Terms Of Reaction Rate Research Articles

Heterogeneous catalytic reactors undergoing chemical deactivation. Part II. Design and analysis: Approach of reactor point effectiveness

AbstractA design method is presented that takes into account the effects of both diffusion and deactivation on hetergeneous catalytic reactors. Improvements that can be made by the method are demonstrated and an efficient methodology is developed by which the design problem as well as associated optimization problems can be solved in a routine manner. The approach is based on the reactor point effectiveness for global rate of reaction in terms of bulk fluid conditions. Some important features emerge from the examples presented.

REACTION MECHANISM OF HALF-CALCINED DOLOMITE DURING SULFIDATION

The kinetics of the sulfidation of half-calcined dolomite have been studied over a wide range of reaction conditions in terms of reaction rate and solid phase structure. The rate of reaction shows a strong dependence on temperature, pellet size, and H2S partial pressure. Analysis of both rate and solid phase structure data indicates that the sulfidation reaction proceeds topochemically on the pellet level with two interfaces below 780° C and with three interfaces above that temperature.

Critical conditions for thermal explosion in reactive viscous flows

Thermal stability of some simple steady viscous flows of a reactive fluid with heated boundaries is investigated; only flows which are known exact solutions of the Navier-Stokes equations are considered. Two approaches are used: (i) a perturbation series about the inviscid solution; and, briefly, (ii) a step-function approximation to the reaction rate term. Results for the critical value of the Frank-Kameneskii parameter and the critical maximum temperature are presented and compared.

Incompatible reactions between some plants and pathogenic bacteria

Incompatible combinations of plant and plant bacteria produce an incompatible reaction at different rates, producing, besides the typical hypersensitive reaction, also darkening, yellowing, and fading. Plants differ in their responsiveness to plant bacteria in terms of reaction rate. Study of incompatible reactions may differentiate the species and strains of plant bacteria if properly explored.

Periodic solutions to systems of reaction-diffusion equations

In this paper, necessary and sufficient conditions are derived for the existence of temporally periodic “dissipative structure” solutions in cases of weak diffusion with the reaction rate terms dominant in a generic system of reaction-diffusion equations ∂c i/∂t = D i▿ 2 c i+Q i( c), where the enumerator index i runs 1 to n, c i = c i( x, t) denotes the concentration or density of the ith participating molecular or biological species, D i is the diffusivity constant for the ith species and Q i( c), an algebraic function of the n-tuple c = (c 1 ,\\3., c n), expresses the local rate of production of the ith species due to chemical reactions or biological interactions.

1] Introductionk

This chapter discusses the complications related to immobilize enzymes while determining of the kinetic constants. The measurement of the kinetic constants of immobilized enzymes may not yield true values equivalent to those obtained in homogeneous reactions, because of the influence of physical factors, such as diffusion. Hence, rate and binding constants should be referred to as apparent Vmax and apparent Km and written as Vmax (app) and Km (app). An operational definition of Km (app) would be that substrate concentration which gives a reaction velocity corresponding to half the Vmax (app). It is suggested that immobilized enzymes generally show less specific activity than the native enzyme. Losses in activity are minimized because more sophisticated procedures have being developed. In characterizing immobilized enzymes, the activity should ideally be expressed as the initial reaction rate in terms of for instances microkatals per milligram of dry immobilized enzyme preparation or any multiple thereof.

Numerical solution of equations arising in energy transport in non-equilibrium chemically reacting systems

A two-point boundary value method is presented for avoiding the numerical instability which can occur when initial value methods are used for solving the non-linear differential equations describing transport of heat and mass in chemically reacting systems near equilibrium. The instability arises primarily owing to the slight computational errors that arise when the forward and reverse reaction-rate terms are large compared to the net rate. The method used to avoid this instability is to convert the continuity of species equation containing the reaction-rate terms into a second order differential equation in such a manner that small errors in computing the reaction rates do not appreciably affect the solution. Though equations leading to this instability have sometimes been referred to in the past as “stiff,” today this term most commonly refers to quite a different situation, namely, a system of equations where at any point the ratio of the largest to smallest eigenvalue is very large. Thus, in this work the term “stiff” will be replaced by “instability.” As an example of how to circumvent this instability, the method is applied to the problem of heat transfer through a reacting gaseous mixture confined in a parallel plate conductivity cell. It is demonstrated that this case presents a more stringent test of the applicability of the method than the case of convective heat transfer.

