Multi-dimension feature analyses for low-liquor ratio yarn bobbin dyeing combining mass transfer modeling

Abstract

Bobbin dyeing is a low-liquor-ratio technique that consumes less water and generates less wastewater. However, the bobbin dyeing is carried out in a closed kettle body, and many interfering factors that affect dyeing rate and dyeing uptake concentration cannot be observed and controlled in real time. Dyeing digitization can be used to achieve real-time monitoring of the amount of dyeing on the yarn bobbin, and is an important means of obtaining information on bobbin dyeing. To achieve digital dyeing, in this paper, a mathematical model for mass transfer based on yarn dyeing principles is established first, describing the process of dyeing from the beginning to equilibrium. Subsequently, by using anti-overlap winding principles and Darcy’s law, the anisotropic permeability in yarn bobbins is calculated. Then, a two-dimensional axisymmetric finite-element model describing the bobbin yarn’s dyeing behavior is constructed. Finally, the flow characteristics and the dye amounts are obtained. The results reveal that unevenness occurs in the initial dyeing stage but is gradually alleviated with dyeing continuation. After 3300 s, dye concentration stabilizes at 2.11 mol/m³ across most areas. Moreover, operating pressures significantly affect the dyeing rate and the time needed to reach dyeing equilibrium. The model proposed in this paper offers real-time prediction of the flow characteristics and the distribution of dye uptake. It is helpful for evaluating different schemes, such as different inlet and outlet pressures, different dyeing times, etc., to improve the uniformity of dyeing.

Keywords

Bobbin dyeing dyeing uptake concentration mass transfer model anisotropic permeability dyeing equilibrium real-time prediction

Nowadays, new manufacturing methods that focus on improving energy efficiency, saving water, and reducing wastewater, pollution, carbon footprint, and resource consumption are receiving more and more attention.^1,2 The textile industry is a major driver of the global economy in many countries, but it is also a major source of environmental pollution.³ Among the various stages of textile production, the dyeing and finishing processes are particularly responsible for ecological damage, accounting for 17% to 20% of global water pollution.⁴ Wastewater from dyeing and finishing processes contains a variety of substances, such as dyes and leveling agents, which require effective treatment.⁵ If not properly managed, untreated wastewater can harm both the natural environment and human health.⁶ As a result, reducing water consumption and minimizing wastewater generation in the dyeing and finishing industry has become a key area of research.

Yarn dyeing is a crucial step in the dyeing and finishing process. Compared with hank dyeing and beam dyeing, which are commonly used in the industry, bobbin dyeing has the following advantages: simple production process, higher productivity, lower energy consumption, less waste, less yarn damage, and small space requirements.⁷ Nevertheless, there are currently some challenges in yarn bobbin dyeing. Yarn bobbin dyeing uses a large number of dyeing auxiliaries and chemical dyes, and produces a large amount of wastewater at the end of the dyeing process. In addition to the problem of wastewater generation, uneven dyeing remains a common problem. This is evidenced by the presence of inconsistencies in coloration within a single yarn bobbin, disparities in coloration between yarn bobbins within the same dyeing vat, and discrepancies in coloration between yarn bobbins in different dyeing vats. When these color differences are substantial, re-dyeing must be carried out,⁸ which will not only increase dyeing costs and reduce the yarn strength, but also increase water usage, leading to a further increase in wastewater production.

Various factors affect the dyeing uniformity of a yarn bobbin, such as the yarn material, winding density, bobbin shape, dye selection, dyeing time, and circulation and pressure of dye liquor, etc.^7,9,10 Nowadays, the problem of even dyeing coloration has attracted the attention of more and more researchers. Vuthiganond et al.,¹¹ Burkinshaw and Salihu,¹² and Schmeißer et al.¹³ experimentally analyzed the effects of various dyeing parameters on dyeing uniformity, including the dyeing temperature, the pH of the dye liquor, the dyeing time, and the use of leveling agents. Their results demonstrated that combining positive and negative cycles, increasing flow rate, reducing yarn bobbin thickness and density, or adding a leveling agent were all helpful in achieving even dyeing of yarn bobbins. In addition to experimental methods, some researchers studied the dyeing of yarn bobbins using mechanism analysis and numerical simulation. They regarded the yarn bobbin as an isotropic porous material and analyzed its internal flow field and mass transfer phenomena.^14,15 Mancusi et al.¹⁶ developed a mathematical model of mass transfer in the bobbin dyeing process to examine the concentration of dye adsorbed on the yarn (CDY). The results showed that circulating the dye liquor led to a more uniform dye distribution. De Souza et al.¹⁷ used the volume averaging method to analyze the dyeing process from microscopic to macroscopic scales. The CDY along the radial direction was calculated and compared with the experimental results. The theoretical calculation results were in good agreement with the experimental results. Zhao et al.¹⁸ adopted the numerical modeling method and finite-difference method, and regarded the yarn bobbin as an isotropic porous medium. The ratio of the average concentration change of the inner and outer layers of the yarn bobbin was calculated as a function of time. Moreno et al.¹⁹ and Chiango²⁰ conducted numerical simulations of the dyeing process for acrylic bobbins and warp dyeing. The flow conditions of the dye liquor inside the yarn bobbins were calculated. These results demonstrated that achieving a good and uniform flow distribution of the dye liquor within the yarn bobbins led to an ideal dyeing level.

