Modeling high-viscosity,melt-phase,condensation polymer reactors with a time-scales analysis

Mathematical models for simultaneous reaction and mass transfer occurring in the manufacture of high-viscosity, condensation polymers are considered. The specific example of polycondensation of polyethylene terephthalate is examined. Reactor performance is estimated by using a kinetic expression modified by an effectiveness factor. The effectiveness factor is correlated against a ratio of two characteristic times, one identified as the time scale of mixing and the other identified as the time scale of reaction. The time scale of mixing is estimated from experimental mixing data, thereby avoiding the use of potentially inaccurate mixing assumptions. In place of reaction experiments, overall reaction rates are generated using a more detailed mixing-cell model. The effectiveness factor correlation is compared against previous models. Because the correlation is based on predicted reaction rates rather than experimentally measured reaction rates, the value of the present work lies in the demonstration of the time scales modeling technique. © 1995 John Wiley & Sons, Inc.     

Bildgebende UV/VIS-Spektroskopie zur Untersuchung des Einflusses der Fluiddynamik auf die Entstehung von Haupt- und Nebenprodukt in schnellen konkurrierenden konsekutiven Gas-Flüssig-Reaktionen

To form many bulk chemicals, gaseous substances must be mixed and reacted with a continuous liquid phase. In this research work, we systematically investigate to what extent the formation of a side product in a fast competing consecutive reaction can be influenced by the time scales of mixing. For this purpose, a Taylor bubble experiment is used, in which the time scales of mixing can be varied systematically and reproducibly. It is shown, that the mixing in the wake of a bubble is of great importance for the formation of by-products.     

A novel approach for describing mixing effects in homogeneous reactors

The Liapunov–Schmidt technique of classical bifurcation theory is used to spatially average the convection–diffusion–reaction (CDR) equations over smaller time/length scales to obtain low-dimensional two-mode models for describing mixing effects due to local diffusion, velocity gradients and reactions. For the cases of isothermal homogeneous tubular, loop/recycle and tank reactors, the two-mode models are described by a pair of coupled balance equations for the mixing-cup (C_m) and spatial average (〈C〉) concentrations. The global equation describes the variation of C_m with residence time (or position) in the reactor, while the local equation expresses the coupling between local diffusion, velocity gradients and reaction at the local scales, in terms of the difference between C_m and 〈C〉. It is shown that the two-mode models have many similarities with the classical two-phase models of heterogeneous catalytic reactors with the concept of transfer between phases being replaced by that of exchange between the two-modes. It is also shown that when the local Damköhler number (ratio of local diffusion to reaction time) is small, the solution of two-mode models approaches the exact solution of full CDR equations, while for fast reactions the two-mode models retain all the qualitative features of the latter. Examples are provided to illustrate the usefulness of these two-mode models in predicting micromixing effects on homogeneous reactions.     

Mixing assessment by chemical probe

Quantification of micro-mixing is a fundamental issue in industrial chemical processes. Local mixing that is not “fast enough” compared with the reaction kinetics reduces the selectivity of the reaction. Micro-mixing can be characterized by chemical probe methods based on observation of a local chemical reaction that results from the competition between turbulent mixing at micro-scales and the reaction kinetics. However, real-world experimental conditions rarely comply with the grounding assumptions of this method. Starting from physical considerations, the present study aims to establish some guidelines for obtaining quantitative information from the chemical probe and for improving the accuracy of the method by an adaptive protocol. For the first aspect, an analytical approach is proposed to define the validity domain based on analysis of the turbulent time scales. For the second purpose, a novel experimental procedure is suggested that entails targeting the concentrations of the chemical species that can provide the optimal conditions for a relevant use of the chemical probe.     

Gasinduziertes Einmischen von Reaktionsstoppern in Behälter mit niedrigviskosen Flüssigkeiten

These studies consist of experiments, physical modelling, and numerical simulations concerning gas-induced injection and mixing of reaction inhibitors in vessels containing low viscosity liquids. The mixing times required to achieve a final mixing quality of 95% were determined by means of a decolorization technique and a probe method (conductivity probes). The mixing time decreases with increasing gas flow rate, liquid level, eccentricity of the gas injection point, and with decreasing liquid viscosity. Based on fundamental physical principles, analytical models were developed which can be used to estimate, with an average precision of ± 13 %, the mixing times measured on three different scales. The results of scale-up were additionally confirmed via numerical simulations. The results of these studies show that gas-induced mixing of reaction inhibitor solutions can represent a reliable safety system for preventing exothermal runaway reactions. Moreover, direct injection of gas may also be used for admixing additives during normal operation of reactors and storage tanks.     

