A stochastic flow model for a tubular solution polymerization reactor.
The engineering of polymerization reactors may constitute a very complex issue, as it depends on countless aspects, such as the polymerization kinetics (reaction mechanism), physical properties related to transport phenomena (heat and mass transfer, mixing), reactor configuration (tubular or tank), and of reactor operation conditions. The integration of these factors defines the macromolecular architecture and the morphological properties of the final product .
The tubular reactor is a very simple piece of equipment because it does not present any movable parts. Besides, the heat transfer capacity of these reactors is very high, because of the large ratio between the heat transfer area and the reactor volume. However, significant radial velocity profiles may develop during the reaction . During bulk and solution polymerizations, the increase of monomer conversion leads to significant increase of the solution viscosity, especially in the vicinities of the reactor walls, where the average residence time is larger. This may cause a significant effect upon product homogeneity and process operation [2, 3] and also may cause the increase of the radial velocity gradient and the reduction of the heat transfer coefficient . To minimize problems associated with the large residence time distributions (RTD), external recycling [4-8] and static mixers  are normally used, to reduce the effects of radical velocity profiles upon process operation.
The quantitative study of the fluid-dynamic behavior of chemical reactors is usually performed through tracer response tests . Basically, a constituent (tracer) that does not interact with the reaction system is added into the reactor feed stream, while its concentration is monitored continuously in the outlet stream. The commonest types of feed perturbations are the pulse, step, and sinusoidal modification of the tracer feed concentration. Fluid-dynamic characterization through dynamic tracer response testing is used thoroughly in polymerization systems. For example, tracer response testing has been used for the evaluation of the performance of extruders [11-14], empty tubular reactors , packed tubular reactors , and loop tubular reactors [4-8].
The performance of empty tubular solution polymerization reactors has been evaluated experimentally trough introduction of step inhibitor concentration perturbations in the feed stream [6, 7, 16-18]. Complex dynamic responses could be observed, including oscillatory behavior and multiple delayed responses [6, 7, 16-18]. As a consequence, a compartmental model was proposed to explain this complex fluid-dynamics, where two concentrical plug flow reactors were assumed to coexist. In the central reactor, it was assumed that the polymer solution is allowed to flow freely, while a viscous layer is formed in the annular reactor, in the proximities of the reactor walls. It is important to emphasize that oscillatory responses could not be explained entirely by deterministic models in the previous papers, as diffusion effects are usually unimportant in these very viscous systems and the reactors are normally very long.
A very simple definition for a stochastic process can be given as a sequence of random variables ordered by an index set . Stochastic phenomena are classically observed in many experimental studies, when different experimental results can be observed even when experiments are performed in very similar conditions . Physical and chemical phenomena are normally interpreted in terms of mass and energy balance models. As such mechanistic models are constituted by deterministic equations, they cannot explain the different observed experimental data when similar experiments are performed. For this reason, several studies assume that the experiments can be subject to stochastic processes [20-27]. For instance, stochastic models have been proposed to explain chemical reactions [22, 24, 25] and mixing of solid particles [20, 21, 23].
A stochastic micromixing model was presented by Too et al.  to describe axial dispersion in tubular reactors. Call and Kadlec  developed a stochastic micromixing model to describe the observed fluctuations of the RTD in flow systems. This work was based solely on simulations, since the "experimental" data were generated by an independent computer program.
In the present work, the fluid-dynamic characteristics of three tubular solution polymerization reactors are investigated both theoretically and experimentally. Three distinct reactor configurations are used: an empty tubular reactor, a tubular reactor packed with continuous mixing elements, and a tubular reactor packed with discontinuous mixing elements. Distinct configurations were analyzed to verify whether the use of mixing elements might cause significant modification of the reactor behavior. Free-radical solution styrene polymerizations initiated by benzoyl peroxide are performed in the tubular reactors at 70 and 85[degrees]C. The experimental system was selected because of the practical industrial importance of tubular styrene polymerizations and because previous studies detected complex flowing patterns for this system [6, 7, 16-18]. The fluid dynamic characterization was performed through introduction of step tracer concentration perturbations in the feed stream. As shown here, the experimental response data present random fluctuations that can be explained in terms of a simple stochastic model, which assumes that plugs of concentrated polymer solution are released at random from the reactor walls from time to time.
