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Ind. Eng. Chem. Res. 1998, 37, 1464-1472
A Model for Foam Devolatilization in an Extruder Chi-Tai Yang† and Theodore G. Smith Department of Chemical Engineering, University of Maryland, College Park, Maryland 20742
David I. Bigio* Department of Mechanical Engineering, University of Maryland, College Park, Maryland 20742
Colin Anolick Central Research and Development Department, DuPont, Experimental Station, Wilmington, Delaware 19880
A model is developed for the design of the devolatilization section in an extruder, which is optimized with respect to foam growth and process parameters. The previously developed bimodal/film model for polymer devolatilization is used to design Lc, the distance from the melt seal to the vacuum port for completion of foaming and breakup in the extruder. The model requires one free parameter, the number of stripper bubbles as a function of important process parameters. A statistical regression method and a backpropagation neural network are used to develop the correlations. The developed correlation along with the bimodal/film model is applied to design a distance, Lc, prior to the vacuum port for completion of foaming and breakup in an extruder. Polyethylene/acrylic acid devolatilization in a 28-mm co-rotating twin-screw extruder is used as a case study for model demonstration. Introduction Devolatilization is one of the most important operations in the manufacturing and compounding of polymers. In many polymer processing operations it is necessary to remove the volatile components from polymer melts or solutions in order to improve product quality, reduce product cost, and eliminate health hazards. The volatile species can be residual monomers, reaction byproducts, or solvents. Devolatilization involves the application of a reduced pressure or vacuum to extract volatile vapors and often the injection of a stripping agent to enhance the devolatilization performance. As a result, the devolatilization process often generates bubbles of the volatile component and the stripping agent. Water and nitrogen are two commonly used stripping agents (Biesenberger and Sebastian, 1983). The various aspects of polymer devolatilization have been reviewed in the literature (Werner, 1980; Biesenberger and Sebastian, 1983; Biesenberger, 1980; Denson, 1983; Albalak, 1996). Most work in equipment design of the extruder for devolatilization has focused on surface renewal of polymer melts in single-screw extruders (Latinen, 1962; Coughlin and Canevari, 1969; Roberts, 1970; Biesenberger, 1980; Biesenberger and Kessidis, 1982; Biesenberger et al., 1990, 1991) and in twin-screw extruders (Collins et al., 1985; Secor, 1986; Biesenberger et al., 1990, 1991). These studies used the penetration model for mass diffusion to describe the mass-transfer mechanism for polymer devolatilization. Their model relied on the ability of the extruders to continuously generate free surface area available for diffusion of the volatile * Author to whom correspondence should be addressed. Telephone: (301) 405-5258. Fax: (301) 314-9477. E-mail:
[email protected]. † Current address: Formosa Plastics Corporation, P.O. Box 320, Delaware City, DE 19706. Telephone: (302) 836-2225.
component from the polymer liquid phase to the polymer-gas interface and into the contiguous gas phase. Scale-up of the extruder performance for polymer devolatilization was based on the machine geometry, screw speed, and throughput. The design for the optimum position and length of the vacuum section for the desired degree of devolatilization was not described. A few papers address the issues of the design of the vacuum section for devolatilization and scale-up of the extruder performance for devolatilization associated with foam formation (Foster and Lindt, 1989, 1990). Foster and Lindt (1989, 1990) studied devolatilization transitioning from a foaming regime to a bubble-free regime in a 20-mm counter-rotating tangential twinscrew extruder. They used a foam simulation/RTD model to describe the foam devolatilization regime and a penetration mass-transfer model to describe the bubble-free diffusion-controlled regime for devolatilization. Their work suggested that if the screw channel did not provide sufficient availability of free volume for the foam to grow, the foaming was constrained and a low effective mass-transfer coefficient for devolatilization was obtained. To achieve a high degree of devolatilization in the heavily foaming regime, the extruder needs to have sufficient free volume in the screw channels to accommodate the unconstrained foaming of the polymer in the vacuum section. Given the extruder configuration and operating conditions, the question arises as to how to design the vacuum section (i.e., position and length of the vacuum port) to ensure that the desired foam volume expansion is achieved. Tukachinsky et al. (1994) reported an experimental study on foam-enhanced devolatilization of a polystyrene melt in a 50-mm vented single-screw extruder by scanning electron microscopy (SEM) and video photography. A special port was designed for venting gases, for observation, for quenching, and for polymer sampling. The port was located three leads downstream
