Calendering of Newtonian and Non-

Newtonian Fluids

By

Muhammad Zahid

CMS No: 8056

Reg. No: F11C09P05001

DOCTOR OF PHILOSOPY

IN

MATHEMATICS

DEPARTMENT OF BASIC SCIENCES RIPHAH INTERNATIONAL UNIVERSITY

ISLAMABAD, PAKISTAN MARCH, 2015

Calendering of Newtonian and Non-

Newtonian Fluids

By

Muhammad Zahid

Supervised by

Dr. Muhammad Afzal Rana

and

Prof. Dr. Tahira Haroon

DEPARTMENT OF BASIC SCIENCES RIPHAH INTERNATIONAL UNIVERSITY

ISLAMABAD, PAKISTAN March, 2015

In the name of Almighty Allah, the Most Gracious, the Ever Merciful

Calendering of Newtonian and Non-

Newtonian Fluids

By

Muhammad Zahid

Supervised by

Dr. Muhammad Afzal Rana

(Department of Basic Sciences, Riphah International University, Islamabad, Pakistan)

Prof. Dr. Tahira Haroon (Department of Mathematics, COMSATS Institute of Information Technology,

Islamabad, Pakistan)

A thesis submitted in the partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPY

IN

MATHEMATICS

DEPARTMENT OF BASIC SCIENCES RIPHAH INTERNATIONAL UNIVERSITY

ISLAMABAD, PAKISTAN

Dedication

Dedicated to My Beloved Parents, My Lovely Wife, My Son Afnan Zahid and to My Daughter Anashrah Zahid

Acknowledgements To begin with the name of Almighty ALLAH, Who inculcated consecration upon me to fulfill the requirements for this thesis. I offer my humblest words of thanks to the Holy Prophet Muhammad (Peace be upon him) who is forever a torch of guidance for humanity. I would like to express my sincere gratitude to my Supervisors Dr. Muhammad Afzal Rana and Prof. Dr. Tahira Haroon for providing me with the opportunity and means to undertake a fundamental study of the behavior of non-Newtonian material in a calendering environment. I am deeply indebted to Prof. Dr. A. M. Siddiqui (Pennsylvania State University, USA) for his guidance, encouragement and patience throughout the course of my studies. He always showed the right way of doing research by giving stimulating ideas and fruitful suggestions. Without his encouragement, patience and insightful suggestion, I could not finish my program so smooth. I, with all my pleasures thanks to Dean Riphah International University and Head of Basic Science Department for their support and especially to my Teachers who give me direction and guidance for completing my studies. It was indeed a rewarding journey for me I am grateful to my parent University COMSATS Institute of Information Technology, Abbottabad for giving me permission to complete this work. It gives me great pleasure to express my profound gratitude to my colleague Dr. Saqib Hussain, Aamir Shahzad, Ossam Chohan, Syed Z. Ali Zaidi and Dr. Muahmmad Ayub for their help and encouragement during the PhD work. Last but not the least my family members being helpful and supportive specially my younger brother Aamir Ghiyyas, my mother and wife. Completing this research would have never been easy without their emotional support. I am very thankful to my wife, her support and encouragement was instrumental to our success in the trials of the past years of which this project was just a small part, and the little angles, my son Afnan Zahid and daughter Anashrah Zahid never be ignored, whom mischievous yet innocent simile fresh me during my research work. I am extremely grateful to my elder brothers Muhammad Sajjad and Muhammad Tahir for their financial support throughout my PhD studies and research.

Muhammad Zahid

Contents

1 Introduction 51.1 Magnetohydrodynamics (MHD) Equations . . . . . . . . . . . . . . . 7

1.1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 81.1.2 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Calendering Geometry and Boundary Conditions . . . . . . . . . . . 11

1.4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Limited Region under Study . . . . . . . . . . . . . . . . . . . 13

1.5 A General Consideration of Lubrication Approximation Theory . . . 141.6 Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6.1 Perturbation Technique . . . . . . . . . . . . . . . . . . . . . 151.6.2 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . 16

2 Influence of Porous Rolls on Newtonian Calendering 172.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . 172.2 Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Detachment Point . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Sheet Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Operating Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Maximum Pressure . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Rate of Strain and Stress . . . . . . . . . . . . . . . . . . . . . 222.3.3 Power Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.4 Adiabatic Temperature . . . . . . . . . . . . . . . . . . . . . . 232.3.5 Roll-Separating Force . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Effect of Magnetohydrodynamics on Newtonian Calendering 293.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 31

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3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 MHD Calendering of Non-Isothermal Viscoplastic Materials 404.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Dimensionless Equations . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Sheet thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Heat Transfer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Operating Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6.1 Roll Separating-force . . . . . . . . . . . . . . . . . . . . . . . 464.6.2 Power Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 474.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Calendering analysis of a third-grade material 565.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Dimensionless Equations . . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Sheet thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.5 Solution for β � 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.5.1 Zero-order Problem . . . . . . . . . . . . . . . . . . . . . . . . 605.5.2 First-order Problem . . . . . . . . . . . . . . . . . . . . . . . . 605.5.3 Zero-order Solution . . . . . . . . . . . . . . . . . . . . . . . . 615.5.4 First-order solution . . . . . . . . . . . . . . . . . . . . . . . . 61

5.6 Operating Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.6.1 Roll-Separating Force . . . . . . . . . . . . . . . . . . . . . . . 635.6.2 Power Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.6.3 Normal Stresses Effect . . . . . . . . . . . . . . . . . . . . . . 64

5.7 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 645.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Calendering of a second-grade material onto a moving porous sheet 706.1 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 716.1.2 Dimensionless Equations . . . . . . . . . . . . . . . . . . . . . 73

6.2 Solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 Coating Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4 Operating Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.4.1 Separation Force . . . . . . . . . . . . . . . . . . . . . . . . . 766.4.2 Power Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 776.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7 Calendering analysis of a third-grade material onto a moving thinsheet 857.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . 857.2 Dimensionless equations . . . . . . . . . . . . . . . . . . . . . . . . . 877.3 Solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.3.1 Zero-order problem and its solution: . . . . . . . . . . . . . . . 887.3.2 First-order problem and its solution: . . . . . . . . . . . . . . 89

7.4 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 927.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8 Conclusions and Future Work 958.1 Summary of Work Undertaken . . . . . . . . . . . . . . . . . . . . . . 958.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

iii

Abstract

In this thesis, a theoretical analysis has been presented for the calendering processof incompressible Newtonian and non-Newtonian materials when they pass throughthe small gap between two counter rotating rolls. The calendering phenomena of amaterial flow between two co-rotating rolls to produce sheets of specific thickness andfinal appearance is an important process in many industries, such as the plastics,paper, rubber and steel industries for the production of rolled sheets of specificthickness and final appearance. In particular, the calendering of molten polymersis a process for the production of continuous sheet or film by squeezing the meltbetween a pair of heated counter-rotating rolls. A bench-top apparatus simulatesthe process. In this study two types of calendering processes are considered. Inthe first process, a molten material is transformed into a sheet by passing througha pair of rollers, whereas in second process a uniform film of liquid is depositedon a moving sheet. These two processes are quite similar, there is a converging-diverging character to the kinematics, and we can expect the dynamics to be similarto that described in the first process. The major difference is in the character of theseparation region, where the material splits and adheres to both moving surfaces.In the first process, it is assumed that the material separates clearly from the roll,whereas, in a second process, it is assumed that the material evenly wets both theroll and the sheet.

The lubrication approximation theory (LAT) is outlined together with its calcu-lations for viscoelastic and viscoplastic materials. Measurements of the gap betweenthe roll surfaces, velocity and pressure profiles, film thickness, roll-separating forceand power input to the rolls were made for a known roll speed and external load.The control of these engineering parameters are of excessive significance during themanufacture process.The goal of this work is to develop various mathematical mod-els to characterize the effect of various materials in the calendering process.

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List of Figures

1.1 Schematic presentation of a typical calendering process . . . . . . . . 61.2 Roller setup in a typical ’I’, inverted ’L’ and ’Z’ type Calenders . . . 71.3 Calendering Geometry (Not to scale) . . . . . . . . . . . . . . . . . . 121.4 Geometrical representation of h(x) . . . . . . . . . . . . . . . . . . . . 13

2.1 Schematic representation for the flow in the gap between two porousco-rotating calender rolls . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Effete of Reynolds number on pressure-gradient distribution . . . . . . 212.3 Solution curve for the detachment point when p = 0 as x → −∞,

fixing Re = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Power input during calendering process . . . . . . . . . . . . . . . . . 232.5 Roll-separating force during calendering process . . . . . . . . . . . . 242.6 Velocity distribution at x = 0 and at x = 0.25 respectively . . . . . . . 262.7 Velocity distribution at x = 0.5 and at x = 0.75 respectively . . . . . . 272.8 Contour graph pattern for the longitudinal velocity distribution during

the calendering process at Re = 0.2 and at Re = 0.8 respectively . . . 272.9 Contour graph pattern for the longitudinal velocity distribution during

the calendering process at Re = 1.2 and at Re = 2.0 respectively . . . 282.10 Contour graph pattern for the longitudinal velocity distribution during

the calendering process at Re = 2.5 . . . . . . . . . . . . . . . . . . . 28

3.1 Geometry of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 The effect of magnetic field on the point where sheet leaves the roll . . 353.3 Velocity distribution at (a) x = 0.0, (b) x = 0.2, (c) M = 0.4 and λ

= 0.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 The effect of M and ε on pressure distribution . . . . . . . . . . . . . 373.5 (a) Power transmitted to the fluid when M = 0.4, λ = 0.38, ε = 0.01.

(b) Separation force for M = 0.4, λ = 0.38, ε = 0.01 . . . . . . . . . 37

4.1 Physical presentation of calendering process between two co-rotatingheated rolls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Geometry of the studied physical model . . . . . . . . . . . . . . . . . 424.3 Effect of MHD on velocity distribution at x = 0, fixing ζ = 0.5 . . . . 48

v

4.4 Effect of MHD on velocity distribution at x = 0.25, fixing ζ = 0.5 . . . 484.5 Effect of MHD on velocity distribution at x = 0.5, fixing ζ = 0.5 . . . 494.6 Effect of MHD on velocity distribution at x = 0.75, fixing ζ = 0.5 . . . 494.7 Effect of viscoplastic parameter ζon velocity distribution at x = 0,

fixing M = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.8 Effect of viscoplastic parameter ζ on velocity distribution at x = 0.25,

fixing M = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.9 Effect of viscoplastic parameter ζon velocity distribution at x = 0.5,

fixing M = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.10 Effect of viscoplastic parameter ζ on velocity distribution at x = 0.75,

fixing M = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.11 Effect of viscoplastic parameter ζ on pressure gradient distribution

fixing M = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.12 Effect of MHD on pressure gradient distribution fixing ζ = 0.5 . . . . 514.13 Effect of viscoplastic parameter ζ on pressure distribution fixing M = 0.5 514.14 Effect of MHD on pressure distribution fixing ζ = 0.5 . . . . . . . . . 514.15 Effect of M on temperature distribution fixing ζ = 0.5 and Br = 0.5 . 524.16 Effect of ζ on temperature distribution fixing M = 0.5 and Br = 0.5 . 524.17 Effect of Br on temperature distribution fixing ζ = 0.5 and M = 0.5 . 524.18 Yielded/unyielded region for viscoplastic fluid during the calendering

process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.19 Contour graph pattern for longitudinal velocity in the calendering vis-

coplastic material at M = 0.5, ζ = 0.5, Br = 0.5 . . . . . . . . . . . . 544.20 Contour graph pattern for longitudinal velocity in the calendering vis-

coplastic material at M = 2, ζ = 2, Br = 0.5 . . . . . . . . . . . . . . 54

5.1 Geometry of the studied physical model . . . . . . . . . . . . . . . . . 575.2 Geometry in dimensionless variables of the physical model . . . . . . 595.3 Effect of β on velocity at x = −0.5 . . . . . . . . . . . . . . . . . . . 655.4 Effect of β on velocity at x = −0.25 . . . . . . . . . . . . . . . . . . . 655.5 Effect of β on velocity at x = 0 . . . . . . . . . . . . . . . . . . . . . 665.6 Effect of β on velocity at x = 0.25 . . . . . . . . . . . . . . . . . . . . 665.7 Effect of β on velocity at x = 0.4 . . . . . . . . . . . . . . . . . . . . 675.8 Effect of β on velocity at x = 0.5 . . . . . . . . . . . . . . . . . . . . 675.9 Effect of β on velocity at x = 0.6 . . . . . . . . . . . . . . . . . . . . 685.10 Effect of β on velocity at x = 1.0 . . . . . . . . . . . . . . . . . . . . 685.11 Effect of β on pressure gradient distribution . . . . . . . . . . . . . . 685.12 Effect of β on pressure distribution . . . . . . . . . . . . . . . . . . . 685.13 Normal stress effect at position x = 0 and at x=0.5, respectively fixing

β = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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6.1 Physical presentation of calendering of a second-grade material ontoa moving porous sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 Geometry of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 726.3 Velocity distribution at x = 0.0 fixing Re = 0.1 . . . . . . . . . . . . . 796.4 Velocity distribution at x = 0.5 fixing Re = 0.1 . . . . . . . . . . . . . 796.5 Velocity distribution at x = 1.0 fixing Re = 0.1 . . . . . . . . . . . . . 816.6 Velocity distribution at x = 0.0 fixing β = 0.1 . . . . . . . . . . . . . . 816.7 Velocity distribution at x = 0.5 fixing β = 0.1 . . . . . . . . . . . . . . 816.8 Velocity distribution at x = 1.0 fixing β = 0.1 . . . . . . . . . . . . . . 816.9 Axial distribution of the pressure gradient fixing Re = 0.1 . . . . . . . 826.10 Axial distribution of the pressure gradient fixing β = 0.1 . . . . . . . 826.11 Separation force for Re = 0.1, β = 0.05 and Q = 1.2414 . . . . . . . 826.12 Power input for Re = 0.1, β = 0.05 and Q = 1.2414 . . . . . . . . . 826.13 Axial distribution of the pressure for β = 0.5, Re = 0.1 and Q = 1.1840 836.14 Axial distribution of the pressure for β = 0.1, Re = 0.5 and Q = 1.24320 83

7.1 Geometry of the studied physical model . . . . . . . . . . . . . . . . . 867.2 Effect of β on velocity at x = 0 . . . . . . . . . . . . . . . . . . . . . 917.3 Effect of β on velocity at x = 0.25 . . . . . . . . . . . . . . . . . . . . 917.4 Effect of β on velocity at x = 0.5 . . . . . . . . . . . . . . . . . . . . 927.5 Effect of β on velocity at x = 0.75 . . . . . . . . . . . . . . . . . . . . 927.6 Effect of β on velocity at x = 1.0 . . . . . . . . . . . . . . . . . . . . 937.7 Effect of β on velocity at x = 2.0 . . . . . . . . . . . . . . . . . . . . 937.8 Effect of β on pressure gradient distribution . . . . . . . . . . . . . . 947.9 Effect of β on pressure distribution . . . . . . . . . . . . . . . . . . . 94

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List of Tables

2.1 Effect of Reynolds number on detachment point, sheet thickness, powerinput and separation force . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Effect of magnetic field on attachment/detachment points power androll-separating force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Effect of ε on power and roll-Separating force . . . . . . . . . . . . . 38

4.1 Effect of viscoplastic parameter ζ on detachment point, sheet thick-ness, roll separation force and power fixing M = 0.5 . . . . . . . . . . 47

4.2 Effect of MHD on detachment point, sheet thickness, Roll-Separatingforce and Power fixing ζ = 0.5 . . . . . . . . . . . . . . . . . . . . . . 53

5.1 The effect of material parameter β on leave-off distance, final sheetthickness, power input and roll-separating force . . . . . . . . . . . . 64

6.1 Effect of non-Newtonian parameter on different rheological parame-ters fixing Re = 0.1, for calendering of second-grade material onto amoving porous sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Effect of Reynolds number on different rheological parameters fixingβ = 0.1, for calendering of second-grade material onto a movingporous sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3 Effect of non-Newtonian parameter on dimensionless volumetric flowrate, separation force, power input and pressure distribution fixingRe = 0.1, for calendering of a second-grade material onto a movingporous sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4 Effect of Reynolds number on dimensionless volumetric flow rate, sep-aration force, power input and pressure distribution fixing β = 0.1,for calendering of a second-grade material onto a moving porous sheet 84

7.1 Effect of non-Newtonian parameter on various rheological parameters 93

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Notations

Br Brinkman numberCP Heat capacityEc Eckert numberF Dimensionless roll-separating forceF Dimensional roll-separating forceh(x) Roll variable curvature lengthH0 Distance between rolls (nip)Hf Entering Sheet thicknessK Consistency indexk Thermal conductivityn Power law indexp Dimensional Hydrodynamic pressureP Modified pressurePw Dimensionless powerPw Dimensional powerPr Prandtl numberQ Dimensional volumetric flow rate per unit widthQ Dimensionless volumetric flow rate per unit widthR Radius of the rollRe Reynolds numberSxy Dimensionless Stress tensorSxy Dimensional Stress tensor∆T Adiabatic temperatureu, v Dimensional velocity components in x- and y-directions respectivelyu, v Dimensionless velocity components in x- and y-directions respectivelyU Rolls velocityW Rolls widthx, y Dimensional Axial and Transversal coordinates respectivelyx, y Dimensionless Axial and Transversal coordinates respectivelyxf Axial coordinate at the inletGeek letters

β Non-Newtonian parameter

ix

θ Dimensionless temperatureθ Dimensional temperatureθ0 Roll temperatureλ Separation pointρ Fluid densityω Angular velocity

x

Preface

The process of transforming a polymer melt into films, webs or sheets by squeezingthe polymer melt between several pairs of rolls which revolve in opposite direc-tions is termed calendering. This process is commonly used for the manufacturingof various PVC sheets, resilient flooring tiles, rubber sheets, rubber tires, leatherclothing, shrink films for packaging, textured or embossed surfaces, etc. The calen-dering process for shaping of materials into sheets and films has been the subject ofinvestigation for many decades now.

Researchers have done extensive work in calendering. This process was intro-duced in the 1830s by Edwin Chaffee and Charles Goodyear in the United States [1],however the first theoretical analysis of calendering was carried out by Ardichvili [2]on the basis of the Reynolds lubrication theory of Newtonian hydrodynamics. Thehydrodynamics theory of calendering as it stands today was initially developed byGaskell [3]. He was the first to analyze the process by developing a mathematicalprocedure in one dimension for Newtonian fluids. He derived a specific extensionfor a purely viscous (non-elastic and time independent) fluid. The notebooks byMcKelvey [4], Middleman [5] and Agassant et al. [6] summarized previous works oncalendering and described models for the calendering of Newtonian and power-lawf1uids. These models based on the lubrication approximations and unidirectionalf1ow, i.e., no f1ow in the widthwise direction. It is reasonable approximation foroperations where the width of the sheet is large, but not true when the width ofthe sheet is narrow. Commercial operations where the sheet width is small are verycommon in the food industry, notably sheeting of dough. Numerical results arepresented by Levine et al. [7], Engmann et al. [8], Mitsoulis et al. [9], Wang etal. [10], Abdallah [11] etc., on calendering of finite width sheets. Ralph [12] hascomputed pressure distribution for Newtonian and Maxwell fluids in the lubricantfilm separating two cylindrical rollers. A comparison of the results indicates thatshear elasticity can have effects of first-order significance on both the magnitudeand form of the pressure distribution function, reducing load-carrying capacity andmaximum shear stress in the film by sizeable amounts. Mitsoulis [13] perform somenumerical test to prove that the lubrication approximation theory (LAT) results canbe used as a good approximation for the detachment point.

