Hydrodynamic modeling of e-textile fabric washing behavior by the Coupled Eulerian

Abstract

The paper examines the washing behavior of fabric by using the finite element method (FEM) along with the Coupled Eulerian–Lagrangian (CEL) approach. Many prototypes of e-textiles with different functions have been developed for various applications in laboratories worldwide, but only a limited number of products exist on the market. The washing process, even for mild wash cycles, damages mainly conductive yarns and electrical contacts on wearable fabrics. A hydrodynamic simulation method is proposed to investigate the mechanical response of fabric during a washing cycle, using the FEM and CEL approaches with the Abaqus finite element solver. The FEM is described with the following inputs: the fabric properties; the Mie–Grüneisen equation of state (EOS) for water; the ideal gas EOS for air; the geometry of the model; the drum spin data; and the boundary conditions. The movement of fabric inside the drum and reaction forces on the drum are utilized to verify the simulations. The fabric movements that are attributed to be the reason for damage in a conductive yarn showed a typical washing response. The frictional dissipation energy results show different regions depending on the motion and interaction of the components inside the drum. Also, the contact forces were determined. These forces can be input for future damage modeling studies. The findings of the study are expected to be used in development phases of reliable e-textile products with an extended life of service and readiness for the market.

Keywords

hydrodynamic simulation e-textiles washing modeling Coupled Eulerian–Lagrangian Method equation of state

Electronic textiles (e-textiles) or smart textiles are known as smart fabrics embedded with electronic components. All those devices are interconnected in a flexible motherboard using conductive yarns, mainly silver-plated polyamide multifilament threads, or tracks deposited on a textile substrate. E-textiles encompass more general concepts, including software, remote databases, data treatment algorithms, decision support systems, IoT (Internet of Things) concepts, and artificial intelligence aspects. Stoppa and Chiolerio¹ expressed typical applications of fabric sensing in biomedical applications, for example, the electrocardiogram (ECG), electroencephalography (EEG), and sensing temperature. Similarly, researchers maintained other usage areas of e-textiles, for example, power generation and storage,² antennas and radio frequency (RF) technology,^3–5 organic light-emitting diodes (OLEDs),⁶ and even accelerometers for sports.⁷

Virkki et al.⁵ claimed that wearable textiles should contain electronic components that are small, lightweight, conformal, reliable, and washable. Nowadays, most wearable e-textile products are made with detachable batteries and/or modules for cleaning or washing. However, as Kang⁸ stated, these electronic components undergo mechanical deformations during manufacturing, daily usage, and washing. To investigate the effect of environmental conditions, Bogan et al.⁹ studied the conductive woven subjected to various ranges of temperature and humidity conditions. They tested the resistivity of conductive yarns for relative humidity from 20RH to 90RH and temperatures from 30℃ to 55℃. Also, scholars evaluated the performance of RF and sensor integrated e-textile wear resistance.^10–12

The mechanical stress, corrosion, temperature, and chemical reactions (wetting, detergents) in the washing process cause harm to the conductive yarns and electrical contacts among the yarns and wearable devices. Among these parameters, mechanical stresses play a major role. The damage in the conductive yarns and thus on e-textiles during the movement of fabrics should be minimized by developing an appropriate washing cycle.

In e-textiles, damages to conductive threads (in the conductive layer) result in an increase of electrical resistance and connection failures of the modules. Thus, often after a few washing cycles, an e-textile becomes unreliable. E-textiles have to be washable repeatedly without losing their electrical conductivity or they will never be ready for the market. For this purpose, many researchers have studied the washing durability of e-textiles experimentally.^13–16 In addition, Tao et al.¹⁷ compared the washability of e-textiles encapsulated by thermoplastic polyurethane (TPU) and a barrier made of latex. Among these studies, mechanical forces applied to the e-textiles during the washing cycle were considered as the main parameter to yield damage to conductive yarns.

As mechanical stresses and fabric movements play a significant role in the washing cycle and are necessary for good washing results, it is essential to have a better understanding of fabric dynamics during washing in order to design wear-resistant and durable e-textiles. Most washing modeling studies focus on a specific aspect of a washing machine to calculate the consumption of energy, water, life cycle assessments, etc.^18–21 Lasic et al.¹⁸ conducted research aimed at modeling the washing process as a whole, but their study does not extend beyond multiple linear regression analysis. There are a few studies conducted on mechanical interactions inside a washing machine. Gorguluarslan et al.²² investigated the performance of the spider of a front-loading washing machine by finite element analysis and fatigue life analysis. They validated the simulation results with physical experimentation. Park and Wassgren¹⁹ used a simplified discrete element model, where spheres represent bundled fabrics to model fabric dynamics in a rotating horizontal drum. They calculated the dissipated energy from bundle contacts as a degree of cleaning.

