Steady-state reaction rate calculation

There was always an induction period of 10 to 20 min before the benzene product reached its steady-state rate of production as detected by the mass spectrometer after the introduction of cyclohexane onto the crystal surface. This is shown in Fig. 22 for several catalyst temperatures. The catalyst was initially at 300 K. When steady-state reaction rates were obtained, the catalyst temperature was rapidly increased (in approximately 30 sec) to 423 K and the reaction rate monitored. This was repeated with heating to 573 and 723 K. The benzene desorbed during rapid heating of the catalyst surface is approximately 1 x 1013 molecules or less and represents only a small fraction of the carbon on the surface. The steady-state reaction rates at a given temperature are the same whether the catalyst was initially at that temperature or another. This induction period coincides with a higher than steady-state uptake of cyclohexane. A mass balance calculation on carbon, utilizing the known... 

Finally, the steady-state reaction rate ks is calculated as the ratio of the current J0 to the reactant population N(xR), that is,... 

Steady-state reaction rates can be calculated on both branches of the slow curve taking into account the following relationships ... 

Results. Figure 8.2 gives steady-state profiles of O2 and CH4 and the corresponding reaction rates calculated with the model for the fixed root system defined in Assumption 9. Net O2 consumption is 460 tLmolm h net CH4 emission is 480tLmolm h the fractions of the O2 and CH4 fluxes through the plant are 0.84 and 0.97, respectively, and the fraction of CH4 oxidized prior to emission is 0.13. These are all credible numbers. 

For untempered systems, pressure relief cannot control the temperature rise caused by the runaway. The reacting mixture will therefore increase in temperature and reaction rate during relief. The steady-state relief sizing calculation needs to be performed at the peak reactipn rate, which is normally close to the end of the reaction. The minimum relief ji system size will be obtained if the calculation is performed at the maximum accumulated pressure. 

Let the concentrations of At, A2, B1( and B2 be constant and let the coverage of surface [ZI] at a certain initial moment be not equal to the coverage that would correspond to a steady-state reaction at the given concentrations. A simple calculation (36) shows that the reaction rate, r, will change in time t, according to the law... 

In the determination of steady state reaction kinetic constants of enzyme-substrate reactions, FABMS also provides some very unique capabilities. Since these studies are best performed in the absence of glycerol in the reaction mixture, the preferred method is that which analyzes aliquots which are removed from a batch reaction at timed intervals. Quantitation of the reactants and products of interest is essential. When using internal standards, generally, the closer in mass the ion of interest is to that of the internal standard, the better is the quantitative accuracy. Using these techniques in the determination of kinetic constants of trypsin with several peptide substrates, it was found that these constants could be easily measured (8). FABMS was used to follow the decrease in the reactant substrate and/or the increase in the products with time and with varying concentrations of substrate. Rates of reactions were calculated from these data for each of the several substrate concentrations used and from the Lineweaver-Burk plot, the values of Km and Vmax are obtained. 

The rate of chain initiation, R can be controlled and calculated by using an initiator with a known rate of decomposition, and known initiator efficiency, e. This correction, e, is needed since only those radicals which escape the solvent cage in which they are formed can react with oxygen to initiate reaction on the substrate. At steady state, the rate of chain initiation = the rate of termination, as shown in equation 6 for an azo-initiator. 

This need not be true in vivo where the concentrations of reactants and their enzymes in some cases are nearly comparable. Under these conditions, the nominal concentration of substrate could be significantly greater than the level of unbound substrate, and the reaction rate calculated with nominal concentrations inserted into the rate law clearly would overestimate the rate observed in vivo (Wright et al., 1992 Shiraishii and Savageau, 1993). This condition does not alter the basic chemical kinetic equations that describe the mechanism, but it does mean that the quasi-steady state assumption (e.g., see Peller and Alberty, 1959 Segel and Slemrod, 1989) may be inappropriate when reaction rates change with time in vivo. 

Since Eq. (8.139) does not depend on either steady-state reaction conditions or a knowledge of [M" "], it is more conyenientto calculate the ratios of various rate constants from DPn data than from Rp data. However, the use of DPn data, like the use of Rp data, does require (if the Mayo equation is used) that one employs data at low conversions so that the monomer concentration does not change appreciably. 

It is implied that we are calculating steady-state surface rates under conditions of adsorption equilibrium, and surface reaction steps are slow compared to, those of adsorption and desorption. 

The equation for the rate of propagation, shown above, contains the term [M ]. It designates radical concentration. This quantity is difficult to determine quantitatively, because it is usually very low. A steady state assumption is therefore made to simplify the calculations. It is assumed that while the radical concentration increases at the very start of the reaction it reaches a constant value almost instantly. This value is maintained from then on and the rate of change of free-radical concentration is assumed to quickly become and remain zero during the polymerization. At steady state the rates of initiation and termination are equal, or Ri = R i=2Xi[M ]. It is possible to solve for [M ], which can then be expressed as ... 

At low oxygen concentrations has been shown to be correlated with steady state electron flow calculated from gas exchange measurements (1). Here we investigate the validity of this relation during photosynthetic induction in dark adapted spinach leaves. Photosynthetic electron flow was estimated from measurements of the rate of oxygen evolution with the photoacoustic technique (2,3,4). Our results indicate that equation [1] holds during induction, but that Op is very small compared to OpQ, i.e. electron flow is almost completely suppressed in "energized open PS2 reaction centers. 

These studies seem to indicate that, for structureless particles, it is most important to understand the dependence of nucleation rate coefficients on cluster size for very small clusters. At the low temperatures appropriate for argon nucleation, the decay rate coefficient for excited clusters for clusters larger than seven or eight monomers becomes essentially zero, and the capture cross section for this size cluster apparently increases very slowly with n. These facts should make it very easy to compute steady-state nucleation rates for argon, provided similar information is available for the rate coefficients for the "quenching" reactions of equation (2), since it may not be necessary to use trajectories to calculate any of these rate coefficients for clusters larger than ten or twelve monomers in size. 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

In a continuous steady state reactor, a slightly soluble gas is absorbed into a liquid in which it dissolves and reacts, the reaction being second order with respect to the dissolved gas. Calculate the reaction rate constant on the assumption that the liquid is semi-infinite in extent and that mass transfer resistance in the gas phase is negligible. The diffusivity of the gas in the liquid is 10" 8 m2/s, the gas concentration in the liquid falls to one half of its value in the liquid over a distance of 1 mm, and the rate of absorption at the interface is 4 x 10"6 kmol/m2 s. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

The utihty stream gets started at operating temperature and flow rate. In the following experiments, the utihty stream is heated so as to initiate the reaction. The main and secondary process tines are fed with water at room temperature and with the same flow rate as one of the experiments. Once steady state is reached, operating parameters are recorded. Process tines are then fed with the reactants, hydrogen peroxide and sodium thiosulfate. At steady state, operating parameters are recorded, and a sample of a known mass of reactor products is introduced in the Dewar vessel. Temperature in the Dewar vessel is recorded until equilibrium is reached, that is, until the reaction ends. This calorimetric method is aimed at calculating the conversion rate at the product outlet and thus the conversion rate in the reactor. The latter is also determined by thermal balances between process inlet and outlet of the reactor. Finally, the reactor is rinsed with water. This procedure is repeated for each experiment... 

This approach consists in estimating heat exchanged by each stream in order to determine the heat provided to the system by the reaction. Actually, at steady state, the heat of reaction wiU lead to a temperature rise of process and utility streams, while the utility stream aims at limiting this increase (cooling effect). The conversion rate, x, is easily calculated by... 

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