Polymer unsteady-state

We have developed a detailed two-phase model for the UNIPOtr process. This model was used to investigate the steady and dynamic characteristics of this important industrial process that creates polymers directly from gaseous components. The reader should develop MATLAB programs to solve for the steady state and the unsteady-state equations of this model. This will enable him or her to investigate... 

Inaba, A. and Kashiwagi, T. A calculation of thermal degradation initiated by random scission. II. Unsteady state radical concentration. European Polymer Journal 1987 23 871. 

Continuons emulsion polymerization is one of the few chemical processes in which major design considerations require the use of dynamic or unsteady-state models of the process. This need arises because of important problems associated with sustained oscillations or limit cycles in conversion, particle number and size, and molecular weight. These oscillations can occur in almost all commercial continuous emulsion polymerization processes such as styrene (Brooks et cl., 1978), styrene-butadiene and vinyl acetate (Greene et cl., 1976 Kiparissides et cl., 1980a), methyl methacrylate, and chloropene. In addition to the undesirable variations in the polymer and particle properties that will occur, these oscillations can lead to emulsifier concentrations too low to cover adequately the polymer particles, with the result that excessive agglomeration and fouling can occur. Furthermore, excursions to high conversions in polymer like vinyl acetate... 

From the viewpoint of prediction of service lives, the photochemical deterioration processes of polymers used as paints and finishes are theoretically analyzed based upon unsteady state dynamics. Theoretical results are compared with experimental data under natural and accelerated exposure. Infrared spectra and scanning micrographs show that the deterioration proceeds continuously inwards from the surface, but differently with the exposure conditions. Parabolic (/t ) law was derived approximately for the increase in the depth of the deteriorated layer of polymers with time. Paying attention to the influence of the deterioration of polymeric finishes, the parabolic law involving a constant term was also derived for the progress of carbonation of concrete. These parabolic laws well predict the progress of deterioration and explain the protective function of finishes on reinforced concrete. 

This report deals with dynamic processes of the deterioration of polymers often used as paints and finishes in housing, and also refers to their influence as the reduction in protective performance on the durability of reinforced concrete. The deterioration processes of polymers by the simiiltaneous action of ultraviolet (UV) light and diffusive oxygen is explained theoretically based upon unsteady state dynamics. The parabolic law (/t" law) is derived for a typical path for the progress of the deterioration of polymers inwards from the surface (l), and compared with some experimental data. The same parabolic law involving a constant term was also derived for the carbonation of concrete, which well explains the retardation effects of finishes on the carbonation (2). 

Outline of the Theoreyical Model. The main assumptions for the unsteady state dynamics are as follows l) Only polymer moleciiles which are raised into excited state by absorbing UV light (photon flux, no wavelength, X) near the absorption band charasteristic of polymers can participate in photochemical reactions (efficiency, n molar concentration, C ). (2) Photochemical reactions are i) depolymerization of activated polymer molecules (first order reaction,... 

The deterioration progresses continuously from the surface, and the depth of the deteriorated layer increases in proportion to the square root of exposure time as shown in equations 7- and 8. This parabolic (/t" ) law was obtained as the natural derivation based upon unsteady state dynamics, assuming the simultaneous action of UV-li t and diffusive oxygen. Figure 12 is the least-square plot based on experimental data from Kubota et al. (2.) It can be seen that many polymers show the deterioration by the power law of exposure time (t n = 0.5 - l.O). The difference between theory and experiment is considered to be due to the complex mechanisms not explictly treated in this theory. 

Fick s first law provides a method for calculation of the steady state rate of diffusion when D can be regarded as constant during the diffusion process, and the concentration is a function only of the geometric position inside the polymer. However, concentration is often a function of time as well as of position. We said Equation 14.9 describes a steady state flow, but how does the system reach this steady state The unsteady state flow, or transient state, is described by Fick s second law. For a one-dimensional diffusion process, this can be written as... 

A partly reversible polymer retention is observed during unsteady-state polymer flow. Mathematical equations describing the unsteady-state polymer flow are presented. 

The effect of residual oil saturation on polymer retention and the polymer retention during the displacement of oil from porous media has not been reported. Although some phenomena [1-11] indicated that more polymer is retained in the first segment of a porous media, the literature lacks quantitative data on the distribution of retained polymer in porous media. The mechanism of polymer retention during unsteady-state flow [11] has not been adequately described. 

