyl chloride) (PVC) 2 8 Poly(ethylene naphthalate) (PEN) 0.5 Polyamide (PA) 0.1 1 ( dry) Poly(vinyl alcohol) (PVAL) 0 .02 (dry) Ethylene vinyl alcohol (EVOH) 0.001 0.01 (dry) Poly(vinylidene chloride) (PVDC) 0.01 0.3 Unsteady State Diffusion The polymer is initially free of the gas being measured unless it is stored in an environment containing the test gas. The polymer must become saturated with the test gas before a constant flow of test gas maybe observed If one face (at x = 0 ) of a membrane is kept at a constant concentration C 1 and the other face (at x = l ) at C 2 and the polymer initially contains no test gas, there is a finite interval of time during which the s teady state condition is set up (Figure 1 1). The solution for concentration as a function of time in the polymer is given by Equation 1 9 (Crank, 1975) ( 1 9) PAGE 24 24 Figure 1 1. Unsteady State Diffusion in a Polymer Film with Uniform Initial Distribution and Surface Concentrations The rate at which gas or other diffusing su bstance emerges from the face x=l of the polymer is given by ( ) which is derived from Equation 1 9. By integrating with respect to t, we obtain the total amount of gas diffusing Q t which has passed through the polymer in time t (Equation 1 10). ( 1 10) T he polymer surface where the test gas emerges C 2 is zero i n the most common experimental setup Based on this simplification Equation 1 10 can be re arranged to ( 1 11) Equation 1 11 approaches the line ( 1 12) The t intercept l 2 /6D is known as the lag time. (Crank, 1975) From the lag time the diffusion coefficient can be estimated ( Figure 1 2 ) PAGE 25 25 Figure 1 2 Concentration versus time curve of a gas in a polymer Permeability U nits The standard unit for permeability, so lutes transporting through solvents, is the same units as diffusion However, when describing gas transmissio n through a polymer, (Equation 1 8 ), units are typically expressed as The advantage of these units is the ability to calculate gas flux under different conditions since the units of practical parameters such as membrane area, thickness, and applied differential pressure are included in the composite units. The disadvantage is that it is difficult to compare the value of permeability coefficients with other types of transport such as dissolved solid solutes and miscible liquids in a given medium (Yasuda, 1975) Table (1 2) PAGE 26 26 Table 1 2. Common Permeability Coefficients Units and Conversion Factors. Before conversion After conversion 1 7.50x10 4 6.57x10 10 10 1 7.50x10 5 6.57x10 9 1.32x10 02 9.87x10 06 8.64x10 3 3.87x10 14 2.90x10 17 2.54x10 3 9.82x10 12 7.37x10 15 6.45x10 1 1.52x10 11 1.1 4x10 14 1 1.54x10 11 1.16x10 14 1.01 1.33x10 3 1 8.75x10 13 3.04x10 14 2.28x10 17 2.x10 3 Adapted from (Yasuda & Stannett, 1975) Oxygen Permeability v er s es Oxygen Transmission Rate (OTR) Oxygen p ermeability is a fundamental property of a material and is independent of thickness. Instruments measure the o xygen transmission rate (OTR) and the permeability coefficients can be determined by multiplying the OTR by the thickness for the typical case of 1 atm partial pressure difference Film manufacturers usually coefficients as shown in Figure ( 1 3 ) Oxygen transmission rate should always be reported at a specified temperature and an oxygen partial pressure difference of 1 atm. PAGE 27 27 Figure 1 3. Gas Transmission Rate properties from a packaging film spec ification sheet for breathable film manufactured by Cryovac Division of Sealed Air Corporation Methods for Measuring OTR There are many ways to measure the O TR of polymer films. The three most common include pressure or volume increase methods, the steady state method, and the dynamic accumulation method. Pressure and Volume Increase Methods A standard for measuring transmission of any gas through a film incl uding oxygen is defined in American Standard Testing Methods (ASTM D 1434) Within this standard there are two sub methods, including a manometric method and a volumetric method. Briefly, a sample is mounted in a gas transmi ssion cell to form a sealed semi barrier between two chambers. One chamber initially contains pure test gas at a specific higher pressure, and the other chamber receives the test gas at a lower pressure. In the manometric method the chamber receiving the t est gas is initially evacuated and transmission of the gas through the film is determined by measuring increase in pressure. In the volumetric method the lower pressure chamber is maintained at atmospheric pressure and gas transmission rate is determined b y measuring change in volume. This method has been cited in the literature (George, 2006) ; (Laohakunjit N., 2004) (Raj, 2005) but is not considered as accurate compared to m ethods specifically created for measuring OTR. PAGE 28 28 Dynamic Accumulation Method Dynamic accumulation method is the main focus for this work. It uses nitrogen to purge the sensor side of the permeation cell prior to the test. Once purged, the chamber valves are sealed. The oxygen enriched side may be left open to the atmosphere, flushed with air using a pump or compressed air or flushed with pure oxygen. Schematic of dynamic accumulation method is depicted in Figure (1 4 ). Figure 1 4 Dynamic Accumulation App aratus Over time oxygen in the oxygen deficient side accumulates and asymptotically approaches the concentration of the oxygen enriched side. This behavior is depicted in Figure (1 5 ) Figure 1 5 Oxygen accumulation over time in purged chamber PAGE 29 29 Steady S tate Method The most commonly cited method for measuring OTR is defined by (ASTM D 3985) (Brennan, Haag, White, & Brown, 1998) (Kim J. G., 2004) (Yasuda, 1975) (Mexis & Kontominas, 2012) Measuring OTR using this method involves a permeation cell where the sample ma terial separates two chambers. Gases at similar absolute pressures and flow rates pass on e ither si de of the sample film. An oxygen rich gas (e.g. air or pure oxyge n) passes on one side, while oxygen deficient gas (e.g. purified nitrogen) passes on the other side of the sample. The oxygen deficient gas is called the carrier gas. The carrier gas stream c arries oxygen that permeated through the film sample area from the oxygen rich gas to an oxygen sensor Since the absolute pressure is the same on both sides of the sample, this method is often referred to as isostatic. Since the oxygen partial pressure di fference is held constant, this method is also referred to as steady state. This isostatic, steady state method is depicted in Figure 1 6 Figure 1 6 Cell arrangement ASTM D 3985 Statement of the P roblem Th e dynamic accumulation method or similar meth ods has been documented in the literature (Ghosh & Anantheswaran, 2001) ; (Kim, 2004) ; (Moyls, Hocking, Beveridge, & Timber, 1992) ; (Mo yls L. 2004) ; (Siro, 2010) with non perforated films Each of these studies, however, used different methods for measuring PAGE 30 30 oxygen concentration. Some studies withdrew gas from the chamber and measured accumulated oxygen co ncentration using a gas chromatograph. In one study (Moyls, Hocking, Beveridge, & Timber, 1992) relatively small samples were withdrawn compared to the chamber volume but only a few samples could be taken In another stud y (Ghosh & Anantheswaran, 2001) the cell was repurged after each measurement resulting in exceedingly long test times. Recently, florescence based oxygen measurement was applied to the dynamic accumulation method for measur ing OTR of non perforated films (Abdellatief & Welt, 2009) (Siro, 2010) which made this method more attractive There is a need to validate the dynamic accumulation by comparing it with the alread y well established steady state method. Once validated, criteria for design of instruments using dynamic accumulation method to measure non perforated films need to be established. Oxygen transmission through a perforation needs to be modeled which could b e possibly used to design an instrument for measuring oxygen transmission rate in perforated films and has application in Modified Atmosphere Packaging. Research Objectives The ma in objectives of this work were 1. Compare Dynamic Accumulation Measurement with a fluorescent oxygen sensor of Oxygen Transmission Rate to Steady State Measurements by measuring the same films using both methods. 2. Mathematically Model and simulate oxygen concentration as a function of position and time in accumulation chamber during m easurement of non perforated films using dynamic accumulation method Once validated this model will be used to establish criteria for instruments measuring OTR of non perforated films using dynamic accumulation method. 3. Mathematically Model and simulate ox ygen concentration as a function of position and time in a dynamic accumulation chamber where oxygen is transmitted through a perforation PAGE 31 31 CHAPTER 2 DESCRIPTIONS OF THE DYNAMIC ACCUMULATION METHOD AND STEADY STATE METHOD Steady State Method Theory of Op eration F low rates of both the oxygen rich and oxygen deficient sides of a polymer film are known as well as the area of the sample and oxygen concentration of the oxygen rich side. The oxygen sensor measures oxygen concentration in the carrier gas side. T heoretically, the oxygen transmission rate is the mass fraction of oxygen in the oxygen deficient side multiplied by the flow rate divided by the sample area. In practice with instruments that use this method only a fraction of the oxygen molecules actuall y hit the sensor. Typically these instruments need to be calibrated against a film of known OTR to compensate for this. Steady State Method Oxygen Sensing Mechanism Oxygen can be detected electrochemically and is generally known as a Clark electrode (Ramamoorthy, Dutta, & Akbar, 2003) (Severinghaus & Astrup, 1986) (Amao, 2003) (Clark, 1959) ASTM D 3985 specifies a coulometric oxygen sensor wh ich is a type of electrochemical sensor The c oulometric sensor generates an electric current in proportion to oxygen entering the sensor. It contains a cadmium anode a graphite cathode and a catalytic platinum surface The cathodic and anodic reactions a re, respectively: O 2 + H 2 O + 2e 2OH ( 2 1 ) Cd + 2OH Cd(OH) 2 + 2e ( 2 2 ) PAGE 32 32 E lectrons create an electrical current which flow across a scaling load resistor from which voltage is measured. V oltage corresponds to the amount of oxygen transmi tting through the film to the sensor. D isadvantages of this method include high capital and operating expense s limited sensor life due to the consumption of the cadmium anode consumption of oxygen at the electrode surface, and sensor sensitivity to conde ab ove refrigeration temperature for most foods. Dynamic Accumulation Method Theory of Operation C alculat ing OTR for this method was developed from the well known permeation relationship (Equation 2 3 ) and assumes the concentration in the accumulation chamber is uniform at any time : ( 2 3 ) Where n is moles of oxy gen, is the