A mixing model for diagnosing reacting tracers and tracing reactants in chemical and biochemical processes

It has previously been shown (Tanner et al., 1985) that biochemical and chemical reaction processes of the Type A → B → C can lead not only to kinetic hysteresis between the rate of formation of C and the concentration of species B in batch processes, but also hysteresis between those variables in a closed, imperfectly mixed (two zone) batch reactor. Furthermore, crossplotting the intermediate reactant, B, in one region of the poorly mixed reactor against B from the other major region leads to a clockwise hysteresis curve which is defined by both its area and the coordinate phase angle. This paper shows that the earlier analysis (Tanner et al., 1985) can be extended to the more general system A → B, where A in one region of the vessel is crossplotted against A measured in the other region. With an initial concentration of A* in one zone equal to its highest concentration, the inscribed area double-valued crossplot of A, uniquely defines the system in terms of the inter-vessel flowrate to reaction rate constant ratio, D/k.     

Autocatalytic reactions in the isothermal,continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C

The prototype, cubic autocatalytic reaction (A + 2B → 3B) forms the basis for the simplest homogeneous system to display “exotic” behaviour. Even under well-stirred, isothermal, open conditions (CSTR) we may find multistability, hysteresis, extinction, ignition and anomalous relaxation times. Allowing for the finite lifetime of the catalyst (B→inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas or mushrooms. Sustained oscillations (stable limit cycles) are also possible. The onset of oscillation corresponds to a change in the character of the stationary-state (from stable focus to unstable focus) and the conditions for this change can be evaluated analytically. The period of the oscillations and their amplitudes increase as the residence time is lengthened. A total of nine different phase-portraits in the a–b plane has been found.The isothermal system is less complex than the exothermic, first-order reaction in a CSTR, but there are strong analogies between the two.     

Numerical study on microstructured reactor with chaotic heat and mass transfer and its potential application for exothermic process

Design of microstructured reactors with thermal control function is investigated through numerical simulation. It consists of one middle channel for handling chemicals and two other channels attached to its top and bottom for cooling purpose. Three designs are examined. Reactor A uses simple straight channels. In reactor B, chaotic flow is applied to the middle channel, and in reactor C chaotic flow is applied to all the three channels. Results show that in comparison with the straight channel, the Nusselt number in current design is greatly improved through chaotic flow. Rapid mixing is also achieved. Potential application of the design for continuous exothermic process is analyzed. For reactor A, it is not workable as the temperature of the chemical solution continuously increases over the channel. In comparison, for both reactors B and C the temperature can be well controlled within the required range. As the coolant flow in reactor C is also chaotic, it provides a higher heat removal capacity.     

Certain preparative aspects of the flow electrolysis on porous electrodes

Two modes of operation of a flow electrolytic reactor are considered, one consisting of a recycle of a solution between the reactor and a tank, and the other consisting of flowing the solution only once through the reactor. The times of electrolysis to a given final degree of conversion for the two modes of operation of a given reactor are calculated and compared. In the second part of the paper the case of a reaction of ece or ec mechanism going on in a flow electrolytic reactor is considered. Using kinetic theory for two consecutive first-order reactions, equations are derived allowing to calculate the value of the flow rate corresponding to the formation of a maximal concentration of an intermediate electrolytic product, and the value of the concentration. Numerical solutions of the equations are given for two porous electrodes differing in length.     

Steady state multiplicity of chemically reacting systems. The method of computation

To analyse the steady state multiplicity of chemically reacting systems of different types it is necessary to find all roots of systems of non-linear equations describing steady states of the systems. This can be solved rather simply in the case of a reducible system to a single equation. We present a numerical method of nonlocal solution for systems of non-linear equations which does not require computing with the Jacobian matrix. The method is illustrated by joint equilibrium absorption of two species for a semi-empirical model of induced inhomogeneity of the catalyst surface and by computation of the steady states of the stirred tank reactor in which the reaction A → B → C occurs.     

Matched asymptotic solutions of the coalescence-redispersion model for unmixed feed stream plug flow reactors

Matched asymptotic expansions are given for the first moments of the concentration distribution function of Curl's coalescence-redispersion population balance model when mixing is faster than reaction. The physical situation considered is an unmixed feed stream tubular reactor with a plug flow residence time distribution in which the single reaction A + nB → P occurs. The asymptotic expansions lead to analytical expressions for species conversion in such reactors and thus obviate the need for numerically tedious Monte Carlo simulations. It is also demonstrated that the perturbation technique applies in a straightforward manner to situations with competing reactions and that the analytical calculations compare favourably with Monte Carlo simulations.     