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from the melt inlet. Under steady-state conditions, they observed that polymer melt foaming takes place only in the starting section of the screw in no more than two leads of its length. Foaming is completed within three screw turns. In other words, foaming starts to subside and collapse after three screw leads (or 3 L/D), which happens to be at the beginning position of the sampling/ vent port. Although the authors did not intend to examine the design of the vacuum section for foamenhanced devolatilization, their observations provide useful information on the design of the extruder configuration and operation conditions for desired foam volume expansion. By applying the free-volume theory in the prediction of diffusion coefficients in polymer-solvent systems, it has been shown that 1.0 wt % of dissolved nitrogen (stripping agent) can enhance the apparent diffusion coefficient of ethylbenzene (volatile component) in polystyrene at 150 °C by 9% (Yang, 1995; Yang et al., 1996c). However, considering the low diffusion coefficients of stripping agents in polymer (on the order of 10-6-10-5 cm2/s), it may take several hours for stripping agents to dissolve in the polymer. Since the average residence time of a stripping agent in an extruder is very short, typically about 5-10 s, there is not sufficient time for a stripping agent to dissolve in the polymer liquid phase. Consequently, a predominant fraction of the stripping agent should stay in the gas phase, either dispersing in the polymer as bubbles or sweeping over the polymer surface. A mathematical model of polymer devolatilization has been developed (Yang, 1995; Yang et al., 1996a,b, 1997a,b), which is based on the single-cell growth model for devolatilization (Amon and Denson, 1984; Powell, 1987). The mathematical model combines a bimodal model for foam growth and a film model for mass transfer. The details of the mathematical formulation for the single-cell growth dynamics, the bimodal model for foam growth, and the film model for mass transfer are summarized in the appendices. The effect of stripper bubbles on foam dynamics and devolatilization performance is examined. We also found that there was a maximum foam volume for a polymer to expand. This critical foam volume change (φcrit) is recognized as an important parameter in the design of the devolatilization section and scale-up of the extruder for devolatilization. Beyond the critical foam volume expansion, the foam ruptures and collapses. The bimodal model for foam growth describes the initial formation and growth of the volatile and stripper bubbles in polymer during devolatilization. When foaming and breakup occur at a critical foam volume expansion in which the polymer looks like an open-cell structure with thin polymer films providing a free path for volatile diffusion and escape, a film model for mass transfer is used to model the devolatilization as mass diffusion of the volatile component from the polymer liquid film to the contiguous gas phase. Our model has shown improvements over the cell model for devolatilization (Denson, 1983; Amon and Denson, 1984; Powell, 1987; Foster and Lindt, 1990) in that (1) it requires a lower initial number of bubbles, which is more physically achievable, and (2) it accounts for the observation of a limiting foam volume for a specific polymer in devolatilization. In addition, our model has demonstrated that the devolatilization section in an extruder needs to be designed with sufficient free channel volume and residence time to allow for the
desired foam volume expansion; otherwise, the foaming is constrained and/or the foam breaks earlier than desired, resulting in a low degree of thermodynamic devolatilization efficiency. The capability of the bimodal/ film model for polymer devolatilization in the prediction of thermodynamic devolatilization efficiency has been confirmed by experimental results (Yang, 1995; Yang et al., 1997b). The bimodal/film model developed and applied to an extruder process does not include shear rate effects. Shear rate affects the devolatilization in many levels including (1) enhanced bubble nucleation due to mechanical stress and stretching (Tukachinsky et al., 1994) and cavitation (Biesenberger and Lee, 1987, 1989), (2) bubble elongation and rupture (Biesenberger and Lee, 1987), and (3) foam compression and collapse (Biesenberger and Lee, 1987, 1989; Tukachinsky et al., 1994). Each of these phenomena should be considered in the devolatilization process as they affect all stages of foaming such as bubble nucleation, growth, coalescence, and breakage. The model has shown improvements and is better than either the pool model or the cell model. The reason the shear rate is not a primary parameter lies in the characteristic times for different parts of the process. In an extruder, our experiments and calculations have shown that when there is rapid foam volume growth, the polymer pool can grow to 4-5 times its original volume on the order of 0.5 s. The shear rate manifests on the process by deforming the polymer pool. Therefore, a viable measure of the characteristic time for shear rate is the time it takes for the pool to rotate once in the screw channel. If the screw speed is 120 rpm, then the screw rotates once every 0.5 s. One can compute that the pool rotates once for every one-andhalf screw rotations. Therefore, the characteristic time for shear rate is 0.75 s. Physically this implies that, as the polymer enters into the devolatilization region, the polymer undergoes a rapid growth due to the supersaturation in chemical potential. It can complete the pool volume growth prior to when the major effects of the shear rate can come into play. It would seem that at that point the three effects described above would still come into play, and whether shear rate enhances or suppresses the devolatilization process would have to be determined. Since the most important variable for the computation of thermodynamic devolatilization efficiency in this model is the foam volume expansion, shear rate seems to only play a secondary role to that parameter. The current research focuses on how the mathematical model can be applied to the modeling and design of the devolatilization section in an extruder based on foam growth (Yang and Bigio, 1996a,b, 1997). The goal of this study is to obtain an expression for the design of the devolatilization section which is optimized for foaming dynamics of the polymer and other important process parameters. Polyethylene/acrylic acid devolatilization in a 3-lobe 28-mm Werner & Pfleiderer (W&P) ZSK-type co-rotating twin-screw extruder is used as a case study to show the design concept and the foam devolatilization modeling in an extruder. The design principles can be applied to the design of the devolatilization section in single-screw extruders and counterrotating twin-screw extruders. A statistical regression method and a backpropagation neural network are used to generate the correlations between the data and the model.