Residence Time Distribution
The RTD can be obtained experimentally through introduction of a tracer pulse on the reactor feed stream. The tracer concentration response, C(t), obtained at the outlet stream of the reactor is the RTD, when presented in terms of a normalized concentration response E(t) :
E(t) = C(t)/[[[infinity].[integral].0] C(t)dt] (1)
Other common type of feed perturbation is the step feed modification of the tracer concentration, which is much easier to perform experimentally. The step response concentration is the accumulated fraction of residence time. Using the standard Danckwerts' notation :
F(t) = [t.[integral] 0] E(t')dt' (2)
Therefore, F(t) is the fraction of molecules that leave the system for residence times that are smaller or equal to t . F(t) can be normalized as:
F(t) = [C(t) - [C.sub.[infinity]]]/[[C.sub.0] - [C.sub.[infinity]]] (3)
Some typical step responses, F curves or F diagrams, can be easily obtained for ideal reactors, as extensively discussed in the open literature .
Three reactors were used in the present work, as already described. The empty tubular reactor bas been used in previous works and is described in detail elsewhere [4-8, 16-18, 28]. Two different packing configurations were also used in order to observe whether the fluid-dynamic characteristics of the reactor were sensitive to packing. In the first case, the continuous packing was constituted by stainless steel wire, with 5 spirals/cm and width of 0.5 mm. The steel wire was placed inside the reactor at random, with the help of a flexible metal rod. Each reactor segment (length of 1 m) contained approximately 2 g of packing (porosity of 90%). In the second case, 10 static discontinuous mixing elements with length of 10.0 cm were introduced regularly along the tubular reactor. The static mixing elements were formed by introducing 1.0 g of stainless steel wool in stainless tubes of internal diameter of 0.4 cm and length of 10.0 cm at random. Mixing elements (porosity of 10%) were connected to each other by 11 empty tubes of length of 1.0 m and internal diameter of 0.4 mm. The three reactors were made of stainless steel with external diameter of 0.63 cm (I.D. 0.4 cm) and were 12.0 m long.
All polymerization runs were performed at two different temperatures: 70 and 85[degrees]C. Unless stated otherwise, the feed monomer and initiator concentrations were always made equal to 40 wt% and 1 wt% respectively, in accordance with previous studies and typical operation conditions used to perform solution polymerizations [4-8, 16-18, 28].
The reagents used were: styrene, provided by Nitriflex, Rio de Janeiro, Brazil (polymer grade with minimum purity of 99.9%); toluene, provided by Vetec, Rio de Janeiro, Brazil (analytical grade with minimum purity of 99.9%); benzoyl peroxide (95% of purity) from Reagen, Rio de Janeiro, Brazil; nitrogen, provided by AGA, Rio de Janeiro, Brazil (99% of purity); cyclohexane provided by Vetec, Rio de Janeiro, Brazil (analytical grade with minimum purity of 99.9%); and benzene, provided by Vetec, Rio de Janeiro, Brazil (analytical grade with minimum purity of 99.9%). All reagents were used as received.
The polymerization unit comprised the feeding section, the tubular reactor, the heating system, and the inline analysis system. The feeding section comprised a feeding tank and a Masterflex pump model 7550-62. The flow was monitored continuously and was controlled by a process computer. The heating system comprised an air blower, an electrical resistance, a variable-voltage controller, and two thermocouples. The thermocouples indicated the temperatures of the hot air (used to control the reactor temperature) at two distinct points, 50 cm apart from each other, and close to the reactor walls. Previous studies indicated that these temperature readings are essentially equal to the reaction temperatures [4-8, 14-16, 28]. The variable-voltage controller and the thermocouples were linked to the process computer, which controlled the reaction temperature through manipulation of the heat released by the electrical resistance. The tubular reactors were coiled in the spiral form with approximately 40 cm of diameter. The reactor was placed inside the heating section, which was isolated from the ambient with corrugated aluminum, glass wool, and asbestos. The in-line analysis was performed by a Digital Anton Paar Density meter (model mPDS 2000) placed at the outlet stream of the reactor. The density meter provided the temperature and the density of outlet flowing system. The precision of the equipment was of 0.001 g/mL. The monomer conversion could be calculated from the density values with the help of an empirical correlation obtained by Vega et al. .
Samples of the reaction product were collected in sampling jars regularly. Known amounts of ethanol were then used to precipitate the polystyrene product, which was analyzed through gel permeation chromatography. The filtrated liquid was analyzed by gas chromatography.
Known amounts of styrene and toluene were measured in test tubes and added into the feed tank. Then, the tank feed was covered and the nitrogen flow was initiated, to agitate the feed mixture and to remove the oxygen. After a few minutes, the initiator was weighed in analytical scales and added into the feed tank. The pump was turned on and the flow calibration procedure was started. A feed flow rate of 2 mL/min was used.