In the sheet calendering process, a uniform film of liquid or material is deposited

1

onto a moving sheet. The Sheet calendering process has many industrial applicationslike manufacturing photographic films, coated products, coating thin sheets, papercoating and magnetic-recording media etc.. This process was introduced theoreti-cally by Greener and Middleman [14]. Following Middleman’s work, a great deal ofeffort was made by numerous researchers to improve the model [14]. The finding till1977 has been summarized by Middleman [5]. Sofou and Mitsoulis [15] employ LATto give numerical results by considering HerschelBulkley model of viscoplasticityand got results for several engineering quantities.

Recently much attention has been given on calendering of non-Newtonian materi-als. Since non-Newtonian materials are encountered in many technological processes,therefore, are significant in both theoretical and industrial contexts. Due to complexnature, the solutions of flow problems for non-Newtonian materials are in generalmore difficult to obtain. This is true not only for exact analytical solutions, but alsofor numerical solutions. However, there are studies available in the literature on thecalendering process in which nonlinear equation regarding non-Newtonian materialis treated successfully[10, 13, 16].

The study of magnetohydrodynamics (MHD) flows has great interest, as theseflows are quite prevalent in nature. Such flows have extensive applications in in-dustry. For example electromagnetic forces can be used to pump liquid metals inthe cooling systems of nuclear power stations, without the need for any movingparts. They can shape the flow of a molten metal and so aid in controlling its shapeonce solidified, and can even levitate and heat a sample of metal to prevent anycontact with (and consequent contamination from) a container. Hartmann [17] wasfirst who studied an incompressible electrically conducting fluid under the action oftransverse magnetic field. Due to extensive applications in processing industry, wehave analyzed MHD material in Chapter 3 and 4.

Heat generation in the processing industry plays an important role during han-dling and shaping the end product. The heat transfer analysis of calendering processis an important area of design of calender machines. The study of this class of heattransfer problems represents a fundamental branch that allows to obtain a betterthermal design and control polymer processing performance. Various workers haveanalyzed the interesting problems regarding the calendering process in this direction[16, 18].

Chapter wise summary of the thesis is as follows:Chapter 1 builds the mathematical prerequisites required for studying calender-

ing process in the theory of a typical calender system particularly the specific areaunder study. We introduce important basic laws encountered in fluid mechanics,such as the laws of conservation of mass, momentum and energy. Different consti-tutive equations, a general consideration of LAT and some important mathematicalmethods in order to solve the governing boundary value problems are also discussed.

Chapter 2 deals with the process of Newtonian material calendering using porous

2

rolls. Exact solutions for velocity field and pressure gradient are presented withoutusing the LAT. Integrated engineering quantities like extreme pressure, power androll-separating force are calculated numerically as well as presented graphically. Itis noted that the transverse mass across the material has great influence on velocity,pressure distributions and detachment point. The work of this Chapter has beensubmitted in European Journal of Mechanics B/Fluids.

The effect MHD on Newtonian calendering process is incorporated in Chapter 3.The LAT is used to simplify the equations of motion. Exact solutions for velocityprofile, pressure gradient, flow rate per unit width, rate of strain, shear stress,maximum shear rate and shear stress at the roll surface are obtained. The value of λ,the distance from the nip to the point where the sheet leaves the rolls, is calculatedusing Newton-Cotes formula along with the regula-falsi method and Simpson’s rule.Numerical results are presented for pressure distribution, Power transmitted by rollto the material, force separating the two rolls are discussed and shown graphically.It is found that the Magnetic field provides the controlling parameter to increaseor decrease power input, roll-separating force and the distance between attachmentand detachment points, which are useful in the calendaring process. This analysishas been published in Journal of Plastic Film and Sheeting, 29(4) 347364,2013.

In Chapter 4 we model the incompressible, non-isothermal flow of MHD vis-coplastic material when it passes through the small gap between two counter rotat-ing rolls. The conservation equations in light of LAT are non-dimensionalized andanalytic solutions for velocity, pressure gradient and temperature distribution arepresented. The value of λ is calculated using Simpson’s 1/3 formula for numericalintegration along with the modify regula-falsi method. Moreover, engineering quan-tities such as extreme pressure, power transmitted by roll and roll-separating forceare computed. The outcome shows that the presence of the viscoplastic parametereffect significantly the velocity field, pressure gradient and temperature distribution.It is found that the magnetic field provides a mechanism to control power input,roll-separating force and the distance between attachment and detachment points,which are very useful in the calendaring process. The work of this Chapter has beenaccepted for publication in Journal of Plastic Film and Sheeting.

Chapter 5 examines the behavior of a non-Newtonian material when it is draggedthrough the small gap between two counter rotating rolls. Equations of continuityand momentum using LAT are solved for velocity and pressure with the help of per-turbation technique. By considering the influence of material parameter, detachmentpoint for the third-grade material in the calendering process is determined. Inte-grated engineering quantities like extreme pressure, power-input and roll-separatingforce are also calculated. It is observed that the third-grade material parameter hasgreat influence on detachment point, velocity and pressure distribution, which areuseful in the calendaring process. The work of this Chapter has been published in

3

Journal of Plastic Film and Sheeting, Vol. 30(4) 345368, 2014Chapter 6 addresses the effectiveness of LAT used to give numeric results in

calendering onto a moving flat porous sheet. The rate of injection at the roll surfaceis assumed equal to the rate of suction at the web. The second-grade material isused, which becomes the Newtonian material model with some suitable variations.Unified quantities of interest has been attained like pressure distribution, coatingthickness, splitting point, stresses, Separation forces and power consumption etc.,are calculated. These results are shown graphically. It is found that material param-eter β and Reynolds number are the devices to control flow rate, coating thicknessseparation points, Separation force, Power input and Pressure. The contents of thisChapter have been accepted for publication in Tribology Transactions.

Chapter 7 analysis the calendering process in order to apply incompressible third-grade material onto a moving thin sheet. The LAT is used to simplify the equationsof motion. Solutions for velocity profile, pressure gradient, flow rate per unit width,and shear stress at the roll surface are obtained by using a regular perturbationmethod. Unified engineering quantities like extreme pressure, separation point, rollseparating force, power transmitted to the material by the roll and coating thicknessare also calculated. It is found that these processing quantities increase significantlyand monotonically as the material parameter increases. The material of this Chapterhas been published in Advances in Applied Mathematics, Springer Proceed-ings in Mathematics and Statistics Volume 87, pp 179-196, 2014.

4

Chapter 1

Introduction

The term “calender” is derived from the Greek word Kylindros (cylinders) andaccording to Webster’s International Dictionary, it means “to press (as cloth, rubber,paper, etc.) between rollers or plates in order to make smooth glossy or glazed or thinsheets”. Basically calendering is a process in which molten material is transformedinto films and sheets by squeezing it between two or more than two co-rotating high-precision rolls. Calenders are used to produce certain surface textures which may berequired for different applications. It has been in use for over a hundred years andwhen first developed it was mainly used for processing rubber. Today, calenderinglines are used to manufacture Polyvinyl chloride (PVC) sheet, floor covering, rubbersheet, and rubber tires. Calender sizes range up to 90 cm in diameter and 250 cm inwidth, with polymer materials up to 4000 kg/h, according to Tadmor and Gogos [19],who give further details of the machinery involved. Roll speeds may be as high as 2m/s for certain thin, flexible films (thickness less than 0.1 mm). Calendering lines arevery expensive in terms of capital investment in machinery. Film and sheet extrusionare inexpensive processes because the capital investment for the extruder itself is onlya fraction of the cost of a calender. However, the high quality and volume capabilitiesof calendering lines make them advantageous for many types of products, specificallyfor temperature sensitive materials. Polyvinyl chloride (PVC) is the major polymerthat is calendered, while elastomers maintain a heavy volume of calendered sheets.New trends seem to move from PVC towards ethylene–styrene interpolymers (ESI)and thermoplastic olefins (TPO) for improved recyclability. It should be furtherhighlighted that polymer blends and composites are also calendered and offer newchallenges due to the complex nature of the materials involved. They are also usedto texture or emboss surfaces. When producing PVC sheet and film, calender lineshave a great advantage over extrusion processes because of the shorter residencetimes, resulting in a lower requirement for stabilizer. This can be cost effective sincestabilizers are a key part of the overall expense of processing these polymers. Figure1.1 presents a typical calender line for manufacturing PVC sheet.

5

Figure 1.1: Schematic presentation of a typical calendering process

A typical calender system is composed of the five following units:

1. Plasticating unit

2. Calender

3. Cooling unit

4. Accumulator

5. Wind-up station

In the plasticating unit (which comprise the internal batch mixer and the strainerextruder) the material is melted and mixed and is fed in a continuous stream betweenthe nip of the first two rolls. In another variation of the process, the mixing maytake place elsewhere, and the material is simply re-heated on the roll mill. Oncethe material is fed to the mill, the first pair of rolls controls the feeding rate, whilesubsequent rolls in the calender standardize the sheet thickness. Most calendersystems have four rolls as does the one in Fig. 1.1, which is an inverted L- or F-type system. Other typical roll arrangements are shown in Fig. 1.2. After passingthrough the main calender, the sheet can be passed through a secondary calenderingoperation for embossing. The sheet is then passed through a chain of chilling rolls

6

Figure 1.2: Roller setup in a typical ’I’, inverted ’L’ and ’Z’ type Calenders

where it is cooled from both sides in an alternating fashion. After cooling, the filmor sheet is wound. One of the major concerns in a calendering system is producing afilm or sheet with a uniform thickness distribution with tolerances as low as ±0.005mm. To reach this, the dimensions of the rolls must be precise. It is also necessaryto compensate for roll bowing resulting from high pressures in the nip region. Rollbowing is a structural problem that can be mitigated by placing the rolls in a slightlycrossed pattern, rather than completely parallel, or by applying moments to the rollends to counteract the separating forces in the nip region.

1.1 Magnetohydrodynamics (MHD) Equations

Magnetohydrodynamics deals with the motion of a flow-able material in the exis-tence of a magnetic field, and it is a mutual interaction between the electro-magneticand material velocity fields. Interest in magnetohydrodynamics phenomena has ex-isted ever since the end of the nineteenth century, especially in the astrophysicscommunity where the work of Cowling [20] and Ferraro [21] can be seen as pioneer-ing work starting the formal theory of MHD on an astrophysical scale. One has towait until the nineteen forties for the maturing interest in magnetohydrodynamicswith the work of Hartmann [17, 22] and Alfven [23], whose contributions are asso-ciated with two unique MHD physical features, namely: the Hartmann layer andAlfven waves. Magnetohydrodynamics differs from hydrodynamics. Flow able ma-terial is considered to be electrically conducting regarding MHD study by virtue ofelectric current flowing in it. The flow able material consist of free charges becauseof which it conducts. The diffusive and thermal effects in magnetohydrodynam-ics equations are so complicated and nonlinear that they need to be approximateddepending upon the physical mechanisms in any specific phenomena. The flow of

7

electrically conducting material is governed by a shortened form of Maxwells equa-tions, along with Ohm’s law and the laws of conservation of mass and momentumin magnetohydrodynamics approximation.

1.1.1 Maxwell’s Equations

These equations are∇ · B = 0, (1.1)

∇×B = µm +1

c2∂E

∂t, (1.2)

∇× E = −∂B∂t, (1.3)

here µm the magnetic permeability and E is the total electric field current. TheEq. (1.1) shows that magnetic flux tube has a constant strength along its length inthe case when no magnetic poles. The Eq. (1.2) demonstrate that magnetic field isproduced by time varying electric field or by the current, however Eq. (1.3) pointout that either time varying field or electric charge can increase the electric fields.The basic assumption of MHD is V0 � c, where c and V0 respectively, are speedof light and initial velocity of the material. One of the consequences of V0 � c, is

that the term1

c2∂B

∂tmay be neglected in Eq. (1.2), another is that the equation of

continuity, which is obtained from the divergence of Eq. (1.2), becomes ∇ · J = 0.

1.1.2 Ohm’s Law

According to Ohms law the current density and the total electric field are propor-tional, and it has the following form:

J = σ (E + V ×B) , (1.4)

where σ is the electrical conductivity of the fluid.We make the following assumptions:

1. All physical quantities are constant throughout the flow field.

2. The magnetic field B is perpendicular to the velocity V and induced magneticfield is small compared with the applied magnetic field so that the magneticReynolds number is small [24].

3. The electric field is assumed to be zero.

8

In view of these assumptions, Lorentz force has the form

J × B = −σB20V. (1.5)

If the strength of the magnetic field is very large, the generalized Ohm’s law ismodified to include the Hall current [25] so that

J +ωeτeB0

J × B = σ

(

E + V× B +1

ene

∇pe

)

, (1.6)

where τe is the electron collision time, ωe is the cyclotron frequency of the electron,e is the electron charge, ne is the number of density of electrons, pe is the electronpressure. It is assumed that ωeτe ≈ O (1) and ωiτi � 1, where ωi and τi are thefrequency of ions and collision time, respectively. In the above equation, the ion-slipand thermoelectric effects are excluded.

1.2 Governing Equations

To study polymer calendering process, it is important first to study conservationequations (mass, momentum, and energy [2, 6, 26]) governing the flow. The flow ofan incompressible material like polymer solutions and melts, at least in conditionswhere they are measured as incompressible for pressures under 100MPa, is governedby

∇ · V = 0, (1.7)

ρ

(

DV

Dt

)

= −∇p+ ∇ · S + f, (1.8)

ρCp

(

Dθ

Dt

)

= k∇2θ + S : ∇V, (1.9)

whereD (·)Dt

=∂ (·)∂t

+ (V · ∇) (·) is the material time derivative and ρ V, p, f,

S, Cp, k and θ respectively are density, velocity, pressure, body force, extra stresstensor, specific heat, thermal conductivity and temperature distribution. Since weare considering MHD as a body force in this dissertation, therefore body force inlight of Eq. (1.5), can take the following form

f = −σB20V. (1.10)

9

1.3 Constitutive Equations

In rheology, a constitutive equation is used to describe how a material behaves whenit is deformed. The above system of conservation Eqs. (1.7-1.9) are not limited onlyto Newtonian materials due to stress tensor S. There exists a relationship among thestress tensor, velocities and velocity gradients which are given by suitable rheologicalconstitutive equations [5, 19, 27, 28]. The general form of constitutive equation is:

T = −pI + S, (1.11)

where p the pressure [Pa], T the total stress tensor [Pa] and S is the deviatoric orshear stress tensor [Pa], and I is the identity matrix.The constitutive equation for an incompressible Newtonian fluid is defined by theequation

S = µA1, (1.12)

in which µ is the constant viscosity [Pa s] and A1 = ∇V+(∇V)T

is the first RivilinEricksen tensor. The rheological constitutive equation that relates the stresses S tothe velocity gradients for the generalized Newtonian model [5, 19, 27, 28] is obtainedby simply replacing the constant viscosity µ by the non-Newtonian viscosity η andis written as

S = η (A1)A1, (1.13)

where the apparent viscosity η (A1) is given by the power-law model [5]

η (A1) = K (A1)n−1 . (1.14)

Most of the materials used in industries for calendering process are non-Newtonianshowing either shear thickening/thinning (existence of yield stress) effects [26]. Inthis thesis, we have used material named viscoelastic materials, including the second-grade, third-grade fluids and viscoplastic the Casson fluid model. The third grademodel has the advantage that it can be reduced to second grade, power law or New-tonian as a special case with appropriate choice of material parameter.This thesis consists of incompressible non-Newtonian material models, namely Cas-son, second-grade and third-order. The rheological equation for an incompressibleCasson material is [29].

S =

{ (

µB + py/√

2π)

2A1, π > πc,(

µB + py/√

2πc

)

2A1, π ≤ πc,(1.15)

where µB is the plastic dynamic viscosity of the non-Newtonian fluid, py is theyield stress of fluid, π is the product of the component of deformation rate withitself, namely, π = A2

1 and πc is the critical value of this product based on thenon-Newtonian model.

10

The rheological equation for extra stress tensor of a second-grade material [30] is ofthe form

S = µA1 + α1A2 + α2A21, (1.16)

where µ is the dynamic viscosity, and αi (i = 1, 2) are the measurable materialconstants. They denote, respectively, the elasticity and cross-viscosity. These ma-terial constants can be determined from viscometric flows of any real fluid, and A2

is the rate of acceleration or second Rivlin-Ericksen tensor and is defined as

A2 =

(

∂

∂t+ (∇ ·V)

)

A1 + A1(∇V) + (∇V)TA1, (1.17)

It should be noted that when α1 = α2 = 0, the constitutive equation of second gradefluid reduces to that for Newtonian fluid.Similarly the rheological equation for extra stress tensor of a third-order material[31] is

S = µA1 + α1A2 + α2A21 + β1A3 + β2(A1A2 + A2A1) + β3(trA

21)A1, (1.18)

here µ is the dynamic viscosity, and αi (i = 1, 2) , and βi (i = 1, 3) are thematerial constants and the Rivlin–Erickson tensors are defined by

An =

(

∂

∂t+ (∇ · V)

)

An−1 + An−1(∇V) + (∇V)TAn−1, n = 1, 2, 3. (1.19)

Eq. (1.18) reduces to classical linearly viscous Newtonian fluid model when allmaterial moduli except µ is zero, and reduces to second grade fluid when βi =0, i = 1, 2, 3.

1.4 Calendering Geometry and Boundary Condi-

tions

Defining the system under study is an important element of modeling. Fig.1.3demonstrates the important features of the Calendering geometry. In Fig. 1.3 x isthe horizontal coordinate, y vertical coordinate, R roller diameter, hi inlet height,he exit height, U surface speed of the rollers, h height between the rollers, xi inletdistance of material and xe exit distance of material.

1.4.1 Boundary Conditions

In order to define the system under consideration the boundaries need to be de-scribed.