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.     

Theoretische Berechnung des Stofftransports in der Umgebung einer Einzelblase

Mass transfer between a spherical bubble and a surrounding fluid has been investigated theoretically. Stationary transfer and constant diameters were assumed. The local concentration around a single bubble as well as local and mean Sherwood numbers were calculated. The velocity distributions were determined in an earlier investigation by numerical solution of the Navier—Stokes equation.The mean Sherwood number for bubbles of constant shape were proved to be a function of two dimensionsless numbers: the Schmidt- number Sc and the product Re Sc of Reynolds and Schmidt-number. Curves fitting the calculated values are lying between two limiting curves. The lower curve is valid for the limiting condition Re → 0 and Sc → ∞ while the upper curve is valid for Re → ∞ and Sc → 0.For bubbles of varying shape the Sherwood number will not follow the upper boundary but will increase considerably beyond this limit. Analysis of the mechanism of mass transfer indicates that this behavior results from periodic and aperiodic deformations of the bubble.     

Mixing effects in the bromination of resorcin

The product distribution from the bromination of resorcin (m-dihydroxybenzene) is influenced by mixing in an experimentally convenient range of concentration and mixing intensity at room temperature. The degree of bromination is insensitive to mixing, a fact which can be explained chemically. Similarly explicable is the high sensitivity of the composition of the isomeric dibromoresorcins; the % 2,4-dibromoresorcin formed has been used to characterise mixing in semi-continuous and continuous stirred tank reactor operation. In the latter case the influences of the concentration and flow rate of the feed, turbine speed and feed positions near the turbine were investigated. Although full reaction kinetics are not available, these reactions are fast and sensitive enough to study micromixing.     

On the dynamics of a stirred tank with consecutive reactions

The pattern of multiplicities of the steady states of a stirred tank reactor in which A → B → C takes place has been known for some time to be complex. We show that even in regions of parameter space for which there is only one steady state the dynamical behavior can be very complicated. Since the number of parameters is very large a complete exploration of the space is impossible, but by careful transection of a small, but significant, region we show that repeated bifurcation of limit cycles leads into a pattern of periodic and chaotic behaviour that returns to a simple limit cycle and then reflects itself.     

The concentration spectrum of the product of a fast bimolecular reaction

The concentration spectrum of the product of a fast chemical reaction of form A + B → P is determined analytically for wave numbers greater than the Kolmogorov wave number in a turbulent reactor model with vortical microscale structure. The Schmidt number is large and the volume of one of the reactants is small. The spectrum first decreases like k⁻¹ as for a passive scalar, then increases linearly with wave number, peaking near the Batchelor wave number from which it drops off like k⁻⁴ for larger wave numbers to finally decay exponentially for wave numbers of the order of the reciprocal of the reaction layer thickness.     

Mixing and fast chemical reaction—II: Diffusion-reaction model for the CSTR

Of several possible representations of inhomogeneity in mixtures and of its influence on chemical reactions, a diffusion-reaction formulation has been developed here to model the mixing-dependent product distributions leaving a CSTR. (A similar treatment of competitive, consecutive reactions in a batch reactor was given earlier[20]). The reactions reported in Parts I and III are conducted using a much larger volume of alkaline, 1-naphthol (A) solution than of diazo salt solution (B). Partial segregation is thus visualised as a dispersion of quiescent, spherical B-rich zones, into which A diffuses and within which all reactions occur. Unsteady-state diffusion-reaction equations thus characterise the concentration changes within these zones. These have been solved numerically with boundary conditions etc. characteristic of the CSTR. In addition to the dimensionless variables determining the product distribution in the chemical regime, a further group (similar to the Thiele modulus and theta number) has a dominant effect in the mixed, diffusional-chemical regime. As various variables are changed, the model predicts effects on the product distribution, which are readily understood.     

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.     

A theoretical analysis of the deactivation of an immobilised enzyme in multi-enzyme reactions in fixed-bed and fluid-bed reactors

The effect of time-averaging on the selectivity to an intermediate R (yR ) and the yield to product S (yR ) in a consecutive reaction A→R→S, occurring in a deactivating fixed-bed reactor, has been examined. Time-averaging smooths the selectivity and increases the yield for high values of the deactivation (λ) and reaction (δ) parameters. The kinetics and dynamics of yR and yS in a deactivating steady-state fluid-bed reactor were also examined. Both non-selective and selective deactivation are considered. The practical implications of the development are considered in terms of an example, in which the values chosen for parameters are arbitrary though reasonable with respect to experimental values.     

