State multiplicity in CSTR-separator-recycle polymerisation systems

This article continues earlier work (Comput. Chem. Eng. 24 (2000) 209) concerning the design and control of isothermal reactor-separator-recycle systems. The multiplicity behaviour of six reaction systems of increasing complexity, from one-reactant, first-order reaction to chain-growth polymerisation, is investigated. Below a critical value of the plant Damkohler number, Da<Da^cr, the only steady state involves infinite flow rates. Feasible steady states become possible if the critical value is exceeded, Da>Da^cr. For one-reaction systems, one stable steady state is born at a transcritical bifurcation. For consecutive-reaction systems, including polymerisation, a fold bifurcation can lead to two feasible steady states. Moreover, the transcritical bifurcation is destroyed when two reactants are involved. If the gel-effect is included, a maximum of four steady states are possible. When multiple steady states exist, the achievable conversion is constrained by the instability of the low-conversion branch. This has practical importance for polymerisation systems when the radicals’ quasi-steady state assumption is not valid or the gel effect is significant.     

The role of kinetics and hydrocarbon distribution in the multiplicities of a two-bed catalytic distillation column for deep hydrodesulfurization

The Damköhler number (Da) is used to research the physical mechanisms leading to multiplicities and the effect of the hydrocarbon distribution on a light gasoil (LGO) deep hydrodesulfurization (HDS) two-bed catalytic distillation column. In doing so, three hydrocarbon mixtures are studied: a real LGO fraction, a synthetic gas oil mixture (SGO), and hexadecane (n-C₁₆). The parametric planes, Da(RR) and Da(B), were obtained for the three mixtures as a function of the reflux ratio, RR, and the bottom flow rate, B. The total sulfur regions (C_s) with respect to Da and RR, and Da and B, were also obtained. RR and B were chosen as bifurcation parameters, since they are frequently used for column operation.Results show that this system is controlled kinetically because the operating point and the multiplicity region lay on the kinetically controlled region (Da ? 1). Results also show that the hydrocarbon distribution plays an important role in the occurrence of multiplicities, as well as exhibiting the importance of choosing a realistic gasoil fraction during process design and lab scale experiments.     

Parametric sensitivity of a CSTR

It is shown that the generalized sensitivity criterion recently developed in the context of thermal explosions and tubular reactors can be easily applied in the case of a CSTR as well. An illustrative example concerning sensitivity analysis of a single nth order irreversible exothermic reaction in a nonadiabatic CSTR is presented. A generalized region of parametric sensitivity is identified where the reactor temperature is parametrically sensitive simultaneously to all the input parameters. Asymptotic analysis for large heats of reaction is performed to investigate limiting behaviour, which leads to the classical Semenov limit in the case of large activation energies. It is shown that parametric sensitivity can occur even when unique steady states exist for all Damköhler number (Da) values. Furthermore, if operating conditions are chosen so as to avoid completely the possibility of parametric sensitivity for all Da, then the possibility of steady-state multiplicity is automatically avoided.     

Elucidating two-phase transport in a polymer electrolyte fuel cell, Part 1: Characterizing flow regimes with a dimensionless group

This paper explores the through-/in-plane characteristics of water transport in the cathode gas diffusion layer (GDL) of a polymer electrolyte fuel cell (PEFC). Theoretical analysis is performed on the non-isothermal two-phase flow under flow channels. A dimensionless group Da (Damkohler number for PEFC operation), defined as the ratio of water generation rate to water vapor-phase removal rate, is formulated to characterize the flow regimes in a PEFC. This group, lumping geometrical parameters and physical properties, compares the water vapor-phase removal capability (via water diffusion and holding capacity) with the rate of water production by the oxygen reduction reaction. We find that this dimensionless group can be used to characterize the non-isothermal, two-phase phenomena: when Da→0, the fuel cell is subjected to single-phase operation; while as Da→∞ we have full two-phase operation. A more precise expression is explored for the dimensionless group at the channel central line, i.e. Da₀: when Da₀>1 the entire cathode GDL–CL (catalyst layer) interface is in two-phase region, whereas part of the interface is free of liquid water for Da₀<1. The latter scenario is the concept that this paper proposes for improving fuel cell water management: the consequent co-occurrence of single- and two-phase flows in the in-plane direction at Da₀<1 is beneficial to avoid severe dryout and flooding. A two-phase transport model, describing the water and heat transport on the PEFC cathode side, is employed to perform a two-dimensional numerical study. Detailed liquid and temperature distributions are displayed. Simulation predictions are in reasonably good agreement with the dimensionless-group analysis.     