A conceptual framework for mixing in continuous chemical reactors

A conceptual framework of some generality for describing mixing in continuous flow systems is developed. The framework is based on the notions of residence time distributions, residual lifetimes, and coalescence and redispersion of fluid elements. A deterministic version of the conceptual framework is derived and shown under appropriate constaints to specialize into three more restricted mixing models which have appeared in the literature. Monte-Carlo simulation is suggested as the most practical general means of implementation. This technique is illustrated in a brief study of the effects of mixing on three simple homogeneous reactions. It is conjectured that the conceptual framework is comprehensive, although the arguments presented in support of this conjecture do not prove rigorously that flow mixing situations can be represented.     

Direct use of mixing data for modeling high-viscosity,melt-phase,condensation polymer reactors

Mathematical models for simultaneous reaction and mass transfer occurring in the manufacture of high-viscosity condensation polymers are considered. Particle tracking experiments are used to estimate convective flow rates and mixing volumes in a disc-ring reactor configuration. These results are incorporated directly into a mixing-cell model without resorting to the use of restrictive assumptions regarding the convective mixing. Both a penetration theory model and a flash evaporation model are used to simulate the transport at the liquid–vapor interface. Although widely used in previous studies, the penetration theory model is ultimately rejected because it underpredicts the overall reactivity. Model results predict interactions between agitation rate, residence time, and the overall reaction rate for commercial-scale systems producing poly(ethylene terephthalate). The model is partially verified by comparison with degassing data. © 1995 John Wiley & Sons, Inc.     

Experimental characterization and multi-scale modeling of mixing in static mixers

Mixing in static mixers is studied using a set of competitive-parallel chemical reactions and computational fluid dynamics (CFD) in a wide range of operating conditions. Two kinds of mixers, a wide angle Y-mixer and a two jet vortex mixer, referred to as Roughton mixer, are compared in terms of reaction yields and mixing times. It is found that the Roughton mixer achieves a better mixing performance compared to the Y-mixer. The effect of flow rate ratio on mixing in the Roughton mixer has been studied as well and it is shown that the mixing efficiency is not affected by the flow rate ratio. Moreover, experimental results and model predictions are in good agreement for all mixer geometries and operating conditions. CFD is used to calculate absolute mixing times based on the residence time in the segregated zone and it is shown that mixing times of less than 1 ms can be achieved in the Roughton mixer. In addition, CFD provides insight in local concentrations and reaction rates and serves as a valuable tool to improve or to scale-up mixers.     

Experimental characterization and multi-scale modeling of mixing in static mixers. Part 2. Effect of viscosity and scale-up

Static micro-mixers are used in precipitation processes to avoid mixing limitations. The mixing performance of these mixers, which are used in this study to mix two streams of different viscosity, is characterized using competitive-parallel chemical reactions and computational fluid dynamics (CFD). This work is an extension of a previous paper where mixing of fluids with equal viscosity has been studied [Lindenberg, C., Schöll, J., Vicum, L., Brozio, J., Mazzotti, M., 2008. Experimental characterization and multi-scale modeling of mixing in static mixers. Chemical Engineering Science 63, 4135-4149]. It is found that the mixing performance in terms of reaction yield and mixing time decreases slightly with increasing viscosity ratio in a two jet vortex mixer (Roughton mixer). In the Y-mixer the trend is the same at low flow rates, but it is the opposite at large flow rates due to a symmetry breaking phenomenon. The Roughton mixer is scaled-up using the CFD model and a linear relationship between scale-up factor and mixing time is observed. Finally, it is shown that mixing times can be described satisfactorily as a function of velocity, jet diameter and viscosity.     

