Modelling of the dyeing process of packed cotton threads using reactive dyes

Transp Porous Med (2007) 68:341–363DOI 10.1007/s11242-006-9046-7

O R I G I NA L PA P E R

Modelling of the dyeing process of packed cottonthreads using reactive dyes

Selene M. A. Guelli Ulson de Souza ·Dilênio Pacheco de Souza · Eduardo AlbertoBorges da Silva · Antônio Augusto Ulson de Souza

Received: 4 October 2005 / Accepted: 18 August 2006 / Published online: 10 October 2006© Springer Science+Business Media B.V. 2006

Abstract The Method of Volume Averaging is used to model the process of dyeingtextile threads on bobbins. This analysis allows one to upscale the relevant informa-tion at the micro-scale, composed of the textile fibres of the thread in contact withthe dyeing bath fluid, to the macro-scale, consisting of the bobbins of threads insidethe equipment. The final mathematical model consists of two equations, one for thefluid phase external to the thread and the other for the fluid phase internal to thethread. In order to solve the partial differential equations obtained in the mathemat-ical model, the authors developed a computation code using the Method of FiniteVolumes. This code utilized a system of generalized coordinates to facilitate applica-tion of the boundary conditions to different bobbin geometries. The numerical resultsfor the kinetics of dyeing packed cotton threads with reactive dyes are comparedto the experimental results obtained in Brazilian textile industries, leading to goodagreement between theory and experiment. This demonstrates that the model devel-oped in this paper is able to predict the operational conditions to be used in the textileindustries, minimizing the consumption of dyes and other auxiliaries necessary for thedyeing process.

Keywords Mathematical modelling · Mass transfer · Method of volume averaging ·Textile · Multiphase · Dyeing · Diffusion · Kinetics

S. M. A. Guelli Ulson de Souza (B) · D. P. de Souza · E. A. Borges da Silva · A. A. U. de SouzaChemical Engineering Department, Federal University of Santa Catarina, P.O. Box 476,Florianópolis, SC CEP 88040-900, Brazile-mail: [email protected]

D. P. de Souzae-mail: [email protected]

E. A. Borges da Silvae-mail: [email protected]

A. A. Ulson de Souzae-mail: [email protected]

342 S. M. A. Guelli Ulson de Souza et al.

Nomenclature

av|γ κ Surface area per unit volume on the micro-scale, [1/m]av|βσ Surface area per unit volume on the intermediary scale, [1/m]bσ Vector closure variables of the σ -region, [m]bβ Vector closure variables of the β-phase, [m]Ci Initial concentration of the η-phase, [kg/m3]CAη Concentration of the η-phase, [kg/m3]CAσ Concentration of the σ -region, [kg/m3]CAβ Concentration of the β-phase, [kg/m3]〈CA〉 Superficial average concentration, [kg/m3]CAβ Spatial derivation concentration of the β-phase, [kg/m3]CAσ Spatial derivation concentration of the σ -region, [kg/m3]〈CA〉γ Intrinsic average concentration of the γ -phase; [kg/m3]⟨CAβ

⟩β Intrinsic average concentration of the β-phase; [kg/m3]〈CAσ 〉σ Intrinsic average concentration of the σ -region; [kg/m3]d′

1 Upper internal diameter of the bobbin, [m]d′

2 Upper external diameter of the bobbin, [m]d′

3 Lower internal diameter of the bobbin, [m]d′

4 Lower external diameter of the bobbin, [m]dβ Velocity-like coefficient of the β-phase, [m/s]dσ Velocity-like coefficient of the σ -region, [m/s]Deff

∣∣γ κ

Effective diffusivity tensor of the γ -phase, [m2/s]Dβ Molecular diffusivity, [m2/s]Dσ Diffusivity of the σ -region, [m2/s]D∗

β Total dispersion tensor, [m2/s]

Deff∣∣∣ββσ Effective diffusivity tensor of the β-phase, [m2/s]

D Hydrodynamic dispersion, [m2/s]h Mass transfer convective coefficient, [m/s]H Height of the bobbin, [m]kr Rate constant for reaction with the fibre, [s−1]kh Pseudo-first-order rate constant of hydrolysis, [s−1]Keq Equilibrium constant, [m]�i Lattice vectors, [m]�β Characteristic length for the β-phase, [m]�γ Characteristic length for the γ -phase, [m]�κ Characteristic length for the κ-phase, [m]�σ Characteristic length for the σ -region, [m]Q Discharge rate of the dyeing bath, [m3/s]N Number of cotton threadsnβσ Unit normal vectorMbobbin Average mass of the bobbins, [kg]RB Liquor ratio, [kg/m3]R′

th Average radius of each cotton thread, [m]rσ Radius of the micro-scale, [m]rω Radius of the intermediary scale, [m]r′′′

h Rate of hydrolysis, [kg/m3.s]sβ Scalar closure variables of the β-phase

Modelling of the Dyeing process 343

sσ Scalar closure variables of the σ -regiont Time, [s]uβ Convective transport vectors of the β-phaseuσ Convective transport vectors of the σ -regionvβ Velocity of the β-phase, [m/s]vβ Spatial derivation velocity of the β-phase, [m/s]V Volume of the dyeing equipment, [m3]Vσ Volume of the micro-scale, [m3]Vω Volume of the intermediary scale, [m3]Vβ Volume of the β-phase, [m3]Vesp

fibre Specific volume of the cotton fibre, [m3/kg]

Greek Symbolsεγ Porosity in the γ -phaseεβ Porosity in the β-phaseεσ Porosity in the σ -region Variable that defines the type of dye (reactive) Variable that defines the type of dye (adsorptive)

1 Introduction

The development of new dyes and textile fibres, together with the necessity for greatercompetitiveness in the textile industry, has driven the development of new dyeing tech-nologies. The main objectives of these technological advancements are to increase theproductivity of the process and to reduce environmental pollution.

These technological advancements are the result of a large number of studies car-ried out in the area of textile dyeing. These studies are largely empirical in nature, andthey investigate the various problems encountered during the dyeing process with aview to improving the process efficiency and the quality of the final textile product.However, the number of studies in the textile area that apply numerical techniques isvery small.