permeation coefficient in molar units, p is partial pressure, A is sample area, l is sample thickness and t is time. Equation 2 3 describes the rate at which oxygen permeates through a sample of known area and thickness under a driving fo rce defined by the partial pressure difference on either side of the sample. It is assumed that the volume of the sensor side of the permeation chamber remains constant With volume and pressure constant, oxygen partial pressure is directly related to mole s of oxygen via Equation 2 4 : (2 4 ) PAGE 33 33 Where V is total chamber volume, R is ideal gas law constant and T is absolute temperature. Substitution of Equation 2 4 into Equation 2 3 provides: (2 5 ) Integrating Equation 2 5 from the beginning of the experime nt (t = 0) to time, t, yields: ( 2 6 ) The quantity is known as the accomp lished oxygen accumulation Equation 2 6 suggests that plotting the accomplished oxygen accumulation versus time should yield a straight line whose slope is pr oportional to OTR (Figure 2 1) once the permeation coefficient is converted from molar units to volumetric units: ( 2 7 ) Equation 2 7 provides actual O TR under any cond itions of driving force. Therefore, standardized OTR is determined by multiplying slope by chamber volume and divided by sample area. PAGE 34 34 Figure 2 1 A ccomplished o xygen accumulation v er s us time Dynamic Accumulation Oxygen Sensing Mechanism Intensity and decay based m easurements The oxygen measurement technique is based upon the fluorescence quenching of an organo metallic fluorescent dye (Kautsky, 1939) (Freeman & Seitz, 1981) immobili zed in a gas permeable hydrophobic polymer. A fluorophore made from ruthenium complexes was used as the dye in this study. When light shine s on the ruthenium dye used in this study each valence electron in the dye absorbs a photon in the blue region of the electromagnetic spectrum and is excited from the ground state to a higher energy state. As the electrons return to the ground state from the higher energy state photons that were absorbed are emitted with less energy and wavelengths within the red region of the spectrum. The process of electrons absorbing photons and going from the ground state to a higher energy state and then returning to the ground state and emitting the same photons at longer wavelengths tha n were previously absorbed at short er wavele ngths is known as photoluminescence Fluorescence is the type of photoluminescence that occurs when blue light shine s on the ruthenium dye. Th e PAGE 35 35 process of fluorescence is depicted in the Jablonski diagram (Figure 2 2) named after Polish physicist Aleksand er Jablonski. Figure 2 2 Jablonski diagram showing the process of fluorescence. The presence of oxygen lowers the intensity of the lifetime of emitted light (photons) which is known as decay time. This process is known as quenching. Optical oxygen s en sors measure intensity of lifetime of emitted light (Mohr & Wolfbeis, 1995) (Vaughan, Baron, & Narayanaswamy, 1996) (McMurray, Douglas, Busa, & Garley, 1994) (Stern & Volmer, 1919) The quenching process depends on the number of collisions between oxygen molecules and excited valence electrons where the energy from the excited valence electron is transferred to oxygen molecules during a collis ion, hence, reducing emission intensity (Peterson, Fitzgerald, & Buckhold, 1984) as well as the fluorescent decay time of the dye (Vanderkooi, Maniara, Green, & Wilson, 1987) The reduction of decay time with presence of oxygen is depicted in Figure 2 3 Excited State PAGE 36 36 Figure 2 3 Fluorescence Decay with and without oxygen. Since this energy transfer mechanism does not consume oxygen, the oxygen content of the enclosed space is not changed by the measurement, an d therefore the OTR measurement is not affected. Changes in emission intensity and decay time are related to the oxygen partial pressure and can be calibrated to determine oxygen concentration. Stern Volmer (Stern & Volmer, 1919) equations used to calibrate intensity based and decay time based oxygen fluorescent sensors (Fischkoff & Vanderkooi, 1975) (Carraway, Demas, Degraff, & Bacon, 1998) (Mc Donagh, MacCrath, & McEvoy, 1998) (Demas, De Graff, & Xu, 1995) (Hartmann, Leiner, & Lippitsch, 1995) (2 8) (2 9) Frequency Modulated Excitation When the valence electrons of a fluorophore are excited with varying light intensity at a constant frequency forming a sinusoidal curve the emitted light has the same frequency with a tim e delay or phase shift (Figure 2 4). Oxygen concentration can be calibrated to this phase shift (Valledor, Campo, Sanchez Barragan, Costa Fernandez, Alvarez, & Sanz Medal, 2001) (Holst, Kosher, Voges & Lubbers, 1995) PAGE 37 37 (Murtagh, Ackley, & Shahriari, 1996) (Berndt & Lakowicz, 1992) Oxygen concentrations are measured using f requency m odulated e xcitation in this work. This method for measuring o xygen concentration was selected because intensity and decay based oxygen measurement are affected by ambient light and drift due to changes in optical paths and photobleaching of the dye. Frequency m odulated e xcitation is not susceptible to drift and less susceptible to photobleaching. Figure 2 4 Sinusoidal Curves for Frequency Modulated Excitation PAGE 38 38 CHAPTER 3 MATHEMATICAL MODELING AND DESIGN CRITERIA FOR THE DYNAMIC ACCUMULATION METHOD WITH NON PERFORATED FILMS Partial Differential Equation and Boundary Conditions for the Dynamic Accumulation System with A Non Perforated Film The dynamic accumulation method is attractive because it is operationally simple, requires consumption of considerably less gas and utilizes a relatively inexpe nsive and robust sensor as compared to the steady state method. While dynamic accumulation experiments are simple to perform, the mathematics involved to extract the OTR value from measured data are more involved than f rom the steady state approach. A ssum ing that the oxygen concentration is uniform throughout the accumulation chamber simplifies the calculation of OTR. For practical purposes this assumption allows for oxygen to be measured at any point in the accumulation chamber. L imits of these assumptio ns must be understood in order to ensure that calculated OTR values are correct. Additionally, better understanding these limits will provide useful guidance for dynamic accumulation instrument designers. Therefore a model was developed describing oxygen transfer from the ambient environment through a non perforated film through the accumulation chamber to the oxygen senso r. A model 2 nd law of diffusion. A schematic of the film and accumulation chamber model is shown in Figure 3 1 PAGE 39 39 Figure 3 1 Diagram of dynamic accumulation chamber with film sample. Oxygen concentration profiles for the film sample with initial and boundary conditions are provided by Equations 3 1, 3 2, 3 3 and 3 4. ( 3 1) Initial c ondition (IC 1) is the initial concentration of oxygen in the film. For our purposes, we assumed that the film was initially free of oxygen: ( 3 2) Boundary c ondition (BC 1) is the concentration of o xygen at outer surface of film, Since the p ermeability coefficient already contains the oxygen solubility coefficient in the polymer, it was ( 3 3) Boundary c ondition (BC 2) states that oxygen emerges from the sample surface at the same rate that it appears in the accumulation chamber: ( 3 4) PAGE 40 40 Oxygen concentration profiles in the accumulation chamber are determined from Equation 3 5 and initial and boundary conditions provided in Equations 3 6, 3 7 and 3 8. ( 3 5) Initial c ondition (IC 2) is the initial oxygen concentration in the accumulation chamber which w as assume d to be zero since the accumulation chamber is purged with n itrogen before the measurement. ( 3 6) Boundary c ondition (BC 3) is the concentration of oxygen at surface of film sample adjacent to space at time, t, and is proportional to the oxygen concentration at the surface of the space adjacent to film sample Again it was assumed the proportionality constant was equal to 1. ( 3 7) Boundary c ondition (BC 4) states that gas does not diffuse through the wall of the accumulation chamber. ( 3 8) Numerical Solution for Mathematical Model of Dynamic Accumulation System with a Non perforated F ilm A model was defined for the dynamic accumulation chamber as a set of 2 linked 2 nd order partial differential eq uations with respect to space and first order with respect to time. Crank Nicholson implicit method was used to estimate the solution to the model. M odel for dynamic accumulation with a non perforated film sample was divided into 5 different types of node s (Figure 3 2) PAGE 41 41 Figure 3 2 Node setup for Dynamic Accumulation Model Mass balances were performed at each node (Equations 3 9 3 1 2 3 1 4 3 1 8 and 3 2 1 ). The concentration at each node was calculated as average of the current concentration and concen tration at next time step (Crank & Nicolson, 1947) Equations for the next time step were determined from mass balances and solved simultaneously for all nodes in a tridiagonal matrix at each time step for the future time step (Equations 3 9 3 1 2 3 1 4 3 1 8 3 2 1 ). Node Type 1 (Surface Node); i =1 ( 3 9 ) w hich can be re arranged to ( 3 1 0 ) w here PAGE 42 42 ( 3 1 1 ) Node Type 2 (Film Nodes); 1< i < N f where N f is the number of nodes in the film ( 3 1 2 ) w hich can be re arranged to ( 3 1 3 ) Node Type 3 (Interface node); i = N f ( 3 1 4 ) w hich can be re arranged to ( 3 1 5 ) w here ( 3 1 6 ) a nd ( 3 1 7 ) Node Type 4 (Space Nodes); N f < i < N f + N s where N s is the number of nodes in the space ( 3 1 8 ) w hich can be re arranged to ( 3 19 ) w here ( 3 2 0 ) Node Type 5 (Bottom Node); i = N f + N s PAGE 43 43 ( 3 2 1 ) w hich can be re arranged to ( 3 2 2 ) Design Cr iteria for Dynamic Accumulation System with Non perforated F ilms This model can be used to generate accumulation chamber oxygen concentratio n gradients (the difference in oxygen concentration between the inner edge of the film and oxygen sensor) for a film of given permeability P, thickness l, on a chamber of given length L, with oxygen diffusion coefficient, D, in air. Oxygen transmission rat e may be calculated from oxygen concentrations predicted at the sensor location within the accumulation chamber via the method described by (Abdellatief & Welt, 2013) When oxygen concentration gradients within the accumulation chamber are insignificant, apparent OTR (OTR apparent ) should be the same as the actual OTR of the film sample used as an input to the model (OTR actual ), where ( 3 23 ) V ariables that determine significance of any expected oxygen concentration gradient in the accumulation chamber are distance (L) between the inner surface of film to the sensor, thickness of film (l), permeability of oxygen in the film (P) and diffusion of oxygen in the accumulation chamber (D). Time scales and a t ime scale ratio may be defined for oxygen permeating through the film