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.     

Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors

In this paper, we perform linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors. We consider three different models of varying dimensionality—the 3D convection-diffusion-reaction (CDR) model is the high dimensional one, and the Liapunov–Schmidt reduction based spatially averaged two-dimensional CDR model and its regularized form are the two low-dimensional ones. For each of these three models, steady state bifurcation diagrams that show the presence of multiple steady states were obtained and the stability of these multiple steady states to transverse perturbations was analyzed using linear stability analysis. Parametric analysis of the steady state bifurcation diagrams shows that for sufficiently large values of transverse Péclet number p, mixing-limited patterns may emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Comparison of the bifurcation diagrams and the stability boundaries of the two low-dimensional models with that of the 3D CDR model reveals that the regularized form of the low-dimensional model has higher accuracy and a larger region of validity than the averaged form and is therefore recommended over the latter.     

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.     

Hydrogen mass transfer and mixing energy effects in SRC-II coal liquefaction reactors

This paper summarizes the experimental work performed on a bench-scale pre-pilot unit for investigating hydrogen mass transfer and mixing energy effects in SRC-II coal liquefaction reactors. Experiments were carried out with an Ireland Mine coal where the effects of mixing energy level (150–1000 rpm), method of hydrogen introduction (preheater flow and direct reactor sparging) and hydrogen treat rate (4 to 6 g of hydrogen/100 g of feed slurry) were investigated. Several runs using Powhatan No. 6 coal were also carried out where the effect of mixing energy level (200–1000 rpm) was investigated. Other run conditions were fixed to correspond to those likely to be used in commercial operation. The experimental results clearly indicated that below a mixing energy level corresponding to 400 rpm a significant cement-like solid deposition within the reactor (hydrogen mass transfer limitation) occurred. Below this mixing energy level the C₅+ liquid yield decreases, and the selectivity of the reaction changes, resulting in an increase in the C₁C₄ yield. This critical mechanical mixing level corresponds to a mixing energy per unit of reactor volume of ≈3500 ergs/cm³ s (350 watts m^?3). For the run conditions employed, increasing the preheater hydrogen flow from 4 to 6 g of $\text{H}{}_2\text{100 g}$ of slurry prevented the formation of solid deposits at a mechanical mixing energy level as low as that corresponding to 200 rpm. Furthermore, the highest C₅+ yield in the entire data set occurred when the preheater hydrogen flow was at the higher level.     

Lateral mixing in trickle bed reactors

Knowledge of lateral mixing is essential to understand heat and momentum transfer parameters in both single-phase liquid and two-phase gas-liquid co-current down flow through packed bed columns. The reactors through which gas and liquid concurrently flow downwards through a bed of catalytic packing are called trickle bed reactors. Experimental data on lateral mixing coefficients from both the heat transfer and radial liquid distribution studies are obtained over a wide range of flow rates of gas and liquid using glass spheres (4.05 and 6.75 mm), ceramic spheres (2.59 mm), and ceramic raschig rings (4 and 6.75 mm) as packing materials covering trickle flow, pulse flow, and dispersed bubble flow regimes. In the present work, an expression for estimation of lateral mixing coefficient (αβ)_L is derived using the data on radial liquid distribution studies. The agreement between the values of (αβ)_L obtained from heat transfer studies and from radial liquid distribution studies using the experimental data shows that there exists an analogy between the heat transfer and radial liquid distribution in packed beds. Since (αβ)_L is an important variable for estimation of various heat and mass transfer parameters, a correlation for (αβ)_L based on present heat transfer study is proposed. The agreement between the (αβ)_L values estimated from the proposed correlation and experimental values is satisfactory with a standard deviation (s.d.) of 0.119.     

