Kim B. McAuley

Dynamic optimization of constrained chemical engineering problems using dynamic programming

Abstract In many chemical engineering process control applications, one frequently encounters differential-algebraic optimization problems. Such optimal control problems are difficult to solve, in general, because of the presence of singular arcs for systems whose Hamiltonian is linear with respect to the control variable. We propose the use of absolute error penalty functions (AEPF) in handling constrained optimal control problems in chemical engineering by posing the problem as a nonsmooth dynamic optimization problem. We show that Iterative dynamic programming (IDP) is a very useful technique for solving constrained dynamic optimization problems without unduly increasing the dimension of the system or the computational burden. A move suppression criterion has been incorporated into the IDP algorithm in order to penalize excessive control moves. To show the efficacy of the method, an analytical (exact) solution of a simple problem is obtained using least squares control theory and compared with results obtained using IDP. Results obtained for other seemingly difficult optimal control problems in chemical engineering compare very favourably with those reported in the optimization and optimal control literature.

Parameter estimation in continuous-time dynamic models using principal differential analysis

Principal differential analysis (PDA) is an alternative parameter estimation technique for differential equation models in which basis functions (e.g., B-splines) are fitted to dynamic data. Derivatives of the resulting empirical expressions are used to avoid solving differential equations when estimating parameters. Benefits and shortcomings of PDA were examined using a simple continuous stirred-tank reactor (CSTR) model. Although PDA required considerably less computational effort than traditional nonlinear regression, parameter estimates from PDA were less precise. Sparse and noisy data resulted in poor spline fits and misleading derivative information, leading to poor parameter estimates. These problems are addressed by a new iterative algorithm (iPDA) in which the spline fits are improved using model-based penalties. Parameter estimates from iPDA were unbiased and more precise than those from standard PDA. Issues that need to be resolved before iPDA can be used for more complex models are discussed.

A COMPARISON OF TWO-PHASE AND WELL-MIXED MODELS FOR FLUIDIZED-BED POLYETHYLENE REACTORS

A steady-state model incorporating interactions between separate bubble and emulsion phases in a fluidized-bed polyethylene reactor was developed by Choi and Ray [Chem. Engng Sci.40, 2261–2279 (1985a)]. Correlations for maximum stable bubble size indicate that bubbles within the bed are considerably smaller than those in their original model. In the paper, the influence of bubble size and superficial velocity on reactor operation are examined. It is shown that bubble size critically influences the rate of heat and mass transfer within the bed, and when the bubbles are as small as those predicted by the maximum stable bubble size correlations, there is little or no resistance to the transfer of monomer and heat between the phases. A simplified well-mixed model is developed to describe reactor operation in the limiting case where there is no difference between bubble and emulsion gas temperatures and concentrations. The differences between the predictions of temperature and monomer concentrations of the two-phase and simplified models are less than 2 or 3 K and 2 mol%, respectively, in the operating range of industrial interest. Therefore, a simple back-mixed model is appropriate for predicting temperature and concentration in the gas phase of industrial fluidized-bed polyethylene reactors.

TEMPERATURE CONTROL OF INDUSTRIAL GAS PHASE POLYETHYLENE REACTORS

Abstract The performance of linear and nonlinear temperature control schemes is assessed for an open-loop unstable gas-phase polyethylene reactor (GPPER), based on speed, damping, robustness and the ability to maintain closed-loop stability in different operating regimes. An existing industrial GPPER model is improved by modelling the temperature states in the external heat exchanger using linear and nonlinear driving force models with varying numbers of heat transfer stages. Differences in heat exchanger models do not produce gain mismatch but do result in phase mismatch. It is shown that the nonlinear error trajectory controller (ETC) exhibits significantly superior responses in terms of speed, damping and robustness compared with an optimally-tuned PID controller. Therefore, substantial benefits could be realized using nonlinear controllers because they can provide good disturbance rejection capabilities and ensure closed-loop stability over a wide range of operating conditions. An approach is presented for tuning ETCs for minimum-phase processes of arbitrary relative degree.

