The use of inequalities in the design and optimisation of complex chemical and metallurgical plants

An interconnected chemical or metallurgical plant is viewed as a combination of separators, reactors and mixers. The operation of these units may be described in terms of non-linear split factors, conversions and yields. The mass balance coefficient matrix may be transformed into an M-Matrix. Ostrowski's inequality may then be used to provide linear inequalities based on the row sums of the transformed coefficient matrix. The production rates of components are “quasi” monotonic with regard to these row sums. The mass balance equations, together with the inequalities may be transformed into linear constraints for use in either Parametric Linear or Non-Linear Programming. If split factors, conversions and yields are independant, then Parametric L. P. may be used directly. If dependant, however, the row sums are useful parameters which may be used to improve objective functions depending monotonically on production rate in Non-Linear systems. The method has been applied to a flotation plant where production is maximised using L. P.     

SIMULATION OF STEADY DISTILLATION PROCESS ACCOMPANIED WITH MULTIPLE CHEMICAL EQUILIBRIUM REACTIONS BASING ON TRANSFORMED VARIABLES

The equations describing the steady state operation of distillation column with simultaneous equilibrium chemical reactions have been transformed into a new set or MESH equations by introducing transrormed composition, flow rate and enthalpy variables. The new set of equations are similar to equations describing traditional non-reactive distillation, and it is general when one or several reactions occur and inert components are present.

An algorithm combining relaxation method and modified Newton-Raphson method is proposed to solve the new equations, the relaxation method is used to estimate initial values, while the modified Newton-Raphson method is used to set solution. The reactive distillation processes of MTBE synthesis with n-butane and the separation of meta- and para-xylene are simulated as numerical examples     

Two-dimensional Fourier transform rheological study on thermosensitivity of poly(N,N-diethylacrylamide) in aqueous solutions

The phase transition of a thermo-responsive polymer, poly(N,N-diethylacrylamide) (PDEA) above its critical overlap concentration (c*) has been studied by two-dimensional Fourier transform (FT) rheology using Large Amplitude Step Shear Oscillation (LASSO). This technique allows the separation of the linear and nonlinear contributions to different relaxation processes and the determination of their time scale and amplitude through the time response of the shear stress after step strain experiments. The interchain interactions increase at the onset of the phase transition at 29 °C, indicated by an increased non-linear contribution at short relaxation times as compared to the single phase condition. During the phase separation of a concentrated solution above the phase transition temperature, the polymer-rich phase can form a transient network created by the hydrophobic interactions between the collapsed polymer chains. The non-linear behavior of a phase-separated system well above the transition temperature (at 33 °C) reflects the stretching of the bridging chain segments between larger aggregated domains and the coalescence of aggregates broken during the step in strain. Relaxation time distributions have been fitted in the LASSO spectra by the nonlinear regularization (NLREG) technique and the relaxation times have been attributed with various linear and non-linear processes below and above the phase transition temperature.     

Aliphatic polyesters as models for relaxation processes in crystalline polymers: 3. Mechanical relaxation in copolymers of adipic acid with 1,6 and 2,5-hexanediols

The shear moduli of a series of the title polyesters spanning a crystallinity range of 0–60% have been measured as a function of temperature at ≈1 Hz using a torsion pendulum. The experimental isochronal temperature scans of G′ and G″ are fitted to phenomenological equations. With only minor adjustments, the same relaxation spectrum parameters as found dielectrically for these polymers (relaxation shape, central relaxation times) fit the mechanical data. Thus, for the β (glass—rubber) relaxation in the amorphous fraction the broadness is very sensitive to the presence of the crystal fraction and becomes increasingly broad as the degree of crystallinity increases. In contrast, the γ process dynamic behaviour is insensitive to the presence of and degree of crystallinity. Unrelaxed and relaxed moduli values are determined for the γ and β processes. A composite model approach is used to determine bounds on the amorphous-phase unrelaxed and relaxed γ and β moduli from the bulk specimen values. As was the case dielectrically, the γ process, in addition to being assigned to the amorphous fraction, has a strength that depends on the diol composition also. The relaxed (γ + β) amorphous-phase rubbery shear modulus is bound reasonably well from application of the composite model to the bulk specimen values and is assigned the value 100 ± 20 MPa at 250 K. It decreases relatively strongly with increasing temperature.     

