Characteristic time reaction

The dimensions and units of the reaction rate constant, k T), depend on the form of function h(C/s). For chemical reactions whose rate expressions are power functions of die species concentrations, one can determine the units of the rate constant. From Eq. 3.3.8, the rate can be written as 

The rate expressions derived above describe the dependence of die reaction rate expressions on kinetic parameters related to the chemical reactions. These rate expressions are commonly called the intrinsic rate expressions of the chemical reactions. However, as discussed in Chapter 1, in many instances, the local species concentrations depend also on the rate that the species are transported in the reaction medium. Hence, the actual reaction rates are affected by the transport rates of reactants and products. This is manifested in two general cases (i) gas-solid heterogeneous reactions, where species diffusion through the pore plays an important role, and (ii) gas-hquid reactions, where interfacial species mass-transfer rate as wen as solubility and diffusion play an important role. Considering the effect of transport phenomena on the global rates of the chemical reactions represents a very difficult task in the design of many chemical reactors. These topics are beyond the scope of this text, but the reader should remember to take them into consideration. 

In the preceding sections, we characterized the rates of chemical reactions in terms of their variation with changes in temperature and their dependency on species compositions. The progress of chemical reactions is a rate process that can be characterized by a time constant Reactions with the same rate expression (say. 

For volume-based reaction rate expressions, the characteristic reaction time, t, is defined by 

The selection of the reference reaction rate, ro, should be done in a way that can be used for all forms of rate expressions. Furflier, since the reaction rate depends on the initial (or inlet) composition, vq should be selected such that it is independent of die specific composition of the reference state (or stream). To define ro, we express the reaction rate at any instance by 

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Under the inherent assumption that the mass fractions of the reactants are not changing, further interesting insights can be obtained by rearranging Eq. (7.22). If the reaction proceeds at a constant rate corresponding to T(), a characteristic reaction time rr can be defined as... 

Quantitatively, these observations can be treated in terms of the characteristic residence time U and the characteristic reaction time tc, introduced by... 

Table 12.1 Calculated characteristic reaction time tc, residence time tr, and the Mikhelson number Mi for the combustion of the stoichiometric methane-air mixture in a combustor with the open-edge V-gutter flame holder of height H and apex angle 60° at the mean inlet velocity Uin. Also presented is the maximum approach-stream velocity Um- Signs and correspond to stabilized flame and unstable flame, respectively... 

The Peclet number, uLjD, when written in the form (L /D)l(Llu) is seen to be a ratio of characteristic dispersion time to characteristic residence time and the Damkohler number can, in similar manner, be considered as a ratio of characteristic residence time, L/u, to characteristic reaction time, l/feCA " [59]. 

As better and better methods for following fast reactions with precision were introduced and exploited, characteristic reaction times faster than a second— times measured in milhseconds (ms, 10 s), or microseconds (ps, 10 s), or nanoseconds (ns, 10 s) and then in picoseconds (ps, 10 s)—were measured through stopped-flow techniques (Chance, 1940), flash photolysis (Norrish and Porter, 1949), temperature-jump and related relaxation methods (Eigen, 1954), and then... 

Rock Type Expt. Characteristic Reaction Time, h Partition Coefficient, s ... 

This equation, sometimes called the test equation in texts on numerical differential equations [13], has an important resemblance to chemical kinetics. Specifically, the rate of disappearance of y is proportional to y itself. As X (i.e., the rate constant) increases, the shorter the characteristic reaction time. The general solution to this problem is obviously... 

A similar study of the reaction of acetylene with iron supported on quartz was made by Maksimov et al. (240). The Mossbauer spectrum before reaction with acetylene was a spectral doublet characteristic of iron silicate. After reaction at 1270 K for 50 sec the sample was quenched to room temperature, and in the subsequent Mossbaucr spectrum a new peak was noted. The intensity of this peak increased with increasing reaction time up to 0.1 hr, after which time the intensity remained constant. In this case, it was only possible to study the rate of this surface reaction using a series of low-temperature quenches, since the characteristic reaction time was the order of time required to obtain the Mossbauer spectrum. 

The modified Stanton criterion compares the characteristic reaction time with the thermal time constant of the reactor. The time can be eliminated from the equations by building the ratio ... 

The reaction system, the experiment procedure, and the analytical method used for the determination of micromixing in the TIJ mixer are the same as those described in the last section of this book but Mahajan et al. correlated their experimental data not with impinging velocity w() but with the jet Reynolds number Re. Also, the researchers employed the measure of increasing both the initial concentration CBo and the reaction temperature to raise the sensitivity of the procedure. The characteristic reaction time constant tK = 200 ms at 25 °C and CBo = 2.5 mM, while rR = 65 ms at 35 °C and CB0 = 4.7 mM, which can be used to bound the micromixing times, rM, no greater than them, respectively. 

Chemical reactions proceed at very different reaction rates, so that typical values of characteristic reaction time in industrial batch reactors range from few hours to... 

P is the ratio of the characteristic reaction time r defined in (4.5) to the characteristic time of heat exchange e defined as... 

R S V characteristic reaction time dimensionless velocity vector... 

In the above Da denotes the Damkohler number as the ratio of the characteristic process time H/V to the characteristic reaction time l/r0. The reaction rate r0 is a reference value at the system pressure and an arbitrary reference temperature, as the lowest or the highest boiling point. For catalytic reactions r0 includes a reference value of the catalyst amount. R is the dimensionless reaction rate R = r/r0. The kinetics of a homogeneous liquid-phase reaction is described in general as function of activities ... 

A prominent trade-off in fixed-bed reactor design concerns the catalyst particle size. What is the basis for the choice of a certain particle size When the catalyst performance is to be optimized, the application of the Thiele model helps to provide an answer (Figure 7). The Thiele modulus accounts for the competition between the chemical reaction and the limitation of transport of reactants by diffusion in a porous catalyst particle. It is defined as the square root of the ratio of the characteristic diffusion time fo = L /D and the characteristic reaction time (r. For a... 

For solubility equilibrium to predict the aqueous concentration of a trace element, thermodynamic equilibrium is required and the solid phases must also be identified. In the case of oxyhydroxides and carbonates, it is reasonable to assume a close approach to equilibrium because the characteristic reaction times of dissolution of these minerals are in the range of a few days to a few hundred years (Bruno et al., 2002). For silicate phases the assumption of thermodynamic equilibrium is more problematic due to the low reaction rates compared to the residence time of waters in hydrosystems. Examples of codes and database used by modelers to calculate the speciation and the solubility of a number of trace elements can be found in Bruno et al. (2002). 

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