1466 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998
Figure 1. Modeling and design of the foam devolatilization in an extruder.
Modeling and Design Figure 1 schematically shows the modeling and design of the devolatilization section in a co-rotating twin-screw extruder by using the bimodal model for foam growth and the film model for devolatilization. The vacuum section for devolatilization usually starts at some distance after the polymer melt seal. The melt seal is formed by passing the polymer through a high-pressure zone, such as by pressure buildup through the reversepitch screw elements in a co-rotating twin-screw extruder. Large-pitch forward-conveying screw elements are used in the vacuum section, resulting in partially filled screw channels in order to accommodate foaming. Design Principles. The purpose of modeling foam devolatilization is to establish an expression for the design of the devolatilization section in an extruder which is optimized for foam dynamics and process parameters. As discussed above and in other literature (Amon and Denson, 1984; Powell, 1987), we proposed a bimodal/film model for devolatilization. Our approach to designing the devolatilization region in an extruder is to generate an expression for the adjustable parameter (the number of stripper bubbles) as the input in the bimodal/film model for fitting the available experimental data. One of the design considerations is to determine a distance (Lc) for completion of foaming and the beginning of breakup prior to the vacuum port. In other words, prior to the beginning of the vacuum port the extruder needs to be designed with sufficient free volume in the screw channels (i.e., a sufficiently low degree of channel fill) in order to allow for the desired foam volume expansion. The issue in screw design in the vacuum section is to determine how many channels (or L/D) are needed for complete foam growth of the polymer. To have a sufficiently low degree of channel fill in the vacuum section to accommodate foaming, polymer throughput needs to be reduced for a given
screw speed. A reduction in throughput may not be economic in satisfying the production requirement. Increasing the throughput would increase the degree of channel fill whereby the desired foam growth is constrained, resulting in a low degree of devolatilization and poor polymer product quality. Thus, the screw design and operating conditions in the devolatilization need to be designed to maximize the devolatilization performance while maintaining the minimum production requirements. A properly designed distance Lc also prevents the vacuum port from being clogged by the polymer due to intensive foaming and/or elastic climbing. Since the desired maximum foam volume expansion is allowed prior to the vacuum port, 80-90% devolatilization should be achieved. Under and after the vacuum port, a lesser degree of foaming and foam collapse occurs which contributes to the final part of devolatilization. A design factor, β, is introduced in the modeling and computation. For instance, one may want to design the Lc such that TDEc ) 0.9TDEf; i.e., β ) 0.9. First, we fit the bimodal/film model to the experimental data with known extruder design and operating conditions. The procedure of the design algorithm for correlation is shown as follows: 1. Choose the critical foam volume (φcrit) and the number of stripper bubbles (Ns). (Note that φcrit is the maximum foam volume for a polymer to grow and sustain in devolatilization and is used in the model for transition from the bimodal model for foam growth to the film model for mass transfer.) 2. For known screw design and experimental conditions, perform the computer simulation to calculate the thermodynamic devolatilization efficiency. 3. Repeat steps 1 and 2 until the computed thermodynamic devolatilization efficiency fits the experimental one at the end of devolatilization (t ) tf). 4. Develop a correlation between Ns and process parameters. On the basis of the correlation developed, we are able to design the vacuum section in an extruder for devolatilization and predict the devolatilization performance within the domain of the process parameters and operating conditions for the system investigated. In other words, we are exploring the capability of our model in designing the devolatilization section and predicting the devolatilization performance. The procedure of the design algorithm for prediction is shown in the following: 1. Specify the process and production requirements, e.g., throughput, temperature, devolatilization efficiency, and design factor. 2. Design an initial screw configuration and other necessary process parameters. 3. Based on the correlation developed, calculate the required initial number of stripper bubbles (Ns). 4. Perform the computation for the current set of screw design and operating conditions. 5. If the desired TDEf at tf is not achieved, change the screw design and operating conditions. Then repeat steps 2-5 until the required conditions are satisfied. 6. Check tc at TDEc, and estimate the Lc. Artificial Neural Networks. Polymer processes involve very complex phenomena (fluid flow, heat transfer, and mass transfer), equipment selection (configurations and operating conditions), and material properties (viscosity, elasticity, and other physical prop-
Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1467
erties). The combination of these factors makes the modeling of polymer processes a challenging task. The correlations between the desired output and the process conditions are often not simple linear relationships. In the foam devolatilization modeling and design algorithm, a correlation between the number of stripper bubbles and process parameters is developed. Based on this correlation, the devolatilization section can be designed and the thermodynamic devolatilization efficiency can be predicted. The correlative relationship between the number of stripper bubbles and process parameters should be complex and nonlinear. We do not have the knowledge regarding the functions or the forms of the equations governing the correlations between the data. Consequently, conventional statistical methods may not satisfactorily regress the data. Artificial neural networks have recently accepted a wide range of applications because of their promise of solving complex and nonlinear problems (Rumelhart and McClelland, 1986; Wasserman, 1989). Computational neural networks have been used in modeling chemical process systems (Bhat et al., 1992; Su, 1992) and predicting materials properties (Sumpter and Noid, 1995; Keshavaraj et al., 1995). The most widely used learning neural network is the backpropagation net (BPN). Neural networks provide the advantages of modeling poorly understood phenomena by training the net repeatedly with sets of input data and their corresponding sets of target values. A neural net consists of an input layer, an output layer, and one or more hidden layers between the input and output layers. The net learns to recognize the patterns in the data and create an internal model of the process governing the data. This internal model then can be used to predict outputs for new sets of inputs. However, this internal model does not provide any information about the underlying mechanism for the process. The neurons in the hidden and output layers perform as summing and nonlinear mapping functions (Keshavaraj et al., 1995). They multiply all inputs and a bias by a weight and then sum the weighted inputs by a processor within the neuron.