Diffusive transport with consecutive catalytic wall reactions

A new complete solution is obtained for the plug flow reactor with consecutive catalytic wall reactions at low Peclet number flow. The significant effects of the Peclet number and the wall reaction kinetics on the concentration field, the Sherwood number, the yield of the intermediate product and the optimal reactor length are established. A useful graph for the design purpose is presented for the determination of the optimal yield of the intermediate product B and the reactor length for Pe = 1.     

Optimal temperature policy in a riser reactor

The nature of the optimal axial temperature distribution in a riser reactor to maximise the yield of the intermediate B, in the reaction sequence A→B→C was investigated by way of the selective oxidation of naphthalene to phthalic anhydride. For this case, imposition of an axial profile could not improve the yield over that obtainable by operating at the optimum isothermal temperature. This result appears to be general, and suggests that optimal yields could be obtained by operation at the optimum isothermal temperature, which should prove easier to implement in practice than a specified temperature distribution.     

Steady state multiplicity behavior of an isothermal axial dispersion fixed-bed reactor with nonuniformly active catalyst

An isothermal, heterogeneous fixed-bed reactor packed with nonuniformly active catalyst pellets where a biomolecular Langmuir-Hinshelwood reaction occurs, is studied using an axial dispersion model. A catalyst activity distribution given by a Dirac delta function, where the active catalyst is deposited at a specific location within the pellet, is considered. This includes the common case of externally coated pellets with external mass transfer resistance. The steady state multiplicity behavior of this reactor, and its limiting cases: CSTR, PFR and pseudohomogeneous axial dispersion, are examined in detail. The nonlinearity of the reaction kinetics provides two sources of multiplicity, through the heterogeneous nature of the reactor and the presence of axial dispersion in the fluid phase. Their roles in determining reactor multiplicity behavior are fully explored. It is shown that this system can admit at most nine steady state solutions. The limiting behavior of the heterogeneous axial dispersion model as Pe → 0 or ∞ is not represented fully by the CSTR or PFR models because of ignition phenomenon. Finally, the effects of mixing on reactor conversion are discussed.     

Concentration forcing of isothermal plug-flow reactors for autocatalytic reactions

The performance of isothermal plug-flow reactors under feed concentration forcing is analysed for improvement in average yield of product B for a homogeneous autocatalytic reaction, A →k1 A₁ →k2 A_, →k3 B, where A₁ or A₂ has an activity influence on the reaction rate constant k₁. The performance of the periodically forced reactor with the autocatalytic reaction is compared with that of the corresponding uncatalysed reaction.     

Blending of irregularly shaped particles

Existing blending theories have mostly been verified in the past by mixing of monosized, smooth, spherical particles. In this study, binary mixing of non-spherical particles with rough surfaces is shown to adhere to diffusional mixing equations as long as the mean diameters of the two fractions (A and B) are identical, i.e.d_A = d_B. When this is not the case, blending rates (diffusional coefficients of blending, D) reduce drastically when the small particles are not of such a size that they will fit into the interstices between the larger particles. It is shown also that the final standard deviation in these cases is many orders of magnitude higher than the predicted random standard deviation. When d_A = d_B, the diffusion coefficient will depend on the magnitude of d_A. The diffusion coefficient is not a function of the ratio of A to B. In blending in a horizontal mixer, D decreases with addition of lubricant. This is not the case in a V-blender, where bulk mass transfer appears to be the controlling step. In all other respects, V-blending is identical, functionally, to the profiles obtained in cylindrical blending.     

Design and control of an olefin metathesis reactive distillation column

This paper considers the design and control of a reactive distillation column in which one reactant is consumed and two products are formed (A?B+C). The volatilities are α_B>α_A>α_C, i.e. the reactant is intermediate boiling between the two products. The metathesis of 2-pentene is considered as the demonstrative example. The column has a single feed of the intermediate boiling reactant. The distillate contains mostly light component and the bottoms mostly heavy.Three designs are considered: the base case (low-conversion/low-pressure), a low-conversion/high-pressure case and a high-conversion/high-pressure case. The base design is obtained from the literature, and the other two steady-state designs are optimized with respect to the total annual cost. All the designs are found to be openloop stable. Five control structures are studied for the base design. Then the best two structures are applied to the remaining two designs. This category of reactive distillation exhibits less challenging problems than other categories since it uses a single feed, which eliminates the need for the control structure to perfectly balance two fresh feeds.Simulation results demonstrate that effective dynamic control is provided by a control structure that uses two temperatures to maintain the purities of both product streams. No internal composition measurement is required. This structure is found to be robust and stable and rejects loads and tracks setpoints very well.