Damkohler number reaction control

Where dissolution or precipitation is sufficiently rapid, the species concentration quickly approaches the equilibrium value as water migrates along the aquifer the system is said to be reaction controlled. Alternatively, given rapid enough flow, water passes along the aquifer too quickly for the species concentration to be affected significantly by chemical reaction. The system in this case is transport controlled. The relative importance of reaction and transport is described formally by the nondimensional Damkohler number, written Da. 

A notable aspect of this equation is that L appears within it as prominently as the rate constant k+ or the groundwater velocity vx, indicating the balance between the effects of reaction and transport depends on the scale at which it is observed. Transport might control fluid composition where unreacted water enters the aquifer, in the immediate vicinity of the inlet. The small scale of observation L would lead to a small Damkohler number, reflecting the lack of contact time there between fluid and aquifer. Observed in its entirety, on the other hand, the aquifer might be reaction controlled, if the fluid within it has sufficient time to react toward equilibrium. In this case, L and hence Da take on larger values than they do near the inlet. 

A high Damkohler number means that the global rate is controlled by mass transfer phenomena. So, the process rate can be rewritten in terms of the Damkohler number and the external effectiveness factor for each reaction order can be deduced, as shown in Table 5.5. In Figure 5.3, the external effectiveness factor versus the Damkohler number is depicted for various reaction orders. It is clear that the higher the reaction order, the more obvious the external mass transfer limitation. For Damkohler numbers higher than 0.10, external mass transfer phenomena control the global rate. In the case of n = 1, the external effec-... 

Here rmix is a characteristic mixing time for the system, while rreac is a characteristic time for the chemical reaction. If the Damkohler number has a value much larger than unity, the process is diffusionally controlled a value much lower than 1 indicates a kinetically controlled process. The terms in the Damkohler number may be defined in different ways, according to the physical characteristics of the system of interest. 

A plot of r t against the DamkOhler number is shown in Fig. 5.50 with the bulk concentration as parameter. On the graph it may be seen that r)e is dependent on both fib and Da, but that three identifiable regions exist. At low values of Da, kinetic control of the reaction is observed and the curves show that rje approaches unity for most substrate concentrations, whilst at high bulk substrate concentrations the... 

The Damkohler numbers are useful measures of the characteristic transport time relative to the reaction time. If the surface Damkohler number (sometimes referred to as the CVD number see reference 7) is large, mass transfer to the surface controls the growth. For small Damkohler numbers, surface kinetics governs the deposition. Similarly, if the gas-phase Damkohler number is large, the reactor residence time is an important factor, whereas if it is small, gas-phase reactions control the deposition. 

The second requirement, for reactions that are not diffusion controlled to reach completion, is that the Damkohler number be larger than 10. The previous discussion and Fig. 11.12 strongly indicate that for SSEs, where at t 0.757 there is an almost vertical ascent of F(t), we use Da = 0.75 t/tr > 10 as the requirement for completion. 

At kinetically controlled reactive conditions (Da = 1), Fig. 4.28(b) shows that the stable node moves into the composition triangle, as in reactive distillation (Fig. 4.27(b)). This point is termed the kinetic arheotrope because its location in the phase diagram depends on the membrane mass transfer resistances and also on the rate of chemical reaction. The kinetic arheotrope moves towards the B vertex with increasing C-selectivity of the membrane. At infinite Damkohler number, the system is chemical equilibrium-controlled (Fig. 4.28(c)), and therefore the arheotrope is located exactly on the chemical equilibrium curve. In this limiting case, it is called a reactive arheotrope . 

Here, we are concerned with intraparticle diffusion controlling a first-order readsorption reaction therefore, the Damkohler number (15), which reflects the ratio of reaction to diffusion rates within catalyst particles, is given by ... 

The effective diffusivity Dn decreases rapidly as carbon number increases. The readsorption rate constant kr n depends on the intrinsic chemistry of the catalytic site and on experimental conditions but not on chain size. The rest of the equation contains only structural catalyst properties pellet size (L), porosity (e), active site density (0), and pore radius (Rp). High values of the Damkohler number lead to transport-enhanced a-olefin readsorption and chain initiation. The structural parameters in the Damkohler number account for two phenomena that control the extent of an intrapellet secondary reaction the intrapellet residence time of a-olefins and the number of readsorption sites (0) that they encounter as they diffuse through a catalyst particle. For example, high site densities can compensate for low catalyst surface areas, small pellets, and large pores by increasing the probability of readsorption even at short residence times. This is the case, for example, for unsupported Ru, Co, and Fe powders. 

Figure 6. Channel length at the same point in time versus Damkohler number. In the transport-controlled regime, the channel length is independent of the reaction rate. As the Damkohler number decreases from 0.1 to 0.01, the length of channels increase (see text for discussion). At Damkohler numbers below about 0.01, dissolution becomes pervasive and the channel does not propagate.

For small Damkohler numbers (Da <3C 1) the mass transport is much faster than the surface reaction itself and therefore the mass transport effect may be ignored. On the other hand, if the Damkohler munber is high (Da 1) the sensorgram profile is completely controlled by the diffusion mass transfer and is it not possible to determine rate constants of the surface reaction. 

Effectiveness factor calculations summarized in Tables 19-1 to 19-5 are consistent with Langmuir-Hinshelwood kinetics, as discussed in this chapter. E is larger and approaches 1 asymptotically in the reaction-controlled regime where the intrapellet Damkohler number is small, and E decreases in the diffusion-controlled regime at large values of A a- These trends are verified by simulations provided in Table 19-1. 

