4796
Ind. Eng. Chem. Res. 1996, 35, 4796-4800
RESEARCH NOTES Modeling and Simulation of the Coagulation Process of Poly(acrylonitrile) Wet-Spinning Sea Cheon Oh,† Young Soo Wang,‡ and Yeong-Koo Yeo*,† Department of Chemical Engineering, Hanyang University, 17 Haengdang-dong, Sungdong-ku, Seoul 133-791, Korea, and Technical Research and Development Center, Hanil Synthetic Fiber Company Ltd., 222 Yangduk-dong, Hoiwon-ku, Masan 630-791, Korea
The coagulation process of PAN (poly(acrylonitrile)) wet-spinning was modeled and simulated based on the numerical analysis of the coagulation of a viscous polymer solution by diffusional interchange with a bath. Experiments were performed with gelled solutions of PAN in nitric acid to determine the diffusion rate of solvent and nonsolvent (water) during the coagulation. The experimental data were analyzed by using equations of diffusion coefficient which are a function of the solvent concentrations of the coagulation bath and the filament. The concentration profile of solvent in moving filament was predicted by solving the diffusion model equation numerically. A simplex method was used in the computation of the parameters of the diffusion equations to minimize the difference between the numerical results and experimental data. Introduction Fiber formation by wet-spinning accounts for a significant fraction of the production of today’s man-made fibers. In this process a viscous polymer solution is extruded through the small holes of a spinneret immersed in a liquid bath. A diffusional interchange between the freshly formed fluid filament and the bath causes the polymer to solidify. This process is called coagulation. Diffusion of solvent from the solidifying filament is an important part of fiber formation by the wet-spinning process and is a dominant factor in the coagulation process. Diffusion of solvent influences fiber morphology and properties by which design of the process is greatly affected. Various experimental and theoretical approaches to this problem have been taken (Paul, 1968; Han and Segal, 1970; Doppert and Harmsen, 1973; Ziabicki, 1976; Wu and Paul, 1978). During coagulation process one or more of the bath components diffuse into the filament while the solvent diffuses out of it. As a consequence of this exchange the polymer precipitates or crystallizes, which is caused by chemical reaction of the polymer or by an excessive buildup of nonsolvent or by both. Recently, much research in the field of acrylic fiber has been focused on the development of an industrial material. However, it should be noted that there are many difficulties of developing the industrial material in the spinning process of commercial fibers due to the problems of the spinning process, such as spinnability and the rheological and diffusional phenomena. In the past the mechanism of the coagulation process in isothermal acrylic fiber formation has been analyzed by using a diffusion model with a constant diffusion coefficient. Therefore, in order to design the spinning * Author to whom correspondence is addressed. Telephone: 82-2-290-0488. Fax: 82-2-291-6216. E-mail:
[email protected]. † Hanyang University. ‡ Hanil Synthetic Fiber Co. Ltd. Telephone: 82-551-903201. Fax: 82-551-90-3209.