Ui,p )
∑XiWi,j - θj
(1)
where Ui,p is the sum total of the inputs for pattern p. Wi,j is the weight associated with the connection from the ith to the jth neuron. The sum total is modified by a commonly used sigmoid transfer function to generate the weighted output.
Si,p )
1 1 + exp(-Ui,p)
(2)
A backpropagation neural net learns by repeatedly adjusting and updating the weights to minimize the sum of squared errors between its predictions and a training data set (Bhat et al., 1992). This procedure is repeated in as many cycles as needed until the error is within a prescribed tolerance. It has forward-flowing information in the prediction mode and backpropagated error corrections in the training mode. Repeated iterations result in a converged set of the weights. The net is trained to identify and recognize the patterns between sets of input data and corresponding sets of target ouputs.
Figure 2. Schematic diagram of the computational paradigm for modeling the polymer devolatilization process.
Figure 3. Simplified diagram of the devolatilization section in the ZSK-28 co-rotating twin-screw extruder for polyethylene/ acrylic acid devolatilization.
Figure 2 shows the schematic diagram of foam modeling in polymer devolatilization. The design principle and computational paradigm are also illustrated. The design principles begin with the materials properties, screw designs, and operating conditions. These can further generate a set of parameters that may influence the devolatilization process. Devolatilization is also subject to the foam dynamics observed and the process/ production requirements specified. Internally, we develop a correlation between the number of stripper bubbles and the set of parameters. Based on this correlation, we can compute the thermodynamic devolatilization efficiency and design the devolatilization section in an extruder. In this study, we use a backpropagation simulator (Rao and Rao, 1993) to solve the complex problems in modeling, design, and development of the polymer devolatilization process in an extruder. A conventional statistical regression method is also used to correlate the data and to compare the results with the backpropagation neural net simulation. Case Study: Polyethylene/Acrylic Acid Devolatilization Polymer System and Extruder Configuration. In this section, we show our model algorithm and design concept for the foam devolatilization modeling in an extruder, based on polyethylene/acrylic acid devolatilization in a W&P ZSK-28 co-rotating twin-screw extruder (Anolick, 1991). A similar concept and modeling can be applied to the design of the devolatilization section in single-screw extruders and counter-rotating twin-screw extruders. Figure 3 shows the simplified schematic diagram of the devolatilization section in the ZSK-28 co-rotating twin-screw extruder. Table 1 lists an example of the operating conditions (Anolick, 1991). In this set of experiments, the 24/24 left-handed reversepitch screw element is used to build pressure and form
1468 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 Table 2. List of Process Parameters for Correlation in the Polyethylene/Acrylic Acid Devolatilization Data
Figure 4. Simulation of polyethylene/acrylic acid devolatilization using the bimodal model for foam growth and the film model for devolatilization. Table 1. Example of the Operating Conditions for Polyethylene/Acrylic Acid Devolatilization in a ZSK-28 Co-rotating Twin-Screw Extruder (Anolick, 1991) condition
value
temperature, T screw speed, NR throughput, m ˘ water, ws vacuum pressure, Pvac initial monomer concentration, C0 thermodynamic devolatilization efficiency, TDE
187 °C 155 rpm 10 lb/h 4.41 wt % 38.6 mmHg 5700 ppm 92%
