A model of rotary spinning process
Citation for published version (APA):
Hlod, A., Ven, van de, A. A. F., & Peletier, M. A. (2010). A model of rotary spinning process. (CASA-report; Vol. 1065). Technische Universiteit Eindhoven.
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EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Mathematics and Computer Science
CASA-Report 10-65
October 2010
A model of rotary spinning process
by
A. Hlod, A.A.F. van de Ven, M.A. Peletier
Centre for Analysis, Scientific computing and Applications
Department of Mathematics and Computer Science
Eindhoven University of Technology
P.O. Box 513
5600 MB Eindhoven, The Netherlands
ISSN: 0926-4507
A model of rotary spinning process
A. Hlod, A.A.F. van de Ven, and M.A. Peletier
Abstract A rotary spinning process is used to produce aramide fibers. In this process
thin jets of polymer solution emerge from the nozzles of the rotating rotor and flow towards the cylindrical coagulator. At the coagulator the jets hit the water curtain in which they solidify forming fibers. The rotary spinning is described by a steady jet of viscous Newtonian fluid between the rotor and the coagulator. The jet model includes the effects of inertia, longitudinal viscosity, and centrifugal and Coriolis forces. For the jet model the specific type of the boundary conditions depends on the balance between the inertia and viscosity in the momentum transfer through the jet cross-section. Based on that we find two possible flow regimes in rotary spinning: 1) viscous-inertial, where viscosity dominates at the rotor and inertia at the coagulator, 2) inertial, where inertia dominates everywhere in the jet. Moreover, there are two situations where spinning is not possible, either due to lack of a steady-jet solution or because the jet wraps around the rotor. Finally, we characterize the parameter space.
1 Rotary spinning process
A rotary spinning device consists of a rotor and a coagulator [2, 4]. Both the coag-ulator and the rotor have the form of a vertical cylinder. A water curtain falls along
A. Hlod
Dept. of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven The Netherlands, e-mail: avhlod@gmail.com
A.A.F. van de Ven
Dept. of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven The Netherlands, e-mail: A.A.F.v.d.Ven@tue.nl
M.A. Peletier
Dept. of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven The Netherlands, e-mail: M.A.Peletier@TUE.nl
2 A. Hlod, A.A.F. van de Ven, and M.A. Peletier
the coagulator’s inner wall; see Figure 1. Inside the coagulator the rotor is placed so
Polymer solution Rotor
Coagulator
Water curtain
Washing & drying
Fig. 1 Rotary spinning process
that the symmetry axes of the rotor and the coagulator coincide. The rotor rotates counterclockwise and has small nozzles in its lateral surface. Hot polymer solution is pumped through the rotor’s nozzles, flows to the coagulator under the influence of Coriolis and centrifugal forces, and hits the water curtain at the coagulator wall. The resulting fiber is transported away by the water, then it is washed and cut into small pieces to get pulp.
The rotary spinning problem was presented at the 48th European Study Group Mathematics with Industry in Delft (2004) [2]. To describe the jet, the string model was used. At that time, the equations were not solved due to the assumption that the jet always leaves the nozzle radially. The second study of the rotary spinning process was done in [4, 5]. It has been shown there that the jet orientation at the nozzle is determined by the jet itself. However, understanding why the jet orientation can not be prescribed at the nozzle was missing.
In this paper we present the analysis of the rotary spinning model [2, 4, 5] with the boundary conditions derived in [3, Chapter 2], and characterize the complete parameter space. Similar configurations without coagulator are studied in [1, 6–14].
2 Model
We model the rotary spinning process by considering the steady jet (see Figure 2) in two dimensions (neglecting the vertical motion due to gravity). The jet moves in a fixed horizontal frame from the nozzle of the rotor to the contact point with
A model of rotary spinning process 3
the coagulator. In a fixed reference frame the rotor rotates counter-clockwise with
se Rrot Rcoag Ω s R(s) vnozzle ΩRcoag s= 0 φ(s) β(s) FC Fc ex ey
Fig. 2 A schematic picture of the rotary spinning process.
angular velocityΩ. The radii of the rotor and the coagulator are Rrot and Rcoag, respectively. The jet is parameterized by its length s with s= 0 at the nozzle and s= sendat the contact with the coagulator. Note that the jet length sendis unknown in advance. The flow velocity at the nozzle is vnozzle, and at the contact with the coagulator the jet sticks to it having the flow velocityΩRcoag. To describe the jet position in the rotating reference frame of the rotor, we use two sets of coordinates: either the polar coordinates R andβ, with the origin at the center of the rotor, or the arc length of the jet s, and the angleφ the tangent to the jet makes with the radial direction; see Figure 2. In the rotating reference frame, two (inertial) body forces act on the jet, i.e the centrifugal, Fc, and Coriolis, FC, force.