Residue curve maps for reactive distillation systems with liquid-phase splitting

A model has been developed to study the effects of chemical kinetics on the residue curve maps (RCM) for reactive distillation systems with liquid phase splitting. In the model, chemical reaction can occur in both or only one of the two liquid phases. The heating policy V/V₀=H/H₀ is applied so that the kinetic effect can be described by a single parameter, the Damköhler number Da. The effects of reaction kinetics on pseudohomogeneous and heterogeneous mixtures have been compared. The properties of their RCMs are the same outside, but are fully different inside the liquid-liquid (L-L) region if they have different chemical equilibrium curves. Inside the L-L region, the chemical equilibrium curve coincides to a unique reactive liquid-liquid tie line in case that the pseudohomogeneous chemical equilibrium curve intersects with the L-L envelope. When the reaction occurs in only one of the two liquid phases, the residue curves inside the L-L region are strongly affected by the L-L envelope, especially at high Da. In the present paper, first an illustrative arbitrary reaction system, and then the reaction of cyclohexene with water to cyclohexanol are analysed with respect to their RCMs.     

Residue curve maps of reactive membrane separation

A batch reactive membrane separation process is analysed and compared with a batch reactive distillation process by means of residue curve maps. In both processes, the chemical reaction takes place (quasi-) homogeneously in the liquid bulk phase and vapour-liquid equilibrium is assumed to be established. Additionally, in the reactive membrane separation process, selective vapour phase permeation through a membrane is incorporated.A model is formulated which describes the autonomous dynamic behaviour of reactive membrane separation at non-reactive and reactive conditions when vacuum is applied on the permeate side. The kinetic effect of the chemical reaction is characterized by the Damköhler number Da, while the kinetic effect of multicomponent mass transfer through the membrane is characterized by the matrix of effective mass transfer coefficients. The process model is used to elucidate the effect of selective mass transfer on the singular points of reactive membrane separation for non-reactive conditions (Da=0), for kinetically controlled reaction (0<Da<∞), and for equilibrium controlled reaction (Da→∞). Scalar, diagonal and non-diagonal mass transfer matrices are considered. As examples, the simple reaction A⇔B+C in ideal liquid phase, and the cyclization of 1,4-butanediol to tetrahydrofurane in non-ideal liquid phase are investigated.     

Theoretical and experimental study on residue curve maps of propyl acetate synthesis reaction

Residue curve maps (RCMs) of propyl acetate synthesis reaction in the batch reactive distillation process are studied. In order to adapt the model equations of residue curve maps to a practicable heating policy, the theoretical analysis and experimental measurements in this paper are carried out isothermally instead of the autonomous heat policy first introduced by Venimadhavan et al. (A.I.Ch.E. Journal 40 (1994) 1814-1824). The chemical equilibrium constant of this reaction is determined by experiments to be 20 within the temperature range 80-110 °C. Using this equilibrium constant, the RCMs predicted by simulation are in good agreement with the experimental measurements. The results show that there is an unstable node branch emerging from the propyl acetate-water edge, moving toward the chemical equilibrium surface with the increasing Damköhler number (Da), and eventually reaching the quaternary reactive azeotrope when Da→∞. Residue curves are measured with initial compositions around the unstable node, and thus the results verify the existence of this reactive azeotrope. Further bifurcation analysis shows that different heat policies will influence the singular points and topology of kinetically controlled RCMs, but not the cases when Da=0 or Da→∞.     

Oxygen transport rate on Rhodococcus erythropolis cultures: Effect on growth and BDS capability

The influence of oxygen transport rate on Rhodococcus erythropolis cultures has been studied in a stirred tank bioreactor under different transport and uptake conditions. Oxygen uptake rate has been measured by applying a modified dynamic method and a kinetic model is proposed, obtaining the kinetic parameter values: specific maintenance and yield coefficients. The volumetric mass transfer coefficients under inert conditions, k_La, and during the bioprocess, K_La, have been determined. The values obtained are different and a biological enhancement factor E, has been considered. These parameters have been predicted by the theoretical model and good agreement with experimental data under the conditions studied has been found. The oxygen limitation has been expressed by a modified dimensionless Damköhler number, Da, the relationship between transport and biological reaction rates. This number decreases with increasing stirrer speed; that is, when mass transport resistance decreases. The efficiency of oxygen utilization can be determined by a film effectiveness factor, η. The effectiveness factor was found to be a strong function of Damköhler number and decrease with increasing Da. Furthermore, oxygen concentration into culture depends on the mass transfer and consumption rates. The theoretical model proposed is able to reasonably predict the evolution of dissolved oxygen concentration with time of cultivation.     

