Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A → B → C reactions

We use a continuation technique for branches of periodic solutions to investigate the oscillatory behavior of a continuously stirred tank reactor with consecutive A → B → C reactions. This continuation technique allows the computation of entire periodic solution branches, including those with limit points and asymptotically unstable solutions. Our computations reveal dynamic phenomena not seen in previous studies of this reactor. The results include response diagrams exhibiting stable and unstable periodic branches that contain multiple limit points. The presence of these points indicates that the reactor may jump from a steady state to a periodic orbit or from one orbit to another. The computations also illustrate interactions of multiple steady state limit points, Hopf bifurcations and infinite periodic bifurcations.     

Autocatalytic reactions in the isothermal,continuous stirred tank reactor: Isolas and other forms of multistability

Autocatalytic reactions are often complicated, and analyses of their behaviour in open systems can seem too particular to permit useful generalisation. We study here the simplest of circumstances (uniform temperatures and concentrations in the isothermal CSTR) and the simplest of reaction schemes: (i) quadratic autocatalysis (A + B→2B); and (ii) cubic autocatalysis (A + 2B→3B). The catalyst B may be stable or have a finite lifetime (B→ inert products). Allowing for this finite lifetime adds another dimension to the interest.The phenomena encountered include multistability, hysteresis, critical extinctions, critical ignitions, and anomalous relaxation times (though infinite values do not arise). Patterns of stationary states as function of residence time can show isolas and mushrooms. All these aspects yield to simple algebraic analysis. The presence of the catalyst B in the inflow can make qualitative differences of a kind parallelled by an additional, non-catalytic reaction of the same stoichiometry (e.g. A→B). Invoking the reversibility of the reactions neither increases nor diminishes their variety, and thermodynamic considerations have little to do with the many different patterns of reactivity displayed.The local stability of the various stationary states has also been characterized. Quadratic autocatalysis shows limited variety (stable node, stable focus); cubic autocatalysis generates all the kinds of stationary state possible in a two-variable system. Again all the algebra is straightforward if not always simple. Sustained oscillatory behavior (limit cycles) also occur.All these remarks relate to isothermal systems, but there are the most striking parallels between isothermal autocatalysis and the exothermic, first-order reaction in the CSTR. Behaviour with an autocatalyst of complete stability corresponds to perfect heat insulation (adiabatic operation) in the non-isothermal, exothermic system.     

Multiplicity of conversion in a cascade of imperfectly stirred tank reactors

The multiple steady states of conversion in the model of non-ideal CSTR's in series have been studied for irreversible, homogeneous exothermic reaction of first order. The influences of the parameters p and n and those of some kinetic parameters on the occurrence of multiple conversions for a given residence time are presented. The existence of steady-state multiplicity of conversion in each CSTR is shown to be independent of n and depen largely on the values of E/RT₀ (activation energy), b/T₀ (heat of reaction), and p. For certain parametric values, more than three steady states can exist in such a model.     

Periodic control of continuous stirred tank reactors—II Cases of a nonisothermal single reactor

The optimal periodic control of a single nonisothermal CSTR in which a single reaction, a parallel reaction or a consecutive reaction is taking place is studied. Both the feed rate and the heat transfer between the reactor content and the immersed coil are periodically controlled. Utility consumption and conversion of the raw material into the waste are included in the performance index together with conversion into the desired product. The results indicate a respectable improvement in the reactor performance.     

Theoretical investigations of dynamic behavior of isothermal continuous stirred tank biological reactors

The dynamic behavior of an isothermal biological CSTR modelled by idealized cell and substrate balance equations has been investigated theoretically in terms of multiplicity and stability of steady states and existence and stability character of limit cycles. Various types of dynamic behavior have been classified in terms of a Damköhler number and two other system parameters. The predicted types of behavior have been illustrated by numerical computation of cell and substrate concentration trajectories.     

Periodic control of continuous stirred tank reactors—I: The pi criterion and its applications to isothermal cases

The pi criterion which is useful for examining if the optimal periodic control is proper is briefly introduced in a somewhat generalized form and then it is applied to optimal periodic control problems of CSTR's in isothermal condition.     