Lysine Retention in Food During Extrusion-Like Processing

ABSTRACT A VAILABLE lysine levels were A measured as a function of process temperature, time, product moisture content, pH, protein level, carbohydrate type and level. The results are reported in terms of acti-vation energy (ca 34000 cal/mole) and reference reaction rate at 100 C (ca 0.24 mg/g sample minute) for an initial zero order loss, and equa-tions which represent the data in latter stages of the loss curve.

Solutions to systems of nonlinear reaction-diffusion equations.

General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ∂ci/∂t = Di▿2ci + Qi(c) subject to boundary conditions of practical interest, where the enumerator index i runs 1 to n, ci = ci(x, t) denotes the concentration or density of the ith participating molecular or biological species, Di is the diffusivity constant for the ith species, and Qi(c), an algebraic function of the n-tuple c = (c1,…, cn), expresses the local rate of production of the ith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals of c which decrease monotonically with time can often be found, as examplified here for Volterra and Verhulst-Volterra n-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reaction-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, lim ⁡ t → ∞ c i ( x , t ) = ĉ i ( x ) , if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl of Q(c) in c-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of “weak diffusion” with the reaction rate terms dominant in the governing equations.

On the extinction of diffusion flames

Detailed analysis of diffusion flames involves the solution of a non-linear differential system. However, by approximating the reaction rate term in the thermodynamic equations by a Dirac delta function, the differential system becomes linear. Hence, the dependence of the flame temperature upon the Damkohler number is investigated. Comparison of the results of the analysis with some observations of flames in the forward stagnation region of a porous cylinder by Tsuji and Yamaoka suggests that a criterion for flame extinction is simply that the flame temperature is less than some critical value. When the analysis is extended to model the flame on the upstream face of a burning droplet of fuel, it is found that the extinction criterion must he modified if the reaction involves a high activation energy.

Rational definition of rate of reaction. General method of expressing rate which leads to no anomalies in systems of changing volume

The conventional way of expressing rate of reaction in terms of rate of change of concentration (e.g. –d[A]/dt) is shown to be inappropriate for systems in which any volume change accompanies the conversion. Thus for such liquid phase reactions, with or without inert solvent, it is demonstrated that the rate of decrease of reactant concentration is not equal to the rate of increase of product concentration, that the apparent external order of reaction depends on whether the rate is expressed in terms of reactant or product, and that the apparent internal order of reaction can not reflect the molecularity of the reaction process. A revised expression for rate of reaction is proposed; this defines the rate at any time as the product of the time derivative of the number of moles of a reaction component and the reciprocal volume of the system at that time, e.g.–(1/Vt)(dA/dt)t. The above anomalies do not arise when rates are expressed in this way; it is therefore recommended that this revised definition of rate be generally employed in all kinetic studies.

Improved assay for catalase based upon steady-state substrate concentration

A modification of the polarographic assay for catalase is described that is based upon automatic titration of a buffered H 2O 2-catalase reaction mixture with a more concentrated H 2O 2 solution such that there is no significant change in the volume of the reaction mixture. The recorded rate of addition of the titrant required to maintain a steady-state substrate concentration yields enzyme activity measurements in terms of actual reaction rate instead of the less satisfactory rate constant for H 2O 2 decomposition, as is the case for most extant assays for catalase. An additional advantage of the new method is that the reaction can be easily carried out at considerably lower temperature and substrate concentrations than can be employed practicably in other types of assays for this enzyme. Both of these features are desirable to minimize the formation of inactive catalase-H 2O 2 complexes. The improved assay works satisfactorily for measuring catalase activity in rat tissue homogenates.