A UV-LEDs based photomicroreactor for mechanistic insights and kinetic studies in the norbornadiene photoisomerization

In this work, we report a simple and inexpensive UV-LEDs based photomicroreactor assembly constructed by commercially available components. The photoisomerization of norbornadiene to quadricyclane was selected to validate this novel photomicroreactor design. Mass transport limitation was eliminated and indicated by dimensionless numbers Fo and Da _II, and photons loss was evaluated considering the absorption and reflection of the microreactor walls. The solvents, photosensitizers, and light sources selections were optimized for achieving better photochemical performance and mechanistic insights. The detailed comparison between the high-pressure mercury arc lamp and the UV-LEDs strip revealed great potential of UV-LEDs as appealing light source for photochemical transformations. Moreover, the reaction mechanism was thoroughly discussed and illustrated by the Jablonski diagram indicating electronic states transitions. According to possible intermediate steps, a kinetic model was proposed with the reaction rate constant being correlated with the photon flux, which is valuable for process optimization and further understanding reaction mechanisms.     

Multiplicity and uniqueness for binary,exothermic reaction in a non-adiabatic continuous stirred tank reactor

Exact multiplicity and uniqueness criteria for steady state in a non-adiabatic continuous stirred tank reactor are studied through simple tangent analysis for binary, exothermic reaction of the type A + bB → Products with rate expression r_A = kC_A^m C_Bⁿ, where A is the limiting reactant. Important parameters for multiplicity criteria are reaction orders m and n, stoichiometric coefficient b, the ratio p of feed concentration of A to that of B, dimensionless activation energy α, dimensionless heat of reaction β, dimensionless heat transfer coefficient γ and dimensionless coolant temperature. Necessary conditions for the system to have multiple exit conversions (temperatures) are defined in the (m, n, b, p, α, β, γ) space. Multiplicity is guaranteed by limiting the dimensionless space time ? in a proper range in addition to the necessary conditions. Effects of various parameters on multiplicity and uniqueness are numerically calculated and graphically represented. Theoretical prediction for multiplicity are further compared with multiplicity data reported in literatures.     

Role of entropy generation on thermal management during natural convection in porous square cavities with distributed heat sources

Material processing by thermal convection may be carried out in an energy efficient way based on the second law of thermodynamics. In the current work, the entropy generation in porous square cavities with distributed heat sources during laminar natural convection has been studied. Four different configurations of discretely heated cavities are considered for the study based on the location of the heat sources on the walls of the cavities. The governing equations are solved using Galerkin finite element method. The entropy generation terms are evaluated using finite element basis sets and the derivatives at particular nodes are estimated based on the functions within adjacent elements. Simulations are performed for the range of Darcy number, Da=10⁻⁶–10⁻³ and Rayleigh number, Ra=10³–10⁶ for various fluids (Prandtl number, Pr=0.015,0.7,10 and 1000). A detailed analysis on the effect of Da on entropy generation due to heat transfer (S_θ) and fluid friction (S_ψ) based on their local distribution in various cases is presented. The maximum values of S_θ are found to occur near the hot–cold junctions while the maximum values of S_ψ are found at various locations on the walls of the cavity depending on the circulation cells in various configurations. Significant S_ψ is also observed in the interior regions due to the friction between counter rotating circulation cells. The dominance of S_ψ is found to be high for higher Pr fluids. The total entropy generation rate (S_total) is found to increase with Da and the average Bejan number (Be_av) is found to be less than 0.5, indicating the dominance of fluid friction irreversibility at higher Da in all cases for various fluids. Finally, the thermal mixing, temperature uniformity has been correlated with S_total and Be_av for all distributed heating cases and the thermal management via enhanced thermal mixing vs optimal entropy production for efficient thermal processing of various fluids in porous media are proposed.     