Effect of Mixing on the Nucleation and Growth of Titania Particles

Aerosol processes that produce titania particles by reacting gaseous precursors (such as titanium tetrachloride) initially must mix the precursor into the oxidizer at elevated temperatures to initiate the formation of product. Oftentimes the rate of reaction is sufficiently large as to be mixing limited. Thus the rate of mixing of the reacting species will control the chemistry and morphological properties of the particles that are produced. The interplay between mixing, nucleation, and growth in these systems is difficult to observe experimentally due to the small time scales that are involved and the spatial limitations of most diagnostics. An alternative approach is direct numerical simulation (DNS). DNS refers to a class of numerical solutions of the three-dimensional time-dependent governing equations for a particular system in which no turbulence modeling assumptions are made. To within the precision of the numerical algorithm, DNS can be thought of as a numerical experiment. Here we apply DNS to the mixing, reaction, nucleation, and growth of titania particles formed from the reaction of titanium tetrachloride with oxygen. The simulation solves for the velocity, species concentration, and eight moments of the particle size distribution using a combination of a pseudospectral method (for the velocity) and a compact finite difference scheme (for all of the scalars). The results show that increasing the rate of mixing increases the rate of particle formation while decreasing the variance in the particle size distribution. However, for a given extent of reaction, poorer mixing leads to larger mean particle sizes and larger standard deviations. The results are most easily interpreted in terms of the reaction volume between the unmixed reactants, where most of the reaction occurs. Based on this analysis, we present rules of thumb for controlling the particle size distribution in aerosol reactors.     

A fast approximation method for computing effectiveness factors with non-linear kinetics

This paper describes an approximation method for the estimation of effectiveness factors in porous catalysts that have reactions with non-linear kinetic models. The reaction rate expression is expressed as a combination of two asymptotic solutions based on a local concentration, leading to an approximate rate function containing four parameters. The effectiveness factors computed with the approximated rate are shown to be close to the values calculated from the real rate function for a variety of non-linear LHHW type models. The advantage of the approximate method is that a given set of four parameters, which can easily be calculated, uniquely identifies an effectiveness factor. This observation enables the construction of a table from which the effectiveness factor can be readily extracted during reactor simulation, thus resulting in a decrease in computational effort.     

Mixing Time Scale Determination in Microchannels Using Reaction Calorimetry

Mixing time scales are derived from heat flux profiles for an instantaneous and exothermic reaction in a commercially available microreactor. A continuous reaction calorimeter, based on numerous heat flux sensors, is used to record spatially resolved heat flux profiles in steady state. Total volumetric flow rate is varied at constant flow rate ratio and the region of main reaction progress is shifted within the microreactor according to the advancement of the mixing process. Secondary flow patterns, induced by Dean mixing elements within the microchannel at higher flow rates, enhance the mixing. Results display a decrease in mixing time at increased flow rates and energy dissipation rate. Additionally, the passive micromixer is evaluated regarding its efficiency.     

搅拌混合中的循环与剪切

搅拌混合中的循环与剪切被定量化,搅拌功率分解为P=Q∑S_k,分解后的循环流量和总剪切强度,以及剪切强度在空间的分布应匹配．混合对偶合反应过程影响的实验结果表明,循环／剪切比适中的搅拌桨能较好地满足偶合反应工艺的需要,其分隔指数最小.用修正的拉格朗日法进行过程模拟,结果表明团块的循环经历影响其局部分隔指数,增大单位体积功耗、循环／剪切比以及循环区与桨叶区的能耗分配比能降低偶合反应的分隔指数．     

An optimal catalyst activation policy for poisoning problems

Pontryagins minimum principle is used to calculate the optimum distribution of active material throughout a single pellet that is uniformly experiencing activity decay. The definition of optimality is based upon balancing catalyst costs against the net return from reactant conversion. The optimal policy is full activation to a fractional depth with an inert core, the depth depending upon the Thiele parameter, poisoning time constant, operating time, and an economic parameter. An isothermal, first order, irreversible reaction is considered.Auxiliary calculations of the effectiveness factor are given for the reaction rate in the nonisothermal pellet with the catalyst distribution found to be optimal in the isothermal case.The results of the pellet calculations are utilized in numerically calculating a uniform activation policy that is optimal for a homogeneously poisoned bed. The definition of optimality is on the same basis as that for the single pellet and the results depend upon the same physical parameters in addition to the time constant for convection in the tube relative to that for diffusion in the pellets. Four regimes of behavior arise depending upon catalyst costs: (i) the reactor cannot be operated economically, (ii) only partial activation policies are economical, a single one being optimal, (iii) both partial and full activation schemes are economical, with one of the former being optimal, and (iv) full activation is optimal but the reactor can be operated economically with partially activated pellets.The results for both the single pellet and the packed tube are viewed as the homogeneous poisoning limit of many practical problems and are believed to reflect the major characteristics of more complicated poisoning mechanisms.     