a melt seal prior to the subsequent vacuum section. The maximum melt pressure across the melt seal is calculated to be about 3.55 MPa. This pressure is used as the initial gas pressure of the stripper bubbles. The 45/ 45 right-handed forward-conveying screw element is used in the vacuum section. The percent channel fill in the vacuum section is 15.3%. The mean residence time in the devolatilization section is about 9 s. The vacuum port position starts at 70 mm after the melt seal; that is, Lc ) 70 mm or is about 2.5 L/D. The corresponding average residence time to the port is roughly 4 s; that is, tc ) 4 s. Comparison of Model with Experiments. Based on the above operating conditions and proper process parameters, the bimodal model is fit for foam growth combined with the film model for devolatilization developed to the experimental data. Figure 4 shows the effects of the number of stripper bubbles and the critical foam volume on the devolatilization. From the above conditions, the bimodal model for foam growth shows that a 3 or 4 times foam volume change is reached very rapidly. The film model for mass transfer is then used to continue the computation to 9 s. The conditions Ns ) 550 no./cm3 of liquid and φcrit ) 3.0 cm3 of gas/cm3 of liquid may well be sufficient to reach 92% devolatilization. At 4 s, devolatilization reaches about 75% and is about 82% of the final devolatilization efficiency (75/92 × 100% ) 82%). In other words, the design factor β ) 0.82. Foaming and breakup occur mostly in the 70-mm lead (or the first 4 s) which contribute to 82% devolatilization. Subsequently, the thin-film-spaced foam cannot withstand the interfacial forces and the shear flow. The foam collapses to a pool-like structure. A diffusional process in the polymer pool contributes to the final 18% of devolatilization. Data Correlation. We have fitted our model to
run no.
T (°C)
ws (wt %)
γ˘ /df (s-1)
ht (s)
Ns (no./cm3 of liquid)
TDEf (%)
1 2 3 4 5 6 7 8 9 10
187 190 252 253 256 191 211 255 232 190
4.41 8.81 8.81 2.20 2.94 5.87 5.15 5.72 5.72 8.47
265.35 264.93 106.53 106.44 70.95 73.46 72.81 155.36 157.36 238.57
9.0 9.0 9.0 9.0 6.0 6.0 5.3 5.8 5.8 8.7
550 500 50 50 100 600 325 50 100 600
92 93 91 96 97 89 88 87 87 93
several experimental data from polyethylene/acrylic acid devolatilization. The process conditions and the corresponding adjusted model parameter (the number of stripper bubbles) are listed in Table 2. The thermodynamic devolatilization efficiency obtained from the experiment for each set of process conditions is also listed. It is noted that the melt temperature is an important factor in determining the diffusion coefficient, which, in turn, dominates the rate of mass transfer. Temperature also influences other physical properties of the polymer such as viscosity and Henry’s law constant. In this work, ws is the concentration of water stripping agent while γ˘ and df are the average shear rate and the degree of fill in the 45/45 forward-conveying screw element, respectively. The ratio of the two values of γ˘ and df is a characteristic for the rate of free surface regeneration, which we believe is an important factor in the incorporation of stripper bubbles into the polymer. Kalyon et al. (1991a,b) reported that, in the processing of thermoplastic elastomers in a co-rotating twin-screw extruder, air bubble entrainment was found in the partially filled channels in contrast to the completely filled channels. The partially filled channels provide a continuous free surface where the polymer melt can contact air. A high surface-to-volume ratio of the melt and the surface renewal rate in the partially filled channels contribute to the incorporation of air into the polymer. ht is the mean residence time in the vacuum zone. Ns is the number of stripper bubbles which is adjustable to fit our model to the experiment. By performing the computation through a range of operating conditions, we are able to correlate the number of stripper bubbles with several important process parameters.
Ns ) function(T,ws,γ˘ /df,th)
(3)
A conventional statistical regression and a backpropagation neural net simulator (Rao and Rao, 1993) are used to correlate the number of stripper bubbles with the process parameters. One statistical regression equation is shown below.
yˆ ) 23 900 - 155.6x1 - 615.6x2 + 14.7x3 1009.2x4 + 0.3x12 + 42.8x22 - 0.04x32 + 52.9x42 (4) where yˆ ) Ns, x1 ) T, x2 ) ws, x3 ) γ˘ /df, and x4 ) ht. The backpropagation simulator utilizes four neurons in the input layer and one neuron in the output layer. One hidden layer is assigned and three neurons are arbitrarily chosen in the hidden layer. The statistical regression method requires a function for correlating the data while the backpropagation neural net requires a learning and pattern recognition procedure.
Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1469 Table 3. Example of Process and Production Requirements in Polyethylene/Acrylic Acid Devolatilization in a ZSK-28 condition
value
temperature, T throughput, m ˘ vacuum pressure, Pvac initial monomer concentration, C0 thermodynamic devolatilization efficiency, TDE design factor, β
220 °C 15 lb/h 40 mmHg 5700 ppm 95% 0.8
We can also correlate the thermodynamic devolatilization efficiency (TDEf) with the corresponding set of process parameters, but we only obtain a single item of information, i.e., TDEf at tf. We lose the useful information obtained by computation through the bimodal/film model. The computation shows the rate of devolatilization with time, i.e., TDE(t) vs t. This information enables the procedure for the design of the devolatilization section as described above. As a result, the correlation between the number of stripper bubbles and the process parameters not only provides the parameter (Ns) necessary to calculate the final thermodynamic devolatilization efficiency (TDEf) but also shows the dynamics of the devolatilization process (TDE(t)) available for the design of the devolatilization section. Design of Devolatilization Section. By applying the correlation developed in eq 3 to obtain the number of stripper bubbles needed for the model, we can design the screw configurations and operating conditions and use our model to predict the devolatilization performance for future experimental design. For instance, Table 3 lists an example of the process and production requirements for polyethylene/acrylic acid devolatilization in a ZSK-28 co-rotating twin-screw extruder. We apply the design algorithm and evaluate what are the screw configurations and operating conditions necessary to satisfy the specifications. Suppose we know the limiting foam volume growth of polyethylene in an extruder is 4 times the initial volume, i.e., φcrit ) 3.0 cm3 of gas/cm3 of liquid. This suggests that the extruder should be operated below 25% channel fill in the vacuum section. First, we use the 45/45 forward-conveying screw element in the vacuum section, operate the extruder at 100 rpm, and inject 4 wt % water. These result in a 6-s mean residence time; 39.15 s-1 average shear rate; and 35.7% channel fill, which provide too small a free volume in the screw channel to accommodate the foam growth of polyethylene. We need to increase the screw speed to reduce the channel fill below 25%. A 145 rpm is found to have a reasonable 24.6% degree of fill. According to the correlation developed, the initial number of stripper bubbles is calculated as 924 no./cm3 of liquid by both methods. The computation shows that TDE ) 96.8% at 6 s. If the design factor is 0.8, then TDEc ) 77.4% and tc ) 2.41 s. We can calculate Lc ) 64.15 mm. In other words, we should locate the vacuum port at a minimum of 2.29 L/D after the melt seal for optimal design of the devolatilization section. Depending on the required design factor, the length Lc can be varied. In our model, we assume that the foam breaks as the maximum foam volume expansion is achieved. The polymer looks like an open-cell web structure with constant liquid films available for mass transfer. In an extruder the shear flow in the screw channels helps break the foam and stretch the polymer films into a
polymer pool with some entrained bubbles in it. Unlike the thin polymer films, the polymer pool does not have a large free surface area for fast rate of mass transfer. Instead, surface renewal in the rotating polymer pool provides the primary mass-transfer mechanism for devolatilization. Thus, our model will overpredict the devolatilization efficiency at the latter stage of devolatilization when foam collapse occurs. One must recognize that foam collapse is a transitional process from a decrease in foaming intensity to a bubble-free polymer pool. At one extreme case, if we assume the foam suddenly breaks into a polymer melt pool beyond Lc, the penetration theory for mass transfer is used to compute devolatilization efficiency at the latter stage of devolatilization rather than the film model. In this case, the bimodal/film model was found to overpredict the devolatilization efficiency by 5-15% more than used by the bimodal model with penetration theory. This only represents one extreme case of the process. The actual overprediction should be less. Although there is a need to modify our model to reflect the complex phenomenon of foam collapse, the current approach shown in this work is reasonable for the purpose of model demonstration and parameter correlation. Conclusions We have schematically shown our concept in the modeling of foam devolatilization and the design and development of the devolatilization section in the extruder. Our objective is to establish a correlation for the design of the devolatilization section which is optimized with respect to foam dynamics and process parameters and to predict the devolatilization performance for a given set of process conditions. On the basis of polyethylene/acrylic acid devolatilization in a W&P ZSK-28 co-rotating twin-screw extruder, we have fitted the bimodal/film model to the experimental data. On the basis of the data fitted over a range of operating conditions, we are able to generate a correlation between the number of stripper bubbles and a list of important process parameters. Both a statistical regression method and a backpropagation neural net approach are used to develop the correlation. We have demonstrated how the correlation developed can aid in the design of the devolatilization section. It suggests the optimal distance (Lc) to arrange the position of the vacuum port in a co-rotating twin-screw extruder for completion of foaming and breakup and for prevention of the vacuum port from being clogged due to intensive foaming and/or elastic climbing of the polymer. Acknowledgment The authors gratefully acknowledge the financial support from DuPont Central Research and Development Department, Wilmington, DE. Nomenclature C ) concentration, g/g df ) degree of fill, % L ) distance, mm m ˘ ) throughput, lb/h N ) number of bubbles per unit liquid volume, no./cm3 of liquid NR ) screw speed, rpm P ) pressure, Pa
