MODELING OF A THIN-LIQUID FALLING-FILM IN ABSORPTION 
COOLING SYSTEMS 
 
 
 
BY 
 
 
 
Kamilu Moradeyo ODUNFA 
 
B.Sc. (Hons) Mechanical Engineering (Ibadan) 
M. Sc. Mechanical Engineering (Ibadan) 
M. Phil. Mechanical Engineering (Ibadan) 
(Matriculation Number: 44113) 
 
 
 
A thesis in the Department of 
MECHANICAL ENGINEERING 
Submitted to Faculty of Technology in partial fulfillment  
of the requirement for the Degree of DOCTOR OF PHILOSOPHY 
of the 
UNIVERSITY OF IBADAN 
 
 
 
 
 
Mechanical Engineering Department, 
University of Ibadan, 
Ibadan. 
 
 
 
 
MARCH 2012 
 
  
ABSTRACT 
 
Absorption refrigeration systems are generally characterized by low Coefficient 
of Performance (COP). Absorption enhancement is an effective way of improving the 
COP of refrigeration systems. Literature is sparse on the use of magnetic field for the 
enhancement of absorption refrigeration systems despite its cheapness and environmental 
friendliness as compared with other enhancement methods. Although the method has 
recently been employed on ammonia solution, its influence on lithium bromide (LiBr) 
and lithium chloride (LiCl) solutions is yet to be fully studied. In this study a numerical 
model for the magnetic field enhancement of the absorption cooling-system using LiBr 
and LiCl solutions was developed and evaluated.   
The flow within the film thickness to the absorber wall was considered as a two-
dimensional steady laminar flow. A Finite Difference model was developed based on 
conservation of mass, momentum, energy equations and mass transport relationship. The 
model was validated using data from the literature on ammonia solution. Standard 
-3 
parameters including absorber wall length (1 m), film thickness (10 m), magnetic field 
-6 -2 2
vacuum permeability (1.257 x 10 kgmA s ), magnetic mass susceptibilities and 
magnetic induction intensities were used for LiBr and LiCl solutions’ modeling. Changes 
in their concentrations, both in the direction of falling film and across its thickness, were 
investigated. Data were analysed using descriptive statistics and Student’s t-test (p=0.05).   
The concentration distribution for ammonia solution within the film thickness was 
not significantly different from results in the literature. For the magnetic induction range 
of 0.0 and 3.0 Tesla, the concentration distribution of LiBr solution in the direction of 
falling film was between 54.9% and 60.0%, while that of LiCl solution ranged between 
39.9% - 45.0%. Meanwhile, across the film thickness and for the same range of magnetic 
induction of 0.0 and 3.0 Tesla, the concentration distribution for LiBr solution was 
between 0.0 and 0.19 and those of LiCl solution were between 0.0 and 0.13. The 
concentration of LiBr solution increased from 0.0 to 4.7 and 0.0 to 21.7 when magnetic 
induction was increased from 0.0 to 1.4 and 0.0 to 3.0 Tesla, respectively. Similarly for 
LiCl solution, increased values of 0.0 to 3.3 and 0.0 to 15.5 were obtained when magnetic 
induction was increased from 0.0 to 1.4 and 0.0 to 3.0 Tesla, respectively. In both cases, 
it implies higher cooling effect. Relative to 0.0 Tesla, the COP of LiBr and LiCl solutions 
ii 
 
  
absorption refrigeration systems was increased by 0.1% when magnetic induction was 1.4 
Tesla, while increment of 0.3% and 0.2% respectively were obtained when magnetic 
induction was 3.0 Tesla. The percentage increments in COP of LiBr solution were not 
significantly different from that of the LiCl solution.  
Magnetic field enhanced the absorption performance in the lithium bromide and 
lithium chloride solutions; hence can be used in typical absorption refrigeration systems.  
 
Keywords: Magnetic field, Refrigeration, Thin-liquid Falling-film, Absorption cooling- 
                   Systems 
Word count: 457  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
iii 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
DEDICATION 
 
To all 
  
Who fear God and acknowledge the power of His might,  
 
The doer of His word 
  
Through our Lord Jesus Christ  
 
This work 
 
Is dedicated 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
iv 
 
  
ACKNOWLEDGEMENT 
 
To God be the glory honor and power for His mercy and kindness who out of His 
marvelous light discovered and set me free from the power of darkness and oppression. 
He clothed my nakedness, opened my eyes and established my going academically, 
professionally and in every other ramification of my life. God has been so wonderful to 
me, He gives me life, strength and sound health even throughout the first frustrating 
seven years of my imprisonment and entanglement in my first aborted M. Phil degree 
programme. He did not leave or forsakes me in my successful M. Phil degree programme 
even up to this end, indeed He is a shelter in time of storm.    
I must appreciate my supervisor Prof. R.O Fagbenle for being so kind to me. In 
my state of academic hopelessness, he assured me of greater hope. For allowing himself 
to be used of the Lord to gather my battered and shattered academic life, I say a big thank 
you Sir. In deed you are a builder of life, especially for the young ones, you are also a 
template of a good Professor; an excellent father academically, professionally and in 
every other facet of human life, I prophesy onto your life Sir, it shall be well with you 
and your family. You will continue to be a vessel of honor in God’s mansion. Throughout 
your life you will not sorrow, the joy of the Lord will continue to be your strength.    
I am grateful to Prof. A. E Oluleye, the present Dean of the Faculty of 
Technology, University of Ibadan. He is such a loving and caring father, a template of a 
Dean in the citadel of learning. He enthusiastically took interest in the progress and 
development of young ones myself inclusive. Words can not be enough to express my 
gratitude to you for this. I can only pray that God in His infinite mercies reward you with 
v 
 
  
blessings. The Lord will satisfy you with long life and sound health, you will live to eat 
the fruits of your labour. 
I must equally express my gratitude to entire staff of the Department of 
Mechanical Engineering, University of Ibadan, whose love and concern to see me to the 
end of this programme was intoxicating. I must specifically thank the immediate and the 
present head of the Department Dr. O. Oluwole and Dr. M.O Oyewola respectively for 
helping me throughout, your constant encouragement on this programme has been of 
tremendous help. God will continue to bless you all. My specific appreciation also goes 
to Dr. Dare, Dr. T.A.O Salau, Engr. R. Abu, Dr. Ismail, Dr. I.F, Odesola and Dr. Fadare 
for their supports in one form or the other. To Dr.(Mrs) F,O Akintayo of Civil 
Engineering Department, I say thank you. The Dean of Postgraduate School Prof. 
Olorunnisola, you are highly appreciated Sir, Dr. Victor Oladokun and all other heads of 
the Departments in the Faculty of Technology, I appreciate you all, God will continue to 
bless you. 
I specifically express my appreciation to the past Faculty Sub-Dean Post graduate 
Dr. Falade of the Department of Food Technology, University of Ibadan. He is such a 
loving and caring colleague, he enthusiastically took interest in my research work. Words 
can not be enough to express my gratitude to you for your assistance and support. I can 
only pray that God in His infinite mercies will reward you with more blessings. The Lord 
will satisfy you with long life and sound health, you will live to eat the fruits of your 
labour. Immediate past and present Faculty Sub-Deans Postgraduate Dr. Ogunjuyigbe 
and Dr. Ewemoje, you are all highly appreciated. My special appreciation also goes to the 
present Sub-Dean Postgraduate School Dr. J.O Babalola of Chemistry Department and 
vi 
 
  
immediate past ASUU Chairman, University of Ibadan, Dr Ademola Aremu; May the 
Lord God continue to lift you all higher and higher.  
I must thank specially my loving wife, an Estate Surveyor per excellent Mrs. 
Victoria Oluwafunmike Odunfa (a.k.a Peace). Indeed you have demonstrated your name 
by allowing yourself to be the channel of the peace of the Lord to my life. I thank you for 
your moral support and continued endurance. You have been a vessel of honor in God’s 
mansion, so you will continue to be, I pray that God will continually make you a light to 
many generations. I am grateful to my God-given children, Victoria, Esther and Victor; in 
deed you are children of signs and wonder to me. The Lord will build around you the 
wall of fire that will continually separate you from the dangers of both day and night. 
You will live to continue to declare the glory of God in the land of the living. I must also 
appreciate Ojo Temitope O, my able research and engineering assistant, you have been so 
wonderful and helpful to me in this work, May the Lord God grant you your heart desire.  
I am also grateful to the present President of the Gospel Fellowship Incorporated 
Bro. Abel Aro for his care and concern, and his spiritual support throughout the period of 
this programme, I say thank you Sir. I am appreciative to the University for the Improved 
Internet Facilities that have made it possible to have a better grip of the subject of study. 
There are many others whom I may not be able to mention, but have contributed in one 
way or the other towards the success of this work. I am grateful to all and wish you 
success in your endeavours.           
 
 
K.M Odunfa 
 
 
 
vii 
 
  
TABLE OF CONTENTS 
ABSTRACT .................................................................................................................. ii-iii 
DEDICATION.................................................................................................................. iv 
ACKNOWLEDGEMENT.... .……………………………………………………….v-vii 
TABLE OF CONTENT……………………….…......................................................... ix                                                                                         
LIST OF FIGURES ..................................................................................................... x-xx 
LIST OF TABLES ................................................................................................... xix-xxi 
LIST OF SYMBOLS .................................................................................................... xxii 
CERTIFICATION ....................................................................................................... xxiii 
CHAPTER ONE:  INTRODUCTION ............................................................................ 1 
    1.1: General Background ................................................................................................ 1 
1.2:Statement of the problem .......................................................................................... 5 
   1.3: Justification /Objectives………………………………………………………......6 
CHAPTER TWO: LITERATURE REVIEW ................................................................ 7 
2.1: Introduction  ............................................................................................................. 7 
    2.1.1: Cooling Technology  ............................................................................................ 7 
    2.1.2: Advantages and disadvantages of vapour compression and Absorption  
              refrigeration………………………………………………………………………7 
 
    2.1.3: Vapour compression System  ............................................................................... 8 
    2.1.3: Vapour compression cycle  ................................................................................... 9 
    2.1.4: Absorption System  ............................................................................................ 11 
    2.1.5: Principle of Absorption System  ......................................................................... 11 
2.2: Working fluids/Refrigerants .................................................................................. 14 
    2.2.1: Various designs of Absorption Refrigeration Cycles………..…………………14 
    2.2.2: Practical Absorption Cycle…………………………………..…………………25 
2.4: Numerical Modeling  ........................................................................................ 26-31 
CHAPTER THREE: METHODOLOGY  ................................................................... 32 
3.1:  Assumptions  ........................................................................................................ 32 
3.2: Governing equations  ............................................................................................. 32 
viii 
 
  
    3.3: Finite Difference formulation of the governing equations…………….…….. ..  36 
    3.4: Formulation of the Magnetic field enhanced finite difference model of the  
           governing equations..…………………………………………………………....39 
3.5: Boundary conditions ……………………………………………………..……..39 
3.6: Solution method………………….……………………………………...…..…..43  
    3.6.1:Computer rogramming…………………………………………………………43 
3.6.2: Main Program …………………………………………………………………44 
3.6.3: Subroutines…………………………………………………………………….45 
    3.6.4:Main Program Flow chart....................................................................................45 
    3.6.5: Subroutine solution flow charts…………………………………………....46-47 
    3.7: Data…………………………………………………………………………..48-50 
CHAPTER FOUR: RESULTS AND DISCUSSION.................................................51 
4.1: General remark on the result presentation.........................................................51 
4.2: Numerical Results in the direction of falling film ......................................51- 52 
    4.3: Tabular and Graphical representation of the literature and present 
          work's result…..…………………………………………….…,………….…..52-61 
        4.4: Numerical Results in the direction of falling film (X) for LiBr and LiCl 
 solutions and their Coefficients of Performance….………………………..63-130 
        4.5: Numerical Results in the direction of the film thickness……..…………133-187 
CHAPTER FIVE: CONCLUSION AND RECOMMENDATION..........................188 
    5.1 Conclusions........................................................................................................188 
    5.2: Recomendations…………………………………………………………189-190 
REFERENCES………………………………………………………………......191-196 
APPENDICES..………………………………………………………………………..197 
APPENDIX A: DEVELOPMENT OF THE MODEL EQUATIONS……………198-203 
APPENDIX B: COMPUTER PROGRAM FOR THE MODEL ……………......204-234  
APPENDIX C:COMPUTER PROGRAM RESULTS….......................................235-266 
APPENDIX D: GAUSSIAN ELIMINATION..……………………..…….….. …267-268 
 
ix 
 
  
LIST OF FIGURES     Page 
 
Figure 2a: Vapour Compression Chiller-three stage Compressor ……….........................9 
Figure.2b: Basic Vapour Compression cycle- Single Stage……………………..……....10 
Figure 2c: Basic Vapour Compression cycle-Two Stage compressure and a flash 
chamber......................................................................................................................10 
Figure 2d: Typical two stage Absorption Chiller ……………………………………….11 
Figure 2e: An Intermittent Absorption Cycle……………………………………………13 
Figure 2f: A continuous absorption refrigeration cycle composes of two processes…....13 
Figure 2g: A Single–effect LiBr/water absorption refrigeration system with a  
solution heat exchanger (HX)…………………………………….……….…………….15 
Figure 2h: Absorption heat transformer Cycle…………………………………………..16 
Figure 2i: A double – effect water/LiBr absorption cycle……………………………….16 
Figure 2j: A double – effect absorption cycle operates with two pressure levels………..17 
Figure 2k: A triple – effect absorption cycle operates at 4 pressure levels……………...17 
Figure 2l: An Absorption Cycle with heat transference from high to low 
 temperature in the generator…………………...………………………………………..18 
 Figure 2m: The cycle with absorber heat recovery uses heat of absorption 
 of preheat the outgoing stream from the absorber to the generator……………………..18 
Figure 2n: A half – effect absorption cycle is a combination of two single  
– effect cycles  but working at different pressure levels………...……………………….19 
Figure 2o: Combined vapour absorption/compression heat pump………………………19 
Figure 2p: Configured Double effect absorption –compression cycle as a heat pump….20 
Figure 2q: A combined cycle proposed by Caccoila et al.(86)………………………….20 
Figure 2r: A resorption cycle proposed by Altenkirch uses two solution circuits………21 
Figure 2s: Solar driven dual cycle absorption employs to different working fluids i.e 
 NH3/water and water/LiBr………………………………………………………21 
Figure 2t: A modified double effect combined ejector – absorption refrigeration cycle..22 
Figure 2u: A combined ejector/absorption system using DMETEG/R22 and  
 DMETEG/R21 as working fluids……………………………………………….22 
Figure 2v: A combined ejector/absorption proposed by Aphornratana and Eames…….23 
x 
 
  
Figure 2w: A combined cycle proposed by Eames and Wu (93)……………………….23 
Figure 2x: An osmotic membrane absorption cycle…………………………………….24 
Figure 2y: The diagram shows a bubble pump in the generator module……………….24 
Figure 2zi: Lithium bromide-water single-stage absorption Refrigeration cycle……....25 
Figure 2zii:Lithium bromide-water two-stage absorption Refrigeration cycle…………25 
Figure 3.1a: 2-d representation of a thin-liquid falling-film……………………….……35 
Figure 3.1b: Finite difference diagram…………………………………………….……37 
Figure 3.2: Temperature boundary conditions analysis…………………………………41 
Figure 3.3: Concentration boundary conditions analysis…………………………….… 42 
Figure 3.4a: Main  program flow-chart.............................................................................45 
Figure 3.4b:Subroutine solution flow-chart......................................................................46 
Figure 3.4c: Subroutine Velocity flow-chart……………………………………………47 
Figure 4.1: Literature (NH3-H2O ) velocity distribution in the film  
Comparison @ 0.0,  1.4 and 3.0Tesla……………………………………..……………55 
Figure 4.1.1: Present work (NH3-H2O) Velocity Distribution in the Film Comparison 
 @ 0.0, 1.4 and 3.0Tesla ..……………………………………………………………...55 
Figure 4.1.2: Literature (NH3-H2O ) Temperature distribution in the film  
Comparison @ 0.0,  1.4 and 3.0Tesla……………………………………..……………58 
Figure 4.1.3: Present work (NH3-H2O) Temp.distribution in the Film Comparison 
 @ 0.0, 1.4 and 3.0Tesla ..……………………………………………………………...58 
Figure 4.1.4: Literature (NH3-H2O ) Concentration distribution in the film  
Comparison @ 0.0,  1.4 and 3.0Tesla……………………………………..……………61 
Figure 4.1.5: Present work (NH3-H2O) Conc. distribution in the Film Comparison 
 @ 0.0, 1.4 and 3.0Tesla ..……………………………………………………………...61 
Figure 4.1.6: LiBr-H2O Velocity Distribution in the Film Comparison @ 0.0, 1.4 
 and 3.0 Tesla……..………….………………………………………………………….64 
Figure 4.1.7: LiBr-H2O Velocity Changes within the Film in X-Direction 
 from 0.0 to 1.4Tesla……………………………………………………………………65 
Figure 4.1.8 : LiBr-H2O % Velocity Changes within the Film in X-Direction 
 from 0.0 to 1.4Tesla……………………………………………………………………66 
Figure 4.1.9: LiBr-H2O Velocity Changes within the Film in X-Direction 
xi 
 
  
 from 0.0 to 3.0 Tesla…..……………………………………………………………….66 
Figure 4.2: LiBr-H2O % Velocity Changes within the Film in X-direction 
 from 0.0 to 3.0Tesla……………………………………………………………………67 
Figure 4.2.1: LiBr-H2O Vel. Distribution in the Film comparison @ the Interface....... 68 
Figure 4.2.2: LiBr-H2O Velocity Changes at the Interface in X-direction  
from 0.0 to .1.4Tesla….………………………………………………………………...70 
Figure 4.2.3: LiBr-H2O % Velocity Changes at the Interface in X-Direction 
 from 0.0 to1.4Tesla…………………………………………………………………….70 
Figure 4.2.4: LiBr-H2O velocity Changes at the Interface in X-Direction  
from 0.0 & 3.0Tesla….…………………………………………………………………71 
Figure 4.2.5: LiBr-H2O % velocity Changes at the Interface in X-Direction 
 from 0.0 to 3.0Tesla…..……………………………………………………………….71 
Figure 4.2.6: LiBr-H2O Temp.Distribution in the Film Comparison @ 0.0Tesla……..73 
Figure 4.2.7: LiBr-H2O % Deviation in Temperature Distribution in the Film 
 Comparison @ 0.0Tesla……………………………………………………………….73 
Figure 4.2.8: LiBr-H2O Temp. Distribution in the Film @ 1.4 Tesla (Pre.Result)……74 
Figure 4.2.9: LiBr-H2O Temp. Distribution in the Film @ 3.0 Tesla(Pre.Result)……..74 
Figure 4.3: LiBr-H2O Temperature Distribution in the Thin-Liquid Smooth  
Falling film (Bulk) @ 0.0,1.4 and 3.0Tesla…………………………….……………...75 
Figure 4.3.1 LiBr-H2O Temperature Distribution at the Interface Comparison 
 @ 0.0 Tesla…….……………………………………………………………………...78 
Figure 4.3.2: LiBr-H2O % deviation in Temperature distribution at the interface 
Comparison@ 0.0Tesla………………………………………………………………..78 
Figure 4.3.3: LiBr-H2O Temperature Distribution at the Interface @ 1.4 Tesla……...79 
Figure 4.3.4: LiBr-H2O Temperature Distribution at the Interface @ 3.0Tesla………79 
Figure 4.3.5: LiBr-H2O Temperature Distribution at the Interface Comparison 
 @ 0.0, 1.4 and 3.0Tesla……………………………………………….……………...80 
Figure 4.3.6: LiBr-H2O Conc. Distribution in the Film Comparison @ 0.0Tesla……82 
Figure 4.3.7: LiBr-H2O % Deviation in Concentration Distribution in the Film. 
Comparison @ 0.0Tesla……………………………………………………………....82 
 Figure 4.3.8: LiBr-H2O Concentration Distribution in the Film @ 1.4Tesla………..83 
xii 
 
  
Figure 4.3.9: LiBr-H2O Concentration Distribution in the Film @ 3.0Tesla…………83 
Figure 4.4: LiBr-H2O Concentration Changes within the Film in X-Direction 
 from 0.0 and 1.4 Tesla………………………………………………………………..85 
Figure 4.4.1: LiBr-H2O % Concentration Changes within the Film 
 in X-direction @ 0.0 and 1.4Tesla…………………………………………………...85 
Figure 4.4.2: LiBr-H2O Concentration Changes within the Film in  
X-direction from 0.0 & 3.0 Tesla…………………………………………………….86 
Figure 4.4.3: LiBr-H2O % Concentration Changes within the Film in  
X-Direction @ 0.0 & 3.0Tesla………………………………………………………..86 
Figure 4.4.4: LiBr-H2O Concentration Distribution at the Interface  
Comparison @ 0.0Tesla……………………………………..………………………..90 
Figure 4.4.5: LiBr-H2O % Deviation in Concentration Distribution 
 at the Interface Comparison @ 0.0Tesla….………………………………………….90 
Figure 4.4.6: LiBr-H2O Concentration Distribution at the Interface @  
1.4Tesla (present Result)……………………………………..………………………91 
Figure 4.4.7: LiBr-H2O Concentration Distribution at the Interface @  
3.0Tesla (Present Result)……………………………………………..………………91 
Figure 4.4.8: LiBr-H2O Concentration Changes at the interface in 
 X-Direction from 0.0 to 1.4 Tesla……………………………………………...…….93 
Figure 4.4.9: LiBr-H2O % Concentration Changes at the interface in  
X-direction from 0.0 to 1.4Tesla……………………………………………………..93 
Figure 4.5: LiBr-H2O Concentration Changes at the interface in 
 X-direction @ 0.0 & 3.0 Tesla………………………………………………………94 
Figure 4.5.1: LiBr-H2O % Concentration Changes at the interface 
 in X-Direction from 0.0 to3.0Tesla……………………………………..…………..94 
Figure 4.5.2: LiBr-H2O Concentration Distribution at interface 
 X-direction @ 0.0, 1.4 & 3.0 Tesla………………………………………………….95  
Figure 4.5.3: LiBr-H2O Concentration Distribution in the Bulk at 
X-direction @ 0.0, 1.4 & 3.0 Tesla…………………………………………………..95 
Figure 4.5.3a: LiCl-H2O velocity distribution in the film comparison @ 0.0, 1.4  
and 3.0Tesla………………………………………………………………………….97 
xiii 
 
  
Figure 4.5.4: LiCl-H2O Temp. Distribution in the Film Comparison @ 0.0Tesla.….98 
Figure 4.5.5: LiCl-H2O % Deviation in Temp. Distribution in the Film 
 Comparison @ 0.0Tesla…………………..………………………………………….99 
Figure 4.5.6: LiCl-H2O Temp.Distribution in the Film @ 1.4 Tesla (Pres Result) ..  99 
Figure 4.5.7: LiCl-H2O Temp.Distribution in the Film @ 3.0 Tesla(Pres Result)…100 
Figure 4.5.8: LiCl-H2O Temp. Distribution in the Thin-Liquid Smooth Falling film 
 (Bulk) @ 0.0,1.4 and 3.0Tesla….………………………………………..………….100 
Figure 4.5.9: LiCl-H2O Conc. Distribution in the Film Comparison @ 0.0Tesla…...102 
Figure 4.6: LiCl-H2O % Deviation in Concentration Distribution in the Film 
 Comparison @ 0.0Tesla……………………………………………………………..102 
 Figure 4.6.1: LiCl-H2O Concentration Distribution in the Film @ 1.4Tesla……….103 
Figure 4.6.2: LiCl-H2O Concentration Distribution in the Film @ 3.0Tesla………..103 
Figure 4.6.3: LiCl-H2O Concentration Distribution in the Film  
Comparison @ 0.0, 1.4 and 3.0 Tesla………………………………………………..104  
Figure 4.6.4: LiCl-H2O Vel. Distribution in the Film comparison @ the Interface…106 
Figure 4.6.5: LiCl-H2O Temp.Distribution at the Interface Comparison@ 0.0Tesla..110 
Figure 4.6.6: LiCl-H2O % deviation in Temperature distribution at the interface 
Comparison@ 0.0Tesla…………………………………………………………........110 
Figure 4.6.7: LiCl-H2O Temperature Distribution at the Interface @ 1.4 Tesla….....111 
Figure 4.6.8: LiCl-H2O Temperature Distribution at the Interface @ 3.0Tesla……..111 
Figure 4.6.9: LiCl-H2O Temp. Distri. at the interface @ 0.0,1.4 and 3.0Tesla105….112  
Figure 4.7: LiCl-H2O Concentration Distribution at the Interface 
 Comparison @ 0.0Tesla…………………………………………………………….115 
Figure 4.7.1: LiCl-H2O % Deviation in Concentration Distribution at  
the Interface Comparison @ 0.0Tesla………………………………………………115 
Figure 4.7.2: LiCl-H2O Concentration Distribution at the Interface 
 @ 1.4Tesla (present Result)……………………………………………………… 116 
Figure 4.7.3: LiCl-H2O Concentration Distribution at the Interface @  
3.0Tesla (Present Result)…………………………………………………………. 116 
Figure 4.7.4: LiCl-H2O Concentration Distribution at the Interface 
 @ 0.0 1.4 and 3.0Tesla…………………………………………………………....117 
xiv 
 
  
Figure 4.7.5: LiCl-H2O Vel. Changes within the Film in X-Direction  
from 0.0 to 1.4 Tesla………………………………………………………………119 
Figure 4.7.6: LiCl-H2O % Velocity Changes within the Film in  
X-Direction from 0.0 to 1.4Tesla…………………………………………………119 
Figure 4.7.7: LiCl-H2O Vel. Changes within the Film in X-Direction 
 from 0.0 to 3.0 Tesla…………………………………………………………….120 
Figure 4.7.8: LiCl-H2O % Velocity Changes within the Film in X-direction  
from 0.0 to 3.0Tesla………………………………………………………………120 
Figure 4.7.9: LiCl-H2O Velocity Changes at the Interface in X-direction  
from 0.0 to 1.4Tesla………………………………………………………………122 
Figure 4.8: LiCl-H2O % Velocity Changes at the Interface in X-Direction 
 from 0.0 to1.4Tesla……………………………………………………………….122 
Figure 4.8.1: LiCl-H2O velocity Changes at the Interface in X-Direction 
 from 0.0 & 3.0Tesla………………………………………………………………123 
Figure 4.8.2: LiCl-H2O % velocity Changes at the Interface in 
 X-Direction from 0.0 to 3.0Tesla………..……………………………………….123 
Figure 4.8.3: LiCl-H2O Temperature Distribution within the  
Bulk @ 0.0, 1.4 and 3.0Tesla……………………………………………………..125  
Figure 4.8.4: LiCl-H2O Temp. Distribution at the interface  
@ 0.0, 1.4 and 3.0Tesla..........................................................................................126  
Figure 4.8.5: LiCl-H2O Conc. Changes within the Film in X-Direction 
 from 0.0 and 1.4 Tesla……………………………………………………………128 
Figure 4.8.6: LiCl-H2O % Concentration Changes within the  
Film in X-direction @ 0.0 and 1.4Tesla……………………………… …………128 
Figure 4.8.7: LiCl-H2O Concentration Changes within the  
Film in X-direction from 0.0 & 3.0 Tesla…………………………………………129 
Figure 4.8.8: LiCl-H2O % Concentration Changes within  
the Film in X-Direction @ 0.0 & 3.0Tesla…………………..……………………129 
Figure 4.8.9: LiCl-H2O Concentration Changes at the interface 
 in X-Direction from 0.0 to 1.4 Tesla…………………………………..…………131 
Figure 4.9: LiCl-H2O % Concentration Changes at the  
xv 
 
  
interface in X-direction from 0.0 to 1.4Tesla………………………..……………131  
Figure 4.9.1: LiCl-H2O Concentration Changes at the interface 
 in X-direction @ 0.0 & 3.0 Tesla………………………..………………………132 
Figure 4.9.2: LiCl-H2O % Concentration Changes at the interface 
 in X-Direction from 0.0 t0 3.0Tesla……………………..……………………....132 
Figure 4.9.3: LiBr-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &1.4Tesla at 0.25m level……………………………..138 
Figure 4.9.4: LiBr-H2O% Velocity Changes in the Direction of  
Film Thickness from 0.0 to 1.4Tesla @ =0.25m level…………………  ……….138  
Figure 4.9.5: LiBr-H2O Velocity distribution in the Direction of 
 Film Thickness @ 0.0 &3.0Tesla at 0.25m level………………………………..139 
Figure 4.9.6: LiBr-H2O %Velocity Changes in the Direction 
 of Film Thickness from 0.0 to 3.0Tesla @ =0.25m…………………………..…139 
Figure 4.9.7: LiBr-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.25m level……………………..140 
Figure 4.9.8: LiBr-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &1.4Tesla at 0.50m level………………….………….141 
Figure 4.9.9: LiBr-H2O %Velocity Changes in the Direction  
of Film Thickness from 0.0 to 1.4Tesla @ =0.50m level………………………...142  
Figure 4.10: LiBr-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at 0.50m level……………………………..142 
Figure 4.10.1: LiBr-H2O% Velocity Changes in the Direction  
of Film Thickness from 0.0 to 3.0Tesla @ =0.50m……………………………...143 
Figure 4.10.2: LiBr-H2O Velocity distribution in the Direction  
of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.50m level………………………143  
Figure 4.10.3: LiBr-H2O Velocity distribution in the Direction  
of Film Thickness @ 0.0 &1.4Tesla at 0.75m level………………………………145 
Figure 4.10.4: LiBr-H2O %Velocity Changes in the Direction 
 of Film Thickness from 0.0 to 1.4Tesla @ =0.75m level…………………….......145  
Figure 4.10.5: LiBr-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at 0.75m level………………………………146 
xvi 
 
  
Figure 4.10.6: LiBr-H2O %Velocity Changes in the Direction  
of Film Thickness from 0.0 to 3.0Tesla @ =0.75m…………………………….146 
Figure 4.10.7: LiBr-H2O Velocity distribution in the Direction  
of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.75m level…………………….147 
Figure 4.10.8: LiBr-H2O Temperature distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.25m level……………….....…148  
Figure 4.10.9: LiBr-H2O Temperature distribution in the Direction 
 of Film Thickness  @ 0.0, 1.4 and 3.0 Tesla at 0.50m level……………………150  
Figure 4.11: LiBr-H2O Temperature distribution in the Direction  
of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.75m level…………..………….151  
Figure 4.11.1: LiBr-H2O concentration distribution in the Direction  
of Film Thickness @ 0.0 &1.4Tesla at X=0.25m level …………………………153 
Figure 4.11.2: LiBr-H2O% concentration changes in the Direction 
 of Film Thickness from 0.0 to 1.4Tesla @ X=0.25m level……………………...153    
Figure 4.11.3: LiBr-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at X= 0.25m level………………………....154  
Figure 4.11.4: LiBr-H2O % concentration Changes in the Direction 
 of Film Thickness from 0.0 to 3.0Tesla @ X= 0.25m level……………………..154  
Figure 4.11.5: LiBr-H2O concentration distribution in the Direction 
 of Film Thickness  @ 0.0, 1.4 and 3.0 Tesla at X= 0.25m level…………………155 
Figure 4.11.6: LiBr-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0 &1.4Tesla at X=0.50m level …………………………156 
Figure 4.11.7: LiBr-H2O % concentration changes in the Direction 
 of Film Thickness from 0.0 to 1.4Tesla @ X=0.50m level………………………157    
Figure 4.11.8: LiBr-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at X= 0.50m level……………………..…..157  
Figure 4.11.9: LiBr-H2O % concentration Changes in the Direction 
 of Film Thickness from 0.0 to 3.0Tesla @ X= 0.50m level…………………….158  
Figure 4.12: LiBr-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at X= 0.50m level………………...158 
Figure 4.12.1: LiBr-H2O concentration distribution in the Direction 
xvii 
 
  
 of Film Thickness@ 0.0 &1.4Tesla at X=0.75m level ………………………..160 
Figure 4.12.2: LiBr-H2O % concentration changes in the Direction 
of Film Thickness from 0.0 to 1.4Tesla @ X=0.75m level……………………..160    
Figure 4.12.3: LiBr-H2O concentration distribution in the Direction  
of Film Thickness @ 0.0 &3.0Tesla at X= 0.75m level……………………..…161  
Figure 4.12.4: LiBr-H2O % concentration Changes in the Direction 
 of Film Thickness from 0.0 to 3.0Tesla @ X= 0.75m level…..……………….161  
Figure 4.12.5: LiBr-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at X= 0.75m level……………….162 
Figure 4.12.6: LiCl-H2O Velocity distribution in the Direction  
of Film Thickness @ 0.0 &1.4Tesla at 0.25m level…………………………...163 
Figure 4.12.7: LiCl-H2O% Velocity Changes in the Direction 
 of Film Thickness from 0.0 to 1.4Tesla @ =0.25m level…..………..………..164  
Figure 4.12.8: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at 0.25m level…………………………..164 
Figure 4.12.9: LiCl-H2O %Velocity Changes in the Direction  
of Film Thickness from 0.0 to 3.0Tesla @ =0.25m……………………………165 
Figure 4.13: LiCl-H2O Velocity distribution in the Direction  
of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.25m level……………….…...165 
Figure 4.13.1: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &1.4Tesla at 0.50m level…………………………...167 
Figure 4.13.2: LiCl-H2O %Velocity Changes in the Direction  
of Film Thickness from 0.0 to 1.4Tesla @ =0.50m level……………………….167  
Figure 4.13.3: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at 0.50m level…………………………….168 
Figure 4.13.4: LiCl-H2O% Velocity Changes in the Direction  
of Film Thickness  from 0.0 to 3.0Tesla @ =0.50m….…………………………168 
Figure 4.13.5: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.50m level…………………….169  
Figure 4.13.6: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &1.4Tesla at 0.75m level…………………………….170 
xviii 
 
  
Figure 4.13.7: LiCl-H2O %Velocity Changes in the Direction 
 of Film Thickness from 0.0 to 1.4Tesla @ =0.75m level………………………171  
Figure 4.13.8: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0 &3.0Tesla at 0.75m level……………………………171 
Figure 4.13.9: LiCl-H2O %Velocity Changes in the Direction 
 of Film Thickness from 0.0 to 3.0Tesla @ =0.75m……………………………172 
Figure 4.14: LiCl-H2O Velocity distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.75m level……………………172 
Figure 4.14.1: LiCl-H2O Temperature distribution in the Direction  
of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.25m level……………….……174  
Figure 4.14.2: LiCl-H2O Temperature distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.50m level…………..…….….175  
Figure 4.14.3: LiCl-H2O Temperature distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at 0.75m level……………..….….177  
Figure 4.14.4: LiCl-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0 &1.4Tesla at X=0.25m level ……………………….178 
Figure 4.14.5: LiCl-H2O% concentration changes in the Direction 
of Film Thickness from 0.0 to 1.4Tesla @ X=0.25m level……………………..179    
Figure 4.14.6: LiCl-H2O concentration distribution in the Direction  
of Film Thickness @ 0.0 &3.0Tesla at X= 0.25m level………………………...179  
Figure 4.14.7: LiCl-H2O % concentration Changes in the Direction 
 of Film Thickness from 0.0 to 3.0Tesla @ X= 0.25m level…………………….180  
Figure 4.14.8: LiCl-H2O concentration distribution in the Direction 
 of Film Thickness @ 0.0, 1.4 and 3.0 Tesla at X= 0.25m level…...……………180 
Figure 4.14.9: LiCl-H2O concentration distribution in the Direction of Film 
 Thickness @ 0.0 &1.4Tesla at X=0.50m level ………………………………...182 
Figure 4.15: LiCl-H2O % concentration changes in the Direction of Film 
 Thickness from 0.0 to 1.4Tesla @ X=0.50m level……………………………..182    
Figure 4.15.1: LiCl-H2O concentration distribution in the Direction of Film 
 Thickness @ 0.0 &3.0Tesla at X= 0.50m level………………………………...183  
Figure 4.15.2: LiCl-H2O % concentration Changes in the Direction of Film 
xix 
 
  
 Thickness from 0.0 to 3.0Tesla @ X= 0.50m level…………………………….183  
Figure 4.15.3: LiCl-H2O concentration distribution in the Direction of Film  
Thickness @ 0.0, 1.4 and 3.0 Tesla at X= 0.50m level…………………………184 
Figure 4.15.4: LiCl-H2O concentration distribution in the Direction of Film 
 Thickness @ 0.0 &1.4Tesla at X=0.75m level ………………………………..185 
 Figure 4.15.5: LiCl-H2O % concentration changes in the Direction of Film 
 Thickness from 0.0 to 1.4Tesla @ X=0.75m level……………………………..186    
Figure 4.15.6: LiCl-H2O concentration distribution in the Direction of Film 
 Thickness @ 0.0 &3.0Tesla at X= 0.75m level………………………………...186  
Figure 4.15.7: LiCl-H2O % concentration Changes in the Direction of Film 
 Thickness from 0.0 to 3.0Tesla @ X= 0.75m level…………………………….187  
Figure 4.15.8: LiCl-H2O concentration distribution in the Direction of Film 
 Thickness @ 0.0, 1.4 and 3.0 Tesla at X= 0.75m level…………………………187 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
xx 
 
  
 
 
LIST OF TABLES 
Table 3.1: Generaral Data …………………………………………………………...….48 
Table 3.2: NH3-H2O Data …………………………………………………………...….48 
Table 3.3: LiBr-H2O Data …………………………………………………………...….49 
Table 3.3: LiCl-H2O Data………………………………………………………….........50 
Table 4.1: NH3-H2O Velocity Distribution in the film Comparison ................………...52 
Table 4.1.1: NH3-H2O Velocity Deviation Analysis……................................................53 
Table 4.1.2: NH3-H2O Velocity changes within the film in X-direction…….................54 
Table 4.1.3: NH3-H2O Velocity T-Test Analysis……..…………………………….......54 
Table 4.1.4: NH3-H2O Temp.  Distribution in the film Comparison ................………...56 
Table 4.1.5: NH3-H2O Temp. Deviation Analysis……....................................................57 
Table 4.1.6: NH3-H2O Temp. T-Test  Analysis……..………………………………......57 
Table 4.1.7: NH3-H2O Conc.  Distribution in the film Comparison ................………...59 
Table 4.1.8: NH3-H2O Conc. Deviation Analysis……....................................................60 
Table 4.1.9: NH3-H2O Conc. T-Test Analysis……..………………………….……......60 
Table 4.2: LiBr-H2O Velocity Distribution in the film Comparison………..………… .63 
Table 4.2.1: LiBr-H2O Velocity changes within the film inX-direction………………...64 
Table 4.2.2: LiBr-H2O Velocity distribution in the film  
in X-direction @ interface…………………………………………….………………...67 
Table 4.2.3: LiBr-H2O Velocity changes within the film  
in X-direction @ interface…………………………………………….………………...69 
Table 4.2.4: LiBr-H2O Temperature Distribution in the film Comparison………….…..72 
Table 4.2.5: LiBr-H2O Temperature Distribution at the interface Comparison…….….. 75 
Table 4.2.6: LiBr-H2O Temp. Deviation Analysis in the bulk and at the Interface…….76  
Table 4.2.7: LiBr-H2O Temp. Deviation Analysis in the bulk and at the Interface….....77  
Table 4.2.8: LiBr-H2O Concentration Distribution in the film Comparison…………….80 
Table 4.2.9: LiBr-H2O Concentration Changes within the film in X-direction………....84 
Table 4.3: LiBr-H2O Concentration Distribution at the interface Comparison…………87 
Table 4.3.1:LiBr-H2O Conc. Deviation Analysis in the bulk and at the Interface……...88 
xxi 
 