The experimental research requires that the dyed yarn should be washed, rewound, and then weaved into a certain shape of fabric. With the assistance of optical instruments, the color values are detected.²¹ It is complex and unfeasible to obtain the color values in real time, and even more impossible to distinguish the color difference between different areas within the whole yarn bobbin. The analytical method, which relies on certain assumptions and experimental data, allows for accurate determination of the CDY. Nevertheless, its derivation is also complex, making it challenging to determine the CDY accurately at any position. Numerical simulation enables prediction of the dyeing process, contributing to significant reductions in power and water consumption. However, challenges currently persist with dyeing simulation. Most studies are mainly focused on the concentration of dye in the dye solution (CDS), ignoring the fact that the color differences appearing in different yarns are caused by the CDY. After all, the higher the CDY is, the deeper the color will be, and vice versa.²² Additionally, in many studies, the yarn bobbin is regarded as an isotropic porous medium. However, the yarn bobbin is formed by winding yarn according to specific principles. It exhibits a non-uniform arrangement in the axial, radial, and circumferential directions, rendering it an anisotropic porous material. Moreover, the yarn dyeing process is a process in which the dyes adsorbed on the yarns gradually increase until equilibrium. Few studies present the CDY during this process by using numerical simulation methods. In this work, mathematical modeling and numerical simulation are used to study the dyeing behavior of anisotropic yarn bobbins, from the initial stage of dyeing to the equilibrium state. This approach not only facilitates better control over dyeing time and improves dyeing quality, but also helps reduce water and electricity consumption, as well as the generation of wastewater, supporting the development trend of green printing and dyeing.

In this paper, based on the yarn dyeing principles, especially the process of the yarn from the beginning of dyeing to equilibrium, a mathematical mass transfer model for bobbin dyeing is established. Considering the winding characteristics of the yarn bobbin, the yarn bobbin is therefore regarded as an anisotropic porous material, and a two-dimensional axisymmetric finite-element model is constructed to deduce the distribution of the CDS and CDY. The results of numerical calculations of the CDY and CDS at different positions inside the dyeing domain are compared and analyzed. The digitization of the bobbin dyeing is bound to provide a basis for achieving intelligent dyeing.

Mathematical modeling for bobbin dyeing

According to the principles of yarn dyeing, a mathematical model for bobbin dyeing will be set up in this section.

Principle of yarn bobbin dyeing

To minimize the discharged amount of water and reduce electrical energy consumption, unidirectional circulation of the dye solution, especially referring to the positive cycle, is used in bobbin dyeing,²³ as illustrated in Figure 1(a) and (b). Yarns are wound around the bobbin. When pressure at the inlet is applied, the dye solution flows through the holes of the bobbin, penetrates the gaps between yarns and the internal pores of yarns, streams into the dyeing jar from the outer layer of the yarn bobbins, and finally flows out from the outlet. The outflowing dye liquor is then deposited in a mixing cylinder and may be mixed as needed, and utilized as a charge for subsequent cycles.

Yarn dyeing mainly comprises three steps: transfer of dyes from the dye liquor to the yarn interface (position 1 in Figure 1(c)), the rapid adsorption of dyes onto the yarn on reaching the yarn interface (position 2 in Figure 1(c)), and the diffusion of dyes into the yarn interior (position 3 in Figure 1(c)).²⁴ In these steps, the transfer and adsorption processes directly influence the dyeing rate and dye amount.

The thickness of the diffusion boundary layer δ in Figure 1(c) is obtained according to the following formula:²⁵

δ = 3 (\frac{D}{e}) \times {(\frac{e d}{| u |})}^{1 / 2}

(1)

where e represents the kinematic viscosity of the dye liquor (m²/s), with magnitude μ/ρ_f; μ represents the dynamic viscosity of the dye liquor (kg/(m·s)); ρ_f represents the density of the dye liquor (kg/m³); D represents the diffusion coefficient of dye in the dye liquor (m²/s); d denotes the diameter of the yarn (m); and u represents the velocity of the dye liquor (m/s).

Dyes must traverse the diffusion boundary layer to reach the yarn surface from the dye liquor. The required time t_k depends on the thickness of the diffusion boundary layer:²⁶

\sqrt{D t_{k}} \propto δ

(2)

When the dye reaches the dye liquor–yarn interface, it interacts with the yarn, resulting in the transfer of dye from the liquor to the yarn. This process involves both the adsorption of dye on the yarn surface and the desorption of dye from the yarn back into the liquor. The total adsorption rate of the yarn R_i is determined by the difference between the adsorption and desorption rates:²⁷

R_{i} = k_{1} c (S - c_{s}) - k_{2} c_{s}

(3)

where k₁c(S − c_s) represents the adsorption rate of the dye (ARD); k₂c_s represents the desorption rate of dye (DRD); c_s and c represent the CDY (mol/m³) and CDS (mol/m³), respectively; S represents the dye saturation value on yarn (mol/m³); and k₁ and k₂ represent the adsorption rate constant (m³/(mol·s)) and the desorption rate constant (s⁻¹), respectively.

According to equation (3), at the early stage of dyeing, the ARD is much higher than the DRD, owing to the high CDS and low CDY. As dyeing progresses, the CDS decreases while the CDY increases. Consequently, the ARD decreases and the DRD increases. Eventually, the ARD will be equal to the DRD, leading to a point where the CDY stabilizes over time. The relationship between the CDS and CDY is illustrated in:²⁸

c_{s} = \frac{K ​ S c}{1 + K c}

(4)

where K is the equilibrium constant (m³/mol), with magnitude k₁/k₂.

This mathematical model, as established in this section, can be utilized to depict the characteristics of yarn bobbin dyeing, so that the dyeing process of yarn bobbins can be investigated.