11

Figure 1.3: Calendering Geometry (Not to scale)

The no slip condition means that at the roll surface, y = h the material velocityis equal to the roller surface linear velocity, U

at y = h, u = U (No Slip at the Roll Surface). (1.20)

There is a line of symmetry at y = 0, this means:

at y = 0, v = 0, (1.21)

at y = 0,∂u

∂y= 0. (1.22)

At the point of inlet (xi, hi), there is a free surface at which the total upward stressis equal to zero

at x = xi, y = hi, Tyy = −p+ Syy = 0 (inlet Condition). (1.23)

At the point of exit (xe, he), there is also a free surface at which the total upwardstress is equal to zero

at x = xe, y = he, Tyy = −p + Syy = 0 (outlet Condition). (1.24)

12

Figure 1.4: Geometrical representation of h(x)

1.4.2 Limited Region under Study

Only the shaded region in Fig. 1.3 is considered. For this region we need to definean expression h (x). Consider Fig 1.4, we have

|AB| = R −(

h−H0

)

, |BC| = R, |AC| = x and |DC| = h(x) −H0.(1.25)

Which implies[

R −{

h (x) −H0

}]2+ x2 = R2, (1.26)

orR −

[

h (x) −H0

]

=√R2 − x2, (1.27)

Yieldsh (x) = R +H0 −

√R2 − x2. (1.28)

Since we are considering only the shaded region (see fig 1.3) where x� R. For thiswe proceed as:

h = h∣

∣

x=0+∂h

∂x

∣

∣

∣

∣

x=0

x+1

2

∂2h

∂x2

∣

∣

∣

∣

x=0

x2 + ... (1.29)

Now differentiating Eq. (1.28) twice w. r. t x, we have

∂h

∂x= x

(

R2 − x2)−1/2

, (1.30)

and∂2h

∂x2=(

R2 − x2)−1/2

+ x2(

R2 − x2)−3/2

. (1.31)

At x = 0, Eqs. (1.28), (1.30) and (1.31) reduces to

h∣

∣

x=0= H0, (1.32)

13

∂h

∂x

∣

∣

∣

∣

x=0

= 0, (1.33)

∂2h

∂x2

∣

∣

∣

∣

x=0

=1

R. (1.34)

Using Eqs. (1.32)-(1.34) in Eq. (1.29), we get

h (x) = H0

(

1 +x2

2H0R

)

. (1.35)

1.5 A General Consideration of Lubrication Ap-

proximation Theory

Assuming that the fluid does not spread as it enters the gap between the rolls asshown in Fig. 1.3, we may write Eqs. (1.7), (1.8) (1.9) for two dimensions, as

∂u

∂x+∂v

∂y= 0, (1.36)

ρ

(

u∂u

∂x+ v

∂u

∂y

)

= −∂p

∂x+∂Sxx

∂x+∂Sxy

∂y+ fx, (1.37)

ρ

(

u∂v

∂x+ v

∂v

∂y

)

= −∂p∂y

+∂Syx

∂x+∂Syy

∂y+ fy, (1.38)

ρCp

(

u∂θ

∂x+ v

∂θ

∂y

)

= k

(

∂2θ

∂x2+∂2θ

∂y2

)

+ Sxx

∂u

∂x+ Sxy

∂u

∂y+ Syx

∂v

∂x+ Syy

∂v

∂y, (1.39)

here Sxy and Syx are the shear stresses, Sxx and Syy represent the normal stresses.Since the flow in the narrow gap is almost parallel so we can suppose that ∂

∂x� ∂

∂y,

v � u, thus we have∂u

∂x+∂v

∂y= 0, (1.40)

0 = −∂p

∂x+∂Sxy

∂y+ fx, (1.41)

0 = −∂p∂y

+∂Syy

∂y+ fy, (1.42)

ρCpu∂θ

∂x= k

(

∂2θ

∂y2

)

+ Sxy

∂u

∂y. (1.43)

14

A Problem that can be described by Eq. (1.40)-(1.43) are said to obey the lubricationapproximation theory [5, 19, 27, 33]. The continuity equation (1.40) may be replacedby its integral form

Q = 2

∫ h(x)

0

udy, (1.44)

where Q is the volume flow rate. To solve the above equation we also need aconstitutive equation that relates stress to rate of strain.

1.6 Methods of Solution

Exact solutions are very rare in the calendering process when solving an equationof motion using non-Newtonian materials. Perturbation and numerical techniquesproved to be a powerful mathematical tools to obtain an approximate solution ofsuch kind of problems whose analytic or exact solutions are not possible or difficultto obtain.

1.6.1 Perturbation Technique

Perturbation methods [34, 35] has been extensively used for obtaining approximateresults for the problems arising in science and technology. The problems containingthe function u (y, ε) can be characterized mathematically by the following differentialequation

L (u, y, ε) = 0, (1.45)

subject to the boundary condition

B (u, ε) = 0, (1.46)

in above equation y represents independent variable and ε is any physical parameterinvolve in a particular problem. Suppose in general, above problem has no exactsolution and ε(ε� 1) for which exact solution or approximate solution of the aboveproblem exist. So one can seeks a series solution of the form

u (y, ε) = u0 (y) + εu1 (y) + ε2u2 (y) + · · · (1.47)

where ui are free of ε and u0 (y) represents the zero-order solution for ε = 0. Onsubstitution of Eq. (1.47) into Eq. (1.45) and Eq. (1.46) and equating the equalpowers of ε, one can get system of equations, which must hold for all values ofε. Since each coefficient is linearly independent, so must vanish individually. Thisobtained a system of equations are simple equations governing un, and can be solvedsuccessively. This perturbation technique has been used in Chapter 5, 7 and 8.

15

1.6.2 Numerical Techniques

There are some complex integrals involve in Chapters 2, 3, 4, 5, 6 and 7 whichcannot be solved analytically. Therefore, we need some suitable numerical algo-rithm. A composite Simpson’s rule for numerical integration was employed for thecomputation of these integrals. The number of spatial discretization for numericalprocedure used was 200. The governing equations for the pressure in the followingchapters was first solved via a composite Simpson’s rule, and to calculate the valueof λ (separation point), the modified regula-falsi and Newton-Raphsons method hasbeen used. The calculated convergence criterion was 10−10. Note that the valida-tion of the algorithm was tested by reproducing the results for Newtonian materials[5, 36].

16

Chapter 2

Influence of Porous Rolls onNewtonian Calendering

In this Chapter, equations of motion for the process of Newtonian material calen-dering with porous rolls are modeled and analyzed. Exact solutions for velocity fieldand pressure gradient are presented without using the LAT. Integrated engineeringquantities like extreme pressure, power-input and roll-separating force are calcu-lated numerically. It is noted that the transverse mass across the fluid has greatinfluence on detachment point, velocity and pressure distributions. Several physicalparameters of engineering interest are discussed and presented graphically.

2.1 Formulation of the Problem

Consider a steady and laminar flow of an isothermal incompressible Newtonian ma-terial, which is dragged through the small gap between two counter rotating rollsof same radii R and same angular velocity U = ωR in order to make a sheet. Itis assumed that the mass is sucked by the upper roll with a velocity v0 and thesame mass is injected from the lower roll with the velocity v0. Separation betweenthe two rolls at the nip is 2H0, and the length of the curved channel formed by therolls is very large compared with the separation at the nip i.e., H << R. With theassumption that the fluid does not spread while entering the gap between the tworolls, we can take flow as two dimensional. Here x is along the flow and y is normalto it (Fig. 2.1). As the physical model is symmetric, therefore we will take upperhalf of the given configuration. We seek the velocity field of the following form

V = [u(x, y), v0] . (2.1)

17

Figure 2.1: Schematic representation for the flow in the gap between two porousco-rotating calender rolls

In view of Eqs. (1.11), (1.12) and (2.1), Eqs. (1.7) and (1.8) becomes

∂u

∂x= 0, (2.2)

ρ

(

u∂u

∂x+ v0

∂u

∂y

)

= −∂p

∂x+ µ

(

∂2u

∂x2+∂2u

∂y2

)

, (2.3)

∂p

∂y= 0. (2.4)

In light of Eqs. (2.2) and (2.4), Eq. (2.3) become

∂2u

∂y2−(v0

ν

) ∂u

∂y=

1

µ

dp

dx, (2.5)

where ν =µ

ρis the kinematic viscosity. Fig. 2.1 shows the representation of the

applied boundary conditions, and has the following mathematical form

u = U on y = h (x) ,∂u

∂y= 0 on y = 0.

(2.6)

Form Fig. 2.1, h (x) is the distance in y direction from the center line to the rollsurface, and is defined by Eqs. (1.35).

18

2.2 Solution of the Problem

In order to solve the isothermal calendering process, we define the following dimen-sionless variables

x =x√

2RH0

, u =u

U, y =

y

H0, p =

√

H0

2R

pH0

µU, h(x) =

h(x)

H0= 1 +

x2

2H0R. (2.7)

Eqs. (2.5) and (2.6) become

∂2u

∂y2− Re

∂u

∂y=dp

dx, (2.8)

u = 1 on y = h (x) ,∂u

∂y= 0 on y = 0,

(2.9)

where Re =v0H0

ν, is Reynolds number and h(x) = h = 1 + x2.

Solution of Eq. (2.8) with the help of boundary conditions (2.9) takes the form

u(x, y) = 1 +1

Re2

dp

dx

[

eRey − eReh +Re (h− y)]

. (2.10)

Thus, u is an explicit function of y and an implicit function of x, through h(x) andp(x). The volumetric flow rate per unit width Q is defined as

Q = ∫h0 udy, (2.11)

where Q =Q

2UH0, Q is the dimensional volumetric flow rate per unit width. Eq.

(2.11) gives

Q = h+1

2Re3

dp

dx

[

2 (1 − hRe) eReh +Re2h2 − 2]

. (2.12)

Solving fordp

dx, we get

dp

dx=

2Re3 (Q− h)

2(1 − hRe)eReh + h2Re2 − 2. (2.13)

If we assume that the sheet leaves the rolls at x = λ with a thickness H and a speedU, we have

Q = 2UH givesH

H0

=Q

2UH0

= Q. (2.14)

19

From Eq. (1.26), we haveH

H0

= 1 + λ2, (2.15)

thereforeQ = 1 + λ2. (2.16)

Using Eq. (2.16) and Eq. (1.26), (2.13) can be written as

dp

dx=

2Re3 (λ2 − x2)

2 {1 − (1 + x2)Re} eRe(1+x2) + (1 + x2)2Re2 − 2. (2.17)

Eq. (2.17) gives zero pressure gradient at x = ±λ, and this fact can be seengraphically from Fig. 2.2. The distance x = λ represents the point where the sheetleaves the rolls. The boundary conditions on pressure are

p =dp

dx= 0, at x = λ

p = 0, as x→ −∞

}

(2.18)

Eq. (2.18) is referred to as the Swift condition [11] and is instrumental in the solutionof the problem.Integrating (2.17) with respect to x, we obtain pressure of the form

p = 2Re3 ∫λx

(λ2 − x2)

2 {1 − (1 + x2)Re} eRe(1+x2) + (1 + x2)2Re2 − 2dx. (2.19)

The velocity profile (2.10) with the help of (2.17) becomes

u (x, y) = 1 + 2Re

[

{(eRey − eRe(1+x2)) +Re(1 + x2 − y)}(λ2 − x2)

2{1 − (1 + x2)Re}eRe(1+x2) + (1 + x2)2Re2 − 2

]

. (2.20)

2.2.1 Detachment Point

The leave-off distance (detachment point) λ, in dimensionless form can be foundfrom Eq. (2.19) as

0 = ∫λ−∞

(λ2 − x2)

2 {1 − (1 + x2)Re} eRe(1+x2) + (1 + x2)2Re2 − 2dx (2.21)

This equation has no analytical solution. Therefore, a numerical solution must befound. We used Simpson’s 3/8 rule to compute the integral and an iterative method(Newton-Raphson method) in order to calculate λ at which pressure vanishes. Itfollows that λ has the value 0.4260. When Re = 0.4. It is found that with an increasein Reynolds number, the point of detachment moves towards the nip region.

20

Figure 2.2: Effete of Reynolds number on pressure-gradient distribution

2.2.2 Sheet Thickness

For λ = 0.4260, we have the maximum sheet thickness to the minimum gap widthratio

Hmax

H0= 1 + λ2 = 1.1814, (2.22)

The thickness of the material Hf , when it touches the roles at the entrance gives

xf =

(

Hf

H0− 1

)1/2

, (2.23)

−xf is the distance where the roll first bite on the polymer.

2.3 Operating Variables

Once λ is found, then all the other quantities of interest are easily available. Vari-ables which of the interest of engineering problems as follows:

21

Figure 2.3: Solution curve for the detachment point when p = 0 as x → −∞, fixingRe = 0.4

2.3.1 Maximum Pressure

The maximum pressure distribution is given by

pmax = 2Re3 ∫−λ−∞

(λ2 − x2)

2 {1 − (1 + x2)Re} eRe(1+x2) + (1 + x2)2Re2 − 2dx, (2.24)

here x = −λ is the point of maximum pressure distribution as shown in Fig. 2.5.

2.3.2 Rate of Strain and Stress

The rate of strain and stress, in non-dimensional form can be computed using Eq.2.20, which become

γyx(x, y) = Syx(x, y) =

[

2Re(λ2 − x2)(eRey − 1)

2{1 − (1 + x2)Re}eRe(1+x2) + (1 + x2)2Re2 − 2

]

,

(2.25)

where γyx(x, y) =γyx(x, y)

U

H0

and Syx(x, y) =Syx(x, y)

U

H0

. An extremum in shear rate

and stress occur at the roll surface at x = 0, where the gap is at minimum, that is

γmax(0, 1) = 2λ2Re

[

(eRe − 1)

2{1 − Re}eRe +Re2 − 2

]

= Smax(0, 1). (2.26)

22

Figure 2.4: Power input during calendering process

2.3.3 Power Input

In order to calculate the power input, we need to integrate the product of the shearstress and the roll surface speed over the surface of the roll. Therefore, by settingy = H0 in Eq. 2.25, we get total power as

Pw = 2 ∫ λ−∞ Syx(x, 1)dx = 4 ∫ λ

−∞ I(Re, λ, x)dx, (2.27)

where

I(Re, λ, x) =Re(

eRe − 1)

(x2 − λ2)

2{1 − (1 + x2)Re}eRe(1+x2) + (1 + x2)2Re2 − 2, (2.28)

Pw =P ∗

w

µWU2is dimensionless power and W is the width of the each roll. Again Eq.

(2.27) has no analytical solution, using Simpson’s 3/8 rule, with 200 partitions theapproximate value of power is Pw = 0.2974, fixing Re = 0.4and λ = 0.4260.

2.3.4 Adiabatic Temperature

The power input has the potential to rise the temperature of the fluid by an amountwhich is, at most, given by an adiabatic energy balance equation

∆T =Pw

ρQCp

. (2.29)

23

Figure 2.5: Roll-separating force during calendering process

2.3.5 Roll-Separating Force

The roll-separating force can be calculated from the relation

F

W= ∫x0

−∞ Syydx. (2.30)

In dimensionless form it becomes

F = −∫λ−xf

pdx. (2.31)

On substitution of Eq. 2.19, above equation takes the form

F = 2Re3 ∫λ−xf

[

∫λx

(x2 − λ2)

2 {1 − (1 + x2)Re} eRe(1+x2) + (1 + x2)2Re2 − 2dx

]

dx , (2.32)

here xf is the point where the roll first bites on the polymer. Eq. (2.32) containsa complicated integral whose exact solution is not possible, therefore Simpson’s 3/8rule is used to calculate its value taking λ = 0.4260 and Re = 0.4, we find that itsnumerical value is F = 0.278010.

It can be noted that the power transmitted by roll to the material, temperaturerise of the fluid and separation force depend upon the Re.

2.4 Results and Discussion

Equations of motion for the process of Newtonian material calendering with porousrolls are modeled and analyzed. The effects of various dimensionless parameters on

24

Table 2.1: Effect of Reynolds number on detachment point, sheet thickness, powerinput and separation force

Re λ H Pw F

0.2 0.4466 1.1994 0.065945 0.3801530.4 0.4260 1.1814 0.006152 0.2780100.6 0.4087 1.1670 0.000877 0.2106630.8 0.3936 1.1549 -0.000216 0.1630451.0 0.3802 1.1445 -0.000088 0.1282851.2 0.3680 1.1354 -0.000186 0.1016981.4 0.3569 1.1273 -0.000045 0.0813991.6 0.3466 1.1201 -0.000121 0.0653271.8 0.3371 1.1136 -0.000056 0.0527692.0 0.3282 1.1077 -0.000107 0.042665

velocity and pressure are investigated graphically. The effect of Reynolds numberRe on various rheological parameters are shown in Table 2.1. It is interesting to notefrom Table 2.1 that by increasing the values of Reynolds number Re, reduces thedetachment point, sheet thickness, roll separation force and magnitude of the powertransmitted by roll to the material. This effect is very important from an industrialpoint of view, that is by increasing or decreasing the value of the Reynolds number,we can really control the sheet thickness and all the engineering quantities involvedduring the calendering process.

The effects of Reynolds number Re on the velocity field are depicted in Figs. 2.6and 2.7 for different values of Re at x = 0, 0.25, 0.5, 0.75. It is observed, as naturallyexpected, that the velocity decreases while sheet moves away from the narrow regionbetween the two co-rotating rolls. Moreover, the velocity decreases with an increasein Reynolds number along the center line for the fix value of x.

In Fig.2.5, dimensionless pressure distribution and numerical calculation of theroll separation force along the domain fixing Re = 0.4 is plotted. It shows theattachment and detachment points. The maximum pressure has been observed atx = −λ during the calendering process. It is noted that, the detachment point isreduced as compared with Newtonian, non-porous rolls calendering process [3]. Theresults of Middleman [5] are recovered when Re → 0. It is also found that with anincrease in Reynolds number the detachment point decreases. The power requiredto turn the roll and pull the sheet has been sketched in Fig. 2.4 for Re = 0.4. Theseparation force and power consumption are computed numerically using compositeSimpson’s 3/8 rule. The domain of interest is divided into 200 subintervals, and thecalculated convergence criterion was 10−10. It has been found that with an increasein Reynolds number we can economize the power, this behavior can also be noted

25

Figure 2.6: Velocity distribution at x = 0 and at x = 0.25 respectively

from Table 2.1.Contour graph pattern for the longitudinal velocity distribution in case of New-

tonian material calendering for altered values of Reynolds number (Re = 0.2, 0.8,1.2, 2.0) are shown in Figs. 2.8 to 2.10. Because of a free surface upstream ofthe incoming sheet, a large flat vortex appears, which almost double the flow raterecirculation in the material bank in the nip region, which becomes smaller as theReynolds number increases, and disappears for Re > 2.0.

2.5 Concluding Remarks

This chapter extends Gaskell’s [3] and Middleman’s [5] analysis for Newtonian calen-dering. Majority of mathematical models for Newtonian/non-Newtonian materialscalendaring are based on the lubrication approximation theory. In this chapter,analytic solutions for the velocity field, the associated shear strain, shear stressand volume flux flow of an incompressible Newtonian fluid between two co-rotatingcylinders are presented without using the lubrication approximation theory.

The major findings of the present study are summarized as follows:

1. Reynolds number provides a mechanism to control the separation region, powerinput and roll-separating force.

2. With the help of Reynolds number one can indeed control the sheet thickness,and this effect is obvious from Table 2.1.

3. Maximum velocity and pressure gradient occur in the nip region.

4. Velocity decreases as sheet moves away from the narrow region.

26

Figure 2.7: Velocity distribution at x = 0.5 and at x = 0.75 respectively

Figure 2.8: Contour graph pattern for the longitudinal velocity distribution duringthe calendering process at Re = 0.2 and at Re = 0.8 respectively

5. When Re → 0, the solutions for Newtonian fluids [3] without porous bound-aries under the same physical conditions are recovered.