Maximal and minimal solutions,effectiveness factors for chemical reaction in porous catalysts

A chemical reaction in a porous catalyst particle is considered. It is shown that in both the isothermal and non-isothermal cases, with either Dirichlet or third kind boundary conditions, two sequences of functions can be obtained, one of which converges to the maximal solution and the other to the minimal solution. Corresponding to these two sequences of functions, two sequences of numbers, which converge to the maximal and the minimal effectiveness factors, can be obtained. It is shown that in the isothermal case, the maximal (minimal) solution has the minimal (maximal) effectiveness factor while in the non-isothermal case, the maximal (minimal) temperature solution has the maximal (minimal) effectiveness factor. Two numerical examples are given to illustrate the results.     

Models of SHS: An overview

The theoretical models of SHS based on lamellar or cellular approximations of the heterogeneous reactive media are comparatively analyzed. It is shown that the ratio of the reaction time to the characteristic time of heat transfer between particles is a decisive parameter for the combustion wave propagation. When the time of reaction is shorter than the time of heat exchange, the combustion occurs in a discrete mode; in the opposite case, a quasi-homogeneous combustion mode occurs. Development of the discrete cellular model does not discard the quasi-homogeneous approach but markedly extends the scope of combustion theory. This extension enables explanation of many old and new experimental results that could not be rationalized within the framework of homogeneous theory.      

Simulation of light-limited algae growth in homogeneous turbulence

A study is performed of the effect of turbulent mixing on algae growth rate under light-limiting conditions. In order to isolate the effect of light fluctuations and to eliminate possible effects of mass transfer and other side effects of mixing, a numerical model of algae growth is employed. The study examines three aspects of turbulent mixing using models of increasing complexity. First, mixing increases the depth of the “mixed layer”—the layer of approximately uniform algae concentration—spreading the algae produced near the surface over a thicker fluid region. We demonstrate that mixed layer depth has no effect on the net algae production rate. Second, mixing causes motion of algae cells across the optical gradient. A single-frequency harmonic model is used to demonstrate the influence of periodic motion on algae production rate for different frequencies and concentration values. The model results explain previous apparently contradictory experimental observations. Significant enhancement of algae production rate with mixing is observed for small values of the ratio of illuminated layer depth to total fluid depth; however, growth rate enhancement saturates beyond a certain mixing frequency. Third, turbulent mixing involves a broad range of fluid time and length scales. A direct numerical simulation of homogeneous turbulence coupled to the algae growth model is used to show that the main conclusions of the single-frequency model can be carried over to a more realistic turbulent flow by appropriate choice of the mixing time scale.     

Die maximal durchmischte Zwillingsschlaufe. Ein erweitertes Schlaufenmodell für isotherme,homogene reaktionen

The total range of reactor performance, that is between the PFR and the whole reactor volume being a stagnant zone, may be represented by the three parameter twin loop model. The two basic assumptions made in the model—plug-flow with no longitudinal mixing in each loop, as well as immediate mixing of the two recycling streams and the inflow on the molecular scale (maximum mixedness)—are fairly well achieved in large-scale reactors. Consequently, such reactors are easy to calculate without scale-up problems. At constant mean residence time the three parameters—total recycle number, partial recycle number, ratio of recycle times—can be varied independently. Thus, the twin loop is rather adaptable and easy to control.     

Overall effectiveness factor of heterogeneous-homogeneous chain reactions with one limited species in a catalytic slurry reactor

The analysis of the overall effectiveness factor for the reaction system involving heterogeneous-homogeneous chain reactions at low concentration of one species is studied. A more general theoretical analysis for estimating the overall effectiveness factor of reaction systems which can be a heterogeneous-homogeneous chain reaction system at low concentration of one species in a slurry reactor is presented, incorporating all the transport parameters. The concepts of overall effectiveness factors for reactions have been extended from the traditional heterogeneous reaction system to the heterogeneous-homogeneous chain reaction system. The effects of reaction mechanism and kinetic models on the overall effectiveness factor are also proposed. Comparisons of the overall effectiveness factor for a slurry reactor with or without considering the reaction mechanism are also obtained.