Mathematical model of mass transfer

In this section, a mathematical model for the mass transfer of yarn bobbin dyeing is constructed based on the dyeing principle. The boundary conditions are then discussed in relation to the actual working conditions.

Specific assumptions

To establish a mathematical model of dye mass transfer in yarn bobbins, the following assumptions are made.^14,19

The dye solution used in this work primarily consists of disperse dyes, water, glacial acetic acid, leveling auxiliaries, and other substances. The addition of these substances may have a certain effect on the overall performance of the dye solution. However, the proportion of these substances is very small, which makes the properties of the whole dyeing solution very close to that of water. Therefore, the dye solution is considered as a Newtonian incompressible fluid.

In this work, the dyeing process of a yarn bobbin at a certain temperature is investigated. Since the volume of dye liquor is much greater than that of the yarn bobbin, the overall change in dye liquor concentration before and after dyeing is relatively small. As a result, the physical properties of the dye liquor, such as density and kinematic viscosity, are assumed to remain constant throughout the process.

The yarn bobbin is treated as an anisotropic porous material. Moreover, during the dyeing process, the volume deformation of the yarn is ignored. Therefore, the porosity and permeability of the yarn bobbin are considered as constants.

Momentum equation

The momentum equation governing the flow of dye liquor through the internal pores of the yarn bobbins is presented as

\begin{array}{l} \nabla \cdot [- P I + \frac{μ}{ε} (\nabla u + {(\nabla u)}^{T}) - \frac{2}{3} μ \frac{(\nabla u) I}{ε}] \\ - (\frac{μ}{κ} + β | u |) u + ρ_{f} g = \frac{ρ_{f}}{ε} \frac{\partial u}{\partial t} \end{array}

(5)

where

\nabla \cdot P I

represents the momentum change due to the pressure of the dye liquor;

\nabla \cdot (\frac{μ}{ε} (\nabla u + {(\nabla u)}^{T}) - \frac{2}{3} μ \frac{(\nabla u) I}{ε})

is the momentum change caused by the viscous force of the dye solution;

(\frac{μ}{κ} + β | u |) u

denotes the momentum change caused by the internal resistance of the porous material; ρ_fg represents the momentum change caused by the gravitational force; ε is the porosity of the yarn bobbins;

\nabla u

is the gradient of the dye liquor velocity; P is the pressure (Pa); κ and β are the permeability (m²) and the inertia resistance coefficient (kg/m⁴) of the yarn bobbin, respectively; g is the acceleration of the gravitational force (m/s²); and t is the dyeing time (s).

Mass transfer

As the dye is transferred from the dye liquor onto yarns, the concentration change in dye liquor can be visualized as

ε \frac{\partial c}{\partial t} + \nabla \cdot (- D \nabla c) + u \cdot \nabla c = - R_{i}

(6)

where ∂c/∂t is the change rate of the CDS with time;

u \cdot \nabla c

is the convective mass transfer term in the dye liquor;

- D \cdot \nabla c

is the mass diffusion term in the dye liquor; and

\nabla c

is the gradient of the CDS.

As described previously, the transfer of dye from the dye liquor onto yarns passes through a diffusion boundary layer with a thickness of δ. Therefore, the total adsorption rate R_i and the dyeing time t have the following relationship

R_{i} = {\begin{array}{l} 0, & 0 < t \leq t_{k} \\ k_{1} c (S - c_{s}) - k_{2} c_{s}, & t > t_{k} \end{array}

(7)

When the dye molecules reach the yarn surfaces, they are adsorbed by the surfaces and diffuse in the yarns. This transfer process is expressed as

\frac{\partial c_{s}}{\partial t} + \nabla \cdot (- D_{s} \nabla c_{s}) = R_{i}

(8)

where ∂c_s/∂t represents the change rate of CDY over time;

\nabla \cdot (- D_{s} \nabla c_{s})

denotes the mass diffusion in the yarn; and D_s represents the diffusion coefficient of the dye within the yarn (m²/s).

Boundary conditions

Prior to the dyeing circulation, yarn bobbins are completely immersed in dye liquor, with an initial concentration of c₀. The flow rate of the dye solution at all positions in the dye field is 0. At this moment, c_s = 0 and ∂c_s/∂t = 0.

During the dyeing circulation, as shown in Figure 1(b), the pressures at the inlet and outlet of the dyeing cylinder are P_in and P_out, respectively. The flow rate of dye liquor entering the dyeing jar inlet is equal to the flow rate at the outlet. Moreover, the amount of dye entering the inlet is equal to the combined amount of the dye leaving the outlet and the dye added to the mixing tank, that is Q_in = Q_out and b_in = b_out + b_add, where Q_in and Q_out represent the inflowing and outflowing flow rates of the dyeing jar, respectively (m³/s); b_in and b_out denote the amounts of inflowing and outflowing dye, respectively (mol); and b_add denotes the amount of dye added to the mixing cylinder (mol).

In practice, dyes are often injected into the mixing cylinder intermittently, rather than continuously. In this case, the CDS flowing through the inlet c_in can be expressed as c_in = c_out, where c_out is the average CDS at the outlet of the dyeing jar (mol/m³).

Multi-dimension feature analyses for yarn bobbin dyeing

The finite-element technique is a powerful numerical method, which is widely used in various engineering and scientific fields.²⁹ This technique allows for detailed and accurate predictions of physical phenomena. Based on the mathematical equations given previously, a finite-element model will be built in this section, to quantitatively calculate the dye changes and the distributions of the dyeing uptake concentration across the yarn bobbin.