Reactant equilibrium constants Kp and affect the forward kinetic rate constant, and all Ki s affect die adsorption terms in the denominator of the Hougen-Watson rate law via the 0, parameters defined on page 493. However, the forward kinetic rate constant does not appear explicitly in the dimensionless simulations because it is accounted for in Ihe numerator of the Damkohler number, and is chosen independently to initiate the calculations. Hence, simulations performed at larger adsorption/desorption equilibrium constants and the same intrapellet Damkohler number implicitly require that the forward kinetic rate constant must decrease to offset the increase in reactant equilibrium constants. The vacant-site fraction on the internal catalytic surface decreases when adsorption/desorption equilibrium constants increase. The forward rate of reaction for the triple-site reaction-controlled Langmuir-Hinshelwood mechanism described on page 491 is proportional to the third power of the vacant-site fraction. Consequently, larger T, s at lower temperature decrease the rate of reactant consumption and could produce reaction-controlled conditions. This is evident in Table 19-3, because the... 

Effect of the Damkohler Number on Conversion in Square Ducts. More conversion is predicted at higher Damkohler numbers because the rate of surface-catalyzed chemical reaction is larger. At a given axial position z, reactant conversion reaches an asymptotic limit in the diffusion-controlled regime, where oo. Actual simulations of I Abuik vs. f at /i = 20 are almost indistinguishable from those when p = 1000. The effect of p on bulk reactant molar density is illustrated in Table 23-5 for viscous flow in a square duct at = 0.20, first-order irreversible chemical reaction, and uniform catalyst deposition. These results in Table 23-5 for the parameter A, as a function of the Damkohler number p can be predicted via equations (23-80) and (23-81) when C = A and... 

Simulations in the Diffusion-Limited Regime p = 10. None of the profiles listed in Table 23-9 is more effective than uniform deposition at any aspect ratio when the Damkohler number p = 10. This is understandable in the diffusion-controlled regime where the rate of chemical reaction on the catalytic surface is not the primary factor that governs the conversion of reactants to products. However, some profiles perform poorly relative to uniform deposition when diffusion of reactants toward the active surface is slow. Differences among the profiles are more pronounced at higher aspect ratios. A qualitative summary at each aspect ratio is provided in Table 23-10. Identification numbers are underlined when... 

Simulations are presented below in tabular and graphical forms when the temperature at the external surface of the pellet is constant at 350 K. The effective thermal conductivity of alumina catalysts is 1.6 x 10 J/cm s K. The chemical reaction is first-order and irreversible and the catalysts exhibit rectangular symmetry. Most important in Tables 27-5 to 27-8 and Figures 27-1 to 27-3, the diffusivity ratio a(0) varies with temperature in the mass transfer equation. This effect was neglected in Tables 27-1 to 27-4. Notice that in all of these tables (i.e., 27-1 to 27-8), numerical simulations reveal that the actual max exceeds I + fi, except when the intrapellet Damkohler number is small enough and 4 a( = 0) > 0 because the center of the catalyst is not reactant starved in the chemical-reaction-rate-controlled regime. 

Fig. 10.5 Bifurcation diagram for methyl formate distillation column with 45 trays, feed at tray 23, reflux ratio 5, Murphree efficiency 0.65, and a kinetically controlled reaction with different relative Damkohler numbers Da, = Da /Da,, ...

Keywords Borghi diagram Combustion Chemical reactions Chemical kinetics Damkohler number Flamelets Ignition Karlovitz number Mixing-controlled... 

Since Luyben identified the snowball effect (Luyben, 1994), the sensitivity of reactor-separator-recycle processes to external disturbances has been the subject of several studies (e.g., Wu and Yn, 1996 Skogestad, 2002). Recent work by Bildea and co-workers (Bildea et al., 2000 and Kiss et aL, 2002) has shown that a critical reaction rate can be defined for each reactor-separator-recycle process using the Damkohler number. Da (dimensionless rate of reaction, proportional to the reaction rate constant and the reactor hold-up). When the Damkohler number is below a critical value, Bildea et al. show that the conventional unit-by-unit approach in Figure 20.15 leads to the loss of control. Furthermore, they show that controllability problems associated with exothermic CSTRs and PFRs are resolved often by controlling the total flow rate of the reactor feed stream. 

As pointed out, the influence of mass transfer on the observed reaction rate depends on the ratio between the characteristic reaction time and the characteristic time for mass transfer. By increasing the temperature, the intrinsic reaction rate increases more strongly (exponentially) than the rates of external and internal mass transfer. Consequently, the Thiele modulus and the second Damkohler number augment with increasing temperature, and transport phenomena become more and more important and will finally control the transformation process. In addition, the temperature dependence of the observed reaction rate will change as indicated. 

For kinetically controlled reactions, an autonomous residue curve expression can be derived in terms of the Damkohler number (Da = ) and for the heating policy... 

At Earth surface conditions, gypsum (CaS04 2H20) dissolution appears to be controlled partly by the rate of Ca and release at the mineral surface and partly by the rate of transport of these species away from the dissolving surface. The second Damkohler number, Eq. (7.18), can be used to determine the relative importance of reaction rate versus diffusion rate for gypsum dissolving into a static solution in a fracture with a width, L, of 1.0 x 10 m... 

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