Intramolecular catalysis—V : Long range effects in steroids. Acetylation rates of some Δ 5-3β-hydroxy steroids

A variety of solvolysis, addition, dissociation, reduction, oxidation, condensation, and enolization reactions are subject to long-range effects either in terms of reaction rate or product geometry. The effects, some of which operate over the entire steroid molecule, have been classified as inductive, electrostatic field, conformational transmission, and direct interaction. Significant long-range effects were found to be absent in the acetylation of a series of Δ 5-3β-ols whose rates were measured by following the change in optical rotation.

Inviscid reacting flow near a stagnation point

A reacting flow free of molecular transport exhibits noteworthy behaviour in the neighbourhood of a blunt, symmetrical stagnation point. A local analytical study using the Lighthill-Freeman gas model reveals the basic structure of such a flow. Chemical activity is found to affect some, but not all, of the local characteristics of the flow. Unaffected are the pressure and velocity fields near the stagnation point, where the pressure varies quadratically and the velocity varies linearly as in an inert flow. In addition, the stagnation point is found to be in chemical equilibrium for all non-zero reaction rates. On the other hand the density, temperature, and concentration fields are affected by the non-equilibrium reactions. The extent of this effect can be predicted on the basis of a reaction parameter that measures the rate of reaction in terms of the velocity gradient at the stagnation point. A rapidly reacting flow (with reaction parameter greater than unity) approaches the stagnation point with vanishing gradients of density and temperature, whereas a slowly reacting flow approaches with infinite gradients. The flow field is represented mathematically by functions that are regular along the body but non-analytic in the normal direction. Numerical computations support the validity of the local closed-form solution, and provide information on the local effects of the chemical history of the flow.

Electrochemical reaction rates in terms of electrochemical affinities and potentials-case of hydrogen overvoltage

On the basis of previously presented ideas, rate expressions are written for the various possible rate-determining steps of the hydrogen overvoltage mechanism in terms of the corresponding electrochemical affinities and potentials. This is done for two types of cases, according to whether the overall electrochemical affinity is concentrated in one step or distributed between two or more steps. The equality of the anodic and cathodic transfer coefficients is shown to imply the local equilibrium H2 ⇌ H + H+ + e− at the actual reaction sites, this step being otherwise the rate-determining one of the Heyrovsky-Horiuti mechanism. The cases of both transfer coefficients equal to 1 or to 1/2 are examined. The presentation is to be regarded as covering only some aspects of a general treatment characterized by the use of Marcelin-De Donder or MD formulae.

A Simple Method of Estimating the Vortex Intensity

a catalytic [CKA(O) = 0] or noncatalytic [(daA/dY)(0) = 0] wall. While the values of CKA(0) and (daA/dY)(0) are determined by the degree of catalytic activity on the wall surface and the integrated effect of the gaseous-reaction rates across the entire boundary layer, the value of (daA/dY)(0) is directly influenced by the gaseous-reaction rates in the vicinity of the wall. Furthermore, since WA/P ^ £e ( 77300 ) w _ 2 for dissociation-recombination in air 2 the tendency of (daA/dY) to increase with Y for recombination-dominated reactions will be greater as the local pressure increases (being a maximum at the stagnation point), and will be most prominant at a given pressure for a recombination-rate temperature-dependence law that assumes o> > 2 for Tw > 300°K. The effect of the reaction-rate term on the temperature profile near the wall (Eq. 3) is opposite to the effect on OLA{ an inflection point in T( Y) will appear in the presence of dissociationdominated chemical nonequilibrium in the gas adjacent to the wall. This qualitative analogy with a pressure-gradient effect proves useful in the interpretation of the effects of dissociationrecombination reactions on the thermodynamic-state profile behavior in the boundary layer.