Ward²¹ built a parametric model of a horizontal-axis (front-loading) washing machine and investigated the motion of the fabric. For each drum rotation, Ward²¹ defined three distinct motions: fabric pulled through the water; fabric lifted out of the water; and fabric impacts water, that is, the fabrics suddenly drop on the water, causing an impact. Ward²¹ modeled this hydrodynamic motion and derived geometric, kinematic, and dynamic similarity equations. However, the authors could not find any modeling of the washing process focusing on the forces applied to the fabric in the literature. The objective of this study is to construct a macro model to computationally determine the forces applied to the fabric during the washing, computationally. Thus, the main mechanism inducing wear in a washing process is modeled to help researchers develop reliable e-textile products with extended service life and the possibility of laundering. Simulation results were compared to the experimental study of Yun et al.²³

The macro-modeling of washing of e-textiles requires a comprehensive model that should consider rubbing, beating, flexing, sliding, and rotational forces with hydrodynamic actions included, temperature and chemical interactions, and most importantly, abrasion in the machine drum. These mechanical and chemical interactions occur at macro dimensions. On the other hand, damage to the silver coating is a micro-dimensional problem. To overcome this issue, we propose combining two separate analyses: hydrodynamic simulations in the macro-scale and damage modeling in the micro-scale. Figure 1 shows the relation between the presented model in this paper and a future study for the damage model. The contact forces, dissipation energy, and fabric motion that are determined from the hydrodynamic simulations in Ansys software could be used as the input values for future damage modeling studies.

Figure 1.

Flow diagram for modeling the durability of e-textiles.

In the present work, the washing behavior of an e-textile product was modeled by using hydrodynamic simulations. In the second section, the modeling techniques, including constitutive equations for the finite element model, fabric properties, equations of state (EOSs) for the water and air, the model geometry, and boundary conditions, were exhibited. The results of the washing simulation were verified by using movement analyses and comparing the force results with the analytical results in the third section. The results section described the verification process, the movement of the fabric along with the drum speed, frictional dissipation energy, and forces. The fourth section gives concluding remarks of the current work, or in other words, the macroscopic model was demonstrated. Lastly, in future work, a microscopic model, to determine damage to a conductive yarn, is explained.

Modeling methodology

A typical front-loading washing machine was taken into consideration to model the movement of the fabric during washing. The movement of the drum, including acceleration and deceleration periods, affects the forces on the fabric. The viscous nature of the water makes the movement much more complex, and thus estimation of the forces on the fabric becomes complicated. Researchers use three different approaches to model the behavior of liquids like water, where they interact with the solids. Fluid–structure interaction (FSI), Coupled Eulerian–Lagrangian (CEL), and Smoothed Particle Hydrodynamics (SPH) are the common finite element methods (FEMs) to model the interaction between fluids and solids. The FSI method is used by the co-execution of structure model, that is, static, dynamic-implicit, and dynamic-explicit with computational fluid dynamics (CFD) or the fluid model. In FSI simulation, the flow equations and the structural equations are solved by the CFD solver and the standard solver, respectively. However, this approach is not effective for problems that exhibit strong physics coupling, as in this study.²⁴ Furthermore, the SPH method is a part of the meshless method, where the nodes are referred to as particles.²⁵ Similarly, the CEL method is used to model large deformations where the mesh is fixed and the Eulerian material flows through the mesh.²⁶ Xiaoying et al.²⁷ compared the CEL and SPH methods during the analysis of a water storage tank of a Chinese nuclear reactor. According to them, the CEL method gives more accurate results considering the stress on the steel tank. Therefore, the CEL method was used in the FEM to define interactions between the drum, fabric, water, and air.

The present study was aimed at obtaining contact forces on the fabric, generated by the movement of the drum. Only one two-dimensional (2D) woven fabric was modeled for the washing simulation. The fabric was modeled as a continuous domain using shell elements, since the fabric thickness is negligible compared to its length and size. Flores and Oñate²⁸ compared the shell elements with the membrane elements. The shell elements consider the bending that is crucial to model the behavior of the fabric during the washing process. Drum movement was taken from the spin-time data of the experimental work of Stamminger et al.²⁹ The finite element model of the washer drum and fabric was analyzed by the CEL approach using Abaqus finite element software. The constitutive equations for the CEL approach, material properties for the fabric, EOSs for water and air, the geometry of the model, and details of the simulation are given below.