Relaxation of the axial strain induced stresses in double–coated optical fibers

The axial strain induced stresses in double‐coated optical fibers are analyzed by the viscoelastic theory. A closed form solution of the axial strain induced viscoelastic stresses is obtained. The viscoelastic stresses are a function of the radii, Young's moduli, relaxation times and Poisson's ratios of the polymeric coatings. If the applied axial strain linearly increases, the induced stresses increase with the time. On the other hand, if the axial strain is fixed, besides the axial stress in the glass fiber, the stresses exponentially decrease with the time. The relaxation of stresses is strongly dependent on the relaxation times of the polymeric coatings. If the relaxation time of the polymeric coating is very long, the viscous behavior of the polymeric coatings will not appear, and the axial strain induced stresses solved by the viscoelastic theory are the same as those solved by the elastic theory. On the other hand, if the relaxation time of the polymeric coating is very short, the relaxation of stresses is very apparent. A compressive radial stress at the interface of the glass fiber and primary coating will result in an increase of the transmission losses, and a tensile interfacial radial stress will possibly produce debonding at the interface of the glass fiber and primary coating. To minimize this interfacial radial stress, the radius, Young's modulus and Poisson's ratio of the polymeric coatings should be appropriately selected, and the relaxation time of the primary coating should be shortened. Finally, the stresses in single‐coated and double‐coated optical fibers are discussed.     

Local thermodynamic models for high pressure process calculations

This paper discusses the local thermodynamic model concept in the context of high pressure chemical processes where equations of state are usually required for performing thermodynamic property evaluations. Functional forms for local K-value models are developed by simplifying the rigorous equations governing fluid phase equilibria. The local models have imbedded adjustable parameters that are evaluated by regression against either experimental data or thermodynamic properties calculated with more rigorous theoretically based models. During the course of the calculations the parameters in the local models are updated periodically, thereby ensuring that the local models continually provide accurate values for thermodynamic properties. Use of the concepts for process calculations is demonstrated in an example problem.     

Broadband dielectric spectroscopy of nanostructured maleated polypropylene/polycarbonate blends prepared by in situ polymerization and compatibilization

The miscibility and molecular dynamics of nanostructured maleated polypropylene (mPP)/polycarbonate (PC) blends prepared by in situ polymerization of macrocyclic carbonates with polypropylene modified with 0.5 wt% of maleic anhydride-reactive groups were investigated over a wide range of frequencies (10⁻²-0.5 × 10⁷ Hz) at different constant temperatures using broadband dielectric spectroscopy and scanning transmission electron microscope (STEM). The molecular dynamics of the glass relaxation process of the blend (α-relaxation process) appeared at a lower temperature range compared with that of the pure PC. This shift in the molecular relaxation process is attributed to the partial miscibility of the two polymer components in the blends as previously confirmed by the morphology via STEM. Nanoscale morphologies with average domain diameters as small as 50 nm were obtained for the different blend compositions studied. The STEM photographs show that the graft mPP-g-PC prefers to locate at the interfaces as previously reported. The relaxation spectrum of pure PC and mPP/PC blends was resolved into α- and β-relaxation processes using the Havriliak-Negami equation and ionic conductivity. The dielectric relaxation parameters, such as relaxation peak broadness, maximum frequency, f_max, and dielectric strength, Δ? (for the α- and β-relaxation processes), were found to be blend composition dependent. The kinetics of the α-relaxation processes of the blends were well described by Vogel-Fulcher-Tammann (VFT) equation. The local process of PC was resolved into two relaxation processes β₁ and β₂, associated with the carbonyl groups' motion and the combined motions of carbonyl and phenylene groups, respectively. Only β₂ shifted to lower frequency in the blend while β₁ was relatively not affected by blending. The electric modulus of the blends was used to get a sufficient resolution of the different relaxation processes in the samples, i.e., α-, β-relaxation processes, ionic conductivity, and interfacial polarization. In addition, the blending method used was found to increase the d.c. conductivity without affecting the charge carrier transport mechanism, making it possible to develop novel polymer blends with tunable dielectric properties and morphology from existing polymers.     

