Flow Modeling for Vacuum Pressure-Based Handling of Porous Electrodes of Lithium-Ion Batteries

Abstract

In lithium-ion battery (LIB) production, limp electrodes are handled gently by vacuum-pressure based handling and transport systems, which generate a fluid flow that propagates through the porous electrode coating during handling. To investigate the limits and material-damaging behavior of vacuum pressure-based handling, it is required to understand how process parameters and electrode qualities affect fluid flow. Questions on how fluid flow reduces electrode quality are insufficiently addressed or modeled. Modeling the electrode and handling system interaction requires knowledge of the effective surface geometry and the volumetric flow rate caused by the pressure difference. In this article, flow through porous electrode coatings during handling is modeled. Experiments demonstrate a flow behavior according to the generalized Darcy’s law. Thus, using Darcy’s law, modeling fluid flow through the electrode improves the exploration of the limits and design of vacuum pressure-based handling and transport of electrodes in LIB production.

1 Introduction

It is cost-intensive to optimize the production of lithium-ion batteries (LIB) through time-consuming experiments. To increase development efficiency, modeling methods are used that provide adequate results for a given quality. This article further increases development efficiency by simplified modeling.

LIBs are composed of electrochemical cells consisting of cathodes and anodes, both called electrodes, with or without separators and any electrolyte. The cell assembly produces the electrode-separator compound (ESC), a stack, a Z-fold, or a roll. Cell production includes handling and transport of the electrodes between their manufacturing and sealing of the ESC. Electrode and LIB quality is affected by handling and transport parameters. Vacuum suction-based adhesion is a gentle way to handle and transport electrodes fast. The background of this article is to model the mechanical stress and damage that occur during handling and transport in the electrode. This article aims to model the interrelationships between the electrode active material (AM) properties, i.e., porosity, and vacuum suction-based adhesion parameters, i.e., pressure and volumetric flow rate during contact, to enable local stress and damage modeling of the AM.

The following describes the structure of the article. This section begins with a brief overview of the field of study. Section 2 provides fundamentals of suction-based handling and transport in cell assembly. It summarizes the modeling of electrode handling and transport. It motivates a flow model for suction-based adhesion of electrodes. Section 3 illustrates the physical problem and presents modeling based on Darcy’s law for vacuum suction cups and effective vacuum surfaces. In Sect. 4, several electrodes are experimentally examined to evaluate the model. Finally, Sect. 5 concludes how the model enhances the design of handling and transport solutions for LIB assembly.

2 Scope and Motivation of Fluid-Dynamic Electrode Model

Electrodes must be gripped securely, fixed reliably, and not damaged during handling and transport [1]. Handling and transport can be carried out through adhesion using vacuum suction at the material and application-specific limitations. Compelling examples in industry and research are introduced in the following.

Adhesion is modeled with uniform pressure distribution, and flow through the electrode is neglected to account for loading in widely used modeling methods. Modeling methods for local damage modeling that would benefit from improved flow modeling are considered below.

Section 2.1 introduces applications for vacuum suction-based handling and transport of electrodes. Section 2.2 analyses methods for modeling stress during electrode transport. Section 2.3 motivates a modeling approach for adhesion during electrode transport to increase development and commissioning efficiency.

2.1 Relevance of Vacuum Suction-Based Handling and Transport of Electrodes in Battery Production

In cell assembly, vacuum suction cups and area grippers are means for adhesion during handling [2]. These grippers adhere the electrode force-locked by a negative pressure difference between the inner and the ambient pressure. This gripping principle is popular for handling air-impermeable materials. However, electrode surfaces are air-permeable, which this article models and demonstrates by experiment. In the following, applications for electrode handling and transport are introduced and illustrated as per Fig. 1.

Fig. 1

Vacuum suction-based handling and transport systems for LIB electrodes

Vacuum suction cups are relatively inexpensive and stand out with high accuracy. The downsides include an increased risk of AM abrasion, electrode sheet absorption, and marks. For vacuum suction cups, the resulting flow through AM is modeled and experimentally evaluated hereafter.