Mass transfer with adsorption in porous media is a subject of great interest toengineers, having applications in many areas, such as, the development of chemi-cal reactors, drying, and dyeing, among others. However, problems in porous mediainvolving the packing of textile threads are poorly addressed in the literature. Con-sidering the economic importance of the textile sector, the dyeing of packed textilethreads is investigated in this study in which a mathematical model of the problem isdeveloped on three length scales.

The model is based on the Method of Volume Averaging (Whitaker 1985, GuelliU. de Souza and Whitaker 2003). The modelling is performed on the micro-scale,composed of the fibres of the textile substrate and the dyeing bath fluid contained inthe pores of this scale, on the intermediate scale, at which the threads on the bobbinare found, together with and surrounded by the dyeing bath fluid, and on the macro-scale formed by the bobbins of threads immersed in the dyeing bath fluid inside theequipment. Figure 1 presents the scales of the process under study. It is reasonable tostate that the thread is composed of small fibres, which are assumed to be rigid andimpermeable.


The Method of Volume Averaging is used to derive the governing differentialequations for the multiphase transport. By means of the application of this method, aset of partial differential equations is obtained and that enables the calculation of theeffective properties, such as the total dispersion tensor for mass transfer problems. Thecomplete development of the mathematical model on the micro-scale, as well as thetheoretical calculation of the effective diffusivity tensor of the thread, are presentedin Ulson de Souza and Whitaker (2003), where it can be seen that the solution to theclosure problem on the micro-scale enables the calculation of the effective diffusivity,and produces a high level of agreement with experimental data.

The intermediate scale mathematical model describes the process of multiphasemass transfer that occurs inside the bobbin of thread, taking into consideration thenature of the phenomena of mass transfer that occur on the micro-scale. The clo-sure problem derived from the intermediate scale allows the calculation of the totaldispersion tensors.

The final equations obtained at the intermediate scale are upscaled to the macro-scale, at which point the bobbin of thread is in contact with the bath fluid of the textiledyeing equipment. Thus, the final equation for the macro-scale incorporates impor-tant information from the intermediate- and the micro-scales, which enables the moreprecise prediction of the textile dyeing kinetics. Much of the upscaling presented inthis paper has been discussed in great detail in earlier studies by Plumb and Whittaker(1990a, b).

2 Transport equations for the intermediate scale

The micro-scale illustrated in Fig. 1 is composed of the fluid phase (γ -phase) insidethe threads, and the fibres (k-phase). The dyeing of fibres occurs at the surface offibres and the reaction and adsorption terms appear only as boundary conditions inγ -phase, as given by Eq. 2.

Equations 1–4 describe the modelling on the micro-scale, including the boundaryconditions:

∂CA

∂t= ∇ · (Dγ ∇CA) + r′′′

h , in the γ -phase (1)

B.C.1 −nγ κ · Dγ ∇CA = kCA + ∂ CAs

∂t, in Aγ k (2)

B.C.2 CA = F(r, t), in Aγ e (3)

C.I CA = G(r), at t = 0 (4)

Equation 1 represents the mass conservation equation of chemical species,applied to processes involving a homogeneous reaction (hydrolysis of reactive dyes),without convective effects (the fluid flow inside of thread is considered negligible).

The variable CA represents the dye concentration in the γ -phase, CAs is thedye concentration at the fibre surface, Dγ is the molecular diffusivity of dye, Aγ kis the interfacial area between the phases γ -k and Aγ e represents the area of theγ -phase at the entrances and exits of control volume illustrated in Fig. 1. The Variablenγ k represents the unit normal vector pointing from the γ -phase towards thek-phase.


lγ lkrs

γ -phase

ω

Bobbin of Thread

Macro-scale

β σ -region

-phase

lβ

rwVw lσ

Intermediate scale

Micro-scale

-region

h -phase

Vσ

k -phase

Fig. 1 Representative drawing of the bobbin of thread on the different scales

In the modelling, an equilibrium model for the linear adsorption isotherm was con-sidered allowing the surface concentration to be related to the bulk concentration. Forother equilibrium models, it is necessary to change the term containing the variable to represent the desired equilibrium relation. In this study, dyes were used whichreact with the fibres, without adsorption, so the reactive term is modelled imposing equals 1 and equals 0. To the cases where only an adsorption process occurs it isnecessary to impose equals 0 and equals 1.

The final equation written on the micro-scale through the application of the Methodof Volume Averaging to Eqs. 1–4 is presented in Eq. 5, where it can be seen that theconvective term is absent. This is because of the low velocity of the dyeing bath fluidinside the thread, which makes the convective term negligible when compared to thediffusive term.

The complete development of mathematical modelling on the micro-scale to obtainEq. 5 is presented in Ulson de Souza and Whitaker (2003) and Guelli U. de Souza andWhitaker (2003), who considered the adsorption and reactive terms, respectively.

In Eq. 5 it can also be seen that two source terms are present, one reactive andthe other adsorptive. The employment of these terms is dependent on the natureof the dye used in the dyeing process. For dyes that adhere to the fibre surface byadsorption, such as basic, acidic and adsorptive dyes, the adsorptive term is used byimposing = 0 and = 1. For the cases in which the dyes react with the fibres, suchas the reactive dyes, the reactive term is used, thus equals 1 and equals 0.


εγ

∂ 〈CA〉γ∂t

= ∇ ·(εγ Deff

∣∣γ κ

· ∇ 〈CA〉γ)

− (

av|γ κ kr + εγ kh) 〈CA〉γ

− av|γ κ

∂

∂t

(Keq 〈CA〉γ )

(5)

The variable 〈CA〉γ represents the intrinsic average concentration of dye in the γ -phase, Deff

∣∣γ κ

the effective diffusivity tensor, t time, εγ the porosity of the γ -phase,av|γ κ the interfacial area by unit volume for the micro-scale, kr the constant for thereaction of the dye with the surface of the fibre, kh the constant of the hydrolysisreaction and Keq the equilibrium constant of the linear isotherm (Revello et al. 2001).

The objective of this section is to develop a model that would be valid for the inter-mediate scale (the region formed by the threads that comprise the bobbin), and to takeinto consideration the relevant information from the process at the micro-scale, whichis given in Eq. 5. This procedure makes the numerical formulation more consistentwith the physical problem, reducing the requirement for empirical information.