  
Table 4.3.2:LiBr-H2O Conc. Deviation Analysis in the bulk and at the Interface……...89  
Table 4.3.3: LiBr-H2O Concentration Changes at the interface in X-direction…………92 
Table 4.3.4: LiCl-H2O Velocity Distribution in the film Comparison…..............………96 
Table 4.3.5: LiCl-H2O Temperature Distribution in the film Comparison………….…..97 
Table 4.3.6: LiCl-H2O Concentration Distribution in the film Comparison………….. 101 
Table 4.3.7: LiCl-H2O Velocity Distribution at the interface Comparison….................105 
Table 4.3.8: LiCl-H2O Temperature Distribution at the interface Comparison………..107 
Table 4.3.9: LiCl-H2O Temp. Deviation Analysis in the bulk and at the Interface…...108  
Table 4.4: LiCl-H2O Temp. Deviation Analysis in the bulk and at the Interface…......109  
Table 4.4.1: LiCl-H2O Concentration Distribution at the interface Comparison………112 
Table 4.4.2: LiCl-H2O Conc.Deviation Analysis in the bulk and at the Interface…….113 
Table 4.4.3: LiCl-H2O Conc.Deviation Analysis in the bulk and at the Interface…….114  
Table 4.4.4: LiCl-H2O Velocity Changes within the film in X-direction…...................118 
Table 4.4.5: LiCl-H2O Velocity Changes within the film in X-direction 
 @ the interface .…………………………………………………………………….....121 
Table 4.4.6: LiCl-H2O Temperature Changes within the film in X-direction…………124 
Table 4.4.7: LiCl-H2O Temperature Changes at the interface Comparison……………125 
Table 4.4.8: LiCl-H2O Concentration Changes within the film in X-direction……..…127 
Table 4.4.9: LiCl-H2O Concentration Changes at the interface in X-direction……….130 
Table 4.5: Percentage increment of coefficient of Performance (COP) @ 1.4 Tesla…136 
Table 4.5.1: Percentage increment of coefficient of Performance (COP) @ 3.0 Tesla.136 
Table 4.5.2: LiBr & LiCl   COP increments at 3.0 Tesla T-Test analysis…………….136 
Table 4.5.3: LiBr-H20: Velocity Changes in the direction of film 
 thickness ( ) @ X=0.25m…………………………………………………………,,..137 
Table 4.5.4: LiBr-H20: Velocity Changes in the direction of film 
 thickness ( ) @ X=0.50m…………………………………………………………….140 
Table 4.5.5: LiBr-H20: Velocity Changes in the direction of 
 film thickness ( ) @ X=0.75m……………………………………………………….144 
Table 4.5.6: LiBr-H20: Temperature Changes in the direction of 
 film thickness ( ) @ X=0.25m……………………………………………………….147 
Table 4.5.7: LiBr-H20: Temperature Changes in the direction of 
xxii 
 
  
 film thickness ( ) @ X=0.50m……………………………………………………….149 
Table 4.5.8: LiBr-H20: Temperature Changes in the direction of 
 film thickness ( ) @ X=0.75m……………………………………………………….150 
Table 4.5.9: LiBr-H20: Concentration Changes in the direction of 
 film thickness ( ) @ X=0.25m……………………………………………………….152 
Table 4.6: LiBr-H20: Concentration Changes in the direction of 
 film thickness ( ) @ X=0.50m………………………………………………………155 
Table 4.6.1: LiBr-H20: Concentration Changes in the direction of 
 film thickness ( ) @ X=0.75m………………………………………………………159 
Table 4.6.2: LiCl-H20: Velocity Changes in the direction of 
 film thickness ( ) @ X=0.25m………………………………………………………162 
Table 4.6.3: LiCl-H20: Velocity Changes in the direction of 
 film thickness ( ) @ X=0.50m………………………………………………………166 
Table 4.6.4: LiCl-H20: Velocity Changes in the direction of  
film thickness ( ) @ X=0.75m……………………………………………………….169 
Table 4.6.5: LiCl-H20: Temperature Changes in the direction 
 of film thickness ( ) @ X=0.25m….………………………………………………..173 
Table 4.6.6: LiCl-H20: Temperature Changes in the direction 
 of film thickness ( ) @ X=0.50m…………………………………………………...174 
Table 4.6.7: LiCl-H20: Temperature Changes in the direction 
 of film thickness ( ) @ X=0.75m…………………………………………………...176 
Table 4.6.8: LiCl-H20: Concentration Changes in the direction 
 of film thickness ( ) @ X=0.25m…………………………………………………...177 
Table 4.6.9: LiCl-H20: Concentration Changes in the direction 
 of film thickness ( ) @ X=0.50m….………………………………………..………181 
Table 4.7: LiCl-H20: Concentration Changes in the direction 
 of film thickness ( ) @ X=0.75m…………………………………………………...184 
 
 
 
 
xxiii 
 
  
 
 
 
       LIST OF SYMBOLS 
                                            Unit     
 μ   = film dynamic viscosity……………………………………… kg/m/s  
 υo   = mean velocity………………………………………………. m/s  
2
 α = thermal diffusivity……..……………………………………… m /s 
 k = thermal conductivity of fluid ……………................................. W/m K 
3
 ρ = liquid density………………………………………………….. kg/m  
2
 D = species diffusivity………………… …………………………. m /s 
-1 
 β = cubic expansivity of fluid …………………………………….. K
0
Tw = dimensional wall temperature………………………………… C 
0
Tin = inlet refrigerant temperature…………………………………. C  
Cin = initial absorbent concentration   ………….............................. % 
Ceq =   equilibrium absorbent concentration……………………….. % 
2
g   = gravity………………………………………………………. m/s  
h0 = mean film thickness….……………………………………… m 
2
ν   = kinematic viscosity of fluid…………………………………. m /s  
Ha = heat of absorption…………………………………………... kJ/kg   
Pv = absorbent vapour pressure……………………………......... mm. Hg 
Ref = film Reynolds number 
 Ґ    = film mass flow rate 
 
                      
 
 
 
 
xxiv 
 
  
   CERTIFICATION 
 
 
 
I certify that this research work was carried out by Mr. K.M. Odunfa in the Department 
of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria. 
 
 
 
 
 
______________ 
Research Supervisor, 
Prof. R. O. Fagbenle. FNSE 
B.S.M.E. (Urbana-Champaign), M.S.M.E. (Iowa State), PhD (Urbana-Champaign), 
R.Engr. 
Professor 
Department of Mechanical Engineering, 
University of Ibadan,  
Ibadan, Nigeria. 
 
 
xxv 
 
 
CHAPTER ONE 
 
INTRODUCTION 
1.1 General Background 
 In recent years, the terms energy conservation and environmental safety have become a 
thing of global concern due to the increasing energy prices, energy security and environmental 
impact of energy prospecting, processing and utilization. In time past, not much emphasis was 
placed on the issue as it is now. The worldwide attention to Climate Change phenomenon and its 
impacts which have been conclusively linked to fossil energy use has prompted the emergence of 
the new technologies in many areas of global economy, such as in cooling system development 
sector. Cooling system basically may be divided into two categories; Vapour compression 
system and sorption system.  Sorption system is further sub-divided into absorption and 
adsorption systems. Vapour compression system involves the use of a compressor for the 
compression process; An absorption system is simply the replacement of the traditional 
compression with a thermo chemical fluid lifting process.  In other words it is the mixture of a 
gas in a liquid, the two fluids present a strong affinity to form a solution, while adsorption is a 
process that occurs when a gas or liquid solute accumulates on the surface of a solid or, more 
rarely, a liquid (adsorbent), forming a molecular or atomic film (the adsorbate). 
 In the manufacturing of cooling machine/system, for the increasing interest in the 
efficient use of energy at minimum environmental cost has necessitated the increased demand for 
absorption refrigeration systems driven by waste heat or solar thermal energy instead of 
conventional systems driven by fossil-energy derived or conventional. The current imbalance of 
energy demand and supply coupled with the environmental degradation in many developing 
countries such as Nigeria has further increased the urgent need for highly efficient and 
sustainable energy technologies.  
 In the absorption process, heat and mass transfer usually take place within a thin-liquid 
falling-film. Heat and mass transfer in thin-liquid falling film absorption process has received the 
attention of many researchers over the years especially in the last two decades.  This is as a result 
of its wider application in many modern devices such as absorption air-conditioners, absorption 
chillers, absorption heat pumps etc Yang and Wood (1992). Absorption enhancement is another 
aspect in this area that has also attracted the attention of researchers. Absorption enhancement is 
an effective way to improve the performance of absorption refrigeration systems. Generally, 
 1 
 
there are three kinds of methods in absorption enhancement (Kim et al.,2006).The first kind  falls 
under the category of mechanical methods, which improve the performance by modifying the 
shape, surface and structure of the heat transfer tubes (Chen et al.,2006). The second kind 
comprise chemical methods which involve the addition of surfactant in the absorbent while the 
third kind is the addition of nano-particles in the absorbing solution e.g Cu, CuO and Al2O3 
nano-particles added into ammonia-water solution (Kim et al., 2007a and 2007b), Fe and Carbon 
nano-tubes (CNT) in lithium bromide-water solution (Yong Tae Kang et al.,2007). Research on 
nano-fluids / nano-particles in absorbent are categorized into five groups(1) stability analysis and 
experiments; (2)property measurement such as thermal conductivity and viscosity;(3)convective 
and boiling heat transfer;(4)mass transfer in binary nanofluids; and (5)theoretical analysis and 
model development.  
However, the effect of magnetic field on absorption refrigeration system is seldom 
mentioned in the literature apart from its established influence on the absorption process in 
ammonia vapour into ammonia-water solution absorption refrigeration system (Niu et al., 2006). 
The magnetic field may therefore also have certain influence on the absorption process in other 
absorption refrigeration systems such as water vapour into lithium bromide-water solution and 
water vapour into lithium chloride-water solution absorption system.  
 In previous studies without any form of enhancement, Andberg (1982), Grossman (1983) 
and Andberg and Vliet (1983) have employed the modeling technique. Modeling is categorized 
into two namely (i) Numerical and (ii) Experimental. 
Numerical methods have been used over many decades and continue to be developed to 
effectively tackle many engineering problems.  Some of these numerical methods include Finite 
Element method, finite difference method, boundary element method, Monte Carlo technique 
and Vortex Method.  These methods may be categorized into two: probabilistic approach and 
deterministic approach. Probabilistic methods generally make constant recourse to random 
numbers, and Monte Carlo and Vortex element techniques fall into this category while Finite 
Element, Finite difference and boundary element techniques are deterministic in nature. 
 In absorption process modeling either unenhanced or enhanced, experimental modeling 
has been extensively employed, while the numerical modeling of the problem has been rather 
difficult due to the presence of waves in the falling liquid film. However, the smooth film 
absorption assumption has allowed successful modeling of the problem even though without any 
 2 
 
form of enhancement. One of the earliest of such models was developed by G. Grossman (1983), 
later followed by other researchers such as Andberg and Vliet (1983) and Yang and Wood, B.D 
(1992).  
 In the area of numerical modeling of enhanced absorption system, Xiao Feng Niu et al. 
(2006) established the mathematical model for magnetic field enhanced absorption process for 
ammonia-water solution on a falling-film. The changes in physical properties of ammonia-water 
solution in absorption, the variation of falling film and the convection in the direction of 
thickness of liquid film were considered in the modeling; Distribution of some parameters in 
falling–film absorption, such as velocity, temperature and concentration; in the application of 
magnetic field was obtained. Numerical results obtained show that magnetic field can improve 
the performance of ammonia-water falling-film absorption, and the absorption strengthening 
effect increases with the enhancement of magnetic induction intensity. The strengthening effect 
is limited within the magnetic field intensity of 0-3 T, but there are trends of increasing 
strengthening effect in stronger magnetic fields. In both unenhanced and enhanced absorption 
system studies, several working fluids have been investigated, some of which are Lithium 
bromide-water (LiBr-H2O), Lithium-Chloride Water (LiCl-H2O) and Ammonia-water (NH3-
H2O) all of which are popularly used in single-stage and advanced absorption air-
conditioning/heat pump technology. 
Xiao Feng Niu et al. (2009) experimentally studied the effect of external magnetic field 
on falling film absorption for ammonia-water system. The study established the following 
findings: (i) An external magnetic field acting in the same direction as falling film has an 
enhancing effect on absorption of ammonia-water process, and the absorption enhancement is 
more greater in stronger magnetic field. When external magnetic field with the same direction as 
the falling film is exerted, several absorption variables, including the concentration of the 
ammonia-water solution after absorption, the outlet temperature of the cooling water, the 
absorption heat and absorption mass in the external magnetic field, are higher than those without 
external magnetic field exerted. Moreover, the four absorption variables increase with the 
increase in magnetic induction intensity, (ii) Not all magnetic fields can enhance the ammonia-
water absorption process. When an external magnetic field acting against the direction of falling 
film is exerted, the absorption variables in magnetic field are all smaller than those in 
conventional absorption without magnetic field exerted. The magnetic field opposing the 
 3 
 
direction of the falling film weakens the absorption of ammonia-water (iii) The absorption can be 
more intense if the external magnetic field is combined with optimal operating conditions. 
Experimental results show that the changes in the outlet cooling water temperature, absorption 
heat and absorption mass with and without external magnetic field exerted, are larger when the 
inlet solution concentration is lower. Larger cooling water flow rate, lower cooling water 
temperature and smaller solution flow rate are beneficial to absorption in magnetic field as they 
do in conventional absorption.  
 The finite difference method has been applied much more than the finite element method 
in the analysis of absorption/adsorption systems. The finite element method sometimes described 
as a versatile and powerful numerical method is a piecewise approximation method. It 
approximates a problem described by a system of differential equations by a number of algebraic 
equations relating a finite number of variables (unknowns). These algebraic equations are then 
solved on the digital computer (for large systems) using an appropriate solution technique. The 
fundamental concept of the method is that any continuous field quantity (e.g. temperature and 
concentration) in a continuum can be approximated by a discrete model composed of a set of 
piecewise continuous parameter functions defined over a finite number of sub-domains. These 
sub-domains which are referred to as finite elements are connected at discrete points called 
nodes. The parameter functions must satisfy specific compatibility and completeness 
requirements. 
 In engineering problems, either probabilistic or deterministic methods could be used 
depending on the degree of accuracy required of the solution in comparison with experimentally 
data or analytic solution. Absorption process enhancement under magnetic field apart for its 
established effect on ammonia-water absorption process (Xiao Feng Niu et al.( 2006), (2009)) as 
earlier mentioned, magnetic field effect is seldom used and to the best of the knowledge of the 
researcher, it has not been used in either lithium bromide-water or lithium chloride-water 
absorption system. This present work therefore investigates the effect of a magnetic field 
enhanced absorption process on a smooth thin-liquid falling-film in a cooling system using 
lithium bromide-water (LiBr-H2O) and (LiCl- H2O) refrigerants/absorbents combinations. A 
model of the problem is first developed from the resulting differential equations and solved using 
the finite difference method. Such an investigation would reveal sections of the absorber that 
 4 
 
need to be redesigned and its material re-specified, e.g for optimal efficiency of refrigerant 
absorption by the absorbent.   
Statement of the Problem 
 In the manufacturing of modern absorption devices such as absorption air-conditioners, 
chillers and heat pumps, the most popularly used refrigerant/absorbent combination or working 
fluids are Lithium-Water Bromide (LiBr-H2O), Lithium Chloride-Water (LiCl-H2O), Ammonia-
water (NH3–H2O), Pentafluoroethane (HFC-125) and Trifluoroethane (HFC-143a). Due to the 
zero ozone depletion potential (ODP) and zero global warming potential (GWP) of NH3-H2O 
and H2O-LiBr) Fagbenle et al. (1994), these two refrigerant/absorbent combination are relatively 
environmentally friendly. Ammonia-water solution is however incompatible with copper, the 
popular tubing material for transporting refrigerants, for this reason, it is less popularly used than 
the other working fluids. 
   The other working fluids such as Halogenated Chlorofluorocarbons (HCFCs), and 
Azeotropic mixtures (R-500 and R-502) that were popularly used in the past have been declared 
non-environmentally friendly following the Montreal Protocol and subsequent London and 
Copenhagen meetings of 1990 and 1992, Fagbenle et al. (1994).  This is because of their high 
Ozone Depletion Potential (ODP) and high Global Warming Potential (GWP) coupled with their 
roles in the green house effect (either directly through the fluids used or indirectly through the 
energy consumption in systems using fossil-energy) and as such they are no more in use.  
Numerical modeling of the absorption process on a thin-liquid falling-film generally has 
been difficult due to complication arising from the presence of waves. The Odunfa (2008), 
approached this issue by considering waves as a second order effect, thereby, appropriating the 
flow as a smooth falling-film. The degree to which ignoring the second-order effect affects the 
results by using a thin-liquid falling-film approximation to the absorption process was 
investigated in his work using lithium bromide-water and lithium chloride-water as working 
fluids. The results obtained then compared well with the existing experimental results. This 
research work thus aims at further investigating and establishing the influence of the magnetic 
field on the absorption process in a smooth thin-liquid falling-film for a cooling system using 
(LiBr-H2O) and (LiCl-H2O) refrigerants/absorbents.      
 5 
 
Justification 
 Energy conservation and environmental concerns are increasingly attracting global 
attentions, due to the increasing energy prices and environmental impact of energy prospecting, 
processing and utilization. As such conventional refrigeration systems are gradually giving way 
to newer technologies such as absorption cooling systems. Since the major concern in absorption 
refrigeration systems hinges largely on the energy conservation, environmental issues and 
efficient cooling system which the present work is to be addressed, hence the justification for this 
work. 
 
Aims and Objectives  
The present work aims and objectives are (i) to develop and evaluate a numerical model 
for the magnetic field enhancement of the absorption cooling systems using lithium bromide 
(LiBr) and lithium chloride (LiCl) water solution. (ii) to establish absorption system‟s 
performance improvement with increment in magnetic induction intensity enhancement on the 
two investigated fluids and (iii) to establish the enhancement of the Coefficient of Performance 
(COP) of both fluids with magnetic field.   
Expected practical application of this work includes the design of efficient absorption 
plant components such as absorbers, evaporators, condensers and generators. Also in this study, 
key factors such as refrigerant/absorbent combination‟s parameters and film velocity which 
influence the improvement level in the absorption performance under magnetic field 
enhancement would be established.  
 
 
 
 
 
 
 
 
 
 
 6 
 
CHAPTER TWO 
 
LITERATURE REVIEW 
2.1: Introduction 
The production of cold temperature has application in many fields of human endeavour, 
e.g. for preservation of perishable products, in the food processing industry, in the air 
conditioning sector and for the preservation of pharmaceutical products. Based on the cooling 
temperature requirement, the applications of the absorption systems/machines can be broadly 
o o
classified into three categories; air-conditioning (8-15 C) for space cooling, refrigeration (0-8 C) 
o
for food and vaccine storage and freezing (< 0 C) for ice making purposes. 
 
2.1.1: Cooling Technology 
Cooling technology is classified into two broad systems; vapour compression and absorption 
cooling systems. Based on the cooling temperature requirement, either of the two classifications 
can be further broken down into the three categories mentioned above. 
2.1.2: Advantages and Disadvantages of Vapour Compression and Absorption  
          Refrigeration 
 i. The amount of power required: The compressor of the vapor compression cycle requires 
large amount of power for its operation and it increases as the size of the refrigeration system 
increases. In case of the vapor absorption refrigeration system, the pump power required to 
circulate the absorber-absorbent fluids is relatively very small. 
ii. Running cost: The vapor compression refrigeration system can run only on electric power, 
and they require large amount of power. In case of the absorption refrigeration system only small 
pump is required whose electric power consumption is generally quite low. Thus the running 
cost of the absorption refrigeration system is much smaller than for the vapor compression 
system. 
iii. Foundations required and noise: The compressor of the vapour compression system is 
operated at very high speeds and with consequent vibrations and noise. It also requires very 
strong foundation to withstand the vibrations. In the absorption refrigeration system, there are no 
 7 
 
major moving parts hence they do not vibrate and make no noise. The absorption refrigeration 
system operates silently with no vibration   
iv. Maintenance: The compressor is a key component of the compression cycle having a number 
of moving parts. Thus the compression system generally requires a lot maintenance attention. In 
the absorption refrigeration system the only moving part is the small pump that fails rarely 
making the system to be robust and requiring little or no maintenance. 
v. Type of refrigerant used and its cost: The refrigerant used by the absorption refrigeration 
system is environmentally friendly; easily and cheaply available. In the case of the vapour 
compression refrigeration system, halocarbons that are been used as the refrigerants are not 
environmentally friendly and also very expensive.  
vi. Leakage of the refrigerant: In the absorption refrigeration system leakage of the refrigerant 
seldom occurs while the refrigerant itself is relatively inexpensive. Thus refrigerant recharging 
costs is minimal. In vapour compression systems, leakages of the refrigerant occurs quite often 
requiring regular refrigerant recharge of the system usually at a relatively high cost.  
vii. Greenhouse effect: Most of the halocarbon refrigerants used in the compression 
refrigeration system has a high GWP. Following the Montreal Protocol, their use has to stop 
completely by the year 2020. In the absorption refrigeration system, refrigerants have very low 
or zero GWP and they are not in any exclusion list. 
2.1.3: Vapour compression System 
Fig. 2a is a typical vapour compression system (Chiller) York model. The following major 
components can be identified in the figure: compressor, condenser coil, evaporator and fans. 
Other minor components such as filter drier, oil line and other accessories are also shown in the 
figure.    
 8 
 
 
 
Fig.2a. Vapour Compression Chiller-three stage Compressor 
 
2.1.4: Vapour compression cycle 
Fig 2b shows the typical schematic diagram of a single stage compression system containing a 
compressor, a condenser, an evaporator and an expansion valve. The two- stage compression 
system shown in fig 2c contains a flash chamber, and two compressors instead of one as in the 
single-stage system. 
 
 9 
 
Condenser
Compressor
Evaporator
Expansion
Valve  
Fig.2b. Basic Vapour Compression cycle- Single Stage 
 
Condenser
Compressor
Compressor
Flash
Chamber
Evaporator
 
Fig.2c. Basic Vapour Compression cycle - Two-stage 
                     Compressor and a flash chamber 
 
 
 
 10 
 
2.1.5: Absorption System 
Fig. 2d is a typical two-stage absorption system (Chiller). It contains two generators (G1& G2), 
two condensers (C1& C2), an absorber, and an evaporator. Other components such as pump, an 
expansion valves and solution heat exchanger (SHE) are also in the figure. The line diagram of 
the two-stage absorption refrigeration cycle appears in fig 2f. 
 
 
Fig. 2d.Typical two-stage Absorption Chiller 
 
2.1.6: Principle of operation of Absorption system 
The working fluid in an absorption refrigeration system is a binary solution consisting of 
refrigerant and absorbent. In the fig. 2e below, two evacuated vessels are connected to each 
 11 
 
other. The left vessel contains liquid refrigerant while the right vessel contains a binary solution 
of absorbent/refrigerant. The solution in the right vessel will absorb refrigerant vapour from the 
left vessel causing pressure to reduce. While the refrigerant vapour is being absorbed, the 
temperature of the remaining refrigerant will reduce as a result of its vaporization. This causes a 
refrigeration effect to occur inside the left vessel. At the same time, solution inside the right 
vessel becomes more dilute because of the higher content of refrigeration absorbed. This is 
called the “absorption process”. Normally, the absorption process is an exothermic process, 
therefore it must reject heat out to the surrounding in order to maintain its absorption capability. 
Whenever the solution cannot continue with the absorption process because of saturation of the 
refrigerant, the refrigerant must be separated out from the diluted solution. Heat is normally the 
key for this separation process. It is applied to the right vessel in order to dry the refrigerant from 
the solution as shown in fig. 2e(b) below. The refrigerant vapour will be condensed by 
transferring heat to the surroundings. With these processes, the refrigeration effect can be 
produced by using heat energy. However, the cooling effect cannot be produced continuously as 
the process cannot be done simultaneously. Therefore, an absorption refrigeration cycle is a 
combination of these two processes as shown in fig. 2f. As the separation process occurs at a 
higher pressure than the absorption process, a circulation pump is required to circulate the 
solution. Coefficient of performance of an absorption refrigeration system is obtained from;  
cooling capacity obtained at evaporator
 COP =  
heat input for the generator  work input for the pump
The work input for the pump is negligible relative to the heat input at the generator; therefore, 
the pump work is often neglected in the evaluation of the COP. 
 
 12 
 
Refrigerant
Solution Refrigerant Solution
QI QI
QL
QH
(a)
            (b)  
Fig. 2e: An intermittent Absorption Cycle 
 
.   (a) Absorption process occurs in right vessel causing cooling effect in the other,          
 (b) Refrigerant separation process occurs 
 
refrigerant separation process
generator condenser
QH QI
absorber evaporator
QI QL
absorption process
 
Fig. 2f: A continuous absorption refrigeration cycle composes of two processes mentioned in the 
earlier figure. 
  
 
 
 13 
 
2.1.7: Working fluids/Refrigerants  
A review of literature carried out on the refrigerants used in air-conditioning and 
refrigerating systems reveals that fully halogenated chlorofluoro carbon (CFCs) and halon 
compounds, carbon tetra chloride (CCL4), 1,1, l-trichloride ethane (Methyl Chloroform 
CH3CCl3) and methyl bromide (CH3Br) are the commonly used refrigerants in the industrial and 
commercial compression cooling systems. Following the scientific findings on the above 
mentioned refrigerants that they have high ozone depletion potential (ODP) and global warming 
potential (GWP), the 1987 Montreal Protocol was made in 1990 and amended in Copenhagen in 
1992 to phase out or ban these refrigerants from usage by the year 2015. Ever since this protocol, 
the search for replacement chemicals has been exclusively in the developed countries, because 
the bulk of the production and consumption of these restricted chemicals is in these countries. 
Furthermore the technical expertise, research infrastructure and the direct and allied economic 
investment are all in these countries. 
 Researchers in this field of study worldwide have intensified efforts towards finding 
alternative refrigerants to the CFC‟s which will not only be environmentally friendly, but which 
will also be energy efficient. A survey of absorption fluids provided by Marcriss et al.(1988) 
suggests that, there are some 40 refrigerant compounds and 200 absorbent compounds available. 
However, the most common working fluids are water lithium bromide (H2O–LiBr), water 
lithium Chloride (H2O–LiCl) and Ammonia–water (NH3–H2O) solutions. Heat and mass transfer 
analysis of cooling systems has been continuously undertaken by researchers such as Yang and 
Wood (1992), Muhsin et al.(2004), Fernandez et al.(2005), Xu et al.(2006) and Gu et al.(2007) in 
order to improve on plant and component design and efficiencies. Water lithium bromide (H2O–
LiBr) and water lithium chloride (H2O-LiCl) solutions remain the most popularly used 
refrigerant-absorbent pairs in absorption cooling systems and as such the present work employs 
these two working fluids, albeit with magnetic field enhancement. 
 
2.1.8: Various designs of absorption refrigeration cycles 
 Pongsid Srikhirin et al. (2001) in his review of absorption refrigeration technologies 
established the following designs of absorption refrigeration cycles: the single-effect absorption 
system (Fig.2g), the absorption heat transformer (Fig.2h), the double-effect water/LiBr 
absorption cycle (Fig.2i), the double-effect absorption cycle operates with two pressure levels 
 14 
 
(Fig.2j), a triple-effect absorption cycle operates at four pressure levels (Fig.2k), the dotted loop 
shows secondary fluid used for transferring heat from high the temperature section in the 
absorber to low temperature section in the generator (Fig.2l), The cycle with absorber heat 
recovery (Fig. 2m),  A half – effect absorption cycle (Fig. 2n), Combined vapour 
absorption/compression heat pump (Fig. 2o),  Double effect absorption –compression cycle, 
(Fig. 2p), A combined cycle proposed by Caccoila et al (Fig.2q, A resorption cycle proposed by 
Altenkirch (Fig. 2r), Solar driven dual cycle absorption (Fig. 2s), A modified double effect 
combined ejector – absorption refrigeration cycle (Fig. 2t),  A combined ejector/absorption 
system using DMETEG/R22 and DMETEG/R21 as working fluids (Fig. 2u), A combined 
ejector/absorption proposed by Aphornratana and Eames (Fig. 2v), A combined cycle proposed 
by Eames and Wu (Fig. 2w), osmotic-membrane absorption cycle (Fig.2x), and diffusion 
absorption refrigeration system DAR) (Fig.2y).  
generator condenser
QH QI
HX
QI absorber evaporator QL
 
 
Fig.2g: A Single–effect LiBr/water absorption refrigeration system with a solution heat 
exchanger (HX) that helps decrease heat input at the generator 
 
 15 
 
QH absorber evaporator QI
HX
generator condenser
QI QI
 
Fig.2h: Absorption heat transformer absorbs waste heat at the generator. Liquid refrigerator is 
pumped to the evaporator to absorb waste heat. High temperature useful heat from the absorber 
is heat of absorption. 
 
 
generator I
QH
HX II Condenser QI
generator II
HX I
QI absorber evaporator QL
 
Fig.2i: A double – effect water/LiBr absorption cycle. Heat released from the condensation of 
refrigerant vapour is used as heat input in generator II. This cycle is operated with 3 pressure 
levels i.e high, moderate and low pressure. 
 
 16 
 
rectifier
QI Condenser generator I generator II QH
rectifier
Heat of
                 HX I Absorption                   HX II
QI evaporator absorber I QI absorber II
 
Fig. 2j: A double – effect absorption cycle operates with two pressure levels. Heat of absorption 
from Absorber II is supplied to the Desorber I for the refrigerant separation process. 
 
QH
1st stage
generator
                HX I
QI
                HX II condenser
                HX III
QI absorber evaporator QL
 
Fig. 2k: A triple – effect absorption cycle operates at four pressure levels. Heat of condensation 
from the higher – pressure stage is used for refrigerant separation in the lower pressure stage. 
 
 17 
 
generator
condenser
QI
secondary
rectifier fluid
QH
condensate
precooler
QL
absorber
QL evaporator
 
Fig. 2l: The dotted loop shows secondary fluid used for transferring heat from high temperature 
section in the absorber to low temperature section in the generator. 
 
QH generator condenser QI
HX
absorber heat
recovery
evaporator QL
QI absorber
Fig. 2m: The cycle with absorber heat recovery uses heat of absorption of preheat the outgoing 
stream from the absorber to the generator. 
 18 
 
QI condenser generator II
QH
HX
QI absorber II generator I
HX
QL evaporator absorber I QI
 
Fig. 2n:  A half – effect absorption cycle is a combination of two single – effect cycles but 
working at different pressure levels. Letting heat source temperature be lower than the minimum 
temperature is necessary for a single – effect cycle working at the same pressure level. 
QI resorber QH resorber
QL desorber QI resorber
(a) (b)
Fig. 2o: Combined vapour absorption/compression heat pump. 
 19 
 
absorber I absorber II QH
heat of absorption compressor
                 HX                  HX
desorber I QI desorber II
 
Fig. 2p:  Double effect absorption –compression cycle is configured as a heat pump. Heat of 
absorption in the first stage will be supplied to the second stage for refrigerant separation. 
 
QH generator condenser QI
                 HX QI absorber
QL
desorber
Fig.2q: A combined cycle proposed by Caccoila et al.(86), employing two combinations of 
working fluids ie. NH4/H2O/KHO.  The rectifier is absent and also the highest pressure is 
decreased. 
 20 
 
QI resorber generator QH
                 HX                  HX
QL desorber absorber QI
 
Fig. 2r: A resorption cycle proposed by Altenkirch uses two solution circuits. 
 
ammonia/ water circuit water/ LiBr circuit
NH3/ Water Water/LiBr
condenser generator
generator
generator Q Solar
absorber
absorber evaporator
evaporator QL
QI
QL
 
Fig. 2s: Solar driven dual cycle absorption employs to different working fluids i.e NH3/water and 
water/LiBr. Heat of absorption and condensation from NH3/water cycle are supplied to the 
generator of water/LiBr cycle. 
 
 21 
 
QH generator I
                 HX
generator II
                 HX
ejector
QI absorber generator QL
Fig. 2t: A modified double effect combined ejector – absorption refrigeration cycle where there 
is no condenser included. 
 
 
QH generator generator QI
HX
condenser condenser QL
QI
Fig. 2u:  A combined ejector/absorption system using DMETEG/R22 and DMETEG/R21 as 
working fluids. The strong solution in the returning leg from generator serves as primary fluid 
and refrigerant vapour from evaporator as second fluid. 
 22 
 
ejector
QH generator condenser QI
HX
QI absorber evaporator QL
Fig. 2v: A combined ejector/absorption proposed by Aphornratana and Eames, was invented. 
High pressure refrigerant vapour from the generator enters the ejector as motive fluid to carry the 
refrigerant vapour from the evaporator.  
 
ejector
Steam
QH
generator
condenser
generator
HX
QL absorber evaporator QL
Fig. 2w: A combined cycle proposed by Eames and Wu (93). The highest solution circuit 
temperature ia maintained at about 18C. So the corrosion problem is alleviated. 
 23 
 
QH
QI condenser
generator
Membrane
absorber
QI
QL condenser
 
Fig. 2x: An osmotic membrane absorption cycle employs heat for refrigerant separation and 
producing pressure within the system. 
generator
condenser QI
absorber evaporator QL
QH
QI
 
Fig. 2y: The diagram shows a bubble pump in the generator module. Heat input to the generator 
is used for both circulation of working fluid and evaporation of refrigeration. 
 
2.1.9: Practical Absorption Cycle 
Fig. 2zi is a basic absorption cycle that uses a lithium bromide and water solution as an absorbent 
and water as refrigerant. The single-stage fundamental absorption refrigeration system (ARS) 
contains a generator, an absorber, a condenser, an evaporator, a solution pump, expansion valves 
or restrictors, and a solution heat exchanger (SHE). The flow path is shown in the figure with 
arrows. 
 24 
 
6
Condenser Generator
5
1
Heat
Out
7 Heat
Exchanger
Restrictor 2
Solution 
Restrictor  Pump
4
8 Vapour
3
Evaporator 9 Spillover Absorber
Low
Heat
Temperature Out
In  
Fig.2zi. Lithium bromide-water single-stage absorption Refrigeration cycle 
 
 
6
Condenser 2 Generator
C2 G2
61
Condenser Generator
G
C 1 51
1
Heat
Heat Exchanger
Out 7
2
Solution 
Restrictor Restrictor  Pump
4
8 Vapour 3
Evaporator 9 AbsorberSpillover
Heat  Out  
 Fig. 2zii. Lithium bromide-water two-stage absorption Refrigeration cycle 
 25 
 
2.3: Numerical Modeling 
 Experimental modeling of heat and mass transfer on thin-liquid falling film in 
unenhanced and enhanced absorption has been extensively studied, while there are relatively few 
studies on numerical modeling of the problem. In experimental unenhanced absorption process 
modeling, Ali et al. (2002) investigated the technical feasibility of driving a lithium chloride-
water solution absorption-cooling unit by a low-temperature heat source (such as solar energy 
using a simple flat-plate collector) for air-conditioning applications. The operating characteristics 
of the unit were extensively investigated and the C.O.P of the unit was found to be 19% as 
against the design value of 21%. Safarik et al. (2004) also carried out an experimental modeling 
on a solar power absorption chiller with low capacity using lithium bromide-water solution as the 
working fluid. The field test carried out at three sites in the summer of 2003 after the prototype 
test in this experimental modeling showed that the absorption chiller performed reliably and 
flexibly over a wide range of external conditions. Abdelmessih et al. (2005) experimentally 
investigated the use of non-traditional absorbent/refrigerant pairs such as ethylene glycol-water 
on a designed and built absorption refrigeration cycle. The investigation was successful in 
replacing traditional hazardous absorbent/refrigerant pairs with ethylene glycol-water pair. The 
working fluids chosen are safe, unlike the commercial absorbent/refrigerant pairs. Yaxiu et al. 
(2007) also examined experimentally a compact solar pump-free lithium bromide absorption 
refrigeration system equipped with a second generator, a falling-film absorber, a falling-film 
evaporator and an efficient luminate thermosiphon elevation tube. The experiment confirmed a 
48.5% increase in the COP. 
  Since the numerical modeling of both unenhanced and enhanced absorption process is 
complicated by the presence of the waves in the falling liquid-film, the smooth falling film 
unenhanced absorption approximation has been more popularly investigated, the earliest of such 
being the work of Grossman and Andberg (1983). This was even considered complicated in 
formulation due to the restriction of the model to the case with the inlet absorbent temperature 
being equal to that of the wall. In the same year Andberg and Vliet (1983) also investigated the 
smooth falling-film absorption under laminar flow using a different model from his research 
model, this was also considered quite sophisticated and somewhat too complicated in 
formulation.  
 