Vacuum effective surfaces inherit multiple openings at low pressure to distribute the load resulting from the motion of electrodes. In the following example applications like vacuum area grippers, vacuum-deflection rolls, vacuum draw-off rolls, and vacuum-conveyor belts are described.

Vacuum area grippers have a high lateral force absorption and a high deposition accuracy. They reduce the risk of damage by distributing the gripping force across the entire electrode [2]. The model presented in the article covers the flow and pressure resulting from the multiple openings of these grippers.

Vacuum draw-off rolls were used to separate electrodes from a pile in the projects KontiBAT and HoLiB at the research group of the authors. An effective vacuum area in the roll (adjustable with suction insert) provides adhesion to orient and accelerate electrodes [3, 4]. This article models the flow resulting from the suction inserts.

Vacuum deflection rolls implemented in the KontiBAT project adhere, deflect and guide the electrodes while maintaining a constant material velocity. Vacuum-deflection rolls have also been used to ensure constant process velocity of continuous web-based electrodes [5]. The model developed in this article is applicable to flow caused by vacuum deflection rolls adhesion area.

Vacuum conveyor belts have been used to transport electrodes [6]. During transport, a lower pressure under the perforated belt ensures the fixation of the electrodes. The model developed in this article applies to vacuum conveyor belts.

In summary, vacuum suction is used in various applications to adhere and guide electrodes in space. The following discusses how the interaction of the applications and electrodes is modeled.

2.2 Earlier Work in Modeling Handling and Transport of Electrodes

In addition to modeling the kinematics of the electrodes, there are approaches capable of modeling effects on the electrode material to derive measures for the design of handling processes. Some approaches are subsequently evaluated.

Finite element method (FEM) has been used for the characterization of mechanical stresses, and their occurrence during handling processes on the electrode surfaces [3, 4]. FEM relies on continuous macroscopic models. The external loads on the electrodes are modeled as uniform surface stresses to represent, e.g., the suction pressure [3]. FEM cannot model the load from volumetric flow through the electrodes, which can be done with the model in this article.

Computational fluid dynamics (CFD)-FEM simulations are used to map the movement of the electrode foils in air-filled space and to model the use of different operating materials. This is particularly suitable for mapping macroscopic processes and determining the effects of forces on the contact points of the electrodes [4]. Modeling the volume flow through the electrode material at the contact point has not been done with CFD-FEM to the authors’ knowledge. This article models the volumetric flow through the electrode AM.

Discrete element method (DEM) models aimed at reproduction of mechanical testing like nanoindentation and on the reproduction of mechanical behavior due contact with a handling system [3]. There is no modeling of mechanical stresses on the electrode at the microscopic level from vacuum suction-based handling and transport with DEM. This article’s modeling enables it.

In summary, none of the existing approaches modeled flow through the electrodes to the authors’ knowledge. If the flow through the electrode material and its effects can be better modeled, the attempts to parameterize the systems could be reduced. This reduction serves the goal of development efficiency and the scope of the article. As a result, the focus of this article is on modeling the flow through the electrode material.

2.3 Discussion

Research aims to increase electrode dimensions for the application in e-mobility. However, this increases the stress at the AM that interacts with the handling and transport system during cell assembly. The industry looks for AM that is easy to handle and has a high energy density because the handling parameters play a decisive role in the competitiveness of production. In the development of new materials, only a few handling properties (e.g., electrode strengths) are taken into account since other interrelationships are missing.

During ramp-up or changeover of production, the equipment is parameterized to electrodes, the speed is increased, the resulting quality is validated, and then the production speed is further increased. In addition, the time the equipment runs continuously is increased to identify long-term effects on the materials. These processes are time-consuming and inefficient.

Knowledge of the interrelationship between the handling parameters and the effects on electrode quality, or the ability to estimate them, would save many resources during development and commissioning. Up to now, there is a lack of investigation of the interrelationship between electrode properties, such as porosity, and process parameters, such as pressure difference and volumetric flow rate of typical vacuum suction-based handling and transport operations in LIB production. This article is the first step to model the effects of macroscopic handling parameters on the electrode microstructure’s quality. The approach begins in modeling the adhesion at the macroscale, to derive flow properties from there, to model the effects on the morphology of the electrodes at the microscale.