and accumulation chamber. Time scales for the space (accumulation chamber) is ( 3 2 4 ) The time scale for the film sample is PAGE 44 44 ( 3 2 5 ) The ratio of the two time scales provides a relative measure of the time required for oxygen to traverse the film sample to the time required to traverse the space between the film and sensor in the accumulation chamber. The time scale ratio is ( 3 2 6 ) The model can be used to show operational limits of the dynamic accumulation method in terms of accumulation chamber dimensions, sample thickness and permeability. To visualize these limits, the O TR ratio (Equation 3 2 7 ) vs time scale ratio (Equation 3 2 6 ) was plotted as a function of accumulation chamber length, L, and film permability, P. ( 3 2 7 ) As gradients become increasingly significant, OTR ratio de viates significantly from unity At some practical limit that depends upon desired resolution of the test the and gradients should be accounted for in estimation of fi lm OTR. Calculating Oxygen Transmission Rates in Dynamic Accumulation Systems with N onuniform Oxygen Concentrations for Non P erforated F ilms To determine OTR for case s where well mixed assumption does not apply average concentration at each time (Equatio n 3 2 8 ), which effectively mixes the chamber mathematically may be used as a technique for estimating film OTR When this technique is incorporated the simplifying assumption of a well mixed accumulation chamber, as described by (Abdellatief & Welt, 2009) may be used to determine OTR. ( 3 2 8 ) PAGE 45 45 CHAPTER 4 MATHEMATICAL MODELING OF OXYGEN TRANSMISSION THROUGH A MICROPERFORATION Application for Perforated Films Perforating films is a method used in the design of packaging of high respiring products ( m any fresh or minimally processed fruits and vegetables) (Fishman, Rodov, & Ben Yehoshua, 1996) (Fonseca, Oliviera, Lino, Brech t, & Chau, 2000) (Paul & Clark, 2002) (Sanz, Perez, Olias, & Olias, 1999) (Gonzales, Ferrer, Oria, & Salvador, 2008) (Oliviera, Fonseca, Ol iviera, Brecht, & Chau, 1998) (Ozdemir, Monnet, & Gouble, 2005) (Silva, Chau, Brecht, & Sargent, 1999) A common way to extend the shelf life of such products is to modify the atmosphere it is pac metabolic rate or rate of consumption of O 2 and rate of production of CO 2 (Kader, Zagory, & Kerbel, 1989) (Beaudry, Cameron, Shirazi, & Dostal Lange, 1992) (Exama, Arul, Lencki, Lee, & Toupin, 1993) (Cameron, Boylanpett, & Lee, 1989) (Amanatidou, Slump, Gorris, & Smid, 2000) (Hertog, Peppelenbous, Evelo, & Tijskens, 1998) (Carlin, Nguyen The, Hilbert, & Chambroy, 1990) (Lee, Haggar, Lee, & Yam, 1991) .Typically environments of O 2 concentrations lower than air and CO 2 concentrations much higher than a ir at refrigeration temperatures is applied (Hertog, Peppelenbous, Evelo, & Tijskens, 1998) (Lee, Haggar, Lee, & Yam, 1991) (Fonseca, Oliviera, Brect, & Chau, 2005) (Joles, Collier Cameron, Shirazi, Petracek, & Beaudry, 1994) Oxygen concentration in the package must not be allowed to go too low or the respiration of the product will shift from aerobic to anaerobic ( Barry Ryan, Pacussi, & O'beirne, 2007) (Joles, Collier Cameron, Shirazi, Petracek, & Beaudry, 1994) (Kim, Luo, Tao, Saftner, & Gross, 2005) (Varoquaux, Albagnac, T he, & Varoquaux, 1996) (Fonseca, Oliviera, Brect, & Chau, 2005) (Kader A. 1989) (Tarasila, Cameron, & Jole, 1994) (Hansen, PAGE 46 46 Sorenson, & Ca ntwell, 2001) (Pesis, Devir, Feygenberg, Arie, Ackerman, & Lichter, 2002) which will spoil the product. Unfortunately many respiring products have oxygen requirements that exceed what can be supplied by even the highest tran smitting films commercially available. Therefore it becomes necessary to incorporate perforations in the packaging of high respiring products to meet these oxygen requirements. Incorporating even small perforations in the packaging material drastically inc reases the oxygen transmission because the diffusion coefficient of oxygen in air is at least 6 orders of magnitude greater than the permeability of low density polyethylene, the highest oxygen transmitting film. (Gonzales, Ferre r, Oria, & Salvador, 2008) Attempts at Modeling Oxygen Transmission T hrough a P erforation Many methods have been applied in the literature to model oxygen transmission in a perforation. Several researchers modeled the oxygen transmission through perfora tions empirically (Emond, Castaigne, Toupin, & Desilets, 1991) (Fonseca, Oliviera, Lino, Brecht, & Chau, 2000) (Oliviera, Fonseca, Oliviera, Brecht, & Chau, 199 8) (Silva, Chau, Brecht, & Sargent, 1999) Unfortunately empirical models can only be applied to the conditions under which they were derived. The most common theoretical model applied in oxygen transmission through a perfor ation used by many (Fishman, Rodov, & Ben Yehoshua, 1996) (Paul & Clark, 2002) (Gonzales, Ferrer, Oria, & Salvador, 2008) (Techavises & Yoshio, 2008) Other theoretical models have been used in the literature. Renault, Souty, and Chambroy modeled gas diffusion from the ambient environment through the perforations using Stephen Maxwell gas laws and i (Renault. P., 1994a) e ffusion for designing a PAGE 47 47 modified atmosphere package with a perforation (Hirata, Makino, Ishikawa, Katsuura & Hasegawa, 1996) Most studies in the literature assume that the concentration inside a perforated package is uniform at any particular time. However when oxygen transmits very rapidly in small localized areas compared to the rest of the package, whic h is what happens in a perforated package, the oxygen concentration inside the package tends not to be uniform. The way to address this issue is to calculate the oxygen concentration at discrete points throughout the package rather than one uniform concent ration in the package. A t least two studies in literature have done this. Rennie and Tavoularis modeled a time dependent and space dependent modified atmosphere package using Finite Element Method with commercialized software (Re nnie & Tavoularis, 2009) Emond et al discretized a modified atmosphere package using the finite difference method. (Emond, Chau, Brecht, & Ngadi, 1998) These studies demonstrate that oxygen concentration in a perforated mod ified atmosphere package is not uniform. Experimental Procedure for Measuring Oxygen Transmission T hrough a Perforation There are 2 experimental procedures known in the literature for measuring oxygen transmission through a perforation, the flow through me thod and the static method. Flow T hrough Method In the flow through method the apparatus is setup as described in ASTM D 3985 (Ghosh & Anantheswaran, 2001) (Gonzalez, Ferrer, Oria, & Salvador, 2 013) Figure 1. The main problem with the flow through method for measuring oxygen transmission through perforations is the difficulty in maintaining the same pressure on both sides of PAGE 48 48 the sample. Even slight differences in pressure between both sides of the sample will cause convective flow through the perforation which will give an inaccurate measurement. Figure 4 1 Schematic representation of the flow through method for measuring oxygen transmission through a perforated film. Static Method In the st atic method a chamber is sealed with a perforated sample and flushed with an inert gas, typically nitrogen (Figure 4 2). Oxygen accumulates in the chamber through the perforat ion and is measured over time. There are some limitations to the static method. T otal volume of all samples extracted has to be insignificant compared to the total volume of the accumulation chamber otherwise accuracy will be significantly affected (Moyls, Hocking, Beveridge, & Timber, 1992) or the chamber has to be re purged after withdrawing each sample resulting in exceedingly long test times (Ghosh & Anantheswaran, 2001) These limitations can be overcome when using a fluorescent oxygen sensor which measures oxygen non in vasively without taking a sample. (Abdellatief & Welt, 2013) PAGE 49 49 Figure 4 2 Schematic representation of the static method for measuring oxygen transmission through a perforated film. Mathematical Model of Dynamic Accumulation Ch amber with a Perforation If the perforation is centered on the end of the chamber the model can be considered 2D axial symmetric with oxygen transfer in the radial and axial directions (Figure 4 3). Figure 4 3 Schematic Profile of Oxygen Accumulation C hamber with centered perforation. PAGE 50 50 The governing differential equation for oxygen transfer in the accumulation equation 4 1 (Crank, 1975) Oxygen delivered from the ambient environment to the accumulation chamber through the perforation can be approximated as a source term (4 1) The following initial and boundary conditions apply (Equations 4 2 4 5). Init ially there is no oxygen anywhere in the chamber (Equation 4 2). (4 2) Oxygen cannot transfer in through the center (Equation 4 3). (4 3) Oxygen cannot transfer out through the wall (Equa tion 4 4). ( 4 4) Oxygen cannot transfer up through the top (Equation 4 5). ( 4 5) Oxygen cannot transfer down through the bottom (Equation 4 6). ( 4 6 ) Average distance the oxygen molecule has to travel from the ambient environment through the perforation into the chamber is greater than the length of the perforation itself. Some studies on gas transmission through perforated films have PAGE 51 51 applied end corr ection terms for an effective diffusion length and got more accurate predictions from their model (Gonzales, Ferrer, Oria, & Salvador, 2008) (Fishman, Rodov, & Ben Yehoshua, 1996) (Paul & Clark, 2002) The reason is the density of the diffusing gas at the end of the perforation varies from the density in the ambient environment due to the end effects of the perforation. This effect is significant for the diameters and thi cknesses of typical micro perforated films used in modified atmosphere packaging. The effective diffusion path length for a narrow tube which resembles stomata in plants is given by Equation 4 7 (Meidner & Mansfield, 1968) Th is equation is derived by the resistance to diffusion at ends of the perforation and through the perforation itself. ( 4 7) Numerical Approximation for Model of Dynamic Accumulation Chamber with a Perforation The solution to Equation 4 1 cannot be obtained analytically and must be approximated numerically by discretizing the accumulation chamber into cylindrical discs in the center surrounded by toroids everywhere else (Figure 5 3 ). Moles of oxygen transferred into the accu mulation chamber are approximated as a source term and is delivered into the top center disc at each time step (Equation 4 8). (4 8) Oxygen is transferred to the adjacent positions by integrating the governing differential equation over the subdomain of each position, known as the Finite Volume Method (Equation 4 9). PAGE 52 52 (4 9) The other approach would be to perform mass balances at each surface of each position. Both procedures yield the same algebra ic equation, which has the form. (4 10) Where the