A novel approach to the design and operation scheduling of heterogeneous catalytic reactors

A number of studies have been conducted to reduce the overall level of catalyst deactivation in heterogeneous catalytic reactors, and improve the performance of reactors, such as yield, conversion or selectivity. The methodology generally includes optimization of the following: (1) operating conditions of the reaction system, such as feed temperature, normal operating temperature, pressure, and composition of feed streams; (2) reactor design parameters, such as dimension of the reactor, side stream distribution along the axis of the reactor beds, the mixing ratio of inert catalyst at each bed; and (3) catalyst design parameters, such as the pore size distribution across the pellet, active material distribution, size and shape of the catalyst, etc. Few studies have examined optimization of the overall catalyst reactor performance throughout the catalyst lifetime, considering catalyst deactivation. Furthermore, little attention has been given to the impact of various configurations of reactor networks and scheduling of the reactor operation (i.e., online and offline-regeneration) on the overall reactor performance throughout the catalyst lifetime. Therefore, we developed a range of feasible sequences of reactors and scheduling of reactors for operation and regeneration, and compared the overall reactor performance of multiple cases. Furthermore, a superstructure of reactor networks was developed and optimized to determine the optimum reactor network that shows the maximum overall reactor performance. The operating schedule of each reactor in the network was considered further. Lastly, the methodology was illustrated using a case study of the MTO (methanol to olefin) process.     

Multi-mode low-dimensional models for non-isothermal homogeneous and catalytic reactors

A systematic procedure based on the Liapunov-Schmidt method of bifurcation theory is used to derive low-dimensional models for different types of non-isothermal homogeneous, catalytic and coupled homogeneous-heterogeneous reactors. These low-dimensional models are described by multiple concentration and temperature modes (variables), each of which is representative of a physical scale of the system. These “multi-mode models” capture mass and thermal micromixing as exchange of material and energy, respectively, between the modes (scales). The multi-mode models retain all the parameters and most of the qualitative features of the full convection-diffusion-reaction equations. While in the limit of vanishingly small local heat and mass diffusion times, they reduce to the classical ideal pseudo-homogeneous reactor models, they are also capable of capturing the mixing or mass (and/or heat) transfer-limited asymptotes for the case of fast reactions. We illustrate the usefulness of the multi-mode models in predicting mixing and selectivity effects on reactor performance and the influence of local transport effects on reactor runaway and bifurcation behavior for the case of non-isothermal homogeneous and catalytic reactors.     

Catalyst mixing in bubble column slurry reactors

The reported experimental data of Pandit and Joshi (1984) on axial and radial steady-state catalyst concentration in a semibatch bubble column slurry reactor is interpreted by the dispersion model. The elliptic partial differential equation with its associated boundary conditions is solved analytically for catalyst concentration by the method of separation of variables. The proposed model adequately fits the experimental data.     

Longitudinal mixing in horizontal rotary drum reactors

The longitudinal mixing of sodium carbonate (soda) (mean particle size 137 μm) has been investigated in a laboratory horizontal rotary drum reactor 250 mm in diameter and 600 mm long. The distributions of residence times have been estimated by means of a pulse of a small amount of sodium bicarbonate. The concentration of the tracer at the reactor discharge has been evaluated by thermal decomposition of the samples and by measuring their weight loss.The hold-ups, the mean residence times and the variances of the distribution of residence times were evaluated as a function of the rotational speed of the reactor and the feed rate of the particles. By applying a dispersion model, the Peclet numbers were evaluated from the standardized variances and plotted as a function of the feed rate and rotational speed. The mean residence time of the particles was calculated by means of an extended model of Vahl and Kingma. They correspond with the experiments.     

A rigorous approach to photochemicals reactors

The modelling of photochemical reactors has been rigorously treated describing the radiation field within the realm of radiant energy transfer. This kind of approach causes a heavy integro-differential problem to arise any time the reacting species is the absorbing one while the differential nature of the model equations is maintained in the case of photosensitized reactions in a purely absorbing medium. An annular photoreactor has been considered for this last situation and the influence on the absorption process of the most parameters has been investigated for both the cases of a reactor shielded and unshielded by a reflector.     

A new approach for scale-up of bubble column reactors

The reactor of choice for the Fischer-Tropsch synthesis is a slurry bubble column. One of the few disadvantages of bubble columns is the difficulties associated with their scale-up. The latter is due to complex phases’ interactions and significant back-mixing.     