Constitutive equations

The Eulerian elements can bear large deformations without any elemental distortion in the mesh during simulation.³⁰ Therefore, water and air were introduced to the model as the Eulerian domain. On the other hand, the Lagrangian domain was used to describe the fabric in the FEM.

Eulerian constitutive equations are expressed using spatial time derivatives. In contrast, the mathematical description of the Lagrangian model uses material time derivatives. The relationship between material and spatial time derivatives is expressed as follows^31–33

\frac{D Φ}{Dt} - \frac{\partial Φ}{\partial t} - w_{i} . (\nabla Φ) = 0

(1)

where Φ is the coupled solution variable and w is the velocity of the material.

Further, the governing equations for fluid dynamics, the conservation of mass, momentum, and energy are respectively³³

\frac{D ρ}{Dt} + ρ \nabla . w_{i} = 0

(2)

ρ \frac{Dw}{Dt} - \nabla . σ_{ij} - ρ B = 0

(3)

\frac{DE}{Dt} - \nabla . (σ_{ij} . w_{i}) - ρ B . w_{i} = 0

(4)

where ρ is the density, σ_ij is the Cauchy stress tensor, B is the body force, and E is the total energy per unit volume.

Fabric material properties

In the model, three different material properties for fabric, water, and air were given as input to the Abaqus software. The drum was modeled as linear elastic with a Young’s modulus of 210,000, Poisson’s ratio of 0.3, and density of 7.8 g/cm⁴.

Yun et al.²³ presented a washing performance analysis conducted on different type of fabrics. They tracked the movement of these fabrics during the washing process to evaluate the washing efficiency.²³

The present simulation is performed by using a plain-weave cotton fabric having similar physical properties in both dry and wet states to the C3-Fabric represented in Yun et al.’s study.²³ The tracking record of the fabric was used to validate the hydrodynamic simulations.

In order to simulate wet conditions of the fabric, the mass per unit area of the fabric must be measured under wet conditions. After remaining in water for 1 hour, the mass/unit area became 422 g/m², according to the NF EN 12127 standard.³⁴ Also, the fabric has a wet thickness of 0.44 mm ± 0.02 mm, measured experimentally, according to the NF EN ISO 5084 standard.³⁵ The fabric density was calculated through the areal density of the fabric in the wet condition as 0.96 g/cm⁴ and this was given as the input.

We used a cantilever apparatus to determine the flexural properties of the fabric. The test was performed with a sample size of 250 mm length and 25 mm in width. In this test, the flat sample bends under its own weight and the characterizing length of the sample, called the cantilever length, corresponding to the length of the free part of the fabric when it touches the inclined surface at 41.5° relative to the horizontal plane. This fabric length is employed in formula (5) to identify the flexural rigidity per unit width of the fabric

G = \frac{(\frac{p \times l^{3}}{8})}{(\frac{tan θ}{cos θ / θ 22})}

(5)

where G is the flexural rigidity in terms of per unit width, θ is the angular deflection of the cantilever end (41.5o), p is the weight per unit area, and l is the length of the cantilever.

The measured cantilever length for the simulated fabric was about 35 mm, indicating a flexural rigidity of 0.023 N.mm. The obtained value is employed to identify the Young’s modulus of the shell element using Equation (6).³⁶ So, the Young’s modulus of the shell is determined based on flexural rigidity

G = \frac{E \times t^{3}}{12}

(6)

E = \frac{12 \times G}{t^{3}}

(7)

where E is the Young’s modulus and t is the thickness of the fabric. Therefore, using Equation (7), the Young’s modulus of the fabric was calculated as E = 3.26 MPa.

Equation of state

The EOS was utilized to model water and air. The Mie–Grüneisen EOS was used to model the behavior of water. The most common form of the Mie–Grüneisen EOS is linear energy³⁷

p - p_{H} = Γ \times ρ \times (E_{m} - E_{H})

(8)

where p_H is the Hugoniot pressure, E_m is the internal energy per unit mass, ρ is the current density, E_H is the Hugoniot specific energy, and Γ is the Grüniesen ratio (Equation (9))³⁷

Γ = Γ_{0} \times \frac{ρ_{0}}{ρ}

(9)

where Γ₀ is a material constant and ρ₀ is the reference density. In an actual washing process, the water will be not just water but rather a wash liquor consisting of water and detergent. It should be noted that the physical properties of the water are affected by the detergent and temperature. For simplicity in the simulations, the model was constructed with just room temperature water. The properties used as inputs for the FEM are exhibited in Table 1.

Table 1.