The mathematical model of the intermediate scale is formed by two equations: oneequation for the fluid phase, comprising the dyeing bath fluid external to the threadand inside the bobbin, known as the β-phase; and another for the σ -region, whichconsists of the fluid phase in the interior of the thread and the fibres that comprise it,as shown in Fig. 1 (Quintard and Whitaker 1993).

The equation valid for the σ -region is that developed for the micro-scale, where theaverage concentration on the micro-scale, 〈CA〉γ , when transformed into an equationfor the intermediate scale, becomes punctual, CAσ , due to the difference in scale, asshown in Eq. 6.

〈CA〉γ = CAσ (6)

The equation for the β-phase is written from the equation of conservation of thechemical species, where the chemical species of interest is the dye. On the interme-diate scale, convection cannot be neglected, since on this scale, and in contrast tothe previous example, the fluid velocity is high, making the convective mechanismimportant.

The aim of this study is to develop a mathematical model that represents thethread dyeing process using various types of dyes. For the specific case of a reactivedye, hydrolysis is a very important consideration. Therefore, a hydrolysis reaction rate,r′′′

h , is included in the β-phase to take into consideration the hydrolysis of the dye.The governing equation for the β-phase and its boundary conditions are presented

below:

∂CAβ

∂t+ ∇ · (

vβCAβ

) = ∇ · (Dβ∇CAβ

) + r′′′h in the β-phase (7)

B.C.1 nσβ · Dσ ∇CAσ = h(CAβ − CAσ

)at the interface β–σ (8)

B.C.2 − nβσ · Dβ∇CAβ = nσβ · Dσ ∇CAσ at the interface β–σ (9)

where CAβ is the point concentration of the β-phase, CAσ the point concentrationof the σ -region, vβ the velocity of the dyeing bath fluid inside the bobbin, nβσ thenormal vector at the interface β–σ , Dβ the molecular diffusivity of the dye, Dσ theeffective diffusivity of the dye inside the thread and h the convective coefficient ofmass transfer.


The boundary condition given by Eq. 8 states that the flux at the interface β–σ isa function of the difference between the concentration in the β-phase and that in theσ -region. Meanwhile, the boundary condition for Eq. 9 indicates the equivalent massflux at the interface β–σ , that is, the flux of dye that leaves the β-phase is equal to theflux that enters the σ -region.

The Method of Volume Averaging is applied in such a way as to specify the averageconcentrations in the β-phase and in the σ -region.

Equation 10 for the β-phase can be written as

∂⟨CAβ

⟩

∂t+ ∇ · ⟨

vβCAβ

⟩ = ∇ ·⎛

⎜⎝Dβ∇ ⟨

CAβ

⟩ + Dγ

Vω

∫

Aβσ

nβσ CAβdA

⎞

⎟⎠

+ 1Vω

∫

Aβσ

nβσ · Dβ∇CAβdA + ⟨r′′′

h

⟩(10)

where Vω represents the control volume on the intermediate scale, and correspondsto the sum of the volume of the β-phase and of the σ -region.

The average surface concentration is defined by Whitaker (1999) as

〈CA〉 = 1Vω

∫

Vβ

CAdV (11)

and the rate of hydrolysis is represented by Eq. 12.⟨r′′′

h

⟩ = −〈kh CA〉 (12)

The spatial average theorem (Homes and Whitaker 1985) is applied together with theno-slip condition at interface β–σ , to obtain

∂⟨CAβ

⟩

∂t+ ∇ · ⟨

vβCAβ

⟩ = ∇ ·⎛

⎜⎝Dβ∇ ⟨

CAβ

⟩ + Dγ

Vω

∫

Aβσ

nβσ CAβdA

⎞

⎟⎠

+ 1V ω

∫

Aβσ

nβσ · Dβ∇CAβdA − 〈kh CA〉 (13)

The superficial average concentration is transformed into the intrinsic averageconcentration in the β-phase

⟨CAβ

⟩β , through Eq. 14:⟨CAβ

⟩ = εβ

⟨CAβ

⟩β (14)

with,

εβ = Vβ

Vω

(15)

where εβ is the porosity of the β-phase.Applying the spatial decomposition proposed by Gray (1975), we arrive at Eqs. 16

and 17, for the punctual concentration and for the velocity, respectively,

CAβ = ⟨CAβ

⟩β + CAβ (16)

vAβ = ⟨vβ

⟩β + vβ (17)


The variable⟨vβ

⟩β is the intrinsic average velocity of the β-phase, vβ is the punctualvelocity of the β-phase, CAβ and vβ are the spatial deviations of the concentrationand the velocity, respectively.

Considering that the σ -region is rigid, which enables us to state that εβ is indepen-dent of time, Eq. 13 can be written in the form of Eq. 18, where Aβσ represents thearea of the interface β–σ .

εβ

∂⟨CAβ

⟩β

∂t+ ∇ ·

(εβ

⟨vβ

⟩β ⟨CAβ

⟩β)= ∇ ·

[Dβεβ∇ ⟨

CAβ

⟩β]

+∇ ·⎛

⎜⎝

Dβ

Vω

∫

Aβσ

nβσ CAβdA

⎞

⎟⎠

+ 1Vω

∫

Aβσ

nβσ · Dβ∇CAβdA

−∇ · ⟨vβCAβ

⟩ − kh εβ

⟨CAβ

⟩β (18)

Equation 18 is obtained by setting the terms⟨vβ

⟩β and⟨CAβ

⟩βas equal to zero, as

discussed and proven by Quintard and Whitaker (1994a).The same procedure adopted for the spatial smoothing of the punctual equation

for the β-phase, is adopted for the σ -region. Therefore, the integral of the volumeis applied together with the spatial averaging theorem in Eq. 5, resulting, after somemathematical transformations, in Eq. 19.