Exact criteria for uniqueness and multiplicity of an nth order chemical reaction via a catastrophe theory approach

The development of a simple, generalized technique for the exact determination of regions of unique and multiple solutions to certain nonlinear equations via a catastrophe theory-implicit function theorem approach, is presented. The application of this technique to the nth order chemical reaction in the nonadiabatic and adiabatic CSTR yields exact, explicit bounds for all n ≥ 0. To our knowledge, this is the first report of exact, explicit bounds for these systems, except for n = 0, 1 for the adiabatic CSTR, and n = 1 for the nonadiabatic CSTR. For the nonadiabatic CSTR, these bounds show that the higher the reaction order, the smaller the region in parameter space for which multiplicity can occur for all γ and x_2c, (dimensionless activation energy and coolant temperature, respectively). This behavior is similar to that reported by Van den Bosch and Luss[1] for the adiabatic CSTR. The zeroth order reaction in the nonadiabatic CSTR exhibits more complex behavior and assumes characteristics of both high and low reaction orders insofar as increasing and/or decreasing the uniqueness space, in comparison to all other n > 0.An exact implicit bound between regions of uniqueness and multiplicity is also derived for the nth order reaction in a catalyst particle with an intraparticle concentration gradient and uniform temperature, and is fully demonstrated for the first order reaction. In addition, explicit criteria, sufficient for uniqueness and multiplicity of the catalyst particle steady state, stronger than those of Van den Bosch and Luss, are also developed by combining the present technique with bounds suggested by these authors.     

Uniqueness and multiplicity criteria for an nth order chemical reaction

New and very strong criteria are presented for a priori prediction of the conditions for which the steady-state lumped parameter model of an nth order chemical reaction (n ≥ 0) in an adiabatic CSTR has either a unique or multiple solutions. The criteria show that the higher the order of the reaction the smaller is the region in the parameters space for which multiplicity can occur.New uniqueness and multiplicity criteria are developed also for an nth order reaction in a porous catalyst using a model, which accounts for intraparticle concentration gradients, while assuming a uniform intraparticle temperature different from the ambient one. The region in the parameters space for which steady state multiplicity can occur for this model is smaller than that for a corresponding lumped model, which ignores the intraparticle concentration gradients.     

Low-dimensional models for describing mixing effects in reactive extrusion of polypropylene

Spatially averaged low-dimensional models based on Liapunov-Schmidt technique of bifurcation theory have been developed to study mixing effects in peroxide-induced reactive extrusion of polypropylene degradation. The two-dimensional convection-diffusion-reaction equations for each species and the energy balance equation have been averaged in the transverse direction to obtain low-dimensional models that describe transverse (local) mixing effects on conversion, average molecular weight and temperature distribution in a reactive extruder channel with asymmetric thermal boundaries. Our models predict that incomplete local mixing due to velocity distribution, backflow and transverse diffusion may significantly reduce the conversion (by more than 50%) in a reactive extruder, compared to a plug-flow case. Our analysis further reveals that beyond a transition value of Damköhler number (Da), the overall reaction occurs in the mixing-limited regime, where the conversion and the average molecular weight of the polymer melt are determined only by the dimensionless local mixing time (which, in turn, depends on the screw speed) and are independent of Da. Increased Graetz number (i.e. slow transverse thermal diffusion) decreases the polymer-melt temperature and reduces conversion, while increase in screw speed increases viscous heat generation resulting in higher exit temperature accompanied by reduced conversion and produces off grade high molecular weight (low melt flow index) product when the mixing effect dominates the temperature effect.     

Evaluation of the shrinking‐core model for examining the kinetics of film formation in a reactive latex blend

We prepared reactive latex blends from two copolymer latices comprised of n‐butyl methacrylate (n‐BMA) with acetoacetoxyethyl methacrylate and n‐BMA/dimethylaminoethyl methacrylate to study the kinetics of film formation. We generated thin films by blending equal weights of the two latices. The films were then cured at temperatures ranging from 50 to 90°C. The extent of the crosslinking reaction was calculated from the crosslink density, which was determined from swelling measurements of the films in toluene. The shrinking‐core model, a diffusion/reaction model, which was originally derived for combustion reactions of coal particles, was adopted to calculate the diffusion coefficient (D_e) and reaction rate constants from the extent of the reaction with time data. This model system exhibited a diffusion‐controlled regime above 70°C and a reaction‐controlled regime at temperatures below 70°C. In the reaction‐controlled regime, the shrinking‐core model predicted D_e for the system, which was in agreement with literature values for n‐BMA. In the diffusion‐controlled regime, the model predicted a lower apparent value for D_e but with an activation energy that was close to that obtained for n‐BMA. The model was also used to examine the kinetics of the crosslinking reaction. The kinetic rate constants for the crosslinking reaction were also determined. The activation energy for the crosslinking reaction was 18.8 kcal/mol, which compared reasonably with the activation energy of 22.8 kcal/mol determined for the reaction between the functional monomers as small molecules. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 101: 3659–3665, 2006     