27

Figure 2.9: Contour graph pattern for the longitudinal velocity distribution duringthe calendering process at Re = 1.2 and at Re = 2.0 respectively

Figure 2.10: Contour graph pattern for the longitudinal velocity distribution duringthe calendering process at Re = 2.5

28

Chapter 3

Effect of Magnetohydrodynamicson Newtonian Calendering

In this chapter, an analysis has been presented for the calendering process of in-compressible MHD Newtonian material. The lubrication approximation is used tosimplify the equations of motion. Exact solutions for velocity profile, pressure gra-dient, flow rate per unit width, rate of strain, shear stress, maximum shear rate andshear stress at the roll surface are obtained. The value of λ, the distance from thenip to the point where the sheet leaves the rolls, is calculated using Newton-Cotesformula along with the regula-falsi method. Numerical results are presented forpressure distribution, power transmitted by roll to the material and force separatingthe rolls using Newton-Cotes formula with regula-falsi method and Simpson’s rulefor different values of magnetic parameter, M , and the corresponding values of λ.Some results are shown graphically. It is found that the magnetic field provides thecontrolling parameter to increase or decrease power transmission, separation forceand the distance between attachment and detachment points, which are useful inthe calendaring process.

3.1 Problem Formulation

Consider laminar and steady flow in the presence of an electrically conducting in-compressible Newtonian fluid which is dragged through the small gap between twocounter rotating rolls of the same radii R to produce a sheet. The upper roll isrotating anti-clockwise while the lower roll is rotating clockwise with the same an-gular velocity U and their separation at the nip is 2H0. It is assumed that boththe rolls are of infinite length and the length of the curved channel formed by therolls are very large as compared with the separation at the nip, i.e., H0 � R. It isalso assumed that the fluid does not spread as it enters the gap between the rolls,hence the flow can be taken as two dimensional. x is the direction of flow and y

29

Figure 3.1: Geometry of the problem

perpendicular to the roll axis (the x-axis). If u and v are the velocity componentsin x and y directions, respectively, then the boundary conditions are defined in Eq.(2.6)For two dimensional flow, let us consider a velocity profile of the following form

V = [u(x, y), v(x, y)] . (3.1)

For Eq. (3.1) continuity equation is defined in Eq. (1.36). Eq. (1.8) in light ofabove equation become

∂u

∂x+∂v

∂y= 0, (3.2)

ρ

(

u∂u

∂x+ v

∂u

∂y

)

= −∂p

∂x+ µ

(

∂2u

∂x2+∂2u

∂y2

)

− σB20 u, (3.3)

ρ

(

u∂v

∂x+ v

∂v

∂y

)

= −∂p∂y

+ µ

(

∂2v

∂x2+∂2v

∂y2

)

− σB20 v. (3.4)

The cross-channel dimension of the flow domain is very small as compared to theradii of curvatures of the two rolls, therefore, we can apply lubrication theory ap-proximation. We can assume that the velocity gradients in the direction parallel tothe boundaries are asymptotically small compared with those across the flow do-main, which is true only when the shape of the flow domain is varying very slowly.Therefore, it is reasonable to assume that

∂

∂x� ∂

∂y,

v � u andu (x, y) only

(3.5)

30

Then above Eqs. (3.2 - 3.4) become

∂u

∂x+∂v

∂y= 0, (3.6)

0 = −∂p

∂x+ µ

(

∂2u

∂x2+∂2u

∂y2

)

− σB20 u, (3.7)

0 = −∂p∂y

· (3.8)

In view of Eqs. (3.6) and (3.8), Eq. (3.7) becomes

∂2u

∂y2− σ

µB2

0 u =1

µ

dp

dx· (3.9)

Eq. (3.9) is identical to [5] in the absence of magnetic field, i.e., when B0 = 0.

3.2 Solution of the Problem

Defining the dimensionless parameters

x =x√

2RH0

, u =u

H0, y =

y

H0, p =

pH0

µU, h =

h

H0. (3.10)

Eq. (3.9) in non-dimensional form becomes

∂2u

∂y2−M2u = ε

dp

dx, (3.11)

where

M =

√

σ

µB0H0, and ε=

√

H0

2R.

Solution of Eq. (3.11) with the help of boundary conditions (2.9) becomes

u(x, y) =

(

1 +ε

M2

dp

dx

)

coshMy

coshMh− ε

M2

dp

dx. (3.12)

Thus, u is an explicit function of y and an implicit function of x, through h(x) andp(x).

Flow rate per unit width. Eq. (2.11) gives

Q =

[(

1 +ε

M2

dp

dx

)

tanhMh

M− hε

M2

dp

dx

]

. (3.13)

31

Solving fordp

dx, we obtain

dp

dx=M2

ε

Q− tan (M(1 + x2))

Mtan (M(1 + x2))

M− (1 + x2)

. (3.14)

If we assume that the sheet leaves the rolls with a thickness H and a speed U wehave, Q = 2UH,

H

H0

=Q

2UH0

= Q. (3.15)

Letx2

2RH0

= λ2 at h(x) = H, then from Eq. (2.11)

H

H0= 1 + λ2. (3.16)

Therefore, Eq. (3.14) becomes

dp

dx=M2

ε

1 + λ2 − tan (M(1 + x2))

Mtan (M(1 + x2))

M− (1 + x2)

. (3.17)

Eq. (3.17) gives zero pressure gradients at λ2 =tanh (M (1 + x2))

M− 1.

The distance λ =

(

tanh (M (1 + x2))

M− 1

)1

2

represents the point where the sheet

leaves the rolls and λ = −(

tanh (M (1 + x2))

M− 1

)1

2

is the point of maximum

pressure, where the rolls bite on the polymer.Pressure is obtained by integrating Eq. (3.17)

p =M2

ε

∫ λ

−∞

1 + λ2 − tan (M(1 + x2))

Mtan (M(1 + x2))

M− (1 + x2)

dx. (3.18)

The pressure distribution has a maximum value at

λ = −(

tanh (M (1 + x2))

M− 1

)1

2

32

given by

pmax =M2

ε

∫ −λ

−∞

1 + λ2 − tan (M(1 + x2))

Mtan (M(1 + x2))

M− (1 + x2)

dx. (3.19)

It is difficult to find the analytical solution of Eq. (3.18) with respect to the boundarycondition defined in Eq. (2.18). Newton-Cotes formula is applied to compute theintegral then regula-falsi method in order to calculate λ at which pressure vanishes.Thus, we obtain

p = 0 at λ = 0.38 keeping M = 0.4. (3.20)

For this value of λ we have the maximum sheet thickness to the minimum gap widthratio

Hmax

H0= 1 + λ2 = 1.1444. (3.21)

The velocity profile is obtained with the help of Eqs. (3.17) and (3.12) as

u =coshMy

cosh (M(1 + x2))+

tanh(M(1 + x2)) −M(1 + λ2)

tanh(M(1 + x2)) −M(1 + x2)

[

1 − coshMy

cosh(M(1 + x2))

]

,

(3.22)and flow rate per unit width (2.11) becomes

Q =tanhMh

M+

tanh(M(1 + x2)) −M(1 + λ2)

tanh(M(1 + x2)) −M(1 + x2)

[

h − tanhMh

M

]

. (3.23)

The non-dimensional rate of strain and stress are given by

γyx(x, y) = Syx(x, y) =du

dy

=

[

λ2 − x2

tanh(M (1+x2))M

− (1 + x2)

]

M sinhMy

cosh(M(1 + x2)), (3.24)

where γyx(x, y) =¯γ yx(x, y)

U

H0

and Syx(x, y) =Syx(x,y)

µU

H0

· The maximum shear rate and

shear stress, at the roll surface, ((x, y) = (0, 1)), where the gap is minimum, are

γmax(0, 1) = M2λ2

[

tanhM

tanhM −M

]

=Smax(0, 1). (3.25)

Power transmitted [5] by roll to the material is calculated by integrating the productof the shear stress and the rolls surface speed over the surface of rolls which isobtained by setting y = H0, as

Pw =2

ε

∫ λ

−∞Sxy (x, 1) dx, (3.26)

33

here Pw =P ∗

w

µWU2is the dimensionless power and where W is width of the rolls.

Using (3.24), we get

Pw =2

ε

∫ λ

−∞

[

M2(x2 − λ2) sinhM

sinh(M(1 + x2)) −M(1 + x2) cosh(M(1 + x2))

]

dx

=2

ε

∫ λ

−∞I(M,λ, x)dx. (3.27)

Using Simpson’s rule to approximate the complicated integral∫ λ

−∞ I(M,λ, x)dx, keep-ing M = 0.4 and λ = 0.38, we get

∫ λ

−∞I(M,λ, x)dx = 0.0117. (3.28)

Therefore Eq. (3.27) can be written as

Pw =0.0234

ε. (3.29)

This power input has the potential to raise the temperature of the fluid by an amountwhich is, at most, given by an adiabatic energy balance given by Eq. (2.29).Upon integrating the normal stress, the force separating the two rolls can be calcu-lated. Within the lubrication approximations, we may ignore the viscous contribu-

tion 2µ∂u

∂ythat appears in the normal stress component. Therefore upon integration

of pressure p, the roll separation force can be found as

F =−1

ε

∫ λ

−∞p (x) dx. (3.30)

Using Eq. (3.18) we can write

F =1

ε2G(λ,M), (3.31)

where

G(λ,M) =

∫ λ

−xf

[

M2

∫ λ

x

tanh(M(1 + x2)) −M(1 + λ2)

tanh(M(1 + x2)) −M(1 + x2)dx

]

dx

xf is the point where the roll bites the polymer. G(λ, M) is a complicated function,Simpson’s rule is used to calculate its value which is 0.3515, for λ =0.38, M = 0.4,we have

F =0.3515

ε2. (3.32)

Thus, the power transmitted by roll to the material causes rise in temperature of

the material. The separation force depend upon the ratio ε =

√

H0

2R.

34

Figure 3.2: The effect of magnetic field on the point where sheet leaves the roll

3.3 Results and Discussion

In this chapter, we analyzed the calendering processes for incompressible MHDNewtonian fluid. The lubrication approximation is used to simplify the equationsof motion. Exact solutions for velocity profile, pressure gradient, flow rate perunit width, rate of strain, shear stress, maximum shear rate and shear stress atthe roll surface are obtained. Numerical calculations were carried out for pressuredistribution, power transmitted by roll to the material, force separating the tworolls using Newton-Cotes formula with regula-falsi method and Simpson’s rule fordifferent values of magnetic parameter, M, and the corresponding values of λ, as λdependents on M. Numerical values of dimensionless leave-off distance λ for differentvalues of M are given in Table 3.1. These results are also verified using finite elementmethod and omitted here for the sake of space saving. Some results are showngraphically.

The effect of magnetic field M on the point from the nip where sheet leaves theroll, λ can be seen in Fig. 3.2. With an increase in M , it has been observed that λdecreases.

The effect of M on velocity is sketched in Fig. 3.3, at different positions in thedirection of flow. The maximum velocity is observed at the nip x = 0 in Fig. 3.3(a).It decreases with an increase in M. Fig 3.3(b) shows the effect of M on velocity atx = 0.2. Fig. 3.3(c) is showing velocity profiles at different axial positions, keepingM = 0.4 and λ = 0.38. The velocity profiles are parabolic type facing down for0 ≤ x < 0.4. However it faces up for x ≥ 0.4.

The graphs of Eq. (3.18) for pressure distribution are presented in Figs. 3.4(a)

35

Figure 3.3: Velocity distribution at (a) x = 0.0, (b) x = 0.2, (c) M = 0.4 and λ =0.38

and 3.4(b). In Fig. 3.4(a) pressure decreases with an increase in M keeping ε = 0.01.In Fig. 3.4(b) we observe the effect of the ratio ε on pressure keeping M = 0.4 andλ = 0.38. It decreases with an increase in ε.

Power transmitted to the fluid can be seen in Fig. 3.5(a) for M = 0.4, λ = 0.38,ε = 0.01 and its magnitude is 2.34. The distribution of the force separating the tworolls within the region from where rolls bite the polymer to leave off point whenM = 0.4, λ = 0.38, ε = 0.01 can be observed in Fig. 3.5(b). The magnitude ofseparating force for these values is 3515.

The effect of M on separating force and power transmitted is given in Table 3.1.,which is showing the dependence of magnetic parameter, M on points where therolls bite on the polymer, where the sheet leaves the rolls, power transmitted to thefluid and force separating the two rolls. Table 3.2, is providing the effect of ε onthe power transmitted to the fluid and the roll-separating force. It is noticed thatpower, separation force decrease with an increase in ε.

It is noticed from the Table 3.1 that the attachment/detachment points, powerand separation force strongly depends upon the magnetic field. An increase in mag-netic field causes decrease in separation force and power required for manufacturingsheet. The Same effect of magnetic field is also observed in the distance betweenthat attachment and detachment points. Magnetic field provides a mechanism to

36

Figure 3.4: The effect of M and ε on pressure distribution

Figure 3.5: (a) Power transmitted to the fluid when M = 0.4, λ = 0.38, ε = 0.01.(b) Separation force for M = 0.4, λ = 0.38, ε = 0.01

37

control dimensionless leave-off distance λ. Power and separation force entirely de-pend upon the ε, the ratio of H0 (separation at the nip) and R (diameter of therolls). It provides a simple device to save power and separation force by decreasingthe diameter of rolls.

Table 3.1: Effect of magnetic field on attachment/detachment points power and roll-separating force

M -λ λ Power Roll-Separating Force

0.1 -2.45 0.47 7.82 54450.2 -1.88 0.45 6.36 48800.3 -1.42 0.42 5.38 41990.4 -1.12 0.38 2.34 35150.5 -0.88 0.33 -0.70 27840.6 -0.70 0.25 -9.25 18940.7 -0.50 0.15 -10.10 1023

Table 3.2: Effect of ε on power and roll-Separating force

ε Power Roll-separating Force0.01 2.34 35150.03 0.78 3900.05 0.46 1400.07 0.33 710.09 0.26 43

3.4 Conclusion

We observed that:

1. Magnetic field provides the controlling parameter to increase or decrease powertransmission, separation force and the distance between attachment and de-tachment points,

2. Magnetic field economies the power and separation force,

3. we can control power and separation force by dimensionless parameter ε, theratio of H0 (separation at the nip) to R (diameter of the roll),

4. λ, the distance from the nip where sheet leaves the roll decreases with anincrease in magnetic field M,

38

5. with the increasing value of M, velocity profile decreases,

6. by increasing M pressure increases,

7. by increasing the gap between the rolls, pressure decreases,

8. power transmitted to the fluid decreases with an increase in M,

9. with an increase in ε power transmitted to the fluid and separation forcedecreases,

10. separation force decreases with an increase in M,

11. when M → 0 results of [5] are recovered.

39

Chapter 4

MHD Calendering ofNon-Isothermal ViscoplasticMaterials

This chapter aims to investigate the flow of non-Newtonian viscoplastic materialby passing it between two counter-rotating rolls. Viscoplastic materials are thosematerials, that behave as enormously high viscosity materials when submitted tolow stresses and that flow when submitted to stresses higher than a yield stressvalue. Usually, they also show shear thinning behavior. The experimentally observedphenomena of shear-thinning (pseudoplasticity) or shear-thickening may be modeledby purely viscous constitutive equations which do not account for effects such asnormal stresses differences, time dependence or memory [37]. Barnes [38] and SouzaMendes and Dutra [39] point out that some of these models also describe a yieldstress, i.e., a level of shear stress under which the material would behave as highlyviscous. In fact, the yield stress is a model for the behavior of some structuredliquids that present a dramatic change in mechanical properties in a small rangeof stress. These fluids are called viscoplastic. Fluids such as molten chocolate,xanthan gum solutions, blood, wastewater sludge, muds, and polymer solutionspresent viscoplastic shear-thinning features.

We have modeled the incompressible, non-isothermal flow of a MHD viscoplasticfluid when they pass through the small gap between two counter rotating rolls.The conservation equations in light of LAT are non-dimensionalized and analyticsolutions for velocity, pressure gradient and temperature distribution are presented.The value of λ, the distance from the nip to the point where the sheet leaves therolls, is calculated using Simpson’s 1/3 formula along with the modify regula-falsimethod. Moreover, integrated engineering quantities like extreme pressure, power-input and roll-separating force are also calculated numerically, which are useful forengineers. Analysis shows that the insertion of the viscoplastic parameter effect

40

Figure 4.1: Physical presentation of calendering process between two co-rotatingheated rolls

substantially the velocity field, pressure gradient and temperature distribution. It isfound that the magnetic field provides a mechanism to control power transmission,separation force and the distance between attachment and detachment points, whichare important for the calendaring process.

4.1 Formulation

Consider the steady, non-isothermal two-dimensional flow of an incompressible vis-coplastic material, which is dragged through the small gap between two counterrotating heated rolls as to make a sheet. Each roll has a radius R and angularvelocity U = ωR. Their separation at the nip is H0 . The point −xf is that locationwhere the material first bites the rolls, which is known. The roll surfaces are keptat a constant temperature θ0 and θ1 is the fluid temperature, therefore θ1 < θ0, asshown in Fig. 4.2.

The basic equations governing the flow of viscoplastic material are continuityEq. (1.7), momentum Eq. (1.8) and energy Eq. (1.9). The extra stress tensor forviscoplastic (Casson) material and the rheological equation of extra stress tensor(S) for an isotropic and incompressible flow of a Casson fluid is given by Eq. (1.15).

As the physical model is symmetric, therefore we will take upper half of theabove sketched configuration. The x-axis is in the direction of flow and y-axis isnormal to it as shown in Fig. 4.2. The length of the curved channel formed by therolls are very large compared with the separation at the nip i-e, Ho � R, hence the

41

Figure 4.2: Geometry of the studied physical model

flow can be taken two dimensional, therefore from Eq. (3.1) we can write

A1 =

(

2∂u∂x

∂u∂y

+ ∂v∂x

∂u∂y

+ ∂v∂x

2∂v∂y

)

. (4.1)

Using Eq. (4.1) and setting ζ = µB

√2πc

/

py as the viscoplastic Casson parameter,along with continuity Eq. (1.36) and energy Eq. (1.39), Eqs. (1.8) and (1.9) in viewof Eqs (1.15) and (3.1) lead to the following equations

ρ

(

u∂u

∂x+ v

∂v

∂y

)

= −∂p

∂x+∂Sxx

∂x+∂Syx

∂y− σB2

0 u, (4.2)

ρ

(

u∂v

∂x+ v

∂v

∂y

)

= −∂p∂y

+∂Sxy

∂x+∂Syy

∂y− σB2

0 v, (4.3)

where Sxy = Syx and , Sxx, Syy respectively are shear and normal stresses.The essential dynamic events happen in the nip section, and spreading in ±x

direction the roll surfaces are nearly parallel. Then it is reasonable to assume thatv � u and ∂

∂x� ∂

∂y. In order to obtain the characteristic scale for the velocity and

pressure, we conduct in brevity an order of magnitude analysis, with the help ofLAT we can identify the following scales for x, y and u, x ∼ Lc, y ∼ H0, u ∼ U .Then from the Eq. (1.36), we obtain vc

U∼ H0

Lc� 1, showing that the order of mag-

nitude of a transversal velocity, vc, is smaller than the longitudinal velocity, where

Lc =√

2RH0 and ∆θc ∼√

2RH0

µ0

ρf cp

UH0

respectively are the longitudinal characteristic

length and temperature.