Modelling the effects of imperfect mixing on the performance of anaerobic reactors for sewage sludge treatment

An approach to the modelling of suspended-growth anaerobic digestion systems based on the assumption of an incompletely mixed reactor is presented. The mathematical model developed describes the dynamic behaviour of anaerobic sludge digesters under non-ideal mixing conditions. The microbial kinetic model for the anaerobic digestion of waste-activated sludge distinguishes the processes of death and lysis of activated sludge cells, hydrolysis of particulate material, fermentation of soluble substrates, volatile fatty acids utilisation and methane formation. The interaction of two microbial groups is considered, i.e. acid-formers and methanogens. Their growth is assumed to depend on Monod kinetics for the substrates. Death and lysis, hydrolysis and biomass decay are described by first order reactions. The biokinetic expressions were linked to a simple mixing model which considered the reactor volume split into two sections: the flow-through and the retention regions. The transfer of material between regions was assumed to be limited. Deviations from an ideal completely mixed regime were represented by changing the relative volume of the flow-through region (α) and the turnover time of material in the vessel (τ). The dynamic model described the effects of the retention time and reactants' distribution, resulting from the mixing condition, on process performance. Computer simulations under different conditions showed a considerable decline in methane production and treatment efficiency due to incomplete mixing. The COD removal efficiency increased by extending the retention time and the degree of mixing. The evaluation of the impact of the mixing parameters showed that α has a far more significant effect on the performance of anaerobic digestion than τ does. Nevertheless, both are important and the overall efficiency is a complex function of both parameters. The results obtained confirm and emphasise the importance of considering mixing when simulating anaerobic digestion, calculating process conversion efficiency, and during anaerobic reactor design. © 1998 SCI.     

Simulating the effects of mixing on the performance of unpremixed flow chemical reactors

A parallel three environment model is developed to describe the effects of incomplete mixing on the performance of continuous flow chemical reactors fed separately by feedstreams which may have arbitrary flowrate and arbitrary residence time distribution. Unpremixed reactants are assumed to first enter partially segregated entering environments and subsequently transfer to partially mixed leaving environments at a rate and in amounts defined by an environment transfer function. Mixing in the environments is stimulated stochastically by Monte Carlo techniques. Calculations based on a kinetic model of the Michaelis-Menten type for arbitrarily selected pairs of residence time distributions suggest that performance is strongly influenced by the degree of overlap of the distributions and the method of feeding the reactants. The model is shown to compare directly with published experimental data obtained from a “jet-stirred” reactor and an equivalence found between the models' parameters and those of previously developed micromixing models.     

A population balance approach to describing bulk attrition

Attrition arising from mechanical damage during processing has been studied in annular shear cells, these having the ability to vary readily the testing stress and shear strain. A population balance approach has been deployed to analyse the evolution of the size distribution with strain which bases its arguments on the kinetic theory of grinding. This incorporates the ideas of a selection function and a breakage function. The extent of attrition is determined in terms of three parameters, one related to the selection function, one to the breakage function, and additionally one which allows for the balance of fracture and abrasion. The product size distribution of the Gaudin-Schuhmann form is consistent with experimental findings from annular shear cells. Whatever reasonable physical assumptions are made about the form of the selection and breakage functions, and of the balance between fracture and abrasion, particle size distributions of the same form arise. From comparisons with several materials, the index in the breakage function is consistent with the particle fracture pattern observed experimentally.     

Liquid phase mixing in trayed bubble column reactors

The compartmentalization of conventional bubble columns by perforated trays constitutes a very effective method to reduce the liquid backmixing. The effect of tray design and operating conditions on the overall liquid mixing was studied in a bench-scale trayed bubble column. The extent of liquid backmixing in the column was investigated in light of liquid-phase tracer response experiments. In average, a three fold reduction in the liquid backmixing was achieved in the trayed column as compared to the column without the trays. Moreover, the tray open area and the superficial liquid velocity were found to have the strongest effects on the liquid backmixing. The N-CSTR with Backmixing Model was found to match the experimental tracer response curves better than the Axial Dispersion Model.     

Liquid flow and mixing in concentric tube air-lift reactors

Liquid velocity and mixing were measured in two concentric tube air-lift reactors (ALR) of 30 and 300 dm³ (nominal volume). The influences of the geometrical design and the reactor scale were studied as a function of gas flow rates. The mixing in the 30 dm³ ALR, which had an enlarged cross-sectional area in the gas separator region, indicates that in this geometrical configuration most of the mixing occurs in this region. It is demonstrated that the location of the injection and measuring points influence the measurement of mixing time in air-lift reactors. Correlations for pressure drop at the bottom, gas hold-up and mass transfer coefficients, which were published in a previous paper, are extended to include the effect of the ratio of downcomer-to-riser cross-sectional areas.