Yarn bobbin winding microunit

The yarn bobbin winding process involves both axial and tangential movements along the bobbin. Each revolution of the bobbin creates a spiral loop. When the yarn reaches the upper or lower end faces of the yarn bobbin, it will wind around the opposite axis, forming a foldback point on that end face. Continuous winding is achieved by repeating this cycle. The main winding parameters in this process are illustrated in Figure 2(a). An inappropriate selection of winding parameters might lead to yarn overlap, which can increase the yarn breakage rate, reduce yarn capacity, and cause difficulties in subsequent processes, such as warping and weaving.

The key to preventing the overlapping of adjacent yarns is to control the position of the foldback point.³⁰ According to the theory of anti-overlapping winding of yarn bobbins, the winding ratio G is the number of revolutions of the yarn bobbin in one winding cycle of the yarn movement, whose value can be calculated as

G = \frac{2 n W}{v_{z}}

(9)

where n is the rotational speed of the bobbin (r/s), n = v_x/πϕ₁; W is the height of the yarn bobbin (m); and v_z is the axial component of the winding rate v (m/s), v_z = vsinα, where v is the length of yarn wound per second (m/s).

As shown in Figure 2(a), α is the yarn winding angle (rad); ϕ₁ is the inner diameter of the yarn bobbin (m); and v_x is the tangential component of the winding rate v (m/s), v_x = vcosα. Therefore, equation (9) can be rewritten as

G = \frac{2 W}{π ϕ_{1} \tan α}

(10)

For a given yarn bobbin, both W and ϕ₁ are constant values. Thus, the winding ratio G solely depends on the winding angle α. To prevent overlap between adjacent yarns, the winding ratio must be adjusted. This adjustment involves selecting the nearest integer to the calculated value as the base winding ratio and determining the non-integer portion. The correction of the non-integer part of the winding ratio is determined in relation to the number of foldback points of the yarn bobbin, M. The initial selection of M is

M = \frac{π ϕ_{1} \sin α}{l}

(11)

where l is the minimum spacing between yarns (m).

The adjacent integer portion of M is selected as the initial foldback point value. A suitable correction value for the decimal part of the winding ratio can be found in the commonly used anti-overlap winding ratio correction value table.³⁰ This correction value is then added to the base winding ratio to determine the value closest to the value calculated in equation (10). This corrected winding ratio is then used to calculate the winding angle α and the minimum yarn distance l. The comparison of parameters before and after anti-overfitting correction is given in Table 1.

Table 1.

Comparison of parameters before and after anti-overfitting correction

Parameter	Symbol	Initial value	Corrected value
Internal diameter of yarn bobbin	ϕ ₁	6 × 10⁻² m	6 × 10⁻² m
Yarn bobbin height	W	1.65 × 10⁻¹ m	1.65 × 10⁻¹ m
Minimum yarn distance	l	6 × 10⁻³ m	5.24 × 10⁻³ m
Winding angle	α	π/12 rad	0.085π rad
Winding ratio	G	6.537	6.778
Number of foldback points	M	9.481	9

Permeability of yarn bobbin

As illustrated in Figure 2(a), a microunit of the wounded yarn bobbins is extracted with a thickness of 1.2 × 10⁻³ m and dimensions of 2.0 × 10⁻³ m in length and height. Because of the special spiral winding method described previously, significant differences are present along the axial and radial directions. This will directly affect the magnitude of the resistance in all directions when the liquid flows through this unit.

Permeability is an important indicator for measuring the ease of flow inside the yarn bobbin. Its magnitude is related to such factors as the porosity between the yarns, the tightness of the yarns, and the yarn winding angle, but it has nothing to do with the properties of the medium through which it flows. To quantify the permeability of the dye liquor within the yarn bobbin, a finite-element model is developed for the winding microunit. Water is chosen as the medium at a temperature of 293.15 K; the velocity inlet and pressure outlet are used as the boundary conditions. The pressure drops through the microunit at different inlet velocities are calculated using steady-state calculations, as illustrated in Figure 2(b). According to Darcy’s law,³¹ permeability is directly proportional to the kinematic viscosity and the thickness through which the fluid passes, and inversely proportional to the value of the primary term of the velocity–pressure drop fitting curve. Therefore, the permeability of the winding microunit for a yarn bobbin in the radial direction, κ₁, is 1.97 × 10⁻⁹ m² and the permeability in the axial direction, κ₃, is 3.91 × 10⁻⁹ m².

Finite-element solution for yarn bobbin dyeing

COMSOL software will be used to solve the mathematical model of bobbin dyeing by employing the finite-element method.

Extraction of dyeing field

Figure 3(a) shows, schematically, the internal structure of a specific type of small yarn bobbin dyeing machine. To minimize the liquor ratio, the dye solution is only added to the horizontal plane of the end cover, ensuring that yarns are completely immersed in the dyeing liquid. Considering that the influence area of the mandrel and bobbin on the flow field is relatively small,²⁰ the mandrel and bobbin are not involved in this study. Therefore, the entire yarn bobbin and all the areas through which the solution flows are axisymmetric bodies. To reduce the calculation scale, a two-dimensional symmetric plane is extracted, as illustrated in Figure 3(b). The inlet and outlet of the dye field are the pressure inlet and the pressure outlet, respectively. The line in the center is its axis of symmetry. All boundaries are standard non-slip wall surfaces, except for the inlet, outlet, and symmetry axis, and the boundaries of the yarn bobbin.