42

Based on the above discussion, Eqs. (4.2) and (1.39) are transformed to

µ

(

1 +1

ζ

)

∂2u

∂y2− σB2

0 u =dp

dx, (4.4)

0 = k∂2θ

∂y2+ µ

(

1 +1

ζ

)(

∂u

∂y

)2

. (4.5)

Since identical rolls have constant temperature θ0, and are rotating with the sameangular speed U , then appropriate boundary conditions are

{

u = U, θ = θ0 on y = h(x),∂u∂y

= 0, ∂θ∂y

= 0 on y = 0.(4.6)

In above equation h(x) is the y distance from the center line to the roll surface,defined in Eq. (1.35).

4.2 Dimensionless Equations

Based on the previous LAT analysis, the following non-dimensional quantities canbe defined :

x = x√2RH0

, u = uU, y = y

H0

, p =√

H0

2R

pH0

µU, Q = Q

2UH0

, Sxy = SxyH0

µU,

Sxx = SxxH0

µU, Syy = SyyH0

µU,M =

√

σµB0H0, h(x) = h(x)

H0

= 1 + x2, θ = θ−θ0

∇θ0

.(4.7)

Introducing the dimensionless variables defined by previous relationship (4.7), Eqs.(4.4) - (4.6) yield

(

1 +1

ζ

)

∂2u

∂y2−M2u =

dp

dx, (4.8)

∂2θ

∂y2− Br

(

1 +1

ζ

)(

∂u

∂y

)2

= 0, (4.9)

u = 1 on y = h(x),θ = 1 on y = 1,

∂u∂y

= 0, ∂θ∂y

= 0 on y = 0.(4.10)

and Br = U2

Cp∆θc× µCp

k= EcPr, in which Ec and Pr are respective Eckert and

Prandtl numbers and Br is the Brinkman number.Moreover, the standard conditions for pressure in the calendering analysis has thefollowing forms:

{

dp

dxx=λ= p(x = λ) = 0,

p(x = −xf ) = 0.(4.11)

43

Along with above equation, the dimensionless form of the volumetric flow rate hasthe following form

Q = 1 + λ2 = ∫1+x2

0 udy, (4.12)

where λ is the leave-off distance, which is related to the exiting sheet thickness inthe calendering process by the relationship defined in the next subsection. From theclassical calendering analysis [3, 4], for Newtonian and non-Newtonian polymeric

liquids, there are two flow regions, one in which the pressure increases

(

dp

dx> 0

)

for (−xf ≤ x ≤ −λ), and one in which the pressure decreases (dp

dx< 0) for

(−λ ≤ x ≤ λ). In the following section, the dimensionless velocity and pressureprofiles are obtained for each region.

4.3 Sheet thickness

Once λ is found, then all other quantities of interest are freely available. The exitingsheet thickness H from an infinite reservoir is given by:

H

H0= 1 + λ2, (4.13)

where the entering sheet thickness can be calculated form Eq. (2.23).

4.4 Velocity Distribution

In order to find the velocity, pressure and detachments point of the viscoplasticsheet, we find an exact answer to Eqs. (4.8) and (4.12). The solution for Eqs. (4.8)subject to Eq. (4.10), is given by

u(x, y) =1

M2

(

M2 +dp

dx

) cosh(

My√γ

)

cosh(

Mh√γ

) − dp

dx

, (4.14)

where γ = 1 + 1ζ.

We note from Eq. (4.14) that u is an explicit function of y and an implicitfunction of x, through h(x) and p(x). For dimensionless flow rate, using Eq. (4.14)in Eq. (4.12) we arrive at

Q =1

M2

[√γ

M

(

M2 +dp

dx

)

tanh

(

Mh√γ

)

− dp

dxh

]

. (4.15)

44

Now applying the boundary condition, given by Eq. (4.11), into Eq. (4.15), weobtain

Q =

√γ

Mtanh

(

M (1 + λ2)√γ

)

. (4.16)

Substituting Eq. (4.16) into Eq. (4.15), and after some simplifications, one can getan explicit expression for dp/dx as

dp

dx= M2

tanh

(

M(1+λ2)√

γ

)

− tanh

(

M(1+x2)√

γ

)

tanh(

M (1+x2)√γ

)

− M√γ

(1 + x2)

. (4.17)

This equation is valid for −xf ≤ x ≤ λ, where λmust be determined. On integratingEq. (4.17) one can get pressure distribution of the form

p (x) = M2

∫ x

−xf

tanh

(

M(1+λ2)√

γ

)

− tanh

(

M(1+x2)√

γ

)

tanh(

M (1+x2)√γ

)

− M√γ

(1 + x2)

dx. (4.18)

The unknown integral in Eq. (4.18) cannot be solved analytically. Therefore, weneed some suitable numerical algorithm. A composite Simpson’s 1/3 rule for numer-ical integration was employed for the computation of these integrals. The numberof spatial discretization for numerical procedure used was 200. Now to calculate λat which we assume that P0 → 0 as x → −∞, a modified regula-falsi method hasbeen used. The calculated convergence criterion was 10−10and we get λ = 0.4634fixing M = 0.5 and ζ = 0.5. Keeping in mind that−xf is known. For this value ofλ we have the maximum sheet thickness to minimum gap width ratio is 1.2137.The velocity profile is obtained by substituting Eq. (4.17) in to Eq. (4.14)

u(x, y) =cosh

(

My√γ

)

cosh(

Mh√γ

) +

cosh(

My√γ

)

cosh(

Mh√γ

) − 1

tanh

(

M(1+λ2)√

γ

)

− tanh

(

M(1+x2)√

γ

)

tanh(

M (1+x2)√γ

)

− M√γ

(1 + x2)

.

(4.19)The results of Middleman [5] are retrieved for ζ → ∞ and M → 0.

45

4.5 Heat Transfer Analysis

Now using the velocity distribution (4.19) into Eq. (4.9), then Eq. (4.9) takes theform

d2θ

dy2− γBr

cosh(

My√γ

)

cosh(

Mh√γ

) +1

M2

dp

dx

cosh(

My√γ

)

cosh(

Mh√γ

) − 1

2

= 0, (4.20)

wheredp

dxis define in Eq. (4.17), now solving Eq. (4.20) subject to Eq. (4.10), we

get

θ (x, y) = γBr

−M√γ

(

1 + 1M2

dp

dx

)

2 cosh(

M (1+x2)√γ

)

2[

M2

γ

(

1 − y2)

− cosh2

(

My√γ

)

+ cosh2

(

M√γ

)]

.

(4.21)Again we can retrieve temperature distribution results for Newtonian fluid by settingζ → ∞ and M → 0.

4.6 Operating Variables

After the velocity, temperature, pressure distribution are established, then all en-gineering quantities are immediately available. These engineering quantities arecalculated in the succeeding manners.

4.6.1 Roll Separating-force

The roll-separating force F is defined by

F =

∫ λ

−∞p(x)dx, (4.22)

where F = FH0

µURW,F is the dimensional roll separating force per unit width W.

4.6.2 Power Input

The power transmitted by roll to the material is calculated by integrating the prod-uct of shear stress and the roll surface speed over the surface of the roll, which isobtained by setting y = 1 in Sxy.

Pw =

∫ λ

−∞Sxy(x, 1)dx, (4.23)

46

here Pw = Pw

µWU2 is the dimensionless power and the dimensionless stress tensor Sxy

is defined by

Sxy =

(

1 +1

ζ

)

∂u

∂y, (4.24)

where

∂u

∂y=

M√γ

1 +

tanh

(

M(1+λ2)√

γ

)

− tanh

(

M(1+x2)√

γ

)

tanh(

M (1+x2)√γ

)

− M√γ

(1 + x2)

sinh(

My√γ

)

cosh(

M (1+x2)√γ

) · (4.25)

With the help of Eq. (4.25) and setting y = 1, the Eq. (4.23) gives power.

Table 4.1: Effect of viscoplastic parameter ζ on detachment point, sheet thickness,roll separation force and power fixing M = 0.5

ζ λ H/H0 F Pw

0.1 0.4640 1.2153 5.1479 0.33390.3 0.4637 1.2150 1.9727 0.33260.5 0.4634 1.2147 1.3378 0.33140.7 0.4631 1.2144 1.0661 0.33040.9 0.4629 1.2142 0.9150 0.32911.1 0.4628 1.2141 0.8187 0.32771.3 0.4626 1.2139 0.7525 0.32701.5 0.4625 1.2139 0.7073 0.32611.7 0.4624 1.2138 0.6665 0.32542 0.4623 1.2137 0.6246 0.3244

4.7 Results and Discussion

We begin presenting our results by showing the velocity distribution obtained forviscoplastic material at different positions in the calendering process for differentvalues of Hartman number M and the viscoplastic parameter ζ. In Figs. 4.3-4.6graphs for the velocity distribution at x = (0, 0.25, 0.5, 0.75) for different values ofHartman number M, fixing viscoplastic parameter ζ at 0.5 are presented. It hasbeen observed that Hartman number reduces the velocity curve, as expected. TheHartman number M is due to the Lorentz force, which opposes the flow. That iswhy the velocity u (x, y) is a decreasing function of the Hartman number M. Thegraphs for the velocity distribution at x = (0, 0.25, 0.5, 0.75) for different values of

47

Figure 4.3: Effect of MHD on velocitydistribution at x = 0, fixing ζ = 0.5

Figure 4.4: Effect of MHD on velocitydistribution at x = 0.25, fixing ζ = 0.5

viscoplastic parameter ζ by fixing Hartman number at 0.5 are sketched in Figs. 4.7-4.10. It has been found that the velocity decreases with an increase in viscoplasticparameter ζ, and this decrement is more pronounced at the center line. By increasingthe value of viscoplastic parameter or decreasing the yield stress velocity decreases.

The effects of the viscoplastic parameter ζ and the Hartman number M onvarious rheological parameters are shown in Table 4.1. and 4.2 respectively. Itis interesting to note from Table 4.1 that by increasing the values of viscoplasticparameter ζ, fixing M = 0.5, reduces the detachment point, sheet thickness, powerand roll-separating force. Similarly, it is noted from Table 4.2 that with an increasein the value of magnetic parameter M, fixing ζ = 0.5, increases the power and allthe other quantities of engineering interest have the same behavior as was observedin Table 4.1.

The graphs of pressure gradient distributions are sketched in Figs. 4.11 and4.12 for several values of the Hartman number M and viscoplastic parameter ζ.Staring at x = λ, and moving towards the negative x -axis along the calender gap,it has been perceived that the pressure gradient turn into negative and attain itsminimum value at x = 0, beyond the nip region pressure gradient evenly increaseback to zero at x = −λ and then carry on to attain its maximum value and thenpressure gradient will decline until the attachment point is reached at x = −xf . It isworth to mention that for a certain value of the detachment point, λ, fixing M = 0.5and increasing the value of ζ tends to spread out the size of the field. Viscoplasticparameter ζ has a great effect at x = 0, since at this instant shear stress has itshighest negative value, which can be obtained from Eq. (4.4). On the other handit has smaller effects at x = ±λ, for the reason that a flat velocity profiles at this

48

Figure 4.5: Effect of MHD on velocitydistribution at x = 0.5, fixing ζ = 0.5

Figure 4.6: Effect of MHD on velocitydistribution at x = 0.75, fixing ζ = 0.5

point are obtained, with respect to an unyielding gesture of the sheet.The pressure distribution along the rolls are shown in Figs. 4.13 and 4.14 for

different values of the Hartman number M and viscoelastic parameter ζ. Startingfrom separation point x = λ and moving towards the negative x -axis the pressuredistribution rises monotonically and reaches its highest value at x = −λ, and thenreduces till the attachment position for a predetermined sheet thickness is attained.The viscoplastic parameter ζ and Hartman number M effect the pressure signifi-cantly atx = −λ. Effects of MHD, viscoplastic parameter ζ and Brinkman numberBr on temperature profile are shown in the Figs. 4.15-4.17 respectively. It is foundthat the temperature increases as M and ζ are increased, on the other hand by in-creasing Br number, temperature decreases. The presentation of contour graphs areshown in Figs. 4.19 and 4.20, for different values of Hartman number, viscoplasticparameter and Brinkman number.

Viscoplastic materials have the features of viscous material and plastic solidsbecause of the yield stress. The yielded and unyielded regions separated by the yieldline y1, are shown in Fig. 4.18, such regions have also been shown schematically byworkers [3, 40] during the calendering process. From Fig. 4.18, we observed thatthe incoming and the leaving sheets have a persistent thickness and has a plugvelocity like a plastic solid sheet. So, they establish the unyielded regions (gray)right before and after entering into contact with the rolls. Likewise from Fig. 4.18the plug velocity occurs at ±λ. Since the material first touches the rolls at −xf , asclear from Fig. 4.18, the material near to the rolls becomes yielded (blue) becauseof the softness. This is because of the shearing motion of the rollers. The central

49

Figure 4.7: Effect of viscoplastic pa-rameter ζon velocity distribution at x =0, fixing M = 0.5

Figure 4.8: Effect of viscoplastic pa-rameter ζ on velocity distribution atx = 0.25, fixing M = 0.5

Figure 4.9: Effect of viscoplastic pa-rameter ζon velocity distribution at x =0.5, fixing M = 0.5

Figure 4.10: Effect of viscoplastic pa-rameter ζ on velocity distribution atx = 0.75, fixing M = 0.5

50

Figure 4.11: Effect of viscoplastic pa-rameter ζ on pressure gradient distri-bution fixing M = 0.5

Figure 4.12: Effect of MHD on pressuregradient distribution fixing ζ = 0.5

Figure 4.13: Effect of viscoplastic pa-rameter ζ on pressure distribution fix-ing M = 0.5

Figure 4.14: Effect of MHD on pressuredistribution fixing ζ = 0.5

51

Figure 4.15: Effect of M on temperaturedistribution fixing ζ = 0.5 and Br =0.5

Figure 4.16: Effect of ζ on temperaturedistribution fixing M = 0.5 and Br =0.5

Figure 4.17: Effect of Br on temperature distribution fixing ζ = 0.5 and M = 0.5

52

Table 4.2: Effect of MHD on detachment point, sheet thickness, Roll-Separating forceand Power fixing ζ = 0.5

M λ H/H0 F Pw

0.1 0.4641 1.2153 1.4251 0.01360.3 0.4639 1.2152 1.3962 0.12050.5 0.4634 1.2147 1.3378 0.33140.7 0.4623 1.2137 1.2528 0.63680.9 0.4606 1.2121 1.1471 1.01621.1 0.4583 1.2100 1.0289 1.44551.3 0.4555 1.2074 0.9065 1.90991.5 0.4523 1.2045 0.7866 2.36331.7 0.4487 1.2013 0.6742 2.81902 0.4627 1.1959 0.5256 3.4715

speed rises, whereas the shear-rate reduces, this will allow soften or yield material tobecome solid or unyielded. The core speed is equal to that of roll speed at x = −λ,and this core speed varies for different rolls, creating a crest and remain totallyunyielded. Core speed starts to increase when material flow through the unyieldedregion at x = −λ, this act will also increase the shear forces, the material closer torolls become softened (yielded regions, blue). This effect will continue till x = 0,where we have the maximum core velocity and minimum core thickness. Beyondthe small gap, one can see the reverse process and at x = λ the unyielded or solidcore spread out forming a crest once again, the calendered sheet leaves the rolls atthis point. This result is also pointed out by [3, 40].

4.8 Conclusions

A theoretical model of viscoplastic non-isothermal calendering process has beendeveloped. The lubrication approximation theory has been used to simplify theequations of motion and exact solutions for velocity and temperature distributionsare derived. Various engineering quantities are also calculated. The outcome of thepresent chapter can be summarized as follows:

1. The velocity decreases as viscoplastic parameter ζ and Hartman number Mincreases.

2. Temperature increases as M and ζ are increased, on the other hand by in-creasing Br number temperature decreases.

53

Figure 4.18: Yielded/unyielded region for viscoplastic fluid during the calenderingprocess

Figure 4.19: Contour graph pattern forlongitudinal velocity in the calenderingviscoplastic material at M = 0.5, ζ =0.5, Br = 0.5

Figure 4.20: Contour graph pattern forlongitudinal velocity in the calenderingviscoplastic material at M = 2, ζ = 2,Br = 0.5

54

3. By increasing the values of viscoplastic parameter ζ and fixing M = 0.5,reduces the detachment point, sheet thickness, roll separation force and powertransmitted by roll to the material.

4. With an increase in the value of magnetic parameter M, by fixing ζ = 0.5,increases the power.

5. By increasing the values of magnetic parameter M and fixing ζ = 0.5, reducesthe detachment point, sheet thickness and roll separation force.

6. Shear thinning/thickening phenomena is observed

7. Viscoplastic parameter ζ causes to enhance sheet thickness, power input, androll separation force

8. Magnetic parameter M and viscoplastic parameter ζ provides a mechanismto control sheet thickness, power input, roll separating force and leave-offdistance.

9. The yielded and unyielded regions has been identified.

10. The results of Middleman [5] can be recovered as a special case of this chapterwhen ζ → ∞ and M → 0.

55

Chapter 5

Calendering analysis of athird-grade material

In this chapter, the study of a non-Newtonian material when it is dragged throughthe small gap between two counter rotating rolls is carried out. Equations of conti-nuity and momentum using LAT are solved for velocity and pressure with the helpof perturbation technique. By considering the influence of material parameter, de-tachment point for the third-grade material in the calendering process is determined.The leave-off distance is expressed in terms of eigenvalue problem. Integrated engi-neering quantities like extreme pressure, power-input and roll-separating force arealso calculated. It is observed that the material parameter has great influence ondetachment point, velocity and pressure distribution, which are useful in the calen-daring process.

5.1 Governing Equations

Consider an isothermal, incompressible laminar flow of a third-grade material whenthere is no body force are Eq. (1.7) and (1.8), and the extra stress tensor S for athird grade material and Rivlin–Erickson are given in Eqs. (1.18) and (1.19). Weshall not consider the model defined by Eq. (1.18) as an approximation to the simplematerial in the sense of a retardation, but consider it to be an exact model in thesense described by Fosdick and Rajagopal in [31]. We require that the Clausius-Duhem inequality hold and that the specific Helmholtz free energy be a minimumwhen the material is locally at rest, which leads to the following restrictions on thematerial coefficients:

µ ≥ 0, α1 ≥ 0, β1 = β2 = 0, β3 ≥ 0, −√

24µβ3 ≤ α1 + α2 ≤√

24µβ3. (5.1)

The model of Eq. (1.18) thus reduces to

S = µA1 + α1A2 + α2A21 + β3(trA

21)A1. (5.2)

56

Figure 5.1: Geometry of the studied physical model

If we substitute Eqs. (1.7), (1.19) and (5.2) in Eq. (1.8), we find that the Eq. (1.8)becomes

−∇p + µ∇2V + α1[∇2Vt + ∇2(∇×V) ×V + ∇(V · ∇2V + 14|A1|2)]

+(α1 + α2)∇ · A21 + β3A1∇|A1|2 + β3|A1|2∇2V = ρDV

Dt,

(5.3)

where ∇2 is Laplacian operator and subscript t denotes partial derivative with re-spect to time t.