Precipitation of asphaltenes from solvent-diluted heavy oil and thermodynamic properties of solvent-diluted heavy oil solutions

Ratios of n-heptane (hep) to toluene (tol) affect the solubility of the asphaltenes in heavy oil extraction processes. Consequently phase changes and time after mixing n-heptane and heavy oil in toluene are important for understanding produced emulsions. The kinetics of phase change when n-heptane is added to toluene-diluted heavy oils, and the thermodynamic properties of partially deasphalted heavy oils were studied. The methods used were monitoring precipitation in time using light microscopy, quantitative asphaltenes analysis by near infrared spectroscopy, refractive index and densities measurements, and calculated solubility parameters of mixtures. At critical mass ratios of hep/tol from 1.37 to 2.0 in diluted heavy oil the precipitated asphaltene particles were observed under the microscope after lag times from 2 h to instantly. Lag times were longer at low initial oil concentration. The floc growth time decreased as heavy oil concentration in toluene increased. The growth patterns in time appeared as dots to beads (strings) to clusters (fractal-like flocs). Final wt% precipitated asphaltenes vs. mass fraction (hep+tol)/heavy oil followed sigmoidal relationships. Curves showing wt% soluble asphaltenes vs. mass fraction hep/tol after 24 h initially followed the same shape as time zero curves and diverged at the onset ratios of hep/tol. Slope for precipitated asphaltenes vs. solubility parameters curve showed a break at 16.4 MPa^1/2. Linear correlations were established for concentrations of soluble asphaltenes in residual oils and density, for refractive index and density and for refractive index and solubility parameter. The latter correlation was in accordance with Lorenz-Lorentz theory. These equations provided a means by which oil density, refractive index and solubility parameter can be predicted when these measurements are difficult to measure practically.     

Identification of parameters in systems of ordinary differential equations using quasilinearization and data perturbation

Given a set of observed data for a particular physical phenomenon, the problem of computing the “best fit” parameters for the mathematical model describing the phenomenon is a common problem in process or reaction mechanism identification. If the mathematical model comprises a set of non-linear ordinary differential equations, this leads to a non-linear boundary value problem. A very powerful way of attacking this class of problem uses an adaptation of the Newton-Raphson-Kantorovich procedure, called quasilinearization, which regards the non-linear problem as the limit of a sequence of linear problems. Starting from an initial trial solution, convergence if it does occur, occurs rapidly; further, convergence is assured if the initial guess is “close enough” to the true solution. The difficulty of making a good initial guess, a serious limitation of the method in the past, can in principle be overcome by the algorithm proposed. When a given vector may not be within the domain of convergence of the original problem, it must be within the domain of convergence of some other derived problem. The latter may then be perturbed towards the original problem in a finite number of steps. In the case of process identification, new data points are derived; these are subsequently adjusted until they coincide with the original data. The algorithm has been successfully applied to several examples from recent chemical engineering literature.     

Concentration and temperature profiles for complex reactions in porous catalysts

A method of computation for stationary concentration and temperature profiles in porous catalysts was developed. These calculations can be used with highly complex reactions and with any type of rate equation or transport model. The differential equations are transformed by the collocation method to non-linear algebraic equations. These are solved by a method which uses the time-derivatives of the variables. The method finds stationary points for all problems which can be formulated as ? = f(y) and is demonstrated on two examples.     

Transverse NMR relaxation of water in wood

The transverse NMR relaxation times of hydrogen nuclei of water absorbed in white spruce sapwood [Picea glauca (Meunch) Voss] were measured for moisture contents in the range from 5 to 176%. The spin echo amplitudes resulting from the Carr–Purcell sequence decay nonexponentially suggesting the possibility of at least two different relaxation times for water in wood. A simplified structural model of the wood–water mixture is used to estimate the rates of chemical exchange at room temperature of hydrogen nuclei between various sites in the system. The high-resolution NMR line shape is discussed briefly in terms of this proposed model.     

Modeling of density evolution of PLA under ultra-high pressure/temperature histories

The density evolution of a semi-crystalline Poly(lactic acid) under ultra/high pressure histories in the region of alpha relaxation has been studied. It was found that the degree of crystallinity does not change in the temperature and pressure ranges under concern and consequently it is assumed that the volume relaxation kinetics involves only the amorphous part of the polymer. The scaling law for relaxation times, recently proposed by Casalini and Roland, has been utilized in the framework of KAHR (Kovacs Aklonis Hutchinson and Ramos) phenomenological theory in order to model the volume relaxation behavior. With this approach the model contains only two fitting parameters, (namely the relaxation time in the reference state, τ_g, and the fractional exponent, β, that describes the dispersion of the alpha relaxation) instead of the five parameters of the classical KAHR theory. The model predicts accurately both the density evolution of the PLA during the sterilization and Pasteurization treatments simulated in a pressure dilatometer and the density changes occurring in the PLA films utilized as packaging material in the HP industrial processes.     