Temperature increases associated with polymerization of irradiated PAG dosimeters

Polyacrylamide gel (PAG) dosimeters show considerable promise as three-dimensional dosimeters for the verification of complex dose distributions associated with conformal therapy. However, the potential of PAG dosimeters has not yet been borne out in clinical practice and it is apparent that basic investigations of these dosimeters are still required. The polymerization reactions in PAG dosimeters are exothermic and the heat given off by the reactions may influence polymerization reaction kinetics. We report the results of in situ measurements of local temperature increases in irradiated PAG resulting from heat generated by the radiation-induced exothermic polymerization reactions. Temperature changes proportional to the absorbed dose were observed in the irradiated gels, reaching a maximum of 12 degrees C under high-dose conditions, depending on the thermal boundary conditions. This has practical implications, for example, using small vials of PAG to calibrate large phantoms may not be appropriate since temperature differences during irradiation between the calibration vials and phantom may alter the morphology and quantity of the polymer formed, even when irradiated to the same dose. The inhibition of radiation-induced polymerization associated with low-level oxygen contamination is manifested by a delay in the onset of temperature rise during irradiation. The observed temperature changes are used to estimate the percentage conversion of double bonds from the bis/acrylamide monomers by polymerization reactions.

Nitroxide-mediated radical polymerization of styrene in miniemulsion: model studies of alkoxyamine-initiated systems

Abstract A mathematical model has been developed to describe the behavior of the nitroxide-mediated miniemulsion polymerization (NMMP) of styrene initiated by alkoxyamine initiators. The model includes mechanisms describing reactions in the aqueous and organic phases, particle nucleation, the entry and exit of oligomeric radicals, and the partitioning of nitroxide and styrene between the aqueous and organic phases. The influence of nitroxide partitioning on the polymerization kinetics was examined by modeling systems initiated by the alkoxyamines BST and hydroxyl-BST; BST and hydroxyl-BST are benzoylstyryl radicals terminated by the nitroxides TEMPO and 4-hydroxyl-TEMPO, respectively. Predicted monomer conversions, number average molecular weights and polydispersities were in agreement with experimentally measured values. Simulations and mathematical analysis showed that the rate of styrene NMMP is not strongly influenced by the partitioning properties of TEMPO and 4-hydroxyl-TEMPO because of the complex interaction between reaction equilibrium, phase equilibrium, termination and thermal initiation. However, in the absence of styrene thermal initiation, nitroxide partitioning was found to have a significant influence on the polymerization kinetics. The model was also used to make quantitative estimates of: the population of active and dormant polymer radicals derived from both alkoxyamine initiators and thermal initiation; the population of dead polymer chains; and the number molecular weight distributions of living and dead polymer chains.

Nitroxide partitioning between styrene and water

Research into nitroxide-mediated radical polymerization (NMRP) performed in emulsions and miniemulsions has progressed significantly over the past several years. However, our knowledge of the conditions during polymerization (e.g., the nitroxide concentrations in the aqueous and organic phases) is incomplete, and as such we have yet to achieve a clear understanding of the mechanisms involved in these processes. To better understand the conditions present in heterogeneous NMRP, we measured the partition coefficients of 2,2,6,6-tetramethylpiperidinyl-1-oxy (TEMPO), 4-hydroxy-TEMPO, and 4-amino-TEMPO between styrene and water from 25 to 135 °C. Experiments were performed in a 250-mL Parr reactor that was equipped for the simultaneous sampling of the aqueous and organic phases. Aqueous-phase and organic-phase nitroxide concentrations were measured with ultraviolet–visible spectrophotometry. Experiments were also performed at 135 °C in the presence of hexadecane (costabilizer), polystyrene, and sodium dodecylbenzenesulfonate (surfactant) to determine the effects of the miniemulsion polymerization recipe ingredients on the partitioning of TEMPO and 4-hydroxy-TEMPO. On the basis of the measured partition coefficients (expressed as the ratio of the nitroxide concentration in the organic phase to the nitroxide concentration in the aqueous phase), 4-hydroxy-TEMPO was the most hydrophilic of the nitroxides investigated, followed by 4-amino-TEMPO and TEMPO. Hexadecane, polystyrene, and sodium dodecylbenzenesulfonate did not have a significant influence on the partitioning of these nitroxides at 135 °C. Experiments with ethylbenzene instead of styrene demonstrated that thermally generated radicals were not responsible for the observed temperature effects on the measured partition coefficients.