S0888-5885(96)00261-8 CCC: $12.00
process for the production of new material, the diffusion experiment is necessarily required. This is why in this work the coagulation of an acrylic fiber was studied to develop the diffusion model which can be used, in general, for a PAN-HNO3 spinning system by using the equations of the diffusion coefficient given as a function of the solvent concentrations of the coagulation bath and filament. Most of the experiments in the past were performed with gelled rods that undergo no internal motions or dimension. In this work experiments on the spinning process were performed to determine the diffusion rate of solvent and nonsolvent during coagulation. Nitric acid was used as a solvent for the polymer. The coagulation bath consisted of a mixture of nitric acid and water. The concentration profile of solvent in a moving filament was computed by using a MOL (method of lines) method (Schiesser, 1991) which is well-known as a method to solve the partial difference equation with effect. The numerical solutions of each diffusion model equation were compared to experimental data. A simplex method (Edgar and Himmelblau, 1988) was used to determine the parameters of each equation of diffusion coefficient by minimizing the difference between the numerical results and experimental data. This is convenient in case the construction of the objective function is difficult. Theoretical Analysis When a spinning solution is brought into contact with a mixture of solvent and nonsolvent of the coagulation bath, a diffusional interchange occurs between the two phases. During coagulation nonsolvent enters and solvent leaves the polymer phase. After a sufficient length of time, no more exchange between the two phases takes place and an equilibrium state is achieved. A knowledge of the diffusion rates associated with coagulation is required both for practical applications and for elucidation of the coagulation mechanisms. For a cylinder of infinite length where diffusion occurs in © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4797
(
)
Q ∂C D ∂C b ∂2C ) +D 1+ πF ∂Z φ ∂φ C∞ ∂φ2
(7)
At r ) 0, the term (D/φ)(∂C/∂φ) in eq 7 is indeterminant (0/0) since ∂C/∂r ) 0 from boundary condition. In order to overcome the indeterminacy, L’Hospital’s rule can be applied to eq 7 to give
(
the radial direction, Fick’s second law for diffusion is given by
∂C ∂C 1 ∂ ) rD ∂t r ∂r ∂r
(
)
(1)
For accuracy a mass-transfer boundary condition at the filament-bath interface must be used. However, the required mass-transfer coefficient for the PAN-HNO3 system was not available, and the use of an inexact mass-transfer coefficient in the analysis of diffusion phenomena results in a wide discrepancy with experimental data. So an infinite concentration boundary condition was used as a boundary condition at r ) R. The solution of eq 1 has to satisfy the following boundary conditions for a cylinder with radius R:
C ) C∞ ∂C )0 ∂r
for 0 e r e R for t g 0 for t g 0
at t ) 0 at r ) R at r ) 0
(2)
To solve eq 1, one has to solve for the axial velocity v(z) as a function of distance along the spinning way, the concentration profile of solvent in filament and rheological force. The velocity at the position of maximum jet swell and the physical constant needed to solve v(z) are difficult to determine, so a simplified model such as assuming a constant filament diameter was used. Nondimensional cylinder radius φ and bath length Z are defined as, respectively
Z ) vzt
(3)
φ ) r/R
(4)
Assuming that the diffusion coefficient depends on the concentration, we have from eq 1
∂D ∂2C Q ∂C D ∂C ) +D1+ πF ∂Z φ ∂φ ∂C ∂φ2
(
)
(5)
In the above equation Q is the mass flow rate given by Q ) πR2vzF. The dependence of the diffusion coefficient on concentration can be represented by a nonlinear equation with respect to the concentration. For this purpose three related equations of the diffusion coefficient were used in this work. First, if we define the equation of the diffusion coefficient as
( )
C D ) a exp b C∞ then eq 5 becomes
(8)
Second, if the equation of the diffusion coefficient is defined as
a
D)
C ) C0
)
b ∂2C Q ∂C )2 D+D πF ∂Z C∞ ∂φ2
Figure 1. Schematic diagram of the experimental system.