Mie–Grüneisen equation of state parameters for water³⁷

	ρ (kg/m³)	C₀ (m/s)	s	Γ₀
Water	1000	1490	1.79	1.65

Moreover, the air was modeled using the ideal gas equation. Hugoniot experimental data frequently represents the relationship between the shock waves and particle velocities. This relationship is usually written in the form

P + P_{A} = ρ \times R \times (T - T_{z})

(10)

where P_A is the ambient pressure, ρ is the density of air, R is the gas constant, T is the current temperature, and T_z is absolute zero.³⁹

Table 2 represents the ideal gas EOS parameters for air used in the simulations. The parameters for a temperature of 298 K were used in the model since the experiments were conducted at room temperature.

Table 2.

Ideal gas equation of state parameters for air³⁹

	ρ (g/cm³)	R (J/kgK)	P_A (kPa)	T_z (K)	T₀ (K)	C_v (J/kgK)
Air	1.293 × 10^–3	287	101.325	0	288.4	1003.5

Geometry of the FEM

A large front-loading washing machine with a capacity of 13.5 kg is modeled for the simulation. To compare the movement of fabric with Yun et al.’s work,²³ the geometry of the drum was selected according to the dimensions of a Samsung Electronics WW-HF135UV washing machine. The diameter of the drum is 518.2 mm, while the depth is 418.2 mm, with a water capacity of 6 liters. Figure 2 shows the drum geometry and drawing of the fin geometry.

Figure 2.

Drum and fin geometry used in the model.

The fin geometry in the drum was drawn based on the study of Yun et al.²³ Three fins were placed 120° apart in the drum.

The behavior of a 100% cotton fabric substrate was simulated in this study. Instead of complex geometry, only one simple geometry fabric was used in the simulations to avoid errors and the expense of computation time. Yun et al.'s study²³ considers only one 400 mm × 800 mm fabric; so that we can compare the results, the size of the fabric for the simulation was selected as 400 mm × 800 mm.

To define water and air regions, a dummy part was drafted and a discrete field of water attributed inside this part. Similarly, another discrete field was constructed using the regions of the Eulerian part outside of the dummy part for the air region. The dummy part was then excluded from the simulation. Further, a mesh sensitivity analysis was conducted to determine optimum mesh size for each component in the Verification of the simulations section. Figure 3 shows the assembly and mesh densities.

Figure 3.

Assembly and the mesh densities for the finite element method.

Simulation procedure

Abaqus software was used to simulate the washing process and to determine the forces applied to the fabric. Although the implicit solver could have reduced the processing time slightly, the CEL method requires the dynamic-explicit solver. Therefore, the analysis was conducted using the explicit solver. Cotton fabric properties were defined to demonstrate an e-textile product. It is well-known that fabric is composed of fibers assembled in long length yarns interlaced together. This multiscale structure makes the fabric discontinuous. However, for the sake of simplicity in the simulations, fabric was taken into consideration as a continuum domain.

Further, the thickness could be ignored against its two in-plane dimensions. Therefore, the fabric was modeled as a continuum 2D element. E-textile fabrics have low out-of-plane flexural rigidity, which can induce wrinkles or folds even at low stress. The membrane elements do not consider the bending behavior, so they do not permit one to simulate the wrinkling phenomena²⁸ and, thus, the fabric movement inside the drum. Considering the limitations in membrane elements in the present study, the fabric was modeled as a “shell.” Another particularity of the fabric structure is anisotropy induced by the presence of the yarns only in two main directions.

The fabric used in the simulations has a balanced structure, so orthotropic behavior could be considered. However, to simplify the present preliminary simulation with the governance of the out-of-plane bending response, the fabric shell element was considered as orthotropic with the same Young’s modulus and Poisson’s ratio in all directions. The Young’s modulus was obtained by the experimental cantilever test described in the Fabric material properties section since the fabric motion was attained by its bending behavior rather than the tensile response. The Poisson’s ratio of the fabric was selected as 0.3, according to the study of Sun et al.⁴¹ The fabric thickness was measured as 0.44 mm and this was assigned as the thickness of the shell element.