Eq. (5.3) reduces to classical linearly viscous Newtonian material model whenall material moduli except µ are zero, and reduces to second grade material whenβ3 = 0.

5.2 Problem Formulation

Here in this chapter, we consider two dimensional flow of an incompressible, isother-mal, laminar, steady flow of a third-grade material, which is dragged through thenarrow region between the two co-rotating cylinders of the same radii R in order tomake a sheet. The rest of the physical parameters are same as defined in section4.1, as shown in Fig. 5.1.

Using the velocity profile (3.1) in Eqs. (1.7) and (5.3) and introducing

ψ =∂v

∂x− ∂u

∂y, (5.4)

57

d =ρ

2q2 − α1(u∇2u+ v∇2v) −

(

3α1 + 2α2

4

)

D + p, (5.5)

where

D = |A1|2 = tr(A1AT1 ) =

[

4

(

∂u

∂x

)2

+ 2

(

∂u

∂y+∂v

∂x

)2

+ 4

(

∂v

∂y

)2]

, (5.6)

andq2 = u2 + v2, (5.7)

then Eqs. (1.7) and (5.3) reduce to

∂u

∂x+∂v

∂y= 0, (5.8)

∂d

∂x+v(−ρψ+α1∇2ψ)+µ

∂ψ

∂y+β3

∂

∂y(ψD)−2β3

{(

∂u

∂x

∂

∂x+∂v

∂x

∂

∂y

)

D

}

= 0, (5.9)

∂d

∂y− u(−ρψ + α1∇2ψ)− µ

∂ψ

∂x− β3

∂

∂x(ψD) − 2β3

{(

∂u

∂y

∂

∂x+∂v

∂y

∂

∂y

)

D

}

= 0.

(5.10)Since the most imperative dynamics events happen in the nip region. Using the LATanalysis as define previously in chapter 4, above Eqs. (5.8)-(5.10) are transformedto

∂u

∂x+∂v

∂y= 0, (5.11)

∂2u

∂y2+

2β3

µ

∂

∂y

(

∂u

∂y

)3

=1

µ

∂P

∂x, (5.12)

∂P

∂y= 0, (5.13)

where the modified pressure P is defined as

P = p− (α1 + α2)

(

∂u

∂y

)2

. (5.14)

Since the two rolls are identical and rotating with the same angular speed U , theirappropriate boundary conditions are defined in Eq. (2.6).

58

Figure 5.2: Geometry in dimensionless variables of the physical model

5.3 Dimensionless Equations

As from the order of magnitude investigation carried out earlier, let us define thefollowing dimensionless variables:

x =x√

2RH0

, u =u

U, y =

y

H0, P =

√

H0

2R

PH0

µU, β =

2β3U2

µH0, h(x) =

h(x)

H0.

(5.15)Using the non-dimensional variables defined by above Eq. (5.15), Eq. (5.12) yield

∂2u

∂y2+ β

∂

∂y

(

∂u

∂y

)3

=dP

dx, (5.16)

The boundary conditions for Eq. (5.16) are defined in Eq. (2.9). Moreover, thenormal conditions for the study of pressure and flow rate are define in Eq. (4.11)and Eq. (4.12).

5.4 Sheet thickness

Once λ is found, then all other quantities of interest are readily available. Theexiting sheet thickness H from an infinite reservoir is given by Eq. (4.13), wherethe entering sheet thickness can be calculated form Eq. (2.23).

59

5.5 Solution for β � 1

In order to calculate the detachment point of the material sheet, an asymptoticsolution has been conducted to Eqs. (5.16), (2.9), (4.11) and (4.12), for this aregular perturbation method is used and setting β as a perturbation parameter, thefollowing series solution can be propose

u(x, y) = u0(x, y) + βu1(x, y) + ..., (5.17)

P (x) = P0(x) + βP1(x) + ..., (5.18)

Q = Q0 + βQ1 + ..., (5.19)

λ = λ0 + βλ1 + ..., (5.20)

where u0, P0, Q0 and λ0 are the zero-order solutions and signify the Newtonianresults [3, 5] whereas u1, P1, Q1 and λ1 are the modifications till first-order solutionand comprise the involvement of the non-Newtonian effect. By using Eqs. (5.17)–(5.20) into Eqs. (5.16), (2.9), (4.11) and (4.12) and gathering terms of the similarpower of β, the set of problems can be obtained as follows:

5.5.1 Zero-order Problem

dP0

dx=∂2u0

∂y2, for − xf ≤ x ≤ λ0, (5.21)

Q0 = 1 + λ20 = ∫1+x2

0 u0dy. (5.22)

The boundary conditions for above equations are

∂u0

∂y= 0 at y = 0,

u0 = 1 at y = 1 + x2,dP0

dx= P0 = 0 at x = λ0.

(5.23)

5.5.2 First-order Problem

dP1

dx=∂2u1

∂y2+

∂

∂y

(

∂u0

∂y

)3

, (5.24)

Q1 = 2λ0λ1 = ∫1+x2

0 u1dy. (5.25)

60

For solving Eqs. (5.24) and (5.25), we want the zero-order solutions of (5.21) and(5.22). The first order boundary conditions are

∂u1

∂y= 0 at y = 0,

u1 = 0 at y = 1 + x2,dP1

dx= P1 = 0 at x = λ1.

(5.26)

5.5.3 Zero-order Solution

The solutions for Eqs. (5.21) and (5.22), for predetermined sheet thickness are

u0 = 1 +1

2

(

dP0

dx

)

[

y2 − (1 + x2)2]

, (5.27)

withdP0

dx= −3

(λ20 − x2)

(1 + x2)3for − xf ≤ x ≤ λ0, (5.28)

and

P0(x) =3

8

x2(1 − 3λ20) − 1 − 5λ2

0

(1 + x2)2x+ (1 − 3λ2

0)(

tan−1 x− tan−1 xf

)

+(xf

2 + 1) (1 − 3λ20) − 2(1 + λ2

0)

(xf2 + 1)2 xf

. (5.29)

From Eq. (5.29), if we assume that P0 → 0 as x → −∞ we get λ0 = 0.4751.Keeping in mind that xf is known. For this value of λ0 we have the maximum sheetthickness to minimum gap width ratio is 1.226.

The zeroth order velocity profile is obtained by substituting Eq. (5.28) into Eq.(5.27)

u0(x, y) = 1 +3

2

(x2 − λ20)

(1 + x2)3

[

y2 −(

1 + x2)2]

. (5.30)

We must emphasize that the solutions given by Eqs. (5.27)-(5.30) were obtained inprevious works [3, 5, 16].

5.5.4 First-order solution

Using Eq. (5.30) in Eq. (5.24) and integrating the resulting equation twice andusing the boundary conditions (5.26), we get the following first order solution

u1(x, y) =27

4

(λ20 − x2)

3

(1 + x2)9

[

y4 −(

1 + x2)4]

+1

2

dP1

dx

[

y2 −(

1 + x2)2]

. (5.31)

61

To determine the first-order flow, we substitute Eq. (5.31) into Eq. (5.25), and get

Q1 =27

5

(x2 − λ20)

3

(1 + x2)4 − 1

3

dP1

dx

(

1 + x2)3, (5.32)

here Q1 involves unknown pressure gradient dP1/dx. Flow rate Q1 and dP1/dx canbe obtained by applying the boundary condition given in Eq. (5.26). Thus, anexplicit expression for dP1/dx is

dP1

dx=

81

5

[

(x2 − λ20)

3

(1 + x2)7 − (λ21 − λ2

0)3

(1 + x2)3 (1 + λ21)

4

]

. (5.33)

Eq. (5.33) is valid for −xf ≤ x ≤ λ1, where λ1 must be determined as a part ofproblem. The pressure distribution is obtained by integrating Eq. (5.33), hence

P1(x) =81

5

∫ x

−xf

[

(x2 − λ20)

3

(1 + x2)7 − (λ21 − λ2

0)3

(1 + x2)3(1 + λ2

1)4

]

dx. (5.34)

The dimensionless leave-off distance λ1 may be found from above equation, since ithas been assumed that P1 = 0 at x = −xf . Therefore Eq. (5.34) can be written as

∫ λ1

−xf

[

(x2 − λ20)

3

(1 + x2)7 − (λ21 − λ2

0)3

(1 + x2)3(1 + λ2

1)4

]

dx = 0. (5.35)

Integration Eq. (5.34) and applying the boundary conditions of Eq. (5.26), we havethe following form of pressure distribution

P1(x) = 0.0129 tan−1(x) − (6.0750λ61 − 4.1137λ4

1 + 0.9285λ21 − 0.0699)

tan−1(x)

(1 + λ21)

4

+

[ −2.4860

(1 + x2)6 +4.5669

(1 + x2)5 − 2.3084

(1 + x2)4 +0.0069

(1 + x2)3 +0.0086

(1 + x2)2 +0.0129

(1 + x2)

]

x

−

(

4.0500

(1 + x2)2 +6.0750

(1 + x2)

)

λ61 −

(

2.7424

(1 + x2)2 +4.1137

(1 + x2)

)

λ41

+

(

0.6190

(1 + x2)2 +0.9285

(1 + x2)

)

λ21 −

(

0.0466

(1 + x2)2 +0.0699

(1 + x2)

)

x

(1 + λ1)4

−0.0129 tan−1 (xf) + (6.0750λ61 − 4.1137λ4

1 + 0.9285λ21 − 0.0699)

tan−1 (xf)

(1 + λ21)

4

62

−[

−2.4860

(xf2 + 1)

6 +4.5669

(xf2 + 1)

5 − 2.3084

(xf2 + 1)

4 +0.0069

(xf2 + 1)

3 +0.0086

(xf2 + 1)

2 +0.0129

(xf2 + 1)

]

(xf )

+

(

4.0500

(xf2 + 1)2 +

6.0750

(xf2 + 1)

)

λ61 −

(

2.7424

(xf2 + 1)2 +

4.1137

(xf2 + 1)

)

λ41

+

(

0.6190

(xf2 + 1)

2 +0.9285

(xf2 + 1)

)

λ21 −

(

0.0466

(xf2 + 1)

2 +0.0699

(xf2 + 1)

)

(xf )

(1 + λ1)4

(5.36)From Eq. (5.36), if we assume that P1 → 0 as x → −∞ we get λ1 = 0.3336. Keepingin mind that xf is known.

The 1st order velocity profile is obtained by substituting Eq. (5.33) into Eq.(5.31)

u1(x, y) =27

4

(λ20 − x2)

3

(1 + x2)9

[

y4 −(

1 + x2)4]

+81

10

(x2 − λ20)

3

(1 + x2)7

− (λ21 − λ2

0)3

(1 + x2)3 (1 + λ21)

4

[

y2 −(

1 + x2)2]

.

(5.37)Combining the solutions at each order of approximation yields the solutions up tofirst order for velocity, pressure gradient and pressure.

5.6 Operating Variables

After the velocity, temperature, pressure distribution are established, then all en-gineering quantities are immediately available. These engineering quantities arecalculated in the succeeding manners.

5.6.1 Roll-Separating Force

The roll-separating force F in dimensionless form can be calculated using Eq. (4.22).

5.6.2 Power Input

The power transmitted to the material by roll is calculated by integrating the prod-uct of shear stress and the roll surface speed over the surface of roll which is obtainedby setting y = 1 in Eq. (5.37)

Pw =

∫ λ

−∞

[

∂u

∂y+ β

(

∂u

∂y

)3]

dx, (5.38)

63

here Pw = Pw

µWU2 is the dimensionless power and

∂u

∂y= 3

(x2 − λ20)

(1 + x2)3 y+β

(

81

5

(

(x2 − λ20)

3

(1 + x2)7 − (λ21 − λ2

0)3

(1 + x2)3 (1 + λ21)

4

)

y + 27(λ2

0 − x2)3

(1 + x2)9y3

)

.

(5.39)With the help of Eq. (5.39) and setting y = 1, Eq. (5.38) gives power.

5.6.3 Normal Stresses Effect

Eq. (5.14) in dimensionless form can be written as

p (x, y) = P (x, y) + α

(

du

dy

)2

, (5.40)

where α =(2α1 + α2)U

µH0

√

H0

2R.

By using Eq.(5.17) and Eq.(5.18) in above equation, one can easily find normalstresses.

Table 5.1: The effect of material parameter β on leave-off distance, final sheet thick-ness, power input and roll-separating force

β λ H/H0 F Pw

0.01 0.4784 1.2289 0.0062 0.55270.03 0.4851 1.2353 0.0114 0.55420.05 0.4917 1.2418 0.0174 0.55570.07 0.4984 1.2484 0.0216 0.55720.09 0.5051 1.2551 0.0243 0.55870.1 0.5084 1.2585 0.0253 0.55940.3 0.5751 1.3308 0.0747 0.57440.5 0.6419 1.4120 0.1267 0.59010.7 0.7086 1.5021 0.1761 0.60680.9 0.7753 1.6011 0.2210 0.62481.0 0.8087 1.6539 0.2422 0.6322

5.7 Results and discussions

In this chapter, we analyze the calendering process for incompressible third-gradematerial. The lubrication theory is used to simplify the equations of motion. The

64

Figure 5.3: Effect of β on velocity atx = −0.5

Figure 5.4: Effect of β on velocity atx = −0.25

numeric results for the flow rate Q, separation point λ, the exit sheet thicknessH/H0, the power input, and roll separation force are presented in Table 5.1. TheTable 5.1 is generated for various values of β. For the physical point of view, anincrease in the third-grade parameter β corresponds to shear thickening effect andan increase in the third-grade parameter β results in an increase in viscosity of thematerial. More viscous material diffuses more momentum. Consequently, magnitudeof velocity decreases. This fact is obvious from Figs. 5.3-5.10. It is noted from Table5.1 that sheet thickness, power input as well as roll separation force increases withan increase in β. This was physically expected because of the shear thickening effect.

In Figs. 5.3-5.10 the dimensionless velocity u(x, y) is presented in terms of thetransversal coordinate y for various values of β. The velocity profiles are simplifiedat eight different locations of x. In Fig. 5.3 and 5.4 the velocity is graphed in theinterval −xf ≤ x ≤ −λ, subject to the position near the entry at which pressuregradient is positive. It can seem that velocity decreases with the material parameterto a certain value of λ, and this increase is more prominent in the nip region. Incontrast, at the locality of the rolls, velocity reduces as compare to Newtonianmaterial. The velocity distribution is also sketched in −λ ≤ x ≤ λ, velocity increasessoftly near the rolls, and decreases at the central position when β is increased aspresented in Figs. 5.5-5.9.

Figs. 5.11 shows the graphs of pressure gradient in terms x for certain valuesof the material parameter β =(0.0, 0.25, 0.5, 0.75, 0.9), and two different valuesof the detachment point, λ =(0.4751, 0.3336). Symmetric profiles about the nippoint are obtained. The pressure gradient is negative at x = 0, and increases sym-

65

Figure 5.5: Effect of β on velocity atx = 0

Figure 5.6: Effect of β on velocity atx = 0.25

metrical about this point, attain maximum value and then decreases exponentiallyand reaches to zero value at x = ±4. Moreover, an increase in β causes to increasepressure gradient. Also, material parameter has a major effect on pressure gradientat x = 0 because the pressure has a maximum absolute value at this point. Thisfact is obvious from Eqs. (5.28) and (5.33). Whereas, pressure gradient has no effectat x = ±λ, since we have flat velocity profiles at this point.

The graphs for the pressure distribution are presented in Fig. 5.12. Starting fromzero value at x = λ, the pressure distribution rises monotonically up to its extremevalue at x = −λ, and then decreases till the attachment point is reached. Bychanging the material parameter β, the pressure is significantly affected at x = −λ.Likewise, an increase in the value of non-Newtonian parameter β results in to extendthe length of attachment point as can be seen from the Table 5.1. Fig. 5.13 showsthe normal stress effects at different positions of the calendering process keepingβ = 0.5 for different values of α . It has been observed the with an increase in αat the nip normal stresses increases, whereas an opposite behavior is observed atx = 0.5.

5.8 Concluding Remarks

The finding of the present chapter can be summarized as follows:

1. Shear thickening phenomena is observed

2. Shear thickening causes to enhance sheet thickness, power input, and roll

66

Figure 5.7: Effect of β on velocity atx = 0.4

Figure 5.8: Effect of β on velocity atx = 0.5

separation force

3. Material parameter β provides a mechanism to control sheet thickness, powerinput, roll separating-force and leave-off distance

4. Sheet thickness, power input, roll separating force and leave-off distance fornon-Newtonian material are larger than those of Newtonian material

67

Figure 5.9: Effect of β on velocity atx = 0.6

Figure 5.10: Effect of β on velocity atx = 1.0

Figure 5.11: Effect of β on pressuregradient distribution

Figure 5.12: Effect of β on pressure dis-tribution

68

Figure 5.13: Normal stress effect at position x = 0 and at x=0.5, respectively fixingβ = 0.5

69

Chapter 6

Calendering of a second-gradematerial onto a moving poroussheet

In this chapter, the LAT is used to provide numerical results when a porous sheetis dragged through the small gap between two counter rotating porous rolls. Therate of injection at the roll surface is assumed equal to the rate of suction on theweb. The second-grade material is used, which becomes the Newtonian materialmodel with some suitable variations. Unified quantities of interest has been attainedlike pressure distribution, coating thickness, splitting point, stresses, Separationforces and power consumption etc., are calculated. Some of these results are showngraphically. It is found that material parameter β and Reynolds number are thedevices to control flow rate, coating thickness separation points, Separation force,Power input and Pressure.

In this process a uniform film of liquid is deposited on a moving sheet. Thisprocess is similar to the calendering process as discussed in chapters 2-5. Likematerial calendering, there is a converging-diverging character to the kinematics,and we can expect the dynamics to be similar to that described in the materialcalendering process. The major difference is in the character of the separation region,where the fluid splits and adheres to both moving surfaces. In material calenderinganalysis it is assumed that the fluid separates clearly from the roll, whereas in thesheet calendering analysis, it is assumed that the material evenly wets both the rolland the sheet.