3 A Model for Vacuum Suction Flow Through Electrodes

A model for vacuum suction flow through electrodes based on the generalized form of Darcy’s law is presented in this chapter. The prerequisites and principles for the flow description in porous media are described in the following subsections. Section 3.1 models the flow through electrodes for a vacuum suction cup. Section 3.2 models the flow through electrodes for effective vacuum surfaces.

3.1 A Flow Model for Vacuum Suction Cup Gripper

One can use generalized Darcy’s law to model the flow, i.e., the pressure and velocity distribution through the electrode from a vacuum suction cup gripper. Darcy’s law relates the volumetric flow rate Q with the pressure difference $\Delta p$ over the porosity $\phi$ and permeability K of porous media [7]. One assumption is made: the flow through the electrode is assumed to be two-dimensional, with a small channel height h compared to the inner radius $r_i$, $h<<r_i$, creating a radial plane flow channel as per Fig. 2.

Fig. 2

Flow channel (left), parameter study of pressure and velocity (middle), determination relation for electrode permeability (right) for vacuum suction cups

The radial flow channel is characterized by the boundary conditions for pressure $p_i$, $p_0$ at the vacuum suction cup’s outer and inner radius $r_i$, $r_0$. The integration of generalized Darcy’s law and identification of the pressure difference $\Delta p=p_0-p_i$ yields pressure distribution through the electrode

$$\begin{aligned} p(r)=p_i+\frac{\Delta p}{\log (r_0/r_i)}\log (r/r_i). \end{aligned}$$

(1)

For a given pressure difference, (1) yields a model of the pressure through the electrode, as done for three different vacuum suction cups in Fig. 2. It can be seen that the pressure increase varies slightly with the geometry of the vacuum suction cups. From Eq. (1) and generalized Darcy’s law, one gains the radial velocity

$$\begin{aligned} u_r(r)= -\frac{K}{\phi \cdot \eta \cdot r}\frac{\Delta p}{\log (r_0/r_i)}. \end{aligned}$$

(2)

A parameter study of (2) for different vacuum suction cups (see Fig. 2) illustrates velocity of the flow. The highest flow velocity is at $r_i$, the lowest at $r_0$. Also, one can see that small vacuum suction cups have a higher average local radial velocity than bigger ones. For evaluation of the model, one can measure the velocity resulting from the pressure difference. Since the volumetric flow rate Q, is more comfortable to measure; it is handy to integrate Eq. (2) to

$$\begin{aligned} Q =\frac{2\cdot \pi \cdot K \cdot h}{\eta } \frac{\Delta p}{\log (r_0/r_i)}. \end{aligned}$$

(3)

3.2 A Flow Model for Vacuum Effective Surfaces

The flow model through electrodes for effective vacuum surfaces uses potential analysis of flow. Potential flow models can be applied to viscous flows between closely spaced plates, which applies to vacuum adhering of electrodes. Moreover, a constant fluid density $\rho _\textrm{F}$ is assumed. The electrode plane is understood as a complex numerical plane with $z=x+\textrm{i}\,y\in \mathbb {C}$ (as illustrated in Fig. 3). In the plane, the potential is represented as a real part $\varPhi$ of a holomorphic function $f(z)=\varPhi (x,y)+\textrm{i}\,\varPsi (x,y)$, the imaginary part $\varPsi$ its resulting flow direction. Where real and imaginary parts of the complex velocity potential satisfy the Laplace equation in the plane.

Fig. 3

Illustration of flow channel (left), parameter study of pressure and velocity through electrode from a vacuum effective surface with 30 circular openings (right)

From f, assuming a homogeneous permeability K of the electrode, one gets a relation for the pressure distribution of multiple openings $p_n(z_n=x_n+iy_n)$ for any vacuum effective surfaces with circular suction areas. Each opening into which is flowing a quantity $Q_{s,n}$, fluid per unit AM thickness per unit time, contributes to the pressure distribution [7]. From these assumptions one can determine the velocity distribution in x, y direction $u_{x},u_{y}$ as well as in its average value u in flow direction.

$$\begin{aligned} u = -\frac{1}{\phi \cdot 2\pi } \sum _n Q_{s,n}\frac{1}{|z-z_n|} \end{aligned}$$

(4)

If all $Q_{s,n}$ are known Eq. (4) allows to model u, with porosity $\phi$, through the electrode AM. From a practical point of view, it is interesting to model the fluxes’ values $Q_{s,n}$ with generalized Darcy’s law and solve a linear system of equations for known opening pressures $p_n$. Since $\Delta p$ is a design parameter of vacuum handling and transport system, and the separate openings are often connected to the same vacuum-pressure reservoir, the openings are considered to have the pressure $p_i$.