Figure 3 describes the assembly of the FEM used in the simulations. Initially, the fabric was located on the water. The fabric had neither displacement nor rotation restrictions throughout the analysis. On the other hand, the drum was rotated around its main axis during the washing cycle. The rotation speed, in other words, spin speed, varies with a predefined program associated with the fabric material type. Indeed, the variation of the spin speed with the cycle time generates three movement types for the fabric inside the drum: sliding, falling, and rotating, which control the washing efficiency.⁴² Hence, the displacement of the drum was restricted in all directions as well as the rotation, except on its main axis: the Z-axis. Only the angular velocity of the drum about the Z-axis, shown in Figure 3, was controlled during the analysis as a function of step time. The spin speed–time diagram presented in the experimental work of Stamminger et al.²⁹ was adapted to the simulation by adjusting the maximum spin speed to 46 rpm according to the experimental study of Yun et al.²³ The reason for setting the maximum spin speed to 46 rpm was to compare the fabric motion path during the simulation with that of the experimental observation. The applied spin speed–time diagram for the drum movement is presented in Figure 4. A reference point was designated to control the body motion of the drum. Rotational boundary conditions with the tabular amplitude values shown in Figure 4 were given as input.

Figure 4.

Spin speed applied to the drum as a function of step time.

As explained in the introduction section, the main motions of fabric during the washing cycle are beating, falling, and rubbing. These motions, along with the hydrodynamic actions, are the results of acceleration and deceleration of the drum. Thus, the study of Stamminger et al.²⁹ gave explicit data to set the acceleration. Further, no restrictions were described for the water inside the drum since the interaction between the surface of the drum inhibits leakage of water. Consequently, in the present simulation, the fabric and water were free inside the drum, and their movement was controlled by the gravitational, centrifugal, and drum-friction forces. The gravity load was applied to the whole model as 9.81 m/s².

The fabric was meshed using element S4R (reduced integration, hourglass control, and finite membrane strain with second-order accuracy). The elements in the thickness direction were introduced with the Simpson thickness integration rule and five integration points considering the study of Sari Sarraf et al.⁴³ Eulerian elements were used to specify water and air in the model as EC3D8R (an eight-node linear Eulerian brick, reduced integration, hourglass control). The drum was modeled as linear elastic with the following inputs: a Young’s modulus of 210,000 MPa and a Poisson’s ratio of 0.3.

General contact for the whole model, which can only be used in the explicit analyses, was utilized. The friction was defined both in tangential (penalty) and normal directions. Hes et al.⁴⁴ and Kothari and Gangal⁴⁵ claimed that the friction coefficient for the self-contact of fabric could be selected as 0.35 for 100% moisture regain to describe the wet conditions. The friction coefficient of the test fabric was selected as “0.35” by considering those studies.

Also, a mesh sensitivity analysis was conducted, starting from the element size of 50 mm to 5 mm. The findings were compared in the results section, and the mesh density can be seen in Figure 3.

The simulation was conducted through a workstation with the following configuration: Intel Xeon 6136 (2*CPU) with 48 threads, 128 GB Memory, 512 GB M2 SSD, Nvidia Quadro K6000 (VGA). The simulation was divided into 48 different domains and parallel computed by using 24 threads of each CPU.

Results and discussion

Verification of the simulations

The finite element model of the washing simulation, including water, air, drum, and fabric, must be verified. For this purpose, two different validation methods have been executed: comparing the movement of the fabric and comparing the analytically determined forces applied to the fabric.

Mesh sensitivity analysis

To validate the simulation, the movement of the fabric in the washing machine was compared with the results of the experimental study conducted by Yun et al.²³

Yun et al.²³ tracked the movement pattern of the fabric’s center of gravity during the washing to examine washing efficiency. The fabric movement was recorded with a high-speed camera, XS-4 (X-StreamTM, IDT Co., Ltd), and the path of the fabric was traced from recordings, using the outline-tracker functions of TEMA Motion (Image Systems Co. Ltd). Figure 5(a) shows the fabric movement diagram obtained by Yun et al.²³ Similarly, in the current study, the movement of the fabric’s center of gravity through the simulation was tracked.

Figure 5.

(a) Fabric center of gravity path during the washing cycle tracked using a high-speed camera, carried out by Yun et al.²²; (b) Test 1 finite element method (FEM) results; (c) Test 2 FEM results; d) Test 3 FEM results. Note: the drum rotation was counterclockwise.

It is commonly known that the mesh size affects the accuracy of the FEM results and the required computation time. Hence, a mesh sensitivity analysis was carried out to determine the optimal mesh size for each component. Five different mesh density scenarios were tested to reveal the optimal mesh size. Table 3 shows the mesh densities tested for each component.

Table 3.

Different mesh density scenarios tested in the finite element method

	Eulerian region (mm)	Drum (mm)	Fabric (mm)
Test 0	50	50	25
Test 1	40	40	15
Test 2	20	30	10
Test 3	10	20	5
Test 4	5	10	3

Figure 6 shows the effect of mesh size on the required computation time and stable time increment. It should also be noted that in the dynamic-explicit analysis, a smaller mesh size increases the processor time slightly. The decrease in the stable time increment has boosted the computation time, as expected.

