**Modelling, Simulation and Control of the Dyeing Process**

**Contents **

Woodhead Publishing Series in Textiles ix

Preface xvii

Definition of terms xxi

Nomenclature xxv

1 Introduction to dyeing and dyehouse automation 1

1.1 Introduction 1

1.2 Factors affecting dyeing quality 6

1.3 Practical difficulties involved in dyeing process control 8

1.4 Package dyeing machinery 10

1.5 Difficulties in package dyeing 15

1.6 Automation in the dyehouse 21

1.7 References 27

2 Principles underlying the dyeing process 31

2.1 Introduction 31

2.2 Dye transport from the bulk solution to the fi bre surface 31

2.3 Adsorption of dyes by textile fi bres 35

2.4 Diffusion of the dye into the interior of the fi bre 44

2.5 Rate of dyeing 48

2.6 References 50

3 Dye transport in fluid systems 54

3.1 Introduction 54

3.2 Fluid properties in perspective 54

3.3 Flows in porous media 58

3.4 Convective mass transfer in fluid system 64

3.5 Dye transfer in dyeing 70

3.6 References 79

4 Developing theoretical models of dyeing 82

4.1 Introduction 82

4.2 System description and basic assumptions 85

4.3 Development of mathematical model 90

4.4 References 99

5 Solving dynamic equations in dye transport 100

5.1 Introduction 100

5.2 Numerical methods 100

5.3 Summary of the model equations 105

5.4 Numerical solutions of equations 106

5.5 References 112

6 Simulation of the dyeing process 114

6.1 Introduction 114

6.2 Fluid fl ow behaviour in dyeing 114

6.3 Dynamic behaviour of mass transfer in package dyeing 127

6.4 Conclusions from simulation results 150

6.5 References 153

7 Principles of control in dyeing processes 154

7.1 Introduction 154

7.2 Introduction to process control 155

7.3 Control algorithms 157

7.4 Analog and digital processing 164

7.5 Practical application in a pilot- scale operation 165

7.6 pH measurement and control in a dyeing system 181

7.7 The feed- forward control strategy 194

7.8 Acknowledgement 204

7.9 References 205

8 Measurement and control of dyeing 206

8.1 Introduction 206

8.2 Real- time measurement of dye concentration 208

8.3 Data processing and tools to analyse and interpret real- time dye monitoring data 218

8.4 Conclusion 220

8.5 References 220

Index 223

**Preface **

Colour is one of the most important criteria in the production and retail of textiles. It significantly affects sales volumes, and is also directly related to consumer satisfaction. Fashion and market demands require that textile dyers be able to respond to shade changes in a relatively short time period. The need to minimise the time of dyeing while ensuring the standards are maintained necessitates the use of sophisticated control strategies. This requires a careful monitoring and control of parameters that affect the interaction of dye with fi bres in the dyebath and is therefore one of the most important aspects of production in the textile supply chain. This book aims to introduce the development and application of reliable monitoring and control systems for the dyeing of textile fi bres, with a view to obtaining dyeing processes that are both economical and environmentally sound. Various influential parameters, including temperature, fl uid fl ow and direction, pH and dye concentration in the dyebath, in the pre-set and real-time exhaustion control of conventional (all in) as well as step-wise addition of dyebath components such as dyes and auxiliaries, are discussed, and their role in controlling dye exhaustion and levelness of dyes is elucidated. The modelling aspect of the book mainly concentrates on batch dyeings, specifically package or beam dyeing processes, since they allow chemical engineering principles, applied to packedbed reactors, to be emulated in textile dyeing. Such models can, relatively easily, be modified for application to other types of machinery or processes.

The application of pre-set exhaustion profiles – linear, quadratic or exponential in shape – in the dyeing of textile fibres, and the advantages and disadvantages associated with the use of these methods, are also examined. Calculation of the rate of dye exhaustion in these processes based on various mathematical models is explained. Feed-forward control algorithms are introduced and their application to determining the desired rates of dye exhaustion in the dyebath based on dyebath conditions is described. Modelling of dyebath control in the portion-wise addition of dye or chemicals to the dyebath, integration dyeing methods, is also investigated. A major impediment to the commercialisation of this approach has been the capital cost and the potential need to redesign dyeing machinery. However, with the significant reductions in the cost of components, including spectrophotometers and increased computational power available at low cost, this approach is gaining momentum. Indeed, a number of companies in the United States and the Far East (China, Taiwan, Korea, etc.) are already utilising state-of-the-art technology to improve their right-every-time production dyeing rates.

With this in mind, this book is divided into eight chapters. Chapter 1 introduces the concept of textile dyeing, based on a brief review of various types of dyes and fibres and the interaction that takes place between dyes and fibrous assemblies. The mode of controlling dyeing processes based on conventional time-temperature profiles, as well as those based on controlling the rate of dye exhaustion in real time, is introduced. Package dyeing processes and parameters influencing the adsorption and diffusion of dyes within packages are then described. Various adsorption isotherms and factors that affect the dyeing of fibres are examined. Different control strategies based on assessing the amount of dye in the dyebath are then briefly assessed.