coefficients a in a out a p a up and a down depend on the position, diffusion coefficient and time step size. As a result of toroids and discs in the discretized system varying i n size it becomes necessary to convert the algebraic equations in terms of moles (Equation 4 11) which ensures conservation of mass. (4 11) Since the volumes are known, Equation 4 11 can be rearranged so that the left side is i n terms of moles of each position and corresponding adjacent position for the next time step with the corresponding coefficients. The volumes are incorporated into the right side of the equation, b i,j along with all the other corresponding known values. (4 12) Constant f is typically set to 0, 1, or 0.5 and gives the weight of the current time step to the next time step. If f = 0 then the oxygen concentration at a particular position is only dependent on the oxygen concentrati on of the position and adjacent positions at the current time step and the equation becomes fully explicit. The advantage of an explicit scheme is that it does not require a solution of simultaneous equations however it is less accurate for a particular ti me step than other schemes and requires a smaller time step to be numerically stable. If f = 1 then the oxygen concentration at a particular PAGE 53 53 position is only dependent on the oxygen concentration of the position and adjacent positions at the next time step and the equation becomes fully implicit. The advantage of a fully implicit scheme is that it allows a larger time step and is more accurate than an explicit scheme for a particular time step at the expense of having to solve simultaneous equations. If f = 0.5 then the oxygen concentration at a particular position is equally dependent on the oxygen concentration of the position and adjacent positions at the current time step and next time step. This scheme is known as the Crank Nicolson scheme and also allo ws for larger time step but also requires a solution of simultaneous equations. It is considered the most accurate of the 3 schemes for a particular time step size (Crank & Nicolson, 1947) and is used in this study. The r igh t side of equation 4 12, b i,j is dependent on known values and is a function of the constant f, constants a in a out a p a up and a down and the current moles of oxygen at the position and adjacent positions and the ratio of their volumes to the volume of the position. Equation 4 12 has 5 unknowns which gives a sparse matrix of dimension (Nr 2 xNz 2 ) when all positions are solved simultaneously. Where Nr is the number of positions in the radial dimension and Nz is the number of positions in the axial dimensio n. When solving this (Nr 2 xNz 2 ) matrix with a direct method like Gaussian elimination, it requires on the order of (Nr 2 xNz 2 ) 3 operations for each time step. For Parabolic Partial Differential Equations dependent on time and space, the time step can be spl it by the number of spatial dimensions (2 in this case) and each spatial dimension can be solved separately but sequentially. This reduces the sparse matrix into a series of tridiagonal matrixes, (Equations 4 13 and 4 14 ), one for each dimension. Results PAGE 54 54 f rom solving one dimension are substituted into the next dimension in order to cancel out the bias in each dimension. This method is known as the Alternating Direction Implicit Scheme and reduces the order of operations in this problem from (Nr 2 xNz 2 ) 3 with a direct elimination method to (2xNr 2 xNz 2 ). The axial half of equation 4 12 is (4 13) Moles of oxygen calculated for each position in Equation 4 13 can be substituted into radial half of the time step which is (4 14) There are three different types of positions for both axial and radial dimensions. In the ax ial dimensions there is the top interior (between top and bottom ), and bottom positions, with corresponding coefficients a up a down a p an d right side of Equation 4 13, b i,j (Table 4 1 ). In the rad ial dimension there is the center interior (between center and wall ), and wall positions, with corresponding coefficients a in a out a p ,(Table 4 b 1) and right side Equation 4 14, b i,j (Table 4 b 2). Table 4 1 Coefficients a up a down and, a p in the axial direction Right side of equation 12, b i,j in axial direction Top Interior Bottom a up 0 a down 0 a p (1+a down ) (1+a up +a down ) (1+a up ) PAGE 55 55 Table 4 1 Continued b i,j Top Interior Bottom Table 4 2 Coefficients a in a out and, a p in the radial direc tion. Right side of equation 12, b i,j in radial direction Center Interior Wall a in 0 a out 0 a p (1+a out ) (1+a in +a out ) (1+a in ) Table 4 2 Continued b i ,j Center Interior Wall Simulation of Perforation Model Computation Time Minimization Strategies The r ight hand side, b i,j of Equations 4 13 & 4 14 and coef ficient a p must have the opposite sign from coefficients a in a out a up and a down for the simulation to be PAGE 56 56 simulation becomes unstable. As the simulation progresses and the o xygen he simulation. An algorithm can be i s increased incrementally by a specified percentage this percentage so it meets the criteria again and is within this specified percentage of the maximum possible value. Simulation Convergence versus Computational Cost For the simulation to converge number of discretized positions has to be increased until the solution does not change at any particular position and time. This is when the numeri cal solution is an exact representation of what the analytical solu tion would be if one existed. When the number of discretized positions increases, the number of computations per time step increases rapidly as well as the maximum possible time step decrea ses significantly. By doubling the number of discretized positions in each dimension for this particular problem the computation time can increase by as much as a factor of 200 per simulation. A way to alleviate this problem and maintain accuracy is to sta rt with a highly refined discretization to capture the initial rapid changes in the accumulation chamber. Once the oxygen concentration changes slowly after oxygen has reached every discretized point in the chamber, discretized control volumes can be merge d with each other and expanded until the desired time step can be reached. When oxygen transmission through the perforation is approximated as a source term there is a limit to how fine the accumula tion chamber PAGE 57 57 can be discretized. T op center control volu me cann ot have a radius less than the radius of the perforation itself. When control volumes get small enough in the simulation the top center control volume c ould possibly take an amount of oxygen from the perforation such that the oxygen concentration i n that control volume exceed s ambient concentration which is physically impossible PAGE 58 58 CHAPTER 5 EXPERIMENTAL SETUP AND PROCEDURES Comparison of Steady State Method to Dynamic Accumulation Three film samples were chosen from our laboratory stock to provide a broad range of OT R for measurement comparisons. Approximate expected OTR ranges for film samples are shown in Table 5 1: Table 5 1 Expected OTR range of samples tested. Description Approximate OTR Range (cc/m 2 /day) High Barrier 10 Me dium Barrier 1000 Low Barrier 10000 Steady State Measurements Steady state measurements were performed in accordance with ASTM D 3985 using a Mocon Oxtran 2/20 (Mocon, Inc., Minneapolis, MN). The t emperature was set to 23C and the instrument was set to convergence mode. A total of 6 measurements O xtran tester whereas 12 measurements were made on the moderate and high transmitter films on the ST modules. The reason for the difference in number of measurements was an apparent failure of one of the testing chambers on the MH unit which required extensive service by the manufacturer. Dynamic A ccumulation Measurements Dynamic accumulation experiments were performed using perme ation cells and florescence oxygen detection equipment from Oxysense, Inc. (Oxysense Model 301, Dallas, TX). The sensor side of the cell had an area of transfer of 16.62 cm 2 and a volume of 8.3 cc. Initially the cell was purged with compressed commerciall y pure PAGE 59 59 nitrogen. For the high transmitter film the, oxygen enriched side was fed with compressed air. For the low and moderate transmitter films compressed commercially pure oxygen was used as the circulation gas. Oxygen concentration in the accumulation c hamber was measured and recorded periodically during the test. O xygen Transmission Rate was subsequently calculated as previously described Statistical Analysis and Comparison of Steady State Measurements and Dynamic Accumulation Measurements of OTR For e ach method and each film, mean OTRs, standard deviations, and 95% confidence intervals using a t test were calculated. Differences in mean OTR at a 95% confidence interval between steady state and dynamic accumulation method for each film were also determi ned using an independent 2 sample t test. Validation of Mathematical Model for Dynamic Accumulation Method for Measuring Oxygen Transmission through Non P erforated F ilms A model for Dynamic Accumulation Method for Measuring Oxygen Transmission through n on p erforated films The number of each type of node run in the simulation is shown in Table 5 2. Table 5 2 Number of each type of node Type Number of Nodes 1 Surface 1 2 bulk film 99 3 film space interface 1 4 space 98 1 5 sensor O xygen Transmission Rate of a film (100 Gauge OPP/3.5 mil LLDP) was measured experimentally with an ambient concentration of 100% oxygen at ambient pressure and 23 C using the dynamic accumulation method. The accumulation chamber had a volume of 7 cc and transfer area of 16.62 cm 2 Experimentally measured film PAGE 60 60 permeability, thickness, accumulation chamber length (5x10 3 m) a diffusion coefficient of oxygen in air at 23 C (2.05x10 5 m 2 /s) (Fuller, Schettler, & Giddings, 1966) and ambient concentration of %100 O 2 were used as inputs for the model. The experiment was run for about 6 hours and oxygen concentration was recorded approximately ev ery 1500 seconds. The model was run to simulate the 6 hour experiment using 5 x 10 4 time steps. Data generated by the model was used to calculate OTR. O xygen Transmission Rate calculated from model data was compared with OTR calculated from experimental dat a. Measured concentration verses time from the experiment was compared with curves generated by the model. Simulati on of Mathematical Model for Dynamic Accumulation Method for Measuring Oxygen Transmission T hrough Hypothetical Non P erforated F ilms Differen t films each having a thickness of 1 mil (2.54x10 5 m) with OTR of 1,000 cc/m 2 /day (2.94 x 10 13 m 2 /s), 10,000 cc/m 2 /day (2.94 x 10 12 m 2 /s), and 100,000 cc/m 2 /day (2.94 x 10 11 m 2 /s) were simulated using the model in chambers of varying lengths with assoc iated chamber volumes as shown in Table 5.3 Each chamber was assumed to have a gas transfer area of 50 cm 2 and diffusion coefficient of oxygen in air at 23 C of 2.05x10 5 m 2 /s Table 5 3 Hypothetical Chambers simulated in model L(m) 1.5X10 3 0.01 0.1 1 10 100 1. 