Chemically Frozen Boundary Layers with Catalytic Surface Reaction

When a fluid boundary layer containing a single reactant develops along a solid surface which acts chemically as a distribution for this species, its steady-state concentration along the surface, and, hence, the reaction rate distribution depends both upon the true surface chemical kinetics and a combination of convective and diffusive transport through the fluid. Considered here is a simple approximate method for studying this streamwise distribution of reaction rate in terms of the shear and temperature distributions along the surface. The method is based upon a treatment of the concentration integral equation which initially parallels Liepmann's recent treatment of the energy integral equation but which ultimately leads to a differential equation for the concentration distribution of reactant along the surface. This approach to mass transfer problems, which is intended to bring out clearly the role of the governing catalytic parameter as well as several interesting properties of solutions, is then compared with alternate techniques, in particular with that of Chambre and Acrivos.~ The reactant concentration is considered to be so small that constant diffusion coefficient, D, viscosity coefficient, n, Prandtl Number for diffusion (Schmidt Number), PTD = n/(pD), can be assumed, and the fluid is taken to be incompressible, with catalyzed reaction taking place only at the wall. Thus, the hydrodynamic field is unaltered by the surface reaction, and only the effect of small surface temperature differences on the kinetics of the wall reaction—i.e., the effective sink strength—are taken into account. Extension to the case of several reactants is discussed, and an interpretation of the governing catalytic parameter as a ratio of characteristics times is given.

II. One-dimensional laminar flame theory for temperature-explicit reaction rates

Several exact solutions are given for flames with normal diffusion and with reaction rate explicit in temperature. It is shown that, when the eigenvalue is defined in terms of the temperature-mean reaction rate, its value depends only on the position of the centroid of the reaction rate function. Various approximate theories are reviewed and it is shown that they may be arranged in order of merit: Wilde (best), Adams, Kármán-Penner, Zeldovich-Frank-Kamenetsky, Corner. The Wilde approximate method is developed to provide general relations for the effect of reactant and radical diffusion coefficients on the flame velocity. The few checks possible show good agreement with exact solutions. No theory has been developed for diffusion with bimolecular reactions in stoichiometric mixtures.

Stress Relaxation of Natural and Synthetic Rubber Stocks

Abstract 1. The complete decay of stress in the rubbers studied, held at constant elongation, appeared to involve the rupturing of a definite bond, either at some point along the molecular chain or at the cross-linking bond put in by vulcanization. In the case of a Hevea rubber gum stock the data could be fitted very well by ordinary reaction-rate theory, leading to the conclusion that the free energy of activation required for breaking the bond is 30.4 kcal. per mole of bonds. This result was found to be practically independent of the elongation, and of the presence of carbon black in a Hevea rubber tread stock. This is to be compared to a strength of about 90 kcal. per mole for the C—C bond. 2. In the case of other rubbers (Buna-S, Butaprene-N, Neoprene-GN, and Butyl) the activation free energy for breaking the bond did not vary by more than ±2.0 kcal. per mole from that of Hevea rubber. However, these differences were quite definite. For example, the relaxation of stress in GR-S was slower than in Hevea; a small difference in energy corresponding to a 2:1 ratio in the respective times of decay. 3. The effect of temperature on the relaxation of stress appeared to be of the general type characteristic of chemical reactions. By use of the ordinary formula for expressing rate of reaction in terms of energy of activation, one could predict very closely the behavior of the stress-log time curves at different temperatures. 4. Natural rubber and GR-S vulcanized with paraquinone dioxime and lead dioxide showed relaxation curves very similar to those of the sulfur vulcanized stocks. 5. Relaxation experiments in an ordinary air atmosphere and in an atmosphere of commercial nitrogen showed no appreciable differences. 6. Examination of stretched rubber bands in which the stress had decayed nearly completely (at 100° C) gave no evidences of gross oxidation, such as would make the rubber bands sticky or hard, or of surface deterioration. At higher temperatures, however, the rubber could be observed getting sticky, and then brittle. Specimens in which the stress had completely decayed showed very low tensile strength (by hand test). 7. Antioxidant added to a sulfur-stabilized Buna-S stock caused a definite retardation of the rate of relaxation. 8. Comparison of the results of these experiments with previously recorded observations in the literature indicated that the chemical reaction which ruptured the rubber structure and caused the decay of stress in these experiments (and concomitantly a lowering of tensile strength) was an oxidation of the rubber by small amounts of oxygen, the reaction rate being independent of the oxygen pressure in the range between that present in an ordinary air atmosphere and in a commercial nitrogen atmosphere. 9. The tests suggested a convenient and accurate laboratory method of determining the oxidizability of natural and synthetic rubber stock designed for service.