Dimensionless reaction time

Observe that aok has units of reciprocal time so that aokt is dimensionless. The grouping OQkt is the dimensionless rate constant for a second-order reaction, just as kt is the dimensionless rate constant for a first-order reaction. Equivalently, they can be considered as dimensionless reaction times. For reaction rates governed by Equation (1.20),... 

Fig. 13. Conversion f versus dimensionless reaction time 8 for piston flow reactor with drop size distribution zero-order drop conversion.

The dimensionless time r in Example 2.5 does not depend on the arbitrarily chosen fbatch- Instead, fbatch cancels out in the conversion from t to t so that t is scaled by a natural time constant for the system,. This effectively eliminates ki as a parameter while scaling by fbatch does not. Whether the reactor is batch or continuous, it is always possible to use the reciprocal of a rate constant as a characteristic time. The quantity has units of time when ki is a first-order rate constant. Thus kt is a dimensionless reaction time. Similarly a kit is a dimensionless reaction time when... 

Some of the above dimensionless parameters are readily interpreted. II9 is the Arrhenius number and IIio gives the impact of Ts on ATwr-rill indicates the extent of reaction. II12 is the Prandtl number. II5II7 describes the total rotation during reaction time At. II5II6 is a dimensionless reaction time. HsIIn is the Nusselt number and 112116 is the fourth Damkohler number. The third Damkohler number is nlnsne/ninsrivIlH. 1141171114/1112 is the Reynolds number in terms of uj. And lastly, 118/114 provides a power ratio for the reactor. 

Figure 7.4. Monomer conversion as a function of the dimensionless reaction time for the emulsion poiymerization of vinyl acetate in (a) a single continuous stirred tank reactor and (b) a tube-stirred tank reactor series [54],...

Find the value of the dimensionless batch reaction time, kfthatch, that maximizes the concentration of B for the following reactions ... 

In order to derive specific numbers for the temperature rise, a first-order reaction was considered and Eqs. (10) and (11) were solved numerically for a constant-density fluid. In Figure 1.17 the results are presented in dimensionless form as a function of k/tjjg. The y-axis represents the temperature rise normalized by the adiabatic temperature rise, which is the increase in temperature that would have been observed without any heat transfer to the channel walls. The curves are differentiated by the activation temperature, defined as = EJR. As expected, the temperature rise approaches the adiabatic one for very small reaction time-scales. In the opposite case, the temperature rise approaches zero. For a non-zero activation temperature, the actual reaction time-scale is shorter than the one defined in Eq. (13), due to the temperature dependence of the exponential factor in Eq. (12). For this reason, a larger temperature rise is foimd when the activation temperature increases. 

In order to show how specific guidelines for the reactor layout can be derived, the maximum allowable micro-channel radius giving a temperature rise of less than 10 K was computed for different values of the adiabatic temperature rise and different reaction times. For this purpose, properties of nitrogen at 300 °C and 1 atm and a Nusselt number of 3.66 were assumed. The Nusselt number is a dimensionless heat transfer coefficient, defined as... 

In Table 1.4, the characteristic time-scales for selected operations are listed. The rate constants for surface and volume reactions are denoted by and respectively. Furthermore, the Sherwood number Sh, a dimensionless mass-transfer coefficient and the analogue of the Nusselt number, appears in one of the expressions for the reaction time-scale. The last column highlights the dependence of z p on the channel diameter d. Apparently, the scale dependence of different operations varies from dy f to (d ). Owing to these different dependences, some op-... 

In Table 17.2, fA (for the reaction A products) is compared for each of the three flow reactor models PFR, LFR, and CSTR. The reaction is assumed to take place at constant density and temperature. Four values of reaction order are given in the first column n = 0,1/2,1, and 2 ( normal kinetics). For each value of n, there are six values of the dimensionless reaction number MAn = 0, 0.5, 1, 2, 4, and °°, where MAn = equation 4.3-4. The fractional conversion fA is a function only of MAn, and values are given for three models in the last three columns. The values for a PFR are also valid for a BR for the conditions stated, with reaction time t = t and no down-time (a = 0), as described in Section 17.1.2. 