70

Figure 6.1: Physical presentation of calendering of a second-grade material onto amoving porous sheet

6.1 Mathematical Modeling

6.1.1 Problem Formulation

We consider an incompressible isothermal, laminar, steady second-grade material toapply a thin liquid film on a moving porous sheet when it passes through two co-rotating rolls, as shown in Fig. 6.1. Since the physical model is symmetric, thereforewe will take upper half of the above given configuration. The roll is rotating anticlockwise with an angular velocity U while the plane is moving linearly with thesame velocity as that of roll and their separation at the nip is H0. The length of thecurved channel formed by the roll and the plane are very large compared with theseparation at the nip i-e, H0 � R, hence the flow can be taken as two dimensional.In view of velocity field (3.1), the Eq. (1.7) and (1.8) in light of Eq. (1.16) reducesto

∂u

∂x+∂v

∂y= 0, (6.1)

∂d

∂x+ v(−ρψ + α1∇2ψ) + µ

∂ψ

∂y= 0, (6.2)

∂d

∂y− u(−ρψ + α1∇2ψ)− µ

∂ψ

∂x= 0, (6.3)

where

ψ =∂v

∂x− ∂u

∂y,

71

Figure 6.2: Geometry of the problem

d =ρ

2q2 − α1(u∇2u+ v∇2v) −

(

3α1 + 2α2

4

)

D + p,

D = |A1|2 = tr(A1AT1 ) =

[

4

(

∂u

∂x

)2

+ 4

(

∂v

∂y

)2

+ 2

(

∂u

∂y+∂v

∂x

)2]

,

andq2 = u2 + v2.

In case of second-grade material, we consider α1 < 0 and α1 + α2 6= 0 [32].

We note from the geometry of the problem that the most imperative dynamicsevents happen in the nip region. The distance between the roll and the web at thenip, H0, is so small as to be negligible in comparison with length and radius of theroll. Then it is reasonable to assume that the flow is nearly parallel, so that thegeneral movement of the material is mainly in the x-direction, the velocity of thematerial in the y-direction is small. It is also assumed that the roll and plane both areporous, that is, mass is sucked by the cylinder with a velocity v0 and the same mass isinjected from plane with the velocity v0, where v0 is a constant which represents thesuction (v0 > 0) or injection velocity (v0 < 0). To construct the characteristic scalesfor the velocity and pressure, we perform an order of magnitude analysis, for whichwe set the following scales for x, y and u, x ∼ Lc, y ∼ H0, u ∼ U . From the law

72

of conservation of mass Eq. (6.1), and taking into account the above relationship,we obtain vc

U∼ H0

Lc� 1, and the longitudinal characteristic length is given by

Lc =√

2RH0, From Eq. (6.2), a dominant balance between the pressure and viscous

terms permit us to obtain the characteristic pressure, given by pc ∼√

2RH0

µ0

ρfcp

U

H0.

Based on the above discussion Eqs. (6.1)-(6.3) become

∂u

∂x= 0, (6.4)

ρv0∂u

∂y= −∂p

∂x+ µ

∂2u

∂y2+ α1v0

∂3u

∂y3, (6.5)

0 = −∂p∂y

+ 2 (2α1 + α2)∂u

∂y

∂2u

∂y2, (6.6)

where the modified pressure P is defined by

P = p− (2α1 + α2)

(

∂u

∂y

)2

. (6.7)

The final equation of motion will then be

ρv0∂u

∂y= −dP

dx+ µ

∂2u

∂y2+ α1v0

∂3u

∂y3. (6.8)

If we assume that both roll and sheet are moving with the same speed U , thenappropriate boundary conditions are

u = U at y = h(x),u = U at y = 0,∂u

∂y= 0 at y = 0.

(6.9)

here, h(x) is the y distance from the plane to the roll surface, which is explicitlygiven in Eq. (1.35).

6.1.2 Dimensionless Equations

Based on the LAT analysis carried in the previous section, consider the followingnon-dimensional variables:

x =x√RH0

, u =u

U, y =

y

H0, P =

PH3

2

0

µUR1

2

,

β =2α1v0

µH0, h(x) =

h(x)

H0, Re =

h(x)v0

υ.

(6.10)

73

The Eqs. (6.5), (6.9) and (1.35), take the forms respectively.

β∂3u

∂y3+∂2u

∂y2− Re

∂u

∂y=dP

dx, (6.11)

u = 1 at y = h(x),u = 1 at y = 0,∂u

∂y= 0 at y = 0,

(6.12)

where h(x) = 1 +x2

2.

Along with Eq. (6.11), the following mass balance equation in dimensionless formis needed

Q =

∫ h(x)

0

u(y)dy, (6.13)

where Q =Q

UWH0, Q is the dimensional flow rate per unit width.

6.2 Solution of the problem

The exact solution of Eq. (6.11) subject to boundary conditions given by

u(x, y) = 1 − 1

Re

(

dP

dx

)

[

y +K1

(

K2 −K3eG1y −K4e

G2y )]

, (6.14)

where

K1 =1

G1 (eG2h − 1) −G2 (eG1h − 1), K2 =

(

eG2h − eG1h)

+ h (G1 −G2) ,

K3 =(

eG2h − 1)

−G2h,K4 = G1h−(

eG1h − 1)

G1 =−1 +

√1 + 4βRe

2β, G2 =

−(

1 +√

1 + 4βRe)

2β.

At this point, the pressure and the pressure gradient are unknown. However, we canperform a mass balance by using Eqs. (6.12) and (6.14) to compute the volumetric

flow rate and solving it fordP

dx, we arrive at

dP

dx=

2G1G2Re (Q− h)

2K1 [G2K3(eG1h − 1) +G1K4(eG2h − 1) −G1G2K2h] −G1G2h2. (6.15)

74

Combining Eqs. (6.14) and (6.15), we can write

u(x, y) = 1 − 2G1G2 (Q− h)[

y +K1

(

K2 −K3eG1y −K4e

Gy2

)]

2K1 [G2K3(eG1h − 1) +G1K4(eG2h − 1) −G1G2K2h] −G1G2h2.

(6.16)Upon integration Eq. (6.15), we find

P = 2Re

∫ x

−x0

G1G2 (Q− h)

2K1 [G2K3(eG1h − 1) +G1K4(eG2h − 1) −G1G2K2h] −G1G2h2dx.

(6.17)If Re → 0 and β → 0, then Eqs. (6.14), (6.15) and (6.17) reduces to

u(x, y) = 1 +1

2

dp

dx

[

y2 − hy]

, (6.18)

dp

dx= 12

h −Q

h3, (6.19)

p = 12

∫ x

−x0

h−Q

h3dx. (6.20)

We must emphasize that the above three equations was obtained in previous work[5, 36].

6.3 Coating Thickness

Thickness of the entering sheet can be calculated by

Hf = 1 +x2

f

2, (6.21)

It will be a big quantity when we have an infinite reservoir. Since we are assumingthat liquid splits evenly, so we can write that

Q = 2UH,

and in dimensionless formH

H0=Q

2, (6.22)

which is the resulting exit coating thickness. The investigation is still incomplete atthis stage, the relation between Q (flow rate), and the xs (separation point) needto be establish. We can see from Fig. 6.2 that the film splits evenly, and thereforeseparation point is

(

xs,12h(xs)

)

, at this point the velocity and pressure goes to zero.This supposition can be acknowledged because the sheet and the roll has the similar

75

velocity. This will provides xs in terms of Q. In case of Newtonian and viscoelasticmaterials the relation connecting separation point xs, flow rate λ admits the valueof one of these parameters in terms of other [5, 36]. As in case of second-gradematerial we can’t find explicitly xs as a function of Q. To find the separation pointxs we have to use some numerical algorithm.

6.4 Operating Variables

After the velocity, temperature, pressure distribution are establish, then all engineer-ing quantities are immediately available. These engineering quantities are calculatedin the succeeding manners:

6.4.1 Separation Force

The roll-separating force F , is defined by

F =

∫ xs

−∞p(x)dx, (6.23)

where F =FH0

µURW, F is the dimensional separation force per unit width W .

6.4.2 Power Input

The power transmitted to the material by roll is calculated by integrating the prod-uct of shear stress and the roll surface speed over the surface of roll which is obtainedby setting y = H0, as

Pw =

∫ xs

−∞Sxy(x, 1)dx, (6.24)

here Pw =Pw

µWU2is the dimensionless power and Sxy =

SxyH0

µUthe dimensionless

stress tensor defined by

Sxy =du

dy+β

2

d2u

dy2. (6.25)

With the help of Eq. (6.16) and setting y = 1, the Eq. (6.25) becomes

Sxy (x, 1) =2G1G2 (Q− h)

[

K1K3G1eG1

(

1 + βG1

2

)

+K1K4G2eG2

(

1 + βG2

2

)

− 1]

2K1 [G2K3(eG1h − 1) +G1K4(eG2h − 1) −G1G2K2h] −G1G2h2.

(6.26)

76

Using Eq. (6.26) in Eq. (6.25), to obtain

Pw =

∫ x1

−∞

2G1G2 (Q− h)[

K1K3G1eG1

(

1 + βG1

2

)

+K1K4G2eG2

(

1 + βG2

2

)

− 1]

2K1 [G2K3(eG1h − 1) +G1K4(eG2h − 1) −G1G2K2h] −G1G2h2dx.

(6.27)The power input has the potential to raise the temperature of the material by anamount which at most, is given by an adiabatic temperature rise.

6.5 Results and Discussion

In this Chapter, we analyze the roll coating process for incompressible second-gradematerial. The LAT is used to simplify the equations of motion. The unknown in-tegrals in Eqs. (6.17), (6.23) and (6.27) cannot be solved analytically. Therefore,we need some suitable numerical algorithm. A composite Simpson’s rule for numer-ical integration was employed for the computation of these integrals. The numberof spatial discretization for numerical procedure used was 200. The governing Eq.(6.17) for the pressure was first solved via a composite Simpson’s rule, and to calcu-late the value of separation point, the modified regula-falsi method has been used.The calculated convergence criterion was 10−10. Note that the validation of thealgorithm was tested by reproducing the results for Newtonian fluid [5, 36]. Theresults for different rheological parameters are presented in Table 6.1 and Table 6.2.The Table 6.1 is generated for various values of β fixing Re = 0.1 while in Table 6.2non-Newtonian parameter β is fixed (β = 0.1) and Re is varied. It is found thatthe highest coating thickness is 0.62545 for Q = 1.2509, β = 0.01 and Re = 0.1for which the maximum separation point xs = 2.3402 is observed. It is noted thatcoating thickness decreases with an increase in β, the minimum coating thicknesshas been observed at β = 50. Whereas an opposite behavior is observed in Table6.2. With an increase in Re (inertia forces is dominating viscous force) the coatingthickness, volume flow rate as well as separation point increases. The maximumcoating thickness has been observed at Re = 5. It is also noted form Table 6.2, thatwhen the Reynolds number exceed the value 2 the separation point does not reaches.Physically it seems that effect of inertial force is so large that the separation pointdoes not reach. Infect separation force decrease with an increase in Reynolds num-ber and eventually vanish for Re → 2. It is worth to mention that the maximumrange of Re is 0 to 3, beyond this point no separation point is observed. Similarlyfor β the maximum range is 0-50, beyond this point separation point remains thesame.

The results for the dimensionless velocity distributions are shown in Figures 6.3to 6.8. Figs. 6.3-6.5 gives velocity profiles at different positions fixing Re = 0.1and varying β = 0.01, 0.05, 0.1, 0.5, 1.0. It is observed that with an increase in βvelocity of the fluid decrease at different positions of the roll coating process. Figures

77

Table 6.1: Effect of non-Newtonian parameter on different rheological parametersfixing Re = 0.1, for calendering of second-grade material onto a moving poroussheet

β Q H/H0 xs

0.01 1.2509 0.62545 2.34020.02 1.2485 0.62425 2.32930.03 1.2461 0.62305 2.31840.04 1.2437 0.62185 2.30770.05 1.2414 0.62070 2.29720.06 1.2390 0.61950 2.28670.07 1.2367 0.61835 2.27650.08 1.2344 0.61720 2.26240.09 1.2322 0.61610 2.25660.10 1.2300 0.61500 2.24700.20 1.2110 0.60550 2.21660.30 1.1981 0.59905 2.11740.40 1.1897 0.59485 2.09450.50 1.1840 0.59200 2.08590.60 1.7099 0.58995 2.08460.70 1.1777 0.58850 2.08710.80 1.1747 0.58735 2.09110.90 1.1729 0.58645 2.09591.00 1.1715 0.58575 2.10102.00 1.1654 0.58270 2.14123.00 1.1634 0.58170 2.16314.00 1.1625 0.58125 2.17635.00 1.1619 0.58090 2.184950.0 1.1572 0.57860 2.2180

78

Figure 6.3: Velocity distribution at x =0.0 fixing Re = 0.1

Figure 6.4: Velocity distribution at x =0.5 fixing Re = 0.1

6.6 to 6.8 depicts velocity profiles at different positions fixing β = 0.1 and varyingRe = 0.01, 0.05, 0.1, 0.5, 1.0. It is noted that velocity increases with an increase inReynolds number due to dominance of inertia force over viscous force.

Numerical finding for pressure gradient distribution are depicted in Figs 6.9and 6.10, whereas the results for pressure distribution has been sketched in Figs.6.13 and 6.14. It is observed that pressure gradient increases with an increase inReynolds number and second grade parameter β. Pressure distribution decreasewith an increase in Reynolds number as shown in Table 6.4., whereas an oppositebehavior is observed in the case when second grade parameter β increases as shownin Table 6.3. Note that pressure distribution in Tables 6.3 and 6.4 is calculated overcertain range of x.

The graphs for separation force and power input are sketched in Fig. 6.11 and6.12, and their numerical values for different parameters are shown in Table 6.3 and6.4. It is observed that separation force and power input strongly depends upon βand Re. An increase in β causes increase in separation force as well as power input,whereas by increasing Reynolds numbers separation force decrease and power inputincreases.

We have analyzed the calendering process for the steady second grade fluid; thisstudy is valid only in case of porous boundaries. If boundaries are solid then wewill have no effect of the second-grade parameter. This case will simply reduce toNewtonian material, and therefore the results will not further be accurate for thesecond-grade material.

79

Table 6.2: Effect of Reynolds number on different rheological parameters fixing β =0.1, for calendering of second-grade material onto a moving porous sheet

Re Q H/H0 xs

0.01 1.22882 0.61441 2.238960.02 1.22888 0.61444 2.239660.03 1.22896 0.61448 2.240400.04 1.22906 0.61453 2.241200.05 1.22918 0.61459 2.242050.06 1.22931 0.61465 2.242950.07 1.22947 0.61473 2.243910.08 1.22963 0.61481 2.244890.09 1.22981 0.61490 2.245940.10 1.23000 0.61500 2.247020.20 1.23247 0.61623 2.260100.30 1.23563 0.61781 2.276920.40 1.23925 0.61962 2.297320.50 1.24320 0.62160 2.321360.60 1.24738 0.62369 2.349240.70 1.25174 0.62587 2.381300.80 1.25621 0.62810 2.418040.90 1.26076 0.63038 2.460191.00 1.26538 0.63269 2.508812.00 1.31072 0.65536 2.940253.00 1.34990 0.67495 .......4.00 1.38163 0.69081 .......5.00 1.40688 0.70344 .......

80

Figure 6.5: Velocity distribution at x =1.0 fixing Re = 0.1

Figure 6.6: Velocity distribution at x =0.0 fixing β = 0.1

Figure 6.7: Velocity distribution at x =0.5 fixing β = 0.1

Figure 6.8: Velocity distribution at x =1.0 fixing β = 0.1

81

Figure 6.9: Axial distribution of thepressure gradient fixing Re = 0.1

Figure 6.10: Axial distribution of thepressure gradient fixing β = 0.1

Figure 6.11: Separation force for Re =0.1, β = 0.05 and Q = 1.2414

Figure 6.12: Power input for Re = 0.1,β = 0.05 and Q = 1.2414

82

Figure 6.13: Axial distribution of thepressure for β = 0.5, Re = 0.1 andQ = 1.1840

Figure 6.14: Axial distribution of thepressure for β = 0.1, Re = 0.5 andQ = 1.24320

6.6 Conclusion

In the present Chapter, LAT was used for the procedure of calendering of a second-grade material onto a moving porous sheet. The major findings of the present studyare summarized as follows:

1. Reynolds number and volume flow rate provide a mechanism to control coatingthickness as well as separation point.

2. Material parameter is a device to control flow rate, coating thickness separationpoints, Separation force, Power input and Pressure.

3. Reynolds number also works as a device to optimize the size of the roll (radiusor diameter).

4. With an increase in Reynolds number Separation force and pressure distribu-tion decreases.

5. With an increase in Reynolds number Power input increases.

6. By increasing second grade parameter β Separation force, power input andPressure increases.

7. Inertia force play dominant role in coating thickness, separation force, powerand pressure.

83

Table 6.3: Effect of non-Newtonian parameter on dimensionless volumetric flowrate, separation force, power input and pressure distribution fixing Re = 0.1, forcalendering of a second-grade material onto a moving porous sheet

β Q F Pw P

0.01 1.2509 0.5468 1.0503 0.44980.05 1.2414 0.5608 1.1043 0.56180.10 1.2300 0.5912 1.1709 0.59200.50 1.1840 1.4807 1.6152 1.48631.00 1.1715 3.3415 2.0773 3.35972.00 1.1654 7.2893 2.8919 7.33543.00 1.1634 11.323 3.6377 11.3994.00 1.1625 15.340 4.3586 15.4455.00 1.1619 19.406 5.0536 19.54150.0 1.1572 204.29 36.911 205.67

Table 6.4: Effect of Reynolds number on dimensionless volumetric flow rate, sepa-ration force, power input and pressure distribution fixing β = 0.1, for calendering ofa second-grade material onto a moving porous sheet

Re Q F Pw P

0.01 1.22882 0.60886 1.13108 0.610120.05 1.22918 0.60691 1.14726 0.608090.10 1.23000 0.59123 1.17099 0.592110.50 1.24320 0.25922 1.42765 0.256401.00 1.26538 -0.15975 2.81257 -0.165712.00 1.31072 -0.57438 2.63256 -0.58206

8. The result of Middleman [5, 36] are retrieved when Re → 0 and β → 0.

84

Chapter 7

Calendering analysis of athird-grade material onto amoving thin sheet

In this chapter, an attempt is made to study the calendering process in order to applyan incompressible third-grade material onto a moving thin sheet. The LAT is usedto simplify the equations of motion. Solutions for velocity profile, pressure gradient,flow rate per unit width, and shear stress at the roll surface are obtained by usingregular perturbation method. Unified engineering quantities like extreme pressure,separation point, roll separating force power transmitted to the material by roll andcoating thickness are also calculated. It is found that these processing quantitiesincrease significantly and monotonically as the material parameter increases.

7.1 Formulation of the Problem

Consider an isothermal, incompressible, laminar and steady flow of a third-gradematerial between a cylindrical roller of radius R and a flat thin sheet. Roll isrotating counter clockwise with angular velocity ω, and sheet is also moving witha constant velocity U in positive x-direction. The roll and the sheet have a gap atthe nip, H0. At location x = −xb, polymer first bites the plate, as shown in Fig.6.2. The velocity of the roll takes the form U = ωR, hence the flow can be taken astwo dimensional.

We begin with the LAT analysis that the most imperative dynamics events hap-pen in the nip region. In that region, and extending to either side by a distance ofthe order of x0, the roll surfaces are nearly parallel. Then it is reasonable to assumethat v � u and ∂

∂x� ∂

∂y. The material moves in the x -directionand there is no

velocity in y-direction. Thus, Eq. (1.7) implies ∂u/∂x = 0, which means u = u(y).