εσ εγ

∂ 〈CAσ 〉σ∂t

= ∇ · (Dσ · εσ ∇ 〈CAσ 〉σ ) + ∇ ·

⎛

⎜⎝

Dσ

Vω

·∫

Aβσ

nβσ · CAσ dA

⎞

⎟⎠

+ 1Vω

∫

Aσβ

nσβ · Dσ · ∇ CAσ dA − (

av|γ κ k + εγ kh)εσ 〈CAσ 〉σ

− av|γ κ Keqεσ

∂

∂t〈CAσ 〉σ (19)

where 〈CAσ 〉σ is the intrinsic average concentration of the σ -region, Dσ is equal toεγ Deff

∣∣γ κ

and εσ is the porosity of the σ -region.The point concentration present in the interfacial mass flux from Eq. 19 is substi-

tuted by the spatial decomposition of the CAσ , to give Eq. 20,

εσ εγ


= ∇ · (Dσ · εσ ∇ 〈CAσ 〉σ ) + ∇ ·

⎛

⎜⎝

Dσ

Vω

·∫

Aβσ

nβσ · CAσ dA

⎞

⎟⎠

+ 1Vω

∫

Aσβ

nσβ · Dσ ∇ CAσ dA − (

av|γ κ k + εγ kh)εσ 〈CAσ 〉σ

−∇εσ · Dσ ∇ 〈CAσ 〉σ − av|γ κ Keqεσ

∂

∂t〈CAσ 〉σ (20)


The same procedure described above is adopted for the β-phase, giving

εβ

∂⟨CAβ

⟩β

∂t+ ∇ ·

(εβ

⟨vβ

⟩β ⟨CAβ

⟩β)= ∇ ·

[Dβεβ∇ ⟨

CAβ

⟩β]

+∇ ·⎛

⎜⎝

Dβ

Vω

∫

Aβσ

nβσ CAβdA

⎞

⎟⎠

+ 1Vω

∫

Aβσ

nβσ · Dβ∇ CAβdA

−∇εβ · Dβ∇ ⟨CAβ

⟩β

−∇ · ⟨vβCAβ

⟩ − kh εβ

⟨CAβ

⟩β (21)

To complete the modelling of the dyeing process on the intermediate scale it isnecessary to calculate CAβ and CAσ , for which the closure problem is solved on thisscale.

2.1 Closure problem on the intermediate scale

In order for Eqs. 20 and 21 to be written in their final form, that is, where all of theterms are written as a function of the intrinsic average concentration, it is necessary todevelop the closure problem to determine CAβ and CAσ . This procedure is performedto obtain the transport equations for CAβ and CAσ , which are then resolved over arepresentative region.

The equations for the spatial deviation of the concentrations in the β-phase andσ -region are obtained through the subtraction of the equation for the punctual con-centration from the equation of the intrinsic average concentration, following thedefinition of the spatial deviation shown in Eq. 16.

The equation for the spatial deviation of the concentrations in the β-phase isobtained by the subtraction from Eq. 21 of Eq. 7, which allows us to write

∂CAβ

∂t+ ∇ ·

[vβCAβ −

(⟨vβ

⟩β ⟨CAβ

⟩β)]− ε−1

β

⟨vβ

⟩β ⟨CAβ

⟩β · ∇εβ

= ∇ · (Dβ∇CAβ

) −(ε−1β Dβ∇ ⟨

CAβ

⟩β)· ∇εβ − ε−1

β ∇ ·⎡

⎢⎣

Dβ

Vω

∫

Aβσ

nβσ CAβdA

⎤

⎥⎦

−ε−1β

Vω

∫

Aβσ

nβσ · Dβ∇CAβdA + ε−1β ∇ · ⟨

vβCAβ

⟩ − khCAβ (22)

The second term on the left side of Eq. 22 is developed using the definitions givenby Eqs. 16 and 17, to obtain

∇ ·(vβCAβ − ⟨

vβ

⟩β ⟨CAβ

⟩β)= vβ · ∇CAβ + vβ · ∇ ⟨

CAβ

⟩β

+ε−1β

⟨vβ

⟩β ⟨CAβ

⟩β · ∇εβ (23)


The fourth term on the right-hand side of Eq. 22 describes the mass flux of thespecies of interest from the fluid phase to the solid phase, at the interface β–σ . Substi-tuting Eq. 23 in Eq. 22 and eliminating the terms that cancel, we arrive at

vβ · ∇CAβ + vβ · ∇ ⟨CAβ

⟩β

= ∇ · (Dβ∇CAβ

) − ε−1β

Vω

∫

Aβσ

nβσ · Dβ∇CAβdA − khCAβ (24)

The complete analysis together with all the steps necessary to obtain Eq. 24 isexplained in detail in Guelli U. de Souza and Whitaker (2003).

The spatial decompositions of the concentrations in the β-phase and σ -region aresubstituted in Eq. 9, obtaining the boundary condition as shown in Eq. 25. Assumingthe model to be spatially periodic, one can write Eq. 26.

−nβσ · Dβ∇CAβ + nβσ · Dσ · ∇CAσ

B.C.1 in Aβσ

= nβσ · Dβ∇ ⟨CAβ

⟩β − nβσ · Dσ · ∇ 〈CAσ 〉σ(25)

B.C.2 CAβ (r + �i) = CAβ (r) , i = 1, 2, 3 (26)

To obtain the governing equation for CAσ , Eq. 5 is subtracted from Eq. 20 and,adopting a procedure similar to that carried out in the β-phase, one has

∇ · (Dσ · ∇CAσ

) = ε−1σ

Vω

∫

Aσβ

nσβ · Dσ · ∇CAσ dA + (

av|γ κ k + εγ kh)

CAσ

+ av|γ κ Keq∂CAσ

∂t(27)

Also, the condition of periodicity is assumed for the region σ , which is representedby Eq. 28.

B.C.3 CAσ (r + �i) = CAσ (r) , i = 1, 2, 3 (28)

2.2 Closure variables

To solve the equations of the closure problem on the intermediate scale, a characteris-tic function is proposed. This function is represented by Eqs. 29 and 30 in the β-phaseand σ -region, respectively.

CAβ = bβ · ∇ ⟨CAβ

⟩β − sβ

(⟨CAβ

⟩β − 〈CAσ 〉σ)

+ ϕ (29)

CAσ = bσ · ∇ 〈CAσ 〉σ − sσ

(⟨CAβ

⟩β − 〈CAσ 〉σ)

+ (30)

where bβ , bσ , sβ , sσ , ϕ and are the closure variables for both the phases.


Substituting Eqs. 29 and 30 in Eqs. 24–28, one obtains two closure problems for theβ-phase and σ -region.