 26 
 
Yang and Wood (1992) investigated a numerical model of the absorption process on a 
smooth liquid falling-film in lithium bromide water and a lithium chloride water solution system. 
Yang et al. (1992) developed a numerical absorption model of a simple smooth-film of LiCl-H2O 
and LiBr-H2O systems with Reynolds number of 2.7, 27 and 100 using finite difference solution 
approach. The solutions obtained were similar to the results of the earlier work of Grossman et 
al. (1983) and generally agreed with available experimental data. Argiriou et al. (2004) 
conducted a numerical investigation on a prototype low capacity solar assisted lithium bromide 
absorption heat pump coupled with a sub-floor system using the commercial simulator known as 
TRNSYS. The results indicated that the estimated energy savings against a conventional cooling 
system using a compression type heat pump was in the range of 20-27%.  Xu et al. (2006) 
simulated an absorption process in an advanced energy storage system. The latest of the 
modeling in this area was on a two-stage absorption chiller driven at two-temperature levels 
using thermodynamic modeling technique by Gustavo et al. (2007). The study established that 
0
the machine can operate in summer as a double-stage chiller driven by heat at 170 C from 
0
natural gas, as a single-stage chiller driven by heat at 90 C from solar energy, or simultaneously 
in combined mode at both temperatures. It also established the capability of operating in winter 
0
in “double-lift” mode for heating with a driving heat at 170 C from natural gas. 
Ghaddar et al. (1996) modeled solar lithium bromide absorption system performance in 
Beirut using a simulated computer program. The results shows that for each ton of refrigeration it 
2
is required to have a minimum collector area of 23.3m  with  an optimal water storage tank 
capacity ranging from 1000 to 1500 litres for the system to operate solely on solar energy for 
about seven hours a day. The energy use in cooling was also found to be of function of solar 
collector area and storage tank capacity. Based on the economic assessment performed on the 
current cost of conventional cooling system, it was also found that the solar cooling system is 
marginally competitive only when combined with domestic water heating.   
Bruno et al (2004) modeled Ammonia-water-sodium hydroxide mixtures absorption 
refrigeration plant using a commercial process simulator “Aspen Plus 2003”. It was found that 
the system performance is notably increased (lower driving temperature and higher COP). 
Fernandez et al. (2005) also modeled an absorption processes taking place in Ammonia-water 
absorption system using finite difference approach. The simultaneous heat and mass transfer set 
 27 
 
of nonlinear differential equations were solved using the above mentioned approach. The results 
established the expected typical range of values of xvb < z < ∞.or – ∞ < z < xLb and xLb < z < xvi 
for mass transfer against temperature variations in different components of the plant such as 
absorber and evaporator where x, z, b, L, i and v are defined as ammonia molar concentration, 
ammonia to net molar flux transferred ratio, bulk conditions, liquid, liquid-vapour interface and 
vapour respectively.  
Icksoo et al. (2006) developed a water-lithium bromide absorption process model over a 
horizontal tube using finite difference approach. The model predicts that significant absorption 
takes place in the drop formation regime with a considerable variation of temperature and mass 
fraction. Xu et al. (2006) simulated aqueous lithium bromide (LiBr-H2O) advanced energy 
storage system using finite difference method. The result predicts the dynamic characteristics 
and performance of the system, including the temperature and concentration of the working 
fluid, the mass and energy in the storage tanks, the compressor intake mass or volume flow rate, 
discharge pressure, compression ratio, power and consumption work, the heat loads of heat 
exchanger devices in the system and so on. The result also indicated that the integrated 
coefficient of performance (COPint) of the system was 3.09 and 3.26 respectively as against the 
expected value of 3.0 under the two storage strategies, while the isentropic efficiency of water 
vapour compressor was set as 0.6. These results were found to be very helpful in understanding 
and evaluating the system as well as for system design, operation and control. 
Gustavo et al. (2007) studied a two-stage LiBr-water absorption chiller driven at two 
temperature levels using thermodynamic modeling technique. The study established that the 
0
machine can operate in summer as a double-stage chiller driven by heat at 170 C from natural 
0
gas, as a single-stage chiller driven by heat at 90 C from solar energy, or simultaneously in 
combined mode at both temperatures. It also established the capability of operating in winter in 
0
“double-lift” mode for heating with a driving heat at 170 C from natural gas. Balghouthi et al. 
(2007) conducted both experimental and numerical modeling on solar water-lithium bromide 
0 0
absorption air conditioning in Tunisian climatic conditions (36  Latitude and 10  longitude, 
2
400cal/cm day average solar irradiation, and 3700h/year and 350 total insolation period and 
sunny days per year respectively) using the TRNSYS and „EES‟ Engineering Equation Solver 
programs in the study. The model established that absorption solar air-conditioning system was 
suitable under Tunisian conditions. Despite the initial high cost and almost zero maintenance 
 28 
 
cost, this system could help to minimize fossil fuel-based energy use, reduce electricity demand 
on the national grid especially at peak demand periods in summer and eliminate the use of CFCs.  
In experimental enhanced absorption process study, Wen-long Cheng et al. (2003) 
investigated experimentally the effect of additive on falling film absorption of water vapour in 
aqueous LiBr. The experimental results showed that small amounts of additive can enhance the 
heat transfer of absorption process significantly, and the enhancement degree is influenced by 
additive concentration and Reynolds number. Based on a dimensionless analysis of the Navier-
Stokes equations applied to the falling film absorption process, a new dimensionless parameter, 
surface renewal number Rn was introduced, and a semi-empirical equation of enhancement 
factor of additive was obtained, which shows that the enhancement effect of additive on Nusset 
number of absorption process is determined by the absorption Marangoni number Ma, the 
surface Marangoni number MaA, the surface renewal number Rn, the adsorption number ƞ, and 
the Reynolds number Re. It was proved that the semi-empirical equation can agree with the 
experimental results well by introduction of the parameters related to surface tension into the 
equation. The study concluded as follows: i.There is an optimum additive concentration in which 
the enhancement effect of additive is strongest, ii.The Marangoni number Ma, the surface 
Marangoni number MaA, and the surface renewal number Rn enlarge the enhancement of the heat 
transfer during absorption, iii. The adsorption number ƞ reduces the heat transfer of absorption, 
and iv. The enhancement factor decreases as the Reynolds number increases. Yong Tae Kang et 
al. (2006) in their experimental study also obtained the following results: (i). The vapour 
absorption rate increases with increasing solution mass flow rate and the concentration of Fe 
nanoparticles and CNT. The effect of coolant mass flow rate on the vapour absorption rate is not 
significant under the experimental conditions, (ii). The heat transfer rate increases with 
increasing the solution mass flow rate while it is not much affected by the concentration of 
nanoparticles, (iii). The mass transfer enhancement is much more significant than the heat 
transfer enhancement in the binary nanofluids with Fe nanoparticles and CNT, and (iv). The 
mass transfer enhancement from the CNT (average 2.16 for 0.01 wt % and average 2.48 for 0.1 
wt %) becomes higher than that from the Fe nanoparticles (average 1.71 for 0.01 wt % and 
average 1.90 for 0.1 wt %). Therefore, the CNT is a better candidate than Fe nanoparticles for 
performance enhancement in H2O/LiBr absorption system. 
 29 
 
In numerical enhanced absorption refrigeration study, Staicovici et al. (2005) modeled 
water-lithium bromide absorption/generation processes in a Marangoni Convection (applied 
practical method by the thermal absorption technology in the past decades to significantly 
improve the absorption process) Cell using the Two-Point theory (TPT) of mass and heat 
transfer. The model established the capability of (TPT) approach in the Marangoni convection 
assisted water-lithium bromide absorption process following the successful modeling of the 
ammonia-water absorption process. It also confirms Marangoni convection basic mechanism 
explanation in the case of the water-lithium bromide medium. Xohar et al. (2005) investigated 
the influence of diffusion in the ammonia-water diffusion absorption refrigeration (DAR) cycle 
configuration on the system performance using a computer simulator known as „EES‟ 
Engineering Equation Solver. The result reveals that DAR cycle without condensate sub-cooling 
shows higher COP of 14-20% compared with the DAR cycle with the condensate sub-cooling, 
0
but it occurs at higher evaporator temperature of about 15 C.  
Niu et al. (2006) performed a numerical analysis of falling film ammonia-water 
absorption in a magnetic field using a computer program known as TDMA due to the tri-
diagonal matrix formation of the equations after discretization. It was found that when the 
magnetic induction intensity at the solution‟s inlet was 3Tesla (T), the increment in concentration 
of ammonia-water solution at outlet was 1.3% and the absorbability increased by 5.9%. The COP 
of the absorption refrigeration system increased by 4.7% and the decrement in circulation ratio 
was 8.3%. This establishes a positive effect on the ammonia-water falling film absorption to 
some degree. 
From the above review, it is evident that the finite difference approach has received much 
attention in the existing literature thereby establishing its popularity and reliability. In addition 
the review also established some works done in enhanced absorption refrigeration study most 
especially using additives or nano-fluids / nano-particles in both ammonia-water and water-
lithium bromide (H2O-LiBr) solution. However to the best of knowledge of the researcher, only 
magnetic field enhanced ammonia-water falling film absorption has been worked upon while 
none on this type of enhancement has been done on either water-lithium bromide (H2O-LiBr) or 
water-lithium chloride (H2O-LiCl) pairs.   This present work therefore uses the finite difference 
method in an application of the falling film magnetic field enhanced absorption model to the 2-D 
flow over a vertical flat surface. The two working fluids to be investigated under this magnetic 
 30 
 
field enhancement are lithium chloride-water (LiCl-H2O) and lithium bromide-water (LiBr-H2O) 
pairs. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 31 
 
CHAPTER THREE 
 
METHODOLOGY 
 
3.1  Assumptions 
In developing the governing equations for this flow modeling the absorption process in a 
smooth thin-liquid film, the following assumptions are made. 
i. The flow is a fully developed steady laminar flow as shown in fig. 3.1a hence velocity (v) 
in Y-direction is zero 
ii. The fluid properties are constant and not varying with temperature and concentration. 
iii. The mass rate of vapour absorbed is very small compared to the solution flow rate such 
that the film thickness and flow velocities can be treated as constant. 
iv. Heat transfer in the vapor phase is negligible. 
v. Vapor pressure equilibrium exists between the vapour and liquid at the interface. 
vi. The Peclet numbers are large enough such that the diffusion in the flow direction can be 
neglected. 
vii. Diffusion thermal effects are negligible. 
viii. The magnetic induction intensity descends linearly along the flow of falling-film. 
ix. The shear stress at the liquid–vapor interface is negligible  
  
The model coordinate system is as shown in Fig. 3.1a Yang and Wood (1992) where U is the 
velocity in the film in X-direction. 
 
 
3.2: Governing equations 
Niu et al. (2006) in his study of magnetic field enhancement effect on absorption process 
of ammonia-water solution, the first of its kind, utilized model governing equations (1a) 
through (1e). The study was done on non-smooth thin-liquid falling film. The equations 
are as follows; the heat transfer energy equation (1a), continuity equation (1b), 
momentum equation (1c) and quality or mass transfer equation (1e).   
 32 
 
T T   T 
C pu  C  p v      (1a) 
x y y  y


 
u v
  0      (1b) 
x y
u u   u 
u  v     g  mag   (1c) x y y  y 
     
u  v  D m    (1d) x y y  y


Where  mag  (Li, G.D (1999)) in the above equation is the magnetic force which the 
falling-film solution experienced per unit volume. It is in the direction of downward 
vertically. 
 B
2
 mag     (1e)   where   is the magnetic mass susceptibility of 
l.0
either Lithium bromide and or Lithium chloride water solution, B is the magnetic 
induction intensity, l is the length of falling-film flows, and  0  is the vacuum‟s 
permeability. 
Odunfa (2008) utilized the following heat transfer, mass transfer, continuity equation and 
the velocity field equation in smooth thin-liquid falling film (Bird et al (1960) and Yang 
and Wood (1992)). The study was done on a smooth thin-liquid falling film. The 
equations are as follows; the energy conservation equation (2a), continuity equation (2b), 
conservation of linear momentum equation Bird et al (1960) (2c) and quality or mass 
transfer equation (2d).   
 
 
 33 
 
T T   T 
C  pu  C p v       (2a) x y y  y 
 
u v
  0      (2b) 
x y
 23 y  y  
U  V    0 2       (2c) 2  h0  h

 0  
     
u  v  D m    (2d) 
x y y  y  
A closer inspection of the utilized set of equations in the two cases mentioned above i.e non 
smooth falling-film Niu et al. (2006) and smooth thin-liquid falling-film Odunfa (2008) reveals 
that the equations in both cases are similar, only with the exception of equation (1c) of the first 
case and equation (2c) in the latter one. The first case non-smooth falling-film but enhanced 
magnetically, while the second case is smooth falling-film, but not enhanced magnetically. The 
present work is on a smooth thin-liquid falling-film which is to be enhanced magnetically. 
Towards achieving this, a magnetic field enhanced velocity field model equation in smooth thin 
liquid falling-film, using mass transport relationship was developed as shown in appendix A. 
This developed enhanced velocity model equation (xxi) in the appendix A was used to replace 
equation (2c), thus the model set of magnetic enhanced velocity, heat and mass transfer 
equations on a smooth thin-liquid falling-film corresponding to the coordinate system shown in 
fig. 3.1a will now be: 
T  2T
 u   = 0     (3a) 
x y 2
u
  0       (3b) 
x
 2u v B2
 3 0  g   0    (3c) 
y 2 h20 l0
  2
u  Dm  0      (3d) 
x y 2
 34 
 
The final developed model magnetic enhanced velocity field, heat and mass transfer equations on 
a smooth thin-liquid falling-film in a cooling system are: 
 2u v B2
 3 0  g   0    (4a) 
y 2 h20 l0
T  2T
u   = 0     (4b) 
x y 2
  2
u  Dm  0      (4c) 
x y 2
where    is the magnetic mass susceptibility, B   is the magnetic induction intensity, l  is the 
length of falling-film flows, 0  is the vacuum permeability, T is temperature,   is concentration 
(absorbent),   is thermal diffusivity, Dm is species diffusivity, V0  is the average velocity within 
the film thickness and h0  is the film thickness 
y
h0
u p
v
g
x
                                     
 
Fig. 3.1a:  2-d representation of a thin-liquid falling-film  
 35 
 
Boundary Conditions  
At    x = 0;   u = uin , T = Tin and      equil     - (5a) 
At    y = 0; (non permeable wall); 

u = 0, T  Tw  ,   0            - (5b) 
y
T 
At y  h0 ;  K  Dm H a     equil (T, Pv)   (5c) 
y y
 u 
At the vapour-liquid interface, y  h ;   0  0, Pv  Ts , s   const.   
 y  y0
 Where H a  = Heat of absorption, Tw = Wall temperature Pv  = Vapour pressure and  
equil(T , Pv )  = equilibrium concentration at the interface temperature and ambient vapour 
pressure. 
3.3: Finite Difference formulation of the governing equations 
The general finite difference formulation or approximation of the first derivative of a function 
F(x, y) with respect to x is given as 
dF F(x  x, y)  F(x, y)
      - (6a) and 
dx x
2 F(x  x  x, y)  F(x  x, y) F(x  x, y)  F(x, y)
d F ( d dF
  ( )  x x   -   (6b)  
2
dx dx dx x
Similarly the first derivative of a function F(x, y) with respect to y is also given as  
dF F(x, y  y)  F(x, y)
        -  (7a) and    
dy y
                 
 
 36 
 
F(x, y  y  y)  F(x, y  y) F(x, y  y)  F(x, y)
2 
d F d dF y y
 ( )   -  (7b) 
2
dy dy dy y
 
    
i, j+l 
y     
   i-l,j  i,j i+1,j 
  i,j-1  
                        
       x  
                  
                       Fig. 3.1b:   Finite difference diagram      
         
Using a central finite difference approximation in fig. 3.1b then equation (6b) become 
2
d F F(x  x, y)  2F(x, y)  F(x  x, y)     (8a) 
2 2
dx x
F  2F  F
i1, j i, j i1, j
 =       (8b) 
2
x
Similarly equation (7b) become 
 
d 2 F F(x,y  y)  2F(x, y)  F(x, y  y)
     (9a) 
dy 2 y 2
 
F  2F  F
i, j1 i, j i, j1
 =       (9b)  
2
y
 
Relating the above derived equation to the heat transfer equation (1) 
 
F  F F  2F  F
T 2T i1, j i, j i, j1 i, j i, j1
U  = U  -    (10) 
x y2 x 2y
 If x  = y   =   h 
 
 37 
 
T 2T
U   = U F  
2 i1, j
 Fi, j h  -  F  2F  F  0  (10b) 
x y  i, j1 i, j i, j1
 
For    F T       
i, j i, j
 
T 2T
U   =   U T   i1, j Ti, j h  Ti, j1  2TI ,J Ti, j1  0   10c 
x y2
 
Therefore   U T T  i1, j i, j h  Ti, j1  2T T I ,J i, j1  0   - 10d 
  
Similarly for the same laminar flow, relating the derived equations (8a – 9b) to the mass transfer 
equation (3d) 
  2 Fi1, j  Fi, j Fi, j1  2Fi, j  Fi, j1
u  Dm =  u  D2 m        (11a) x y x y 2
again if  x  = y   =  h 
  2
u  Dm = uF   i1, j  Fi, j  Dm Fi, j1  2Fi, j  Fi, j1   - (11b) 
x y 2
     
   Fi, j   i, j   
 
  2
u  D   = u   m 2 i1, j i, j  Dm i, j1  2i, j i, j1   = 0   - (11c) x y
                              
  u   i1, j i, j  Dm i, j1  2i, j i, j1  =   0  -   (11d) 
 
Equations (12a and 12b) depict the basic general finite difference equations for heat and mass 
transfer.   
uT T  i1, j i, j  Ti, j1  2Ti, j T i, j1  = 0    - (12a) 
ui1, j  i, j  D  m i, j1  2i, j i, j1  = 0    - (12b) 
 38 
 
3.4: Formulation of the magnetic field enhanced finite difference model of the  
        governing equations 
  
The thin liquid falling-film thickness is far too small compared to the entire length of the 
absorber wall. The film thickness is similar to the boundary layer thickness in heat transfer 
analysis, therefore the above general finite difference formulation of the governing equations 
cannot be applied for the internal thin film regime. Therefore the required model finite difference 
formulation of the governing equations will be as follows:  
For the model, x  is not equaly  , therefore 
 
 2u v  2 Fi, j1  2Fi, j F0 i, j1 v0 
2
  3  g   [ ] 3  g   0   (13a) 
y 2 h2 l0 y
2 h2 l0 0 0
For  Fi, j  ui, j  
2 2 u v  2 3v y  2
 3 0  g   (u 0 2 2
2 2 i, j1
 2ui, j  ui, j1)   gy  y  0  
y h l h20 l0 0 0
 
3v y 2  2
Therefore, (u 0 2 2i, j1  2ui, j  ui, j1)   gy  y  0               (13b) 
h2 l0 0
 
T  2T
 2u  = u[y ] F  F    [x ]F  2F  F   0   
x y 2  i1, j i, j  i, j1 i, j i, j1
         (13c) 
For    F T       
i, j i, j
T  2T 2  
u  =   u [y ] T   i1, j Ti, j  Ti, j1  2Ti, j Ti, j1 x  0  (13d) 
x y 2
 
2
Therefore   u [y ] T T i1, j i, j Ti, j1  2Ti, j T i, j1 x  0  - (14a) 
 
  2
u  Dm = uy
2 Fi1, j  Fi, j  Dm xFi, j1  2Fi, j  Fi, j1  (14b) 
x y 2  
   Fi, j   i, j    
 
 39 
 
  2 2
u  Dm   = uy    2 i1, j i, j  Dm i, j1  2i, j i, j1 x  = 0 (15a) x y
      
2
Therefore uy i1, j    i, j  Dm i, j1  2i, j i, j1 x  = 0  -   (15b) 
 
Equations (12b) and (13b); the finite difference model equations satisfied at each node in the 
interior face of the domain R for heat and mass transfer.  
3v y 2  2
(ui, j1  2ui, j  ui, j1) 
0  gy 2  y 2  0   - (16a) 
h2 l0 0
uy2 T   i1, j Ti, j  Ti, j1  2Ti, j Ti, j1 x  0   - (16b) 
uy2    i1, j i, j  Dm i, j1  2i, j  i, j1 x  = 0    -  (16c) 
 
3.5: Boundary conditions  
At the boundary, usually the parameters such as temperature and concentration are 
known (Dirichlet conditions) or the boundary is considered to be perfectly insulated (Newmann 
or Adiabatic conditions). Insulated boundaries are handled by developing boundary 
element/nodal equations. In this model, Newmann or Adiabatic boundary condition was used 
along the absorber wall. Figure 3.2 shows a half element in the model lying on an insulated left 
boundary of the smooth film thickness portion of the absorber. The net heat flowing into this 
element must be equal to zero when considering steady-state condition in the film. In the 
boundary shown in Fig 3.2, the quantity of heat flowing into the three faces of the half element in 
time dt is given by the following equations:  
 
 40 
 
x
dq ydq
i+1 3 1
dq dq
1 2
i dq
2
dq
i-1 3
x
y
 
Fig. 3.2 Temperature boundary conditions analysis 
 
From fig. 3.2, the heat balance equation for the half element is 
 dq1 + dq2 +dq3   =   0       (17)
 
The complete boundary difference equations (xxiii to xxvi) in this study appearing in Appendix 
A was applied, thus  
(Left)  T i1, j + 2T i, j1 + Ti-1,j   - 4T i , j  =  0     (18a) 
(Right)  T i1, j + 2T i, j1+ T i1, j - 4T i , j  =   0     (18b) 
(Upper) T i, j1  + 2T i 1, j + T i, j1 - 4T i , j  =   0    (18c) 
(Lower)  T i, j1  + 2T i1, j  + T i, j1 - 4T i , j  =   0     (18d) 
Thus, in obtaining a solution for the steady state heat flow in the smooth film absorption 
medium, Eq. (17) must hold at every interior grid point and Eqs. (18a) through (18d) must be 
 41 
 
satisfied at any appropriate known or insulated boundaries. Gaussian elimination methods or 
Gauss-Seidel iteration method could be employed to solve the equations accurately for the 
number of grid-points required. 
For the steady state mass flow in the smooth film-absorption medium shown in figure 3.3, 
similar approach was employed to develop boundaries equations for the unknown or insulated 
boundaries as shown in the figure. 
x
dm ydm
3 1
i+1
dm dm1 2
i dm
2
dm3
i-1
x
y
 
Fig. 3.3 Concentration boundary conditions analysis 
 
The net mass flowing into this element must be equal to zero when considering steady-state 
condition in the liquid-film. The quantity of mass flowing into the three faces of the half-element 
in time dt are given by the following equations.  
                       
From fig. 3.3 the mass balance equation for the half-element is 
 
dm1    +   dm2    +  dm3  =   0    (19) 
 
 42 
 
The complete boundary difference equations (xxviii to xxxi) in this study  appearing in Appendix 
A was applied, thus  
 (Left)    i1, j  2i, j1 i1, j  4i, j  0      (20a) 
 (Right)    i1, j  2i, j1 i1, j  4i, j  0      (20b) 
(Upper)    i, j1  2i1, j i, j1  4i, j  0         (20c) 
(Lower)    i, j1  2i1, j i, j1  4i, j  0      (20d) 
 
 
Thus, in obtaining a solution for the steady state mass flow in the smooth film absorption 
medium, Equation (19) must hold at every interior grid point and Eqs. (20a) through (20d) must 
be satisfied at any appropriate known or insulated boundaries. 
Equations (17) and (19) are for the interior grid points in the smooth film absorption medium 
while equations (18a) through (18d) and equations (20a) through (20d) are for the boundary 
conditions in heat and mass transfer respectively. The development of these boundary equations 
took due cognisance of the inlet conditions to the absorber, absorber wall and the given condition 
in the thin-liquid falling-film/gas or vapour interface. The two immediate adjacent nodes to each 
of the corner nodes of the medium are taken into consideration in the program to determine the 
unknowns at the corner nodes.  
3.6: Solution Method 
As earlier mentioned, there are many solution techniques available that are being used for 
solving the global finite difference matrix equations 12b and 13b coupled with the boundary 
equations 15 and 20 generated. Two of the most widely used methods are Gauss-Seidel iteration 
and Gaussian elimination method as modified by Paynes and Iron and reported by Okon, 
(1990).The modified Gaussian elimination method employed is given in Appendix D. 
3.7: Computer Programming 
The finite difference method is used to solve the governing equations (1) & (2) as given in the 
form of finite difference equations (16a) & (16b).  The solution technique used is Gaussian 
 43 
 
elimination scheme as modified by Paynes and Iron and reported by Okon, (1990) on the digital 
computer.  The computer program and the subroutines are written in FORTRAN 90 language. 
Main Program 
The program shown in Appendix A solves equations (16a) & (16b) using modified Gaussian 
elimination scheme.  The flow chart is shown in Fig. 3.4a. This main program utilizes two (2) 
different subroutines. These subroutines are written to execute various steps involved in applying 
the finite difference scheme.  The problem data are introduced into the program in the “data 
block”, where the input parameters can be easily modified to suit any case study.  The main 
program, after generating the global matrix, calls subroutine “solution1”, before calling 
subroutine “solution2”.  After calling a subroutine solution the results were finally “displayed. 
Subroutines 
These are sub-programs written to execute various steps involved in applying the finite 
difference method using Gaussian elimination scheme.  They are called by the main program. 
Below are the subroutines employed to execute various steps in the main program; 
Solution1 & Solution 2 
These are the core sub-routines, they also perform similar functions. These subroutines perform 
their functions after the implementation of the boundary conditions in the global domain. 
Solution1 generates the temperature profile of the domain, while solution2 takes care of 
concentration profile within the domain. 
DATA 
There is a well-known cliché in the computer world-GIGO an acronym to mean “Garbage in 
Garbage out” i.e. the quality of the output is no better than the quality of the input.  In order to 
obtain high quality output of temperature and concentration profiles in the domain, the input data 
utilized was obtained from the literature as was used for the available experimental and 
numerical modelings. The data utilized from the literature are as shown in Table 3.1(General 
data), Table 3.2 (LiBr – H2O) and Table 3.3 (LiCl – H2O) respectively. 
 
 
 
 
 44 
 
Fig. 3.4a: Main Program Flow Chart 
 
 
START
READ DIMENSIONS AND 
MATERIALS PROPERTIES
GAUSS-SIEDEL SCHEME TO SOLVE
THE GENERATED GLOBAL MATRIX.
CALL SOLUTIONS
X
STOP & END
 
 
 
 
 
 
 
 
 
 
 
 45 
 
Fig. 3.4b: Subroutine Solution Flow Chart 
 
No Is
X N< NMAX
Yes ?
Assign initial temp . Print out iteration number 
&c onc . and number of points at 
Yes
distribution
which accuracy check fails
Is K=0
Call Subroutine 
?
Velocity
Y Print out tem.p & 
conc. in the film thickness No
N=0
RETURN Print out iteration number 
and number of points at 
K=0 which accuracy check fails
No
DO nJ=1, 5
DO nI=2,12
No
Is J= 5
? 2 2
Is No T  T c  c
J=1 Yes u   & u D .m
? x y
2
x 2y
Yes No
Is I = IB
Yes ?
Is I = IA T Dm HYes c c k T  a ,  
? y k y y Dm Ha y
UTEMP=UT No Left UTEMP=UT
UCONC=UC UCONC=UC
Right 
c
T Tw ,  0 .y
DIFF=UTEMP-U(I,J)
DIFF=UCONC-C(I,J)
Is
| DIFF|<e No
?
Yes K=K+1
n
U(I,J)=UTEMP
C(I,J)=UCONC
DO loop
N=N+1
 
 
 
 
 
 
 
 
 46 
 
Fig. 3.4c: Model Subroutine Velocity Flow Chart 
 
 
No Is
N<NMAX
?
Yes
Print out iteration number 
Yes
Y and number of points at 
which accuracy check fails
Is K = 0
Assign Initial Velocity . ?
Print out Velocy in the film 
distribu.t ion
thic. kness
No
N = 0 RETURN
Print out iteration number 
and number of points at 
K= 0 which accuracy check fails
No
DO nJ= 1,5
DO nI=2, 12
No
Is J=5
? 2
Is No  u VO B
2
J= 1 3  g  .
Yes 2 2
?  y h  lu
Yes
Is I =I B No
Is ?Yes
I=IA vYes  0 .
? y
UVEL= VT No Left
UVEL= VT 
Right 
.
V=0
DIFF= UVEL-U(I,J)
Is
| NoDIFF
?
Yes K=K +1
n
U(I ,J)= UVEL
DO loop
N =N+1
 
 
 
 
 
 47 
 
 
DATA 
Table 3.1: General; Niu Xiaofeng et al. [2006] 
--------------------------------------------------------------------------------------------------------------------- 
β magnetic induction intensity at the inlet of solution        = 0, 1.4 and 3.0 Tesla 
 magnetic mass susceptibility LiBr-H2O &LiCl-H2O solutions = 0.0000343, 0000243 
-1
 uin Initial velocity              = 0.362 ms    
--------------------------------------------------------------------------------------------------------------------- 
 
 
Table 3.2: NH3-H2O Data Niu Xiaofeng et al. [2006] 
--------------------------------------------------------------------------------------------------------------------- 
-4 -1 -1 
 film dynamic viscosity   = 4 x 10 kgm s
-6 -2 2
0  vacuum Permeability    =  1.257 x 10 kgmA s  
-4 -1
 o  mean velocity    =  3.15 x 10 ms  
-3 
    Liquid density    = 127kgm  
-1 -1
 K   Thermal conductivity   = 176Wm K  
o
Tw    Wall Temperature   = 35 C 
o
Tin   Inlet Temperature   = 35 C 
Cin   Initial absorbent Conc.   = 20% 
Ceq Equilibrium absorb. Con   = 20% 
-2
g     Gravity     = 9.8ms  
-3
ho   Mean film thickness   = 1.00 x 10 m 
Ha  Heat of absorption   = 3466kJ/kg 
Pv.  Absorbent Vapour Pressure  = 7.02mm.Hg 
Ref  Film Reynolds number   = 100 
-1 -1
I   Film mass flowrate    = 0.01kgm s  
--------------------------------------------------------------------------------------------------------------------- 
 
 
 
 
 
 
 
 48 
 
Table 3.3: LiBr-H2O Data Yang et al. [1992] 
--------------------------------------------------------------------------------------------------------------------- 
-4 -1 -1 
 film dynamic viscosity   = 4 x 10 kgm s
-6 -2 2
0  vacuum Permeability    =  1.257 x 10 kgmA s  
-4 -1
 o  mean velocity    =  3.15 x 10 ms  
2 -1
   Thermal diffusivity   = k/cp = 0.155m s  
-3 
    Liquid density    = 127kgm  
-1 -1
 K   Thermal conductivity   = 176Wm K  
-10 2 -1
D    Species diffusivity   = 1 x 10  m s  
o
Tw    Wall Temperature   = 35 C 
o
Tin   Inlet Temperature   = 44.44 C 
Cin   Initial absorbent Conc.   = 60% 
Ceq  Equilibrium absorb. Con   = 60% 
-2
g     Gravity     = 9.8ms  
-3
ho   Mean film thickness   = 1.00 x 10 m 
Ha  Heat of absorption   = 3466kJ/kg 
Pv.  Absorbent Vapour Pressure  = 7.02mm.Hg 
Ref  Film Reynolds number   = 100 
-1 -1
I   Film mass flowrate    = 0.01kgm s  
--------------------------------------------------------------------------------------------------------------------- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 49 
 
Table 3.4: LiCl-H2O Data Yang et al. [1992]  
--------------------------------------------------------------------------------------------------------------------- 
-4 -1 -1
 film dynamic viscosity   = 4 x 10 kgm s  
-5 -1
 o  mean velocity    =  3.15 x 10 ms  
2 -1
   Thermal diffusivity   = k/cp = 0.155m s  
-3
    Liquid density    = 1000kgm  
-1 -1
 K   Thermal conductivity   = 176Wm K  
-9 2 -1
D    Species diffusivity   = 2.0 x 10  m s  
o
Tw    Wall Temperature   = 30 C 
o
Tin Inlet Temperature    = 35 C 
Cin Initial absorbent Conc.   = 45% 
Ceq Equilibrium absorb. Con   = 35.8% 
-2
g    Gravity     = 9.8ms  
-3
ho  Mean film thickness   = 1.74 x 10 m 
Ha  Heat of absorption   = 3466kJ/kg 
Pv.  Absorbent Vapour Pressure  = 9.2mm.Hg 
Ref  Film Reynolds number   = 100 
-1 -1
I   Film mass flow-rate   = 0.01kgm s  
--------------------------------------------------------------------------------------------------------------------- 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
CHAPTER FOUR 
 
COMPUTATIONAL RESULTS AND DISCUSSION 
 
4.1 General remarks on result presentation 
The continuity, momentum, energy and species mass transport equations indicated in the 
previous section were coded in a computer algorithm using FORTRAN programming coding 
language. The code was run on a personal computer with sufficient memory facilities to carry out 
the simulation exercise. The model run was initially executed using ammonia-water so as to 
validate the developed model code before using the main two working fluid pairs namely lithium 
bromide-water and lithium chloride-water. The parameters utilized in the literature by Andberg 
(1982), Yang et al (1992) and Niu Xiaofen (2006) and (2009) as shown in Tables 3.1, 3.2, 3.3 
and 3.4 in two cases have been used. The domain area was divided into 13 x 5 mesh evenly 
spaced in both the direction of falling (x) and in the direction of film thickness (y) in this thesis. 
The tabulated and plotted graphical figures of the model results are presented side by side with 
that of the literature where available as shown in the following tables and figures. The results 
cover the velocity, heat and mass analysis in both the direction of falling film and that of the film 
thickness direction.  
 
4.2 Numerical Results in the direction of falling film (X) for Ammonia-water, X = Absorber   
      length 
The nodal parameters distribution obtained in the direction of falling within the film thickness in 
the bulk for Ammonia-water (NH3-H2O) solution compares well with the existing literature. For 
instance Tables 4.1, 4.1.2, 4.1.4 and 4.1.7 and Figs 4.1 to 4.1.5 represent the distribution of 
average velocity, Temperature and concentration of the falling-film within the bulk in X-axis or 
direction of the falling film at various magnetic induction intensities in NH3-H2O solution. Both 
the table and figures established that the velocity at stronger magnetic field is larger than that of 
the weaker one. This established parameter distribution within the bulk when compared with the 
existing literature results in ammonia-water solution had the percentage velocity deviation at 0.0, 
1.4 and 3.0 Tesla ranges between -1.01 to 0.15, -1.26 to 0.00 and -1.40 to 0.00 respectively. 
Tables 4.1.1 and 4.1.2 represent the velocity deviation analysis, upon comparison with the 
-3 -3
literature results, it has standard velocity deviations ( ) of 1.49 X 10  at 0.0Tesla, 1.78 X 10  at 
-3
1.4Tesla and 1.90 X 10  at 3.0Tesla. The T-Test analysis results established that the velocity, 
 51 
 
Temperature and concentration distributions for ammonia-water was not significantly difference 
(p=0.05) from published results in literature as shown in the Tables 4.1.3, 4.1.6, 4.1.9 and 
Figures 4.1, 4.1.1, 4.1.2, 4.1.3, 4.1.4 and 4.1.5.  
4.3: Tabular and Graphical representation of the literature and present work’s result  
Table 4.1: NH3-H2O Velocity Distribution in the film Comparison 
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
Result Result Devia Result Result Devia- Result Result Devia-
 @ 0.0 @ 0.0 -tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
* * *  
-6
0 or 10  0.3620 0.3620 0.00 0.3620 0.3620 0.00 0.3620 0.3620 0.00 
-5.5
10  0.3660 0.3660 0.00 0.3665 0.3663 -0.06 0.3680 0.3667 -0.35 
-5
10  0.3700 0.3700 0.00 0.3710 0.3704 -0.16 0.3720 0.3713 -0.19 
-4.5
10  0.3750 0.3740 -0.27 0.3751 0.3744 -0.19 0.3790 0.3756 -0.92 
-4
10  0.3790 0.3776 -0.37 0.3810 0.3784 -0.68 0.3840 0.3798 -1.09 
-3.5
10  0.3840 0.3813 -0.70 0.3860 0.3822 -0.99 0.3890 0.3837 -1.36 
-3
10  0.3880 0.3850 -0.77 0.3900 0.3859 -1.05 0.3920 0.3875 -1.15 
-2.5
10  0.3920 0.3885 -0.89 0.3940 0.3895 -1.14 0.3950 0.3911 -0.99 
-2
10  0.3960 0.3920 -1.01 0.3980 0.3930 -1.26 0.4000 0.3944 -1.40 
-1.5
10  0.3970 0.3953 -0.43 0.4000 0.3964 -0.90 0.4020 0.3976 -1.10 
-1
10  0.3980 0.3986 0.15 0.4020 0.3997 -0.57 0.4035 0.4006 -0.72 
-0.5
10  0.4020 0.4019 0.03 0.4040 0.4029 -0.27 0.4050 0.4034 -0.40 
0
10  0.4050 0.4050 0.00 0.4060 0.4060 0.00 0.4060 0.4060 0.00 
*    Niu Xiaofeng et al. [2006] 
 52 
 
Table 4.1.1:NH3-H2O Velocity Deviation Analysis 
Absorber Bulk 
length X 
(m) Lit. Presen Devia- Lit. Present. Devia- Lit. Present  Devia-
 Result t tion Result Result tion Result Result tion 
 @ 0.0 Result @ 1.4 @ 1.4 @ 3.0 @3.0 
Tesla @ 0.0 Tesla Tesla Tesla Tesla 
* Tesla * * 
-6
0 or 10  0.3620 0.3620 0.0000 0.3620 0.3620 0.0000 0.3620 0.3620 0.0000 
-5.5
10  0.3660 0.3660 0.0000 0.3665 0.3663 0.0002 0.3680 0.3667 0.0013 
-5
10  0.3700 0.3700 0.0000 0.3710 0.3704 0.0006 0.3720 0.3713 0.0007 
-4.5
10  0.3750 0.3740 0.0010 0.3751 0.3744 0.0007 0.3790 0.3756 0.0034 
-4
10  0.3790 0.3776 0.0014 0.3810 0.3784 0.0026 0.3840 0.3798 0.0042 
-3.5
10  0.3840 0.3813 0.0027 0.3860 0.3822 0.0038 0.3890 0.3837 0.0053 
-3
10  0.3880 0.3850 0.0030 0.3900 0.3859 0.0041 0.3920 0.3875 0.0045 
-2.5
10  0.3920 0.3885 0.0035 0.3940 0.3895 0.0045 0.3950 0.3911 0.0039 
-2
10  0.3960 0.3920 0.0040 0.3980 0.3930 0.0050 0.4000 0.3944 0.0056 
-1.5
10  0.3970 0.3953 0.0017 0.4000 0.3964 0.0036 0.4020 0.3976 0.0044 
-1
10  0.3980 0.3986 -0.0006 0.4020 0.3997 0.0023 0.4035 0.4006 0.0029 
-0.5
10  0.4020 0.4019 0.0001 0.4040 0.4029 0.0011 0.4050 0.4034 0.0016 
0
10  0.4050 0.4050 0.0000 0.4060 0.4060 0.0000 0.4060 0.4060 0.0000 
*     Niu Xiaofeng et al. [2006] 
 
 
 
 
 
 53 
 
Table 4.1.2:NH3-H2O Velocity Deviation Analysis 
  Result @ 0.0 Tesla Result @ 1.4 Tesla Result @ 3.0 Tesla 
 
Absorber     Mean Dev.     Mean     Mean 
  
length X Square Dev. Dev.
(m) Square Square 
  
-6 -6 -6
 Devia- Mean X 10  Devia- Mean X 10  Devia- Mean X 10  
tion. tion. tion. 
-6
0 or 10  0.0000 0.0000 0.0000 
0.0013 1.69 0.0022 4.84 0.0029 8.41 
-5.5
10  0.0000 0.0002 0.0013 
0.0013 1.69 0.0022 4.00 0.0029 2.56 
-5 
10 0.0000 0.0013 1.69 0.0006 0.0022 2.56 0.0007 0.0029 4.84 
-4.5
10  0.0010 0.0007 0.0034 
0.0013 0.09 0.0022 2.25 0.0029 0.25 
-4 
10 0.0014 0.0013 0.01 0.0026 0.0022 0.16 0.0042 0.0029 1.69 
-3.5
10  0.0027 0.0038 0.0053 
0.0013 1.96 0.0022 2.56 0.0029 5.76 
-3 
10 0.0030 0.0013 2.89 0.0041 0.0022 3.61 0.0045 0.0029 2.56 
-2.5
10  0.0035 0.0045 0.0039 
0.0013 4.84 0.0022 5.29 0.0029 1.00 
-2 
10 0.0040 0.0013 7.29 0.0050 0.0022 7.84 0.0056 0.0029 7.29 
-1.5
10  0.0017 0.0036 0.0044 
0.0013 0.16 0.0022 1.96 0.0029 2.25 
-1 
10 -0.0006 0.0013 3.61 0.0023 0.0022 0.01 0.0029 0.0029 0.00 
-0.5
10  0.0001 0.0011 0.0016 
0.0013 1.44 0.0022 1.21 0.0029 1.69 
0 
10 0.0000 0.0013 1.69 0.0000 0.0022 4.84 0.0000 0.0029 8.41 
∑     29.05     41.13     46.71 
 
-3 
@ 0.0 Tesla, Std. deviation ( ) = 2.9.05x106 /13  = 1.49 X 10
6 -3  
@1.4 Tesla, Std. deviation ( ) = 41.13x10 /13   = 1.78 X 10
6 -3 
@3.0 Tesla Std. deviation ( ) = 46.71x10 /13   = 1.90 X 10
 
Table 4.1.3:NH3-H2O Velocity T- Test analysis 
Independent Samples Test
Levene's Test f or
Equality  of  Variances t-test  for Equality  of  Means
95% Conf idence
Interv al of  the
Mean Std.  Error Dif f erence
F Sig. t df Sig. (2-tailed) Dif f erence Dif f erence Lower Upper
Velocity  @ 0.0 Tesla Equal variances
.023 .881 .234 24 .817 .001292 .005526 -.010114 .012698
assumed
Equal variances
.234 23.992 .817 .001292 .005526 -.010114 .012698
not assumed
Velocity  @ 1.4 Tesla Equal variances
.084 .775 .382 24 .706 .002192 .005745 -.009665 .014050
assumed
Equal variances
.382 23.942 .706 .002192 .005745 -.009667 .014051
not assumed
Velocity  @ 3.0 Tesla Equal variances
.018 .893 .507 24 .617 .002908 .005740 -.008939 .014754
assumed
Equal variances
.507 23.961 .617 .002908 .005740 -.008940 .014755
not assumed
 
 54 
 
 
 
 
 
Fig.4.1: NH3-H2O Velocity Distribution in the film at 0.0, 1.4 and 3.0Tesla  
  
 
Fig.4.1.1: NH3-H2O Velocity Distribution in the film at 0.0, 1.4 and 3.0Tesla  
 
 55 
 
 
 
 
 
 
 
 
Table 4.1.4:NH3-H2O Temperature Distribution in the film Comparison 
        BULK           
Absorber Lit.  Present  Devia Lit.  Present  Lit.  Present  Devia- 
  
length X Result Result tion.  Result Result Devia- Result Result tion.  
 