Sampling and Chromatography
The cyclohexane or benzene outlet concentrations of the filtrated liquid were measured off line in an Autosystem gas chromatograph, with capillary column of melted silica Carbowax 20 M with length of 50 m and internal diameter of 0.25 mm. The gas carrier was helium. The temperature was initially maintained at 50[degrees]C for 4 min and later a heating rate of 6[degrees]C/min was imposed to reach the temperature of 140[degrees]C. The detector used was based on flame ionization.
Tracer Step Perturbation of Pure Solvent
The tracer (benzene) perturbation of pure solvent (toluene) was performed at normal polymerization conditions to evaluate the importance of the polymer species on the final results. After reading density, the benzene concentration was suddenly increased from 0 to 10% in volume in the feed tank. The outlet concentration was obtained by gas chromatography.
Tracer Step Perturbation of Polymeric Systems
The reactor flow characterization was performed through step tracer response testing with cyclohexane or benzene. First, the reaction was stabilized at the desired reaction conditions, as indicated by the in-line density meter. Then, the feed concentration was suddenly changed, by adding a known volume of tracer (10% of feed tank volume) into the feed tank and mixing vigorously. Experiments were also performed by recycling the output stream of the polymerization experiments, after addition of inhibitor (hydroquinone), to evaluate the importance of the reaction condition on the observed results.
The obtained F(t) curves were normalized and confined in the interval [0, 1], since some F(t) values did not stabilize at the limiting value of 1 after long experimental times. In these cases, it was arbitrarily assumed that the concentration [C.sub.[infinity]] in Eq. 3 was equal to the largest concentration value observed along the experiment.
RESULTS AND DISCUSSION
Figures 1 and 2 show that the empty tubular reactor behaves as a typical PFR reactor. In spite of the low Reynolds number (Re [approximately equal to] 20) that indicates laminar flow, the tracer concentration increases suddenly near the mean residence time. The blank experiments, performed with constant tracer concentration (as in the case of benzene in solvent, curve 4), indicate that the behavior of the tracer concentrations is very stable and that sampling treatment and chromatographic measurements are reproducible. Similar behavior was observed in all blank experiments performed in the empty and packed reactors, regardless of the polymer concentration of the flowing mixture . This clearly indicates that the selected analytical procedures are appropriate for the analysis of polymer and tracer concentrations.
[FIGURE 1 OMITTED]
Figures 1 and 2 show that, in the presence of the polymer species, the F(t) curves present much more complex behavior. At first, there is an apparently slower growth of the tracer concentration, showing the existence of more intense mixture modes, that is, the existence of deformed velocity profiles. But the most outstanding characteristic of these curves is the apparent oscillatory behavior of the tracer concentrations. These fluctuations cannot be credited to measurement errors, as replicate measurements (analyses of different samples obtained at the same time) confirm the obtained tracer concentration values, and reflect a different fluid dynamic behavior of the polymeric system.
It is interesting to emphasize some points here. First of all, no regular flow model is able to explain the observed behavior for the F(t) curves. Any flow model that admits continuous and decreasing variation of velocity profiles from the center to the tube wall results in a regular and growing profile for F(t). The second point is that F(t) oscillations are clearly limited to a relatively narrow range close to value 1, independently of the analyzed condition. The experimental range of fluctuations is reproducible. The third point is that the reactor temperature and the particular tracer component used to perform the experiments do not seem to exert significant influence on the analysis of the experimental data. Finally, as discussed below, the introduction of the packing material provokes very pronounced increase of the perturbations associated with the measurement of F(t), which indicates that oscillatory responses are related to the fluid-dynamic characteristics of the developed flow.
The oscillatory behavior of the F(t) curves shown in Figs. I and 2 can be assigned to: (1) experimental errors; (2) errors in the tracer concentration analysis; and (3) complex fluid-dynamic behavior of the polymeric system.
The standard deviation of the tracer concentration analysis is equal to 0.03 g/L, as obtained through replication of tracer concentration measurements for all samples. By assuming that [C.sub.[infinity]] is constant in Eq. 3, then the standard deviation for evaluation of F(t) is equal to [29, 30]
[error.sub.F(x)] = 0.03/([C.sub.[infinity]] - [C.sub.0]) (4)
Therefore, the experimental error for evaluation of F(t) is different for each experiment. As shown in Figs. 3 and 4, the oscillatory behavior cannot be explained by measuring experimental errors. This is confirmed by independent inline measurement of the density of the flowing mixture and of average molecular weights of polymer samples, which are also subject to oscillatory behavior when the initiator feed concentration is changed. Therefore, it is assumed here that the observed oscillatory behavior is caused by the complex fluid-dynamic behavior of the flowing mixtures.
