3D network simulations of paper structure - Innventia.com

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PAPER PHYSICS et al. 2007; Switzer et al. 2004; Lindström, Uesaka 2008. Pulp fibers were modeled as multilink chains of rigid cylindrical segments with circular cross-section, immersed into Newtonian liquid. Fiber chains can bend and twist due to joints between segments. Different cases of liquid motion were considered, including simple drainage under gravity and external pressure, (see Miettinen et al. 2007; Switzer et al. 2004), and complex water movement in different sections of roll-blade formers, (Lindström, Uesaka 2008; Lindstrom et al., 2009). These models suffer some disadvantages such as the limitations of rigid segments which do not permit the same fiber bending, the output networks have low density and higher thickness. However, these methods can be used for generating pulp webs of low consistency, and thus can simulate forming situations corresponding to the wet web before the wet pressing section in the paper machine. The main goal of the present work is to develop a numerical approach to build a 3D artificial hand sheet and machine sheet fiber networks with prescribed thickness and apparent density, consisting of collapsed and non-collapsed fibers, fines and fillers. We can take a simulated structure made by using multilink chains of rigid cylindrical segments as fibers and abstract the 3D structure of the formed paper sheet. The abstraction is by using the center lines of the fibers and redecorating their thicknesses and widths to develop a fiber model with prismatic elements comprising the fibers. The fibers are also considered to be hollow to allow for the lumens and can be compressed in a wet pressing simulation to generate a ‘final’ structure. In the following, we will describe our method of generating the paper structure and show how it is applied to a random ‘handsheet’ structure and a realistic, simulated sheet formed using a twin wire roll former with blades. The structures are then analyzed for their mechanical properties (in this case, the elastic modulus). Fiber network Generation Hand Sheet Formation A set of objects is defined for the simulation, which includes objects of three types: fibers, fines or fillers. These are characterized by their geometrical parameters which include length, width and thickness, their distributions and type of cross-section i.e. collapsed or non-collapsed, as in Fig 1. Fiber type 1a has only one finite element in thickness direction. This type was usually used to describe fines and fillers. Fiber types 1b and 1c have arbitrary number of elements in all three directions (length, width, thickness). Non-collapsed fiber type 1d has always one element in the thickness of fiber wall and arbitrary number of elements in other two directions. It is possible to include curl and kink of fibers by suitable modifications of these elementary structures, although this was not implemented in the current simulation. The first step of the simulation is to generate an initial structure. Handsheet formation can be simulated by simply placing fibers (or objects) randomly in space (a) (c) (b) (d) Fig 1. Configurations of fibers used in the model Frequency 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 Hardwood Pulp 0,5 0,9 1,4 2,0 2,7 Fiber length, mm Fig 2. Examples of fiber length distributions for softwood and hardwood pulps according to their distribution in the furnish. An example of such distributions for hard wood and soft wood pulps are shown in Fig 2. The second step is location of centers of objects in Z direction. Each new object has Z coordinate taking into account all objects already generated and located below this particular one. All points of such line have the same Z-coordinate. The third step is the generation of finite element grid in each object according to type of object cross-section and its dimensions. The topology of finite elements (shape and local numbering of nodes) should correspond to finite element software used in the following steps. In our case this was the 8-node hexahedron solid element in the LS- DYNA Explicit program. Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 257
PAPER PHYSICS Fig 4. ZD view of simulated sample structure after forming in paper machine z max z z caliper Fig 3. An example of initial configuration of fiber network (all four fiber configurations and filler particles are presented) The fourth step is trimming out the part of finite element grid outside prescribed sample dimensions. A sample of the finite element network generated after the fourth step is shown in Fig 3. And the fifth, the last step, is the calculation of fiber deposition velocities. All points of each particular object have the same initial velocity in vertical direction. The velocity is calculated from the deposition height and deposition time. Machine Sheet Formation This method of fiber network generation is described in detail by Lindström, Uesaka, 2008. Fibers are represented as sets of rigid segments connected by joints with bend and torsion stiffness. Mass and moment of inertia of segments correspond to solid circular cylinders. Initially all fibers are generated randomly inside a prescribed volume, which is filled by a viscous incompressible fluid. The motion of the fluid is governed by the threedimensional Navier–Stokes equations. Equations of motion of fiber segments includes inertia forces, hydrodynamic drag forces from moving liquid and fiberto-fiber contact forces. The slurry is running through different sections of paper machine, where different boundary conditions are applied (Lindström, Uesaka, 2008; Lindstrom et al., 2009). The final low consistency pulp mat is used as initial input data for the network compression model. Each circular fiber is replaced by the collapsed or non-collapsed fiber, Fig 1, and is divided into a set of finite elements. A mechanical material model (elastic, elastic-plastic, visco-elastic-plastic) is assigned for each particular fiber. The wet fiber flexibility is a fiber parameter that has been investigated extensively in the past and can be used to determine a representative fiber modulus. Since wet fibers are substantially plastic, it is necessary to define plasticity parameters in addition to the elasticity. Fibers are anisotropic and their deformation is considerably complex. However, for an initial simulation such as this, we used a simple isotropic bilinear elastic-plastic model with the modulus determined from the wet fiber flexibility data in the literature [Paavilainen, Luner, 1986; Abitz et al., 1985]. An example of machine-generated network is shown in Fig 4. 258 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 t load t rest t unload Fig 5. Upper rigid plate movement history during mat compression Fiber network Compression The simulation of fiber mat compression was performed with the explicit finite element method, [see Zhong 1993 and Hallquist 2005]. We used LS-DYNA, a program for Nonlinear Dynamic Analysis of Structures in Three Dimensions, Version 971 (Hallquist 2005). The 8 node hexahedron elements with three displacements in each node were used to represent fiber network. The central finite difference scheme was used for the numerical integration of the equations of motion. Fiber-to-fiber interactions during simulation were calculated from balance of nodal inertia forces, external forces, internal forces and contact forces. All contact couples were found automatically during simulation. (This method of contact handling was made possible by using a keyword *CONTACT_AUTOMATIC_GENERAL). To obtain the fiber mat with prescribed thickness and density, the network was compressed between two rigid plates. The non-moving plate is located below the previously generated fiber network. The moving plate with prescribed motion in vertical direction (Fig 5) is initially located on the top of the mat. Fiber settling and compression are performed during a loading time , which was equal to 1-2 ms. A stabilization time is need to decrease dynamic effects in the compressed fiber mat which was generally set to 1 ms. An unloading time is also needed to remove the upper plate and to give the fiber mat possibility to expand in thickness direction due to internal stresses in fibers. The unloading time was 0.1-1.0 ms usually. The results of some 3D network simulations are presented in Fig 6-9. The initial fibers configuration from Fig 3 after compression is shown in Fig 6. A larger scale model with about 1500 fiber of the mixture of hardwood and softwood pulps is presented in Fig 7. Results of simulation of fiber network with high content of mineral filler (25% weight fraction) are shown in Fig 8 and 9. A layer created by fillers is observed in the pictures. The example of machine-made fiber network after compression is shown in Fig 10. The colors serve to differentiate the fibers in the network. t
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