(m) @ 0.0 @ 0.0 @ 1.4 @ 1.4 tion.  @ 3.0 @ 3.0 
 Tesla Tesla Tesla Tesla Tesla Tesla 
* * * 
-6 
0 or 10 35.00 35.00 0.00 35.00 35.00 0.00 35.00 35.00 0.00 
-5.5 
10 32.00 32.00 0.00 32.00 32.00 0.00 32.00 32.00 0.00 
-5 
10 29.00 30.00 -1.00 29.00 30.00 -1.00 29.00 30.00 -1.00 
-4.5 
10 27.00 28.00 -1.00 27.00 28.00 -1.00 27.00 28.00 -1.00 
-4 
10 26.00 26.00 0.00 26.00 26.00 0.00 26.00 26.00 0.00 
-3.5 
10 25.00 24.00 1.00 25.00 24.00 1.00 25.00 24.00 1.00 
-3 
10 23.00 23.00 0.00 23.00 23.00 0.00 23.00 23.00 0.00 
-2.5 
10 22.00 21.00 1.00 22.00 21.00 1.00 22.00 21.00 1.00 
-2 
10 21.00 20.00 1.00 21.00 20.00 1.00 21.00 20.00 1.00 
-1.5 
10 18.00 18.00 0.00 18.00 18.00 0.00 18.00 18.00 0.00 
-1 
10 17.00 17.00 0.00 17.00 17.00 0.00 17.00 17.00 0.00 
-0.5 
10 16.00 16.00 0.00 16.00 16.00 0.00 16.00 16.00 0.00 
0 
10 15.00 15.00 0.00 15.00 15.00 0.00 15.00 15.00 0.00 
*     Niu Xiaofeng et al. [2006] 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 56 
 
Table 4.1.5:NH3-H2O Temperature Deviation Analysis 
  Result @ 0.0 Tesla Result @ 1.4 Tesla Result @ 3.0 Tesla 
 Absorber                   
length X 
(m) 
 
 Devia Mean Mean Devia Mean Mean Devia- Mean Mean 
 
tion. Deviati tion. Deviati tion. Deviation
  
on. on. Square 
Square Square 
                   
-6 
0 or 10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
-5.5 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
-5 
10 -1.00 0.0769 1.16 -1.00 0.0769 1.16 -1.00 0.0769 1.16 
-4.5 
10 -1.00 0.0769 1.16 -1.00 0.0769 1.16 -1.00 0.0769 1.16 
-4 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
-3.5 
10 1.00 0.0769 0.85 1.00 0.0769 0.85 1.00 0.0769 0.85 
-3 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
-2.5 
10 1.00 0.0769 0.85 1.00 0.0769 0.85 1.00 0.0769 0.85 
-2 
10 1.00 0.0769 0.85 1.00 0.0769 0.85 1.00 0.0769 0.85 
-1.5 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
-1 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
-0.5 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
0 
10 0.00 0.0769 0.01 0.00 0.0769 0.01 0.00 0.0769 0.01 
∑     4.92     4.92     4.92 
@ 0.0 Tesla, Std. deviation ( ) = 4.92 /13    = 0.62 
@ 1.4 Tesla, Std. deviation ( ) = 4.92 /13  = 0.62 
@3.0 Tesla, Std. deviation ( ) = 4.92 /13     = 0.62 
 
Table 4.1.6:NH3-H2O Temperature T- test analysis 
Independent Samples Test
Levene's Test f or
Equality  of  Variances t-test  for Equality  of  Means
95% Conf idence
Interv al of  the
Mean Std.  Error Dif f erence
F Sig. t df Sig. (2-tailed) Dif f erence Dif f erence Lower Upper
Temperature Equal variances
.029 .865 .031 24 .975 .077 2.479 -5.039 5.193
result at 0.0 Tesla assumed
Equal variances
.031 23.980 .975 .077 2.479 -5.039 5.193
not assumed
Temperature Equal variances
.029 .865 .031 24 .975 .077 2.479 -5.039 5.193
result at 1.4 Tesla assumed
Equal variances
.031 23.980 .975 .077 2.479 -5.039 5.193
not assumed
Temperature Equal variances
.029 .865 .031 24 .975 .077 2.479 -5.039 5.193
result at 3.0 Tesla assumed
Equal variances
.031 23.980 .975 .077 2.479 -5.039 5.193
not assumed
 
 57 
 
 
 
Fig. 4.1.2: NH3-H2O Temperature Distribution in the film from Lit. Result 
 
 
Fig. 4.1.3: NH3-H2O Temperature Distribution in the film from Present Result 
 58 
 
 
 
 
 
 
 
 
 
Table 4.1.7:NH3-H2O Concentration Distribution in the Film Comparison 
        BULK           
Absorber Lit.  Present  Devia- Lit.  Present  Devia- Lit.  Present  Devia-
   
length X Result Result tion.  Result Result tion.  Result Result tion.  
(m) @ 0.0 @ 0.0 @ 1.4 @ 1.4 @ 3.0 @ 3.0 
 Tesla Tesla Tesla Tesla Tesla Tesla 
* * * 
-6 
0 or 10 0.2000 0.2000 0.0000 0.2000 0.2000 0.0000 0.2000 0.2000 0.0000 
-5.5 
10 0.2200 0.2101 0.0099 0.2220 0.2102 0.0118 0.2250 0.2104 0.0146 
-5 
10 0.2300 0.2207 0.0093 0.2320 0.2211 0.0109 0.2400 0.2215 0.0185 
-4.5 
10 0.2500 0.2320 0.0180 0.2520 0.2326 0.0194 0.2620 0.2332 0.0288 
-4 
10 0.2700 0.2440 0.0260 0.2720 0.2448 0.0272 0.2800 0.2457 0.0343 
-3.5 
10 0.2900 0.2567 0.0333 0.2920 0.2578 0.0342 0.3000 0.2589 0.0411 
-3 
10 0.3050 0.2702 0.0348 0.3100 0.2716 0.0384 0.3200 0.2730 0.0470 
-2.5 
10 0.3190 0.2845 0.0345 0.3200 0.2863 0.0337 0.3300 0.2880 0.0420 
-2 
10 0.3300 0.2998 0.0302 0.3320 0.3020 0.0300 0.3410 0.3040 0.0370 
-1.5 
10 0.3380 0.3160 0.0220 0.3400 0.3186 0.0214 0.3500 0.3211 0.0289 
-1 
10 0.3500 0.3334 0.0166 0.3550 0.3365 0.0185 0.3610 0.3394 0.0216 
-0.5 
10 0.3550 0.3518 0.0032 0.3600 0.3555 0.0045 0.3650 0.3590 0.0060 
0 
10 0.3600 0.3600 0.0000 0.3680 0.3650 0.0030 0.3750 0.3700 0.0050 
*     Niu Xiaofeng et al. [2006] 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 59 
 
 
Table 4.1.8 NH3-H2O Concentration Deviation Analysis 
  Result @ 0.0 Tesla Result @ 1.4 Tesla Result @ 3.0 Tesla 
Absorber Devia- Mean  Mean Devia- Mean  Mean Devia- Mean  Mean 
length X tion.  Devia- tion. Devia- tion. Devia-
   
(m) tion. tion. tion.
 Square Square Square 
-4   -4 -4   X 10   X 10     X 10  
               
-6 
0 or 10 0.0000 0.0183 3.35 0.0000 0.0195 3.80 0.0000 0.0250 6.25 
-5.5 
10 0.0099 0.0183 0.71 0.0118 0.0195 0.59 0.0146 0.0250 1.08 
-5 
10 0.0093 0.0183 0.81 0.0109 0.0195 0.74 0.0185 0.0250 0.42 
-4.5 
10 0.0180 0.0183 0.00 0.0194 0.0195 0.00 0.0288 0.0250 0.14 
-4 
10 0.0260 0.0183 0.59 0.0272 0.0195 0.59 0.0343 0.0250 0.86 
-3.5 
10 0.0333 0.0183 2.25 0.0342 0.0195 2.16 0.0411 0.0250 2.59 
-3 
10 0.0348 0.0183 2.72 0.0384 0.0195 3.57 0.0470 0.0250 4.84 
-2.5 
10 0.0345 0.0183 2.62 0.0337 0.0195 2.02 0.0420 0.0250 2.89 
-2 
10 0.0302 0.0183 1.42 0.0300 0.0195 1.10 0.0370 0.0250 1.44 
-1.5 
10 0.0220 0.0183 0.14 0.0214 0.0195 0.04 0.0289 0.0250 0.15 
-1 
10 0.0166 0.0183 0.03 0.0185 0.0195 0.01 0.0216 0.0250 0.12 
-0.5 
10 0.0032 0.0183 2.28 0.0045 0.0195 2.25 0.0060 0.0250 3.61 
0 
10 0.0000 0.0183 3.35 0.0030 0.0195 2.72 0.0050 0.0250 4.00 
∑     20.27     19.60     28.40 
@ 0.0 Tesla, Std. deviation ( ) = 20.27x104 /13 1.25x102  
@ 1.4 Tesla, Std. deviation ( ) = 19.60x104 /13 1.23x102  
@3.0 Tesla, Std. deviation ( ) = 28.40x104 /13 1.48x102  
Table 4.1.9 NH3-H2O Concentration T- test analysis 
Independent Samples Test
Levene's Test f or
Equality  of  Variances t-test  for Equality  of  Means
95% Conf idence
Interv al of  the
Mean Std.  Error Dif f erence
F Sig. t df Sig. (2-tailed) Dif f erence Dif f erence Lower Upper
Concentration Equal variances
.015 .904 .857 24 .400 .0183 .0213 -.0257 .0623
@ 0.0 Tesla assumed
Equal variances
.857 23.993 .400 .0183 .0213 -.0257 .0623
not assumed
Concentration Equal variances
.013 .911 .888 24 .383 .0195 .0219 -.0258 .0647
@ 1.4 Tesla assumed
Equal variances
.888 23.993 .383 .0195 .0219 -.0258 .0647
not assumed
Concentration Equal variances
.009 .924 1.113 24 .277 .0250 .0225 -.0214 .0713
@ 3.0 Tesla assumed
Equal variances
1.113 23.994 .277 .0250 .0225 -.0214 .0713
not assumed
 
 60 
 
 
Fig. 4.1.4: NH3-H2O Concentration Distribution in the film from Lit. Result 
 
 
 
Fig. 4.1.5: NH3-H2O Concentration Distribution in the film from Present Result 
 
 
 
 
 
 61 
 
4.4 Numerical Results in the direction of falling film (X) for Lithium bromide and Lithium  
      Chloride Solution, X = Absorber length. 
 
Tables 4.2 to 4.2.3, 4.3.4, 4.3.7, 4.4.4 and 4.4.5 and Figs 4.1.6 to 4.2.5 and 4.5.3, 4.7.5 to 4.8.2 
represent the distribution of average velocity of the falling-film both in the bulk and the interface 
in the X-axis or direction of the falling film at various magnetic induction intensities in both 
Lithium bromide and Lithium chloride solutions. Both the tables and figures establish that the 
velocity at stronger magnetic field is larger than that of the weaker one. This is because the 
magnetic field force solution suffered is in the same direction with the gravity and increases with 
the growing of magnetic induction intensity. Moreover, it is found that the velocity u increases 
gradually at a fixed magnetic induction intensity, which may be attributed to the resultant force 
of magnetic force and gravity in the direction of falling both within the bulk and at the interface. 
However, as the frictional force decreases and the increase in falling velocity is slowed down 
within the bulk. The percentage velocity changes for LiBr-H2O within the bulk in X-direction 
when magnetic intensities was increased from 0.0 to 1.4 Tesla increased from 0.00 – 1.15 up to 
2/3length of the absorber wall before decrease to 0.0 at the solution surface. When increased 
from 0.0 to 3.0 Tesla, the percentage velocity change increased from 0.00 to 4.81 to the same 
0
absorber length and finally decreases to 0.0 percent at the end of the absorber length (10 m). For 
LiCl solution, the percentage velocity changes within the bulk in X-direction when magnetic 
intensity was increased from 0.0 to 1.4 Tesla increases from 0.00 to 0.85 up to the 2/3 length of 
absorber wall and decreases down to 0.0 at the end of the absorber length (1 m). When increased 
from 0.0 to 3.0 Tesla it increases from 0.0 to 3.44 up to the middle length of the absorber and 
finally decreases to 0.0 at the end of the absorber length (1 m). 
 
 
 
 
 
 
 
 
 62 
 
4.4: Tabular and Graphical representation of the literature and present work results  
Table 4.2: LiBr-H2O Velocity Distribution in the film  
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia Result Result Devia Result Result 
 @ 0.0 @ 0.0 @ 1.4 @ 1.4 @ 3.0 @3.0 Devia 
Tesla Tesla Tesla Tesla Tesla Tesla 
-6
0 or 10  N/A 0.3620  N/A 0.3620  N/A 0.3620  
-5.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-5
10  “ 0.3702  “ 0.3726  “ 0.3840  
-4.5
10  “ 0.3742  “ 0.3774  “ 0.3879  
-4
10  “ 0.3780  “ 0.3818  “ 0.3944  
-3.5
10  “ 0.3818  “ 0.3860  “ 0.3997  
-3
10  “ 0.3854  “ 0.3898  “ 0.4039  
-2.5
10  “ 0.3889  “ 0.3933  “ 0.4070  
-2
10  “ 0.3924  “ 0.3965  “ 0.4090  
-1.5
10  “ 0.3957  “ 0.3994  “ 0.4099  
-1
10  “ 0.3989  “ 0.4019  “ 0.4097  
-0.5
10  “ 0.4020  “ 0.4041  “ 0.4084  
0
10  N/A 0.4050  N/A 0.4060  N/A 0.4060  
** Absorber length (m) 
 
 63 
 
0.420
0.410
0.400
0.390
Present Result @ 0.0 Tesla
0.380
Present Result @ 1.4 Tesla
Present Result @ 3.0 Tesla
0.370
0.360
0.350
1 3 5 7 9 11 13 15
Absorber length (m)
Fig. 4.1.6:  Velocity Distribution in the Film at 0.0, 1.4 and 3.0Tesla for LiBr- H2O 
 
Table 4.2.1: LiBr-H2O Velocity Changes within the film in X-direction, X = Absorber length. 
Absorber Bulk 
length X 
(m) Present Present.  % Present Present % Changes 
 
Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.3620 0.3620 0.00 0.3620 0.3620 0.00 
-5.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-5
10  0.3702 0.3726 0.63 0.3702 0.3840 2.75 
-4.5
10  0.3742 0.3774 0.85 0.3742 0.3879 3.68 
-4
10  0.3780 0.3818 1.01 0.3780 0.3944 4.33 
-3.5
10  0.3818 0.3860 1.11 0.3818 0.3997 4.70 
-3
10  0.3854 0.3898 1.15 0.3854 0.4039 4.81 
 64 
Velocity (m/s)
 
-2.5
10  0.3889 0.3933 1.13 0.3889 0.4070 4.65 
-2
10  0.3924 0.3965 1.06 0.3924 0.4090 4.25 
-1.5
10  0.3957 0.3994 0.93 0.3957 0.4099 3.61 
-1
10  0.3989 0.4019 0.76 0.3989 0.4097 2.72 
-0.5
10  0.4020 0.4041 0.53 0.4020 0.4084 1.60 
0
10  0.4050 0.4060 0.25 0.4050 0.4060 0.25 
 
 
 
Fig. 4.1.7:  Velocity Changes within the Film in X-Direction at 0.0 & 1.4 Tesla, X = Absorber 
length. 
 65 
 
 
Fig. 4.1.8: % Velocity Changes within the Film in X-Direction from 0.0 & 1.4Tesla, X = 
Absorber length. 
 
Fig. 4.1.9: Velocity Changes within the Film in X-Direction at 0.0 & 3.0 Tesla, X = Absorber 
length. 
 66 
 
 
 
Fig.4.2: % Velocity Changes within the Film in X-direction from 0.0 & 3.0Tesla, X = Absorber 
length. 
 
Table 4.2.2: LiBr-H2O Velocity Distribution in the film  
Absorber Interface 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 Result Result Devia Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 -tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
-6
0 or 10  N/A 0.3620  N/A 0.3620  N/A 0.3620  
-5.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-4.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-4
10  “ 0.3662  “ 0.3674  “ 0.3718  
 67 
 
-3.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-3
10  “ 0.3662  “ 0.3674  “ 0.3718  
-2.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-2
10  “ 0.3662  “ 0.3674  “ 0.3718  
-1.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
-1
10  “ 0.3662  “ 0.3674  “ 0.3718  
-0.5
10  “ 0.3662  “ 0.3674  “ 0.3718  
0
10  N/A 0.4050  N/A 0.4060  N/A 0.4060  
 
Fig. 4.2.1: LiBr-H2O Velocity Distribution in the Film at the Interface 
 
 
 
 68 
 
Table 4.2.3: LiBr-H2O Velocity Changes within the film in X-direction at the interface, X = 
Absorber length. 
Absorber Interface  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.3620 0.3620 0.00 0.3620 0.3620 0.00 
-5.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-4.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-4
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-3.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-3
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-2.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-2
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-1.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-1
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
-0.5
10  0.3662 0.3674 0.35 0.3662 0.3718 1.53 
0
10  0.4050 0.4060 0.25 0.4050 0.4060 0.25 
 
 69 
 
 
 
Fig. 4.2.2: LiBr-H2O Velocity Changes at the Interface in X-direction at 0.0 and 1.4Tesla, X = 
Absorber length. 
 
 
 
 
Fig. 4.2.3: % Velocity Changes at the Interface in X-Direction from 0.0 to1.4Tesla, X = 
Absorber length. 
 70 
 
 
Fig. 4.2.4: Velocity Changes at the Interface in X-Direction from 0.0 & 3.0Tesla, X = Absorber 
length. 
 
Fig. 4.2.5:  % velocity Changes at the Interface in X-Direction from 0.0 to 3.0Tesla, X = 
Absorber length. 
 
 
 71 
 
Table 4.2.4: LiBr-H2O Temperature Distribution in the film Comparison 
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  44.44 44.44 0.00 N/A 44.44  N/A 44.44  
-5.5
10  43.50 44.40 2.07 “ 44.40  “ 44.40  
-5
10  43.20 43.45 0.58 “ 43.45  “ 43.45  
-4.5
10  43.00 42.52 -1.12 “ 42.52  “ 42.52  
-4
10  41.50 41.61 0.27 “ 41.61  “ 41.61  
-3.5
10  39.50 40.72 3.09 “ 40.72  “ 40.72  
-3
10  38.00 39.85 4.87 “ 39.85  “ 39.85  
-2.5
10  36.60 39.00 6.56 “ 39.00  “ 39.00  
-2
10  36.40 38.16 4.84 “ 38.16  “ 38.16  
-1.5
10  36.00 37.35 3.75 “ 37.35  “ 37.35  
-1
10  35.50 36.55 2.96 “ 36.55  “ 36.55  
-0.5
10  35.10 35.77 1.91 “ 35.77  “ 35.77  
0
10  35.00 35.00 0.00  35.00   35.00  
 72 
 
 
Fig. 4.2.6: LiBr-H2O Temperature Distribution in the Film Comparison at 0.0Tesla 
 
 
Fig. 4.2.7: % Deviation in Temperature Distribution in the Film Comparison at 0.0Tesla 
 
 73 
 
 
Fig. 4.2.8: Temperature Distribution in the Film at 1.4 Tesla (Present Result) 
 
Fig.4.2.9: Temperature Distribution in the Film at 3.0 Tesla(Present Result)  
 
 74 
 
 
Fig.4.3: Temperature Distribution in the Thin-Liquid Smooth Falling Film (Bulk) at 0.0,1.4 and 
3.0Tesla 
 
Table 4.2.5: LiBr-H2O Temperature Distribution in the film Comparison 
Absorber Interface 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  44.44 44.44 0.00 N/A 44.44  N/A 44.44  
-5.5
10  44.44 44.35 -0.20 “ 44.35  “ 44.35  
-5
10  44.44 43.50 -2.12 “ 43.50  “ 43.50  
-4.5
10  44.44 42.65 -4.03 “ 42.65  “ 42.65  
-4
10  44.44 41.80 -5.94 “ 41.80  “ 41.80  
-3.5
10  44.00 40.95 -6.93 “ 40.95  “ 40.95  
 75 
 
-3
10  42.00 40.10 -4.52 “ 40.10  “ 40.10  
-2.5
10  39.50 39.25 -0.63 “ 39.25  “ 39.25  
-2
10  38.00 38.40 1.05 “ 38.40  “ 38.40  
-1.5
10  37.00 37.55 1.49 “ 37.55  “ 37.55  
-1
10  36.50 36.70 0.55 “ 36.70  “ 36.70  
-0.5
10  36.00 35.85 2.36 “ 35.85  “ 35.85  
0
10  35.00 35.00 0.00  35.00   35.00  
*** Yang et al.(1992) 
 
 
 
Table 4.2.6: LiBr-H2O Temperature Deviation Analysis at the Bulk and Interface 
  BULK INTERFACE 
       
Absorber Lit. Present   Lit. Result Present   
length X Result Result Devia- @ 0.0 Result Devia- 
(m) @ 0.0 @ 0.0 tion. Tesla*** @ 0.0 tion. 
 Tesla *** Tesla  Tesla   
-6 
0 or 10 44.44 44.44 0.00 44.44 44.44 0.00 
-5.5 
10 43.50 44.40 0.90 44.44 44.35 0.09 
-5 
10 43.20 43.45 0.25 44.44 43.50 0.94 
-4.5 
10 43.00 42.52 0.48 44.44 42.65 1.79 
-4 
10 41.50 41.61 0.11 44.44 41.80 2.64 
-3.5 
10 39.50 40.72 1.22 44.00 40.95 3.05 
-3 
10 38.00 39.85 1.85 42.00 40.10 1.90 
-2.5 
10 36.60 39.00 2.40 39.50 39.25 0.25 
-2 
10 36.40 38.16 1.76 38.00 38.40 -0.40 
-1.5 
10 36.00 37.35 1.35 37.00 37.55 -0.55 
-1 
10 35.50 36.55 1.05 36.50 36.70 -0.20 
-0.5 
10 35.10 35.77 0.67 36.00 35.85 0.15 
0 
10 35.00 35.00 0.00 35.00 35.00 0.00 
*** Yang et al.(1992) 
 
 76 
 
Table 4.2.7: LiBr-H2O Temperature Deviation Analysis at the Bulk and Interface 
  BULK @ 0.0 Tesla INTERFACE @ 0.0 Tesla 
Absorber Deviation. Mean Mean Deviation. Mean Mean 
  
length X Deviation Deviation. Deviation Deviation.
(m) Square Square 
 
   
-6
0 or 10  
0.0000 0.9262 0.86 0.0000 0.7431 0.55 
-5.5
10  
0.9000 0.9262 0.00 0.0900 0.7431 0.43 
-5 
10 0.2500 0.9262 0.46 0.9400 0.7431 0.04 
-4.5
10  
0.4800 0.9262 0.20 1.7900 0.7431 1.10 
-4 
10 0.1100 0.9262 0.67 2.6400 0.7431 3.60 
-3.5
10  
1.2200 0.9262 0.09 3.0500 0.7431 5.32 
-3 
10 1.8500 0.9262 0.85 1.9000 0.7431 1.34 
-2.5
10  
2.4000 0.9262 2.17 0.2500 0.7431 0.24 
-2 
10 1.7600 0.9262 0.70 -0.4000 0.7431 1.31 
-1.5
10  
1.3500 0.9262 0.18 -0.5500 0.7431 1.67 
-1 
10 1.0500 0.9262 0.02 -0.2000 0.7431 0.89 
-0.5
10  
0.6700 0.9262 0.07 0.1500 0.7431 0.35 
0 
10 0.0000 0.9262 0.86 0.0000 0.7431 0.55 
∑     7.11     17.39 
 
 
 
 
@ The Bulk, Std. deviation ( ) = 7.11/13   = 0.7394 
@ The Interface, Std. deviation ( ) = 17.39 /13    = 1.1565 
 
 77 
 
 
Fig4.3.1:  Temperature Distribution at the Interface Comparison at 0.0 Tesla 
 
 
Fig. 4.3.2: %deviation in Temperature distribution at the interface Comparison at  0.0Tesla 
 
 78 
 
 
Fig4.3.3:  Temperature Distribution at the Interface at 1.4 Tesla 
 
 
Fig4.3.4:  Temperature Distribution at the Interface at 3.0Tesla 
 
 79 
 
 
Fig4.3.5:  Temperature Distribution at the Interface at 0.0, 1.4 and 3.0Tesla 
Table 4.2.8: LiBr-H2O Concentration Distribution in the film Comparison 
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  0.600 0.600 0.00 N/A 0.600  N/A 0.600  
-5.5
10  0.600 0.59545 -0.76 “ 0.59547  “ 0.59551  
-5
10  0.600 0.59090 -1.52 “ 0.59094  “ 0.59107  
-4.5
10  0.600 0.58635 -2.28 “ 0.58642  “ 0.58667  
-4
10  0.600 0.58180 -3.03 “ 0.58191  “ 0.58229  
 80 
 
-3.5
10  0.600 0.57724 -3.79 “ 0.57739  “ 0.57792  
-3
10  0.600 0.57267 -4.56 “ 0.57287  “ 0.57355  
-2.5
10  0.599 0.56810 -5.16 “ 0.56834  “ 0.56917  
-2
10  0.590 0.56352 -4.49 “ 0.56380  “ 0.56476  
-1.5
10  0.575 0.55892 -2.80 “ 0.55924  “ 0.56032  
-1
10  0.567 0.55432 -0.01 “ 0.55465  “ 0.55582  
-0.5
10  0.559 0.54970 0.07 “ 0.55005  “ 0.55126  
0
10  0.545 0.54500 0.00 “ 0.54500  N/A 0.54500  
*** Yang et al.(1992) 
  
4.5: Coefficient of Performance (COP) for LiBr-water absorption Refrigeration 
Coefficient of performance (COP) of an absorption refrigeration system is obtained from;  
 coolingcapacityobtainedatevaporator 
COP     
 heatinputforthegenerator  workinputforthepump


   Or 
Concentrationattheoutlet 
COP     
 Concentrationatthein let 
The work input for the pump is negligible relative to the heat input at the generator; therefore, 
the pump work is often neglected for the purposes of analysis. 
From Table 4.2.9 above 
0.549
At 0.0 Tesla, COP =      =    0. 915 
0.600
0.551
At 3.0 Tesla, COP =         =   0. 918 
0.600
Increment = 0.918 – 0. 915 = 0.003 = 0.3%  
 81 
 
 
Fig. 4.3.6: LiBr-H2O Concentration Distribution in the Film Comparison at 0.0Tesla 
 
 
Fig 4.3.7: % Deviation in Concentration Distribution in the Film at 0.0Tesla 
 
 82 
 
 
Fig. 4.3.8: LiBr-H2O Concentration Distribution in the Film at 1.4Tesla 
 
 
 
Fig. 4.3.9: LiBr-H2O Concentration Distribution in the Film at 3.0Tesla 
 
 83 
 
Table 4.2.9: LiBr-H2O Concentration Changes within the film in X-direction, X = Absorber 
length. 
Absorber Bulk  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.600 0.600 0.000 0.600 0.600 0.000 
-5.5
10  0.59545 0.59547 0.003 0.59545 0.59551 0.010 
-5
10  0.59090 0.59094 0.010 0.59090 0.59107 0.029 
-4.5
10  0.58635 0.58642 0.012 0.58635 0.58667 0.055 
-4
10  0.58180 0.58191 0.020 0.58180 0.58229 0.084 
-3.5
10  0.57724 0.57739 0.026 0.57724 0.57792 0.118 
-3
10  0.57267 0.57287 0.035 0.57267 0.57355 0.154 
-2.5
10  0.56810 0.56834 0.042 0.56810 0.56917 0.188 
-2
10  0.56352 0.56380 0.050 0.56352 0.56476 0.220 
-1.5
10  0.55892 0.55924 0.057 0.55892 0.56032 0.250 
-1
10  0.55432 0.55465 0.060 0.55432 0.55582 0.271 
-0.5
10  0.54970 0.55005 0.064 0.54970 0.55126 0.284 
0
10  0.54500 0.54500 0.000 0.54500 0.54500 0.000 
 
 84 
 
 
Fig 4.4: Concentration Changes within the Film in X-Direction at 0.0 and 1.4 Tesla, X = 
Absorber length. 
 
 
Fig 4.4.1: % Concentration Changes within the Film in X-direction at 0.0 and 1.4Tesla, X = 
Absorber length. 
 85 
 
 
Fig 4.4.2: Concentration Changes within the Film in X-direction at 0.0 & 3.0 Tesla, X = 
Absorber length. 
 
 
 
Fig 4.4.3: % Concentration Changes within the Film in X-Direction at 0.0 & 3.0Tesla, X = 
Absorber length. 
 86 
 
 
Table 4.3: LiBr-H2O Concentration Distribution in the film Comparison 
Absorber Interface 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  0.599 0.600 0.17 N/A 0.600  N/A 0.600  
-5.5
10  0.599 0.59500 -0.67 “ 0.59502  “ 0.59507  
-5
10  0.599 0.59004 -1.50 “ 0.59008  “ 0.59019  
-4.5
10  0.599 0.58513 -2.32 “ 0.58518  “ 0.58535  
-4
10  0.599 0.58025 -3.13 “ 0.58032  “ 0.58054  
-3.5
10  0.591 0.57542 -2.64 “ 0.57550  “ 0.57577  
-3
10  0.580 0.57063 -1.62 “ 0.57072  “ 0.57105  
-2.5
10  0.570 0.56587 -0.73 “ 0.56598  “ 0.56636  
-2
10  0.560 0.56116 0.21 “ 0.56128  “ 0.56171  
-1.5
10  0.550 0.55648 1.18 “ 0.55662  “ 0.55710  
-1
10  0.548 0.55185 0.70 “ 0.55200  “ 0.55253  
-0.5
10  0.546 0.54725 0.23 “ 0.54741  “ 0.54799  
0
10  0.545 0.54500 0.00 N/A 0.54500  N/A 0.54500  
*** Yang et al.(1992) 
 
 
 
 
 
 
 87 
 
Table 4.3.1: LiBr-H2O Concentration Deviation Analysis at the Bulk and Interface 
  BULK INTERFACE 
       
Absorber Lit. Present   Lit. Present   
length X Result Result Devia- Result Result Devia- 
(m) @ 0.0 @ 0.0 tion. @ 0.0 @ 0.0 tion. 
 Tesla Tesla  Tesla Tesla   
 *** *** 
-6 
0 or 10 0.600 0.6000 0.0000 0.599 0.600 -0.0010 
-5.5 
10 0.600 0.5955 0.0045 0.599 0.595 0.0040 
-5 
10 0.600 0.5909 0.0091 0.599 0.590 0.0089 
-4.5 
10 0.600 0.5864 0.0136 0.599 0.585 0.0138 
-4 
10 0.600 0.5818 0.0182 0.599 0.580 0.0187 
-3.5 
10 0.600 0.5772 0.0227 0.591 0.575 0.0155 
-3 
10 0.600 0.5727 0.0273 0.580 0.571 0.0093 
-2.5 
10 0.599 0.5681 0.0309 0.570 0.566 0.0041 
-2 
10 0.590 0.5635 0.0264 0.560 0.561 -0.0011 
-1.5 
10 0.575 0.5589 0.0160 0.550 0.556 -0.0064 
-1 
10 0.567 0.5543 0.0126 0.548 0.552 -0.0038 
-0.5 
10 0.559 0.5497 0.0093 0.546 0.547 -0.0012 
0 
10 0.545 0.5450 0.0000 0.545 0.545 0.0000 
*** Yang et al.(1992) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 88 
 
Table 4.3.2: LiBr-H2O Concentration Deviation Analysis at the Bulk and Interface 
  BULK @ 0.0 Tesla INTERFACE @ 0.0 Tesla 
Absorber Deviation. Mean Mean Deviation. Mean Mean 
  
length X Deviation Deviation. Deviation Deviation.
(m) Square Square 
 -4 -4
X 10  X 10  
-6
0 or 10  
0.0000 0.01425 2.03 -0.0010 0.00468 0.32 
-5.5
10  
0.0045 0.01425 0.95 0.0040 0.00468 0.00 
-5 
10 0.0091 0.01425 0.27 0.0089 0.00468 0.18 
-4.5
10  
0.0136 0.01425 0.00 0.0138 0.00468 0.83 
-4 
10 0.0182 0.01425 0.16 0.0187 0.00468 1.97 
-3.5
10  
0.0227 0.01425 0.71 0.0155 0.00468 1.17 
-3 
10 0.0273 0.01425 1.70 0.0093 0.00468 0.21 
-2.5
10  
0.0309 0.01425 2.77 0.0041 0.00468 0.00 
-2 
10 0.0264 0.01425 1.48 -0.0011 0.00468 0.33 
-1.5
10  
0.0106 0.01425 0.13 -0.0064 0.00468 1.23 
-1 
10 0.0126 0.01425 0.03 -0.0038 0.00468 0.72 
-0.5
10  
0.0093 0.01425 0.25 -0.0012 0.00468 0.35 
0 
10 0.0000 0.01425 2.03 0.0000 0.00468 0.22 
∑     12.51     7.54 
 
 
4 -2
@ The Bulk, Std. deviation ( )         12.51x10 /13   = 0.9809 X 10  
4 -2 
@ The Interface, Std. deviation ( )  7.54x10 /13     = 0.7614 X 10
 89 
 
 
Fig 4.4.4: LiBr-H2O Concentration Distribution at the Interface Comparison at 0.0Tesla 
 
 
 
Fig 4.4.5: % Deviation in Concentration Distribution at the Interface at 0.0Tesla 
 
 90 
 
 
Fig 4.4.6: Concentration Distribution at the Interface at 1.4Tesla (present Result) 
 
 
 
Fig 4.4.7:Concentration Distribution at the Interface at 3.0Tesla (Present Result) 
 
 91 
 
Table 4.3.3: LiBr-H2O Concentration Changes within the film in X-direction, X = Absorber 
length. 
Absorber Interface  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.600 0.600 0.000 0.600 0.600 0.000 
-5.5
10  0.59500 0.59502 0.003 0.59500 0.59507 0.012 
-5
10  0.59004 0.59008 0.007 0.59004 0.59019 0.025 
-4.5
10  0.58513 0.58518 0.009 0.58513 0.58535 0.038 
-4
10  0.58025 0.58032 0.012 0.58025 0.58054 0.050 
-3.5
10  0.57542 0.57550 0.014 0.57542 0.57577 0.061 
-3
10  0.57063 0.57072 0.016 0.57063 0.57105 0.074 
-2.5
10  0.56587 0.56598 0.019 0.56587 0.56636 0.087 
-2
10  0.56116 0.56128 0.021 0.56116 0.56171 0.098 
-1.5
10  0.55648 0.55662 0.025 0.55648 0.55710 0.111 
-1
10  0.55185 0.55200 0.027 0.55185 0.55253 0.123 
-0.5
10  0.54725 0.54741 0.029 0.54725 0.54799 0.135 
0
10  0.54500 0.54500 0.000 0.54500 0.54500 0.000 
 
 92 
 
 
Fig 4.4.8: Concentration Changes at the interface in X-Direction from 0.0 to 1.4 Tesla, X = 
Absorber length. 
 
Fig 4.4.9: Concentration Changes at the interface in X-direction from 0.0 to 1.4Tesla, X = 
Absorber length. 
 93 
 
 
Fig 4.5: Concentration Changes at the interface in X-direction at 0.0 & 3.0 Tesla, X = Absorber 
length. 
 
Fig 4.5.1: % Concentration Changes at the interface in X-Direction from 0.0 to 3.0Tesla, X = 
Absorber length. 
 
 
 94 
 
Fig 4.5.2: LiBr-H2O Concentration Distribution at the interface in X direction at 0.0 1.4 
and 3.0Tesla, X = Absorber length. 
 