In dimensionless terms, there is a critical value for S (Damkohler number) that makes ignition possible. From Equation (4.23), this qualitatively means that the reaction time must be smaller than the time needed for the diffusion of heat. The pulse of the spark energy must at least be longer than the reaction time. Also, the time for autoignition at a given temperature T is directly related to the reaction time according to Semenov (as reported in Reference [5]) by... 

Note that the dimensionless parameter PAR is the ratio of the residence time, L/v, and the reaction time for an nth-order reaction, 1 /kCA0. ... 

Fig. 7.2. Thermal or flow diagram for the first-order non-isothermal reaction (FON1) in a non-adiabatic CSTR the rate curve R and the flow line L both depend on the dimensionless residence time, but their intersections still correspond to stationary-state solutions—and tangen-cies to points of ignition or extinction. Note that R has a non-zero value at zero conversion. Exact numerical values correspond to 0ai = 10, t, = tn = A ...

Fig. 6.34. The dimensionless reaction rate as a function of time for asymmetric (a) and symmetric (b) cases. Notations and parameters as in Fig. 6.32.

The equation for the time development of macroscopic concentrations formally coincides with the law of mass action but with dimensionless reaction rate K(t) = K(t)/ AnDr ) which is, generally speaking, time-dependent and defined by the flux of the dissimilar particles via the recombination sphere of the radius tq, equation (5.1.51). Using dimensionless units n(t) = 4nrln(t), r = t/tq, t = Dt/r, and the condition of the reflection of similar particles upon collisions, equation (5.1.40) (zero flux through origin), we obtain for the joint correlation functions the equations (6.3.2), (6.3.3). Note that we use the dimensionless diffusion coefficients, a = 2k, IDb = 2(1 — k), k = Da/ Da + Dq) entering equation (6.3.2). 

Fig. 6.48. The role of non-equilibrium charge screening in eliminating the Coulomb catastrophe the dimensionless reaction rate vs time. Dotted curve - the Debye theory (no screening and similar particle correlation) broken curves - the solution of kinetic equations incorporating these correlations but neglecting screening full curves, screening is taken into account. Parameters L = 5, Da = Db. Curves 1 to 3 correspond to dimensionless concentrations ...

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

The introductory example may be reworked using the Gamma distribution, since the special case given there is n = 1. Let c(x, 0) = Cogn( ) where C0 is the total initial concentration. Let the first order rate constant be k(x) = kx and make time dimensionless as kt. This reaction time or intensity of reaction—severity of reaction as the oil people have it—is really the Dam-kohler number, Da, for the reactor, with t the time of reaction if it is a batch reactor or the residence time if a PFTR. Thus... 

The dimensionless yield depends on the mobility of the reaction but we must decide on a measure of this mobility. We turn first to the rate coefficient of the reaction at the maximum temperature k(Tm), for brevity denoted km. The dimensionless yield can, however, depend only on a dimensionless criterion. The rate coefficient of a bimolecular reaction has the dimensions (time)-1 (concentration)-1. Hence we obtain a dimensionless criterion of the mobility if we multiply the rate coefficient by the reaction time r (which depends on the cooling rate) and the concentration. As concentration unit we have already chosen above the equilibrium concentration at the theoretical temperature [NO]. The dimensionless criterion of the mobility thus has the form... 

We adduce qualitative considerations to find the form of the function / in formula (8.2) for two limiting cases, when the mobility of the reaction is very low and when it is very high. The first case is realized in mixtures with a low fuel content. The dimensionless yield NO/[NO] is then small throughout the process and decomposition of the nitric oxide may be neglected. The quantity of nitric oxide is directly proportional to the rate coefficient and to the reaction time. Neglecting decomposition of the nitric oxide we can write... 

Practical application of the one-dimensional theory developed to the calculation of the effects of losses on the detonation velocity is limited by the fact that even at the limit the reaction time is small and heat transfer and braking do not cover the entire cross-section of the tube. At the same time, in the vast majority of cases, long before the limit is reached one observes the so-called spin—a spiral-like or periodic propagation of detonation which is not described by our theory. Some thoughts are given concerning the dimensionless criteria on which the spin depends. 

By dividing the reaction time by the thermal time constant, one obtains a dimensionless number, the modified Stanton criterion ... 

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