Determination of simulation parameters

In this study, the yarn bobbin has an inner diameter of 60 mm, an outer diameter of 120 mm, and a height of 160 mm. The porosity of the yarn bobbin is measured by the drainage method to be 0.627. The dyes used are dispersion dyes with a molar mass of 331.32 g/mol. The initial concentration of the dye solution is determined according to the literature,³² and is 6.36 mol/m³. The dyeing temperature is 353.15 K. The density and dynamic viscosity of the dye liquor at this temperature are 1020 kg/m³ and 3.551 × 10⁻⁴ Pa·s, respectively. The diffusion coefficient of the dyes in the dye liquor is 1.14 × 10⁻⁷ m²/s, as referenced in related literature.¹⁹ The diffusion coefficient of the dyes in the yarn is calculated using Hill’s formula.³³ Considering the polyester yarn dyeing process with dispersed red dye reported in the literature,³² the diffusion coefficient of the dyes on the yarn is obtained as 2.46 × 10⁻¹² m²/s.

When dispersion red dye is applied to polyester yarn, it follows Langmuir adsorption behavior. The saturation value of the dye on the yarn is 2.146 mol/m³ and the equilibrium constant is 12.65 m³/mol.³² Combined with the up-dyeing rate curves of the dyes at different dyeing temperatures, the adsorption rate constant can be obtained as 5.44 × 10⁻⁶ m³/(mol·s). The desorption rate constant can then be calculated from the equilibrium constant and the adsorption rate constant. The value of the desorption rate constant is 4.3 × 10⁻⁷ s⁻¹. The key parameters used in the dyeing simulation are summarized in Table 2.

Table 2.

Dyeing parameters required for numerical simulation

Parameter	Symbol	Value	Unit
Initial concentration of dye in dye solution	c ₀	6.36	mol/m³
Dye liquor density	ρ _f	1020	kg/m³
Dynamic viscosity of dye liquor	μ	3.551 × 10⁻⁴	Pa·s
Diffusion coefficient of dye in dye liquor	D	1.14 × 10⁻⁷	m²/s
External diameter of yarn bobbin	ϕ ₂	0.16	m
Yarn bobbin porosity	ɛ	0.627	–
Yarn bobbin radial permeability	κ ₁	1.97 × 10⁻⁹	m²
Yarn bobbin axial permeability	κ ₃	3.91 × 10⁻⁹	m²
Yarn diameter	d	2 × 10⁻⁴	m
Diffusion coefficient of dye on yarn	D _s	2.46 × 10⁻¹²	m²/s
Dye saturation value on yarn	S	2.146	mol/m³
Adsorption rate constant	k ₁	5.44 × 10⁻⁴	m³/(mol·s)
Desorption rate constant	k ₂	4.3 × 10⁻⁵	s⁻¹
Inlet pressure	P _in	293292	Pa
Outlet pressure	P _out	284393	Pa
Dyeing temperature	T	353.15	K
Molar mass of dye	M _r	331.32	g/mol
Gravitational acceleration	g	9.8	m/s²

Model calculation method in COMSOL

During the dyeing process, under the effect of a pressure difference between the inlet and outlet, the dye liquor will form a flow field in the dye field. The dyes diffuse and transfer to various areas with the flow of the dye liquor, forming a concentration field in the dye liquor. When the dye liquor reaches the internal pores of the yarn bobbin, the dyes in the dye liquor are adsorbed on the yarn in its vicinity, forming the variation in the CDY. In this process, both the CDS and the CDY are constantly influencing each other.

Many problems concerning the coupling of multi-physics fields can be solved using COMSOL.^34,35 So, for the solution of the bobbin dyeing, to describe the relationship between fluid velocity, pressure, the dyeing uptake concentration, and its changing process, Brinkman’s equation is used to solve equation (5), the dilute matter transfer module in porous media is used to solve equation (6), and partial differential equations of a general form are used to solve equation (8). These three equations are coupled and the entire fluid field is meshed. The mesh used is checked for mesh irrelevance and meets the accuracy requirements. A transient calculation method is used to calculate a series of results, from 0 to 4200 s.

At this time, the concentration of the dye liquor in the vicinity of the area decreases, thereby affecting the concentration field of the dye liquor. Figure 4 is a flow chart for dynamic multi-dimensional feature bobbin dyeing.

Result verification

Bobbin dyeing is carried out in a closed container, making it impossible to measure the CDY or the CDS in real time. To examine the accuracy of the dyeing model established in this paper, the numerical results will be compared with the data from the literature. The verification results are presented next.

The internal flow velocity of the yarn bobbin is first compared. The dye velocities calculated in this paper are compared with those extracted from a beam dyeing domain by Chiango et al..³⁶ Both the velocity and radius are purposely made dimensionless to achieve a normal comparison. The comparison results are shown in Figure 5(a), where u_max is the maximum flow velocity in the yarn area, r₂ is the outer radius of the yarn bobbin, and r₁ is the inner radius of the yarn bobbin.

In Figure 5(a), the comparison demonstrates a strong consistency between the calculated flow velocity results within the dyeing domain obtained in this study and those from Chiango et al.,³⁶ with a maximum difference of 11.96%. The velocity gradually decreases along the radial direction, and the rate of velocity change stabilizes as the outer surface of the yarn bobbin is approached.

The CDY is also compared with the study of Mancusi et al.,¹⁶ as shown in the Figure 5(b). The ratio of maximum to minimum CDY (DDF) varies with time. It can be seen that, after 1000 s, DDF decreases with the increase in time, and finally stabilizes near 1. This proves that the difference between the internal CDY and the external CDY decreases as the dyeing time progresses. It also indicates that the color difference on the yarn bobbin decreases as the dyeing time advances. This tends to agree with the trend discussed in the next section. However, the differences between the calculated results in this paper and the specific values in the references can be attributed to variations in the different concentration of the dye liquor, the different cycle speeds, and the different yarn dyeing parameters.