For the pressure model, it is convenient to introduce a Green’s function G. Which is defined as a solution of Laplace’s equation, symmetrical in two points $(x,y),(x', y')$ (sometimes called mirror charges), possessing a logarithmic singularity when $(x,y)=(x', y')$ and vanishing when (x, y) is a point on the boundary $\partial S$ of the region in question [7]. When G is found, the pressure distribution for one circular vacuum-pressure region on a rectangular region can be calculated as

$$\begin{aligned} p(x, y) = -\frac{1}{2\pi }\int _{\gamma }p_b(x',y')\frac{\partial G}{\partial n'}(x,y,x',y')ds. \end{aligned}$$

(5)

In Eq. (5), $p_b$ is the value of p at the region’s boundary. The line integral elements are denoted by ds, $n'$ is the exterior normal to ds, and the integral extends over the whole boundary $\partial S$. Since Laplace equations allow superposition of their solutions due to their linearity, one can solve Eq. (5) for multiple circular openings of an effective vacuum surface. For that, one creates a linear combination of all pressure distributions $p_{all}=\sum c_j \cdot p_j(x,y)$ and adjusts the constants of each term according to the inner and ambient pressure at the boundaries. With the resulting pressure distribution $p_{all}$, one can then derive a formulation for the respective velocity distribution $u_{all}$ over the surface. One can model the velocity values with a known permeability K of the electrode material.

An example for $p_{all}$ and $u_{all}/K$ has been calculated for 30 circular areas of suction, which is illustrated in Fig. 3 in a boundary of 5 to 6 cm, representing the surface of a suction insert, that could be used in a draw-off roll similar to that in KontiBAT or HoLiB [3, 4]. The pressure near the 30 suction areas with diameters of 3.5 mm is almost as big as the assumed inner pressure and increases faster with a shorter distance to the boundary.

4 Evaluation of the Model

An experiment is conducted to evaluate that vacuum-based handling of porous electrode AM follows generalized Darcy’s law. In addition, the measured date gain permeabilities of the reference electrodes and tune the presented flow model. Section 4.1 introduces the experimental setup and measures to reduce recorded data. Section 4.2 discusses the results of the experiment.

4.1 Experimental Setup and Data Reduction

Within the scope of the investigation of the influence of the gripping parameters, pressure difference $\Delta p$, suction surface geometry $r_i$, $r_0$ on the surface quality were examined. In the experiment, the proposed models were examined according to Eq. (3). For this purpose, samples of the reference electrodes were placed on the vacuum suction cup under different $\Delta p$ while measuring the volumetric flow rate Q. For three different vacuum suction cups, a $\Delta p$ of $\approx 30$ mbar and $\approx 200$ mbar were chosen, according to the resolvable range of the following sensory.

The thermal flow sensor Festo SFAH measures the volumetric flow rate Q. The differential pressure sensor module Beckhoff AEM3712 detects the pressure difference $\Delta p$ between the pressure of the fluid $p_i$, and the ambient pressure $p_0$.

Four reference anodes and four reference cathodes were cut into six squared samples, with edge length $2\cdot r_0$ of the vacuum suction cup. Each sample was placed on a vacuum suction cup a the test rig. A pressure difference was applied, and the volumetric flow rate was recorded. Afterward, the samples’ and current collectors’ thickness were measured with the micrometer screw gauge. The measured thicknesses were used to determine the thickness of the electrode material h, necessary for the proposed modeling (Eq. (3)) and sketched in Fig. 2.