 
Fig 4.5.3: LiBr-H2O Concentration Distribution in the film at 0.0 1.4 and 3.0Tesla  
 95 
 
Table 4.3.4: LiCl-H2O Velocity Distribution in the film  
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 Result Result Devia Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 -tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla  Tesla Tesla Tesla Tesla 
 
-6
0 or 10  N/A 0.3620  N/A 0.3620  N/A 0.3620  
-5.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-5
10  “ 0.3702  “ 0.3719  “ 0.3775  
-4.5
10  “ 0.3742  “ 0.3765  “ 0.3840  
-4
10  “ 0.3780  “ 0.3808  “ 0.3897  
-3.5
10  “ 0.3818  “ 0.3849  “ 0.3946  
-3
10  “ 0.3854  “ 0.3887  “ 0.3987  
-2.5
10  N/A 0.3889  N/A 0.3922  N/A 0.4019  
-2
10  “ 0.3924  “ 0.3955  “ 0.4044  
-1.5
10  “ 0.3957  “ 0.3985  “ 0.4060  
-1
10  “ 0.3989  “ 0.4013  “ 0.4068  
-0.5
10  “ 0.4020  “ 0.4038  “ 0.4068  
0
10  N/A 0.4050  N/A 0.4060  N/A 0.4060  
 
 96 
 
 
Fig 4.5.3:  LiCl-H2O Velocity Distribution in the Film at 0.0, 1.4 and 3.0Tesla 
 
Table 4.3.5: LiCl-H2O Temperature Distribution in the film Comparison 
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  35.00 35.00 0.00 N/A 35.00  N/A 35.00  
-5.5
10  35.00 34.74 0.00 “ 34.74  “ 34.74  
-5
10  35.00 34.49 0.00 “ 34.49  “ 34.49  
-4.5
10  35.00 34.23 -2.20 “ 34.23  “ 34.23  
-4
10  35.00 33.98 -2.90 “ 33.98  “ 33.98  
-3.5
10  35.00 33.73 -3.60 “ 33.73  “ 33.73  
 97 
 
-3
10  35.00 33.48 -4.34 “ 33.48  “ 33.48  
-2.5
10  34.80 33.24 -4.48 N/A 33.24  N/A 33.24  
-2
10  33.90 32.99 -2.68 “ 32.99  “ 32.99  
-1.5
10  32.10 32.75 2.03 “ 32.75  “ 32.75  
-1
10  32.00 32.51 1.59 “ 32.51  “ 32.51  
-0.5
10  31.50 32.27 2.44 “ 32.27  “ 32.27  
0
10  30.00 30.00 0.00 N/A 30.00  N/A 30.00  
*** Yang et al.(1992) 
 
 
Fig 4.5.4: LiCl-H2O Temperature Distribution in the Film Comparison at 0.0Tesla 
 
 98 
 
 
Fig. 4.5.5: % Deviation of LiCl-H2O Temperature Distribution in the Film at 0.0Tesla 
 
 
 
Fig. 4.5.6: Temperature Distribution in the Film at 1.4 Tesla 
 
 99 
 
 
 
Fig 4.5.7: Temperature Distribution in the Film at 3.0Tesla 
 
 
 
Fig. 4.5.8: LiCl-H2O Temperature Distribution in the Film at 0.0, 1.4 and 3.0Tesla 
 
 
 100 
 
Table 4.3.6: LiCl-H2O Concentration Distribution in the film Comparison 
Absorber Bulk 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 
Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  0.450 0.450 0.00 N/A 0.4500  N/A 0.4500  
-5.5
10  0.450 0.4453 -1.04 “ 0.4453  “ 0.4454  
-5
10  0.450 0.4406 -2.08 “ 0.4407  “ 0.4408  
-4.5
10  0.450 0.4360 -3.12 “ 0.4361  “ 0.4362  
-4
10  0.450 0.4313 -4.15 “ 0.4314  “ 0.4317  
-3.5
10  0.450 0.4267 -5.18 “ 0.4268  “ 0.4272  
-3
10  0.448 0.4221 -5.79 “ 0.4222  “ 0.4227  
-2.5
10  0.440 0.4174 -5.13 N/A 0.4177  N/A 0.4183  
-2
10  0.440 0.4128 -6.18 “ 0.4131  “ 0.4138  
-1.5
10  0.430 0.4082 -5.07 “ 0.4085  “ 0.4093  
-1
10  0.420 0.4036 -3.91 “ 0.4039  “ 0.4047  
-0.5
10  0.390 0.3990 2.30 “ 0.3993  “ 0.4001  
0
10  0.358 0.3580 0.00 N/A 0.3580  N/A 0.3580  
*** Yang et al.(1992) 
 
 
 
 
 101 
 
Coefficient of Performance (COP) for LiCl-water absorption Refrigeration 
From Table 4.3.6 above 
0.399
At 0.0 Tesla, COP =     = 0. 887 
0.450
0.400
At 3.0 Tesla, COP =     = 0.889 
0.450
Increment = 0.889 – 0.887 =  0.002= 0.2% 
 
Fig. 4.5.9: LiCl-H2O Concentration Distribution in the Film Comparison at 0.0 Tesla 
 
 
Fig. 4.6: LiCl-H2O % Deviation Concentration Distribution in the Film at 0.0 Tesla 
 102 
 
 
Fig. 4.6.1: LiCl-H2O Concentration Distribution in the Film at 1.4 Tesla 
 
 
 
Fig. 4.6.2: Concentration Distribution in the Film at 3.0 Tesla 
 
 
 
 103 
 
 
 
Fig. 4.6.3: LiCl-H2O Concentration Distribution in the film at 0.0 1.4 and 3.0Tesla  
 
Tables 4.2.8 to 4.3.3, 4.3.6, 4.4.1 to 4.4.3, 4.4.8 to 4.4.9 and Figures 4.3.6 to 4.5.2 and 4.5.9 to 
4.6.3, 4.7 to 4.7.4 and 4.8.5 to 4.9.2 are the average concentration distribution both within the 
bulk and the interface along the falling orientation in different magnetic fields for both LiBr and 
LiCl solution. It can been seen that increase of magnetic induction intensity causes an increase in 
the concentration of LiBr-water and LiCl-water solution which indicates that magnetic field 
enhances the absorption process. In addition, the concentration is decreasing constantly along the 
falling film. The concentration distribution within the bulk and at the interface when compare 
with the existing literature where it exists or available, it compares well. The percentage 
deviation within the bulk and interface in LiBr-water solution at 0.0Tesla ranges between -5.16 - 
0.07 and -3.13 – 1.18 while for the case of 1.4 and 3.0 Tesla, there was no existing literature 
results to compare with. For LiCl-water solution, the percentage deviation ranges between -6.18 
to 2.30 and 0.0-12.50 respectively with no existing literature results available for comparison in 
the case of 1.4 and 3.0 Tesla. 
For LiBr-H2O, when magnetic intensity increases from 0 to 1.4Tesla, the percentage 
concentration changes within the bulk in the X-direction is from 0.0 to 0.064 at the end of the 
absorber. When the magnetic intensity increases from 0.0 to 3.0 Tesla, percentage concentration 
 104 
 
increases from 0.0 to 0.284 at the end of the absorber. For LiCl-water solution, when magnetic 
intensity increases from 0 to 1.4Tesla, the percentage concentration changes within the bulk in 
the X-direction is from 0.0 to 0.078 at the end of the absorber. When the magnetic intensity 
increased from 0 to 3.0Tesla, percentage concentration increases from 0.0 to 0.293 at the end of 
the absorber length. This observation accounts for the condensation of the refrigerant vapour 
towards the end of the absorber length.   
 
 
 
Table 4.3.7: LiCl-H2O Velocity Distribution in the film  
Absorber Interface 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 Result Result Devia Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 -tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
-6
0 or 10  N/A 0.3620  N/A 0.3620  N/A 0.3620  
-5.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-4.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-4
10  “ 0.3662  “ 0.3671  “ 0.3702  
-3.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-3
10  “ 0.3662  “ 0.3671  “ 0.3702  
-2.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-2
10  “ 0.3662  “ 0.3671  “ 0.3702  
 105 
 
-1.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
-1
10  “ 0.3662  “ 0.3671  “ 0.3702  
-0.5
10  “ 0.3662  “ 0.3671  “ 0.3702  
0
10  N/A 0.4050  N/A 0.4060  N/A 0.4060  
 
 
 
 
 
 
Fig. 4.6.4:  Velocity Distribution at the Interface  
 
 
 
 
 
 
 
 
 
 
 106 
 
Table 4.3.8: LiCl-H2O Temperature Distribution in Comparison at the interface 
Absorber Interface 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 Result Result Devia- Result Result Devia- Result Result Devia- 
  @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  36.00 35.00 -2.78 N/A 35.00  N/A 35.00  
-5.5
10  36.25 38.25 5.52 “ 38.25  “ 38.25  
-5
10  37.00 37.50 1.35 “ 37.50  “ 37.50  
-4.5
10  37.60 36.75 -2.26 “ 36.75  “ 36.75  
-4
10  38.00 36.00 -5.26 “ 36.00  “ 36.00  
-3.5
10  38.20 35.25 -7.72 “ 35.25  “ 35.25  
-3
10  38.20 34.50 -9.69 “ 34.50  “ 34.50  
-2.5
10  37.00 33.75 -8.78 N/A 33.75  N/A 33.75  
-2
10  36.00 33.00 -8.33 “ 33.00  “ 33.00  
-1.5
10  34.50 32.25 -6.52 “ 32.25  “ 32.25  
-1
10  33.00 31.50 -4.55 “ 31.50  “ 31.50  
-0.5
10  31.90 30.75 -3.61 “ 30.75  “ 30.75  
0
10  30.00 30.00 0.00 N/A 30.00  N/A 30.00  
*** Yang et al.(1992) 
 
 
 
 
 
 
 
 107 
 
Table 4.3.9: LiCl-H2O Temperature Deviation Analysis at the Bulk and Interface 
  BULK INTERFACE 
Absorber       
length X Lit. Result Present   Lit. Present   
(m) @ 0.0 Tesla Result Deviation. Result Result Deviation. 
 *** @ 0.0 @ 0.0 @ 0.0 
Tesla  Tesla*** Tesla   
-6 
0 or 10 35.00 35.00 0.00 36.00 35.00 1.00 
-5.5 
10 35.00 34.74 0.26 36.25 38.25 -2.03 
-5 
10 35.00 34.49 0.51 37.00 37.50 -0.50 
-4.5 
10 35.00 34.23 0.77 37.60 36.75 0.85 
-4 
10 35.00 33.98 1.02 38.00 36.00 2.00 
-3.5 
10 35.00 33.73 1.27 38.20 35.25 2.95 
-3 
10 35.00 33.48 1.52 38.20 34.50 3.70 
-2.5 
10 34.80 33.24 1.56 37.00 33.75 3.25 
-2 
10 33.90 32.99 0.91 36.00 33.00 3.00 
-1.5 
10 32.10 32.75 -0.65 34.50 32.25 2.25 
-1 
10 32.00 32.51 -0.51 33.00 31.50 1.50 
-0.5 
10 31.50 32.27 0.77 31.90 30.75 1.15 
0 
10 30.00 30.00 0.00 30.00 30.00 0.00 
*** Yang et al.(1992) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 108 
 
Table 4.4: LiCl-H2O Temperature Deviation Analysis at the Bulk and Interface 
  BULK @ 0.0 Tesla INTERFACE @ 0.0 Tesla 
Absorber Deviation. Mean Mean Deviation. Mean Mean 
  
length X (m) Deviation Deviation. Deviation Deviation.
 Square Square 
   
-6 
0 or 10
0.00 0.5715 0.33 1.00 1.4708 0.22 
-5.5 
10
0.26 0.5715 0.10 -2.03 1.4708 12.26 
-5 
10 0.51 0.5715 0.00 -0.50 1.4708 3.88 
-4.5 
10
0.77 0.5715 0.04 0.85 1.4708 0.39 
-4 
10 1.02 0.5715 0.20 2.00 1.4708 0.28 
-3.5 
10
1.27 0.5715 0.49 2.95 1.4708 2.19 
-3 
10 1.52 0.5715 0.90 3.70 1.4708 4.97 
-2.5 
10
1.56 0.5715 0.98 3.25 1.4708 3.17 
-2 
10 0.91 0.5715 0.11 3.00 1.4708 2.34 
-1.5 
10
-0.65 0.5715 1.49 2.25 1.4708 0.61 
-1 
10 -0.51 0.5715 1.17 1.50 1.4708 0.00 
-0.5 
10
0.77 0.5715 0.04 1.15 1.4708 0.10 
0 
10 0.00 0.5715 0.33 0.00 1.4708 2.16 
∑     6.17     32.56 
 
 
 
@ The Bulk, Std. deviation ( )         6.17 /13   = 0.69
@ The Interface, Std. deviation ( )   32.56 /13     = 1.58 
 109 
 
 
 
Fig 4.6.5: Temperature Distribution at the Interface Comparison at 0.0Tesla 
 
 
Fig 4.6.6: % deviation Temperature Distribution at the Interface at 0.0Tesla 
 110 
 
 
 
Fig. 4.6.7: LiCl-H2O Temperature Distribution at the Interface at 1.4Tesla 
 
 
Fig. 4.6.8: LiCl-H2O Temperature Distribution at the Interface at 3.0 Tesla 
 111 
 
 
Fig.4.6.9: LiCl-H2O Temperature Distribution at the interface at 0.0, 1.4 and 3.0Tesla  
 
Table 4.4.1: LiCl-H2O Concentration Distribution in the film Comparison 
Absorber Interface 
length X 
(m) Lit. Present  % Lit. Present. % Lit. Present % 
 Result Result Devia- Result Result Devia- Result Result Devia- 
 @ 0.0 @ 0.0 tion @ 1.4 @ 1.4 tion @ 3.0 @3.0 tion 
Tesla Tesla Tesla Tesla Tesla Tesla 
*** 
-6
0 or 10  0.400 0.450 12.50 N/A 0.4500  N/A 0.4500  
-5.5
10  0.402 0.44486 10.66 “ 0.44487  “ 0.44491  
-5
10  0.405 0.43978 8.59 “ 0.43980  “ 0.43989  
-4.5
10  0.410 0.43476 6.04 “ 0.43479  “ 0.43492  
-4
10  0.414 0.42979 3.81 “ 0.42984  “ 0.43000  
-3.5
10  0.414 0.42488 2.63 “ 0.42494  “ 0.42514  
 112 
 
-3
10  0.414 0.42003 1.46 “ 0.42010  “ 0.42034  
-2.5
10  0.400 0.41523 3.81 “ 0.41531  “ 0.41559  
-2
10  0.390 0.41049 5.25 “ 0.41058  “ 0.41089  
-1.5
10  0.380 0.40580 6.79 “ 0.40590  “ 0.40625  
-1
10  0.370 0.40116 8.42 “ 0.40128  “ 0.40166  
-0.5
10  0.365 0.39658 8.65 “ 0.39671  “ 0.39712  
0
10  0.358 0.35800 0.00 N/A 0.35800  N/A 0.35800  
*** Yang et al.(1992) 
 
Table 4.4.2: LiCl-H2O Concentration Deviation Analysis at the Bulk and Interface 
  BULK INTERFACE 
Absorber       
length X Lit. Present   Lit. Present   
(m) Result Result Deviation. Result Result Deviation. 
 @ 0.0 @ 0.0 @ 0.0 @ 0.0 
Tesla*** Tesla  Tesla*** Tesla   
-6 
0 or 10 0.450 0.4500 0.0000 0.400 0.4500 -0.0500 
-5.5 
10 0.450 0.4453 0.0047 0.402 0.4449 -0.0428 
-5 
10 0.450 0.4406 0.0095 0.405 0.4398 -0.0347 
-4.5 
10 0.450 0.4360 0.0140 0.410 0.4348 -0.0247 
-4 
10 0.450 0.4313 0.0187 0.414 0.4298 -0.0157 
-3.5 
10 0.450 0.4267 0.0233 0.414 0.4249 -0.0108 
-3 
10 0.448 0.4221 0.0259 0.414 0.4200 -0.0060 
-2.5 
10 0.440 0.4174 0.0226 0.400 0.4152 -0.0152 
-2 
10 0.440 0.4128 0.0272 0.390 0.4105 -0.0204 
-1.5 
10 0.430 0.4082 0.0218 0.380 0.4058 -0.0258 
-1 
10 0.420 0.4036 0.0164 0.370 0.4012 -0.0311 
-0.5 
10 0.390 0.3990 -0.0090 0.365 0.3966 -0.0315 
0 
10 0.358 0.3580 0.0000 0.358 0.3580 0.0000 
***Yang et al.(1992) 
 
 
 
 113 
 
Table 4.4.3: LiCl-H2O Concentration Deviation Analysis at the Bulk and Interface 
  BULK @ 0.0 Tesla INTERFACE @ 0.0 Tesla 
Absorber Deviation. Mean Mean Deviation. Mean Mean 
  
length X Deviation Deviation. Deviation Deviation.
(m) Square Square 
 
-4 -4
X 10  X 10  
-6 
0 or 10
0.0000 0.01347 1.81 -0.0500 -0.0237 6.92 
-5.5 
10
0.0047 0.01347 0.77 -0.0428 -0.0237 3.65 
-5 
10 0.0095 0.01347 0.16 -0.0347 -0.0237 1.21 
-4.5 
10
0.0140 0.01347 0.00 -0.0247 -0.0237 0.01 
-4 
10 0.0187 0.01347 0.27 -0.0157 -0.0237 0.64 
-3.5 
10
0.0233 0.01347 0.97 -0.0108 -0.0237 1.66 
-3 
10 0.0259 0.01347 1.55 -0.0060 -0.0237 3.13 
-2.5 
10
0.0226 0.01347 0.83 -0.0152 -0.0237 0.72 
-2 
10 0.0272 0.01347 1.89 -0.0204 -0.0237 0.11 
-1.5 
10
0.0218 0.01347 0.69 -0.0258 -0.0237 0.04 
-1 
10 0.0164 0.01347 0.09 -0.0311 -0.0237 0.55 
-0.5 
10
-0.0090 0.01347 5.05 -0.0315 -0.0237 0.61 
0 
10 0.0000 0.01347 1.81 0.0000 -0.0237 5.62 
∑     15.89     24.87 
 
 
4 -2 
@ The Bulk, Std. deviation ( )         15.89x10 /13   = 1.11 X 10
4 -2 
@ The Interface, Std. deviation ( )  24.87x10 /13     = 1.38 X 10
 
 
 114 
 
 
 
Fig. 4.7: Concentration Distribution at the Interface Comparison at 0.0 Tesla 
 
 
Fig.4.7.1: % Deviation of Concentration at the Interface at 0.0Tesla 
 115 
 
 
Fig.4.7.2: Concentration Distribution at the Interface  at 1.4 Tesla 
 
 
 
Fig.4.7.3: Concentration Distribution at the Interface at 3.0 Tesla 
 
 
 116 
 
 
 
Fig. 4.7.4: LiCl-H2O Concentration Distribution at the Interface at 0.0 1.4 and 3.0Tesla 
 
 
Temperature Distribution within the bulk at various magnetic intensity  
As seen in Tables 4.2.4 to 4.2.7, 4.3.5, 4.3.8 to 4.4, 4.4.6 to 4.4.7 couple with Figs 4.2.6 to 4.3.5, 
4.5.4 to 4.5.8, 4.6.5 to 4.6.9 and 4.8.3 to 4.8.4 the average temperature changes varies very 
slightly with various values of the magnetic induction, and the temperature of solution film is 
higher at the inlet than within the bulk at the beginning of absorption process. For the case of 
LiBr-H2O the driving force of mass transfer is relatively greater at the inlet of solution, a great 
amount of absorbing heat is generated. On the other hand, the cooling water‟s temperature is 
higher at the inlet of solution because of counter current flow. Both these contribute to the higher 
temperature at the entrance of falling solution. The temperature distribution within the bulk and 
interface for the two working fluids under consideration when compare with the existing 
available literature results, compare well. The percentage deviation for LiBr-H2O at 0.0Tesla 
ranges from -1.12-6.56 in the bulk and -6.93 – 2.36 at the interface while at 1.4 and 3.0 Tesla 
there was no existing literature results for comparison. For LiCl-H2O at 0.0 Tesla percentage 
 117 
 
deviation ranges from -4.48 to 2.44 in the bulk and -9.69 to 5.52 at the interface while at 1.4 and 
3.0 Tesla, there was also no existing literature results both experimental and numerical for 
comparison with. For both LiBr-H2O and LiCl-H2O solution, there was no percentage 
temperature change in X-direction at both the bulk and interface.          
 
Table 4.4.4: LiCl-H2O Velocity Changes within the film in X-direction, X = Absorber 
length. 
Absorber Bulk 
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.3620 0.3620 0.00 0.3620 0.3620 0.00 
-5.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.09 
-5
10  0.3702 0.3719 0.46 0.3702 0.3775 1.96 
-4.5
10  0.3742 0.3765 0.62 0.3742 0.3840 2.63 
-4
10  0.3780 0.3808 0.74 0.3780 0.3897 3.09 
-3.5
10  0.3818 0.3849 0.82 0.3818 0.3946 3.36 
-3
10  0.3854 0.3887 0.85 0.3854 0.3987 3.44 
-2.5
10  0.3889 0.3922 0.85 0.3889 0.4019 3.34 
-2
10  0.3924 0.3955 0.80 0.3924 0.4044 3.06 
-1.5
10  0.3957 0.3985 0.72 0.3957 0.4060 2.61 
-1
10  0.3989 0.4013 0.60 0.3989 0.4068 1.99 
-0.5
10  0.4020 0.4038 0.44 0.4020 0.4068 1.20 
0
10  0.4050 0.4060 0.25 0.4050 0.4060 0.25 
   
 118 
 
 
Fig. 4.7.5: Velocity Changes within the Film in X-direction at 0.0 &1.4Tesla, X = Absorber 
length. 
 
Fig.4.7.6: % Velocity Changes within the Film in X-direction from 0.0 to 1.4Tesla, X = 
Absorber length. 
 119 
 
 
Fig.4.7.7: Velocity Changes within the Film in X-direction at 0.0 & 3.0 Tesla, X = Absorber 
length. 
 
Fig.4.7.8: % Velocity Changes within the Film in X-direction at 0.0 & 3.0Tesla, X = Absorber 
length. 
 
 
 120 
 
Table 4.4.5: LiCl-H2O Velocity Changes within the film in X-direction, X = Absorber 
length. 
Absorber Interface  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.3620 0.3620 0.00 0.3620 0.3620 0.00 
-5.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-4.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-4
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-3.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-3
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-2.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-2
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-1.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-1
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
-0.5
10  0.3662 0.3671 0.26 0.3662 0.3702 1.10 
0
10  0.4050 0.4060 0.25 0.4050 0.4060 0.25 
  
 
 
 121 
 
 
Fig4.7.9: Velocity Changes at the Interface in X-direction at 0.0 & 1.4 Tesla, X = Absorber 
length. 
 
Fig 4.8: % Velocity Changes at the Interface in X-direction at 0.0 & 1.4Tesla, X = Absorber 
length. 
 
 
 122 
 
 
Fig 4.8.1: Velocity Changes at the Interface in X-direction at 0.0 & 3.0Tesla, X = Absorber 
length. 
 
 
 
Fig 4.8.2: % Velocity Changes at the Interface in X-direction at 0.0 & 3.0 Tesla, X = Absorber 
length. 
 
 123 
 
Table 4.4.6: LiCl-H2O Temperature Changes within the film in X-direction, X = Absorber 
length. 
Absorber Bulk  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  35.00 35.00 0.00 35.00 35.00 0.00 
-5.5
10  34.74 34.74 0.00 34.74 34.74 0.00 
-5
10  34.49 34.49 0.00 34.49 34.49 0.00 
-4.5
10  34.23 34.23 0.00 34.23 34.23 0.00 
-4
10  33.98 33.98 0.00 33.98 33.98 0.00 
-3.5
10  33.73 33.73 0.00 33.73 33.73 0.00 
-3
10  33.48 33.48 0.00 33.48 33.48 0.00 
-2.5
10  33.24 33.24 0.00 33.24 33.24 0.00 
-2
10  32.99 32.99 0.00 32.99 32.99 0.00 
-1.5
10  32.75 32.75 0.00 32.75 32.75 0.00 
-1
10  32.51 32.51 0.00 32.51 32.51 0.00 
-0.5
10  32.27 32.27 0.00 32.27 32.27 0.00 
0
10  30.00 30.00 0.00 30.00 30.00 0.00 
 
 
 124 
 
 
Fig. 4.8.3: Graph of Temperature Changes within the Film in X-direction, X = Absorber length. 
 
 
Table 4.4.7: LiBr-H2O Temperature Changes within the film in X-direction, X = Absorber 
length. 
Absorber Interface  
length X 
(m) Present Present.  % Present Present % 
 Result Result Changes Result Result Changes 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 in X-dir 
Tesla Tesla Tesla Tesla 
-6
0 or 10  35.00 35.00 0.00 35.00 35.00 0.00 
-5.5
10  38.25 38.25 0.00 38.25 38.25 0.00 
-5
10  37.50 37.50 0.00 37.50 37.50 0.00 
-4.5
10  36.75 36.75 0.00 36.75 36.75 0.00 
-4
10  36.00 36.00 0.00 36.00 36.00 0.00 
-3.5
10  35.25 35.25 0.00 35.25 35.25 0.00 
 125 
 
-3
10  34.50 34.50 0.00 34.50 34.50 0.00 
-2.5
10  33.75 33.75 0.00 33.75 33.75 0.00 
-2
10  33.00 33.00 0.00 33.00 33.00 0.00 
-1.5
10  32.25 32.25 0.00 32.25 32.25 0.00 
-1
10  31.50 31.50 0.00 31.50 31.50 0.00 
-0.5
10  30.75 30.75 0.00 30.75 30.75 0.00 
0
10  30.00 30.00 0.00 30.00 30.00 0.00 
 
 
Fig. 4.8.4: Graph of Temperature Changes at the Interface in X-direction, X = Absorber length. 
 
 
 
 
 126 
 
Table 4.4.8: LiCl-H2O Concentration Changes within the film in X-direction, X = Absorber 
length. 
Absorber Bulk  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.450 0.4500 0.00 0.450 0.4500 0.00 
-5.5
10  0.4453 0.4453 0.005 0.4453 0.4454 0.014 
-5
10  0.4406 0.4407 0.011 0.4406 0.4408 0.032 
-4.5
10  0.4360 0.4361 0.018 0.4360 0.4362 0.060 
-4
10  0.4313 0.4314 0.026 0.4313 0.4317 0.090 
-3.5
10  0.4267 0.4268 0.035 0.4267 0.4272 0.124 
-3
10  0.4221 0.4222 0.045 0.4221 0.4227 0.164 
-2.5
10  0.4174 0.4177 0.053 0.4174 0.4183 0.196 
-2
10  0.4128 0.4131 0.061 0.4128 0.4138 0.230 
-1.5
10  0.4082 0.4085 0.069 0.4082 0.4093 0.260 
-1
10  0.4036 0.4039 0.074 0.4036 0.4047 0.283 
-0.5
10  0.3990 0.3993 0.078 0.3990 0.4001 0.293 
0
10  0.3580 0.3580 0.00 0.3580 0.3580 0.00 
 
 
 127 
 
 
Fig.4.8.5: Concentration Distribution within the Film in X-direction at 0.0 & 1.4 Tesla, X = 
Absorber length. 
 
 
Fig.4.8.6: %  Concentration Changes within the Film in X-direction at 0.0 & 1.4Tesla, X = 
Absorber length. 
 128 
 
 
 
 
Fig.4.8.7: Concentration Changes within the Film in X-direction at 0.0 & 3.0Tesla, X = Absorber 
length. 
 
 
Fig.4.8.8: % Concentration Changes within the Film in X-direction at 0.0 & 3.0T, X = Absorber 
length. 
 129 
 
Table 4.4.9: LiCl-H2O Concentration Changes within the film in X-direction 
Absorber Interface  
length X 
(m) Present Present.  % Present Present % Changes 
 Result Result Changes Result Result in X-dir 
@ 0.0 @ 1.4 in X-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
-6
0 or 10  0.450 0.4500 0.00 0.450 0.4500 0.00 
-5.5
10  0.44486 0.44487 0.002 0.44486 0.44491 0.011 
-5
10  0.43978 0.43980 0.005 0.43978 0.43989 0.025 
-4.5
10  0.43476 0.43479 0.007 0.43476 0.43492 0.037 
-4
10  0.42979 0.42984 0.012 0.42979 0.43000 0.049 
-3.5
10  0.42488 0.42494 0.014 0.42488 0.42514 0.061 
-3
10  0.42003 0.42010 0.017 0.42003 0.42034 0.074 
-2.5
10  0.41523 0.41531 0.019 0.41523 0.41559 0.087 
-2
10  0.41049 0.41058 0.022 0.41049 0.41089 0.098 
-1.5
10  0.40580 0.40590 0.025 0.40580 0.40625 0.111 
-1
10  0.40116 0.40128 0.030 0.40116 0.40166 0.125 
-0.5
10  0.39658 0.39671 0.033 0.39658 0.39712 0.136 
0
10  0.35800 0.35800 0.00 0.35800 0.35800 0.000 
 
 
 130 
 
 
Fig.4.8.9: Concentration Changes at the Interface in X-direction from 0.0 to 1.4 Tesla, X = 
Absorber length. 
 
 
Fig.4.9: % Concentration Changes at the Interface in X-direction at 0.0 & 1.4 Tesla, X = 
Absorber length. 
 131 
 
 
 
 
Fig.4.9.1: Concentration Changes at the Interface in X-direction at 0.0 & 3.0Tesla, X = Absorber 
length. 
 
Fig. 4.9.2: % Concentration Changes at the Interface in X-direction from 0.0 to 3.0 Tesla, X = 
Absorber length. 
 
 132 
 
4.5: Numerical Results in the direction of the film thickness ( )   
One may also be interested in the velocity, temperature and concentration distributions in 
the direction of the smooth film thickness when enhanced with magnetic intensity. Below are the 
results of the investigation. The velocity along the liquid film thickness direction at sections 
X=0.25m, 0.50m and X=0.75m in three different magnetic induction intensities are shown in 
Tables 4.5.3 to 4.5.5, 4.6.2 to 4.6.4 and Figures 4.9.3 to 4.10.7 and 4.12.6 to 4.14 respectively. A 
dimensionless parameter of y/ is used as the abscissa. It is seen from Tables 4.5.3 to 4.5.5, 4.6.2 
to 4.6.4 and Figures 4.9.3 to 4.10.7 and 4.12.6 to 4.14 that with the increase of the magnetic 
induction (  ), the velocity increases at X= 0.25m, 0.5m up to at X= 0.75m for both LiBr and 
LiCl-water solution. This could be explained as follows; At the inlet section, absorption just 
began in an intense way and the film thickness is very thin, the absorption enhancement effect by 
magnetic field is obvious, turbulence in direction of thickness therefore are more intense due to 
the above mentioned reasons. The turbulence becomes weaker with the increase of the film 
thickness and the absorption enhancement. Hence, the velocity v is higher in stronger   at 
X=0.50 and 0.75m respectively. Velocity variations along the thickness direction at the three 
selected sections have the tendency of increasing from absorber wall to the vapour-liquid 
interface as established. This indicates that the refrigerants vapour in the two working fluids 
permeates towards the inner solution from the vapour-liquid interface. When the liquid film 
drops, the increase in velocity is slowed down. It shows that the absorption has been weakening 
gradually towards the outlet of the solution resulting in the lower concentration at X= 0.5, 0.75m 
than at X=0.25m as shown in the Tables 4.5.9 to 4.6.1, 4.6.8 to 4.7 and Figures 4.11.1 to 4.12.5 
and 4.14.4 to 4.15.8 
For the case of temperature distributions in LiBr-H2O solution, there was no noticeable 
temperature change in the temperature apart from the sudden temperature rise at y/ = 0.33 
within the bulk at absorber wall length X= 0.25, 0.5 and 0.75m levels, perhaps as a result of high 
inlet temperature than that of absorber wall temperature. However for LiCl-H2O solution, there 
was no sudden up-rise of temperature observed at y/ = 0.33 within the bulk at absorber wall 
length X= 0.25, 0.5 and 0.75m levels like that of the LiBr-H2O solution. This might be due to the 
same inlet temperature as that of the absorber wall temperature. Tables 4.5.6 to 4.5.8, 4.6.5 to 
4.6.7 and Figures 4.10.8 to 4.11 and 4.14.1 to 4.14.3 also show that with the increase of   from 
0.0T to 1.4T and from 0.0T to 3.0T velocity increases gradually at X=0.25, 0.5 and 0.75m level 
 133 
 
of the absorber wall length. Concentration distribution for both LiBr-H2O and LiCl-H2O solution 
also increases as the values of   increase from 0.0T to 1.4T and from 0.0T to 3.0Tesla. It also 
increases from the wall length up to y/ = 0.58 film thickness before undergoing a reduction at 
the interface. Tables 4.5.3 to 4.5.5, 4.6.2 to 4.6.4 and Figures 4.9.3 to 4.10.7 and 4.12.6 to 4.14 
also show the percentage velocity changes in the film thickness direction for both LiBr-H2O and 
LiCl-H2O solution at X=0.25, 0.5 and 0.75m. The percentage velocity changes when   was 
increased from 0.0 to 1.4T also increase from 0.0 suddenly up to 0.85 before gradually dying 
down to 0.71 at the interface. When the magnetic induction was increased from 0.0 to 3.0T, 
percentage velocity change rises from 0.0 suddenly to 2.22 and reduced down to 1.86 at the 
interface. This result confirms the establishment of magnetic field enhancement effect on the two 
working fluids. For the case of Temperature, there was no percentage changes in Y-direction at 
X=0.25m, 0.5m and 0.75m absorber length for the two working fluids.  Tables 4.5.9 to 4.6.1, 
4.6.8 to 4.7 and Figures 4.11.1 to 4.12.5 and 4.14.4 to 4.15.8 show the percentage concentration 
changes in the film thickness ( ) direction (Y) at X=0.25, 0.5 and 0.75m level for both LiBr–
H2O and LiCl–H2O solutions. At X=0.25m level percentage concentration changes for LiBr-H2O 
solution when magnetic induction was increased from 0.0 to 1.4T rises from 0.00 suddenly to 
0.67 due to the intense of the absorption rate as a result of the magnetic field force and gradually 
reduced to 0.26 at the interface while that of 0.0T to 3.0T also suddenly rises from 0.0 to 1.77as a 
result of the same above mentioned reasons and gradually dying down to 0.67 at the interface. A 
closer look at the Tables shows that the absorption rate increases tremendously when the 
magnetic induction increase from 0.0 to 1.4T and up to 3.0T respectively. Thus the result 
establishes positive influence of the magnetic field enhancement on the working fluids. This 
same remark also applies to the LiCl-H2O solution. For Temperature distribution in LiCl–H2O 
solution shown in Tables 4.5.6 to 4.5.8, 4.6.5 to 4.6.7 and Figures 4.10.8 to 4.11 and 4.14.1 to 
4.14.3, there was no percentage Temperature change. This, in addition to the earlier mentioned 
reason might also be due to non-inclusion of heat in the magnetism effect. Tables 4.5 to 4.5.2 
summarized the COP analysis of both the LiBr and LiCl solutions absorption Refrigeration 
systems.   
 