Two-Fold Diffusion Kinetics of Oxygen Re-Equilibration in Donor-Doped BaTiO3

Oxygen (nonstoichiometry) re-equilibration kinetics was examined, via a conductivity relaxation method, on 1 m/o donor (La)-doped BaTiO₃ (nominal composition, Ba_0.99La_0.01Ti_0.9975O₃) against oxygen partial pressure in the range of −16O2/ atm)≤ 0 at 1200°C. The kinetics has been found to be onefold with a single relaxation time or twofold with two different relaxation times depending on oxygen activity. It is attributed to the relative contributions depending on oxygen activity of fast relaxation of the oxygen sublattice and sluggish relaxation of the cation sublattices: their revelations are determined by the ratio of oxygen vacancy to cation vacancy (say ,     ) concentrations and their relaxation times. Their chemical diffusion coefficients have been extracted from the relaxation times as     and     depending on oxygen activity, which are in good agreement with the reported values. Defect diffusivities are subsequently determined by using the thermodynamic factors of component O and Ti, which have been evaluated from the equilibrium conductivity isotherm, as     and     .     

Nuclear magnetic relaxation in poly(N-amyl maleimide) and poly(N-dodecyl) maleimide

Longitudinal relaxation times in the laboratory (T₁) and rotating (T_1ρ) frames have been measured for poly(N-amyl maleimide) and poly(N-dodecyl maleimide). The results are compared with information previously published using dielectric relaxation techniques. A total of three relaxation processes have been detected corresponding to methyl rotation, motion within the substituent alkyl chain and out-of-plane deformation of the maleimide ring. An attempt to fit the T₁ data in the dodecyl derivative to theoretical equations is described.     

Monitoring of photopolymerization kinetics and network formation by combined real-time near-infrared spectroscopy and ultrasonic reflectometry

For in situ monitoring of fast changes of shear modulus and chemical conversion during UV radiation curing an ultrasonic (US) reflection method was combined with real-time near infrared (NIR) spectroscopy. The combined setup has been applied to study photopolymerization of different resins as acrylates, epoxy acrylates, acrylated polyurethanes and cationic epoxy resins in order to achieve a deeper understanding of the interdependence of reaction kinetics and changes of mechanical/rheological properties. The simultaneously recorded conversion–time and modulus–time curves allow differentiating between mass and diffusion controlled polymerization regime. Light curing and dark curing phases are indicated by two distinct regions in the conversion–time curves. By rescaling the curing time by chemical conversion modulus–conversion curves were constructed, which are described by combining a viscoelastic relaxation model with the conversion dependence of relaxation times. The NIR-US setup was used to study the influence of chemical composition and curing conditions on the polymer network formation.     

Rotating frame 1H n.m.r. spin-lattice relaxation studies of modified isotactic polypropylene

Rotating frame ¹H n.m.r. spin-lattice relaxation times T_1ϱ have been used to study relaxation processes in partially crystalline polypropylene and polypropylene modified by a copolymer of ethylene and alkylaminoacrylate. Measurements were carried out within the temperature range of 250–420 K. Three relaxation times T_1ϱ were detected over the whole temperature range. α_c relaxation in the crystalline regions of the polymer, relaxation processes β(U) and β(L), associated with an apparent double glass transition, and α_a relaxation process, which was ascribed to a free reorientation of the whole chains in the amorphous regions, were observed by temperature dependences of these relaxation times. In addition to the α_c relaxation, another relaxation process with relatively low molecular mobility was found in crystalline regions. The influence of the polymer modifier on the observed relaxation processes greatly depends on its amount.     