At the beginning of the suction, the vacuum suction cup’s available volume of the movable circular bellow is emptied, as shown by volumetric flow rate measurements (illustrated for a cathode sample in Fig. 4). With the identification of the asymptotic volumetric flow rate ($\dot{Q} \approx 0$), the average volumetric flow through the electrode is determined. Since $\Delta p$ in the regions of $\dot{Q} \approx 0$ is considered to be almost constant, the pressure difference $\Delta p(\dot{Q} \approx 0)$ is also averaged, and the standard deviation is formed.

Fig. 4

Pressure difference and volumetric flow rate over time (left), volumetric flow rate over pressure difference of reference anode (right)

Subsequently, the averaged asymptotic volumetric flow rates are plotted over the averaged pressure difference as per Fig. 4. A linear least-squares regression is performed for each vacuum suction cup. The average dynamic viscosity on the measurement day was $18.2 \pm 0.1\cdot 10^{-6}\,\text {Ns}/\text {m}^2$. The size of the electrode permeability K is calculated from the gradient of the fits, as per Eq. (3) In that case, the cathode’s permeability is $K\approx 1773\pm 230\,\text {mD}$ and for the anode $K\approx 2320\pm 613\,\text {mD}$, neglecting the values for the ESS-20 vacuum suction cup.

4.2 Results and Discussion

In an experiment, three vacuum suction cups with different geometries were applied at different pressure differences to electrode samples while measuring the volumetric flow rate simultaneously. The calculated regressions of the measured volumetric flow over the pressure difference (as shown for the anode as per Fig. 4) increase with the pressure difference. It is noticeable that, contrary to the expected course of the permeability determination curve (Fig. 2), the measured curve of the smallest vacuum suction cup (ESS-20) is significantly steeper. This is attributed to the discontinuities on the vacuum suction cup’s contact surface, which increase the distance to the contact surface. The slopes of VASB-40 and VASB-55 are positioned relative to each other as expected as per Eq. (2). The asymptotic volumetric flow rates are often above the regression curve in the low-pressure range. This may be related to the differential pressure sensor’s low measuring accuracy at low measured values.

The characteristic regression curves support the modeling approach over Darcy’s law. As the sensors are comparably inexpensive and part of industrial practice, they are measures to determine the permeabilities of electrodes to model the flow for various handling geometries.

With the model, one can determine K of a reference electrode from relatively simple geometry, e.g., a vacuum suction cup, based on Eq. (3). From there, using the approach for modeling vacuum effective surfaces in Sect. 3.2, one can estimate the average local velocities and pressures in similar electrodes resulting from vacuum effective surface geometry, like a surface area gripper, a deflection roll, a conveyor belt, or a draw-off roll.

5 Conclusion

The authors began manual development and validation to improve cell assembly processes. In order to identify, e.g., handling limits, parameters such as pressure difference were varied, and their impact on the electrode surface and LIB quality was investigated. Identification of sources of damage to electrodes resulting from vacuum-effective surfaces is laborious and expensive. Modeling approaches offer the possibility of reducing the time for experiments and increasing development efficiency.

For numerical modeling of damages from the interaction of electrode and handling and transport system, knowledge of the effective surface’s geometry and the volumetric flow rate resulting from the applied pressure difference are required.

In this article, a pressure and velocity distribution model for the flow through electrodes during handling and transport was developed for vacuum suction cups and vacuum-area effective surfaces. It models pressure and velocity based on the porosity and permeability of the electrode, the pressure difference, and the handling system’s geometry. The models in this article enable the creation of a simulation model to represent the interaction of electrode materials and fluids.

The presented model improves the development of electrode handling and transport processes at several levels. The model maps local effects on the electrode from the vacuum effective surface geometry. Thus, the model allows identifying critical areas of the electrodes for effect characterization (e.g., electromagnetic or electrochemical), which cuts non-valuable experiments. In addition, the model can be used to derive the effects of handling and transport on electrodes via measurements of porosity and permeability as early as the design of the electrodes in the laboratory. In addition, the model can be used to check whether a leakage from a non-optimal contact situation is present during handling and transport.

Our experience shows that modeling local stress and damage of the AM from handling and transport is possible. The presented approach allows to model the local aerodynamic and adhesion load to the AM particles in conjunction with other techniques, but the process must be optimized. The local stress and damage modeling remain to be discussed as it is beyond the scope of this article.