The chemistry and physics of polyacrylamide gel dosimeters: why they do and don't work

Three factors that prohibit widespread clinical use of polyacrylamide gel (PAG) dosimeters are polymerization after irradiation ceases, formation of additional polymer near the edges of irradiated zones, and monomer toxicity. Polymerization can occur long after irradiation ceases because polymeric radicals cannot diffuse and terminate with other radicals. Small monomer molecules can diffuse toward trapped polymeric radicals and polymerize. Edge enhancement occurs because acrylamide and bisacrylamide diffuse from regions of high concentration (where the radical concentration is low) to adjacent regions where monomer concentration is low and the radical concentration is high. As monomers diffuse into the irradiated zone, they are polymerized by radicals near the edge. Acrylamide is a neurotoxin and suspected carcinogen that can be absorbed through the skin and inhaled, making it an undesirable monomer for polymer gel dosimetry. Less toxic monomers, such as n-vinyl formamide, should give similar dosimetry results, with less concern for safety. When selecting alternative monomers and cross-linkers for polymer gel dosimetry, it would be advantageous to choose larger molecules that diffuse more slowly, resulting in less edge enhancement. Larger molecules should also lead to improved safety, because they are less easily absorbed through the skin and are less easily vaporized and inhaled.

Interfacial Mass Transfer in Nitroxide-Mediated Miniemulsion Polymerization

Nitroxide-mediated polymerization (NMP) represents a viable way to produce polymers with predictable molecular weights (MWs) and well-defined architectures. Such polymers have potential application in the fabrication of commercially valuable block copolymers, drug delivery systems, nanotechnology, adhesives, surfactants, and polymer compatibilizers. The distinguishing feature of NMP is the use of a nitroxide, such as 2,2,6,6-tetramethylpiperidine-N-oxyl (TEMPO), to reversibly deactivate active polymer radicals to give a dormant alkoxyamine molecule (see Scheme 1). In Scheme 1, ka is the alkoxyamine dissociation rate coefficient, and kd is the radical deactivation rate coefficient. At 125 8C, the equilibrium coefficient for the reversible deactivation of polystyrene radicals by TEMPO has been measured to be Keq1⁄4 ka/kd1⁄4 2.1 10 11 M. Thus, the chemical equilibrium of the radical deactivation reaction favors the formation of the dormant alkoxyamine. As a result, the concentration of active polymer radicals in NMP systems is lower than in conventional free-radical polymerization systems, which reduces the likelihood Full Paper: A mathematical model has been developed to describe the interfacial mass transfer of TEMPO in a nitroxide-mediated miniemulsion polymerization (NMMP) system in the absence of chemical reactions. The model is used to examine how the diffusivity of TEMPO in the aqueous and organic droplet phases, the average droplet diameter and the nitroxide partition coefficient influences the time required for the nitroxide to reach phase equilibrium under non-steady state conditions. Our model predicts that phase equilibrium is achieved quickly (< 1 10 4 s) in NMMP systems under typical polymerization conditions and even at high monomer conversions when there is significant resistance to molecular diffusion. The characteristic time for reversible radical deactivation by TEMPO was found to be more than ten times greater than the predicted equilibration times, indicating that phase equilibrium will be achieved before TEMPO has an opportunity to react with active polymer radicals. However, significantly longer equilibration times are predicted, when average droplet diameters are as large as those typically found in emulsion and suspension polymerization systems, indicating that the aqueous and organic phase concentrations of nitroxide may not always be at phase equilibrium during polymerization in these systems. Influence of droplet phase TEMPO diffusivity, DTEMPO,drop, on the predicted organic phase concentration of TEMPO. Macromol. Theory Simul. 2002, 11, 953–960 953

Fundamentals of Polymer Gel Dosimeters

The recent literature on polymer gel dosimetry contains application papers and basic experimental studies involving polymethacrylic-acid-based and polyacrylamide-based gel dosimeters. The basic studies assess the relative merits of these two most commonly used dosimeters, and explore the effects of tetrakis hydroxymethyl phosphonium chloride (THPC) antioxidant on dosimeter performance. Polymer gel dosimeters that contain THPC or other oxygen scavengers are called normoxic dosimeters, because they can be prepared under normal atmospheric conditions, rather than in a glove box that excludes oxygen. In this review, an effort is made to explain some of the underlying chemical phenomena that affect dosimeter performance using THPC, and that lead to differences in behaviour between dosimeters made using the two types of monomer systems. Progress on the development of new more effective and less toxic dosimeters is also reported.