Results

Color differences produced in the dyeing process are mainly caused by the difference in the concentration of dye taken up, which is closely related to the internal flow field of the yarn bobbin. Therefore, to capture the changes of dyes on yarns and deduce the dyeing behavior inside the whole dyeing jar under specific working conditions, in this section, the flow field and dye concentration distribution inside the yarn bobbin will be analyzed.

Characteristics in dyeing field

According to the calculation conditions and parameters set previously, the flow within the jar is mainly oriented along the radial and axial directions inside the yarn bobbin. As the dyeing proceeds, both pressure and velocity within the jar stabilize. Figure 6(a) presents the pressure contour map and velocity cloud illustrating this stable state.

Figure 6(a) indicates that both the velocity and the pressure are the greatest at the position near the inlet along the central axis of the cylinder. They both gradually decrease after the dye liquor flows into the yarn bobbin. This phenomenon is due to the hindering effect of the yarn winding structure on liquid flow, causing changes in flow direction and a gradual reduction in the velocity and pressure of the dye liquor. Furthermore, Figure 6(a) demonstrates that the distribution of pressure contours is more concentrated at smaller yarn bobbin radii and becomes increasingly sparse as the radius increases. Additionally, notable pressure unevenness occurs along the axis at smaller or larger yarn bobbin radii. Moreover, a significant disparity is observed in the distribution of dye liquor velocity and pressure within the upper and lower parts of the yarn bobbin.

To quantify the differences in CDY, lines 1 to 6 and points 1 to 5 within the yarn bobbin are chosen along the yarn bobbin axis and radius direction, respectively. The coordinates of these lines or points are shown in Figure 6(b). The velocities extracted along different lines are shown in Figure 7.

As illustrated in Figure 7(a), the velocity distribution in the radial direction exhibits a consistent trend along lines 1 to 3: the dye liquor velocity decreases as the radius increases, with velocities ranging between 0.06 m/s and 0.20 m/s. The difference in dye liquor velocity at the same radius and different heights is minimal near the yarn bobbin and in the middle area of the yarn bobbin (radius, 30–68 mm). However, in the outer region of the yarn bobbin (radius, 68–80 mm), the velocity is significantly lower near the rounded corners than in other areas at the same radius.

As illustrated in Figure 7(b), the velocity in most areas of the yarn bobbin remains relatively stable across different heights at the same radius. However, only the regions near the upper surface of the uppermost yarn bobbin and the lower surface of the lowermost yarn bobbin exhibit more significant changes with increasing height (with the amplitude of change reaching 0.90 m/s). Furthermore, the disparity between the dye liquor velocity at the junction of two yarn bobbins and other regions at the same radius also increases as the radius increases.

Distribution of CDY

The CDY and its distribution directly reflect the amount of dye uptake by yarns, and the differences eventually lead to yarn color differences. Thus, it is particularly crucial to study the CDY.

Figure 8 shows cloud maps illustrating the distribution in the CDY at different dyeing times. During the early stages of dyeing, only a minimal amount of dye is adsorbed on the inside of bobbins and in the regions near the upper end cover interfaces. However, as dyeing progresses, the CDY throughout the whole dyeing field gradually increases, and the yarn bobbin exhibits uneven CDY along the radius and axial directions. Radial non-uniformity mainly appears along the radius at all heights. Axial unevenness is mainly noticeable near the rounded corners of the yarn bobbins and in the proximity of the upper end cover.

To further analyze the variation in the CDY, the CDY along line 1 over time was extracted, as illustrated in Figure 9(a), revealing a pattern of higher CDY on the inner side and lower CDY on the outer side along the radial direction. With the increase in dyeing time, the disparity reduces. This trend arises because, initially, the dye liquor predominantly enters from the inner side of the yarn bobbin, where its velocity is higher and the CDS is greater. Consequently, the dyeing rate in the inner region of the yarn bobbin is greater than that in the outer region, causing the inner part to be dyed first. As the dyeing process progresses, the dyeing rate in the inner region reduces, leading to a gradual reduction in the variation rate in dye concentration. Moreover, dyeing continues in the middle and outer regions at a larger dyeing rate than in the inner region. After 3000 s, the CDY at each point on line 1 hardly changes any more. It can be confirmed that dyeing equilibrium has been reached, and the ARD is almost equal to the DRD.

For a detailed analysis of the variation in CDY across different regions, six curves, from line 1 to line 6, are extracted at a dyeing time of 1000 s, as illustrated in Figure 9(b) and (c).

Figure 9(b) shows that, in the inner region of the yarn bobbins (radius, 30–68 mm), the CDY remains uniform across different heights at the same radius. However, in the vicinity of the outer region (radius, 68–80 mm), significant disparities in the CDY occur among different heights at the same radius, with the maximum difference reaching 0.9 mol/m³. As discussed previously, the velocity variation along lines 1 to 3 is insignificant in the central area and inner region (radius, 30–68 mm) of the yarn bobbin. At any given radius, the initiation time of dyeing and the CDS are similar across different heights, resulting in an insignificant difference in the CDY at the same radius. Conversely, a noticeable velocity disparity exists along lines 1 to 3 in the outside region of the yarn bobbins (radius, 68–80 mm). Near the outer part of the yarn bobbin (radius, 68–80 mm), the velocity differences along lines 1 to 3 are substantial. Regions with lower velocity exhibit thicker diffusion boundary layers, leading to delayed initiation of dyeing and a lower CDY.