1470 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 Pvac ) vacuum pressure, mmHg Si,p ) weighted output in the output layer t ) time, s or h ht ) mean residence time, s T ) temperature, °C TDE ) thermodynamic devolatilization efficiency, % Ui,p ) sum total of the inputs in the hidden layer w ) weight fraction, g/g Wi,j ) weight in the backpropagation net Xi ) input data in the backpropagation net
[(
) ]
∂C ˜ 1 ∂ R ˜3 ) 9 2/3 y˜ + ∂t˜ ∂y˜ Πv Π v
C ˜ avg ) 1 -
Greek Symbols β ) design factor, dimensionless γ˘ ) shear rate, s-1 θ ) bias φcrit ) critical (maximum) foam volume fraction, cm3 of gas/ cm3 of liquid Subscripts
4/3
∂C ˜ ∂y˜
(13)
1 (P ˜ R ˜ 3 - 1) ΠvΠF g
(14)
R ˜ (t˜)0) ) 1
(15)
P ˜ g(t˜)0) ) 1
(16)
C ˜ (y˜ )0,t˜) ) P ˜g
(17)
C ˜ (y˜ ,t˜)0) ) 0
(18)
∂C ˜ (y˜ )1,t˜) ) 0 ∂y˜
(19)
Appendix B: Bimodal Model for Foam Growth
f ) final 0 ) initial s ) stripping agent
t R02/D
(6)
R ˜ ) R/R0
(7)
P ˜ g ) Pg/Pg0
(8)
It has been shown that the stripping agent does not have sufficient time to dissolve in the polymer due to the low diffusion coefficient of the stripping agent in the polymer and the short residence time (typically 5-10 s) in the stripper injection zone in the extruder (Yang, 1995; Yang et al., 1996c). A predominant fraction of the stripping agent stays in the gas phase, either dispersing in the polymer as bubbles or sweeping over the polymer surface. Along with this finding, we assume that the stripping agent only disperses in the polymer as bubbles of a uniform size. A bimodal model for foam growth is developed which takes into account the formation and growth of volatile bubbles and stripper bubbles (Yang, 1995; Yang et al., 1996a,b, 1997a,b). In addition to the bubble population of the volatile component, a second bubble population of the stripping agent is incorporated into the polymer. The creation of stripper bubble sites is assumed to be instantaneous. Each population has a characteristic number and size of bubbles. As with the cell model, each stripper bubble is also regarded as an isolated system such that no mass transfer of the stripper molecules outside the polymer film occurs during bubble growth. In the numerical computation the polymer volume is redistributed to the volatile bubbles and the stripper bubbles by an equal polymer film thickness at each time iteration. Following the development of the cell model for bubble growth, at any time the bubble volume and polymer volume in a cell is related by
(9)
4 4 4 πS 3 ) πRv3 + πVv 3 v 3 3
(20)
(10)
4 4 4 πS 3 ) πRs3 + πVs 3 s 3 3
(21)
Appendix A: Analysis of a Single Cell Growth The growth dynamics of a single spherical bubble in a finite pool of a Newtonian fluid under isothermal conditions is summarized in this section. The analysis is similar to the work of Amon and Denson (1984) and Powell (1987). The dimensionless governing equations have been rederived by Yang (1995), in which the dimensionless bubble radius (R ˜ ) is scaled with respect to the initial bubble radius (R0). The cell model assumes a constant bubble population in the fluid. Each cell consists of a spherical bubble and a finite liquid shell surrounding it. The spherical bubble only grows in the radial direction within the liquid shell. The dimensionless variables are defined as follows:
y˜ )
S03
˜t )
V ˜L )
y - R03
(5)
VL S03
-
R03
C ˜ ) C/C0
The governing equations in dimensionless form with their initial and boundary conditions are listed in the following:
(
)
Πσ tη V ˜L dR ˜ -P ˜f - 4 ) 0 (11) 3 R ˜ tD V dt ˜ ˜L + R ˜ /Πv
P ˜g - 2
(
) |
ΠF 2 d R ˜3 (P ˜ gR ˜ 3) ) 9 1/3R ˜ y˜ + dt˜ Πv Π v
2/3
∂C ˜ ∂y˜
y˜ )0
(12)
where Vv and Vs are the polymer volumes in the volatile bubbles and the stripper bubbles divided by 4π/3, respectively. The conservation of the polymer volume is described as follows:
4 4 1 ) Nv πVv + Ns πVs 3 3
(
)
(
)
(22)
where Nv and Ns are constant numbers of the volatile bubbles and the stripper bubbles, respectively. At each time step, the polymer volumes in the volatile and stripper bubbles (Vv and Vs, respectively) are redistrib-
Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1471
uted such that the thickness (δ) of the polymer film is the same in the volatile and stripper bubbles. That is
Sv - Rv ) δ ) Ss - Rs
(23)
or
(Rv3 + Vv)1/3 - Rv ) δ ) (Rs3 + Vs)1/3 - Rs (24) Vv and Vs are solved simultaneously and substituted into the governing equations for cell growth. The calculated bubble radii are the inputs for solving the polymer volumes in the next time iteration for the bubbles to grow. Appendix C: Film Model for Mass Transfer From observation of the foaming experiments, there is a maximum foam volume for the foam to grow and sustain in which foaming and surface breakup occur (Yang, 1995; Yang et al., 1996a,b, 1997a,b). The polymer becomes an open-cell structure, with thin polymer films providing a free path for the volatile to escape. Beyond this point, the devolatilization is not achieved through diffusion of the volatile component into the volatile and stripper bubbles. Instead, mass transfer occurs by diffusion of the volatile component from the polymer films to the contiguous gas phase. The mass-transfer problem of sorption and desorption through a polymer has been investigated in the literature (Crank and Park, 1968). Based on simple sorption and desorption kinetics, mass transfer of a volatile component through a polymer film can be considered as diffusion of the component through the surfaces of a plane sheet of constant thickness, δ. The governing equation for mass transfer and the corresponding initial and boundary conditions are shown below.