 
 
 
 134 
 
 
 
 
4.5.1: COP analysis of LiBr and LiCl solutions absorption Refrigeration systems 
 
Coefficient of Performance (COP) for LiBr-water absorption Refrigeration 
From Table 4.2.9 above 
0.549
At 0.0 Tesla, COP =       = 0.9152 
0.600
0.5501
At 1.4 Tesla, COP =     = 0.9166 
0.600
0.551
At 3.0 Tesla, COP =       = 0.918 
0.600
Increment at 1.4 Tesla= 0.9166 – 0.9152 = 0.0014 = 0.1%  
Increment at 3.0 Tesla = 0.9186 – 0.9152 = 0.0034 = 0.3%  
 
Coefficient of Performance (COP) for LiCl-water absorption Refrigeration 
From Table 4.3.6 above 
0.339
At 0.0 Tesla, COP =      = 0.8867 
0.450
0.3993
At 1.4 Tesla, COP=     = 0.8873 
0.450
0.400
At 3.0 Tesla, COP =      = 0.8890 
0.450
Increment at 1.4Tesla = 0.8873 – 0.8867 = 0.0006 = 0.1% 
Increment at 3.0Tesla = 0.889 – 0.887 = 0.002       = 0.2% 
 
 
 
 
 
 
 
 
 
 
 
 135 
 
 
 
 
Comparison of COP increment of LiBr and LiCl Solutions at 1.4 and 3.0Tesla 
 
Table 4.5: Percentage increment of coefficient of Performance (COP) @ 1.4 Tesla 
 LiBr Solution % increment LiCl Solution % increment 
of COP at 1.4 Tesla of COP at 1.4 Tesla 
 0.1 0.1 
  
Table 4.5.1: Percentage increment of coefficient of Performance (COP) @ 3.0 Tesla 
 LiBr Solution % increment LiCl Solution % increment 
of COP at 3.0 Tesla of COP at 3.0 Tesla 
 0.3 0.2 
 
Table 4.5.2: LiBr & LiCl   COP increments at 3.0 Tesla T-Test analysis 
 
Independent Samples Test 
  Levene's Test 
for Equality of 
Variances t-test for Equality of Means 
  
Sig. 95% Confidence Interval of 
(2- Mean Std. Error the Difference 
  F Sig. t df tailed) Difference Difference Lower Upper 
Percentage Equal 
Increment of variances 113.293 .000 1.123E16 24 .000 .1000 .0000 .1000 .1000 
COP @ 3.0 assumed 
Tesla 
Equal   
variances 
1.123E16 23.361 .000 .1000 .0000 .1000 .1000 
not 
assumed 
 
 
 
 
 
 
 136 
 
4.5.3: Tabular and Graphical representation of the results in the direction of film thickness  
Table.4.5.3: LiBr-H20: Velocity Changes in the direction of film thickness ( ) @ X=0.25m 
                                  BULK         
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.00 0.00 0.0000 0.0000 0.00 
0.08 0.0113 0.0116 2.72 0.0113 0.0127 12.24 
0.17 0.0222 0.0228 2.63 0.0222 0.0248 11.84 
0.25 0.0325 0.0333 2.55 0.0325 0.0362 11.48 
0.33 0.0421 0.0431 2.49 0.0421 0.0468 11.18 
0.42 0.0508 0.0520 2.43 0.0579 0.0563 10.93 
0.50 0.0585 0.0599 2.39 0.0585 0.0648 10.73 
0.58 0.0650 0.0666 2.35 0.0650 0.0719 10.56 
0.67 0.0704 0.0720 2.33 0.0704 0.0777 10.43 
0.75 0.0744 0.0762 2.30 0.0744 0.0821 10.34 
0.83 0.0772 0.0789 2.29 0.0771 0.0851 10.29 
0.92 0.0785 0.0803 2.29 0.0785 0.0866 10.26 
1.00 0.0785 0.0803 2.29 0.0785 0.0866 10.26 
 
 
 
 
 
 137 
 
 
Fig.4.9.3: Graph of Velocity Changes in the Direction of Film Thickness at X=0.25m 
 
 
Fig. 4.9.4: Graph of % Velocity Changes in the Direction of Film Thickness at X= 0.25m 
 138 
 
 
Fig.4.9.5: Graph of Velocity Changes in the Direction of Film Thickness at X= 0.25m 
 
 
Fig.4.9.6: Graph of % Velocity Changes in the Direction of Film Thickness at X=0.25m 
 
 139 
 
 
Fig.4.9.7: Graph of Velocity Distribution in the Direction of Film Thickness at X=0.25m 
 
 
 
Table 4.5.4: LiBr-H20: Velocity Changes within the direction of film thickness ( ) at 
X=0.50m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.000 0.0000 0.00 
0.08 0.0113 0.0116 2.72 0.0113 0.0126 12.24 
0.17 0.0222 0.0228 2.63 0.0222 0.0248 11.83 
0.25 0.0325 0.0330 2.55 0.0325 0.0362 11.48 
0.33 0.0421 0.0431 2.49 0.0421 0.0468 11.18 
0.42 0.0508 0.0520 2.43 0.0508 0.0564 10.93 
 140 
 
0.50 0.0585 0.0599 2.39 0.0585 0.0647 10.72 
0.58 0.0650 0.0666 2.35 0.0650 0.0719 10.56 
0.67 0.0704 0.0720 2.33 0.0704 0.0777 10.43 
0.75 0.0744 0.0762 2.31 0.0744 0.0821 10.34 
0.83 0.0772 0.0789 2.29 0.0772 0.0851 10.29 
0.92 0.0785 0.0803 2.29 0.0785 0.0866 10.26 
1.00 0.0785 0.0803 2.29 0.0785 0.0866 10.26 
 
 
Fig.4.9.8: Graph of Velocity of Changes within the Film in the Direction of Film Thickness at 
X=0.50m 
 
 
 141 
 
 
Fig.4.9.9: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
X= 0.50m 
 
 
Fig.4.10: Graph of  Velocity Changes within the Film in the Direction of  Film Thickness at 
X=0.50m 
 142 
 
 
 
Fig.4.10.1: Graph of % Velocity Change within the Film in the Direction of Film Thickness at 
X=0.50m 
 
 
Fig.4.10.2: Graph of Velocity Distribution in the Direction of Film Thickness at X =0.50m 
 
 
 
 143 
 
Table 4.5.5: LiBr-H20: Velocity Changes within the direction of film thickness ( ) at 
X=0.75m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.000 0.0000 0.00 
0.08 0.0113 0.0116 2.72 0.0113 0.0126 12.24 
0.17 0.0222 0.0228 2.63 0.0222 0.0248 11.83 
0.25 0.0325 0.0330 2.55 0.0325 0.0362 11.48 
0.33 0.0421 0.0431 2.49 0.0421 0.0468 11.18 
0.42 0.0508 0.0520 2.43 0.0508 0.0564 10.93 
0.50 0.0585 0.0599 2.39 0.0585 0.0647 10.72 
0.58 0.0650 0.0666 2.35 0.0650 0.0719 10.56 
0.67 0.0704 0.0720 2.33 0.0704 0.0777 10.43 
0.75 0.0744 0.0762 2.31 0.0744 0.0821 10.34 
0.83 0.0772 0.0789 2.29 0.0772 0.0851 10.29 
0.92 0.0785 0.0803 2.29 0.0785 0.0866 10.26 
1.00 0.0785 0.0803 2.29 0.0785 0.0866 10.26 
 
 
 144 
 
 
Fig.4.10.3: Graph of Velocity Changes within the Film in the direction of Film Thickness at X= 
0.75m 
 
 
Fig.4.10.4: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
X= 0.75m 
 145 
 
 
Fig.4.10.5: Graph of Velocity Changes within the Film in the Direction of Film Thickness at 
X=0.75m 
 
 
 
Fig.4.10.6: Graph of % Velocity Change within the Film in the Direction of Film Thickness at 
X=0.75m 
 
 
 146 
 
 
Fig.4.10.7: Graph of Velocity Distribution in the Direction of Film Thickness at X=0.75m 
 
 
 
Table 4.5.6: LiBr-H20: Temperature Changes in the direction of film thickness ( ) at 
X=0.25m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 35.00 35.00 0.00 35.00 35.00 0.00 
0.08 35.19 35.19 0.00 35.19 35.19 0.00 
0.17 35.19 35.19 0.00 35.19 35.19 0.00 
0.25 35.19 35.19 0.00 35.19 35.19 0.00 
0.33 37.55 37.55 0.00 37.55 37.55 0.00 
0.42 35.19 35.19 0.00 35.19 35.19 0.00 
 147 
 
0.50 35.19 35.19 0.00 35.19 35.19 0.00 
0.58 35.19 35.19 0.00 35.19 35.19 0.00 
0.67 35.19 35.19 0.00 35.19 35.19 0.00 
0.75 35.19 35.19 0.00 35.19 35.19 0.00 
0.83 35.19 35.19 0.00 35.19 35.19 0.00 
0.92 35.19 35.19 0.00 35.19 35.19 0.00 
1.00 35.19 35.19 0.00 35.19 35.19 0.00 
 
 
Fig.4.10.8: Graph of Temperature Changes within the Film in the Direction of Film Thickness at 
X=0.25m 
 
 
 
 
 
 148 
 
 
Table 4.5.7: LiBr-H20: Temperature Changes in the direction of film thickness ( ) at 
X=0.50m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 35.00 35.00 0.00 35.00 35.00 0.00 
0.08 35.13 35.13 0.00 35.13 35.13 0.00 
0.17 35.13 35.13 0.00 35.13 35.13 0.00 
0.25 35.13 35.13 0.00 35.13 35.13 0.00 
0.33 36.70 36.70 0.00 36.70 36.70 0.00 
0.42 35.13 35.13 0.00 35.13 35.13 0.00 
0.50 35.13 35.13 0.00 35.13 35.13 0.00 
0.58 35.13 35.13 0.00 35.13 35.13 0.00 
0.67 35.13 35.13 0.00 35.13 35.13 0.00 
0.75 35.13 35.13 0.00 35.13 35.13 0.00 
0.83 35.13 35.13 0.00 35.13 35.13 0.00 
0.92 35.13 35.13 0.00 35.13 35.13 0.00 
1.00 35.13 35.13 0.00 35.13 35.13 0.00 
 
 
 149 
 
 
Fig.4.10.9: Graph of Temperature Changes within the Film in the Direction of Film Thickness at 
X= 0.50m 
 
Table 4.5.8: LiBr-H20: Temperature Changes in the direction of film thickness ( ) at 
X=0.75m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 35.00 35.00 0.00 35.00 35.00 0.00 
0.08 35.06 35.06 0.00 35.06 35.06 0.00 
0.17 35.06 35.06 0.00 35.06 35.06 0.00 
0.25 35.06 35.06 0.00 35.06 35.06 0.00 
0.33 35.85 35.85 0.00 35.85 35.85 0.00 
0.42 35.06 35.06 0.00 35.06 35.06 0.00 
0.50 35.06 35.06 0.00 35.06 35.06 0.00 
 150 
 
0.58 35.06 35.06 0.00 35.06 35.06 0.00 
0.67 35.06 35.06 0.00 35.06 35.06 0.00 
0.75 35.06 35.06 0.00 35.06 35.06 0.00 
0.83 35.06 35.06 0.00 35.06 35.06 0.00 
0.92 35.06 35.06 0.00 35.06 35.06 0.00 
1.00 35.06 35.06 0.00 35.06 35.06 0.00 
 
 
Fig.4.11: Graph of Temperature Changes within the Film in the Direction of Film Thickness at 
X=0.75m 
 
 
 
 
 
 
 
 151 
 
Table 4.5.9: LiBr-H20: Concentration Changes in the direction of film thickness ( ) at 
X=0.25m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.0000 0.0000 0.00 
0.08 0.0859 0.0879 2.33 0.0859 0.0948 10.31 
0.17 0.1483 0.1512 1.96 0.1483 0.1611 8.65 
0.25 0.1949 0.1982 1.71 0.1949 0.2095 7.47 
0.33 0.2303 0.2338 1.52 0.2303 0.2455 6.60 
0.42 0.2575 0.2610 1.38 0.2575 0.2728 5.96 
0.50 0.2784 0.2819 1.27 0.2784 0.2936 5.48 
0.58 0.2943 0.2978 1.18 0.2943 0.3093 5.11 
0.67 0.3061 0.3095 1.13 0.3061 0.3209 4.85 
0.75 0.3145 0.3179 1.08 0.3145 0.3291 4.67 
0.83 0.3198 0.3232 1.06 0.3198 0.3344 4.55 
0.92 0.3224 0.3258 1.04 0.3224 0.3369 4.50 
1.00 0.3224 0.3258 1.05 0.3224 0.3369 4.50 
 152 
 
 
Fig.4.11.1: Graph of Concentration Changes within the Film in the direction of Film Thickness 
at X= 0.25m 
 
 
Fig.4.11.2: Graph of % Concentration Changes within the Film in the direction of Film 
Thickness at X= 0.25m 
 153 
 
 
Fig.4.11.3: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.25m 
 
Fig.4.11.4: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at X= 0.25m 
 154 
 
 
 
Fig.4.11.5: Graph of Concentration Distribution in the Direction of Film Thickness at X=0.25m 
 
Table 4.6: LiBr-H20: Concentration Changes in the direction of film thickness ( ) at 
X=0.50m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.0000 0.0000 0.00 
0.08 0.0123 0.0129 4.72 0.0123 0.0150 21.72 
0.17 0.0367 0.0381 3.96 0.0367 0.0433 18.04 
0.25 0.0633 0.0655 3.46 0.0633 0.0731 15.50 
0.33 0.0884 0.0911 3.06 0.0884 0.1005 13.64 
0.42 0.1105 0.1135 2.77 0.1105 0.1240 12.27 
 155 
 
0.50 0.1291 0.1324 2.55 0.1291 0.1437 11.25 
0.58 0.1443 0.1478 2.38 0.1443 0.1595 10.49 
0.67 0.1561 0.1597 2.26 0.1561 0.1717 9.94 
0.75 0.1648 0.1684 2.19 0.1648 0.1806 9.56 
0.83 0.1705 0.1741 2.13 0.1705 0.1863 9.32 
0.92 0.1732 0.1769 2.10 0.1732 0.1892 9.20 
1.00 0.1732 0.1769 2.10 0.1732 0.1892 9.20 
 
 
Fig.4.11.6: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.50m 
 156 
 
 
Fig.4.11.7: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 1.4Tesla 
 
Fig.4.11.8: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.50m 
 157 
 
 
Fig.4.11.9: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 3.0Tesla 
 
Fig.4.12: Graph of Concentration Distribution in the Direction of Film Thickness at X=0.50m 
 158 
 
 
Table 4.6.1: LiBr-H20: Concentration Changes in the direction of film thickness ( )  
X=0.75m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.0000 0.0000 0.00 
0.08 0.0018 0.0019 6.82 0.0018 0.0024 34.09 
0.17 0.0091 0.0096 6.08 0.0091 0.0116 28.29 
0.25 0.0206 0.0216 5.21 0.0206 0.0255 24.14 
0.33 0.0339 0.0355 4.63 0.0339 0.0411 21.17 
0.42 0.0474 0.0494 4.18 0.0474 0.0564 18.97 
0.50 0.0599 0.0622 3.86 0.0599 0.0703 17.35 
0.58 0.0708 0.0733 3.60 0.0708 0.0822 16.14 
0.67 0.0796 0.0824 3.42 0.0796 0.0918 15.28 
0.75 0.0864 0.0892 3.28 0.0864 0.0990 14.67 
0.83 0.0909 0.0938 3.21 0.0909 0.1038 14.30 
0.92 0.0931 0.0960 3.16 0.0931 0.1062 14.11 
1.00 0.0931 0.0960 3.17 0.0931 0.1062 14.11 
  
 
 159 
 
 
Fig.4.12.1: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.75m 
 
 
Fig.4.12.2: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 1.4Tesla 
 160 
 
 
Fig.4.12.3: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.75m 
 
 
Fig.4.12.4: Graph of % Concentration Changes within the Film Thickness at 0.0 & 3.0 Tesla 
 
 161 
 
 
Fig.4.12.5: Graph of Concentration Distribution in the Direction of Film Thickness at X=0.75m 
 
 
Table 4.6.2: LiCl-H20: Velocity Changes in the direction of film thickness ( ) at X=0.25m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.000 0.0000 0.00 
0.08 0.0113 0.0115 1.88 0.0113 0.0122 8.69 
0.17 0.0222 0.0226 1.82 0.0222 0.0240 8.40 
0.25 0.0325 0.0331 1.76 0.0325 0.0351 8.15 
0.33 0.0421 0.0428 1.72 0.0421 0.0454 7.94 
0.42 0.0508 0.0516 1.68 0.0508 0.0547 7.76 
 162 
 
0.50 0.0585 0.0594 1.65 0.0585 0.0629 7.62 
0.58 0.0650 0.0661 1.62 0.0650 0.0699 7.50 
0.67 0.0704 0.0715 1.60 0.0704 0.0756 7.41 
0.75 0.0744 0.0756 1.59 0.0744 0.0799 7.35 
0.83 0.0772 0.0784 1.58 0.0772 0.0828 7.31 
0.92 0.0785 0.0797 1.57 0.0785 0.0842 7.29 
1.00 0.0785 0.0797 1.57 0.0785 0.0842 7.29 
 
 
Fig.4.12.6: Graph of Velocity Changes within the Film in the Direction of Film Thickness at X 
=0.25m 
 
 163 
 
 
Fig.4.12.7: Graph of Velocity Changes within the Film in the Direction of Film Thickness at 0.0 
& 1.4 Tesla 
 
Fig.4.12.8: Graph of Velocity Changes within the Film in the Direction of Film Thickness at 
X=0.25m 
 
 
 164 
 
 
Fig.4.12.9: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
0.0 & 3.0Tesla 
 
 
Fig.4.13: Graph of Velocity Distribution in the Direction of Film Thickness at X=0.25m 
 
 
 165 
 
Table 4.6.3: LiCl-H20: Velocity Changes in the direction of film thickness ( ) at X=0.50m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.000 0.0000 0.00 
0.08 0.0113 0.0115 1.88 0.0113 0.0122 8.70 
0.17 0.0222 0.0226 1.82 0.0222 0.0240 8.40 
0.25 0.0325 0.0331 1.76 0.0325 0.0351 8.15 
0.33 0.0421 0.0428 1.71 0.0421 0.0454 7.94 
0.42 0.0508 0.0517 1.68 0.0508 0.0547 7.76 
0.50 0.0585 0.0594 1.65 0.0585 0.0629 7.62 
0.58 0.0650 0.0661 1.62 0.0650 0.0699 7.50 
0.67 0.0704 0.0715 1.60 0.0704 0.0756 7.41 
0.75 0.0744 0.0756 1.59 0.0744 0.0799 7.35 
0.83 0.0772 0.0784 1.58 0.0772 0.0828 7.31 
0.92 0.0785 0.0797 1.57 0.0785 0.0842 7.29 
1.00 0.0785 0.0797 1.57 0.0785 0.0842 7.29 
 
 166 
 
 
Fig.4.13.1: Graph of Velocity Changes within the Film in the Direction of Film Thickness at 
X=0.50m 
 
 
 
Fig.4.13.2: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
0.0 & 1.4Tesla 
 167 
 
 
Fig.4.13.3: Graph of Velocity Changes within the Film in the Direction of Film Thickness at 
X=0.50m 
 
 
 
Fig.4.13.4: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
0.0 & 3.0 Tesla 
 
 168 
 
 
Fig.4.13.5: Graph of Velocity Distribution in the Direction of Film Thickness at X=0.50m 
 
 
Table 4.6.4: LiCl-H20: Velocity Changes in the direction of film thickness ( ) at X=0.75m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.000 0.0000 0.00 0.000 0.0000 0.00 
0.08 0.0113 0.0115 1.88 0.0113 0.0122 8.70 
0.17 0.0222 0.0226 1.82 0.0222 0.0240 8.40 
0.25 0.0325 0.0331 1.76 0.0325 0.0351 8.15 
0.33 0.0421 0.0428 1.71 0.0421 0.0454 7.94 
0.42 0.0508 0.0517 1.68 0.0508 0.0547 7.76 
 169 
 
0.50 0.0585 0.0594 1.65 0.0585 0.0629 7.62 
0.58 0.0650 0.0661 1.62 0.0650 0.0699 7.50 
0.67 0.0704 0.0715 1.60 0.0704 0.0756 7.41 
0.75 0.0744 0.0756 1.59 0.0744 0.0799 7.35 
0.83 0.0772 0.0784 1.58 0.0772 0.0828 7.31 
0.92 0.0785 0.0797 1.57 0.0785 0.0842 7.29 
1.00 0.0785 0.0797 1.57 0.0785 0.0842 7.29 
 
 
Fig.4.13.6: Graph of Velocity Changes within the Film in the Direction of Film Thickness at 
X=0.75m 
 
 170 
 
 
Fig.4.13.7: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
0.0 & 1.4 Tesla 
 
 
Fig.4.13.8: Graph of Velocity Changes within the Film in the direction of Film Thickness at 
X=0.75m  
 
 
 171 
 
 
Fig.4.13.9: Graph of % Velocity Changes within the Film in the Direction of Film Thickness at 
0.0 & 3.0Tesla 
 
 
Fig.4.14: Graph of Velocity Distribution in the Direction of Film Thickness at 0.75m 
 
 
 
 
 
 172 
 
Table 4.6.5: LiCl-H20: Temperature Changes in the direction of film thickness ( ) at 
X=0.25m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 35.00 35.00 0.00 35.00 35.00 0.00 
0.08 30.17 30.17 0.00 30.17 30.17 0.00 
0.17 30.17 30.17 0.00 30.17 30.17 0.00 
0.25 30.17 30.17 0.00 30.17 30.17 0.00 
0.33 30.17 30.17 0.00 30.17 30.17 0.00 
0.42 30.17 30.17 0.00 30.17 30.17 0.00 
0.50 30.17 30.17 0.00 30.17 30.17 0.00 
0.58 30.17 30.17 0.00 30.17 30.17 0.00 
0.67 30.17 30.17 0.00 30.17 30.17 0.00 
0.75 30.17 30.17 0.00 30.17 30.17 0.00 
0.83 30.17 30.17 0.00 30.17 30.17 0.00 
0.92 30.17 30.17 0.00 30.17 30.17 0.00 
1.00 32.25 32.25 0.00 32.25 32.25 0.00 
 
 
 
 
 
 173 
 
 
Fig.4.14.1: Graph of Temperature Distribution in the Direction of Film Thickness at X=0.25m 
 
Table 4.6.6: LiCl-H20: Temperature Changes in the direction of film thickness ( ) at 
X=0.50m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 35.00 35.00 0.00 35.00 35.00 0.00 
0.08 30.11 30.11 0.00 30.11 30.11 0.00 
0.17 30.11 30.11 0.00 30.11 30.11 0.00 
0.25 30.11 30.11 0.00 30.11 30.11 0.00 
0.33 30.11 30.11 0.00 30.11 30.11 0.00 
0.42 30.11 30.11 0.00 30.11 30.11 0.00 
0.50 30.11 30.11 0.00 30.11 30.11 0.00 
 174 
 
0.58 30.11 30.11 0.00 30.11 30.11 0.00 
0.67 30.11 30.11 0.00 30.11 30.11 0.00 
0.75 30.11 30.11 0.00 30.11 30.11 0.00 
0.83 30.11 30.11 0.00 30.11 30.11 0.00 
0.92 30.11 30.11 0.00 30.11 30.11 0.00 
1.00 31.50 31.50 0.00 31.50 31.50 0.00 
 
 
Fig.4.14.2: Graph of Temperature Changes within the Film in the Direction of Film Thickness at 
X=0.50m 
 
 
 
 
 
 
 175 
 
Table 4.6.7 LiCl-H20: Temperature Changes in the direction of film thickness ( ) at 
X=0.75m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 35.00 35.00 0.00 35.00 35.00 0.00 
0.08 30.05 30.05 0.00 30.05 30.05 0.00 
0.17 30.05 30.05 0.00 30.05 30.05 0.00 
0.25 30.05 30.05 0.00 30.05 30.05 0.00 
0.33 30.05 30.05 0.00 30.05 30.05 0.00 
0.42 30.05 30.05 0.00 30.05 30.05 0.00 
0.50 30.05 30.05 0.00 30.05 30.05 0.00 
0.58 30.05 30.05 0.00 30.05 30.05 0.00 
0.67 30.05 30.05 0.00 30.05 30.05 0.00 
0.75 30.05 30.05 0.00 30.05 30.05 0.00 
0.83 30.05 30.05 0.00 30.05 30.05 0.00 
0.92 30.05 30.05 0.00 30.05 30.05 0.00 
1.00 30.75 30.75 0.00 30.75 30.75 0.00 
 
 
 
 
 176 
 
 
Fig.4.14.3: Graph of Temperature Changes within the Film in the Direction of Film Thickness at 
X=0.75m 
 
Table 4.6.8: LiCl-H20: Concentration Changes in the direction of film thickness ( ) at 
X=0.25m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.0000 0.0000 0.00 
0.08 0.0593 0.0603 1.64 0.0593 0.0638 7.47 
0.17 0.1034 0.1050 1.39 0.1034 0.1101 6.35 
0.25 0.1369 0.1386 1.22 0.1369 0.1445 5.53 
0.33 0.1627 0.1645 1.09 0.1627 0.1707 4.92 
0.42 0.1827 0.1845 0.99 0.1827 0.1909 4.47 
 177 
 
0.50 0.1982 0.1999 0.91 0.1982 0.2063 4.12 
0.58 0.2100 0.2118 0.86 0.2100 0.2181 3.86 
0.67 0.2186 0.2206 0.81 0.2186 0.2269 3.67 
0.75 0.2251 0.2269 0.77 0.2251 0.2331 3.55 
0.83 0.2292 0.2309 0.77 0.2292 0.2371 3.46 
0.92 0.2311 0.2329 0.76 0.2311 0.2391 3.42 
1.00 0.2311 0.2329 0.76 0.2311 0.2390 3.42 
 
 
Fig.4.14.4: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.25m 
 178 
 
 
Fig.4.14.5: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 1.4Tesla 
 
 
Fig.4.14.6: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.25m 
 179 
 
 
Fig.4.14.7: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 3.0Tesla 
 
 
Fig.4.14.8: Graph of Concentration Distribution in the Direction of Film Thickness at X=0.25m 
 180 
 
 
Table 4.6.9: LiCl-H20: Concentration Changes in the direction of film thickness ( ) at 
X=0.50m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.0000 0.0000 0.00 
0.08 0.0078 0.0081 3.33 0.0078 0.0090 15.47 
0.17 0.0238 0.0244 2.82 0.0238 0.0269 13.13 
0.25 0.0417 0.0427 2.45 0.0417 0.0464 11.35 
0.33 0.0588 0.0601 2.19 0.0588 0.0648 10.10 
0.42 0.0742 0.0756 2.00 0.0742 0.0809 9.14 
0.50 0.0872 0.0888 1.83 0.0872 0.0946 8.43 
0.58 0.0979 0.0997 1.73 0.0979 0.1057 7.88 
0.67 0.1064 0.1082 1.64 0.1064 0.1144 7.49 
0.75 0.1126 0.1144 1.58 0.1126 0.1207 7.20 
0.83 0.1167 0.1185 1.54 0.1167 0.1249 7.04 
0.92 0.1187 0.1205 1.53 0.1187 0.1269 6.95 
1.00 0.1187 0.1205 1.52 0.1187 0.1269 6.94 
 181 
 
 
Fig.4.14.9: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.50m 
 
Fig.4.15: Graph of % Concentration Changes within the Film in the Direction of Film Thickness 
at 0.0 & 1.4Tesla 
 182 
 
 
Fig.4.15.1: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.50m 
 
Fig.4.15.2: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 3.0Tesla 
 
 183 
 
 
Fig.4.15.3: Graph of Concentration Distribution in the Direction of Film Thickness at X=0.50m 
 
Table 4.7: LiCl-H20: Concentration Changes in the direction of film thickness ( ) at 
X=0.75m 
                                          BULK 
Y/  Present Present.  % Present Present % Changes 
Result Result Changes Result Result in Y-dir 
@ 0.0 @ 1.4 in Y-dir @ 0.0 @3.0 
Tesla Tesla Tesla Tesla 
0.00 0.0000 0.0000 0.00 0.0000 0.0000 0.00 
0.08 0.0010 0.0011 4.85 0.0010 0.0013 24.27 
0.17 0.0055 0.0057 4.21 0.0055 0.0066 20.33 
0.25 0.0127 0.0131 3.71 0.0127 0.0149 17.52 
0.33 0.0213 0.0220 3.29 0.0213 0.0246 15.52 
0.42 0.0301 0.0310 3.02 0.0301 0.0343 14.03 
 184 
 
0.50 0.0384 0.0395 2.76 0.0384 0.0434 12.89 
0.58 0.0457 0.0469 2.58 0.0457 0.0512 12.05 
0.67 0.0518 0.0530 2.45 0.0518 0.0577 11.42 
0.75 0.0563 0.0577 2.38 0.0563 0.0625 11.01 
0.83 0.0594 0.0608 2.32 0.0594 0.0658 10.74 
0.92 0.0610 0.0624 2.28 0.0610 0.0674 10.60 
1.00 0.0610 0.0624 2.30 0.0610 0.0674 10.62 
 
 
Fig.4.15.4: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.75m 
 
 185 
 
 
Fig.4.15.5: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 1.4Tesla 
 
Fig.4.15.6: Graph of Concentration Changes within the Film in the Direction of Film Thickness 
at X=0.75m 
 186 
 
 
Fig.4.15.7: Graph of % Concentration Changes within the Film in the Direction of Film 
Thickness at 0.0 & 3.0Tesla 
 
 
Fig.4.15.8: Graph of Concentration Distribution in the Direction of Film Thickness at 0.75m 
 
 
 187 
 
CHAPTER FIVE 
CONCLUSIONS AND RECOMMENDATIONS 
 
5.1: Conclusions 
  
 The absorption process modeling of a smooth thin-liquid falling-film in a cooling system 
using Lithium bromide-water and Lithium chloride-water solutions in a magnetic field 
enhancement medium has been undertaken. The physical model for the magnetic field 
enhancement of the absorption cooling-system using lithium bromide water and lithium chloride 
water has been developed. The ensuing equations were developed from the conservation laws for 
mass, momentum, energy, concentration or species and the mass transport equations. The 
changes in physical properties of the two working fluids in the absorption process, the variation 
of falling film within the smooth thin-liquid film thickness along falling and the convection in 
the direction of thickness of liquid film were considered in the modeling. Distributions of 
parameters in falling-film absorption, such as velocity, temperature and concentration in the 
application of magnetic field were obtained. The distributions of the parameters were obtained in 
both the direction of falling-film and across the film thickness.   
 The numerical results obtained show that magnetic field can improve the performance of 
lithium bromide-water and lithium chloride-water falling film absorption, and the absorption 
strengthening effect increases with the enhancement of magnetic intensity. In this work the 
strengthening effect is limited within the magnetic field intensity of 0-3Tesla, but there are trends 
of absorption strengthening effect increasing more in stronger magnetic fields. Macroscopic 
magnetic field force was introduced in the mathematical model to reflect the influence of 
magnetic field on lithium bromide water and lithium chloride water absorption, while the 
microcosmic impact of magnetic field was not considered in this work.  
 
 
 
 
 
 
 
 188 
 
5.2: Recommendations  
 This work has limited its scope of study to only two working fluid-pairs i.e lithium 
bromide and lithium chloride solutions out of numerous available absorption working fluids 
pairs. A few of such additional working fluid pairs are Ammonia-water solution, HFC-
dimethylethylenurea [DMEU], NH3-SrCl2, ethylene glycol solution and Ammonia-water sodium 
hydroxide mixtures, although the literature survey indicated that of all these absorption working-
fluid pairs, the working fluid pairs used in the present study were rated the best. It has also 
limited its scope in the adopted solution method used to only finite difference method out of all 
the available numerical methods such as finite element method, finite volume, boundary element 
method, Monte Carlo technique, vortex method etc; due to the fact that finite difference had been 
proved reliable from the literature and quite adequate for the flow under consideration.  
Out of the three kinds of methods of absorption enhancement and five different 
categories of researchable areas on nano-fluids/nano-particles in absorbent form of absorption 
enhancement study which falls under the third kind of enhancement, magnetic field form of 
enhancement is seldom mentioned in the literature, this work has limited its scope to this seldom 
mentioned form of absorption enhancement in the investigation. 
With earlier mentioned objectives of this work, the followings recommendations are being made. 
 The model developed in this work for the magnetic field enhancement of the absorption 
cooling-system using lithium bromide-water and lithium chloride-water has been 
established and therefore recommended for usage in investigating the magnetic field 
effect on other available working fluids in absorption refrigeration. 
 Improvement in the performance of lithium bromide-water and lithium chloride-water 
falling film absorption, and absorption strengthening increasing with the enhancement of 
magnetic induction intensity has also been established. 
 Positive effect of macroscopic magnetic field force on the two working fluids used was 
also demonstrated. 
 The Coefficient of Performance (COP) of LiBr and LiCl solutions absorption 
refrigeration systems was established to have increased by 0.3% and 0.2%, respectively, 
when magnetic induction was 3.0 Tesla.      
 
 
 189 
 
The following areas are therefore worthy of further research work: 
 Experimental validation of the developed model for the magnetic field enhancement of 
the absorption cooling-system using lithium bromide-water and lithium chloride-water 
 Investigation of microcosmic impact of magnetic field on lithium bromide-water and 
lithium chloride-water absorption refrigeration. 
 Development of magnetic field enhancement absorption process model using any other 
absorption refrigeration working fluids apart from lithium bromide, lithium chloride-
water and ammonia-water solutions. 
 Adoption of any other numerical solution method such as finite element method, finite 
volume, boundary element method, Monte Carlo technique, vortex method etc on 
absorption enhancement modeling study.     
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 190 
 
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 196 
 
 
APPENDICES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 197 
 
APPENDIX A  
 
DEVELOPMENT OF THE MODEL EQUATIONS 
 
a. Development of Magnetic field enhanced velocity field equation in smooth thin 
liquid falling film using mass transport relationship 
 
in
entum
 
Mom
by flo
w
ibutio
n
 distr
 y
ity
Veloc x
y
Vx ( y )
tum in
 by
omenM rt
us tra
nspo
visco
m flux
omen
tu
M
tribut
ion
dis Ø
for
ce
y 
 ravityx(y) G
L
 of
Direc
tion
h o y Gra
vity
um ntum 
out b
omen
t e
M Mom
ow  trans
port
t by fl ou
s
ou visc
 
Fig. 1 
Momentum balances over a thin “shell” of fluid for a steady-state flow is given as 
 
{Rate of momentum in} – {rate of momentum out} + {sum of forces acting on systems} = 0  1 
 
From Fig. 1  
 
rate of x- momentum in    (LW)( ( yx ) |y   i 
across surface at y 
 
 rate of x- momentum out    (LW)( ( yx ) |y + y   ii 
across surface at  y + y  
 
rate of x-momentum in     (Wy Vx) VX  |x  = 0  iii 
across surface at x = 0 
 
 198 
 
rate of x-momentum out   (Wy Vx) VX  |x  = L  iv 
across surface at x = L 
 
gravity force acting    (LWy ) g cos   v 
on fluid 
 
Substitute equations (i) to (v) in the momentum balances equation 1  
 
(LW)( ( yx ) |y - (LW)( ( yx ) |y + y + (Wy Vx) VX  |x  - (Wy Vx) VX  |x + (LWy ) g cos  = 0    
Vx is the same at x = 0 as it is at x = L 
 
Dividing through with LWy and taken the limit as  y  approaches zero 
 
 yx | yy  yx | y 
lim     g cos       vi 
 y 
y - 0 
 

 yx  g cos         vii 

 
Integrating 
 
 yyx  g cos C1         viii 
B. C.  1: at y = 0,   yx  0  
 
Substituting  
 
 yx  gycos         ix 
 
For Newtonian fluid, the momentum flux is related to the velocity gradient according to 
 
v
 yx  u
x          x 
y
 
Substitute for   yx  
v  g cos 
Therefore  x   y        xi                                      y   
0
In this model   is 0 , cos0 1 
v  g 
Therefore         x   y        xii                            y   
u  g 
Since vx  u , Therefore    y      xiia y   
 199 
 
gh2 3V
Mean velocity in the liquid film V = oo  or     g 
0  
3 h2o
Substitution for  g  in equation (xiia) 
u V
Therefore  3 O y        xiii 
y h2o
Second derivative 
 2u V
 3 o    Implies 
y 2 h2o
 2u V
 3 o  0          xix 
y 2 h2o
This is the velocity field equation in the thin film. The magnetic force in the falling film is given as mag .  
In this model the magnetic force coupled with the body weight acts in the direction of the falling-film. 
Therefore the complete magnetic enhanced velocity field equation will be  
 2u V
 3 o  g  mag  0          xx 
y 2 h2o
2
 
mag =  is the magnetic force which the falling-film solution experienced per unit volume 
lo
Finally, Magnetic enhanced Model Equation is 
 
 2
2
u V 
 3 o

 g   0        xxi 
y 2 h2 lo o
 
Equation (xxi) is the developed model magnetic field enhanced velocity field equation in the smooth thin 
liquid film  
 
b. Development of Boundary conditions/equations 
 
Temperature 
Fig 3.2, the quantity of heat flowing into the three faces of the half element in time dt is given by 
the following equations:  
 
 200 
 
x
dq ydq
i+1 3 1
dq dq
1 2
i dq
2
dq
i-1 3
x
y
 
Fig. 2 Temperature boundary conditions analysis 
khd T khd (Ti1, j Ti, j )
dq1   = ( ) 1 dt   =  ( ) dt
2 x i 2 2 h  
T (Ti,J 1 Ti, j )
dq2    = khd dt      =    khd dt     (xxii) 
y 1 h
j
2
khd T khd
dq3   =  ( ) dt   =    ( )   
2 x 1i 2
2
Substituting Eq. (xxii) into the heat balance equation for the half element 
 dq  1 + dq2 +dq3   =   0 
yields 
 (Left)  T i1, j + 2T i, j1 + Ti-1,j   - 4T i , j  =  0     (xxiii) 
Following the same procedure above, the difference equations which apply on the right, upper 
and lower could be obtained. 
(Right)  T i1, j + 2T i, j1+ T i1, j - 4T i , j  =   0     (xxiv) 
(Upper) T i, j1  + 2T i 1, j + T i, j1 - 4T i , j  =   0    (xxv) 
 201 
 
(Lower)  T i, j1  + 2T i1, j  + T i, j1 - 4T i , j  =   0     (xxvi) 
Concentration 
x
dm ydm
3 1
i+1
dm dm1 2
i dm
2
dm3
i-1
x
y
 
Fig. 3 Concentration boundary conditions analysis 
The net mass flowing into this element must be equal to zero when considering steady-state 
condition in the liquid-film. The quantity of mass flowing into the three faces of the half-element 
in time dt are given by the following equations.                
hd  hd  i1, j  i, j 
dm1    = ( ) dt  =  ( ) dt    
2 x 1 2 h
i
2
  i, j1  i, j 
dm2   = hd  dt =  hd  dt     (xxvii)  
y 1 h
j
2
hd  hd  i, j  i1, j 
dm3   =   - ( ) dt = - ( ) dt  
2 x 1 2 h
i
2
Substituting Eq. (xxvii) into the mass balance equation for the half-element 
dm1    +   dm2    +  dm3  =   0 
yields    
 202 
 
 (Left)  i1, j  2i, j1 i1, j  4i, j  0       (xxviii) 
Following the same procedure above, the difference equations which apply on the right, upper 
and lower could be obtained. 
(Right)      i1, j  2i, j1 i1, j  4i, j  0       (xxix) 
(Upper)    i, j1  2i1, j i, j1  4i, j  0          (xxx) 
(Lower)    i, j1  2i1, j i, j1  4i, j  0       (xxxi) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 203 
 
 
APPENDIX B  
 
Source Code 
 
C: LiBr-H2O Parameter distribution in the direction of falling 
C: ABSORPTION PROCESS MODELING ON A LIQUID FALLING FILM  
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC                                                                                                                                
C                                                                                                                                       C 
C           Ph.D FINITE DIFFERENCE MODEL                                                   C 
C                                                                                                                              C                                                    
C                             FOR                                                                C 
C                            C                                                                                                                                                                                            
C           ABSORPTION PROCESS MODELING ON A                                     C                                       
C                                                                           C 
C                                                                           C 
C                               LIQUID FALLING FILM                                                    C  
C                C  
C                                                                                       C 
C                                  K.M ODUNFA (44113)                                                             C 
C                                                                                                                                      C                                          
C                                      SUPERVISED BY                                 C 
C                                PROFESSOR FAGBENLE, R O                             C 
C                           C  
C                                        JANUARY 2009                                   C  
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 
 
            COMMON/XL1/AVISC,DENS,TMMSUC, FMAG,AMAGIND,VT,VB,AGRAV 
C   COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(13),YCO(5) 
 
  OPEN(UNIT = 4,FILE = 'FINAL RESULT.OUT') 
 OPEN(UNIT = 6,FILE = 'LiBr Data.IN',STATUS='OLD') 
C 
C         INPUT THE PROBLEM DATA 
C 
 READ(6,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
  2        FORMAT(2I3,I4,F6.2,F6.0,2F6.4,2F6.4,F8.5,F10.7,F5.2) 
            WRITE(*,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
 READ(*,2) 
            READ(6,*)RH,RL 
 105     FORMAT(2(F8.5,1X)) 
            WRITE(*,105)RH,RL 
 READ(6,*)VELM,VELM1,FILMT,AMAGIND,AGRAV 
 204 
 
 102     FORMAT(3(F9.6,1X),F3.1, F5.3) 
            WRITE(*,102)VELM,VELM1,FILMT,AMAGIND, AGRAV  
            READ(6,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
 104     FORMAT(6(F9.5,1X)) 
   WRITE(*,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
  READ(6,*)NEX,NEY 
 205     FORMAT(2(I3,1X)) 
            WRITE(*,205)NEX,NEY 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
      DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101 CONTINUE 
C 
C COMPUTE TEMPERATURE DISTRIBUTION IN THE FILM 
C 
 CALL SOLUTION1 
C 
C COMPUTE CONCENTRATION DISTRIBUTION IN THE FILM 
 
 CALL SOLUTION2 
C 
            STOP  
 END 
C 
 SUBROUTINE VELOCITY 
            COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c    COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(13),YCO(5) 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
 205 
 
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101 CONTINUE 
 
 WRITE(4,3) 
3 FORMAT(47H1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,13 
 DO 4 J=1,5 
 A=I 
4 V(I,J)=VB+(VT-VB)*(A-1.)/12. 
 N=0 
5 K=0 
 DO 16 J=1,5 
 DO 16 I=2,12 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UVEL = VT 
 GO TO 14 
C8 UTEMP = (V(I+1,J)+2.*V(I,J+1)+V(I-1,J))/4. 
8 UVEL = 0.0 
 GO TO 14 
9 IF(J-5)13,10,13 
10 IF(I-IB)12,11,11 
11 UVEL = VT 
 GO TO 14 
c12 UTEMP=(V(I+1,J)+2.*V(I,J-1)+V(I-1,J))/4 
c 
12       V(I,J)= V(I,J+1) 
 GO TO 14 
13 UVEL = ((V(I+1,J)+V(I-1,J)+(3*VELM/FILMT**2)*DY**2 
      .+DY**2*DENS*AGRAV+((DENS*TMMSUC*AMAGIND*AMAGIND) 
     ./0.000001257)*DY**2))/2.0  
C 
14 DIFF=UVEL-V(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 V(I,J)=UVEL 
C     write(*,*)((V(I,J),I=2,11),J=1,5) 
C read(*,*) 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
 206 
 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
21        FORMAT(///10F9.6) 
 DO 22 M= 1,13 
 I=14-M 
22 WRITE(4,21)(V(I,J),J=1,5) 
            RETURN 
            END 
 
C 
 SUBROUTINE SOLUTION1 
 COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(13),YCO(5),U(13,5) 
c S(100) 
C 
 DY= RL/NEY 
  DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
  NNT=NX*NY 
c 
            DO 102 I = 1,NX 
  DO 102 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  102 CONTINUE 
 
C COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
            CALL VELOCITY 
C 
 WRITE(4,3) 
3 FORMAT(47H1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,13 
 DO 4 J=1,5 
 A=I 
4 U(I,J)=UB+(UT-UB)*(A-1.)/12. 
 