The axial dispersion and power law models cannot represent the experimental data , since they are unable to describe the experimental oscillatory behavior of F(t). In these cases, estimation of power law exponents leads to different exponents at different experiments (0.17-0.25), which can indicate a change of the convective flow behavior. Besides, the Peclet numbers estimated with the help of a standard axial dispersion model generally lies in the range 40-50, indicating that the flow resembles the PFR model .
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
On the basis of the previous results, it is assumed here that it is possible to describe the flow characteristics of the flowing system by combining a PFR model with a source of random perturbations, as described below.
A Stochastic Model
The tracer response tests presented here show random oscillatory behavior in the proximities of F(t) = 1, which cannot be explained in terms of the experimental errors and is increased when packing is added. Therefore, it seems plausible to assume some sort of stochastic perturbation affecting F(t) values. This type of explanation has already been proposed to explain RTD in other systems [31-33].
[FIGURE 4 OMITTED]
A simple model that can be used to explain the observed fluctuations in the F(t) curve can be built by assuming the existence of different flow regions, as in two concentric tubes (layers). It is assumed here that the solution layer that is closer to the wall flows more slowly and that some portions of this layer are eventually dragged by the central layer, which flows much faster (see Fig. 5). It is assumed that the external layer constitutes an almost stagnant layer that flows much slower than the average flow velocity. This layer is periodically (and randomly) dragged by the tensions induced by the flowing central layer. During the perturbation, it is assumed that the tracer flows primarily in the central layer, although tracer concentrations oscillate in the exit stream because of the dragging effect of the external layer. Similar fluctuations are normally observed in the extraction petroleum industry, during the extraction of multiphase gas--oil--water mixtures, and are taken into consideration in the dynamic models used to project oil equipments [34, 35].
The model is implemented here by assuming that the mean flow velocity ([v.sub.m]) is constant and that the velocities of the central layer ([v.sub.m1]) and of the external layer ([v.sub.m2]) are subject to random fluctuations. It is also assumed that internal and external flows follow the plug flow behavior.
According to the proposed model, three parameters are required to completely describe the flow:
* the thickness of the stagnant layer ([phi]);
* the characteristic period of the random fluctuations of the flow velocity: [DELTA]t;
* the maximum allowed flow velocity of the external layer: [v.sub.m2].
[FIGURE 5 OMITTED]
The flow velocities of the distinct layers must obey the global mass balance of the reaction system. By assuming that there is no volume contraction:
[A.sub.1][v.sub.m1] + [A.sub.2][v.sub.m2] = [A.sub.T][v.sub.m] (5)
where [A.sub.i] is the transversal flow area of the different layers, the index 1 represents the internal tube, and the index 2 represents the external layer. [v.sub.mi] are the flow velocities in these two hypothetical layers. Rearranging:
[v.sub.m] = [[A.sub.1]/[A.sub.T]] x [v.sub.m1] + [[A.sub.2]/[A.sub.T]] x [v.sub.m2] = [[phi].sub.1][v.sub.m1] + [[phi].sub.2][v.sub.m2] (6)
where [phi] = [[phi].sub.1] is the percentage of the transversal tube area that is occupied by the faster layer. Therefore, the velocity of the slow layer does not exceed the value of the mean global velocity [v.sub.m]. The mean flow velocity of the external layer can be calculated in the model by multiplying the average flow velocity by a random number, generated by the Park and Miller algorithm . The At value can be defined with good precision if F(t) is monitored continuously for a long time. However, the value of 30 min, the double of the smallest interval of sample retreat, was assumed here.
Distinct characteristics of the simulated F(t) curves can be associated with distinct model parameters. The amplitudes of the oscillations are related primarily with the parameter [phi]; the larger the [phi] values, the larger the amplitude of the oscillations. The length of time for observation of oscillations is related with the maximum flow velocity of the external layer; the smaller [v.sub.m2], the longer time required for stabilization. The frequency of the oscillatory response is related with [DELTA]t. Figure 5 clarifies this point graphically.