0 X10 3 1. 0 X10 4 V(cc) 7.5 50 500 5x10 3 5x10 4 5x10 5 5x10 6 5x10 7 The OTR ratios were calculated for each film on each chamber. The OTR ratio versus time ratio which corresponds to chamber length was plotted for each film. For the 100m chamber with film permeability of 10,000 cc mil/m 2 /day (2.94 x 10 12 m 2 /s) the OTR apparent was compared with the OTR actual and the OTR calculated from the PAGE 61 61 average con centration OTR average The average concentration was calculated by adding all the concentrations in the space nodes (Equation 5 1 ). ( 5 1) Measuring Oxygen Transmission T hrough a Perforation Static method was used with fluorescent oxygen sensor to measure oxygen transmission through a pipe. A chamber 7.125 inches in length was constructed from schedule 40 PVC pipe. At the ce nter of one end of the chamber a perforated disc was mounted and on the other end at the center a fluorescence oxygen sensor (Oxysense Inc.) was mounted (Figure 5 1). Figure 5 1 Schematic Representation of Experimental Oxygen Accumulation Chamber The ch amber was kept at room temperature (23 C) and purged with nitrogen and initially contained no oxygen. Oxygen was allowed to accumulate in the chamber and oxygen concentration was measured at the sensor approximately every 15 minutes until concentration at the sensor exceeded 5%. Oxygen transmi ssion through 3 different perforation diameters was measured. The perforations consisted of stainless steel disks that measured 0.003 inches thick with precision orifices (100, 205, 249 m)(Lenox Laser, Glen Arm, MD). Each perforation was measured 3 times. PAGE 62 62 Numerical Simulation of Oxygen Transmission T hrough a Perforation Numerical model for oxygen transmission through a perforation was simulated in 2 method s. In one method the accumulation chamber including the perforation was simulated with commercial soft ware (COMSOL Multiphysics) that applies the Finite Element Method The other method software was written in Microsoft Visual Basic for Applications (Appendix B Model of Oxygen Transmission Through a Perforation into an Accumulation Chamber ) where only th e accumulation chamber was simulated using the Finite Volume Method with the perforation being approximated as a source term at each time step Table 5 4 gives parameters and dimensions inputted into each model. Predicted oxygen concentrations at the senso r versus time for both models were compared to experimentally measured oxygen concentration at the sensor versus time. Table 5 4 Dimensions and parameters inputted into the model Parameter or Dimension Value Length of Chamber (L) 18.0975 cm Average Radi us of Chamber (R) 2.56137 cm Ambient O 2 Concentration at 23 C 8.60 x10 6 mol/cm 3 Diffusion Coefficient of O 2 in air at 23 C 0.204911638 cm 2 /s (Fuller, Schettler, & Giddings, 1966) Finite Element Method In the first method model was drawn with 2 dimensional geometry centered on the perforation and revolved around the axially dimension Both perforation and se tting in COMSOL Multiphysics (Figure 5 2). Accumulation chamber with each perforation was simulated for a tim e corresponding to each experiment. Oxygen was delivered to the top perforation meshes directly from the ambient environment and then distributed to surrounding meshes via the Finite Element Method. Dimensions and parameters from PAGE 63 63 Table 5 4 and perforation radius, effective diffusion length, and simulation times from Table 5 5 converted to SI units were entered into COMSOL. Figure 5 2 Accumulation chamber and perforation discretized into meshes by COMSOL Multiphysics Finite Volume Method In the second m et hod the accumulation chamber without a perforation was discretized into concentric cylindrical disc surrounded by concentric toroids (Figure 5 3). Figure 5 3 Oxygen accumulation chamber discretized into concentric cylindrical discs surrounded by toroid s Oxygen is delivered to the top center control volume through the perforat ion estimated by equation 4 8. Oxygen is then distributed to surrounding cylindrical discs and toroids via the Finite Volume Method. The algorithm for maintaining the maximum possib 1% of the maximum possible Perforation radius, effective diffusion length, simulation times, initial discretization and re discretization is given in Table 5 5. It was found through trial and error that when the number of radial positions exceeded a ratio of the average radius of the chamber to 1 times the radius of the perforation in the radial PAGE 64 64 dimension, the concentration in the top center node exceeded ambient concentrati on. From this experience Nr in the initial discretization is the nearest integer less than the ratio just described (Equation 5 2) (5 2) Number of axial positions used is the nearest integer less th an 3 times the number of radial positions (Equ ation 5 3) This number was used to have an aspect ratio as close as possible to 1. (5 3) After Oxygen has reached all the discretized points and oxygen concentration at the bottom wall is changing less than 1% between tim e steps which indicates that oxygen is changing by less than 1% everywhere else, the accumulation chamber is rediscretized. Radial and axial control volumes are reduced by half. If either dimension has an odd number of positions the extra position is inco rporated into the adjacent position and the last position i n the new discretization contains three positions of the previous discretization for that dimension. All the moles of oxygen and volumes of the control volumes in the previous discretization are ad ded to the control volume they are incorporated into in the next discretization. This process is repeated until the number of nodes in the radial position is less than or equal to 10. This re discretization gave a reasonable time step which resulted in a r easonable run time. Re discretization was run for the remainder of the simulation. Table 5 5 Perforation radius, effective diffusion length, simulation times, initial discretization and re discretization used in the second method. Precision Orifice Perf oration Radius Discretization (NrxNz) Re Discretization (NrxNz) Effective Diffusion Length Simulation Time PAGE 65 65 250 m 0.0124475 cm 137x479 8x29 0.027125 cm 9.17 h 200 m 0.0102445 cm 166x581 10x36 0.023712 cm 10.00 h 100 m 0.0050065 cm 341x1193 10x37 0.015 484 cm 24.44 h CHAPTER 6 RESULTS AND DISCUSSION Comparison of OTR measurements from Steady State Method with those from Dynamic Accumulation Method Oxygen transmission rates for steady state measurements and dynamic accumulation measurements of non perfor ated films are given in Tables A 1 through A 3 and Tables A 4 through A 6 respectively. Oxygen concentration verses time data for dynamic accumulation measurements of non perforated films are given in Tables A 7 through A 36. Accomplished Oxygen Accumulati on versus time for Dynamic Accumulation Measurement of Non perforated films are given in Figures A 1 through A 30. Statistics of m easured OTR of each film using both dynamic accumulation and steady state method are summarized in Table 6 1 Bar graphs with 95% confidence intervals for each film show visually that both methods give similar values ( Figure 6 1 ) Table 6 1 Measured OTR values using dynamic accumulation and steady state methods. Dynamic Accumulation Steady State Samples OTR ( 95%CI) Sample s OTR ( 95%CI) (n) (cc/m 2 /day) (n) (cc/m 2 /day) Low Transmitter 6 6.8 0.5 6 7.4 0.3 Moderate Transmitter 12 840 30 12 780 70 High Transmitter 12 9700 200 12 9600 700 PAGE 66 66 Figure 6 1 Mean OTR and 95% Confidence Interval for A) low transmitter, B) moder ate transmitter, and C) high transmitter film for both the dynamic accumulation and the steady state method Table 6 2 Differences of means at a 95% Confidence Interval between the Dynamic Accumulation method and the Steady State Method. dynamic accumul ation steady state 95% CI 95% CI Samples mean difference Lower Bound Upper Bound (n) (cc/m 2 /day) Low Transmitter 6 0.6 1.3 0 Moderate Transmitter 12 60 31 140 High Transmitter 12 100 660 950 Tables 6 2 fails to conclude there is a sign ificant difference in mean OTR between dynamic accumulation method and steady state method for any of the 3 films at a 95% confidence interval. This gives quantitative evidence that statistically the dynamic accumulation method for measuring OTR gives simi lar values as the steady state meth od for films that have OTRs within the range used in this study PAGE 67 67 Validation of Mathematical Model for Dynamic Accumulation with Non perforated F ilms Film (100 Gauge OPP/3.5 mil LLDP) measured experimentally using the dyna mic accumulation method had an OTR of 925 cc/m 2 /day, which corresponds to permeability of 1.22x10 12 m 2 /s. Thickness was 1.14x10 4 m (4.5 mils). Data from the experiment and output from model are shown in Figure 6 3 Oxygen Transmission Rate estimated from the model was 922 cc/m 2 initially degassed, but actual samples were stored in air, therefore, the first .03 days of simulated data were neglected to remove the lag phase as the simulated film loa ded with oxygen. Figure 6 2 V alidation of model of dynamic accumulation with non perforated films Dynamic Accumulation Model Simulations with Non P erforated F ilms OTR ratio versus chamber length for 3 hypothetical films is shown in Figure 6 2 OTR ratio deviates from unity by approximately 2% at time ratios of 9.91x10 3 9.91x10 4 and 9.91x10 5 for the 100,000, 10,000, and 1000 cc/m 2 /day films respectively. Table 6 3 illustrates a n extreme example for the case of a high transmitting film with OTR = PAGE 68 68 10,000 cc O 2 /m 2 /day mounted on an accumulation chamber of 100m in length. Table 6 3 shows th a t the simplifying assumption would fail to provide the correct OTR value (OTR apparent = 1884 versus OTR actual = 10,000). Mathematically mixing the chamber pro vides a much better approximation of the correct answer (OTR average = 9600 versus OTR actual = 10,000). The Differences between mathematically mixed OTR (OTR average ) and actual OTR are attributed to the coarseness of nodes used in the simulation. As more nodes are used, average OTR approaches actual OTR at the expense of more computational time. Table 6 3 F ilm permeability 10,000 cc mil/m2/day (2.94 x 10 12 m2/s) measured on 100 m hypothetical chamber OTR actual (cc/m 2 /day) OTR apparent (cc/m 2 /day) OTR aver age (cc/m 2 /day) 10,000 1884.4 9583.2 Figure 6 3 OTR ratio vs time ratio PAGE 69 69 Figure 6 3 demonstrates that for all commercially available films dynamic accumulation chambers have considerable flexibility in their design. Chamber length s would have to b e impractically long before the assumption of uniform concentration in the accumulation chamber is not valid for any commercially available film Validation of Mathematical Model for Oxygen