85

Figure 7.1: Geometry of the studied physical model

Then, the continuity Eq. (1.7) is satisfied identically, the material derivativeDV /Dtvanishes and the momentum Eq. (1.8) reduces to −∇p+ divS = 0, where S is theextra stress tensor for third-grade material defined in Eq. (5.2). This leads Eq.(1.8) in component form as

dSxy

dy− ∂p

∂x= 0, (7.1)

dSyy

dy− ∂p

∂y= 0, (7.2)

where

Sxy =du

dy+ 2β3

(

du

dy

)3

, and Syy = (2α1 + α2)

(

du

dy

)2

. (7.3)

On introducing generalized pressure P

P (x, y) = p (x, y) − (2α1 + α2)

(

du

dy

)2

. (7.4)

Using Eqs. (7.3) and (7.4), we find that Eqs. (7.1) and (7.2) takes the form

µd2u

dy2+ 2β3

d

dy

(

du

dy

)3

=∂P

∂x, (7.5)

∂P

∂y= 0. (7.6)

A consequence of the Eq. (7.6) is that P can be a function of x alone. ThereforeEq. (7.5) can be written as

µd2u

dy2+ 2β3

d

dy

(

du

dy

)3

=dP

dx, (7.7)

86

It is assumed that roll and plane moving with the same linear speed U = U , thenappropriate boundary conditions are

{

u = U on y = h(x),u = U on y = 0,

(7.8)

here, h(x) is explicitly given in Eq. (1.35).

7.2 Dimensionless equations

Base on the LAT analysis carried out previously, following dimensionless variablesare defined as:

x =x√RH0

, u =u

U, y =

y

H0, P =

√

H0

2R

PH0

µU, β =

2β3U2

µH0, h(x) =

h(x)

H0, Q =

Q

UWH0.

(7.9)Then Eqs. (7.7) and (7.8 becomes

∂2u

∂y2+ β

∂

∂y

(

∂u

∂y

)3

=dP

dx, (7.10)

u = 1 on y = h(x), (7.11)

u = 1 on y = 0, (7.12)

where h(x) = 1 + x2

2.

Assuming zero pressure at separation point also at entry, we can write

P (x = xs) = 0, (7.13)

P (x = −xb) = 0, (7.14)

where xsand xb are the separation and attachment points respectively, and dimen-sionless volumetric flow rate is defined in Eq. (6.13).

7.3 Solution of the problem

Applying the regular perturbation technique with β � 1 as the perturbation pa-rameter and writing the following expansions:

u(x, y) = u0(x, y) + βu1(x, y) + ... , (7.15)

P (x) = P0(x) + βP1(x) + ... , (7.16)

87

Q = Q0 + βQ1 + ... , (7.17)

xs = xso + βxs1+ ..., (7.18)

by introducing the relationships (7.15)–(7.18) into Eqs. (7.10)–(7.14) and (6.13),and comparing the coefficients of similar power of non-Newtonian parameter β, onecan get the set of problems as follows:

7.3.1 Zero-order problem and its solution:

dP0

dx=∂2u0

∂y2, (7.19)

Q0 = ∫h(x)0 u0 (x, y) dy. (7.20)

Along with the boundary conditions

u0 = 1 at y = 0, (7.21)

u0 = 1 at y = h(x), (7.22)

P0 = 0 at x = xs0, (7.23)

where u0, P0, Q0 and xs0are the zero-order solutions, which signify the Newtonian

results [2, 3].The solution for Eqs. (7.19) and (7.20) subject to boundary conditions (7.21)-(7.23)is given by:

u0(x, y) = 1 − 1

2

dP0

dx

[(

1 +x2

2

)

y − y2

]

, (7.24)

The pressure gradient distribution in Eq. (7.24) can be found by using principle ofconservation of mass, and has the following form

dP0

dx= 12

(

1 + x2

2

)

−Q0

(

1 + x2

2

)3 , (7.25)

upon integration we find that

P0(x) = 12

∫ x

xb

(

1 + x2

2

)

−Q0

(

1 + x2

2

)3 dx. (7.26)

In above equation Q0 is the zeroth-order volumetric flow rate.

88

Now using Eq. (7.25) in Eq. (7.24), we get

u0(x, y) = 1 − 6

(

1 + x2

2−Q0

(

1 + x2

2

)3

)

(

(1 +x2

2)y − y2

)

. (7.27)

If we assume xb → −∞ in Eq. (7.26), we get

P0(x) =

(

6 − 92Q0

)

x

1 + x2/2− 3Q0x

(1 + x2/2)2 +

(

12√2− 9Q0√

2

)

tan−1

(

x√2

)

+6π√

2

(

1 − 3Q0

4

)

,

(7.28)which is the zeroth order pressure. Since at the separation point

(

xs0, 1

2h(xs0

))

, thevelocity must vanish by symmetry requirements. So one can easily find that

Q0 =1

3

(

1 +1

2x2

s0

)

. (7.29)

If in Eq. (7.28), x is replaced by xs0from above relation and since P0 = 0 at x = xs,

then Eq. (7.28) becomes a transcendental equation in Q0 and whose numericalsolution is 1.33.

The above obtained solutions are same as mentioned by [5, 31].

7.3.2 First-order problem and its solution:

dP1

dx=∂2u1

∂y2+

∂

∂y

(

∂u0

∂y

)3

, (7.30)

Q1 = ∫ h(x)0 u1(x, y)dy. (7.31)

For solving Eqs. (7.30) and (7.31), we need the zeroth-order solution. The firstorder boundary conditions are

at y = 0, u1 = 0, (7.32)

at y = h(x), u1 = 0, (7.33)

at x = xs1, P1 = 0, (7.34)

here u1, P1, Q1 and xs1are the modifications till first-order terms, which contains

the impact of the non-Newtonian effect.Using Eq. (7.27) in Eq. (7.30) then integrating twice and using the boundary

conditions (7.32)-(7.33), we get the following first order solution

u1(x, y) = −1

2

dP1

dx

(

(

1 +x2

2

)

y −(

1 +x2

2

)2)

+27

(

1 + x2

2− λ0

)3

(

1 + x2

2

)5

1 −(

1 + x2

2− 2y

1 + x2

2

)4

.

(7.35)

89

In order to determine Q1, we substitute Eq. (7.35) into Eq. (7.31), in light of Eq.(7.34), we get

Q1 =108

(

1 + x2

2−Q0

)3

5(

1 + x2

2

)4 −

(

1 + x2

2

)3

12

dP1

dx. (7.36)

The above equation can be written as

dP1

dx=

1296(1 + x2

2−Q0)

3 − 60Q1

(

1 + x2

2

)4

5(

1 + x2

2

)7 . (7.37)

Upon integration Eq. (7.37), we find

P1(x) =

∫ x

−xb

1296(1 + x2

2−Q0)

3 − 60Q1

(

1 + x2

2

)4

5(

1 + x2

2

)7 dx. (7.38)

The 1st order velocity profile is obtained by substituting Eq. (7.37) into Eq. (7.35)

u1(x, y) = 2(

y2 − hy)

[

324(h −Q0)3 − 15Q1h

4

5h7

]

+ 27(h−Q0)

3

h5

[

1 −(

h− 2y

h

)4]

.

(7.39)The study is yet not complete at this stage, since we need a relation between Q, andxs. We can see from Fig. 6.2 that the film splits evenly, and therefore separationpoint is

(

xs1,12h(xs1)

)

, at this point the velocity and pressure goes to zero. Thissupposition can be acknowledged because the sheet and the roll has the similarvelocities. This will provides xs in terms of Q1 and has the following form:

Q1 =18

5

(

1 + x2s

2−Q0

)3

(

1 + x2s

2

)4 . (7.40)

Again, if we assume xb → −∞ in Eq. 7.38, we get

P =−√

2γ1

640

(

37422 tan−1

(

x√

2√2

)

+ 18711π

)

− 18711

160 (2 + x2)6

(

γ1x11 + γ2x

9 + γ3x7

+γ4x5 + γ5x

3 + γ6x

)

,

(7.41)where

γ1 =1

11

(

11Q30 − 36Q2

0 + 40Q0 +160

189Q1 −

320

21

)

,

90

Figure 7.2: Effect of β on velocity atx = 0

Figure 7.3: Effect of β on velocity atx = 0.25

γ2 =34

3Q3

0 +8

11

(

51Q20 −

170

3Q0 −

680

567Q1 +

1360

63

)

,

γ3 =(

66Q20 − 216Q0 + 115

) 4Q0

5+

256

7

(

95

891Q1 − 2

)

,

γ4 =

(

4496Q0 −161856

11

)

Q20

35+ (281Q0 − 104)

128

77+

2560

297Q1,

γ5 =10672

63Q3

0 −64

231

(

2001Q20 +

2138

1Q0 +

920

27Q1 −

2224

3

)

,

γ6 =32

77

(

793

3Q3

0 − 772Q20 + 744Q0 +

800

81Q1

)

− 2048

21.

Again following the same procedure as in zeroth-order solution, one can find thenumerical solution for first order volumetric flow rate as

Q1 = −0.1258. (7.42)

Using value of Q0 and Q1 in Eq. (7.40), one can easily found that xs1 = −0.0941.Combining the solutions at each order of approximation yield the solutions up tofirst order for velocity, pressure gradient and pressure.Once the separation point is establish, then all engineering quantities are immedi-ately available. These engineering quantities are calculated in a similar way as werecalculated in chapter number 6.

91

Figure 7.4: Effect of β on velocity atx = 0.5

Figure 7.5: Effect of β on velocity atx = 0.75

7.4 Results and discussions

In this chapter, calendering process for incompressible third-grade material onto amoving thin sheet has been analyzed. The lubrication theory is used to simplifythe equations of motion. The numeric results for Q and xs, the exit sheet thicknessH/H0, the power input, and roll separation force are presented in Table 7.1 fordifferent values of β. It is perceived that by increasing β viscosity of the fluidincreases and velocity decreases which corresponds to shear thickening effect. Thisfact is obvious from Figs. (7.2)-(7.7). It is also noted that with an increase in βcoating thickness decreases, on the other hand magnitude of roll separation forceand power input increase. This was physically expected because of shear thickeningeffect.

In Figs. (2-6) the dimensionless velocity u(x, y) is presented for different valuesof β at six different positions of x. We can see that velocity decreases with anincrease in the β for a particular value of Q, and this increase is more prominent atthe nip region.

The effects of pressure gradient and pressure distribution are sketched in Figs.(7.7)-(7.8). It has been observed that by decreasing the numeric value of β, pressureincrease and higher and pressure gradient graphs will be steeper. Similarly, bydecreasing β, the final coating thickness as well as flow domain will increase. Sincethe curves for the pressure gradient are getting sheer adjacent to small gap H0, sothe fluid behaves more solid-like. The distortions in the flow region become minor,and needs more pressures and domain to flow and deform.

92

Table 7.1: Effect of non-Newtonian parameter on various rheological parameters

β Q H/H0 xs F Pw

0.01 1.3320 0.6660 2.4477 0.8721 -0.68600.03 1.3295 0.6647 2.4452 0.7728 -0.42450.05 1.3270 0.6635 2.4427 0.6737 0.10110.07 1.3245 0.6622 2.4401 0.5749 0.71480.09 1.3220 0.6610 2.4376 0.4763 1.31600.1 1.3207 0.6603 2.4364 0.4270 1.60110.3 1.2955 0.6477 2.4112 -

0.544645.670

0.5 1.2704 0.6352 2.3861 -1.4926

577.84

0.7 1.2452 0.6226 2.3609 -2.4165

2781.9

0.9 1.2200 0.6100 2.3357 -3.3163

8594.3

Figure 7.6: Effect of β on velocity atx = 1.0

Figure 7.7: Effect of β on velocity atx = 2.0

93

Figure 7.8: Effect of β on pressure gra-dient distribution

Figure 7.9: Effect of β on pressure dis-tribution

7.5 Conclusions

In the present study, the main finding can be summarized as follows:

1. The results of [1, 2] were verified and extended, and New procedure presentedand give novel results for third-grade material.

2. Shear thickening and thinning behaviors are observed.

3. Reducing the third grade parameter β leads to

(a) increase coating thickness,

(b) increase in separation point and volumetric flow rate,

(c) increase in flow domain,

(d) increase in velocity.

94

Chapter 8

Conclusions and Future Work

8.1 Summary of Work Undertaken

This dissertation confirms the utility of LAT for viscoelastic and viscoplastic mate-rials when they dragged through the small gap between two counter rotating rolls.Numerical results for separation point and sheet thickness has been derived usingLAT. Quantities of engineering interest like maximum pressure, power input, roll-separating force, and internal temperature are calculated on the basis of LAT. Thecurrent thesis is a first step in the direction of a better understanding of calenderingand rolling of viscoelastic and viscoplastic materials.

The problems considered are presented in chapters 2 to 7. The problem of influ-ence of porous rolls on Newtonian calendering can be looked in chapter 2, whereaschapter 3 presents the effects of MHD on Newtonian calendering. In Chapter 4, theexact solutions for MHD non-isothermal viscoplastic material during the calender-ing process are established. The calendering analysis of a third-grade material ismodeled and solved in chapter 5. The application of a second-grade and third-gradematerial onto a moving thin sheet for the final appearance has been established inChapters 6 and 7.

The major observation regarding the effects of non-dimensional parameters onthe velocity, leave-off distance, sheet thickness, extreme pressure, roll- separatingforce, power input and temperature distributions in the present analysis can besummarized as follows:

1. In case of porous rolls effects on Newtonian calendering, Reynolds numberprovides a mechanism to control the separation region, power input and rollseparation force, and one can indeed control the sheet thickness.

2. In case of porous rolls effects on Newtonian calendering Maximum velocityand pressure gradient occurs at the nip region and velocity decrease as sheetmoves away from the narrow region.

95

3. It has been noted from the influence of MHD on Newtonian calendering thatthe magnetic field provides the controlling parameter to increase or decreasepower input, separation force and distance between attachment and detach-ment points and its economies the power and separation force.

4. It is noted that the transverse magnetic field decelerates the material motion.

5. While considering viscoplastic materials, it is observed that shear-thinningshrinks the large vortices existing within the fluid bank and removes them forenormously shear-thinning materials.

6. The velocity and temperature decreases as viscoplastic parameter and Hart-man number increases.

7. With an increase in viscoplastic parameter sheet thickness increases.

8. Maximum of the flow region due to shear motion of the roll is yielded. Un-yielded area can be seen only in the large region of the material bank.

9. A small and limited unyielded region near the centerline has been seen as vis-coplastic parameter increases. Quantities of engineering interest like extremepressure, power input and roll-separating force are increased by increasingviscoplasticity.

10. In case of third-grade material, it was found that shear thinning increases thefinial sheet thickness, while the shear thickening reduces it.

11. By increasing Brinkman number temperature decreases in viscoplastic mate-rials.

12. It has been found that the velocity decreases with an increase in viscoplasticparameter and this decrement is more pronounced at the center line, similarlyby increasing the value of viscoplastic parameter or decreasing the yield stress,velocity decreases.

13. It has been perceived that the viscoplastic parameter has a greater effect atx = 0, since at this instant shear stress has its highest negative value. Onthe other hand it has smaller effects at x = ±λ, for this reason flat velocityprofiles at this point are obtained, with respect to an unyielding gesture of thesheet.

14. In case of calendering analysis of a third-grade material, it is observed from aphysical point of view that by increasing third-grade parameter shear thick-ening decreases, this will result in to rise the viscosity of the material. Moreviscous material diffuses extra momentum. Subsequently, velocity of the ma-terial decreases.

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15. It can be seem regarding calendering of a third-grade material that velocitydecreases with the third-grade parameter to a certain value of the detachmentpoint, and this increase is more prominent in the nip region. In contrast, atthe locality of the rolls, velocity reduces as compare to Newtonian material.Also the velocity increases softly near the rolls, and decreases at the centralposition when third-grade parameter is increased.

16. In calendering of a third-grade material it has been perceived that by increasingthird-grade parameter causes to increase pressure gradient. This parameteralso has a major effect on pressure gradient at x = 0, because the pressure hasa maximum absolute value at this point. Whereas, pressure gradient has noeffect at x = ±λ, since we have flat velocity profiles at this point.

17. By changing the third-grade parameter in calendering analysis, the pressure issignificantly affected at x = −λ. Likewise, an increase in the value of materialparameter results in to extend the length of attachment point.

18. In calendering of a second-grade material onto a moving porous sheet it isnoted that the coating thickness decreases with an increase in the second-gradeparameter, the minimum coating thickness has been observed for β = 50.

19. With an increase in Re (inertia force is dominating viscous force) the coatingthickness, volume flow rate as well as separation point increases. In caseof calendering of a second-grade material onto a moving porous sheet, themaximum coating thickness has been observed at Re = 5. It is also noted thatwhen the Reynolds number exceeds the value 2, the separation point does notreach. Physically, it seems that the effect of inertial force is so large that theseparation point does not reach.

20. It is detected that by increasing second-grade parameter velocity of the ma-terial decreases at different positions of the roll coating process. It is alsonoted that with an increase in Reynolds number velocity increases due to thedominance of inertia force over viscous force.

21. It has been observed that the pressure gradient increases by increasing Reynoldsnumber and second-grade parameter, on the other hand pressure distributiondecreases.

22. An increase in the second-grade parameter causes to increase in separationforce as well as power input, whereas by increasing Reynolds numbers separa-tion force decreases and power input increases.

23. In case of calendering of a third-grade material onto a moving porous sheet, itis perceived that by increasing third-grade parameter viscosity of the material

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increases and velocity decreases which corresponds to shear thickening effect.It is also noted that with an increase in third-grade parameter coating thicknessdecreases, on the other hand magnitude of roll separation force and powerinput increases. This was physically expected because of shear thickeningeffect.

24. It has been observed regarding calendering of a third-grade material onto amoving thin sheet that lesser the third-grade parameter, higher the maximumpressure and steeper the pressure gradient graphs. Similarly, smaller the third-grade parameter, bigger will be exit coating thickness and the flow domain.

The present results offer a good reference for researchers/engineers working on cal-endering process and can be used for production purposes.

8.2 Future Work

The main development of this thesis must be an experimental assessment of theresults. The procedure for such a project was considered beyond the scope of thework. The solutions shown in this dissertation must be advantageous not onlyin foreseeing thicknesses of calendered sheet and roll power required, but also incomputing the interactions between rolls deflections, roll speed, and instabilities inreservoir feedstuff for keeping unvarying finial height. The two normal extensions ofthe present work are the prediction of the roll pressure distribution and consideringthe two-dimensional model. For this one have to divide the non-uniform reservoirinto segments of unbroken width. In future, effect of slip condition on all abovemodels in this thesis will be considered and more complicated rheological modelssuch as Fourth-grade, Oldroyd, Burgers and Carreau materials etc., will be solvednumerically.

Finally, new challenges in the area of calendering elastomer and polymer blendsand composites must be addressed. Virtually no calendering studies have been donefor the latter, due to their complicated nature, may present extra difficulties duringthe process. It is expected that this work will be a significant step in the directionof calendering.

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