Figure 9(c) shows that, in the small-radius region of the yarn bobbins, differences in the CDY are most pronounced near the upper end face. In the central region, variations occur in the CDY where the two yarn bobbins connect, but this area represents a smaller proportion. In the large-radius region, the area of disparity in the CDY is similar to that in the central region, but it constitutes a larger proportion (≈78.8%).

To analyze the variation in the CDY over time, the CDY at points 1 to 5 is extracted, as illustrated in Figure 10(a). The trends at all five points are nearly the same. In the initial dyeing period, the CDY remains constant at 0. Then, it increases rapidly, and finally stabilizes. During the initial dyeing period, the dye has to pass through the diffusion boundary layer to reach the yarn surface, where the CDY is 0. As the dye diffuses to the yarn interface, dye from the dye liquor begins to transfer onto yarns, driven by the difference between the CDS and the CDY. Thus, the CDY gradually increases. With prolonged dyeing duration, the CDY continues to increase, while the CDS progressively reduces, since no new dye is added in the mixing cylinder. Eventually, yarn adsorption reaches equilibrium. The CDY stabilizes at 2.11 mol/m³.

Figure 10(b) shows the dyeing rate over time at points 1 to 5. The dyeing rate of the yarns represents the rate of change in the CDY with time. In different locations, all points exhibit a dyeing rate of 0, then a sudden increase, and finally a constant decrease. When the dye reaches the junction of the yarn and the dye liquor but has not yet been adsorbed onto the yarn, the CDS is high while the CDY remains at 0. Dye in the dye liquor is then adsorbed onto the yarn, driven by the difference between the CDS and the CDY. Adsorption is instantaneous, occurring within a picosecond. The CDY steeply increases for some time afterwards, accompanied by an instantaneous rise in the ARD, leading to maximization of the dyeing rate and the emergence of a mutation point. As the dyeing time continues to increase, the CDS gradually decreases while the CDY increases, resulting in a decline in the ARD and a rise in the DRD. Ultimately, the dyeing rate is almost 0, presenting a dyeing equilibrium state.

Figure 10(a) highlights that there is also a difference in the time at which the dyeing of the yarn starts along the yarn bobbin radius, for the same height (points 1 to 3). This discrepancy arises from the higher dye liquor velocity close to the yarn bobbin inner area, resulting in a thinner boundary layer and a shorter diffusion time for the dye to reach the yarn surface. Additionally, there are discrepancies in the initial dyeing rate: the CDS is higher in the inner yarn bobbin, owing to the earlier onset of dyeing for these yarns. Hence, when yarns in the outer yarn bobbin start to be dyed, those in the inner yarn bobbin have already undergone a period of dyeing. This leads to a lower CDS in the outer yarn bobbin compared with the inner yarn bobbin. The ARD during the initial stage of dyeing is significantly lower on the outer yarn bobbins than on the inner side.

Considering that, in practical production, the outer yarn bobbins are prone to uneven dyeing, particularly at the edges and corners, the CDY and dyeing rate are extracted at three points (points 3 to 5) along the exterior of the yarn bobbin. These three points denote the middle, bottom, and top regions of the outer yarn bobbin, respectively. Figure 10(a) also illustrates that at the same moment, the CDY is the highest in the middle of the outer yarn bobbin, followed by the area below the outer yarn bobbin. It is the lowest in the region above the outer yarn bobbin. As depicted in Figure 10(b), the variation in dyeing rate (points 3 to 5) is the primary cause of the significant differences in the outer yarn bobbin, particularly near the rounded corners. As the dyeing process progresses, the disparity in CDY in the outermost part gradually reduces.

Once the dyeing concentration at a given point has remained constant for a sufficient period, it can be concluded that the point has reached equilibrium. This is represented by a specific time point, known as the balance point. By employing the aforementioned calculation, the relationship between velocity and balance point for points 1 to 5 is derived, as illustrated in Figure 11.

It is found that the balance point decreases as the flow rate of the dye liquor increases. When all areas of the yarn bobbin reach the balance point, the dyeing of the whole yarn bobbin reaches equilibrium. At this stage, the CDY and the CDS do not change with time at any point. The corresponding time is called the dyeing balance point. From this, it can be seen that the dyeing balance point of the whole yarn bobbin depends mainly on the position of the largest balance point, i.e., the position where the flow rate of dye liquor is the smallest. When the dyeing balance is achieved at this position, the dyeing equilibrium of the whole yarn bobbin is reached. In actual industrial production, the dyeing balance point plays a critical role in yarn dyeing. It helps operators manage dyeing time efficiently, ensuring that the dyeing machine operates as briefly as possible while still achieving uniform dyeing. This also can reduce electricity consumption and dyeing cost.

Changes in average CDS

As the dyeing continues, not only does the CDY change, but the CDS also changes. Figure 12 illustrates the behavior of the average CDS with time.

As shown in Figure 12, the average CDS remains constant, at 6.36 mol/m³, for a certain period before decreasing continuously. After approximately 3400 s, the average CDS stabilizes and no longer changes. This is due to the dyeing process: the yarn bobbin is initially immersed in a highly concentrated dyeing field. During the early stages of dyeing (0–215 s), the average CDS remains essentially constant because of the presence of the yarn diffusion boundary. As the dye liquor continues to circulate, dye reaches the interface between the yarns and dye liquor, causing the CDS to decrease. As dyeing progresses, the difference between CDS and CDY reduces, and the rate at which the CDS declines also reduces. When equilibrium is achieved, the CDS reaches a steady state and remains at 4.843 mol/m³ thereafter.