The system describing the stationary jet follows from the conservations of mass and momentum. We scale the flow velocity v with respect toΩRcoag, both R and
s with respect to Rcoag. Then the system is fully described by three dimensionless
numbers B= 3ν/(ωR2
coag), R0= Rrot/Rcoag, Dr= vnozzle/(ΩRcoag), whereνis the
kinematic viscosity of the fluid. The resulting system is
ξ′(s) = cos(φ(s))R(s)/v(s), (1)
ξ(s)φ′(s) = −R(s)sin(φ(s))/v(s) − sin(φ(s))ξ(s)/R(s) + 2, (2)
v′(s) = (v(s)2
−ξ(s)v(s))/B, (3)
R′(s) = cos(φ(s)),β′(s) = − sin(φ(s))/R(s), (4)
together with the boundary conditions
v(0) = Dr, v(send) = 1, R(0) = R0,β(0) = 0, R(send) = 1,
4 A. Hlod, A.A.F. van de Ven, and M.A. Peletier
Here, v(s) is the flow velocity in the jet, ξ(s) represents the momentum transfer
through the jet cross-section. The boundary conditions for the jet orientationφ(s)
are determined by the sign ofξ(s); see [3, Chapter 2].
From (1) it follows thatξ(s) is a strictly increasing function implying that there
are three possibilities for the sign ofξ(s):
• The first situation isξ(s) > 0 everywhere in the jet, and then inertia dominates
everywhere in the momentum transfer through the jet cross-section. We call this jet flow regime inertial. In this case the jet must be aligned with the radial nozzle direction,φ(0) = 0.
• In the second situationξ(s) changes sign from negative to positive, and viscosity
dominates near the nozzle and inertia near the coagulator in the momentum trans-fer through the jet cross-section. We call this jet flow regime viscous-inertial. In this case we can not prescribe any boundary condition for the jet orientation. However, from (2), it follows that at the point s0whereξ(s0) = 0 the jet should be aligned with the direction of the resulting external force acting on the jet, yielding
φ(s0) = arcsin(2v(s0)/R(s0)). (5)
• In the third situation ξ(s) < 0 everywhere in the jet, and viscosity dominates
everywhere in the momentum transfer through the jet cross-section. We call this jet flow regime viscous. For viscous jet we require tangency with the coagulator
φ(send) =π/2. However, the viscous jet situation is not possible in the current
setup because of the following argument.
The border between the parameter regions for the viscous and viscous-inertia jets should satisfy the conditionξ(send) = 0. Then from (5) and the boundary conditions it follows that sin(φ(send)) = 2 leading to a contradiction.
The following solution strategy is suggested. For the equations (1)-(2) we find a first integral
sin(φ(s))ξ(s) =R(s)
2
− C1
R(s) ,
where C1= R0for the inertial jet, C1= R(s0) for the viscous-inertial jet. Then sys-tem is solved using the shooting method.
The parameter space is described by the three dimensionless numbers R0, Dr, and B. In Figure 3 we present the partitioning of the parameter space in the R0, Dr-plane for different B. In this Dr-plane we observe the regions of inertial jet, viscous-inertial jet, and two regions of nonexistence of a jet solution. For the parameter regions ”no solution 1” the cause of nonexistence is that the jet does not reach the coagulator and wraps round the rotor. For the parameter regions ”no solution 2” the jet reaches the coagulator, but the flow velocity at the coagulator can not be matched, indicating the unsteady jet. In the region ”no solution” both causes of nonexistence are possible. The borders between the viscous-inertial and inertial regimes are obtain from the conditionξ(0) = 0 and the borders of the nonexisting
regions are obtained using monotonicity properties of the jet solution together with the conditionφ(send) =π/2.
A model of rotary spinning process 5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 D r R0 B= 0.01 inertial jet viscous-inertial jet no solution 1 no solution 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 D r R0 B= 0.15 inertial jet viscous-inertial jet no solution 1 no solution 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 D r R0 B= 0.2617 inertial jet no solution 1 no solution 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 D r R0 B= 0.4 inertial jet no solution
Fig. 3 Parameter regions in the R0, Dr-plane for different B.
For B close to zero (see the plot for B= 0.01) the region of the inertial jet
occupies almost the whole plane except a narrow area near Dr= 0. The
viscous-inertial jet region forms long and narrow strip along the line Dr= 0. The
nonex-istence regions ”no solution 1” and ”no solution 2” are small areas concentrated near the points Dr= 0, R0= 0, and Dr = 0, R0= 1, respectively. With increasing B (see the plot for B= 0.15) the border of the inertial jet region rises, the
viscous-inertial jet region becomes higher and shorter, and the nonexistence regions grow. For B= 0.2617 the viscous-inertial jet regions is just disappeared (shrinked to one
curve). For larger B> 0.2617 (see the plot for B = 0.4) the viscous-inertial jet is
absent, the inertial jet region becomes smaller and the nonexistence region ”no so-lution” expands.
3 Conclusions
The steady jet model in rotary spinning configuration is studied. The jet is described by a system of ODE’s on an unknown domain. The scaled system describing the jet is characterized by three dimensionless parameters. For the jet in rotary spinning we distinguish three situations: the inertial jet, the viscous-inertial jet, and a steady jet solution does not exist. There are two causes of the nonexistence of the jet between the rotor and the coagulator 1) the jet wraps around the rotor, and 2) the flow velocity at the coagulator can not be reached, indicating unsteady jet.
6 A. Hlod, A.A.F. van de Ven, and M.A. Peletier
Acknowledgements The authors would like to acknowledge Teijin Aramid, a part of the Teijin
group of companies.
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