∂2C ∂C )D 2 ∂t ∂x
(25)
t ) 0, any x; C ) C0
(26)
t > 0, x ) (δ/2; C ) Ce
(27)
I.C.: B.C.s:
To express the rate of mass transfer in terms of thermodynamic devolatilization efficiency (TDE), the above equation is transformed as
(
TDE(t) % ) 1 -
)
Cf(t) - Ce C0 - Cf(t) × 100% ) × C0 - Ce C0 - Ce 100% (28)
where Cf(t), C0, and Ce are the volatile concentration at the final time t, initial, and equilibrium conditions, respectively. δ is the polymer film thickness, and D is the diffusion coefficient of the volatile component in the polymer. Literature Cited Albalak, R. J., Ed. Polymer Devolatilization; Marcel Dekker: New York, 1996. Amon, M.; Denson, C. D. A Study of the Dynamics of Foam Growth: Analysis of the Growth of Closely Spaced Spherical Bubbles. Polym. Eng. Sci. 1984, 24, 1026-1034.
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1472 Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 Tukachinsky, A.; Talmon, Y.; Tadmor, Z. Foam-Enhanced Devolatilization of Polystyrene Melt in a Vented Extruder. AIChE J. 1994, 40, 670-675. Wasserman, P. D. Neural Computing: Theory and Practice; Van Nostrand Reinhold: New York, 1989. Werner, H. Devolatilisation of Plastics; Verein Deutscher Ingenieure VDI-GmbH: Dusseldorf, Germany, 1980. Yang, C.-T. A Study of Trace Devolatilization of Polymers. Ph.D. Thesis, University of Maryland, College Park, MD, 1995. Yang, C.-T.; Bigio, D. I. Design of the Vacuum Section and Modeling of Polymer Foam Devolatilization in Screw Extruders. In Compounding ’96; Executive Conference Management: Plymouth, MI, 1996a; Session 4, Paper 2, pp 1-15. Yang, C.-T.; Bigio, D. I. Foam Devolatilization Modeling in an Extruder. In AIChE Annual Meeting; American Institute of Chemical Engineers: New York, 1996b; Paper 72f. Yang, C.-T.; Bigio, D. I. Analysis and Modeling of Polymer Devolatilization in Screw Extrusion and Compounding Processes. In Annual Technical Conference Proceedings; Society of Plastics Engineers: Brookfield, CT, 1997; pp 156-161. Yang, C.-T.; Smith, T. G.; Bigio, D. I.; Anolick, C. A Model of Polymer Devolatilization. In Annual Technical Conference
Proceedings; Society of Plastics Engineers: Brookfield, CT, 1996a; pp 350-355. Yang, C.-T.; Bigio, D. I.; Anolick, C.; Smith, T. G. Polymer Trace Devolatilization: Foam Experiments and Modeling. In AIChE Annual Meeting; American Institute of Chemical Engineers: New York, 1996b; Paper 213b. Yang, C.-T.; Bigio, D. I.; Smith, T. G. Prediction of Diffusion Coefficients in Polymer-Solvent Systems Using Free-Volume Theory. Chem. Eng. Sci. 1996c, submitted for publication. Yang, C.-T.; Smith, T. G.; Bigio, D. I.; Anolick, C. Polymer Trace Devolatilization: I. Foaming Experiments and Model Development. AIChE J. 1997a, 43, 1861-1873. Yang, C.-T.; Smith, T. G.; Bigio, D. I.; Anolick, C. Polymer Trace Devolatilization: II. Case Study and Experimental Verification. AIChE J. 1997b, 43, 1874-1883.
Received for review May 8, 1996 Revised manuscript received September 29, 1997 Accepted December 19, 1997 IE9602623