Problem Ia:

vβ + vβ · ∇ bβ = Dβ∇2bβ − ε−1β uβ − khbβ , in the β-phase (31)

B.C.1 −nβσ · ∇bβ = nβσ in Aβσ (32)

B.C.2 bβ (r + �i) = bβ (r) i = 1, 2, 3 (33)

⟨bβ

⟩β = 0 (34)

Problem Ib:

Dσ ∇2bσ = ε−1σ uσ +

(av|γ κ k + εγ kh

)bσ

+ av|γ κ Keq∂ bσ

∂ t, in the σ -region (35)

B.C.1 −nβσ · ∇bσ = nβσ in Aβσ (36)

B.C.2 bAσ (r + �i) = bAσ (r) , i = 1, 2, 3 (37)

〈bσ 〉σ = 0 (38)

Problem IIa:

vβ · ∇ sβ = Dβ∇2sβ − ε−1β av|βσ h − khsβ , in the β-phase (39)

B.C.1 nβσ · ∇sβ = 0 in Aβσ (40)

B.C.2 sβ (r + �i) = sβ (r) , i = 1, 2, 3 (41)

⟨sβ

⟩β = 0 (42)

Problem IIb:

Dσ ∇2sσ = −ε−1σ av|βσ h +

(av|γ κ k + εγ kh

)sσ

+ av|γ κ Keq∂ sσ

∂ t, in the σ -region (43)


B.C.1 nβσ · ∇sσ = 0 in Aβσ (44)

B.C.2 sσ (r + �i) = sσ (r) , i = 1, 2, 3 (45)

〈sσ 〉σ = 0 (46)

where uβ from Eq. 31 is defined by Eq. 47 and uσ from Eq. 35 is defined by Eq. 48.

uβ = 1Vω

∫

Aβσ

nβσ · Dβ∇ bβdA (47)

uσ = − 1Vω

∫

Aσβ

nσβ · Dσ ∇ bσ dA (48)

The term av|βσ h is defined by Eq. 49.

av|βσ h = 1Vω

∫

Aβσ

nβσ · Dβ∇ sβdA = − 1Vω

∫

Aβσ

nσβ · Dσ ∇ sσ dA (49)

2.3 Final equation for the intermediate scale

To obtain the average equations for the β-phase and σ -region in the closed form, Eqs.29 and 30 are substituted in the equations for the calculation of the intrinsic averageconcentration, Eqs. 20 and 21, giving Eqs. 50 and 55, respectively.

εβ

∂⟨CAβ

⟩β

∂t+ ∇ ·

(εβ

⟨vβ

⟩β ⟨CAβ

⟩β)− ∇ ·

[dβ

(⟨CAβ

⟩β − 〈CAσ 〉σ)]

− uβ · ∇ ⟨CAβ

⟩β

= ∇ ·(εβD∗

β∇ ⟨CAβ

⟩β)− av|βσ h

(⟨CAβ

⟩β − 〈CAσ 〉σ)

− kh εβ

⟨CAβ

⟩β − ∇εβ · D∇β

⟨CAβ

⟩β (50)

where D∗β is the total dispersion tensor in β-phase defined by the equation,

D∗β = Deff

∣∣∣ββσ + D (51)

with Deff∣∣∣ββσ being the effective diffusivity tensor in the β-phase defined by:

εβDeff∣∣∣ββσ = εβDβI + Dβ

Vω

∫

Aβσ

nβσ bβdA (52)

and D the hydrodynamic dispersion tensor defined by Eq. 53.

D = − ⟨vβbβ

⟩β (53)

The additional coefficient of velocity generated by the difference between the con-centrations in the β-phase and σ -region, dβ , is defined by Eq. 54. The additionalcoefficient of velocity generated by the concentration gradient in the β-phase, uβ , is


given by Eq. 47 and the convective coefficient of mass transfer, h, is defined by Eq.49.

dβ = −Dβ

Vω

∫

Aβσ

nβσ sβdA (54)

The final average equation for the calculation of the concentration of dye insidethe thread on the bobbin (σ -region) is given by

εσ εγ


+ ∇ ·[dσ

(⟨CAβ

⟩β − 〈CAσ 〉σ)]

+ uσ · ∇ 〈CAσ 〉σ

= ∇ · (D∗

σ · εσ ∇ 〈CAσ 〉σ ) + av|βσ h(⟨

CAβ

⟩β − 〈CAσ 〉σ)

−(

av|γ κ k + εγ kh)εσ 〈CAσ 〉σ − av|γ κ Keqεσ

∂

∂t〈CAσ 〉σ (55)

where D∗σ is the effective diffusivity tensor in the σ -region defined by:

D∗σ = Dσ I + Dσ

Vσ

∫

Aβσ

nβσ · bσ dA (56)

and dσ is defined by Eq. 57.

dσ = Dσ

Vω

∫

Aβσ

nβσ sσ dA (57)

Equations 50 and 55 present some classical coefficients of transport, such as totaldispersion tensor D∗

β , and the convective coefficient of mass transfer, h. On the otherhand, the coefficients dβ and uβ are not classical and do not appear in the empiricalequations used in the literature. These additional terms indicate that the convectiveterm is altered by the presence of a concentration gradient, a phenomenon that hasbeen observed in the case of dispersion in pores with adsorption on the surface of asolid, as reported by Paine et al. (1983) and Edward et al. (1991).

Studies carried out by Quintard and Whitaker (1994b) show the situations in whichthese terms become negligible.

3 Transport equations for the macro-scale

The objective of this section is to develop a model that is valid for the region formedby the whole bobbin of thread (ω-region) and for the dyeing bath fluid surroundingit, limited to the interior of the equipment, as shown in Fig. 1.

The development of the mathematical model on this last scale is necessary in orderto obtain the dye concentration in the dyeing bath fluid outside of the bobbin as afunction of time, because as the dyeing process is carried out, the dye is transferredfrom the bath to the fibre, with the hydrolysis effect for reactive dyes being an aspectthat must be calculated. This dye lost from the dyeing bath fluid is not replaced, sincethis process occurs by the dye exhaustion process (bath). Thus, the dye concentrationprofile in the bath fluid is an important aspect of the dyeing process, with the predic-tion, through simulation, of the behaviour of this parameter being of great interest toindustry.


Fig. 2 Diagram of equipmentfor the dyeing of threads onbobbins

Axis of symetry

ηAC

H

V

Dyeingbath

EastWest

North

SouthIR ER

fAC η

Threadbobbin

Recirculation pump

X

Z

The equations for the average concentrations on the intermediate scale, whentransformed into equations for the macro-scale, become punctual concentrations,due to the difference of scale, with the new notations presented being given inEqs. 58 and 59.