 N=0 
5 K=0 
 DO 16 J=1,5 
 DO 16 I=2,12 
 207 
 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UTEMP = UT 
 GO TO 14 
C8 UTEMP = (U(I+1,J)+2.*U(I,J+1)+U(I-1,J))/4. 
8 UTEMP = 35 
 GO TO 14 
9 IF(J-5)13,10,13 
10 IF(I-IB)12,11,11 
11 UTEMP = UT 
 GO TO 14 
12        UTEMP=(U(I-1,J)+0.85) 
 GO TO 14 
C 
            UVEL=V(I,J) 
C 
13 UTEMP=(ALPHA*DX*(U(I+1,J)+U(I-1,J))+(UVEL*DY*DY*U(I-1,J))/ 
      .(2*ALPHA*DX-(UVEL*DY*DY))) 
 
 
14 DIFF=UTEMP-U(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 U(I,J)=UTEMP 
C 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
C 
21 FORMAT(///10F7.2) 
 DO 22 M= 1,13 
 I=14-M 
22 WRITE(4,21)(U(I,J),J=1,5) 
    RETURN 
 END 
C 
 SUBROUTINE SOLUTION2 
 COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10), 
 208 
 
C COMMON/XL4/V(11,5),S(110) 
 DIMENSION XCO(14),YCO(10),C(14,10),CSTORE(50,20),DIFFD(50, 
 &20) 
 Open(unit=273,file='SalauOdunfasolution2.out') 
 
 DY= RL/NEY 
  DX= RH/NEX 
  NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
 
            DO 103 I = 1,NX 
            DO 103 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
 103  CONTINUE 
  
C     COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
 
            CALL VELOCITY 
  
 do 121 i=1,5 
 do 121 j=1,13 
c write(4,*) 'when',v(i,j) 
 121  continue 
 
 WRITE(4,3) 
3 FORMAT(47H1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 
 DO 4 J=1,5 
 DO 4 I=1,13 
 A=I 
C IF(J.EQ.5)GOTO 4 
  C(I,J)=5 
  
4          CONTINUE 
 
 
 
7 DO  116  J=1,5 
 DO  116  I=1,13 
  
 IF ( I.EQ.1) C(I,J)=CT 
 IF(I.EQ.1) GOTO 6 
 IF(I.EQ.13)C(I,J) = CB 
 209 
 
 IF(I.EQ.13) GOTO 6 
C IF(J.EQ.1) C(I,J)=CB+(CT-CB)*(A-1.)/10 
C IF(J.EQ.1) GOTO 14 
 IF(J.EQ.5) CONC= C(I,J)+0.0035 
 IF(J.EQ.5) GOTO 6 
 6 CSTORE(I,J)=C(I,J) 
116      CONTINUE 
 DO 16 J=1,5 
 DO 16 I=2,12 
 Do 275 k=13,1,-1 
            WRITE(273,274)k,(V(k,kj),kj=1,5) 
274 Format(i3,2x,5(f12.5,2x)) 
275 Continue 
c READ(*,*) 
c WRITE(*,*)I,J 
C 
13    c(i,j)=(sdifu*dx*(c(1+1,j)+c(1-1,j))+(v(i,j)*dy*dy*c(i-1,j))/ 
     .(2*sdifu*dx+(v(i,j)*dy*dy))) 
 
c READ(*,*) 
16 CONTINUE 
 DO 22 J=1,5 
 DO 22 I=1,13 
  
 
 DIFFD(I,J)=ABS( CSTORE(I,J)-C(I,J)) 
22  CONTINUE 
 
 DIFF= DIFFD(1,1) 
 DO  12  J =1,5 
 DO  12  I =1,13 
 
 IF(DIFFD(I,J).LT.DIFF)DIFF=DIFFD(I,J) 
 12      CONTINUE 
 
c WRITE(*,*)'DFF',DIFF,EPSI 
c READ(*,*) 
       
 IF(DIFF.LE.EPSI)GOTO 18 
 K=K+1 
  
 GOTO 7 
C DO 101 N=1,11 
C I=12-N 
C101 WRITE(*,*)(V(I,J),J=1,5) 
 
 210 
 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
C GOTO 5 
20 WRITE(4,19)N,K 
21 FORMAT(///10F10.5) 
 DO 232 M= 13,1,-1 
 I=14-M 
C IF(C(1,5))EQ((C(1,4)+C(2,5))/2)GOTO 22 
232 WRITE(4,274)i,(C(I,J),J=1,5) 
 RETURN 
 END 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 211 
 
 
C: LiBr-H2O: Parameter's distribution in the direction of film thickness 
C: ABSORPTION PROCESS MODELING ON A LIQUID FALLING FILM   
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC                                                                                                                                
C                                                                                                                                      C 
C           Ph.D FINITE DIFFERENCE MODEL                                                  C 
C                                                                                                                             C                                                    
C                             FOR                                                               C 
C                           C                                                                                                                                                                                            
C           ABSORPTION PROCESS MODELING ON A                                   C                                       
C                                                                          C 
C                                                                          C 
C                               LIQUID FALLING FILM                                                   C  
C               C  
C                                                                                      C 
C                                  K.M ODUNFA (44113)                                                            C 
C                                                                                                                                     C                                          
C                                      SUPERVISED BY                                C 
C                                PROFESSOR FAGBENLE, R O                            C 
C                          C  
C                                        JANUARY 2009                                  C  
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c   COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(5),YCO(13) 
 
  OPEN(UNIT = 4,FILE = 'FINAL RESULT.OUT') 
 OPEN(UNIT = 6,FILE = 'LiBr2Data.IN',STATUS='OLD') 
C 
C     INPUT THE PROBLEM DATA 
C 
 READ(6,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
  2        FORMAT(2I3,I4,F6.2,F6.0,2F6.4,2F6.4,F8.5,F10.7,F5.2) 
            WRITE(*,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
 READ(*,2) 
            READ(6,*)RH,RL 
 105     FORMAT(2(F8.5,1X)) 
            WRITE(*,105)RH,RL 
 READ(6,*)VELM,VELM1,FILMT,AMAGIND,AGRAV 
 102     FORMAT(3(F9.6,1X),F3.1, F5.3) 
           WRITE(*,102)VELM,VELM1,FILMT,AMAGIND, AGRAV  
            READ(6,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
 104     FORMAT(6(F9.5,1X)) 
   WRITE(*,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
 212 
 
  READ(6,*)NEX,NEY 
 205     FORMAT(2(I3,1X)) 
            WRITE(*,205)NEX,NEY 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101    CONTINUE 
C 
C COMPUTE TEMPERATURE DISTRIBUTION IN THE FILM 
C 
 CALL SOLUTION1 
C 
C COMPUTE CONCENTRATION DISTRIBUTION IN THE FILM 
 
 CALL SOLUTION2 
C 
            STOP  
 END 
C 
 SUBROUTINE VELOCITY 
            COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c    COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(5),YCO(13) 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
 213 
 
  101    CONTINUE 
 
 WRITE(4,3) 
3 FORMAT(47H1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,5 
 DO 4 J=1,13 
 A=I 
4 V(I,J)=VB+(VT-VB)*(A-1.)/4. 
 N=0 
5 K=0 
 DO 16 J=1,13 
 DO 16 I=2,4 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UVEL = VT 
 GO TO 14 
C8 UTEMP = (V(I+1,J)+2.*V(I,J+1)+V(I-1,J))/4. 
8 UVEL = 0.0 
 GO TO 14 
9 IF(J-13)13,10,13 
10 IF(I-IB)12,11,11 
11 UVEL = VT 
 GO TO 14 
c12 UTEMP=(V(I+1,J)+2.*V(I,J-1)+V(I-1,J))/4 
c 
12        V(I,J)= V(I,J+1) 
 GO TO 14 
13 UVEL = ((V(I,J+1)+V(I,J-1)+(3*VELM/FILMT**2)*DY**2 
     .+DY**2*DENS*AGRAV+((DENS*TMMSUC*AMAGIND*AMAGIND) 
     ./0.000001257)*DY**2))/2.0  
c 
14 DIFF=UVEL-V(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 V(I,J)=UVEL 
C     write(*,*)((V(I,J),I=2,11),J=1,5) 
C read(*,*) 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
21        FORMAT(10F9.6) 
 DO 22 M= 1,5 
 214 
 
 I=6-M 
22 WRITE(4,21)(V(I,J),J=1,13) 
            RETURN 
            END 
 
C 
 SUBROUTINE SOLUTION1 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(5),YCO(13),U(5,13) 
c S(100) 
C 
 DY= RL/NEY 
  DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
  NNT=NX*NY 
c 
            DO 102 I = 1,NX 
  DO 102 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  102    CONTINUE 
 
C COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
            CALL VELOCITY 
C 
 WRITE(4,3) 
3 FORMAT(47H1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,5 
 DO 4 J=1,13 
 A=I 
4 U(I,J)=UB+(UT-UB)*(A-1.)/4. 
 
 N=0 
5 K=0 
 DO 16 J=1,13 
 DO 16 I=2,4 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UTEMP = UT 
 GO TO 14 
C8 UTEMP = (U(I+1,J)+2.*U(I,J+1)+U(I-1,J))/4. 
 215 
 
8 UTEMP = 35 
 GO TO 14 
9 IF(J-5)13,10,13 
10 IF(I-IB)12,11,11 
11 UTEMP = UT 
 GO TO 14 
c12 UTEMP=(U(I+1,J)+2.*U(I,J-1)+U(I-1,J))/4 
12        UTEMP=(U(I-1,J)+0.85) 
 GO TO 14 
C 
            UVEL=V(I,J) 
13 UTEMP=(ALPHA*DX*(U(I+1,J)+U(I-1,J))-(UVEL*DY*DY*U(I-1,J))/ 
     .(2*ALPHA*DX-(UVEL*DY*DY))) 
 
c 
14 DIFF=UTEMP-U(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 U(I,J)=UTEMP 
C 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
C 
21 FORMAT(10F7.2) 
 DO 22 M= 1,5 
 I=6-M 
22 WRITE(4,21)(U(I,J),J=1,13) 
    RETURN 
 END 
C 
 SUBROUTINE SOLUTION2 
 COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
c COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,S,NMAX 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10), 
C COMMON/XL4/V(11,5),S(110) 
 DIMENSION XCO(10),YCO(13),C(13,10),CSTORE(50,20),DIFFD(50, 
 &20) 
 Open(unit=273,file='SalauOdunfasolution2.out') 
 
 216 
 
 DY= RL/NEY 
  DX= RH/NEX 
  NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
 
            DO 103 I = 1,NX 
            DO 103 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
 103     CONTINUE 
  
C     COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
 
            CALL VELOCITY 
  
 do 121 i=1,13 
 do 121 j=1,5 
c write(4,*) 'when',v(i,j) 
 121     continue 
 
 WRITE(4,3) 
3 FORMAT(47H1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 
 DO 4 J=1,13 
 DO 4 I=1,5 
 A=I 
C IF(J.EQ.5)GOTO 4 
  C(I,J)=0.5965 
  
4          CONTINUE 
 
7 DO  116  J=1,13 
 DO  116  I=1,5 
  
 IF ( I.EQ.1) C(I,J)=CT 
 IF(I.EQ.1) GOTO 6 
 IF(I.EQ.5)C(I,J) = CB 
 IF(I.EQ.5) GOTO 6 
C IF(J.EQ.1) C(I,J)=CB+(CT-CB)*(A-1.)/10 
C IF(J.EQ.1) GOTO 14 
 IF(J.EQ.13) CONC= C(I,J)+0.0035 
 IF(J.EQ.13) GOTO 6 
 6 CSTORE(I,J)=C(I,J) 
116      CONTINUE 
 217 
 
 DO 16 J=1,13 
 DO 16 I=2,4 
 Do 275 k=13,1,-1 
           WRITE(273,274)k,(V(k,kj),kj=1,13) 
274 Format(i3,2x,13(f12.5,2x)) 
275 Continue 
c READ(*,*) 
c WRITE(*,*)I,J 
 
13    c(i,j)=(sdifu*dx*(c(1+1,j)+c(1-1,j))+(v(i,j)*dy*dy*c(i-1,j))/ 
     .(2*sdifu*dx+(v(i,j)*dy*dy))) 
 
c write(*,*)I,J,'conc',c(i,j) 
c READ(*,*) 
16 CONTINUE 
c          WRITE(*,*)'OUTTT'  
 DO 22 J=1,13 
 DO 22 I=1,5 
  
 
  DIFFD(I,J)=ABS( CSTORE(I,J)-C(I,J)) 
22  CONTINUE 
 
 DIFF= DIFFD(1,1) 
 DO  12  J =1,13 
 DO  12  I =1,5 
 
 IF(DIFFD(I,J).LT.DIFF)DIFF=DIFFD(I,J) 
 12       CONTINUE 
 
c WRITE(*,*)'DFF',DIFF,EPSI 
c READ(*,*) 
       
 IF(DIFF.LE.EPSI)GOTO 18 
 K=K+1 
  
 GOTO 7 
C DO 101 N=1,11 
C I=12-N 
C101 WRITE(*,*)(V(I,J),J=1,5) 
C          READ(*,*) 
 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
 218 
 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
C GOTO 5 
20 WRITE(4,19)N,K 
21 FORMAT(10F10.5) 
 DO 232 M= 5,1,-1 
 I=6-M 
C IF(C(1,5))EQ((C(1,4)+C(2,5))/2)GOTO 22 
232 WRITE(4,274)i,(C(I,J),J=1,13) 
 RETURN 
 END 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 219 
 
 
C: LiCl-H2O Parameters distribution in the direction of falling 
C: ABSORPTION PROCESS MODELING ON A LIQUID FALLING FILM  
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC                                                                                                                                
C                                                                                                                                      C 
C           Ph.D FINITE DIFFERENCE MODEL                                                  C 
C                                                                                                                             C                                                    
C                             FOR                                                               C 
C                           C                                                                                                                                                                                            
C           ABSORPTION PROCESS MODELING ON A                                   C                                       
C                                                                          C 
C                                                                          C 
C                               LIQUID FALLING FILM                                                   C  
C               C  
C                                                                                      C 
C                                  K.M ODUNFA (44113)                                                            C 
C                                                                                                                                     C                                          
C                                      SUPERVISED BY                                C 
C                                PROFESSOR FAGBENLE, R O                            C 
C                          C  
C                                        JANUARY 2009                                  C  
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(13),YCO(5) 
 
  OPEN(UNIT = 4,FILE = 'FINAL RESULT.OUT') 
 OPEN(UNIT = 6,FILE = 'LiCl Data.IN',STATUS='OLD') 
C 
C   INPUT THE PROBLEM DATA 
C 
 READ(6,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
  2        FORMAT(2I3,I4,F6.2,F6.0,2F6.4,2F6.4,F8.5,F10.7,F5.2) 
            WRITE(*,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
 READ(*,2) 
            READ(6,*)RH,RL 
 105     FORMAT(2(F8.5,1X)) 
            WRITE(*,105)RH,RL 
 READ(6,*)VELM,VELM1,FILMT,AMAGIND,AGRAV 
 102     FORMAT(3(F9.6,1X),F3.1, F5.3) 
            WRITE(*,102)VELM,VELM1,FILMT,AMAGIND, AGRAV  
            READ(6,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
 104     FORMAT(6(F9.5,1X)) 
   WRITE(*,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
  READ(6,*)NEX,NEY 
 220 
 
 205     FORMAT(2(I3,1X)) 
            WRITE(*,205)NEX,NEY 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101    CONTINUE 
C 
C COMPUTE TEMPERATURE DISTRIBUTION IN THE FILM 
C 
 CALL SOLUTION1 
C 
C COMPUTE CONCENTRATION DISTRIBUTION IN THE FILM 
 
 CALL SOLUTION2 
C 
            STOP  
 END 
C 
 SUBROUTINE VELOCITY 
            COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(13),YCO(5) 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101    CONTINUE 
 
 221 
 
 WRITE(4,3) 
3 FORMAT(47H1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,13 
 DO 4 J=1,5 
 A=I 
4 V(I,J)=VB+(VT-VB)*(A-1.)/12. 
 N=0 
5 K=0 
 DO 16 J=1,5 
 DO 16 I=2,12 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UVEL = VT 
 GO TO 14 
C8 UTEMP = (V(I+1,J)+2.*V(I,J+1)+V(I-1,J))/4. 
8 UVEL = 0.0 
 GO TO 14 
9 IF(J-5)13,10,13 
10 IF(I-IB)12,11,11 
11 UVEL = VT 
 GO TO 14 
c12 UTEMP=(V(I+1,J)+2.*V(I,J-1)+V(I-1,J))/4 
c 
12       V(I,J)= V(I,J+1) 
 GO TO 14 
13 UVEL = ((V(I+1,J)+V(I-1,J)+(3*VELM/FILMT**2)*DY**2 
     .+ DY**2*DENS*AGRAV+((DENS*TMMSUC*AMAGIND*AMAGIND) 
     ./0.000001257)*DY**2))/2.0  
14 DIFF=UVEL-V(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 V(I,J)=UVEL 
C     write(*,*)((V(I,J),I=2,11),J=1,5) 
C read(*,*) 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
21        FORMAT(///10F9.6) 
 DO 22 M= 1,13 
 I=14-M 
22 WRITE(4,21)(V(I,J),J=1,5) 
            RETURN 
 222 
 
            END 
C 
 SUBROUTINE SOLUTION1 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10) 
 DIMENSION XCO(13),YCO(5),U(13,5) 
C 
 DY= RL/NEY 
  DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
  NNT=NX*NY 
c 
            DO 102 I = 1,NX 
  DO 102 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  102    CONTINUE 
 
C COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
            CALL VELOCITY 
C 
 WRITE(4,3) 
3 FORMAT(47H1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,13 
 DO 4 J=1,5 
 A=I 
4 U(I,J)=UB+(UT-UB)*(A-1.)/12. 
 
 N=0 
5 K=0 
 DO 16 J=1,5 
 DO 16 I=2,12 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UTEMP = UT 
 GO TO 14 
C8 UTEMP = (U(I+1,J)+2.*U(I,J+1)+U(I-1,J))/4. 
8 UTEMP = 35 
 GO TO 14 
9 IF(J-5)13,10,13 
10 IF(I-IB)12,11,11 
11 UTEMP = UT 
 GO TO 14 
 223 
 
c12 UTEMP=(U(I+1,J)+2.*U(I,J-1)+U(I-1,J))/4 
12        UTEMP=(U(I-1,J)+0.75) 
 GO TO 14 
C 
            UVEL=V(I,J) 
C 
13 UTEMP=(ALPHA*DX*(U(I+1,J)+U(I-1,J))+(UVEL*DY*DY*U(I+1,J))/ 
     .(2*ALPHA*DX-(UVEL*DY*DY))) 
c 
14 DIFF=UTEMP-U(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 U(I,J)=UTEMP 
C 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
C 
21 FORMAT(///10F7.2) 
 DO 22 M= 1,13 
 I=14-M 
22 WRITE(4,21)(U(I,J),J=1,5) 
    RETURN 
 END 
C 
 SUBROUTINE SOLUTION2 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(14,10), 
 DIMENSION XCO(13),YCO(10),C(13,10),CSTORE(50,20),DIFFD(50, 
 &20) 
 Open(unit=273,file='SalauOdunfasolution2.out') 
 
 DY= RL/NEY 
  DX= RH/NEX 
  NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
 
           DO 103 I = 1,NX 
           DO 103 J = 1,NY 
 224 
 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
 103     CONTINUE 
  
C     COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
 
            CALL VELOCITY 
  
 do 121 i=1,5 
 do 121 j=1,13 
c write(4,*) 'when',v(i,j) 
 121      continue 
 
 WRITE(4,3) 
3 FORMAT(47H1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 
 DO 4 J=1,5 
 DO 4 I=1,13 
 A=I 
C IF(J.EQ.5)GOTO 4 
 C(I,J)=0.358 
c          V(I,J)=VB+(VT-VB)*(A-1.)/10. 
  
4         CONTINUE 
 
7 DO  116  J=1,5 
 DO  116  I=1,13 
  
 IF ( I.EQ.1) C(I,J)=CT 
 IF(I.EQ.1) GOTO 6 
 IF(I.EQ.11)C(I,J) = CB 
 IF(I.EQ.11) GOTO 6 
C IF(J.EQ.1) C(I,J)=CB+(CT-CB)*(A-1.)/10 
C IF(J.EQ.1) GOTO 14 
 IF(J.EQ.5) CONC= C(I,J)+0.0040 
 IF(J.EQ.5) GOTO 6 
 6 CSTORE(I,J)=C(I,J) 
116      CONTINUE 
 DO 16 J=1,5 
 DO 16 I=2,12 
 Do 275 k=13,1,-1 
            WRITE(273,274)k,(V(k,kj),kj=1,5) 
274 Format(i3,2x,5(f12.5,2x)) 
275 Continue 
c READ(*,*) 
 225 
 
c WRITE(*,*)I,J 
C 
13    c(i,j)=(sdifu*dx*(c(1+1,j)+c(1-1,j))+(v(i,j)*dy*dy*c(i-1,j))/ 
     .(2*sdifu*dx+(v(i,j)*dy*dy))) 
 
c write(*,*)I,J,'conc',c(i,j) 
c READ(*,*) 
16 CONTINUE 
c          WRITE(*,*)'OUTTT'  
 DO 22 J=1,5 
 DO 22 I=1,13 
  DIFFD(I,J)=ABS( CSTORE(I,J)-C(I,J)) 
22  CONTINUE 
 
 DIFF= DIFFD(1,1) 
 DO  12  J =1,5 
 DO  12  I =1,13 
 
 IF(DIFFD(I,J).LT.DIFF)DIFF=DIFFD(I,J) 
 12       CONTINUE 
 
c WRITE(*,*)'DFF',DIFF,EPSI 
c READ(*,*) 
       
 IF(DIFF.LE.EPSI)GOTO 18 
 K=K+1 
  
 GOTO 7 
C DO 101 N=1,11 
C I=12-N 
C101 WRITE(*,*)(V(I,J),J=1,5) 
C         READ(*,*) 
 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
C GOTO 5 
20 WRITE(4,19)N,K 
21 FORMAT(///10F10.5) 
 DO 232 M= 13,1,-1 
 I=14-M 
C IF(C(1,5))EQ((C(1,4)+C(2,5))/2)GOTO 22 
232 WRITE(4,274)i,(C(I,J),J=1,5) 
 RETURN 
 226 
 
 END 
C: LiCl2-H2O Parameter's distribution in the direction of film thickness 
C: ABSORPTION PROCESS MODELING ON A LIQUID FALLING FILM 
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC                                                                                                                                
C                                                                                                                                      C 
C           Ph.D FINITE DIFFERENCE MODEL                                                  C 
C                                                                                                                             C                                                    
C                             FOR                                                               C 
C                           C                                                                                                                                                                                            
C           ABSORPTION PROCESS MODELING ON A                                   C                                       
C                                                                          C 
C                                                                          C 
C                               LIQUID FALLING FILM                                                   C  
C               C  
C                                                                                      C 
C                                  K.M ODUNFA (44113)                                                            C 
C                                                                                                                                     C                                          
C                                      SUPERVISED BY                                C 
C                                PROFESSOR FAGBENLE, R O                            C 
C                          C  
C                                        JANUARY 2009                                  C  
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 
 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(10,14) 
 DIMENSION XCO(5),YCO(13) 
 
  OPEN(UNIT = 4,FILE = 'FINAL RESULT.OUT') 
 OPEN(UNIT = 6,FILE = 'LiCl2 Data.IN',STATUS='OLD') 
C 
C    INPUT THE PROBLEM DATA 
C 
 READ(6,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
  2        FORMAT(2I3,I4,F6.2,F6.0,2F6.4,2F6.4,F7.5,F10.7,F5.2) 
            WRITE(*,2)IA,IB,NMAX,UT,UB,CT,CB,VT,VB,AVISC,TMMSUC,EPSI 
 READ(*,2) 
            READ(6,*)RH,RL 
 105     FORMAT(2(F8.5,1X)) 
            WRITE(*,105)RH,RL 
 READ(6,*)VELM,VELM1,FILMT,AMAGIND,AGRAV 
 102     FORMAT(3(F9.6,1X),F3.1, F5.3) 
            WRITE(*,102)VELM,VELM1,FILMT,AMAGIND, AGRAV  
            READ(6,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
 104     FORMAT(6(F9.5,1X)) 
   WRITE(*,*)ALPHA,TAMAC,SDIFU,HITAB,CONEQ,DENS 
 227 
 
  READ(6,*)NEX,NEY 
 205     FORMAT(2(I3,1X)) 
            WRITE(*,205)NEX,NEY 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101    CONTINUE 
C 
C COMPUTE TEMPERATURE DISTRIBUTION IN THE FILM 
C 
 CALL SOLUTION1 
C 
C COMPUTE CONCENTRATION DISTRIBUTION IN THE FILM 
 
 CALL SOLUTION2 
C 
            STOP  
 END 
C 
 SUBROUTINE VELOCITY 
            COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(10,14) 
 DIMENSION XCO(5),YCO(13) 
C 
 DY= RL/NEY 
 DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
 NNT=NX*NY 
C    
            DO 101 I = 1,NX 
 DO 101 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  101    CONTINUE 
 228 
 
 
 WRITE(4,3) 
3 FORMAT(47H1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,5 
 DO 4 J=1,13 
 A=I 
4 V(I,J)=VB+(VT-VB)*(A-1.)/4. 
 N=0 
5 K=0 
 DO 16 J=1,13 
 DO 16 I=2,4 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UVEL = VT 
 GO TO 14 
C8 UTEMP = (V(I+1,J)+2.*V(I,J+1)+V(I-1,J))/4. 
8 UVEL = 0.0 
 GO TO 14 
9 IF(J-13)13,10,13 
10 IF(I-IB)12,11,11 
11 UVEL = VT 
 GO TO 14 
c12 UTEMP=(V(I+1,J)+2.*V(I,J-1)+V(I-1,J))/4 
 
12        V(I,J)= V(I,J+1) 
 GO TO 14 
13 UVEL = ((V(I,J+1)+V(I,J-1)+(3*VELM/FILMT**2)*DY**2 
     .+DY**2*DENS*AGRAV+((DENS*TMMSUC*AMAGIND*AMAGIND) 
     ./0.000001257)*DY**2))/2.0  
c 
14 DIFF=UVEL-V(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 V(I,J)=UVEL 
C     write(*,*)((V(I,J),I=2,11),J=1,5) 
C read(*,*) 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
21        FORMAT(10F9.6) 
 DO 22 M= 1,5 
 I=6-M 
 229 
 
22 WRITE(4,21)(V(I,J),J=1,13) 
C22 WRITE(4,*)(V(I,J),J=1,13) 
 
           RETURN 
           END 
 
C 
 SUBROUTINE SOLUTION1 
 COMMON/XL1/AVISC,DENS, TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(10,14) 
 DIMENSION XCO(5),YCO(13),U(5,13) 
C 
 DY= RL/NEY 
  DX= RH/NEX 
 NX=NEX+1 
 NY=NEY+1 
  NNT=NX*NY 
c 
            DO 102 I = 1,NX 
  DO 102 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
  102    CONTINUE 
 
C COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
            CALL VELOCITY 
C 
 WRITE(4,3) 
3 FORMAT(47H1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 DO 4 I=1,5 
 DO 4 J=1,13 
 A=I 
4 U(I,J)=UB+(UT-UB)*(A-1.)/4. 
 
 N=0 
5 K=0 
 DO 16 J=1,13 
 DO 16 I=2,4 
 IF(J-1)9,6,9 
6 IF(IA-I)7,7,8 
7 UTEMP = UT 
 GO TO 14 
C8 UTEMP = (U(I+1,J)+2.*U(I,J+1)+U(I-1,J))/4. 
8 UTEMP = 35 
 230 
 
 GO TO 14 
9 IF(J-13)13,10,13 
10 IF(I-IB)12,11,11 
11 UTEMP = UT 
 GO TO 14 
c12 UTEMP=(U(I+1,J)+2.*U(I,J-1)+U(I-1,J))/4 
12        UTEMP=(U(I-1,J)+0.75) 
 GO TO 14 
C 
            UVEL=V(I,J) 
C 
13 UTEMP=(ALPHA*DX*(U(I+1,J)+U(I-1,J))-(UVEL*DY*DY*U(I-1,J))/ 
     .(2*ALPHA*DX-(UVEL*DY*DY))) 
c 
14 DIFF=UTEMP-U(I,J) 
 IF(ABS(DIFF)-EPSI)16,15,15 
15 K=K+1 
16 U(I,J)=UTEMP 
C 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
 GOTO 5 
20 WRITE(4,19)N,K 
C 
21 FORMAT(10F7.2) 
 DO 22 M= 1,5 
 I=6-M 
22 WRITE(4,21)(U(I,J),J=1,13) 
    RETURN 
 END 
C 
 SUBROUTINE SOLUTION2 
 COMMON/XL1/AVISC,DENS,TMMSUC,FMAG,AMAGIND,VT,VB,AGRAV 
 COMMON/XL2/RH,RL,NEX,NEY,UT,UB,EPSI,CT,CB,NN,NMAX 
 COMMON/XL3/NX,NY,IB,IA,VELM,FILMT,ALPHA,TAMAC,SDIFU,V(10,14), 
 DIMENSION XCO(10),YCO(13),C(13,10),CSTORE(50,20),DIFFD(50, 
 &20) 
 Open(unit=273,file='SalauOdunfasolution2.out') 
 
 DY= RL/NEY 
  DX= RH/NEX 
  NX=NEX+1 
 231 
 
 NY=NEY+1 
 NNT=NX*NY 
 
            DO 103 I = 1,NX 
            DO 103 J = 1,NY 
 NNC= (J-1)*NX+I 
 YCO(NNC)=(J-1)*DY 
 XCO(NNC)=(I-1)*DX 
 103     CONTINUE 
  
C     COMPUTE VELOCITY DISTRIBUTION IN THE FILM 
 
            CALL VELOCITY 
  
 do 121 i=1,13 
 do 121 j=1,5 
c write(4,*) 'when',v(i,j) 
 121     continue 
 
 WRITE(4,3) 
3 FORMAT(47H1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION//) 
 
 DO 4 J=1,13 
 DO 4 I=1,5 
 A=I 
C IF(J.EQ.5)GOTO 4 
 C(I,J)=0.446 
  
4          CONTINUE 
 
7 DO  116  J=1,13 
 DO  116  I=1,5 
  
 IF ( I.EQ.1) C(I,J)=CT 
 IF(I.EQ.1) GOTO 6 
 IF(I.EQ.5)C(I,J) = CB 
 IF(I.EQ.5) GOTO 6 
C IF(J.EQ.1) C(I,J)=CB+(CT-CB)*(A-1.)/10 
C IF(J.EQ.1) GOTO 14 
 IF(J.EQ.13) CONC= C(I,J)+0.0040 
 IF(J.EQ.13) GOTO 6 
 6 CSTORE(I,J)=C(I,J) 
116      CONTINUE 
 DO 16 J=1,13 
 DO 16 I=2,4 
 Do 275 k=13,1,-1 
 232 
 
           WRITE(273,274)k,(V(k,kj),kj=1,13) 
274 Format(i3,x,13(f12.5,x)) 
275 Continue 
c READ(*,*) 
c WRITE(*,*)I,J 
 
C 
13    c(i,j)=(sdifu*dx*(c(1+1,j)+c(1-1,j))+(v(i,j)*dy*dy*c(i-1,j))/ 
     .(2*sdifu*dx+(v(i,j)*dy*dy))) 
 
c write(*,*)I,J,'conc',c(i,j) 
c READ(*,*) 
16 CONTINUE 
c          WRITE(*,*)'OUTTT'  
 DO 22 J=1,13 
 DO 22 I=1,5 
  
 
 DIFFD(I,J)=ABS( CSTORE(I,J)-C(I,J)) 
22  CONTINUE 
 
 DIFF= DIFFD(1,1) 
 DO  12  J =1,13 
 DO  12  I =1,5 
 
 IF(DIFFD(I,J).LT.DIFF)DIFF=DIFFD(I,J) 
 12       CONTINUE 
 
c WRITE(*,*)'DFF',DIFF,EPSI 
c READ(*,*) 
       
 IF(DIFF.LE.EPSI)GOTO 18 
 K=K+1 
  
 GOTO 7 
C DO 101 N=1,11 
C I=12-N 
C         READ(*,*) 
 
 N=N+1 
 IF(N-NMAX)17,20,20 
17 IF(K)18,20,18 
18 WRITE(4,19)N,K 
19 FORMAT(7H IT NO.,I4,4H K= ,I4) 
C GOTO 5 
20 WRITE(4,19)N,K 
 233 
 
21 FORMAT(///10F10.5) 
 DO 232 M= 5,1,-1 
 I=6-M 
C IF(C(1,5))EQ((C(1,4)+C(2,5))/2)GOTO 22 
232 WRITE(4,274)i,(C(I,J),J=1,13) 
 RETURN 
 END 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 234 
 
APPENDIX  C 
COMPUTER RESULTS 
 
LiBr-H2O  
IN THE DIRECTION OF FALLING 
 
LiBr-H2O 
At B= 0.0 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 IT NO.   1 K=   22 
 IT NO.   2 K=   11 
 IT NO.   3 K=   11 
 IT NO.   4 K=   11 
 IT NO.   5 K=   11 
 IT NO.   6 K=   11 
 IT NO.   7 K=   11 
 IT NO.   8 K=   11 
 IT NO.   9 K=   11 
 IT NO.  10 K=   11 
 IT NO.  11 K=   11 
 IT NO.  12 K=   11 
 IT NO.  13 K=   11 
 IT NO.  14 K=   11 
 IT NO.  15 K=   11 
 IT NO.  16 K=   11 
 IT NO.  17 K=   11 
 IT NO.  18 K=   11 
 IT NO.  19 K=   11 
 IT NO.  20 K=   11 
 IT NO.  21 K=   11 
 IT NO.  22 K=   11 
 IT NO.  23 K=   11 
 IT NO.  24 K=   11 
 IT NO.  25 K=   11 
 IT NO.  26 K=   11 
 IT NO.  27 K=   11 
 IT NO.  28 K=   11 
 IT NO.  29 K=   11 
 IT NO.  30 K=   11 
 IT NO.  31 K=   11 
 IT NO.  32 K=   11 
 IT NO.  33 K=   11 
 235 
 
 IT NO.  34 K=   11 
 IT NO.  35 K=   11 
 IT NO.  36 K=   11 
 IT NO.  37 K=   11 
 IT NO.  38 K=   11 
 IT NO.  39 K=   11 
 IT NO.  40 K=   11 
 IT NO.  41 K=   11 
 IT NO.  42 K=   11 
 IT NO.  43 K=   11 
 IT NO.  44 K=   11 
 IT NO.  45 K=   11 
 IT NO.  46 K=   11 
 IT NO.  47 K=   11 
 IT NO.  48 K=   11 
 IT NO.  49 K=   11 
 IT NO.  50 K=   11 
 IT NO.  51 K=   11 
 IT NO.  52 K=   11 
 IT NO.  53 K=   11 
 IT NO.  54 K=   11 
 IT NO.  55 K=   11 
 IT NO.  56 K=   11 
 IT NO.  57 K=   11 
 IT NO.  58 K=   11 
 IT NO.  59 K=   11 
 IT NO.  60 K=   11 
 IT NO.  61 K=   11 
 IT NO.  62 K=   11 
 IT NO.  63 K=   11 
 IT NO.  64 K=   11 
 IT NO.  65 K=   11 
 IT NO.  66 K=   11 
 IT NO.  67 K=   11 
 IT NO.  68 K=   11 
 IT NO.  69 K=   11 
 IT NO.  70 K=   11 
 IT NO.  71 K=   11 
 IT NO.  72 K=   11 
 IT NO.  73 K=   11 
 IT NO.  74 K=   11 
 IT NO.  75 K=   11 
 IT NO.  76 K=   11 
 IT NO.  77 K=   11 
 IT NO.  78 K=   11 
 IT NO.  79 K=   11 
 236 
 
 IT NO.  80 K=   11 
 IT NO.  81 K=   11 
 IT NO.  82 K=   11 
 IT NO.  83 K=   11 
 IT NO.  84 K=   11 
 IT NO.  85 K=   11 
 IT NO.  86 K=   11 
 IT NO.  87 K=   11 
 IT NO.  88 K=   11 
 IT NO.  89 K=   11 
 IT NO.  90 K=   11 
 IT NO.  91 K=   11 
 IT NO.  92 K=   11 
 IT NO.  93 K=   11 
 IT NO.  94 K=   11 
 IT NO.  95 K=   11 
 IT NO.  96 K=   11 
 IT NO.  97 K=   11 
 IT NO.  98 K=   11 
 IT NO.  99 K=   11 
 IT NO. 100 K=   11 
 
 
 
  .362000  .362000  .362000  .362000  .362000 
 
 
 