Transmissions through Perforations Data for Oxygen Accumulation Mea surement versus time is given in Tables A 37 through A 39. Table 6 4 and Figure 6 4 demonstrates model predictions of oxygen concentration at the accumulation chamber sensor can serve as a reasonable approximation with both methods when compared to experim entally measured values. The model becomes slightly more accurate when the perforation is discretized. Approximating the oxygen transmitted from the perforation as a source term can still serve as a reasonable estimate of experimental data and is much simp ler to program and does not require expensive commercial software. Table 6 4 Compar ison of final %O 2 at sensor location achieved by experiment and model predictions using Finite Element (FEM) and Finite Volume Methods (FVM) for each orifice for the times shown in Table 5 5 Precision Orifice Experimental Value Value Predicted by FEM Value Predicted by FVM 250 m 5.17% 4.89% 5.19% 200 m 5.08% 4.69% 4.46% 100 m 5.03% 4.57% 4.09% PAGE 70 70 Figure 6 4 Curve of Oxygen Concentration at Sensor versus time ge nerated by experimental data and model using Finite Element and Finite Volume Method for A ) 250 m, B ) 200 m, and C ) 100 m precision orifice mounted on accumulation chamber. PAGE 71 71 CHAPTER 7 CONCLUSIONS The following Conclusions could be drawn from the result s presented in this work (1) The Dynamic Accumulation Method with fluorescence oxygen sensing can serve as a simpler and cheaper alternative to measuring OTR of non perforated films when compared with the Steady State Method defined by ASTM D3985. (2) A ma thematical model was developed that accurately predicts oxygen concentration as a function of time and position in an accumulation chamber with a non perforated film. (3) This model predicts that dynamic accumulation chambers of reasonable size, manufactur ed for use in the laboratory, would have uniform concentration with OTR levels for any non perforated film commercially available. In the event of non uniform concentration OTR can be calculated from an average oxygen concentration at each time. (4) Two ma thematical models were developed that accurately predict oxygen concentration as a function of time and position in an accumulat ion chamber with a perforation. Oxygen transmission through a perforation into an accumulation chamber modeled by discretizing t he perforation can accurately predict oxygen concentration as a function of position and time but required a complex model or expensive commercial software. Approximating the perforation as a source term predict s oxygen concentration as a function of posit ion and time slightly less accurate than discretizing the perforation but still gives a reasonable approximation and was significantly simpler to code. PAGE 72 72 CHAPTER 8 FUTURE WORK AND RECOMMENDATIONS Additional studies that have been suggested to expand on thi s work include (1) measuring high barrier material at higher pressures to decrease testing times. (2) Adapt the dynamic accumulation method to measure transmission rate of other gases particularly carbon dioxide, through non perforated films, with appropr iate apparatus and sensor. (3) Incorporate the model for oxygen transmission through a perforation developed in this work into a modified atmosphere package with a respiring product. PAGE 73 73 AP PENDIX A OXYGEN TRANSMISSION RATE MEASU R EMENTS Table A 1 Steady State Measurements 10,000 cc/m 2 /day PAGE 74 74 Table A 2 Steady State Measurements 1 ,000 cc/m 2 /day PAGE 75 75 Table A 3 Stead y State Measurements 10 cc/m 2 /day PAGE 76 7 6 Table A 4 Dynamic Accumulation Measurements 10,000 cc/m 2 /day PAGE 77 77 Table A 5 Dynamic Accumulation Measurements 1 ,000 cc/m 2 /day PAGE 78 78 Table A 6 Dynamic Accumulation Measurements 10 cc/m 2 /day PAGE 79 79 Table A 7 Oxygen Concentration versus time Experimental D at a 10,000 cc/m 2 /day Sample 1 OTR = 9306. 7 9 cc/m 2 /day PAGE 80 80 Table A 8 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 2 OTR = 9946.47 cc/m 2 /day PAGE 81 81 Table A 9 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 3 OTR = 1040 5 2 cc/m 2 /day PAGE 82 82 Table A 10 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 4 OTR = 1017 2 7 cc/m 2 /day PAGE 83 83 Table A 11 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 5 OTR = 93 20.0 7 cc/m 2 /day PAGE 84 84 Table A 12 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 6 OTR = 95 38 91 cc/m 2 /day PAGE 85 85 Table A 13 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 7 OTR = 943 5 4 3 cc/m 2 /day PAGE 86 86 Table A 14 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 8 OTR = 9625. 30 cc/m 2 /day PAGE 87 87 Table A 15 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 9 OTR = 1003 3 7 cc/m 2 /day PAGE 88 88 Table A 16 Oxyg en Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 10 OTR = 980 4 14 cc/m 2 /day PAGE 89 89 Table A 17 Oxygen Concentration versus time, Experimental Data 10,000 cc/m 2 /day Sample 11 OTR = 9532. 71 cc/m 2 /day PAGE 90 90 Table A 18 Oxygen Concentrati on versus time, Experimental Data 10,000 cc/m 2 /day Sample 12 OTR = 9531. 87 cc/m 2 /day PAGE 91 91 Table A 19 Oxygen Concentration versus time, Experimental Data 1 ,000 cc/m 2 /day Sample 1 OTR = 888. 7 cc/m 2 /day PAGE 92 92 Table A 19 C ontinued PAGE 93 93 Table A 20 Oxyg en Concentration versus time, Experimental Data 1 ,000 cc/m 2 /day Sample 2 OTR = 889.3 cc/m 2 /day PAGE 94 94 Table A 20 Continued PAGE 95 95 Table A 21 Oxygen Concentration versus time, Experimental Data 1 ,000 cc/m 2 /day Sample 3 OTR = 901.8 cc/m 2 /day PAGE 96 96 Table A 21 Continued PAGE 97 97 Table A 22 Oxygen Concentration versus time, Experimental Data 1 ,000 cc/m 2 /day Sample 4 OTR = 880.4 cc/m 2 /day PAGE 98 98 T a ble A 22 Continued PAGE 99 99 Table A 23 Oxygen Concentration versus time, Experimental Data 1 ,000 cc/m 2 / day Sample 5 OTR = 804. 5 cc/m 2 /day PAGE 100 100 Table A 23 Continued PAGE 101 101 Table A 24 Oxygen Concentration versus time, Experimental Data 1 ,000 cc/m 2 /day Sample 6 OTR = 794.1 cc/m 2 /day PAGE 102 102 Table A 24 Continued PAGE 103 103 Table A 25 Oxygen Concentration versus time, E xperimental Data 1 ,000 cc/m 2 /day Sample 7 OTR = 814. 7 cc/m 2 /day PAGE 104 104 Table A 25 Continued PAGE 105 105 Table A 26 Oxygen Concentration versus time, Experimental Data 1 ,000 cc/m 2 /day 1,000 cc/m 2 /day Sample 8 OTR = 786.9 cc/m 2 /day PAGE 106 106 Table A 26 Continued PAGE 107 107 Table A 27 Oxygen Concentration versus time, Experimental Data 1,000 cc/m 2 /day Sample 9 OTR = 786. 2 cc/m 2 /day PAGE 108 108 Table A 27 Continued PAGE 109 109 Table A 28 Oxygen Concentration versus time, Experimental Data 1,000 cc/m 2 /day Sample 10 OTR = 873.9 cc/m 2 /day PAGE 110 110 T able A 28 Continued PAGE 111 111 Table A 29 Oxygen Concentration versus time, Experimental Data 1,000 cc/m 2 /day Sample 11 OTR = 772.6 cc/m 2 /day PAGE 112 112 Table A 29 Continued PAGE 113 113 Table A 30 Oxygen Concentration versus time, Experimental Data 1,000 cc/m 2 /day Sample 12 OTR = 877. 6 cc/m 2 / day PAGE 114 114 Table A 30 Continued PAGE 115 115 Table A 31 Oxygen Concentration versus time, Experimental Data 10 cc/m 2 /day Sample 1 OTR = 7.0419 cc/m 2 / day PAGE 116 116 Table A 31 Continued PAGE 117 117 Table A 31 Continued PAGE 118 118 Table A 32 Oxygen Concentration versus time, Experimental Data 10 cc/m 2 /day Sample 2 OTR = 7.0622 cc/m 2 / day PAGE 119 119 Table A 32 Continued PAGE 120 120 Table A 32 Continued PAGE 121 121 Table A 33 Oxygen Concentration versus time, Experimental Data 10 cc/ m 2 /day Sample 3 OTR = 6.0786 cc/m 2 /day PAGE 122 122 Table A 33 Continued PAGE 123 123 Table A 33 Continued PAGE 124 124 Table A 34 Oxygen Concentration versus time, Experimental Data 10 cc/m 2 /day Sample 4 OTR = 7.3465 cc/m 2 /day PAGE 125 125 Table A 3 4 Continued PAGE 126 126 Table A 3 4 Continued PAGE 127 127 Table A 35 Oxygen Concentration versus time, Experimental Data 10 cc/m 2 /day Sample 5 OTR = 6.1666 cc/m 2 / day PAGE 128 128 Table A 35 Continued PAGE 129 129 Table A 35 Continued PAGE 130 130 Table A 36 Oxygen Concentration versus ti me, Experimental Data 10 cc/m 2 /day Sample 6 OTR = 7.0922 cc/m 2 / day PAGE 131 131 Table A 36 Continued PAGE 132 132 Table A 36 Continued PAGE 133 133 Table A 37 250 m precision orifice, perforation radius = 1.24475 x 10 4 m PAGE 134 134 Table A 37 C ontinued PAGE 135 135 Table A 37 C ontinued PAGE 136 136 Table A 38 200 m precision orif ice, perforation radius = 1.0244 5 x 10 4 m PAGE 137 137 Table A 38 C ontinued PAGE 138 138 Table A 38 C ontinued PAGE 139 139 Table A 39 100 m precision orifice, perforation radius = 5 .0065 x 10 5 m PAGE 140 140 Table A 39 Continued PAGE 141 141 Table A 39 Continued PAGE 142 142 Table A 39 Continued PAGE 143 143 Table A 39 Continued PAGE 144 144 Figure A 1 Accomplished Oxygen Plots10,000 cc/m 2 /day Sample 1 OTR = 9306.78 cc/m 2 /day Figure A 2 Accomplished Oxy gen Plots 10,000 cc/ m2 /day Sample 2 OTR = 9946.47 cc/m2/day Figure A 3 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 3 OTR = 10405.20 cc/m 2 /day PAGE 145 145 Figure A 4 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 4 OTR = 10172.2 cc/m 2 /day Figure A 5 A ccomplished Oxygen Plots 10,000 cc/m 2 /day Sample 5 OTR = 9320.07 cc/m 2 /day Figure A 6 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 6 OTR = 9538.91 cc/m 2 /day PAGE 146 146 Figure A 7 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 7 OTR = 9435.93 cc/m 2 /day Figure A 8 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 8 OTR = 9625.30 cc/m 2 /day Figure A 9 Accomplished Oxygen Plots10,000 cc/m 2 /day Sample 9 OTR = 10033.70 cc/m 2 /day PAGE 147 147 Figure A 10 Accomplished Oxygen Plots10,000 cc/m 2 /day Sample 10 OTR = 9804 .1 4 cc/m 2 /day Figure A 11 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 11 OTR = 9532.71 cc/m 2 /day Figure A 12 Accomplished Oxygen Plots 10,000 cc/m 2 /day Sample 12 OTR = 9531.81 cc/m 2 /day PAGE 148 148 Figure A 13 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sampl e 1 OTR = 888.7 cc/m 2 /day Figure A 14 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 2 OTR = 889.3 cc/m 2 /day Figure A 15 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 3 OTR = 901.8 cc/m 2 /day PAGE 149 149 Figure A 16 Accomplished Oxygen Plots 1,000 cc/m 2 /d ay Sample 4 OTR = 880 .4 cc/m 2 /day Figure A 17 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 5 OTR = 880.5 cc/m 2 /day Figure A 18 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 6 OTR = 804.4 cc/m 2 /day PAGE 150 150 Figure A 19 Accomplished Oxygen Plots 1,000 c c/m 2 /day Sample 7 OTR = 814 .8 cc/m 2 /day Figure A 20 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 8 OTR = 786.9 cc/m 2 /day Figure A 21 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 9 OTR = 786 .2 cc/m 2 /day PAGE 151 151 Figure A 22 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 10 OTR = 873.9 cc/m 2 /day Figure A 23 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 11 OTR = 772.6 cc/m 2 /day Figure A 24 Accomplished Oxygen Plots 1,000 cc/m 2 /day Sample 12 OTR = 877.6 cc/m 2 /day PAGE 152 152 Figure A 25 Accomplished Ox ygen Plots 10 cc/m 2 /day Sample 1 OTR = 7.0419 cc/m 2 /day Figure A 26 Accomplished Oxygen Plots 10 cc/m 2 /day Sample 2 OTR = 7. 