⟨CAβ

⟩β = CωAβ (58)

〈CAσ 〉σ = CωAσ (59)

A simplified diagram of the dyeing process and of the region that forms the macro-scale is presented in Fig. 2, where the squared region is the calculation domain, that is,the bobbin of threads. Due to the symmetry of the problem only half of the bobbin isconsidered. The western face is the bobbin support, with radius RI; the eastern face,with radius RE, is in contact with the dyeing bath fluid outside of the bobbin, that hasa concentration of CAη, and H is the height of the bobbin.

Through determining the mass balance in the dyeing bath fluid outside of the bob-bin, considering vigorous agitation, is possible to calculate the variation in CAη inrelation to time, as shown in Eq. 60, where Q is the recirculation flux, V the totalvolume of the dyeing bath fluid outside of the bobbin and Cf

Aη the concentration ofdye at the exit of the support, in the same time interval.

dCAη

dt= Q

V

(CAη − Cf

Aη

)(60)

The concentration of dye at the beginning of the process for the β-phase and theσ -region is equal to Ci, that is, the bobbin begins the dyeing process immersed in thedyeing bath fluid external to the bobbin. This concentration is known and depends onthe intensity of colour required for the threads.

The boundary conditions for the concentrations of dye in the σ -region and theβ-phase for the southern and northern faces have a nil concentration gradient, since


there is no flux in this direction. On the other hand, the boundary conditions for theeastern and western faces depend on the direction of flux, which could be from outsidethe bobbin towards the centre or vice-versa. Considering that the flux is directed fromthe outside of the bobbin to the centre, on the eastern face (interface Aηω) we havethe concentration of dye prescribed and equal to the concentration of the dyeing bathfluid, for the β-phase and the σ -region.

The set of final equations that represents the macro-scale model can be given byEqs. 61–70.

dCAη

dt= Q

V

(CAη − Cf

Aη

)(61)

I.C.1 CAη = Ci, t = 0 (62)

εβ

∂CωAβ

∂t+ ∇ ·

(εβ

⟨vβ

⟩β CωAβ

)= ∇ ·

(εβD∗

β∇CωAβ

)

− av|βσ h(

CωAβ − Cω

Aσ

)− kh εβCω

Aβ (63)

I.C.2 CωAβ = Ci, t = 0 (64)

B.C.1 CωAβ = CAη, at interface Aηω, in RE. (65)

B.C.2 ∇CωAβ = 0, at interface Aηω, in RI. (66)

εσ εγ

∂CωAσ

∂t= ∇ · (

D∗σ · εσ ∇Cω

Aσ

) + av|βσ h(

CωAβ − Cω

Aσ

)

−(

av|γ κ k + εγ kh)εσ Cω

Aσ − av|γ κ Keqεσ

∂

∂tCω

Aσ (67)

I.C.3 CωAσ = 0, t = 0 (68)

B.C.3 CωAσ = CAη, at interface Aηω, in RE. (69)

B.C.4 ∇CωAσ = 0, at interface Aηω, in RI. (70)

4 Computational methodology used for the solution of the transport equations

In this section, we present the numerical procedures used to solve the partial differ-ential equations obtained in the mathematical modelling of the process of dyeing ofthreads on bobbins, in order to enable the prediction of the kinetics of textile dyeing.

The macro-scale equations, given by Eqs. 61–70, have been used to simulate thedyeing process of threads in bobbins. These equations have been written in a gener-alized coordinates system that is coincident with the physical domain of the bobbin.


The initial values of variables CωAσ andCω

Aβ , inside the bobbin, are considered equalto zero and equal to the initial concentration of the external bath fluid, Ci, respec-tively. The new values of Cω

Aβ and CωAσ are calculated solving Eqs. 63–70, in the next

time interval. With the new values of CωAβ and Cω

Aσ it is possible to calculate the newvalue for the external dyeing bath fluid concentration, CAη, solving Eqs. 61 and 62.This new value is used for the boundary condition, given by Eqs. 65 and 69, in thenext step of the computational procedure.

The equations of the mathematical model developed were approximated by alge-braic equations, obtained through application of the Method of Finite Volumes, withthe use of the interpolation function WUDS (“Weight Upwind Difference Scheme”),as referred to by Maliska (2004). The discretized equations were implemented in thecomputational algorithm, where the systems of algebraic equations generated weresolved by the MSI (“Modified Strongly Implicit”) method (Schneider and Zedan1981). The computational grid employed in these simulations was structured, with1,600 control volumes.

As in the process of packed threads dyeing, the bobbins can be cylindrical or coni-cal, although in the case of conical bobbins, the system of cylindrical coordinates willnot work with the boundaries of the physical problem. In order to solve this prob-lem and thereby generalise the computational code developed, the equations fromthe mathematical model, for a system of cylindrical coordinates, are transformed tothe system of generalised coordinates, following the procedure described by Ulsonde Souza and Whitaker (2003). The application of this system of coordinates makesthe computational algorithm very general, since this is written in relation to a fixedcomputational domain, allowing a large class of problems to be solved, generatingonly one new system of coordinates, appropriate for the new problem. Therefore, thecomputational code developed for the solution of the process of dyeing of threadson conical and cylindrical bobbins is the same, with only the generation of a newcomputational grid appropriate for the problem under study being necessary.

5 Numerical results of the kinetics of textile dyeing

The mathematical model developed in this study enables the simulation of the dyeingprocess of threads on bobbins with dyes that bind onto the surface of the fibre byreaction or adsorption, this generalisation being one of the contributions of this study.In the modelling, an equilibrium model of linear isotherm was considered. For otherequilibrium models it is necessary to change the terms which contain the variable ,to represent the desired equilibrium relation.

With a view towards corroborating the mathematical model, the process of dyeingcotton threads with the reactive dye “C.I. Reactive Orange 107” was simulated, withthe numerical results being compared to the experimental results obtained in a tex-tile factory. In this study, dyes which react with the fibres were used, so the reactiveterm is modelled imposing equals 1 and equals 0, neglecting the influence of theadsorption process. For the cases where only the adsorption process is important, it isnecessary to impose equals 0 and equals 1.

Table 1 presents the values of the parameters used to simulate the process of dyeingcotton thread with the dye “C.I. Reactive Orange 107”.