(6)
then eq 5 becomes
(
b a C Q ∂C D ∂C ∞ ) + D+ C πF ∂Z φ ∂φ 1-b C∞
(
(9)
C 1-b C∞
(
))
b a C ∞ Q ∂C )2 D+ C πF ∂Z 1-b C∞
(
2
)) 2
∂2 C ∂φ2
∂2C ∂φ2
at φ * 0 (10)
at φ ) 0
(11)
Finally, if we define the equation of the diffusion coefficient as
D)
a
(
)
C 1-b C∞
then we have from eq 5
(
Q ∂C D ∂C ) + D+ πF ∂Z φ ∂φ
(
(
b 2a C∞ C 1-b C∞
))
b 2a C∞ Q ∂C )2 D+ C πF ∂Z 1-b C∞
(
3
(12)
2
))
∂2 C ∂φ2
3
∂2C ∂φ2
at φ * 0 (13)
at φ ) 0
(14)
Equations 9 and 12 are similar to the functional forms of the concentration dependence of the diffusion coefficient for a gas-solid catalyst system using Langmuir’s and Volmer’s isotherm equations, respectively (Garg and Ruthven, 1972). The solutions of these equations were obtained by using the MOL method. The parameters of eqs 6, 9, and 12 were obtained by using the simplex method. The average solvent concentrations in the
4798 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 1. Spinning Conditions Used in the Experiments coagulation bath temp (°C) coagulation bath concn (wt % HNO3) dope flow rate over 15 holes (mL/min) dope concn (wt % polymer) takeup speed (m/min) draw ratio
-3 20-43 0.5-1.5 13.1 10-30 13.5
Table 2. Experimental Results for the Composition in Filaments on the Spinning Conditions coagulation bath concn (wt % HNO3)
dope flow rate (mL/min)
bath length (cm)
20.0
1.5
35.5
1.0
0 10 25 50 100 200 0 10 25 50 100 200 0 10 25 50 100 200
43.0
0.5
composition of filament (wt %) H 2O HNO3 PAN 28.50 60.78 63.74 64.31 64.40 64.69 28.50 45.45 49.33 51.40 52.13 52.75 28.50 39.50 43.06 45.19 45.73 46.19
57.70 19.51 17.35 16.65 16.52 16.21 57.70 35.35 32.18 30.12 29.66 29.05 57.70 42.16 37.58 35.78 35.35 34.91
13.80 19.71 18.91 19.04 19.08 19.10 13.80 19.20 18.49 18.48 18.21 18.20 13.80 18.34 19.36 19.03 18.92 18.90
Figure 2. Comparison of simulation results with experimental data for the coagulation bath concentration 20 wt % HNO3.
Table 3. Parameters of Eqs 6, 9, and 12 Obtained by a Simplex Method eq
a
b
6 9 12
1.4399 × 10-6 0.9167 × 10-6 1.2639 × 10-6
0.2278 0.2449 0.1139
filament were determined by eq 15 with the numerical solutions on the filament radius. N
Cavg )
[∫(i-1)/N2πr dr × Cai ] ∑ i)1 i/N
∫0 2πr dr 1
(15)
where
Cai )
Ci-1 + Ci 2
Experiment Figure 1 describes a schematic diagram of the experimental system for a coagulation process used in the present work. The length of the coagulation bath was 200 cm and the width and depth of the bath were 20 and 40 cm, respectively. For the maintenance of a uniform solvent concentration over the bath length, an aqueous solution of the bath was circulated by the circulation pump. In order to avoid the possibility of interference by heat-transfer phenomena, the coagulation bath was maintained at -3 °C. The storage tank of the polymer solution and the supply lines containing the polymer solution were provided with cooling jackets to keep the solution at the desired temperature of -3 °C. PAN produced in Hanil Synthetic Fiber Co. Ltd. was used as a polymer material. The composition of the spinning solution was the same for all the experiments performed. The weight percent concentrations
Figure 3. Comparison of simulation results with experimental data for the coagulation bath concentration 33.5 wt % HNO3.
of each component in the polymer solution were 13.8% PAN, 57.7% HNO3, and 28.5% H2O. The spinneret used in this experiment contained 15 holes of which each diameter is 0.076 mm. The distances from hole to hole were very accurately uniform. The length of the capillary was just equal to its diameter. The spinning conditions used in this experiment are given in Table 1. The concentration of solvent in filaments was measured by using the following method. The bath solution smeared on a filament surface was carefully removed by using the moisture absorption paper, and filament was dipped in a specified water for a time sufficient to diffuse the solvent from the filament. The solvent concentration in a specified water was measured by means of NaOH titration using phenolphthalein as an indicator. Three or more experiments were performed at the same spinning condition, and the mean solvent concentration of experimental data was used as the solvent concentration in the filament. Results and Discussion The experimental results for the composition in filaments on the spinning conditions are shown in Table 2. Table 3 shows the parameters of the equations of
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4799
Figure 4. Comparison of simulation results with experimental data for the coagulation bath concentration 43 wt % HNO3.