Figure 6.

Computation time and stable time increment for different mesh densities.

The initial mesh size for the Eulerian region and drum was selected as 50 mm while it was 25 mm for the fabric. The mesh size of the drum was chosen for the stability of the contacts between Eulerian and fabric nodes. Test 0 resulted in an error due to the large element size of the Eulerian region. Therefore, the mesh size was decreased for each component in Test 1. However, in Test 1, an intersection was observed between the water region and the drum fins. Also, the initial comparison of the movements between the experimental study and Test 1 showed that the mesh size must be decreased. Similarly, the results of Test 2 showed a contrast when compared to the experimental results. Simulations were repeated with a decreased mesh size to compare the simulation outputs with the experimental study until reaching similar results. There was no significant change in the outputs after decreasing the mesh size for the Eulerian region to 10 mm, for the drum to 20 mm, and the fabric to 5 mm. Thus, the results of Test 4 were very similar to those of Test 3. The cost of processing time (Figure 6) was taken into consideration, and the scenario in Test 3 was utilized in the analyses. Then, the outputs were compared with the findings of Yun et al.²³ The simulation results for each test scenario and experimental results are exhibited in Figure 5.

In a 2D scatter plot, Test 3 showed a similar pattern for the movement of the fabric gravity center to that of the study of Yun et al.²³ According to Figure 5(d), the main motion of the fabric was rotation due to sticking to the drum wall. Also, the fabric slid, that is, kept the position at the fourth quarter of the drum’s circle [0, 250]–[0, −250] with drum rotation. Lastly, the fabric had a falling motion in the right-hand half of the drum’s circle. The path difference between Figures 5(a) and 5(d) could be attributed to the alteration in the acceleration. These motions were not seen exactly in Tests 1 and 2. In Test 1, the falling motion could be observed on the left-hand side of the drum. In Test 2, the falling motion was delayed where it appeared in the left-hand half of the cycle. Thus, the mesh refinement, especially for the Eulerian region, has a considerable influence on the fabric motion inside the drum. The mesh size used in Test 3 was implemented for the study.

Verification of the simulations by comparing applied forces on the fabric

During the washing cycle, the acceleration of the drum leads to the movement of the fabric inside the drum. These movements are generally described as three main motions: rotating, falling, and sliding. These motions generate forces on the fabric, as expected. Yun and Park⁴⁶ claimed that the balance of power could estimate fabric movements in the drum. The centrifugal force, frictional force, and gravitational force are the main forces. Figure 7 displays the forces applied on the drum during a washing cycle.

Figure 7.

Forces generated on the drum.

According to Figure 7, the force equilibrium is governed by the normal force (F_N) and tangential force (F_T) on the drum. Reaction forces (F_N and F_T) on the drum can be determined analytically by using the equations proposed by Yun and Park⁴⁶

F_{N} = (mr ω^{2}) + (mg cos θ)

(11)

F_{T} = | (mg sin θ) - (μ mr ω^{2} + μ mg cos θ) |

(12)

In Equations (11) and (12), m is the mass of the water and fabric: 6.044 kg; g is the gravitational acceleration: 9.81 m/s²; θ is the angle of rotation; r is the radius of the drum: 0.26 m; ω is the angular velocity of drum; and μ is the friction coefficient: 0.35.

The simulations were verified by comparing the analytical and computational results for the contact forces on the drum. The comparison of the forces given in Table 4 reveals that the computational outputs were very similar to those obtained analytically. The error percentage of the computational results did not exceed 5%.

Table 4.

Comparison of analytical and computational forces

			Analytical results [N]		Computational results [N]		Error [%]
	ω [rd/s]	θ [⁰]	F_N	F_T	F_N	F_T	F_N	F_T
t = 15 s	1.12	53.7	37.00	34.84	38.14	33.90	2.6	3.4
t = 40 s	2.87	28.1	64.75	5.27	67.73	5.19	1.1	3.7
t = 90 s	3.56	68.4	40.98	40.79	39.67	39.64	4.7	2.1
t = 150 s	3.97	20.6	79.32	6.90	75.98	7.13	3.6	4.8
t = 240 s	4.61	11.8	90.15	19.43	87.54	20.34	2.4	3.9

Tracking the fabric location inside the drum and comparison of the generated contact forces on the drum showed that the model was able to simulate the washing process with the proposed input values.