The experimental results for the kinetics of dyeing cotton thread with the reactivedye “C.I. Reactive Orange 107” were obtained by the authors in a textile factory in


Table 1 Values of the parameters for the dyeing process of cotton thread with the reactive dye “C.I.Reactive Orange 107”

Parameter Symbol Value Unit

Geometric parametersUpper internal diameter of the bobbin d′

1 0.041 m

Upper external diameter of the bobbin d′2 0.137 m

Lower internal diameter of the bobbin d′3 0.063 m

Lower external diameter of the bobbin d′4 0.159 m

Height of the bobbin H 0.128 mAverage radius of each cotton thread R′

th 1.494 × 10−4 mOperational parametersLiquor ratio RB 1/10 kgfib/lliquorNumber of cotton thread N 24/2 NeSpecific volume of the cotton fibre Vsp

fibre 0.65 cm3/gAverage mass of the bobbins Mbobbin 595.90 gPorosity of the γ -phase εγ 0.772Porosity of the k-phase εk 0.228Porosity of the β-phase εβ 0.259Porosity of the σ -region εσ 0.741Interfacial area per unit volumefor the intermediate scale av|βσ 9920.7 m−1

Pump discharge rate Q 25.84 l/minReynolds’s number Rep 1.10Peclet’s number Pep 2602.0Volume of the dyeing bath V 6.00 m3

Molecular diffusion coefficient Dγ 4.252 × 10−10 m2/sEffective diffusivity tensor in the γ -phase Deff

∣∣γ κ

3.275 × 10−10 m2/s

Total radial dispersion tensor in the β-phase D∗β xx 1.008 × 10−5 m2/s

Total axial dispersion tensor in the β-phase D∗β zz 1.063 × 10−0 m2/s

Effective diffusivity tensor in the σ -region D∗σ 1.105 × 10−10 m2/s

Convective coefficient of mass transfer h 1.67 × 10−6 m/sKinetic and thermodynamic parametersReaction rate constant (before soda addition)

(av|γ κ kr

)3.433 × 10−3 s−1

Reaction rate constant (after soda addition)(

av|γ κ kr)

6.116 × 10−3 s−1

Pseudo-first-order rate constant of hydrolysis(before soda addition) kh 3.429 × 10−4 s−1

Pseudo-first-order rate constant of hydrolysis(after soda addition) kh 3.705 × 10−3 s−1

Equilibrium constant Keq 0a m

a Keq is considered equal to zero due to the influence of adsorption process being neglected, forreactive dyes

this geographical region, at a temperature of 60◦C. These experiments were carriedout using pilot scale dyeing equipment, with a dye capacity of only one bobbin.

The experimental procedure begins with the addition of the bobbin of cotton threadand the heating of 7 l of water in the equipment to the desired temperature, which ismaintained constant throughout the dyeing process. Next, the auxiliary products, suchas salt (NaCl) at 30.0 g/l and detergent at 0.5 g/l are added. At this point the dyeingprocess is begun, which is initially characterised only by the diffusion of the dye withinthe thread. The dyeing bath is maintained under these conditions for 20 min.


γε

γε

γ

γ

D

kDeff

1.0This Study (Cylinders)Sand0.8Mixture of spheres

Carborundum0.6

Sodium chloride

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3 Effective diffusivity on the micro-scale obtained as a function of the volume fraction of theγ -phase. Experimental data obtained from Whitaker (1999)

After 20 min, calcium carbonate (weak base) is added to the dyeing bath fluid ata concentration of 3.0 g/l, increasing the pH of the solution from 6.5 to 7.6. With theaddition of the calcium carbonate and the consequent increase in the pH, the reactionof the dye with the surface of the fibre is initiated. This new solution of the dyeingbath fluid circulates through the bobbin for 15 min. At the end of the second stageof the dyeing process sodium hydroxide (NaOH) is added at 0.74 ml/l from a sodasolution at 50◦C, causing the pH of the dyeing bath fluid to reach 10.7. This alkalinemedium markedly accelerates the reaction of the dye with the fibre and the hydrolysisreaction, completing the dyeing process in only 20 min.

The effective diffusivity tensor in the σ -region is obtained solving two closure prob-lems. The solution of the closure problem on the micro-scale, as presented in Ulsonde Souza and Whitaker (2003) and Guelli U. de Souza and Whitaker (2003), permitsus to calculate the first term on right side of Eq. 56.

The closure problem given on the micro-scale is solved for an array of cylinders,considering an isotropic two-dimensional system. The finite volume methods havebeen used to discretize the governing equations of the closure problem, using theWUDS (Weight Upstream Difference Scheme) scheme to interpolate the variableson the surface of the control volume. The closure problem is solved when the modelhas only chemical reaction, with equals 1 and equals 0. In Fig. 3, the effectivediffusivity on the micro-scale, obtained as a function of the volume fraction of theγ -phase, is shown. In Fig. 3, it is possible to verify a good agreement between ourresults and those obtained by other researchers and experimental data (Whitaker1999). The second term on the right side of Eq. 56 is calculated solving the closureproblem given by Eqs. 35–38.

The total dispersion tensor in the β-phase is obtained solving the closure problemgiven by Eqs. 31–34. Figure 4 illustrates the values of the radial component of the totaldispersion tensor, obtained in this study, considering an in-line cylindrical array, with aporosity of 0.37. In Fig. 4, it is possible to verify a good agreement between the resultsobtained in this study, considering an in-line cylindrical array and the experimental


10-1

100

101

102

103

104

105

100 101 102 103 104

Pe

Edward et al. (1991)

Quintard and Whitaker (1993)

Cylinders in line (This work)β

β

D

D *xx

10-1

Fig. 4 Total dispersion tensor as a function of the Peclet number

data. It is possible to observe that the total dispersion tensor is strongly dependent onthe Peclet number.

The mass transfer convective coefficient can be obtained using correlations givenin the literature or by solving the closure problem given by Eqs. 39–42 and Eqs. 43–46,which will be the subject of our next paper. Given the lack of a correlation availablefor this geometry, in this study, the correlation proposed by Sherwood et al. (1975)was used, which is valid for a fixed bed of cylindrical particles, and gave good results.For the operational conditions used in the pilot dyeing equipment, the value for theconvective coefficient of mass transfer is 1.67 × 10−6 m/s.

