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An Approximate Solution to the Moving Boundary Problem Associated with the Freezing and Melting of Lake Ice Stephen Danforth Foss

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An Approximate Solution to the Moving Boundary Problem Associated with the Freezing and Melting of Lake Ice Stephen Danforth Foss University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Summer 1974 AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE STEPHEN DANFORTH FOSS Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation FOSS, STEPHEN DANFORTH, "AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE" (1974). Doctoral Dissertations. 1064. https://scholars.unh.edu/dissertation/1064 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 75-1655 FOSS, Stephen Danforth, 1941- AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE. University of New Hampshire, Ph.D., 1974 Engineering, chemical Xerox University Microfilms, Ann Arbor, Michigan 48106 THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. AN APPROXIMATE SOLUTION TO THE MOVING BOUNDARY PROBLEM ASSOCIATED WITH THE FREEZING AND MELTING OF LAKE ICE by STEPHEN D. FOSS B.S., University of New Hampshire, 196-4 M.S., University of New Hampshire, 1965 A THESIS Submitted to the University of New Hampshire In Partial Fulfillment of The Requirements for the Degree of Doctor of Philosophy Graduate School Engineering Ph.D. Program Transport Phenomena July, 1974 This thesis has been examined and approved. S J 'Yo j a z. Thesis direq ;or, Stephen S. T. Fan, As so. Prof. of Chemical Engineering William Mosberg, Asso. Prof. of Mechanical En leering £ C Shan S. Kuo, Prof. of Mathematics Russell L. Valentine, Asso. Prof. of Mechanical Engineering Francis Hall, Prof., Institute of Natural and Environmental Resources Q julQu u a ^ ZJ., Date ' ii TABLE OF CONTENTS ACKNOWLEDGEMENTS .................................. V LIST OF TABLES .................................... vi LIST OF FIGURES ...................... vii ABSTRACT...................................... ■___ viii I. INTRODUCTION...................................... 1 II. DESCRIPTION OF THE PROBLEM......................... 13 A. Solution of Moving Boundary Problem Utilizing a Quasi-Steady State Assumption............................... 16 B. Comparison to Westphal’s Solution.......... 18 III. ONE^-DMENSIONAL FREEZING OF ICEWITH CONSTANT HEAT FLUX IN THE WATER L A Y E R ....................... 24 A. Factors Contributing to Heat Flux .......... 26 B. Fundamental Equations ................ 27 C. Solution Utilizing Quasi-Steady State Assumption............................... 28 D. Specific Examples ........................ 29 1. Case I Step drop in ambient air temperature ......................... 30 2. Case II Sinusoidal variation in ambient air temperature .............. 30 IV. DEVELOPMENT OF THE MOVING BOUNDARYPROBLEM TO SIMULATE THE FREEZING AND MELTING CONDITIONS ON A L A K E ......................................... 35 A. Factors Considered in Developing the M o d e l ................................... 35 B. Fundamental Equations .................... 36 C. Determination of the Characteristic Heat Flux in the Water layer................... 4l V. LAKE ICE FORMATION EXPERIMENT...................... 44 A. Great East Lake .......................... 44 iii B. Description of the Experiment ............. 44 C. Supplemental D a t a .......... ,............. 54 VI. ICE DEPTH PREDICTION................................ 65 A. Prediction During January Observation Period................................... 65 B. Prediction of Heat Flux Frcm February Through March D a t a ........................ 65 C. Heat Flux Prediction..................... 67 D. Numerical Procedure ...................... 72 VII. CONCLUSIONS .................................... 74 VIII. NOMENCLATURE....................................... 78 IX. BIBLIOGRAPHY ....................................... 80 X. APPENDICES......................................... 86 A. EXACT ANALYTICAL SOLUTION OF WESTPHAL...... 87 B. SENSIBLE HEAT CONSIDERATIONS.............. 90 C. EFFECTIVE THERMAL CONDUCTIVITY OF S N O W ..... 91 D. AMBIENT AIR TEMPERATURES TAKEN DURING THE FIRST OBSERVATION PERIOD OF JANUARY 7th TO JANUARY 20th, 1974 .......... 92 E. DEFINITION OF CORRELATION COEFFICIENT...... 104 F. COMPUTER PROGRAMS OF ICE MO D E L ............ 106 iv ACKNOWLEDGSVENTS The author is in deep gratitude to his advisor, Professor Stephen S. T. Pan, for his continued guidance and sincere interest throughout the development of this research. Professor Pan’s helpful suggestions for improving the final manuscript was greatly appreciated. The author also wishes to thank his wife, Maureen, for her patience and understanding during the many difficult years of my PhD career. Por this I am deeply indebted. The author wishes to thank Mrs. Brenda Cavanagh for her excellent typing of this manuscript. LIST OF TABLES Table Page I. Deviation of Ice Depth for 'Various Stefan Numbers ....... 23 II. Extinction Factors ................................... 42 III. Ice Depth Measurements at Great East Lake Wakefield, N.H. - Winter 1971! ................................... 52 IV. Snow Depth Observations ............................. 53 V. Average Daily Air Temperatures ........................ 55 VI. Concord Weather lata During First Observation Period 60 VII. Air Temperatures during Second Observation Period....... 6l VIII. Concord Weather lata During Second Observation Period .... 64 vi LIST OP FIGURES Figure Page 1. Simplified Illustration of Air-Ice-Water System .......... 14 2. Comparison of .Approximate Ice Depth Prediction versus Westphal’s Exact Solution ...................... 22 3. Air-Ice-Water System with Constant Heat Flux Existing in the Water Phase ........................... 25 2 4. 5 vs e Step Change Air Temperature with Constant Heat Flux in Liquid Phase ............................. 31 2 5. 6 vs 0 Periodic Air Temperature with Constant Heat Flux in Liquid Phase ............................. 33 6. Air-Snow-Ice-Water Problem............................. 37 7. New Hampshire and Maine .............. ................ 45 8. Great East Lake Wakefield, New Hampshire ................ 46 9. Ice Cover on Great East Lake (March 8, 1974) ............ 47 10. Ice Depth Measurement on Great East lake (March 8, 1974) ...................................... 47 11. Ice Depth Measurement on Great East Lake (March 8, 1974) ...................................... 49 12. Thermally Regulated Cabinet for Temperature Recorder 49 13. Sketch of Ice Depth Measuring Device ................... 50 14. Scatter Diagram of Lake Air Temperature and Concord Air Temperature From January 7th to January 2 0 t h ........ 56 15. Scatter Diagram of Lake Air Temperature and Portland Air Temperature Fran January 7th to January 20th ....... 57 16. Snow-Ice Cover on Great East Lake Together with Predicted Ice Depth Frcm Mathenatical Model Assuming No Heat Flux (First Observation Period) ................ 66 17. Snow-Ice Cover on Great East Lake Together with Predicted Ice Depth Frcm Mathematical Model Assuming No Heat Flux (Second Observation Period) ............... 68 18. Snow-Ice Cover on Great East Lake Together with Predicted Ice Depth Using Calculated Heat Flux (Second Observation Period) ........................... 71 vii ABSTRACT AN APPROXIMATE] SOLUTION TO THE
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