Movement of the fabric

The alteration in the spin speed of the drum leads to acceleration and deceleration. This behavior results in a movement both in the water and the fabric. Also, the effectiveness of washing is highly dependent on these movements. The interaction between the drum walls, fabric, and water not only affects the cleaning but also generates forces on the fabric.

The numerical simulation of the washing process allowed us to estimate the forces applied to the e-textile fabric during laundering. For a better understanding, the positions of water and rotation of the fabric inside the drum are illustrated in Figure 8. Initially, the fabric was set to a position on the water, as seen in Figure 8(a). After the initiation of the analysis, the fabric rolled over and slid into the water, as expected. At the low spin speeds, the fabric stuck to the drum’s internal surface, as demonstrated in Figures 8(b) and (c). Considering the drum spin speed data, represented in Figure 4, the fabric impacted the drum walls with the water (Figures 8(d) and (e)). The main reason behind these impacts was the relatively low spin speed of the drum, that is, the fabric dropped off from the highest position to the lowest position inside the drum. After reaching a certain value of rotational velocity (Figure 8(f)), the fabric rolled over on the fins and the drum walls. This rolling movement is attributed to the centrifugal force and frictional force between the fin and the drum surface and the fabric. When the step time reached 250 s (Figure 8(h)), a decrease in the rotational velocity of the drum led to a collision of the water and fabric with the drum walls. In the last stages of washing simulation, the speed caused the fabric to respond to sliding behavior, as in the initial stage (Figure 8(i)).

Figure 8.

Position of fabric and water during washing simulation.

Frictional dissipation energy

Figure 9 presents the frictional dissipation energy along with the spin speed, and it could be used to evaluate the wear ratio. The frictional dissipation energy considers friction between the fabric–air, air–drum, water–air, fabric–fabric, fabric–water, and fabric–drum. The velocity of the fabric remarkably affects the friction-related energy. According to Figure 9, the dissipation energy has three different regions. In the first region (0–100 s, roughly), the slope of the dissipation energy was highest. After roughly 100 s, the acceleration of the spin shows a decline and the dissipation energy was in the moderate ratio. Although the spin speed reached to its maximum value (roughly 210–300 s), the increase in the dissipation energy remained stable.

Figure 9.

Frictional dissipation energy of the fabric sheet with regard to time.

The characteristic of the three regions was investigated through the simulation results. It was determined from the simulations that the dominant motion within the step time 0–100 s was falling and sliding. In this region, the rotational velocity was not sufficient to generate centrifugal force to stick the fabric to the drum wall. Therefore, the fabric fell down before reaching the highest position inside the drum, and the fabric slid on the drum wall. The repeated action of falling and sliding in the first region is attributed to be the main reason for the highest slope of dissipated energy. The characteristic of the second region is mainly rotating due to high angular velocity, which generated a high centrifugal force. This force maintained the fabric on the drum wall without falling. Also, the sliding motion tends to disappear with increasing spin speed. The main characteristic of the second region is associated with a lower rate of increase of the accumulated dissipated energy between 100 and 210 s. In the final region, deceleration of the drum led to rotation and sliding, thus diminishing the rate of dissipated energy. In this region, sliding was the dominant motion with the deceleration of drum at the end of the study period.

Contact forces and friction

During the simulations, the fabric and water showed typical movements in the washing cycle, including falling, rubbing, and beating.

The fabric rolled over inside the drum with the low rotational speed or, in other words, spin speed. This action could be referred to as rubbing, considering the self-contact of the fabric. The increase in spin speed leads to the falling motion of the fabric. The centrifugal force and friction are strong enough to move the fabric to a higher potential, but this combination is not sufficient to turn it over with the low spin speed. A higher spin speed caused the fabric to repeat rotation as it was attached to the drum sidewall due to the centrifugal force.

The movement of fabric inside the drum showed a relative tangential displacement (sliding) between the surface of the fabric and the internal surface of the drum. Also, energy dissipation occurs due to the air–water, fabric–air, water–air, fabric–fabric, fabric–water, fabric–drum and water–drum interactions. Moreover, during the washing cycle, impacts were caused by the falling of fabric and water on the drum walls. Thus, the fabric deforms, and more of its surface came into contact with the air, water, and drum walls. The contacts between the fabric–drum–water–air led to frictional forces, as expected. The alteration of the magnitude of the normal contact forces and contact-related shear forces on the fabric with regard to time is presented in Figure 10.

Figure 10.

Contact forces on the fabric with regard to spin speed.

Standard abrasion tests for e-textile products are usually carried out under a stable load. However, the washing process generates dynamic loading on the fabric rather than the stable loads in abrasion tests. To evaluate and compare the force outputs of the FEM, a statistical approach should be used.