Table 2 presents the model parameters that are necessary to solve the model equa-tions set.

The numerical simulations are performed under the same conditions as the exper-iments, including with respect to the functioning of the pilot scale dyeing equipment.

As mentioned previously, the reaction of the dye with the surface of the textilefibres only occurs when the calcium carbonate and the sodium hydroxide are addedto the dyeing bath fluid. Therefore, the simulation of the dyeing process is initiatedwith the addition of the calcium carbonate, that is, 20 min after dyeing has begun.

Figure 5 presents the results of the concentration profiles for the dye in the dye-ing bath, CAη simulated for a temperature of 60◦C and an initial dye concentrationof 2.060 g/l and these results are compared to the experimental results. It can beseen from this figure that the dyeing kinetics are altered with the addition of sodiumhydroxide, since besides the increase in the rate of the reaction of the dye with thetextile fibre, the hydrolysis rate of the dye increases substantially. Furthermore, thetheoretical results for the dyeing kinetics show good agreement with the experimentalresults, with the maximum error between the numerical and the experimental resultsbeing 17.2%. This error is calculated through the difference between the concentra-tion of dye obtained numerically and that obtained experimentally, in relation to theconcentration of dye at the 20-min time point.


Table 2 Model parameters

Scale Model equations

Total dispersion tensor β-phase D∗β = Deff

∣∣∣ββσ + DThe total dispersion tensor in the β-phase is obtainedsolving the closure problem given by Eqs. 31–34

Effective diffusivity tensor β-phase εβDeff∣∣∣ββσ = εβDβI + Dβ

Vω

∫

Aβσ

nβσ bβdA

The effective diffusivity tensor in the β-phase is ob-tained solving the closure problem given by Eqs. 31–34

Hydrodynamic dispersion tensorβ-phase

D = − ⟨vβbβ

⟩β

The solution of the closure problem given by Eqs. 31–34 also permits the calculation of the hydrodynamicdispersion tensor in the β-phase

Effective diffusivity tensor σ -region

D∗σ = Dσ I + Dσ

Vσ

∫

Aβσ

nβσ · bσ dA

The effective diffusivity tensor in the σ -region is ob-tained solving two closure problems. The solution ofthe closure problem on micro-scale, as presented inUlson de Souza and Whitaker (2003) and Guelli U.de Souza and Whitaker (2003), permits the calcula-tion of the first term on the right side of the equation.The second term on the right side is calculated solv-ing the closure problem given by Eqs. 35–38

Mass transfer convective coeffi-cient

h

The mass transfer convective coefficient can be ob-tained using correlations in the literature or by solv-ing the closure problem given by Eqs. 39–42 andEqs. 43–46, which will be the subject of our nextpaper. Given the lack of a correlation available forthis geometry, in this study, the correlation proposedby Sherwood et al. (1975) was used, which is validfor a fixed bed of cylindrical particles, and gave goodresults

The experimental results for the concentration of dye in the bath are comparedwith the numerical results in Fig. 6, for an initial dye concentration of 0.408 g/l anda dyeing temperature of 60◦C. This case is also simulated beginning from the first20 min of dyeing, when the concentration of dye in the bath is 0.235 g/l. The numericalresults also showed good agreement with the experimental results, with the greatesterror detected being 11.6%.

In Figs. 5 and 6, it can be seen that a good agreement between the numerical andthe experimental results allows us to validate the proposed model, as well as thecomputational code developed.

An investigation into the parametric sensitivity of the temperature variable wascarried out. Figure 7 shows the influence of temperature on the dyeing kinetics, for40◦C and 60◦C.

The higher the temperature the higher the dye consumption rate, and a reductionof 14.4% in the time necessary for the consumption of all of the dye in the dyeingbath fluid was observed. The temperature is the most important parameter affectingthe dyeing process, it altering the reaction rate. Thus, this variable must be carefullycontrolled.


0.00

0.10

0.20

0.30

0.40

0.50

0.60

20 25 30 35 40 45

( )LgCAηExperimental

NaOH

0.70

Simulated

Time (min)

Fig. 5 Concentration profile for the dye “C.I. Reactive Orange 107” at a temperature of 60◦C andinitial concentration of 2.060 g/l

( )LgC AηNaOH

0.25

0.20 ExperimentalSimulated

0.15

0.10

0.05

0.00 20 25 30 35 40 45

Time (min)

Fig. 6 Concentration profile for the dye “C.I. Reactive Orange 107” at a temperature of 60◦C andan initial concentration of 0.408 g/l

0.6( )LgCAη

NaOH

0.8

0.7

40 o

C

60 o

C 0.5

0.4

0.3

0.2

0.1

0.020 25 30 35 40 45

Time (min)

Fig. 7 Influence of temperature on the dyeing kinetics


6 Conclusion

The mathematical modelling developed in this study represents an important stepforward in relation to the classical models, incorporating all the relevant phenomeno-logical information on the process, from the micro-scale, which is formed by the textilefibres and the dyeing bath fluid surrounding it, to the macro-scale, which comprisesthe bobbin of thread and the dyeing bath fluid contained in the equipment for thedyeing of threads on bobbins.

The final mathematical model comprises three equations, the first describes theconcentration of dye in the dyeing bath fluid outside of the bobbin (macro-scale),the second predicts the concentration of dye in the bath fluid surrounding the thread,but inside the bobbin, and the last equation evaluates the concentration of dye inthe thread. Therefore, the model is capable of calculating the diverse resistances tomass transfer during the dyeing process, making it unnecessary to consider in advanceany kind of resistance that could become important, depending on the operationalconditions of each process. Thus, the final equation for the macro-scale incorporatesimportant information from the intermediate- and the micro-scales, which enablesthe more precise prediction of the textile dyeing kinetics.

The numerical methodology employed in the solution of the mathematical modelproved to be adequate, enabling the reproduction of the dyeing kinetics with goodprecision, when compared to the experimental results obtained together with the tex-tile industry, with the greatest difference found, considering all of the cases studied,being 17.2%. This corroborates the computational code developed, as well as themathematical model.

Acknowledgements The authors acknowledge the financial support given by FINEP/Fundo VerdeAmarelo, PROTEXTIL—Otimização de Processos das Indústrias Têxteis Catarinenses.

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Modelling of the dyeing process of packed cotton threads using reactive dyes

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