A Quasi-steady-state analysis of the dynamics of two-species heterogeneous catalytic reactions

This study demonstrates how low-order algebraic non-linearities that exist in a simple two-component Langmuir-Hinshelwood type reaction kinetics (CO oxidation) are sufficient to produce rate multiplicities and oscillatory steady states (periodic solutions). A singular perturbation analysis is employed wherein certain quasi-steady-state considerations are made which lead to the definition of system manifolds and invariants along which the large-time dynamics of the system can be discerned without recourse to numerical integration. New results, confirmed by simulation, include an explanation for such experimentally observed pathological phenomena as the coexistence of oscillatory and stationary steady states and multi-peak oscillations. It is shown how the existence of oscillatory states and a plausible “buffering” physisorption surface reaction mechanism causing a periodic switching between these states and coexisting stationary ones can give rise to these multi-peak oscillations.     

Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped-parameter systems

Exact criteria are derived for predicting the conditions under which steady-state multiplicity occurs in lumped parameter systems in which either two consecutive or two parallel reactions occur. Most of the qualitative multiplicity features can be determined by considering four simple limiting cases in which only one reaction takes place in the system. The structure of all the possible multiplicity regions in the (Da₁, Da₂) plane is determined for both reaction networks.     

Theoretical analysis of two parallel and consecutive reactions in isothermal symmetrical catalyst pellets using the dusty-gas model

The courses of the simplest multiple reactions such as two parallel reactions and two consecutive reactions proceeding in the isothermal symmetrical ca     

Binary and ternary distillation in a stirred mass transfer cell

A continuous flow stirred cell operating at point conditions has been developed for the study of mass transfer in distillation. Results for the methanol-isopropanol and methanol-ethanol binary systems showed that 85–100% of the resistance to mass transfer resided in the vapour phase, and that the vapour side coefficients correlated well with those obtained in the same cell for gas phase controlled absorption. Flux rates for the ternary system methanol-ethanol-isopropanol were in fair agreement with those predicted from the binary data, using a modified form of the Krishna-Standart theory. It is concluded that the experimental technique shows considerable promise, but requires further development to overcome discrepancies in the interfacial heat balance.     

A dynamic response technique for the estimation of gas—liquid mass transfer coefficients in a stirred vessel

A method is described for evaluating mass transfer coefficients, k_La, in gas sparged stirred vessels. It involves measurement of both liquid and gas concentration responses following a step change in inlet gas concentration, and differs from previously published methods in that the k_La evaluation is independent of the gasresidence time distribution. Results are presented that demonstrate the suitability and consistency of the method for a range of gas mixing levels.A modification, that involves only a knowledge of the early liquid concentration response, is suggested as worthy of further consideration.     

Suspending solid particles in an agitated tank with axial-type impellers

Experimental investigation of the critical impeller speed in systems solids-liquid agitated by axial-type turbine impellers was drawn to more general relations as regards the impact of two more important variables: angle of blade inclination and volume fraction of solids. Both variables manifest existence of separate hydrodynamic regions which are reflected in the respective shape of the theoretically derived correlation function.     

Steady state multiplicity depending on coalescence in liquid—liquid continuous stirred reactors with reaction in the dispersed phase

The influence of drop coalescence and breakup on the existence of multiple steady states is studied for a two-phase stirred isothermal reactor where the chemical reaction in the d?ispersed phase obeys the rate expression ? r = kC/(1 + KC)². The random coalescence model developed by Curl was simulated using a modified Spielman and Levenspiel Monte Carlo technique.For certain range of the coalescence rate, Damköhler number, and dimensionless feed concentration, multiple steady states have been investigated.A special case has also been considered wherein the existence of multiple steady states for finite values of the coalescence rate is contrasted to the unique steady state solution for an infinite coalescence rate.     