For this purpose, the contact forces were extracted from 10 different nodal points, and an average value was used. A statistical approach was used to determine average contact forces for a given confidence level

x = \bar{x} \pm z^{*} \frac{s}{\sqrt{n}}

(13)

In Equation (13), x is the range of samples for a given confidence interval, $\bar{x}$ is the average value of the samples, z* is the confidence level, which is 1.96 for 95%, s is the standard deviation of the samples, and n is the number of the samples.

During the simulation, the normal contact force was calculated using Equation (13) as $37.06 \pm 2.1$ N, with a 95% confidence level. Higher rates of spin speed make a shift in contact force not only in the normal direction but also in the shear. Also, the shear contact force was calculated as $14.26 \pm 0.8$ N, with a 95% confidence level.

According to Figure 10, contact-related shear forces are lower than normal forces. The friction forces and shear forces permit one to evaluate the wear. These values could be used as input for future studies of wear calculations.

It should be expected that increasing contact forces will result in a faster detachment of metallization or abrasion on the conductive yarns. Zaman et al.⁴⁷ studied the abrasion of e-textiles by using Martindale tests. According to their results, linear resistance on the conductive yarns increases linearly with an increasing number of cycles of abrasion testing. According to the study of Zaman et al.,⁴⁸ the increase in resistivity of a conductive yarn after 1000 cycles in Martindale tests could be matched after eight or nine silk washing cycles. Similarly, Rotzler et al.⁴⁹ reported a partial loss of Ag coating in the contacting area after 10 washing cycles. This phenomenon could be explained by the high force values during washing. Also, the force values obtained by hydrodynamic simulations would be used as input values for the wear analysis. The values given in Figure 10 could be used as input values for the Martindale abrasion tests. According to the experiments and/or wear simulations, the washing cycle, especially acceleration and deceleration values, could be re-considered.

Concluding remarks

The washing behavior of fabric was simulated by using Abaqus software. The CEL method was utilized for the hydrodynamic simulations, and the following conclusions were drawn.

A simulation methodology for the hydrodynamic simulations of the washing machine was proposed by applying the CEL method. The methodology described the fabric material properties, the EOSs for water and air, and the boundary conditions.

The proposed model was verified by using the experimental data of Yun et al.²³ Also, the model was sensitive to mesh density. Increasing the mesh density yielded more accurate results. The validation of the model by the motion and analytical forces showed that the model was able to simulate the washing process using the FEM with the proposed input values.

The movement of the fabric in the drum successfully showed typical behavior of a washing cycle that consists of falling, rubbing, and beating. It should be noted that the parameters, that is, the spin speed data, the geometry of the fins, the size of the drum, the number of fabrics, the fabric size, and the amount of water, will affect the fabric’s movement path during the washing cycle.

During hydrodynamic simulations, the normal force and shear forces acting on the fabric were determined computationally. These values could be used as input data for both future damage modeling studies and also wet abrasion tests in laboratories.

Future work

In this study, hydrodynamic simulations were carried out to determine mechanical forces and stresses on fabric. The work would be improved with experimental study data, that is, obtaining the fabric movements and drum spin data using high-speed cameras and accelerometers will increase the validity of the simulations and also provide more detailed and realistic spin speed data. The effect of the fabric size, drum geometry, drum fin geometry, and amount of water added can also be studied in future analyses.

This study should be categorized as macroscopic modeling of a washing process. As future work, a microscopic damage model will be developed to obtain the wear on the conductive yarns. The stress values and contact forces will be used as input values for the modeling study of damage analyses. The damage model may include yarn/ply geometry, experimentally obtained adhesion strength of the conductive layer, mechanical properties of the coating, and the Martindale movements. This complex damage model, along with the hydrodynamic simulations, will be used for the wear modeling of e-textiles. The proposed hybrid model will permit engineers/researchers to estimate wear on e-textiles without additional experiments, which will allow the production of sustainable and stable e-textiles.

Footnotes

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Marie Skłodowska-Curie Actions (Grant No. 644268).

ORCID iDs

T Tetik

RA Yildiz

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Hydrodynamic modeling of e-textile fabric washing behavior by the Coupled Eulerian–Lagrangian method

Abstract

Keywords

Modeling methodology

Constitutive equations

Fabric material properties

Equation of state

Geometry of the FEM

Simulation procedure

Results and discussion

Verification of the simulations

Mesh sensitivity analysis

Verification of the simulations by comparing applied forces on the fabric

Movement of the fabric

Frictional dissipation energy

Contact forces and friction

Concluding remarks

Future work

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References