Fluid motion around and through a porous cylinder

The flow-field and solute transport through and around a porous cylinder is investigated numerically. The range of Reynolds number (based on the cylinder diameter and the uniform sinking rate of the cylinder) considered here is between 1 and 40 with Darcy number (Da) in the range 10^-6?Da?1.5 and porosity in the range 0.629?ε?0.999. The motivation of the present study is the application of flow through porous cylinder extensively applied in nuclear biological chemical filters as well as reduction of carbon fines in filtered water. The influence of Da on the drag coefficient, separation angle, recirculation length, streamline and vorticity pattern are investigated. The drag ratio, defined as the ratio of drag coefficient of porous cylinder to that of solid cylinder, is found to approach zero from unity as Da is increased from 10^-6 to 1.5. The separation point shifts towards the rear stagnation point as Da is increased. The time evolution of the solutal field at different Reynolds number and Darcy number is presented. A long slender concentration plume is found to evolve from the cylinder with decreasing concentration at the outer edge.     

Natural convection and flow simulation in differentially heated isosceles triangular enclosures filled with porous medium

A finite element analysis is performed to investigate the effects of uniform and non-uniform heating of bottom wall on natural convection flows within isosceles triangular enclosures filled with porous medium. The detailed analysis is carried out in two cases depending on various thermal boundary conditions:

(I)

two inclined walls are maintained at constant cold temperature while the bottom wall is uniformly heated;

(II)

two inclined walls are maintained at constant cold temperature while the bottom wall is non-uniformly heated.

The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Darcy number, Da (10^-5?Da?10^-3), Rayleigh number, Ra (10³?Ra?10⁶) and Prandtl number, Pr (0.026?Pr?1000) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It is observed that at small Darcy numbers, the heat transfer is primarily due to conduction irrespective of Pr. As the Darcy number increases, there is a change from conduction dominant regime to convection dominant regime. Flow circulations are also found to be strong functions of Pr at large Da (Da=10^-3) and multiple circulation cells occur at small Pr with Ra=10⁶. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case, but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. As average Nusselt number is same on both the inclined walls, the average Nusselt number for bottom wall is times that of the inclined wall which is well matched in two cases considered for verifying the thermal equilibrium of the system. The correlations are proposed for average Nusselt number as functions of Ra for various Darcy and Prandtl numbers.     

Reaction kinetics for hindered amine/epoxides by DSC

The rate constants for the reaction of two aliphatic hindered amines with phenylglycidyl ether (PGE) and the diglycidyl ether of bisphenol A (DER 332) were determined by differential scanning calorimetry (DSC). The two exothermic peaks which are present in the DSC data result from the consecutive reactions of the primary and secondary amine hydrogens and allow k₁ and k₂ to be determined. The resulting k₁/k₂ ratios obtained for these hindered amine systems are larger than the ratios previously reported for unhindered amine/epoxides.     

Synthesis of hyperbranched polymers via a facile self-condensing vinyl polymerization system - Glycidyl methacrylate/Cp2TiCl2/Zn

A facile self-condensing vinyl polymerization (SCVP) system, the combination of glycidyl methacrylate, Cp₂TiCl₂ and Zn, has been firstly used to prepare novel hyperbranched polymers, consisting of vinyl polymers as the backbone, and cyclic ester polymers (poly(?-caprolactone) or poly(l-lactide)) as the side chains. The polymerizations are initiated by the epoxide radical ring-opening catalyzed by Cp₂Ti(III)Cl which is generated in situ via the reaction of Cp₂TiCl₂ with Zn. The key to success is that the polymerizations can proceed concurrently via two dissimilar chemistries possessing the opposite active initiating species, including ring-opening polymerization (ROP) and controlled/living radical polymerization (CRP). We have demonstrated that this facile one-step polymerization technique can be applied successfully to prepare highly branched polymers with a multiplicity of end reactive functionalities including Ti alkoxide, hydroxyl and vinyl functional groups.