MELTING OF ICE FOR STEFAN PROBLEM USING FINITE DIFFERENCE METHOD NUR SURIANI BINTI Bachelor of Computer Science with Honours (Computational Science) 2019 NUR SURIANI BINTI HUSIN 14/08/2020 DECLARATION OF ORIGINALITY I hereby declare that this research together with all of its content is none other than my work, with the consideration and the exception of research based information and relative materials that were adapted and extracted from other resources which have evidently been quoted or stated respectively. Signed, ……………………………………. NUR SURIANI BINTI HUSIN Faculty of Computer Science and Information Technology, 12 December 2019 Universiti Malaysia Sarawak ACKNOWLEDGEMENT All praises to Allah the Almighty for giving me the strength and patience to complete my final year project. I would like to express my deepest gratitude and thank you to my supervisor, Dr Nuha binti Loling Othman, lecturer in Faculty of Computer Science and Information Technology, UNIMAS, Sarawak for guide me and giving me the encouragement and motivation to complete this project. Also, I would like to thanks to my family for the encouragement, support and assistance given to me in various forms including the financial support I needed to finish this project. Thanks also to my colleagues who also helped and shared information throughout the process of completing this project. Without the help of all of them, it would have been difficult for me to finish this project. TABLE OF CONTENT FORM C ......................................................................................................................................... DECLARATION OF ORIGINALITY ........................................................................................ ACKNOWLEDGEMENT ............................................................................................................ LIST OF FIGURES ..................................................................................................................... 1 ABSTRACT ................................................................................................................................. 9 ABSTRAK .................................................................................................................................. 10 GLOSSARY VARIABLES ....................................................................................................... 11 CHAPTER 1: INTRODUCTION ................................................................................................ 1.1 Overview ............................................................................................................................... 12 1.2 Problem Statement ................................................................................................................. 13 1.3 Scope ..................................................................................................................................... 13 1.4 Objective ................................................................................................................................ 14 1.5 Brief Methodology ........................................................................................................... 14-15 1.6 Significant of project ............................................................................................................. 15 1.7 Project Schedule .................................................................................................................... 16 1.8 Expected Outcome ................................................................................................................. 16 CHAPTER 2: LITERATURE REVIEW .................................................................................... 2.1 Introduction ........................................................................................................................... 17 2.2 Mathematical and Numerical Model of Solidification Process of Pure Metal ................ 18-20 2.3 Freezing of Porous Medium with Water Supply Coupled Stefan Problem ................................... 21-23 2.4 An Application of Stefan’s Problem to the Freezing of the Cylindrical Food-Stuff........ 24-25 2.5 Summary ................................................................................................................................ 26 CHAPTER 3: METHODOLOGY ............................................................................................... 3.1 Introduction ........................................................................................................................... 27 3.2 Modelling Process Methodology ...................................................................................... 28-31 3.2.1 Phase 1: Define Goal .............................................................................................. 28 3.2.2 Phase 2: Characterize Model .................................................................................. 28 3.3.3 Phase 3: Model Formulation ............................................................................. 29-31 CHAPTER 4: IMPLEMENTATION AND ANALYSIS ............................................................. 4.0 Introduction ........................................................................................................................... 32 4.1 Finite Difference Method ...................................................................................................... 32 4.2 Implementation Process ................................................................................................... 33-34 4.3 Result ................................................................................................................................ 35-37 4.3 Conclusion......................................................................................................................................... 38 CHAPTER 5: Conclusion and Future Work ................................................................................. 5.0 Introduction ........................................................................................................................... 39 5.1 Significant of study ............................................................................................................... 39 5.2 Limitation of Study ................................................................................................................ 39 5.3 Improvement in future study ................................................................................................ 40 APPENDIX A ........................................................................................................................ 41-42 Project Schedule for Final Year Project 1 ................................................................................... 41 Project Schedule for Final Year Project 2 ................................................................................... 41 Source Code ................................................................................................................................. 42 REFERENCE............................................................................................................................. 43 LIST OF FIGURES Figure 2.1.1 Solidification scheme (a) pure metal (b) alloys ..................................................... 18 Figure 2.2.2 Temporary state of pure metal solidification process ............................................. 19 Figure 2.3.1 Show that freezing water satured soil ..................................................................... 21 Figure 2.4.1 Schematic model ..................................................................................................... 24 Figure 3.2.3.1 Computational domain of vertical ice layer process ............................................ 28 Figure 4.3.1. Symbols used in the source code ........................................................................... 35 Figure 4.3.2 Initialization in source code .................................................................................... 35 Figure 4.3.3 The graph of phase change in ice cube 1 ............................................................... 36 Figure 4.3.3 The graph of phase change in ice cube 2 ............................................................... 37 8 ABSTRACT Stefan Problem exists in heat and mass transfer in objects under phase changes in many physical processes such as solidification of pure metals, freezing of water, deep freezing of foodstuff this is the processes that involve in which phase changes from solid to liquid or vapor states. The object is assumed to undergo a phase change with a moving boundary that has to be tracked as part of the solution. Various numerical methods used for Stefan Problem. We will use Finite Difference Method to solve of Stefan Problem. 9 ABSTRAK Masalah Stefan terlibat dalam pemindahan haba dan jisim dalam objek yang menjalani perubahan fasa dalam banyak proses fizikal seperti pemejalan logam tulen, pembekuan air, pembekuan dalam barangan makanan dan sebagainya seperti proses yang digunakan di mana fasa perubahan dari keadaan pepejal, cecair atau wap. Objek ini dianggap telah menjalani perubahan fasa dengan sempadan yang bergerak yang perlu dijejak sebagai sebahagian daripada penyelesaian. Pelbagai kaedah berangka yang digunakan untuk masalah Stefan. Kami akan menggunakan kaedah perbezaan terhingga untuk menyelesaikan masalah Stefan 10 Glossary 푢 = 푇푒푚푝푒푟푎푡푢푟푒 푓 = 푓푙푢푥 푑푒푛푠푡푦 푘