Research Papers

System Dynamics of the Open-Draw With Web Adhesion: Particle Approach

[+] Author and Article Information
Sverker Edvardsson1

Department of Natural Sciences, Engineering and Mathematics, FSCN, Mid Sweden University, S-851 70 Sundsvall, Swedensverker.edvardsson@miun.se

Tetsu Uesaka2

Department of Natural Sciences, Engineering and Mathematics, FSCN, Mid Sweden University, S-851 70 Sundsvall, Swedentetsu.uesaka@fpinnovations.ca

A computer C -code is available on request to sverker.edvardsson@miun.se.


Corresponding author.


Present address: FPInnovations PAPRICAN Division, 570 St John’s Boulevard, Pointe Claire, QC, H9R 3J9, Canada.

J. Appl. Mech 77(2), 021009 (Dec 11, 2009) (11 pages) doi:10.1115/1.3197425 History: Received March 10, 2009; Revised June 08, 2009; Published December 11, 2009; Online December 11, 2009

In the present work we propose a particle approach, which is designed to treat complex mechanics and dynamics of the open-draw sections that are still present in many of today’s paper machines. First, known steady-state continuous solutions are successfully reproduced. However, it is shown that since the boundary conditions depend on the solution itself, the solutions for web strain and web path in the open-draw section are generally time-dependent. With a certain set of system parameters, the nonsteady solutions are common. A temporal fluctuation of Young’s modulus, for example, destabilizes the system irreversibly, resulting in the continuous growth of web strain, i.e., break. Finally we exemplify with some strategic draw countermeasures how to prevent a dangerous evolution in the web strain.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

Model of the open-draw section. Paper web is transferred from the first roll to the second roll. We apply a fixed reference system in the present work.

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Figure 2

Topology of a modified Kelvin–Voigt visco-elastic model showing particles interacting through longitudinal stiffness and transverse bending stiffness. Also damping is indicated.

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Figure 3

Elastic bending moment model

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Figure 4

As the bending web recovers its equilibrium, viscous forces (indicated in the figure) act and tend to damp the restoring movement. The z-direction is the normal of the paper web at a given position. In general, this direction varies along the web (fully treated in the present approach).

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Figure 5

Tension dependence on peeling angle. A good fit with Wa′=7800 ergs/cm2(7.8 J/m2) is seen. Experimental values indicated by circles from Fig. 19 and Table 6 in Ref. 31.

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Figure 6

As the bending web moves through the open-draw, air friction acts on the web elements. The local effect is also indicated. The air speed is set to be the same as in Fig. 1(v2=ω2r2). Also an external perturbation from moving air (v⃗) is shown.

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Figure 7

As the web moves on the first roll, a release occurs each time the sum of the forces exerted on node 0 in the normal direction exceeds the fully developed adhesion force a0Wf⃗adh(max). If the forces in the normal direction depend on time, the same would be true for the release angle ϕ1. Since the forces on node 0 depend on the behavior of node 1, which in turn depend on node 2 and so on, the release condition depends on the web solution itself, leading to a complex nonlinear behavior. Also note the possibility of γ≠ϕ (Fig. 1).

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Figure 8

The stationary strain ε in the open-draw. (Top) The result for N=40 nodes. The continuous curve is the continuum solution ε(x) as given below, where x is the distance from the release point along the web (not to be confused with the x-axis in Fig. 1). (Bottom) The continuum result is essentially obtained for N=160 nodes. ε(x) is plotted here as squares to facilitate comparison. The continuum solution ε(x)=(v2−v1)/v1[(1−e−kx)/(1−e−kLw)], where k=(E∗−v2)/νv; (E∗=E/ρ) and ν=30 m2/s.

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Figure 9

The strain rate ε̇ in the open-draw with N=160 nodes. The continuum result is shown as squares: ε̇(x)=k(v2−v1)[e−kx/(1−e−kLw)]; ν=30 m2/s.

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Figure 10

The curvature κi (or κ(x)) in the open-draw with N=160 nodes. κi≡1/Ri≈2 cos(γi/2)/a0.

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Figure 11

N=40. (Top) Total strain at t=0.32 s for various machine speeds. The fitted continuous curve is quadratic in speed. (Middle) The corresponding release angle ϕ1 in deg after 0.32 s. (Bottom) For v2=1700 m/min (and above) the release angle ϕ1 is transient.

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Figure 12

N=40. (Top) Strain evolution at a machine speed of 1700 m/min. (Bottom) The take-off angle ϕ recorded at t=0.32 s (in deg) as a function of machine speed. ϕ is calculated between the tangent line of the release point and a line intersecting a point 25% of Lw (effectively where the web is straight). Also compare with Fig. 1.

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Figure 13

Total strain εT (i.e., strain at the last node) as a function of time. The elastic constant is decreased by 15% at t=5 s. The durations (3 s, 4 s, and 5 s) are then studied after which the elastic constant is restored. N=40.

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Figure 14

The corresponding evolution of the release angle ϕ1(t) for the cases in Fig. 1. N=40.

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Figure 15

Total strain εT as a function of time. The grammage of the paper web entering the open-draw gradually increases to 15% during t=5–7 s. This perturbation is maintained for the durations 6 s, 7 s, 8 s, 9 s, and 10 s after which the original grammage is restored gradually during 2 s. N=40.

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Figure 16

Total strain as a function of time. Changing the draw at strategical timings is shown to be beneficial. N=40.

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Figure 17

The strain response to the elastic modulus perturbation. The perturbation was given between 5 s and 10 s, similar to the case in Fig. 1. After the perturbation is turned on at t=5 s, the web quickly adjusts to a new curvature, ϕ and ϕ1, and the steady-state strain of 3%. As the perturbation is turned off at t=10 s, the web strain temporarily decreases and then returns to steady-state. N=40.



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