  .000000  .366245  .366245  .366245  .366245 
 
 
 
  .000000  .370384  .370384  .370384  .366245 
 
 
 
  .000000  .374418  .374418  .374418  .366245 
 
 
 
  .000000  .378348  .378348  .378348  .366245 
 
 
 
  .000000  .382172  .382172  .382172  .366245 
 
 237 
 
 
 
  .000000  .385891  .385891  .385891  .366245 
 
 
 
  .000000  .389505  .389505  .389505  .366245 
 
 
 
  .000000  .393014  .393014  .393014  .366245 
 
 
 
  .000000  .396418  .396418  .396418  .366245 
 
 
 
  .000000  .399717  .399717  .399717  .366245 
 
 
 
  .000000  .402911  .402911  .402911  .366245 
 
 
 
  .406000  .406000  .406000  .406000  .406000 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   43 
 IT NO.   2 K=    0 
 
 
 
  44.44  44.44  44.44  44.44  44.44 
 
 
 
  35.00  44.40  44.40  44.40  44.35 
 
 
 
  35.00  43.45  43.45  43.45  43.50 
 
 
 
 238 
 
  35.00  42.52  42.52  42.52  42.65 
 
 
 
  35.00  41.61  41.61  41.61  41.80 
 
 
 
  35.00  40.72  40.72  40.72  40.95 
 
 
 
  35.00  39.85  39.85  39.85  40.10 
 
 
 
  35.00  39.00  39.00  39.00  39.25 
 
 
 
  35.00  38.16  38.16  38.16  38.40 
 
 
 
  35.00  37.35  37.35  37.35  37.55 
 
 
 
  35.00  36.55  36.55  36.55  36.70 
 
 
 
  35.00  35.77  35.77  35.77  35.85 
 
 
 
  35.00  35.00  35.00  35.00  35.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
  1        .60000        .60000        .60000        .60000        .60000 
  2        .00000        .59545        .59545        .59545        .59500 
  3        .00000        .59090        .59090        .59090        .59004 
  4        .00000        .58635        .58635        .58635        .58513 
 239 
 
  5        .00000        .58180        .58180        .58180        .58025 
  6        .00000        .57724        .57724        .57724        .57542 
  7        .00000        .57267        .57267        .57267        .57063 
  8        .00000        .56810        .56810        .56810        .56587 
  9        .00000        .56352        .56352        .56352        .56116 
 10        .00000        .55892        .55892        .55892        .55648 
 11        .00000        .55432        .55432        .55432        .55185 
 12        .00000        .54970        .54970        .54970        .54725 
 13        .54500        .54500        .54500        .54500        .54500 
 
 
At B=1.4 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 
 
  .362000  .362000  .362000  .362000  .362000 
 
 
 
  .000000  .367444  .367444  .367444  .367444 
 
 
 
  .000000  .372564  .372564  .372564  .367444 
 
 
 
  .000000  .377361  .377361  .377361  .367444 
 
 
 
  .000000  .381835  .381835  .381835  .367444 
 
 
 
  .000000  .385986  .385986  .385986  .367444 
 
 
 
  .000000  .389814  .389814  .389814  .367444 
 
 
 
  .000000  .393319  .393319  .393319  .367444 
 
 240 
 
 
 
  .000000  .396501  .396501  .396501  .367444 
 
 
 
  .000000  .399360  .399360  .399360  .367444 
 
 
 
  .000000  .401897  .401897  .401897  .367444 
 
 
 
  .000000  .404110  .404110  .404110  .367444 
 
 
 
  .406000  .406000  .406000  .406000  .406000 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   43 
 IT NO.   2 K=    0 
 
 
 
  44.44  44.44  44.44  44.44  44.44 
 
 
 
  35.00  44.40  44.40  44.40  44.35 
 
 
 
  35.00  43.45  43.45  43.45  43.50 
 
 
 
  35.00  42.52  42.52  42.52  42.65 
 
 
 
  35.00  41.61  41.61  41.61  41.80 
 
 
 
 241 
 
  35.00  40.72  40.72  40.72  40.95 
 
 
 
  35.00  39.85  39.85  39.85  40.10 
 
 
 
  35.00  39.00  39.00  39.00  39.25 
 
 
 
  35.00  38.16  38.16  38.16  38.40 
 
 
 
  35.00  37.35  37.35  37.35  37.55 
 
 
 
  35.00  36.55  36.55  36.55  36.70 
 
 
 
  35.00  35.77  35.77  35.77  35.85 
 
 
 
  35.00  35.00  35.00  35.00  35.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
  1        .60000        .60000        .60000        .60000        .60000 
  2        .00000        .59547        .59547        .59547        .59502 
  3        .00000        .59094        .59094        .59094        .59008 
  4        .00000        .58642        .58642        .58642        .58518 
  5        .00000        .58191        .58191        .58191        .58032 
  6        .00000        .57739        .57739        .57739        .57550 
  7        .00000        .57287        .57287        .57287        .57072 
  8        .00000        .56834        .56834        .56834        .56598 
  9        .00000        .56380        .56380        .56380        .56128 
 10        .00000        .55924        .55924        .55924        .55662 
 11        .00000        .55465        .55465        .55465        .55200 
 12        .00000        .55005        .55005        .55005        .54741 
 242 
 
 13        .54500        .54500        .54500        .54500        .54500 
 
 
At B=3.0 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 
 
  .362000  .362000  .362000  .362000  .362000 
 
 
 
  .000000  .371751  .371751  .371751  .371751 
 
 
 
  .000000  .380396  .380396  .380396  .371751 
 
 
 
  .000000  .387933  .387933  .387933  .371751 
 
 
 
  .000000  .394364  .394364  .394364  .371751 
 
 
 
  .000000  .399689  .399689  .399689  .371751 
 
 
 
  .000000  .403908  .403908  .403908  .371751 
 
 
 
  .000000  .407021  .407021  .407021  .371751 
 
 
 
  .000000  .409029  .409029  .409029  .371751 
 
 
 
  .000000  .409930  .409930  .409930  .371751 
 
 243 
 
 
 
  .000000  .409726  .409726  .409726  .371751 
 
 
 
  .000000  .408416  .408416  .408416  .371751 
 
 
 
  .406000  .406000  .406000  .406000  .406000 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   43 
 IT NO.   2 K=    0 
 
 
 
  44.44  44.44  44.44  44.44  44.44 
 
 
 
  35.00  44.40  44.40  44.40  44.35 
 
 
 
  35.00  43.45  43.45  43.45  43.50 
 
 
 
  35.00  42.52  42.52  42.52  42.65 
 
 
 
  35.00  41.61  41.61  41.61  41.80 
 
 
 
  35.00  40.72  40.72  40.72  40.95 
 
 
 
  35.00  39.85  39.85  39.85  40.10 
 
 
 
 244 
 
  35.00  39.00  39.00  39.00  39.25 
 
 
 
  35.00  38.16  38.16  38.16  38.40 
 
 
 
  35.00  37.35  37.35  37.35  37.55 
 
 
 
  35.00  36.55  36.55  36.55  36.70 
 
 
 
  35.00  35.77  35.77  35.77  35.85 
 
 
 
  35.00  35.00  35.00  35.00  35.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
  1        .60000        .60000        .60000        .60000        .60000 
  2        .00000        .59551        .59551        .59551        .59507 
  3        .00000        .59107        .59107        .59107        .59019 
  4        .00000        .58667        .58667        .58667        .58535 
  5        .00000        .58229        .58229        .58229        .58054 
  6        .00000        .57792        .57792        .57792        .57577 
  7        .00000        .57355        .57355        .57355        .57105 
  8        .00000        .56917        .56917        .56917        .56636 
  9        .00000        .56476        .56476        .56476        .56171 
 10        .00000        .56032        .56032        .56032        .55710 
 11        .00000        .55582        .55582        .55582        .55253 
 12        .00000        .55126        .55126        .55126        .54799 
 13        .54500        .54500        .54500        .54500        .54500 
 
LiCl-H2O 
 
At B=0.0 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 IT NO.   1 K=   22 
 IT NO.   2 K=   11 
 245 
 
 IT NO.   3 K=   11 
 IT NO.   4 K=   11 
 IT NO.   5 K=   11 
 IT NO.   6 K=   11 
 IT NO.   7 K=   11 
 IT NO.   8 K=   11 
 IT NO.   9 K=   11 
 IT NO.  10 K=   11 
 IT NO.  11 K=   11 
 IT NO.  12 K=   11 
 IT NO.  13 K=   11 
 IT NO.  14 K=   11 
 IT NO.  15 K=   11 
 IT NO.  16 K=   11 
 IT NO.  17 K=   11 
 IT NO.  18 K=   11 
 IT NO.  19 K=   11 
 IT NO.  20 K=   11 
 IT NO.  21 K=   11 
 IT NO.  22 K=   11 
 IT NO.  23 K=   11 
 IT NO.  24 K=   11 
 IT NO.  25 K=   11 
 IT NO.  26 K=   11 
 IT NO.  27 K=   11 
 IT NO.  28 K=   11 
 IT NO.  29 K=   11 
 IT NO.  30 K=   11 
 IT NO.  31 K=   11 
 IT NO.  32 K=   11 
 IT NO.  33 K=   11 
 IT NO.  34 K=   11 
 IT NO.  35 K=   11 
 IT NO.  36 K=   11 
 IT NO.  37 K=   11 
 IT NO.  38 K=   11 
 IT NO.  39 K=   11 
 IT NO.  40 K=   11 
 IT NO.  41 K=   11 
 IT NO.  42 K=   11 
 IT NO.  43 K=   11 
 IT NO.  44 K=   11 
 IT NO.  45 K=   11 
 IT NO.  46 K=   11 
 IT NO.  47 K=   11 
 IT NO.  48 K=   11 
 246 
 
 IT NO.  49 K=   11 
 IT NO.  50 K=   11 
 IT NO.  51 K=   11 
 IT NO.  52 K=   11 
 IT NO.  53 K=   11 
 IT NO.  54 K=   11 
 IT NO.  55 K=   11 
 IT NO.  56 K=   11 
 IT NO.  57 K=   11 
 IT NO.  58 K=   11 
 IT NO.  59 K=   11 
 IT NO.  60 K=   11 
 IT NO.  61 K=   11 
 IT NO.  62 K=   11 
 IT NO.  63 K=   11 
 IT NO.  64 K=   11 
 IT NO.  65 K=   11 
 IT NO.  66 K=   11 
 IT NO.  67 K=   11 
 IT NO.  68 K=   11 
 IT NO.  69 K=   11 
 IT NO.  70 K=   11 
 IT NO.  71 K=   11 
 IT NO.  72 K=   11 
 IT NO.  73 K=   11 
 IT NO.  74 K=   11 
 IT NO.  75 K=   11 
 IT NO.  76 K=   11 
 IT NO.  77 K=   11 
 IT NO.  78 K=   11 
 IT NO.  79 K=   11 
 IT NO.  80 K=   11 
 IT NO.  81 K=   11 
 IT NO.  82 K=   11 
 IT NO.  83 K=   11 
 IT NO.  84 K=   11 
 IT NO.  85 K=   11 
 IT NO.  86 K=   11 
 IT NO.  87 K=   11 
 IT NO.  88 K=   11 
 IT NO.  89 K=   11 
 IT NO.  90 K=   11 
 IT NO.  91 K=   11 
 IT NO.  92 K=   11 
 IT NO.  93 K=   11 
 IT NO.  94 K=   11 
 247 
 
 IT NO.  95 K=   11 
 IT NO.  96 K=   11 
 IT NO.  97 K=   11 
 IT NO.  98 K=   11 
 IT NO.  99 K=   11 
 IT NO. 100 K=   11 
 
 
 
  .362000  .362000  .362000  .362000  .362000 
 
 
 
  .000000  .366245  .366245  .366245  .366245 
 
 
 
  .000000  .370384  .370384  .370384  .366245 
 
 
 
  .000000  .374418  .374418  .374418  .366245 
 
 
 
  .000000  .378348  .378348  .378348  .366245 
 
 
 
  .000000  .382172  .382172  .382172  .366245 
 
 
 
  .000000  .385891  .385891  .385891  .366245 
 
 
 
  .000000  .389505  .389505  .389505  .366245 
 
 
 
  .000000  .393014  .393014  .393014  .366245 
 
 
 
  .000000  .396418  .396418  .396418  .366245 
 248 
 
 
 
 
  .000000  .399717  .399717  .399717  .366245 
 
 
 
  .000000  .402911  .402911  .402911  .366245 
 
 
 
  .406000  .406000  .406000  .406000  .406000 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   55 
 IT NO.   2 K=   30 
 IT NO.   3 K=   27 
 IT NO.   4 K=   24 
 IT NO.   5 K=   21 
 IT NO.   6 K=   18 
 IT NO.   7 K=   15 
 IT NO.   8 K=   12 
 IT NO.   9 K=    9 
 IT NO.  10 K=    6 
 IT NO.  11 K=    3 
 IT NO.  12 K=    0 
 
 
 
  35.00  35.00  35.00  35.00  35.00 
 
 
 
  35.00  34.74  34.74  34.74  38.25 
 
 
 
  35.00  34.49  34.49  34.49  37.50 
 
 
 
  35.00  34.23  34.23  34.23  36.75 
 
 
 
  35.00  33.98  33.98  33.98  36.00 
 249 
 
  35.00  33.73  33.73  33.73  35.25 
 
 
 
  35.00  33.48  33.48  33.48  34.50 
 
 
 
  35.00  33.24  33.24  33.24  33.75 
 
 
 
  35.00  32.99  32.99  32.99  33.00 
 
 
 
  35.00  32.75  32.75  32.75  32.25 
 
 
 
  35.00  32.51  32.51  32.51  31.50 
 
 
 
  35.00  32.27  32.27  32.27  30.75 
 
 
 
  30.00  30.00  30.00  30.00  30.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
  1        .45000        .45000        .45000        .45000        .45000 
  2        .00000        .44532        .44532        .44532        .44486 
  3        .00000        .44066        .44066        .44066        .43978 
  4        .00000        .43601        .43601        .43601        .43476 
  5        .00000        .43136        .43136        .43136        .42979 
  6        .00000        .42673        .42673        .42673        .42489 
  7        .00000        .42210        .42210        .42210        .42003 
  8        .00000        .41748        .41748        .41748        .41524 
  9        .00000        .41286        .41286        .41286        .41049 
 10        .00000        .40825        .40825        .40825        .40581 
 11        .00000        .40364        .40364        .40364        .40117 
 12        .00000        .39903        .39903        .39903        .39659 
 250 
 
 13        .35800        .35800        .35800        .35800        .35800 
 
At B=1.4 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 
 
  .362000  .362000  .362000  .362000  .362000 
 
 
 
  .000000  .367094  .367094  .367094  .367094 
 
 
 
  .000000  .371929  .371929  .371929  .367094 
 
 
 
  .000000  .376503  .376503  .376503  .367094 
 
 
 
  .000000  .380819  .380819  .380819  .367094 
 
 
 
  .000000  .384874  .384874  .384874  .367094 
 
 
 
  .000000  .388671  .388671  .388671  .367094 
 
 
 
  .000000  .392207  .392207  .392207  .367094 
 
 
 
  .000000  .395485  .395485  .395485  .367094 
 
 
 
  .000000  .398503  .398503  .398503  .367094 
 
 
 251 
 
 
  .000000  .401261  .401261  .401261  .367094 
 
 
 
  .000000  .403760  .403760  .403760  .367094 
 
 
 
  .406000  .406000  .406000  .406000  .406000 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   55 
 IT NO.   2 K=   30 
 IT NO.   3 K=   27 
 IT NO.   4 K=   24 
 IT NO.   5 K=   21 
 IT NO.   6 K=   18 
 IT NO.   7 K=   15 
 IT NO.   8 K=   12 
 IT NO.   9 K=    9 
 IT NO.  10 K=    6 
 IT NO.  11 K=    3 
 IT NO.  12 K=    0 
 
 
 
  35.00  35.00  35.00  35.00  35.00 
 
 
 
  35.00  34.74  34.74  34.74  38.25 
 
 
 
  35.00  34.49  34.49  34.49  37.50 
 
 
 
  35.00  34.23  34.23  34.23  36.75 
 
 
 
  35.00  33.98  33.98  33.98  36.00 
 
 
 252 
 
 
  35.00  33.73  33.73  33.73  35.25 
 
 
 
  35.00  33.48  33.48  33.48  34.50 
 
 
 
  35.00  33.24  33.24  33.24  33.75 
 
 
 
  35.00  32.99  32.99  32.99  33.00 
 
 
 
  35.00  32.75  32.75  32.75  32.25 
 
 
 
  35.00  32.51  32.51  32.51  31.50 
 
 
 
  35.00  32.27  32.27  32.27  30.75 
 
 
 
  30.00  30.00  30.00  30.00  30.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
  1        .45000        .45000        .45000        .45000        .45000 
  2        .00000        .44533        .44533        .44533        .44487 
  3        .00000        .44069        .44069        .44069        .43980 
  4        .00000        .43606        .43606        .43606        .43479 
  5        .00000        .43144        .43144        .43144        .42984 
  6        .00000        .42684        .42684        .42684        .42494 
  7        .00000        .42224        .42224        .42224        .42010 
  8        .00000        .41765        .41765        .41765        .41531 
  9        .00000        .41306        .41306        .41306        .41058 
 10        .00000        .40847        .40847        .40847        .40590 
 11        .00000        .40388        .40388        .40388        .40128 
 253 
 
 12        .00000        .39928        .39928        .39928        .39671 
 13        .35800        .35800        .35800        .35800        .35800 
 
At B=3.0 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 
 
  .362000  .362000  .362000  .362000  .362000 
 
 
 
  .000000  .370146  .370146  .370146  .370146 
 
 
 
  .000000  .377477  .377477  .377477  .370146 
 
 
 
  .000000  .383993  .383993  .383993  .370146 
 
 
 
  .000000  .389695  .389695  .389695  .370146 
 
 
 
  .000000  .394582  .394582  .394582  .370146 
 
 
 
  .000000  .398655  .398655  .398655  .370146 
 
 
 
  .000000  .401914  .401914  .401914  .370146 
 
 
 
  .000000  .404360  .404360  .404360  .370146 
 
 
 
  .000000  .405991  .405991  .405991  .370146 
 
 254 
 
 
 
  .000000  .406808  .406808  .406808  .370146 
 
 
 
  .000000  .406811  .406811  .406811  .370146 
 
 
 
  .406000  .406000  .406000  .406000  .406000 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   55 
 IT NO.   2 K=   30 
 IT NO.   3 K=   27 
 IT NO.   4 K=   24 
 IT NO.   5 K=   21 
 IT NO.   6 K=   18 
 IT NO.   7 K=   15 
 IT NO.   8 K=   12 
 IT NO.   9 K=    9 
 IT NO.  10 K=    6 
 IT NO.  11 K=    3 
 IT NO.  12 K=    0 
 
 
 
  35.00  35.00  35.00  35.00  35.00 
 
 
 
  35.00  34.74  34.74  34.74  38.25 
 
 
 
  35.00  34.49  34.49  34.49  37.50 
 
 
 
  35.00  34.23  34.23  34.23  36.75 
 
 
 
  35.00  33.98  33.98  33.98  36.00 
 
 255 
 
 
 
  35.00  33.73  33.73  33.73  35.25 
 
 
 
  35.00  33.48  33.48  33.48  34.50 
 
 
 
  35.00  33.24  33.24  33.24  33.75 
 
 
 
  35.00  32.99  32.99  32.99  33.00 
 
 
 
  35.00  32.75  32.75  32.75  32.25 
 
 
 
  35.00  32.51  32.51  32.51  31.50 
 
 
 
  35.00  32.27  32.27  32.27  30.75 
 
 
 
  30.00  30.00  30.00  30.00  30.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
  1        .45000        .45000        .45000        .45000        .45000 
  2        .00000        .44537        .44537        .44537        .44491 
  3        .00000        .44078        .44078        .44078        .43989 
  4        .00000        .43624        .43624        .43624        .43492 
  5        .00000        .43172        .43172        .43172        .43000 
  6        .00000        .42722        .42722        .42722        .42514 
  7        .00000        .42274        .42274        .42274        .42034 
  8        .00000        .41825        .41825        .41825        .41559 
  9        .00000        .41376        .41376        .41376        .41089 
 10        .00000        .40925        .40925        .40925        .40625 
 256 
 
 11        .00000        .40472        .40472        .40472        .40166 
 12        .00000        .40014        .40014        .40014        .39712 
 13        .35800        .35800        .35800        .35800        .35800 
 
IN THE DIRECTION OF FILM THICKNESS 
LiBr-H2O 
 
At B = 0.0 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 IT NO.   1 K=   12 
 IT NO.   2 K=    3 
 IT NO.   3 K=    3 
 IT NO.   4 K=    3 
 IT NO.   5 K=    3 
 IT NO.   6 K=    3 
 IT NO.   7 K=    3 
 IT NO.   8 K=    3 
 IT NO.   9 K=    3 
 IT NO.  10 K=    3 
 IT NO.  11 K=    3 
 IT NO.  12 K=    3 
 IT NO.  13 K=    3 
 IT NO.  14 K=    3 
 IT NO.  15 K=    3 
 IT NO.  16 K=    3 
 IT NO.  17 K=    3 
 IT NO.  18 K=    3 
 IT NO.  19 K=    3 
 IT NO.  20 K=    3 
 IT NO.  21 K=    3 
 IT NO.  22 K=    3 
 IT NO.  23 K=    3 
 IT NO.  24 K=    3 
 IT NO.  25 K=    3 
 IT NO.  26 K=    3 
 IT NO.  27 K=    3 
 IT NO.  28 K=    3 
 IT NO.  29 K=    3 
 IT NO.  30 K=    3 
 IT NO.  31 K=    3 
 IT NO.  32 K=    3 
 IT NO.  33 K=    3 
 IT NO.  34 K=    3 
 IT NO.  35 K=    3 
 IT NO.  36 K=    3 
 257 
 
 IT NO.  37 K=    3 
 IT NO.  38 K=    3 
 IT NO.  39 K=    3 
 IT NO.  40 K=    3 
 IT NO.  41 K=    3 
 IT NO.  42 K=    3 
 IT NO.  43 K=    3 
 IT NO.  44 K=    3 
 IT NO.  45 K=    3 
 IT NO.  46 K=    3 
 IT NO.  47 K=    3 
 IT NO.  48 K=    3 
 IT NO.  49 K=    3 
 IT NO.  50 K=    3 
 IT NO.  51 K=    3 
 IT NO.  52 K=    3 
 IT NO.  53 K=    3 
 IT NO.  54 K=    3 
 IT NO.  55 K=    3 
 IT NO.  56 K=    3 
 IT NO.  57 K=    3 
 IT NO.  58 K=    3 
 IT NO.  59 K=    3 
 IT NO.  60 K=    3 
 IT NO.  61 K=    3 
 IT NO.  62 K=    3 
 IT NO.  63 K=    3 
 IT NO.  64 K=    3 
 IT NO.  65 K=    3 
 IT NO.  66 K=    3 
 IT NO.  67 K=    3 
 IT NO.  68 K=    3 
 IT NO.  69 K=    3 
 IT NO.  70 K=    3 
 IT NO.  71 K=    3 
 IT NO.  72 K=    3 
 IT NO.  73 K=    3 
 IT NO.  74 K=    3 
 IT NO.  75 K=    3 
 IT NO.  76 K=    3 
 IT NO.  77 K=    3 
 IT NO.  78 K=    3 
 IT NO.  79 K=    3 
 IT NO.  80 K=    3 
 IT NO.  81 K=    3 
 IT NO.  82 K=    3 
 258 
 
 IT NO.  83 K=    3 
 IT NO.  84 K=    3 
 IT NO.  85 K=    3 
 IT NO.  86 K=    2 
 IT NO.  87 K=    2 
 IT NO.  88 K=    2 
 IT NO.  89 K=    2 
 IT NO.  90 K=    2 
 IT NO.  91 K=    2 
 IT NO.  92 K=    2 
 IT NO.  93 K=    2 
 IT NO.  94 K=    2 
 IT NO.  95 K=    2 
 IT NO.  96 K=    2 
 IT NO.  97 K=    2 
 IT NO.  98 K=    2 
 IT NO.  99 K=    2 
 IT NO. 100 K=    2 
 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 
 .362000 .362000 .362000 
 .000000 .011294 .022194 .032522 .042115 .050824 .058517 .065081 .070426 .074480 
 .077199 .078558 .078558 
 .000000 .011299 .022202 .032534 .042129 .050839 .058532 .065095 .070437 .074490 
 .077205 .078562 .078558 
 .000000 .011303 .022211 .032545 .042142 .050853 .058546 .065109 .070449 .074499 
 .077212 .078565 .078558 
 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 
 .406000 .406000 .406000 
 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   39 
 IT NO.   2 K=    0 
  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44 
  44.44  44.44  44.44 
  35.00  35.19  35.19  35.19  37.55  35.19  35.19  35.19  35.19  35.19 
  35.19  35.19  35.19 
  35.00  35.13  35.13  35.13  36.70  35.13  35.13  35.13  35.13  35.13 
  35.13  35.13  35.13 
  35.00  35.06  35.06  35.06  35.85  35.06  35.06  35.06  35.06  35.06 
  35.06  35.06  35.06 
  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00 
  35.00  35.00  35.00 
 
 
 
 259 
 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000 .60000.60000 
.00000.08595.14838.19499.23041.25759.27847.29437.30619.31457.31991 .32251.32249 
.00000.01231.03669.06335.08846.11057.12922.14441.15624.16491.17057.17335 .17334 
.00000.00176.00907.02058.03396.04745.05996.07083 .07972.08645.09094.09317.09317 
.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500 .54500 
 
At B= 1.4Tesla 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
.362000.362000.362000.362000.362000.362000.362000.362000.362000.362000    
.362000.362000.362000 
.000000.011594.022762.033330.043134.052026.059875.066568.072015.076146 
.078916.080301.080301 
.000000.011598.022771.033342 .043148.052041.059889.066581.072027.076156 
.078923.080304.080301 
.000000.011603.022779.033353.043161.052055.059904 .066595.072038.076165 
.078929.080308.080301 
.406000.406000.406000.406000.406000.406000.406000.406000.406000.406000 
.406000.406000.406000 
 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 IT NO.   1 K=   39 
 IT NO.   2 K=    0 
  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44 
  44.44  44.44  44.44 
  35.00  35.19  35.19  35.19  37.55  35.19  35.19  35.19  35.19  35.19 
  35.19  35.19  35.19 
  35.00  35.13  35.13  35.13  36.70  35.13  35.13  35.13  35.13  35.13 
  35.13  35.13  35.13 
  35.00  35.06  35.06  35.06  35.85  35.06  35.06  35.06  35.06  35.06 
  35.06  35.06  35.06 
  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00 
  35.00  35.00  35.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000 
.00000.08790.15122.19823.23381.26103.28189.29775.30954.31787.32319.32577.32576 
.00000.01287.03810.06548.09109.11354.13242.14775.15967.16840.17409.17688.17687 
.00000.00188.00960.02162.03548.04938.06220.07331.08236.08920.09377.09603.09603 
.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500 
 260 
 
 
At B=3.0Tesla 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
.362000.362000.362000.362000.362000.362000.362000.362000.362000.362000 
.362000.362000.362000 
.000000.012669.024804.036232.046793.056342.064750.071906.077722.082129 
.085083.086559.086559 
.000000.012673.024812.036244.046807.056357.064764.071920.077734.082139 
.085089.086562.086559 
.000000.012678.024821.036255.046821.056371.064779.071933.077745.082148 
.085095.086566 .086559 
.406000.406000.406000.406000.406000.406000.406000.406000.406000.406000 
.406000.406000.406000 
 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 IT NO.   1 K=   39 
 IT NO.   2 K=    0 
  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44  44.44 
  44.44  44.44  44.44 
  35.00  35.19  35.19  35.19  37.55  35.19  35.19  35.19  35.19  35.19 
  35.19  35.19  35.19 
  35.00  35.13  35.13  35.13  36.70  35.13  35.13  35.13  35.13  35.13 
  35.13  35.13  35.13 
  35.00  35.06  35.06  35.06  35.85  35.06  35.06  35.06  35.06  35.06 
  35.06  35.06  35.06 
  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00 
  35.00  35.00  35.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000.60000 .60000 
.00000.09476.16114.20946.24552.27283.29361.30932.32094.32914.33437.33691 .33690 
.00000.01496.04326.07310.10045.12404.14366.15945.17166.18055.18634.18917 .18916 
.00000.00236.01161.02551.04109.05639.07028 .08218.09181.09904.10384.10622.10621 
.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500.54500 .54500 
 
 
 
 
 
 
 
 
 261 
 
LiCl-H2O 
At B=0.0Tesla 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 IT NO.   1 K=   12 
 IT NO.   2 K=    3 
 IT NO.   3 K=    3 
 IT NO.   4 K=    3 
 IT NO.   5 K=    3 
 IT NO.   6 K=    3 
 IT NO.   7 K=    3 
 IT NO.   8 K=    3 
 IT NO.   9 K=    3 
 IT NO.  10 K=    3 
 IT NO.  11 K=    3 
 IT NO.  12 K=    3 
 IT NO.  13 K=    3 
 IT NO.  14 K=    3 
 IT NO.  15 K=    3 
 IT NO.  16 K=    3 
 IT NO.  17 K=    3 
 IT NO.  18 K=    3 
 IT NO.  19 K=    3 
 IT NO.  20 K=    3 
 IT NO.  21 K=    3 
 IT NO.  22 K=    3 
 IT NO.  23 K=    3 
 IT NO.  24 K=    3 
 IT NO.  25 K=    3 
 IT NO.  26 K=    3 
 IT NO.  27 K=    3 
 IT NO.  28 K=    3 
 IT NO.  29 K=    3 
 IT NO.  30 K=    3 
 IT NO.  31 K=    3 
 IT NO.  32 K=    3 
 IT NO.  33 K=    3 
 IT NO.  34 K=    3 
 IT NO.  35 K=    3 
 IT NO.  36 K=    3 
 IT NO.  37 K=    3 
 IT NO.  38 K=    3 
 IT NO.  39 K=    3 
 IT NO.  40 K=    3 
 IT NO.  41 K=    3 
 IT NO.  42 K=    3 
 262 
 
 IT NO.  43 K=    3 
 IT NO.  44 K=    3 
 IT NO.  45 K=    3 
 IT NO.  46 K=    3 
 IT NO.  47 K=    3 
 IT NO.  48 K=    3 
 IT NO.  49 K=    3 
 IT NO.  50 K=    3 
 IT NO.  51 K=    3 
 IT NO.  52 K=    3 
 IT NO.  53 K=    3 
 IT NO.  54 K=    3 
 IT NO.  55 K=    3 
 IT NO.  56 K=    3 
 IT NO.  57 K=    3 
 IT NO.  58 K=    3 
 IT NO.  59 K=    3 
 IT NO.  60 K=    3 
 IT NO.  61 K=    3 
 IT NO.  62 K=    3 
 IT NO.  63 K=    3 
 IT NO.  64 K=    3 
 IT NO.  65 K=    3 
 IT NO.  66 K=    3 
 IT NO.  67 K=    3 
 IT NO.  68 K=    3 
 IT NO.  69 K=    3 
 IT NO.  70 K=    3 
 IT NO.  71 K=    3 
 IT NO.  72 K=    3 
 IT NO.  73 K=    3 
 IT NO.  74 K=    3 
 IT NO.  75 K=    3 
 IT NO.  76 K=    3 
 IT NO.  77 K=    3 
 IT NO.  78 K=    3 
 IT NO.  79 K=    3 
 IT NO.  80 K=    3 
 IT NO.  81 K=    3 
 IT NO.  82 K=    3 
 IT NO.  83 K=    3 
 IT NO.  84 K=    3 
 IT NO.  85 K=    3 
 IT NO.  86 K=    3 
 IT NO.  87 K=    3 
 IT NO.  88 K=    3 
 263 
 
 IT NO.  89 K=    3 
 IT NO.  90 K=    3 
 IT NO.  91 K=    3 
 IT NO.  92 K=    3 
 IT NO.  93 K=    3 
 IT NO.  94 K=    3 
 IT NO.  95 K=    3 
 IT NO.  96 K=    3 
 IT NO.  97 K=    3 
 IT NO.  98 K=    3 
 IT NO.  99 K=    3 
 IT NO. 100 K=    3 
.362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 
.362000 .362000 .362000 
.000000 .011294 .022194 .032522 .042115 .050824 .058517 .065081 .070426 .074480 
.077199 .078558 .078558 
.000000 .011299 .022202 .032534 .042129 .050839 .058532 .065095 .070437 .074490 
.077205 .078562 .078558 
.000000 .011303 .022211 .032545 .042142 .050853 .058546 .065109 .070449 .074499 
.077212 .078565 .078558 
.406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 
.406000 .406000 .406000 
 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   39 
 IT NO.   2 K=    0 
  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00 
  35.00  35.00  35.00 
  35.00  30.17  30.17  30.17  30.17  30.17  30.17  30.17  30.17  30.17 
  30.17  30.17  32.25 
  35.00  30.11  30.11  30.11  30.11  30.11  30.11  30.11  30.11  30.11 
  30.11  30.11  31.50 
  35.00  30.05  30.05  30.05  30.05  30.05  30.05  30.05  30.05  30.05 
  30.05  30.05  30.75 
  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00 
  30.00  30.00  30.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000 .45000 
.00000.05938.10350.13699.16278.18276.19823.21007.21892.22521.22923.23119 .23118 
.00000.00783.02380.04169.05887.07421.08731.09806.10650.11270.11677.11877 .11876 
.00000.00103.00547.01269.02129.03013.03845.04577.05180.05640.05948.06102 .06101 
 264 
 
.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800 .35800 
 
At B= 1.4Tesla 
 
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 
.362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 
.362000 .362000 .362000 
.000000 .011506 .022596 .033095 .042837 .051676 .059479 .066134 .071551 .075661 
.078415 .079793 .079793 
.000000 .011511 .022605 .033106 .042851 .051690 .059493 .066148 .071563 .075670 
.078422 .079796 .079793 
.000000 .011515 .022613 .033118 .042864 .051705 .059508 .066162 .071575 .075679 
.078428 .079799 .079793 
.406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 
.406000 .406000 .406000 
 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   39 
 IT NO.   2 K=    0 
  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00 
  35.00  35.00  35.00 
  35.00  30.17  30.17  30.17  30.17  30.17  30.17  30.17  30.17  30.17 
  30.17  30.17  32.25 
  35.00  30.11  30.11  30.11  30.11  30.11  30.11  30.11  30.11  30.11 
  30.11  30.11  31.50 
  35.00  30.05  30.05  30.05  30.05  30.05  30.05  30.05  30.05  30.05 
  30.05  30.05  30.75 
  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00 
  30.00  30.00  30.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000 .45000 
.00000.06034.10494.13866.16455.18457.20004.21187.22071.22698.23099.23294 .23293 
.00000.00809.02446.04272.06016.07569.08891.09974.10824.11448.11857.12058 .12057 
.00000.00108.00570.01316.02199.03103.03951.04695.05308.05774.06086.06241 .06241 
.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800 .35800 
 
At B= 3.0Tesla 
  
1VEL.DIST ON A FILM BY GAUSS-SEIDEL ITERATION// 
 
 265 
 
.362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 .362000 
.362000 .362000 .362000 
.000000 .012268 .024043 .035151 .045430 .054733 .062933 .069916 .075595 .079899 
.082784 .084227 .084227 
.000000 .012273 .024051 .035162 .045443 .054748 .062947 .069930 .075606 .079909 
.082791 .084230 .084227 
.000000 .012277 .024060 .035173 .045457 .054763 .062962 .069944 .075618 .079918 
.082797 .084233 .084227 
.406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 .406000 
.406000 .406000 .406000 
 
1TEMP DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   1 K=   39 
 IT NO.   2 K=    0 
  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00  35.00 
  35.00  35.00  35.00 
  35.00  30.17  30.17  30.17  30.17  30.17  30.17  30.17  30.17  30.17 
  30.17  30.17  32.25 
  35.00  30.11  30.11  30.11  30.11  30.11  30.11  30.11  30.11  30.11 
  30.11  30.11  31.50 
  35.00  30.05  30.05  30.05  30.05  30.05  30.05  30.05  30.05  30.05 
  30.05  30.05  30.75 
  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00  30.00 
  30.00  30.00  30.00 
 
1CONC DIST ON A FILM BY GAUSS-SEIDEL ITERATION/ 
 
 IT NO.   0 K=    0 
 IT NO.   0 K=    0 
.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000.45000 .45000 
.00000.06377.11001.14450.17072.19085.20632.21811.22689.23311.23708.23901 .23900 
.00000.00903.02689.04639.06476.08093.09459.10571.11439.12074.12490.12694 .12693 
.00000.00128.00657.01489.02456.03431.04336.05123.05766.06254.06580.06742 .06742 
.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800.35800 .35800 
 
 
 
 
 
 
 
 266 
 
APPENDIX D 
 
GAUSSIAN ELIMINATION 
 
Gaussian elimination is direct method for solving sets of simultaneous linear algebraic equations 
of the general form 
 
a11x1 + a12x2 + ……………+ a1nxn = b1 
a21x1 + a22x2 + ……………+ a2nxn = b2 
a31x1 + a32x2 + ……………+ a3nxn = b3   (D.1)   
            ……………    
            …………… 
            …………… 
            …………… 
an1x1 + an2x2 + ……………+ annxn = bn 
where x1, x2, x3 ……….xn are the unknowns and the coefficients aij and bi are all known 
constants. This set of equations can be expressed in matrix form as follows 
   a11    a12 ………….. a1n     x1  b1             
 a11    a12 ………….. a1n       x2  b2 
 ……………………….      .   . 
 ……………………….      .     = .    (D.2) 
 ……………………….      .  . 
 an1    an2……………ann       xn  bn 
 
 
   AX = B   (D.3) 
 
The unknowns are successfully eliminated by algebraic manipulations. The first equation can be 
used to eliminate x1 from the remaining n-1 equations. The modified second equation is then 
used to eliminate x2 from the remaining n-2 equations and so on until the last equation contains 
only xn. Thus xn may be found, followed by all the other unknowns by back substitution. Let the 
1 1
coefficient shown in equation b1 and b2 be given by the notation aij  ai ; 
 267 
 
After the  K th  elimination the modified coefficient are:-  
  a k1  a k kij ij akj  
  bk11  b
k
1 b
k
k     (D.4) 
where  j  =  k,  k+1………..n 
The final set of equations is  
a1 x  a111 1 12x2  ............ a
1
1n xn  b
1
1  
    a22x2  ..................a2n xn  b2   
         +………….+  = … 
         +………+ = ..   (D.5) 
             +…..+ = .. 
    annxn  b  
The unknown are obtained in reverse order 
b n
    xn = 
n    (D.6) 
a nnn
b i
n i
i  ajii ij x j    xn =   (D.7) 
a iii
 
 
 268