0622 cc/m 2 /day Figure A 27 Accomplished Oxygen Plots 10 cc/m 2 /day Sample 3 OTR = 6.0786 cc/m 2 /day PAGE 153 153 Figure A 28 Accomplished Oxygen Plots 10 cc/m 2 /day Sample 4 OTR = 7.3465 cc/m 2 /day Figure A 29 Accomplished Oxygen Plots 10 cc/m 2 /day Sample 5 OTR = 6.1666 cc/m 2 /day Figure A 30 Accomplished Oxygen Plots 10 cc/m 2 /day Sample 6 OTR = 7.09123 cc/m 2 /day PAGE 154 154 AP PENDIX B VISUAL BASIC CODE FOR NUMERICAL SIMULATION Model of Dynamic Accumulation Chamber with a Non Perforated Film Option Explicit 'Inputed variables Dim C_amb As Double, L_f As Double, L_s As Doubl e, t_tot As Double, D As Double Dim P As Double, C_0 As Double N_f As Int eger, N_s As Integer, N_t As Double 'Variables calculated in program from inputed variables Dim lambda_f As Double, lambda_s As Double, dt As Double, t As Double, lambda_fi As Double, Dim lambda_si As Double, dx_f As Double, dx_s As Double Dim Sum_C As D ouble 'Counter Variables Dim i As Double, j As Double, k As Double 'Arrays used in Crank Nicholson Method Dim e() As Double, f() As Double, g() As Double, r() As Double, C() As Double Public Sub Main() GetInputs CrankNicholsonandOutputs En d Sub Public Sub CrankNicholsonandOutputs() 'initializing concentrations For i = 1 To N_f + N_s C(i) = C_0 Next i 'setting up tridiagonal matrix and updating r values at each time step For j = 1 To N_t For i = 1 To N_f + N_s 'First node If i = 1 Then f(i) = 2 (lambda_f + 1) g(i) = lambda_f r(i) = 2 lambda_f C_amb + 2 (1 lambda_f) C(i) + lambda_f C(i + 1) End If PAGE 155 155 'Film nod es If i > 1 And i < N_f Then e(i) = lambda_f f(i) = 2 (lambda_f + 1) g(i) = lambda_f r(i) = lambda_f C(i 1) + 2 (1 lambda_f) C(i) + lambda_f C (i + 1) End If 'Interface Node If i = N_f Then e(i) = lambda_fi f(i) = lambda_fi + lambda_si g(i) = lambda_si r(i) = lambda_fi C( i 1) (lambda_fi + lambda_si) C(i) + lambda_si C(i + 1) End If 'Space Nodes If i > N_f And i < N_f + N_s Then e(i) = lambda_s f(i) = 2 (lambda_s + 1) g(i) = lambda_s r(i) = lambda_s C(i 1) + 2 (1 lambda_s) C(i) + lambda_s C(i + 1) End If 'Last node If i = N_f + N_s Then e(i) = lambda_s f (i) = lambda_s + 1 r(i) = lambda_s C(i 1) + (1 lambda_s) C(i) End If Next i 'Solving tridiagonal matrix at each time step For k = 2 To N_f + N_s 'Decomposition e(k ) = e(k) / f(k 1) f(k) = f(k) e(k) g(k 1) 'Forward Substitution r(k) = r(k) e(k) r(k 1) Next k C(N_f + N_s) = r(N_f + N_s) / f(N_f + N_s) PAGE 156 156 Sum_C = 0 For k = N_f + N_s 1 To 1 Step 1 C(k) = (r(k) g(k) C(k + 1)) / f(k) If k > N_f Or k = N_f Then Sum_C = Sum_C + C(k) End If Next k t = t + dt If j Mod 100 = 0 Then Worksheets("Output ").Cells((j / (N_t / 100)) + 2, 1) = t Worksheets("Output").Cells((j / (N_t / 100)) + 2, 2) = t / 3600 / 24 Worksheets("Output").Cells((j / (N_t / 100)) + 2, 3) = C(N_f + 1) Worksheets("Output").Cells((j / (N_t / 100)) + 2, 4) = Sum_C / N_s Worksheets("Output").Cells((j / (N_t / 100)) + 2, 5) = C(N_f + N_s) End If Next j End Sub Public Sub GetInputs() C_amb = Worksheets("Input").Cells(1, 2) L_f = Worksheets("Input").Cells(2, 2) N_f = Worksheets("Input").Cells(3, 2) L_s = Wor ksheets("Input").Cells(4, 2) N_s = Worksheets("Input").Cells(5, 2) t_tot = Worksheets("Input").Cells(6, 2) N_t = Worksheets("Input").Cells(7, 2) D = Worksheets("Input").Cells(8, 2) P = Worksheets("Input").Cells(9, 2) C_0 = Worksheets("Input").Cells(10, 2) 'Calculating variables used in scheme dt = t_tot / N_t dx_f = (L_f / N_f) dx_s = (L_s / N_s) lambda_f = (P dt) / dx_f ^ 2 lambda_s = (D dt) / dx_s ^ 2 lambda_fi = (P dt) / (dx_f (dx_f + dx_s)) lambda_si = (D dt) / (dx_s (dx_f + dx_s)) 'Dimen sionalizing the matrixes ReDim e(N_f + N_s) ReDim f(N_f + N_s) ReDim g(N_f + N_s) ReDim r(N_f + N_s) PAGE 157 157 ReDim C(N_f + N_s) End Sub Model of Oxygen Transmission T hrough a Perforation into an Accumulation Chamber Option Explicit 'Inputted Variables Dim Cha mberLength As Double, ChamberRadius As Double, holethickness As Double Dim holeradius As Double, AmbientConcentration As Double Nr As Long, Nz As Long Dim pi As Double, totaltime As Double, DiffusionCoefficient As Double, Tolerance As Double Dim e As Dou ble, f As Double, Nl As Double, Nx As Double 'Arrays used 'Locations, Boundaries, Vertical Transfer Areas, and Volumes of Control Volumes Dim r() As Double, z() As Double, radialboundary() As Double, axialbo undary() As Double Dim VerticalTransferArea() A s Double, ControlVolume() As Double 'moles of oxygen and oxygen concentration in Control Volumes Dim CurrentmolesofOxygen() As Double, PreviousmolesofOxygen() As Double, Dim OxygenConcentration() As Double 'Coefficients Derived from finite Volume Method Dim a_in() As Double, a_out() As Double, a_up() As Double, a_down() As Double 'Coefficients substituted in the tridiagonal matrices Dim e_axial() As Double, f_axial() As Double, g_axial() As Double, e_radial( ) As Double Dim f_radial() As Double, g_radial () As Double 'Coefficients resulting from the decomposition and solution of the tridiagonal matrices Dim g_axial_decomp(), r_axial() As Double, x() As Double Dim g_radial_decomp() As Double, r_radial() As Double 'Calculated Variables'Time Variables Dim d t As Double, dr As Double, dz As Double, currenttime As Double Dim numberoftimesteps As Double, rt As Double, zt As Double 'Area of Perforation and Cross Sectional Area of Chamber Dim Areaofhole As Double, ChamberArea As Double 'Mole of Oxygen Delivered to System through the perforation Dim molesofOxygenDelivered As Double Concentration at the sensor Dim SensorConcentration As Double 'Counter Variables Dim i, j, k, n, l, q, h, m, v, p As Long, u As Double Sub Main() GetInput InitialDescreti zationofFiniteCylinder PAGE 158 158 ADI_MethodandOutputs End Sub Sub GetInput() ChamberLength = Worksheets("Input").Cells(1, 2) ChamberRadius = Worksheets("Input").Cells(2, 2) holethickness = Worksheets("Input").Cells(3, 2) holeradius = Works heets("Input").Cells(4, 2) AmbientConcentration = Worksheets("Input").Cells(5, 2) DiffusionCoefficient = Worksheets("Input").Cells(6, 2) totaltime = Worksheets("Input").Cells(7, 2) Nr = Worksheets("Input").Cells(8, 2) Nz = Worksheets("I nput").Cells(9, 2) f = Worksheets("Input").Cells(13, 2) Tolerance = Worksheets("Input").Cells(15, 2) pi = 3.14159265358979 e = 2.71828182845905 'Locations, Boundaries, Horizontal Areas, and Volumes of Control Volumes, and Radi al Layers ReDim r(Nr), z(Nz), radialboundary(Nr), axialboun dary(Nz), ControlVolume(Nr, Nz) ReDim VerticalTransferArea(Nr, Nz) 'moles of oxygen and oxygen concentration in Control Volumes and Radial Layers ReDim OxygenConcentration (Nr, Nz) PreviousmolesofOxygen(Nr, Nz) ReDim CurrentmolesofOxygen(Nr, Nz) 'Coefficients Derived in Finite Volume Method ReDim a_in(Nr), a_out(Nr), a_up(Nz), a_down(Nz) 'Coefficients substituted in the tridiagonal matrices ReDim e_axial(Nr Nz), f_axial(Nr Nz), g_axial(Nr Nz) ReDim e_radial(Nr Nz), f_radial(Nr Nz), g_radial(Nr Nz) 'Coefficients resulting from the decomposition and solution of the tridiagonal matrices ReDim e_axial_decomp(Nr Nz ), f_axial_decomp(Nr Nz), g_axial_decomp(Nr Nz) ReDim r_axial(Nr Nz) e_radial_decomp(Nr Nz), f_radial_decomp(Nr Nz) ReDim g_radial_decomp(Nr Nz), r_radial(Nr Nz), x(Nr Nz) dr = ChamberRadius / (Nr 1) dz = Chamb erLength / (Nz 1) Areaofhole = pi (holeradius ^ 2) ChamberArea = pi (ChamberRadius ^ 2) Nl = Nz l = 0 End Sub Sub InitialDescretizationofFiniteCylinder() 'Location of center of Control Vo lume in Radius PAGE 159 159 r(1) = 0: r(Nr) = ChamberRadius For i = 2 To Nr 1 r(i) = r(i 1) + dr Next i 'Location of center of Control Volume in Height z(1) = 0: z(Nz) = ChamberLength For j = 2 To Nz 1 z(j) = z(j 1) + dz Next j 'Location of Side Boundary of Control Volume radialboundary(1) = r(2) / 2: radialboundary(Nr) = r(Nr) For i = 2 To Nr 1 radialboundary(i) = (r(i + 1) r(i)) / (Log(r(i + 1) / r(i)) / Log(e)) Next i 'Locations of Tops and Bottoms of Control Volume axialboundary(1) = z(2) / 2 : axialboundary(Nz) = z(Nz) For j = 2 To Nz 1 axialboundary(j) = axialboundary(j 1) + dz Next j 'Volume and Vertical Transfer Areas of Control Volumes 'Top Center Cont rol Volume VerticalTransferArea(1, 1) = pi radialboundary(1) ^ 2 ControlVolume(1, 1) = pi radialboundary(1) ^ 2 axialboundary(1) 'Volume of Center Control Volumes For j = 2 To Nz VerticalTransferArea(1, j) = pi radialboundary(1) ^ 2 ControlVolume(1, j) = pi radialboundary(1) ^ 2 (axialboundary(j) axialboundary(j 1)) Next j 'Volume of all other Control Volumes For i = 2 To Nr For j = 1 To Nz PAGE 160 160 VerticalTransferArea(i, j) = pi (radialboundary(i) ^ 2 radialboundary(i 1) ^ 2) ControlVolume(i, j) = pi (radialboundary(i) ^ 2 radialboundary(i 1) ^ 2) (axialboundary(j) axialboundary(j 1)) Next j Next i 'Volume of Radial Layers 'Top Radial Layer Volume RadialLayerVolume(1) = ChamberArea a xialboundary(1) For j = 2 To Nz RadialLayerVolume(j) = ChamberArea (axialboundary(j) axialboundary(j 1)) Next j dt = (radialboundary(1) (r(2) r(1))) / (8 DiffusionCoe fficient) Worksheets("Input").Cells(11, 2) = dt End Sub Sub ADI_MethodandOutputs() 'Starting Time Steps currenttime = 0 'q is counting the total number of time steps q = 0 'n is used for the output and needs to be initialized outside the main Do Loop n = 0 'Initializing Delivery of Oxygen to the System TotalmolesofOxygenDelivered = 0 'Setting all Initial Control Volume Concentrations to zero For j = 1 To Nz For i = 1 To Nr OxygenConcentration(i, j) = 0 CurrentmolesofOxygen(i, j) = 0 PreviousmolesofOxygen(i, j) = 0 Next i Next j 'Th e main time loop Do While currenttime < totaltime PAGE 161 16 1 'I nj ecting Oxygen into the system molesofOxygenDelivered = ((DiffusionCoefficient Areaofhole dt / holethickness) (AmbientConcentration OxygenConcentration(1, 1))) OxygenConcentration(1, 1) = ((CurrentmolesofOxygen(1, 1) + molesofOxygenDelivered) / ControlVolume(1, 1)) PreviousmolesofOxygen(1, 1) = PreviousmolesofOxygen(1, 1) + molesofOxygenDelivered 'Calculating coefficients, Constructing, Decomposing, and Solving Matrices TimeStepAdjustment AxialCoefficientsforAlgebraicEquations RadialCoefficientsforAlgebraicEquations TridiagonalMatrixAxialandDecomposition TridiagonalMatrixRadialandDecomposition MatrixConstantsRadial SolutionRadial MatrixConstantsAxial SolutionAxial u = OxygenConcentration(Nr, Nz) If OxygenConcentration(Nr, Nz) > 1E 3 05 And q > 2 Then If Abs((OxygenConcentration(Nr, Nz) u) / u) < 0.01 Then l = l + 1 ReDescretizationofFiniteCylinder End If End If 'OxygenTransfer SensorConcentration = OxygenConcentration(1, Nz) 'Printing the solution n = n + 1 Worksheets("Output").Cells(n + 2, 1) = dt Worksheets("Output").Cells(n + 1, 2) = currenttime Worksheets("Output").Cells(n + 1, 3) = currenttime / 24 / 3600 Worksheets("Output").Cells(n + 1, 5) = OxygenConcentration(1, 1) Worksheets("Output").Cells(n + 1, 6) = OxygenConcentration(Nr, 1) Worksheets("Output").Cells(n + 1, 7) = SensorConcentration Worksheets("Output").Cells(n + 1, 8) = OxygenConcentration(Nr, Nz) Worksheets("Output").Ce lls(n + 1, 13) = Nr Worksheets("Output").Cells(n + 1, 15) = Nz For i = 1 To Nr For j = 1 To Nz OxygenConcentration(i, j) = CurrentmolesofOxyge n(i, j) / ControlVolume(i, j) PAGE 162 162 Next j Next i currenttime = currenttime + dt Loop End Sub Sub AxialCoefficientsforAlgebraicEquations() 'Axial Coefficients For j = 1 To Nz Select Case j 'Top Control Volumes Case 1 a_up(1) = 0 a_down(1) = (DiffusionCoefficient dt) / (axialboundary(1) (z(2) z(1))) 'Interior Control Volumes Case 2 To Nz 1 a_up(j) = (DiffusionCoefficient dt) / ((z(j) z(j 1)) (axialboundary(j) axialboundary(j 1))) a_down(j) = (DiffusionCoefficient dt) / ((z(j + 1) z(j)) (axialboundary(j) axialboundary(j 1))) 'Bottom Control Volumes Case Nz a_up(Nz) = (DiffusionCoefficient dt) / ((z(Nz) z(Nz 1)) (axialboundary(Nz) axialboundary(Nz 1))) a_down(Nz) = 0 End Select Next j End Sub Sub RadialCoefficientsforAlgebraicEquations() For i = 1 To Nr Select Case i 'Center Control Vol umes Case 1 a_in(1) = 0 PAGE 163 163 a_out(1) = (2 DiffusionCoefficient dt) / (radialboundary(1) (r(2) r(1))) 'Interior Control Volumes Case 2 To Nr 1 a_in(i) = (2 DiffusionCoefficient dt radialboundary(i 1)) / ((r(i) r(i 1)) (radialboundary(i) ^ 2 radialboundary(i 1) ^ 2)) a_out(i) = (2 DiffusionCoefficient dt radialboundary(i) ) / ((r(i + 1) r(i)) (radialboundary(i) ^ 2 radialboundary(i 1) ^ 2)) 'Wall Control Volumes Case Nr a_in(Nr) = (2 DiffusionCoefficient dt radialboundary(Nr 1)) / ((r(Nr) r(Nr 1)) (radialboundary(Nr) ^ 2 radialboundary(Nr 1) ^ 2)) a_out(Nr) = 0 End Select Next i End Sub Sub TridiagonalMatrixAxialandDecomposition() 'Setting up the tridiagonal matrix for the first half of the timestep in the axial direction k = 0 For i = 1 To Nr For j = 1 To Nz k = k + 1 Select Case j Case 1 e_axial(k) = 0 g_axial(k) = (1 / 2) a_down(j) f (ControlVolume(i, j) / ControlVolume(i, j + 1)) f_axial(k) = (1 / 2) (1 + f (a_up(j) + a_down(j))) Case 2 To Nl 1 e_axial(k) = (1 / 2) a_up(j) f (ControlVolume(i, j) / ControlVolume(i, j 1)) g_axial(k) = (1 / 2) a_down(j) f (ControlVolume(i, j) / Contr olVolume(i, j + 1)) f_axial(k) = (1 / 2) (1 + f (a_up(j) + a_down(j))) Case Nl e_axial(k) = (1 / 2) a_up(j) f (ControlVolume(i, j) / ControlVolume(i, j 1)) g_axial(k) = 0 f_axial(k) = (1 / 2) (1 + f (a_up(j) + a_down(j))) End Select Next j PAGE 164 164 Next i 'Decomposition in the axial direct ion g_axial_decomp(1) = g_axial(1) / f_axial(1) For k = 2 To Nr Nl g_axial_decomp(k) = g_axial(k) / (f_axial(k) g_axial_decomp(k 1) e_axial(k)) Next k End Sub Sub TridiagonalMatrixRa dialandDecomposition() 'Setting up the tridiagonal matrix for the second half of the timestep in the radial direction k = 0 For j = 1 To Nz For i = 1 To Nr k = k + 1 Selec t Case i Case 1 e_radial(k) = 0 g_radial(k) = (1 / 2) a_out(i) f (ControlVolume(i, j) / ControlVolume(i + 1, j)) f_radial(k) = (1 / 2) (1 + f (a_in(i) + a_out(i))) Case 2 To Nr 1 e_radial(k) = (1 / 2) a_in(i) f (ControlVolume(i, j) / ControlVolume(i 1, j)) g_radial(k) = (1 / 2) a_out(i) f (Co ntrolVolume(i, j) / ControlVolume(i + 1, j)) f_radial(k) = (1 / 2) (1 + f (a_in(i) + a_out(i))) Case Nr e_radial(k) = (1 / 2) a_in(i) f (ControlVolume( i, j) / ControlVolume(i 1, j)) g_radial(k) = 0 f_radial(k) = (1 / 2) (1 + f (a_in(i) + a_out(i))) End Select Next i Next j g_radial_de comp(1) = g_radial(1) / f_radial(1) For k = 2 To Nr Nz g_radial_decomp(k) = g_radial(k) / (f_radial(k) g_radial_decomp(k 1) e_radial(k)) Next k PAGE 165 165 End Sub Sub MatrixConstantsAxial() k = 0 For i = 1 To Nr For j = 1 To Nz k = k + 1 'The top control volumes If j = 1 Then r_axial(k) = (1 / 2) (( 1 (1 f) a_down(1)) PreviousmolesofOxygen(i, 1) + a_down(1) (ControlVolume(i, 1) / ControlVolume(i, 2)) (1 f) PreviousmolesofOxygen(i, 2)) 'Interior Control Volumes not including top and bottom ElseIf j <> 1 And j <> Nz Then r_axial(k) = (1 / 2) ((1 (1 f) (a_up(j) + a_down(j))) PreviousmolesofOxygen(i, j) + a_up(j) (ControlVolume(i, j) / ControlVolume(i, j 1)) (1 f) P reviousmolesofOxygen(i, j 1) + a_down(j) (ControlVolume(i, j) / ControlVolume(i, j + 1)) (1 f) PreviousmolesofOxygen(i, j + 1)) 'Bottom Control Volumes ElseIf j = Nz Then r_axial(k) = (1 / 2) (( 1 (1 f) a_up(Nz)) PreviousmolesofOxygen(i, Nz) + a_up(Nz) (ControlVolume(i, Nz) / ControlVolume(i, Nz 1)) (1 f) PreviousmolesofOxygen(i, Nz 1)) End If Next j Nex t i End Sub Sub SolutionAxial() 'Solving the tridiagonal matrix for the first half of the timestep in the axial direction 'Forward Substitution r_axial(1) = r_axial(1) / f_axial(1) For k = 2 To Nr Nz r_axial(k) = (r_axial(k) r_axial(k 1) e_axial(k)) / (f_axial(k) g_axial_decomp(k 1) e_axial(k)) Next k PAGE 166 166 'Back Substitution x(Nr Nz) = r_ax ial(Nr Nz) For k = Nr Nl 1 To 1 Step 1 x(k) = r_axial(k) g_axial_decomp(k) x(k + 1) Next k k = 0 'Determining new number of moles in each Control Volume and updating old number of moles in each Control Volume For i = 1 To Nr For j = 1 To Nz k = k + 1 CurrentmolesofOxygen(i, j) = x(k) PreviousmolesofOxygen(i, j) = CurrentmolesofOxygen(i, j) Next j Next i End Sub Sub MatrixConstantsRadial() k = 0 For j = 1 To Nz For i = 1 To Nr k = k + 1 'Center Control Volumes If i = 1 Then r_radial(k) = (1 / 2) ((1 (1 f) a_ out(1)) PreviousmolesofOxygen(1, j) + a_out(1) (ControlVolume(1, j) / ControlVolume(2, j)) (1 f) PreviousmolesofOxygen(2, j)) 'Interior Control Volumes not including Center and Wall ElseIf i <> 1 And i <> Nr Then r_radial(k) = (1 / 2) ((1 (1 f) (a_in(i) + a_out(i))) PreviousmolesofOxygen(i, j) + a_in(i) (ControlVolume(i, j) / ControlVolume(i 1, j)) (1 f) PreviousmolesofOxygen(i 1, j) + a_out(i) (ControlVolume(i, j) / ControlVolume(i + 1, j)) (1 f) PreviousmolesofOxygen(i + 1, j)) PAGE 167 167 'Wall Control Volumes ElseIf i = Nr Then r_radial(k) = (1 / 2) ((1 (1 f) a_in(Nr)) PreviousmolesofOxygen(Nr, j) + a_in(Nr) (ControlVolume(Nr, j) / ControlVolume(Nr 1, j)) (1 f) PreviousmolesofOxygen(Nr 1, j)) End If Next i Next j End Sub Sub SolutionRadial() 'Solving the tridiagonal matrix for the second half of the timestep in the radial direction 'Forward Substitution r_radial(1) = r_radial(1) / f_radial(1) For k = 2 To Nr Nz r_radial(k) = (r_radial(k) r_radial(k 1) e_radial(k)) / (f_radial(k) g_radial_decomp(k 1) e_radial(k)) Next k 'Back Substitution x(Nr Nz ) = r_radial(Nr Nz) For k = Nr Nz 1 To 1 Step 1 x(k) = r_radial(k) g_radial_decomp(k) x(k + 1) Next k k = 0 'Determining concentrations and moles For j = 1 To Nz For i = 1 To Nr k = k + 1 CurrentmolesofOxygen(i, j) = x(k) PreviousmolesofOxygen(i, j) = CurrentmolesofOxygen(i, j) Next i PAGE 168 168 Next j End Sub Sub TimeStepAdjustment() v = 0 Do Do RadialCoefficientsforAlgebraicEquations MatrixConstan tsRadial AxialCoefficientsforAlgebraicEquations MatrixConstantsAxial k = 0 For j = 1 To Nz For i = 1 To Nr k = k + 1 If r_radial(k) > 0 Or r_axial(k) > 0 And l = 0 Then dt = dt / 1.01 v = 1 Exit Do End If Next i Next j dt = dt 1.01 Loop If v = 1 Then Exit Do End If Loop End Sub Sub ReDe scretizationofFiniteCylinder() If Nr <= 10 Or l < 25 Then Exit Sub PAGE 169 169 End If Nx = Int(Nr / 2) Nl = Int(Nz / 2) 'Expanding Control Volumes and adding masses in Expanded Control Volumes For i = 1 To Nx For j = 1 To Nl m = 2 i 1 p = 2 j 1 If 2 i + 1 = Nr And 2 j + 1 = Nz Then CurrentmolesofOxygen(m, p) = CurrentmolesofOxygen(m, p) + Currentmoleso fOxygen(m + 1, p) + CurrentmolesofOxygen(m + 2, p) + CurrentmolesofOxygen(m, p + 1) + CurrentmolesofOxygen(m + 1, p + 1) + CurrentmolesofOxygen(m + 2, p + 1) + CurrentmolesofOxygen(m, p + 2) + CurrentmolesofOxygen(m + 1, p + 2) + CurrentmolesofOxygen(m + 2 p + 2) CurrentmolesofOxygen(i, j) = CurrentmolesofOxygen(m, p) ControlVolume(m, p) = ControlVolume(m, p) + ControlVolume(m + 1, p) + ControlVolume(m + 2, p) + ControlVolume(m, p + 1) + ControlVolume(m + 1, p + 1) + Control Volume(m + 2, p + 1) + ControlVolume(m, p + 2) + ControlVolume(m + 1, p + 2) + ControlVolume(m + 2, p + 2) ControlVolume(i, j) = ControlVolume(m, p) OxygenConcentration(i, j) = (CurrentmolesofOxygen(i, j) / ControlVolume(i, j)) ElseIf 2 i + 1 = Nr And 2 j + 1 <> Nz Then CurrentmolesofOxygen(m, p) = CurrentmolesofOxygen(m, p) + CurrentmolesofOxygen(m + 1, p) + CurrentmolesofOxygen(m + 2, p) + CurrentmolesofOxygen(m, p + 1) + CurrentmolesofOxygen(m + 1, p + 1) + CurrentmolesofOxygen(m + 2, p + 1) CurrentmolesofOxygen(i, j) = CurrentmolesofOxygen(m, p) ControlVolume(m, p) = ControlVolume(m, p) + ControlVol ume(m + 1, p) + ControlVolume(m + 2, p) + ControlVolume(m, p + 1) + ControlVolume(m + 1, p + 1) + ControlVolume(m + 2, p + 1) ControlVolume(i, j) = ControlVolume(m, p) OxygenConcentration(i, j) = (CurrentmolesofOxygen(i, j) / ControlVolume(i, j)) ElseIf 2 i + 1 <> Nr And 2 j + 1 = Nz Then CurrentmolesofOxygen(m, p) = CurrentmolesofOxygen(m, p) + CurrentmolesofOxygen(m + 1, p) + CurrentmolesofOxygen (m, p + 1) + CurrentmolesofOxygen(m + 1, p + 1) + CurrentmolesofOxygen(m, p + 2) + CurrentmolesofOxygen(m + 1, p + 2) CurrentmolesofOxygen(i, j) = CurrentmolesofOxygen(m, p) ControlVolume(m, p) = ControlVolume(m, p) + ControlVolume(m + 1, p ) + ControlVolume(m, p + 1) + ControlVolume(m + 1, p + 1) + ControlVolume(m, p + 2) + ControlVolume(m + 1, p + 2) ControlVolume(i, j) = ControlVolume(m, p) OxygenConcentration(i, j) = (CurrentmolesofOxygen(i, j) / ControlVol ume(i, j)) Else PAGE 170 170 CurrentmolesofOxygen(m, p) = CurrentmolesofOxygen(m, p) + CurrentmolesofOxygen(m + 1, p) + CurrentmolesofOxygen(m, p + 1) + CurrentmolesofOxygen(m + 1, p + 1) CurrentmolesofOxygen(i, j) = CurrentmolesofOxygen(m, p) ControlVolume(m, p) = ControlVolume(m, p) + ControlVolume(m + 1, p) + ControlVolume(m, p + 1) + ControlVolume(m + 1, p + 1) ControlVolume(i, j) = ControlVolume (m, p) OxygenConcentration(i, j) = (CurrentmolesofOxygen(i, j) / ControlVolume(i, j)) End If Next j Next i For i = 1 To Nx If i = 1 Then r(i) 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PAGE 179 179 BIOGRAPHICAL SKETCH Ayman Abdellatief was born in the town of Bellows Fall s, Vermont in 1979. He graduated high school in Tampa, Florida in 1998. After high school he attended the University of Florida and received a B achelor of S cience degree in c hemical e ngineering in December 2002 After completing his undergraduate studies a t the University of Florida, Ayman served one tour of duty in the United States Navy. After completing his tour in 2005 Ayman enrolled in the Agricultural and Biological Engineering Department at the University of Florida and completed his Master of Engi neering degree in May 2008. Ayman did an internship with HEB grocery company in Summer of 2008. Ayman started his doctoral studies in Fall 2009 also in the Agricultural and Biological Engineering at the University of Florida. He received his Ph.D from the University of Florida in the spring of 2014. |