Figure 6. HNO3 concentration on the normalized radial position in the fiber (coagulation bath concentration 33.5 wt % HNO3).
Figure 5. HNO3 concentration on the normalized radial position in the fiber (coagulation bath concentration 20 wt % HNO3).
Figure 7. HNO3 concentration on the normalized radial position in the fiber (coagulation bath concentration 43 wt % HNO3).
diffusion coefficient obtained by the simplex method based on the experimental results of Table 2. In the computation of the parameters of eqs 6, 9, and 12, the initial values of the parameters corresponding a and b are set to 5.0 × 10-6 and 1.0, respectively. Figures 2-4 show results of simulations as well as experimental data at the spinning conditions of Table 1. Simulations were performed based on the parameters of Table 3. As shown in these figures, the simulation using eq 9 gives the best results. Therefore, we can see that the diffusion phenomenon during the coagulation process of PAN wet-spinning is well expressed by eq 9, which is similar to the functional forms using Langmuir’s isotherm equation for the diffusion coefficient in a gas-solid catalyst system. Figures 5-7 show results of simulations on a filament radius for the solvent concentration in a filament at three coagulation bath concentrations of 20, 33.5, and 43 wt % HNO3. From these figures, we can see that the diffusion rate increases as the solvent concentration of the coagulation bath decreases and that the solvent concentration in the filament changes rapidly as the filament surface is approached.
Conclusion The coagulation process of PAN wet-spinning was studied numerically and experimentally. In the numerical simulations, various equations of diffusion coefficient given as a function of the coagulation bath concentration and the concentration of the solvent in a filament were employed. Experiments were performed with gelled solutions of PAN in nitric acid in order to determine the diffusion rate of solvent and nonsolvent during coagulation. The numerical solutions of diffusion model equations for the coagulation process were obtained stably by using the MOL method. The parameters of the equations of diffusion coefficient were obtained by a simplex method. It was found that the diffusion coefficient of the coagulation process of PAN wet-spinning is well expressed by eq 9, which is similar to the functional forms using Langmuir’s isotherm equation for a gas-solid catalyst system. Nomenclature a: a constant b: a constant
4800 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 C: solvent concentration in a filament, wt % HNO3 Ca: average solvent concentration in a filament, wt % HNO3 C∞: solvent concentration of the coagulation bath, wt % HNO3 D: diffusion coefficient, cm2/s N: total section number Q: mass flow rate, g/s R: total filament radius, cm r: filament radius, cm t: time, s vz: velocity in z direction, cm/s Z: coagulation bath length, cm Greek Letters φ: dimensionless number F: density, g/cm3 Subscripts avg: average i: section number
Edgar, T. F.; Himmelblau, D. M. Optimization of Chemical Process; McGraw-Hill: New York, 1988. Garg, D. R.; Ruthven, D. M. The effect of the concentration dependence of diffusivity on zeolite on sorption curves. Chem. Eng. Sci. 1972, 27, 417. Han, C. D.; Segal, L. A Study of Fiber Extrusion in Wet Spinning. II. Effects of Spinning Condition in Fiber Formation. J. Appl. Polym. Sci. 1970, 14, 2999. Paul, D. R. Diffusion During the Coagulation Step of WetSpinning. J. Appl. Polym. Sci. 1968, 17, 383. Schiesser, W. E. The Numerical Method of Lines; Academic Press, Inc.: San Diego, 1991. Wu, W.; Paul, D. R. Time-Lag Technique of Diffusion-Coefficient Measurement in Solution Spinning. Textile Res. J. 1978, 48, 230. Ziabicki, A. Fundamentals of Fibre Formation; Wiley-Interscience: London, 1976.
Received for review May 8, 1996 Revised manuscript received August 23, 1996 Accepted August 24, 1996X IE960261A
Literature Cited Doppert, H. C.; Harmsen, G. J. The Influence of Stretch Ratio on the Rate of Diffusion in a Wet-Spinning Process. J. Appl. Polym. Sci. 1973, 17, 893.
X Abstract published in Advance ACS Abstracts, October 1, 1996.