Chapter 2 presents the basic principles underlying the dyeing process. Some of the more quantitative aspects of dyeing equilibrium and kinetics, including standard affinity of dyes and dyeing rates, are then examined. The change in chemical potential of dye at the standard state from the solution to the fi bre due to the presence of electrolytes is explained. Transport of dye in the dyebath involving sorption is described and various adsorption isotherms are examined. A general dynamic expression of dye sorption by textile fibres is introduced. Diffusion phenomena and models describing the diffusion of dye into textile fibres are then evaluated.

In Chapter 3 the theoretical background for the mechanistic description of fl ow phenomena in open channel and porous media is elucidated. Relevant works are described and the equations governing fl ow are explained. Fundamental concepts of dispersion, convection and diffusion are clarified and models that describe these processes are evaluated. The role of bulk and dispersive flow in dye transfer within a packed-bed medium and the effect of including flow parameters on modelling dye dispersion and diffusion are then evaluated, and various models incorporating fl ow properties are examined.

Chapter 4 develops theoretical models to simulate the dyeing process, starting from brief critical conclusions based on convective dispersion and fluid mechanics models as well as those that describe both the dye transfer and fl ow phenomena during dyeing. Using the theoretical concepts discussed in the previous chapters, the system is described by a set of partial differential equations. Darcy’s law and the Navier–Stokes and Brinkman equations are examined. The boundary and initial conditions are also defined and discussed. The solutions of these models, which describe the dynamic behaviour of the system under given conditions of fl ow rate and given dyeing parameters, are also compared.

Chapter 5 provides an examination of the numerical solutions of the dyeing models that can be applied to different conditions. The application of analytical software applications to solve systems of highly non-linear simultaneous coupled partial differential equations is described. The fi nite difference and finite element methods are introduced. The partition of the fibrous assembly geometry into small units of a simple shape, or mesh, is examined. Different polygonal shapes used to define the element are briefly described. The defined geometries, boundary conditions and mesh of the system are used to determine solutions to the equations of flow or mass transfer models.

In Chapter 6 the results of the simulation of the package dyeing process based on various models are presented. In this section it is explained that, while Darcy’s law assumes that the only driving force for fl ow in a porous medium is the pressure gradient, and the global transport of momentum by shear stress in the fluid is ignored, Brinkman’s equations can be used to extend Darcy’s law to include the viscous transport in the momentum balance and the velocities in the spatial directions as dependent variables. These models, as well as the effect of fl ow velocity and direction, package permeability and dye dosing profiles on dye distribution, are described and their validity examined.

The first part of Chapter 7 introduces the main types of control strategies. The mathematical methods of controlling different dyeing parameters, specifically pH measurement, the effect of various parameters on pH and the development of pH control strategies, are then examined. The performance of different control strategies on the outcome of dyeing, using a pilot package dyeing machine, is also briefly investigated. The control of the dyeing process according to pre-set exhaustion and other profiles is also described. The control strategies shown in this chapter take advantage of basic digital and analog controllers, and readers are advised to consider other sophisticated controllers currently available, as well as more advanced software applications, to facilitate this process.

It is with pleasure that we include Chapter 8, written by Professor Warren Jasper of North Carolina State University and Dr Melih Günay, who cover the implementation of real-time dyebath monitoring in various types of dyeing machinery. They discuss the principles of a dyebath monitoring technology that allows analysis and control of the dyeing process, and use Beer’s law to determine the amount of dye in the dyebath, even in a mixture. The principles of spectral additivity and spectral morphing are then examined, and their application to the calculation of the rate and degree of dye exhaustion in the dyebath is described. They conclude that such measurement technology can aid in troubleshooting root causes in shade reproducibility that can occur from variability in dye strength, the fabric or the dyeing process.

The production of work and the ensuing documentation leading to the writing of this book has taken over two decades. This book would not have been possible without the initial research conducted under the supervision of Dr J.H. Nobbs at the Colour Chemistry Department of Leeds University, UK, in the early 1990s. Dr Nobbs developed a Quick Basic 4.5-based software application to control a pilotscale dyeing machine in Leeds in the late 1980s. The work was extended over a period of 10 years by a number of researchers, including one of the authors of this book. The software incorporated some of the models discussed in certain sections of this book. We have received help from a large number of graduate students, colleagues and friends, who are duly acknowledged. We are also grateful to Dr Juan Lin for assistance in generating some of the figures and Mr M. Zubair for assistance with collection of references.

We believe this book should be a good source of information, not only for academics and researchers but also for coloration professionals, experts and technologists, engineers, textile dyeing machinery designers and allied industries.

Renzo Shamey

North Carolina State University

Xiaoming Zhao

Tianjin Polytechnic University