[FIGURE 6 OMITTED]
Verification and Validation
Tracer response tests were performed for the three analyzed tubular reactors at two different temperatures (70 and 85[degrees]C), using both benzene and cyclohexane as tracers. Random oscillatory behavior was observed in all polymerization experiments (see Figs. 1-4 and 6-8). Given the nature of the stochastic model, parameters [phi], [v.sub.m2], and [DELTA]t are obtained from qualitative analysis of the experimental F(t) curves and model responses.
[FIGURE 7 OMITTED]
The oscillations of the experimental F(t) curves for the empty reactor (Fig. 6 is an example) presented amplitudes in the range from 0.8 to 1.0, which is obtained for [phi] = 0.67-0.81. The oscillatory responses of the F(t) curves did not disappear after 360 min, which indicates that the maximum velocity of the stagnant layer is smaller than 2.4% of the average flow velocity. This probably explains the long concentration drifts observed in previous studies [6, 7, 16, 17]. The period of the oscillatory response is about [DELTA]t = 4-15 min.
[FIGURE 8 OMITTED]
For the tubular reactor packed with continuous mixing elements, the amplitude of the oscillatory response was in the range from 0.6 to 1.0 (Fig. 7 is an example); therefore, [phi] = 0.21-0.96, and the characteristic period of the oscillations was larger ([DELTA]t = 4-20 min). The use of mixing elements accentuates the oscillatory behavior of the F(t) curves, which can be an indication of the enlargement of the stagnant layer.
The previous results are confirmed by results obtained for the reactor with discontinuous packing. The amplitude of oscillations were in the range of 0.5-1.0 at the two temperatures in this case (Fig. 8 is an example) ([phi] = 0.3-0.96). The characteristic period of the oscillations was larger than that obtained for the empty reactor, but smaller than that observed for the reactor with continuous packing. This is very consistent with the assumption that internal mixing elements may provoke the enlargement of stagnant regions in these reactors. As the blank experiments do not lead to oscillatory responses, it may be concluded that the development of concentration gradients (and of viscosity gradients), caused by existence of velocity gradients and large RTD inside the reactor, are fundamental for the development of the complex fluid-dynamic behavior of such reactors.
On the basis of these results, it becomes clear that the introduction of mixing elements may be inadequate for polymerization reactions performed in tubular reactors with low Reynolds number. In these cases, mixing elements may contribute with formation of additional stagnant zones, which may lead to further increase of polymer heterogeneity and cause reactor clogging.
The fluid-dynamic characterization of three distinct tubular polymerization reactors was accomplished through dynamic tracer response F(t) curves. Step tracer responses indicate that complex fluid-dynamic behavior can be obtained during polymerization, and that this behavior may be governed by stochastic phenomena.
Typical observed F(t) curves contain two distinct regions: in the first region, F(t) values increase from 0, while F(t) values start to oscillate when they get closer to 1. The axial dispersion and the power law models can represent the average observed F(t) values, but are unable to represent the oscillatory behavior. In both cases, estimation of model parameters indicates that the flow approaches the behavior of typical plug flow reactors.
A stochastic model is proposed to represent F(t) oscillations. The model depends on three parameters: the thickness of the external stagnant layer, the characteristic frequency of the random perturbations, and the maximum flow velocity of the stagnant layer. Based on the qualitative model behavior, it is possible to estimate that the average velocity of the stagnant layer may be almost 50 times smaller than the velocity of the core fluid. The model is able to represent available experimental data fairly well at distinct operation conditions. Particularly, it seems that packing elements accentuate the oscillatory behavior of the F(t) curves, indicating that mixing elements can cause the enlargement of the stagnant zones. For this reason, addition of mixing elements in tubular polymerization reactors should be performed with caution.
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Ardson dos S. Vianna Jr., (1) Evaristo C. Biscaia Jr., (2) Jose Carlos Pinto (2)
(1) Secao de Quimica (SE/5), Instituto Militar de Engenharia (IME), Praca General Tiburcio 80, CEP 22280-270, Rio de Janeiro, Rio de Janeiro, Brazil
(2) Programa de Engenharia Quimica--COPPE-Universidade Federal do Rio de Janeiro, Cidade Universitaria, C.P 68502 CEP 21945-970 Rio de Janeiro, Rio de Janeiro, Brazil
Correspondence to: Jose Carlos Pinto; e-mail: email@example.com
Contract grant sponsor: Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), Brazil.
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|Author:||Vianna, Ardson dos S., Jr.; Biscaia, Evaristo C., Jr.; Pinto, Jose Carlos|
|Publication:||Polymer Engineering and Science|
|Article Type:||Technical report|
|Date:||Nov 1, 2007|
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