It was earlier reported [11] that using the same polymer and porous media, steady-state or unsteady-state flow can be obtained depending on the flow condition. Our further laboratory studies showed that this phenomenon is commonly observed with a wide variety of water soluble polymers. However, the critical flow parameters of different polymers can vary greatly. 

Although the knowledge on unsteady-state polymer flow is incomplete, a working hypothesis can already be given based on observed phenomenon in the laboratory. This description of unsteady-state polymer flow must be in good agreement with well-established experimental facts, which are as follows ... 

During unsteady-state flow the effluent polymer concentration is constant, and always less than the injected concentration. Generally the effluent polymer concentration is only a few percent less than the influent concentration. 

Figure 3 illustrates the development of an unsteady-state polymer flow. In Figure 3a it is assumed that the polymer molecules have a coiled shape in brine at low linear velocity. The polymer molecules travel on the tracks of the stream lines. The molecules can easily adjust their own shape and their surroundings to the changing velocity field at any location. 

Fig. 3. Schematic illustration of polymer retention during unsteady-state flow.

As we have seen above, the development of unsteady-state flow is concentration dependent. The number of molecules expelled into the dynamic trap in unit time is proportional to the difference between the actual flowing concentration and the critical polymer concentration. This concentration difference, called excess polymer concentration is large at the inlet face, therefore, the rate of polymer buildup is also large. Far from the inlet face, this excess polymer concentration is greatly reduced since the rock has already stripped out a portion of this excess polymer concentration. The rate of polymer buildup will consequently be reduced far from the inlet surface. The experimental verification of this fact will be given in the Discussion. 

Fig. 4. Distribution of flowing polymer concentration in a linear system during unsteady-state flow.

It was earlier pointed out that after the first invasion of a porous body with a polymer solution, the retained polymer shows a characteristic distribution. If, for the first invasion the flow parameters are chosen in such a manner that finally a steady-state flow is attained, this characteristic distribution of retained polymer will not change with time. This function determines a distribution of resistance factors. Then, if the flow rate is increased to a value above the critical velocity, the initial distribution of the resistance factors at this increased velocity shows also a characteristic distribution. Further analysis of the initial resistance factor distribution during unsteady-state flow is given in Appendix B. 

Figure 11 shows the polymer flow and residual resistance factor curve related to the experiments described in Figure 9. Since the effluent concentration did not reach the injected concentration at residual oil saturation, an unsteady-state flow developed.

The unsteady-state polymer flow resistance factors for any moment and distance can be determined from laboratory experiments, provided that the pressures are measured at an adequate number of locations. Combining these experiments with the determination of the flowing polymer concentration at difierent locations, the value of C it. or the absolute quantities of retained polymer can be determined for any location and for any time. 

Suppose the f3 =0.24 from an independent study. The injected polymer concentration is 1200 ppm, and the length of the porous body is 14.9 cm. If the effluent concentration is 1100 ppm (representing a 8.33% polymer loss), the value of C Ht is 1097.1 ppm, applying Equation (8). Using a lower injection rate, the value of is higher. Lowering the injection rate, one can reach a flow velocity at which Ccrit = Q> that is, the unsteady-state flow converts to a steady-state flow. 

The polymer flow resistance factor vs flow rate curve has a minimum in many cases [10]. It was also shown earlier [11] that over a critical flow rate an unsteady-state polymer flow can develop. In such a case, only the first value of the measured resistance factor will be on the resistance factor versus flow rate curve on its imaginary section (Figure 33). 

Steady-state and unsteady-state polymer flow resistance factors at different distances from the injection face as a function of time. 

From unsteady-state heat transfer into a sphere (Bird et al., 1960), = T-Toy (Ti To)=0.99 at the center where the thermocouple is located. If T-To = 2-1 PC, then Ti = roH-(r-7o)/0=lOO+(2-ll)/(O.99)=lO2-llPC, which is also equal to the polymer surface particle temperature or the average polymer particle temperature. 

These equations are then applicable to many aspects of polymer processing, such as cooling material in a mold, heating a sheet, and thermoforming. Fundamentally, they represent cases of unsteady-state heat transfer, and the solutions of Eq. (4-7) and (4-8) can be quite complex. However, generalized solutions for various shapes do exist. The basis for these is to neglect Aq and then to solve the remaining equation ... 

Fig. 4-12 Unsteady-state heat-transfer to polymer chips.

A model [15] based on the schematic shown in Fig. 9-19 was used to analyze the heating of the polymer. The model used an unsteady-state approach with an effective thermal diffusivity (aeffective)-... 

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