To analyze the effect of the difference between inlet and outlet pressure on the dyeing rate, the method of changing the inlet pressure while keeping the outlet pressure unchanged is adopted. The following results are obtained. When the pressure difference between the inlet and outlet is 8.899 kPa, the time required to stimulate the dyeing process is 125 s, and the time needed for dyeing to reach equilibrium is 2620 s; when the pressure difference between the inlet and outlet is increased to 14.899 kPa, the time required to activate the dyeing process is 98 s, and the time needed for dyeing to reach equilibrium is 2380 s; when the pressure difference between the inlet and outlet is decreased to 2.899 kPa, the time required to trigger the dyeing process is 215 s, and the time needed to reach dyeing equilibrium is 3300 s. Therefore, when the pressure difference between the inlet and outlet of the dyeing field increases (decreases) by 67.40%, the dyeing start-up time decreases by 21.60% (increases by 72%) and the dyeing equilibrium time decreases by 9.16% (increases by 27.99%). The higher the pressure difference between the inlet and outlet, the higher the dye liquor velocity inside the yarn bobbin and the faster the exchange of substances between the yarn and the dye liquor; thus, the more quickly the average CDS decreases.

Conclusions

To minimize water consumption and waste generation during yarn dyeing, a mathematical model of yarn dyeing is established in this paper for, it is believed, the first time, combining the principle of yarn dyeing and preventing yarn winding overlap. The method of finite-element simulation is subsequently employed to reproduce the change process in the CDS and CDY. Additionally, the influence of dyeing process parameters, such as dyeing time and the pressure difference between the inlet and outlet on the CDY and dyeing rate, is investigated. The following conclusions are thus drawn.

During the dyeing process, the liquid velocity decreases continuously with the increase in radius, at a given height, with a maximum decrease of 0.14 m/s. For a given radius, the difference between the dye velocity in the region close to the upper cover plate and the rest of the region is as much as 0.90 m/s.

When the dyeing time is 1000 s, the dyeing uptake concentration of the whole yarn bobbin area shows a large unevenness; when the dyeing is carried out to 3000 s, the CDY becomes relatively uniform along the radial direction; after 3300 s, the CDY across most of the area of the yarn bobbins stabilizes, at 2.11 mol/m³, and the dyeing of these areas has reached equilibrium.

The larger the pressure difference is between the inlet and outlet of the dyeing field, the faster are the dyeing start-up time and the equilibrium time for dyeing. When the pressure difference between the inlet and outlet of the dyeing field increases (decreases) by 67.40%, the dyeing start-up time decreases by 21.60% (increases by 72%) and the dyeing equilibrium time decreases by 9.16% (increases by 27.99%).

In conclusion, this study has good applicability to different dyeing machines and provides a solid foundation in the search for process and structural parameters for optimum leveling. However, there might be some limitations in this paper. Therefore, in future work, we intend to measure the dyeing parameters, then carry out dyeing experiments on the yarn bobbins, and finally utilize an effective method to analyze the color difference.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Shandong Provincial Natural Science Foundation (grant number ZR2020ZD22).

ORCID iDs

Bingke Yu

Zhiping Mao

Appendix

b _add

amount of dye molecules added by dosing device (mol)

b _in

amount of inflow of dye molecules at dyeing field inlet (mol)

b _out

amount of outflow of dye molecules at dyeing field outlet (mol)

real-time concentration of dye molecules in dye liquor (mol/m³)

c _in

concentration of dye liquor at inlet of dyeing field (mol/m³)

c _out

concentration of dye liquor at outlet of dyeing field (mol/m³)

c _s

real-time concentration of dye molecules adsorbed on yarns (mol/m³)

c ₀

initial concentration of dye molecules in dye liquor (mol/m³)

diffusion coefficient of dye molecules in dye liquor (m²/s)

D _s

diffusion coefficient of dye molecules on yarns (m²/s)

yarn diameter (m)

kinematic viscosity of dye liquor (m²/s)

winding ratio (–)

gravitational acceleration (m/s²)

equilibrium constant (m³/mol)

k ₁

adsorption rate constant (m³/(mol·s))

k ₂

desorption rate constant (s⁻¹)

minimum spacing between yarns (m)

number of foldback points (–)

M _r

molar mass of dye molecules (kg/mol)

rotational speed of yarn bobbin during winding (r/s)

pressure of dye liquor (Pa)

P _in

inlet pressure of dyeing field (Pa)

P _out

outlet pressure of dyeing field (Pa)

Q _in

dyeing field inlet flow rate of dye liquor (m³/s)

Q _out

dyeing field outlet flow rate of dye liquor (m³/s)

R_i

total adsorption rate (mol/(m³·s))

bobbin yarn radius (m)

r ₁

inner radius of yarn bobbin (m)

r ₂

outer radius of yarn bobbin (m)

dye saturation value on yarn (mol/m³)

dyeing temperature (K)

dyeing time (s)

t_k

time required to pass through diffusion boundary layer (s)

flow rate of dye liquor (m/s)

u _max

maximum flow velocity in yarn area (m/s)

winding rate of yarn bobbin (m/s)

height of yarn bobbin (m)

winding angle of yarn (°)

inertial drag coefficient (kg/m⁴)

yarn diffusion boundary layer thickness (m)

yarn porosity (–)

permeability of yarn bobbin (m²)

κ ₁

radial permeability of yarn bobbin (m²)

κ ₃

axial permeability of yarn bobbin (m²)

dynamic viscosity of dye liquor (kg/(m·s))

ρ _f

density of dye liquor (kg/m³)

ϕ ₁

inner diameter of yarn bobbin (m)

ϕ ₂

outer diameter of yarn bobbin (m)

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