On the global dynamics of an autothermal reactor stabilized by linear feedback control

The global dynamics of an autothermal reactor stabilized at an (open-loop) unstable steady-state by a single proportional controller is studied as a bi     

Effect of particle size distribution on adsorption kinetics in stirred batch systems

The effect of different particle size distributions (PSD's) on kinetic behavior in stirred-batch adsorption systems was investigated by numerically modeling the uptake of p-nitrophenol from aqueous solution by granular activated carbon in a well-stirred batch type of environment. Four different Gaussian PSD'S, two different log-normal PSD's and one case of uniform particle size were studied. The model accounted for both external and internal mass transfer resistances, and for the highly nonlinear equilibrium relation which characterizes the p-nitrophenol/activated carbon system. All isotherm parameters and “base-case” kinetic parameters were experimentally determined values (from the literature). The kinetic parameter values were varied in some cases to assess their influence. Moreover, both infinite-bath and finite-bath situations were evaluated. The modeling results show that the different particle size distributions did not generally give kinetic behavior widely different from the uniform particle size case when an infinite bath was assumed. However, for the more realistic case of a finite bath, the different PSD's gave significantly more distinctive results, especially for fractional mass uptake values greater than about 0.7. It is concluded that, in finite bath experiments, the nature of the adsorbent PSD must be accounted for in interpreting data obtained at fractional mass uptake values greater than about 0.7, unless the adsorbent PSD is reasonably nar     

Transient response of continuous-flow stirred reactors containing heterogeneous systems for catalysis or sorption

Time domain solutions are presented for the transient response of an isothermal CFSR with one-dimensional diffusion and first-order irreversible chemical reaction in catalyst particles or liquid pools of regular geometry and with mass transfer resistance at the fluid-catalyst interface. Three different cases of feed perturbations have been considered, namely, step input, impulse input and the sudden introduction of pellets into the reactor. The solutions have been derived in a general manner, without specifying the particle shape, and are in terms of characteristic functions defined for each of the three particle geometries considered, i.e. slab with sealed edges, cylinder with sealed ends, and sphere. Asymptotic solutions suitable for large Thiele parameters (gf≥3) have also been derived and are particularly convenient when the series solutions converge slowly. Experimental verification of the exact and asymptotic solutions for a specific case of absorption and diffusion in a liquid pool is presented.     

Heat transfer and power measurements in stirred tanks using heat flow calorimetry

A new method is presented for determining the power consumption of an impeller in a stirred tank as well as the inside film heat transfer coefficient.Five working fluids were used. Power consumptions for turbine, Pfaudler and anchor impellers agree well with literature values. Heat transfer coefficieA plant-scale heat transfer coefficient can be estimated rapidly and accurately from a few runs in a bench-scale heat flow calorimeter, without having     

Numerical analysis of a finite cylindrical pellet model in solid-gas reactions

A two-dimensional grain model has been developed to describe solid-gas reactions in a cylindrical pellet of finite length. The grains are assumed to be spherical. Numerical solutions of the model have been compared to infinite cylinder solutions, infinite slab solutions, and the approximate infinite cylinder and infinite slab solutions developed by Szekely et al. (Szekely J., Evans J. W. and Sohn H. Y., Gas—Solid Reactions. Academic Press, New York 1976). Pellet sizes and reactor conditions under which one or more of these solutions is valid have been indicated. The accuracy of the other models depends on whether they are to be used to determine rate constants from initial rate data or whether they are to be used to relate conversion to time.When the Theile-type reaction modulus σ_L = L[3(1 - ε₀)K/D_ear_g]^{$\text{1}\text{2}$}, has a value 2.2 with a cylinder half-length of L = 0.2 cm and a cylinder radius of R_p = 0.8 cm the conversion levels after about 8 min are 40% by an infinite slab model, 50% by the 2-D model and 65% by the approximate infinite slab solution. If one determines a rate constant from the initial slope of the rate curve, then the infinite slab model gives a value 14% high. The approximate infinite slab solution is inappropriate for the latter purpose. Similar results are obtained when the cylinder radius is much smaller than the half-length.Pellets with R_p/L > 4 or <0.25 can be described by infinite slab or infinite cylinder models, respectively, if the diffusional resistance is not large.