Aliphatic polyesters as models for relaxation processes in crystalline polymers: 4. Dielectric relaxation in oriented specimens

To investigate further the effect of the crystal phase on amorphous-phase relaxation, samples of 6-6, 5–7, and 6–10 polyesters have been oriented by extrusion in the solid-state through a tapered die and the dielectric constant and loss measured both parallel and perpendicular to the extrusion direction. The data have been analysed quantitatively in terms of the Cole—Cole phenomenological equation and relaxation times, width parameters and relaxation strengths determined. For the β (glass—rubber relaxation) the relaxation times are shifted 2–3 orders of magnitude to longer times and the relaxation is broadened slightly compared to the unoriented polymers. There also tends to be reduction of overall parallel and perpendicular relaxation strength for the β process on orientation. The γ processes show little effect of orientation on relaxation time and width and do not show reduction of overall relaxation strength. These observations are consistent with crystal displacement further restricting amorphous-phase longer-range segmental motion (β process) but having little effect on the locallized motions associated with the γ process. Both the γ and β processes show anisotropy in relaxation strength. The anisotropy is analysed with a newly developed theory to separate inherent phase anisotropy from apparent anisotropy due to composite or form effects. The average angle made by the lamellar surface normal to the extrusion direction enters as a parameter in the theory. Measurements of relaxed specimen $\text{ε}{}_{\mathrm{}}$ and ε_⊥ values for both the γ and β processes serve to determine both f_a the amorphous-phase orientation function and the average tilt angle for the specimens here. This is possible because for the γ process relaxation strength is small and specimen anisotropy is dominated by inherent phase anisotropy but for the γ + β processes larger relaxation strength leads to specimen anisotropy being strongly influenced by crystal—amorphous phase composite effects. Values of f_a are in the range of 0.05–0.20 for specimens with f_c in the range 0.5–0.9. Indicated lamellar tilt angles are in the range of 24°–80°.     

A computer model for the regenerative bed

The transient behavior of a packed bed of uniform spheres is considered, where heat (or mass) transfer inside the sphere is described by a linear parabolic second order partial differential equation — the standard diffusivity equation. One dimenisonal plug flow is assumed; hence, the fluid behavior is given by a set of first order partial differential equations, with bed axial position and time as the independent variables. The existence of chemical reaction in the fluid makes these equations non-linear. The two systems of equations describing the solid and the gas are coupled by the equations for heat (or mass) transfer at the surface. Numerical methods for solving both the linear (no reaction) and non-linear (with reaction) cases are presented. In the former, the implicit Crank-Nicholson method is used; in the latter the two systems of equations are first decoupled and then solved separately.     

Rotational moulding of metallocene polyethylenes

Abstract

This paper compares the rotomouldability and mechanical properties of conventional polyethylenes with metallocene polymers having a similar melt flowrate or the same density. The rotomouldability has been evaluated in terms of bubble removal, cycle times, and heat stability during the moulding process. The mechanical properties, such as tensile strength and peak impact energy, were measured on parts produced using a variety of moulding conditions. It was found that the metallocene polymers display greater toughness if subjected to the same moulding conditions as the conventional polymers. Conversely, the same toughness can be obtained in both materials if shorter cycle times are used for the metallocenes. This behaviour has been linked to the relaxation behaviour of the molecules in the solid state. Other material parameters, such as the nature of the comonomer, the long chain branching density, and the crystalline morphology were studied by NMR, DSC, and polarising light microscopy. The differences between the molecular structures of the two types of polyethylene can be used to explain why the metallocene materials possess many rheological and other characteristics that are desirable in rotational moulding.     

Precise parameter estimation for chemical batch reactions in heterogeneous medium

A sequential experimental strategy for precise parameter estimation has been used in the case of liquid-liquid dispersions in batch-stirred tank reactors where slow chemical reactions take place. The mathematical model for a batch reaction in a stirred tank reactor is formulated as a system of non-linear differential equations standing for the mass balance of each component. Physical kinetic parameters and chemical kinetic parameters which arise from this model are estimated simultaneously. The estimation problem is posed as a weighted least squares problem and solved by using a standard Levenberg-Marquardt algorithm. In this work, we intend to show how it is possible to develop efficient experimental design strategies that lead to an accurate estimation of the parameters involved in phenomenological models and most particularly in kinetic models. Three design criteria for designing the experiments have been employed in order to increase the precision on the parameter estimates of the model. A standard non-linear sequential quadratic programming method ensures the determination of the operating conditions which define the experimental design. The well-known alkaline hydrolysis of esters in aqueous phase has been treated as a numerical application example.